constrained robust optimal trajectory tracking:...

Constrained Robust OptimalTrajectory Tracking: ModelPredictive Control ApproachesRobuste Optimale Trajektorienfolgeregelung unter Berücksichtigung von Beschränkungen:Ansätze der Modellprädiktiven Regelung

Diplomarbeit von Maximilian BalandatInstitut für Flugsysteme und Regelungstechnik, Prof. Dr.-Ing. Uwe Klingauf

Fachbereich MaschinenbauInstitut für Flugsystemeund Regelungstechnik

Thesis AssignmentWithin the scope of this work, different methods for Robust Model Predictive Control of constrainedsystems (constrained MPC) are to be explored in respect of their applicability to trajectory trackingproblems. To this end, different approaches from current research shall be investigated and comparedregarding their suitability for later use. Promising methods in this context are, among others, Explicit MPCand Tube-Based MPC. The effects of model uncertainties and external disturbances are to be accounted forby using suitable uncertainty models. The approaches shall subsequently be examined in basic simulationexamples, where linear SISO and MIMO systems subject to constraints are to be considered.

AufgabenstellungIm Rahmen dieser Arbeit sollen Methoden zur robusten modellprädiktiven Regelung von beschränktenSystemen (constrained MPC) im Hinblick auf eine Anwendung im Bereich der Trajektorienfolgeregelunguntersucht werden. Hierbei sollen verschiedene Verfahren aus der aktuellen Forschung recherchiert undim Hinblick auf eine spätere Verwendung miteinander verglichen werden. Vielversprechende Verfahrenhierbei sind u.a. Explicit MPC und Tube-Based MPC. Ungenauigkeiten in der Systemmodellierung bzw. derEinfluss von externen Störgrößen sollen durch Verwendung entsprechender Unsicherheitsmodelle Rech-nung getragen werden. Die Ansätze sollen anschließend an einfachen Simulationsbeispielen untersuchtwerden, dabei sind lineare Ein- und Mehrgrößensysteme mit Beschränkungen zu betrachten.

i

DeclarationI hereby declare that I have written this thesis without any help from others, and that no other than theindicated references and resources have been used. All parts that have been drawn from the referenceshave been marked as such. This work has not been presented to any other examination board in any way.

ErklärungHiermit versichere ich, dass ich die vorliegende Arbeit ohne Hilfe Dritter und nur mit den angegebenenQuellen und Hilfsmitteln angefertigt habe. Alle Stellen, die aus den Quellen entnommen wurden, sind alssolche kenntlich gemacht. Diese Arbeit hat in gleicher oder ähnlicher Form noch keiner Prüfungsbehördevorgelegen.

Darmstadt, 22. Juli 2010

Maximilian Balandat

iii

AbstractThis thesis is concerned with the theoretical foundations of Robust Model Predictive Control and itsapplication to tracking control problems. Its first part provides an introduction to MPC for constrainedlinear systems as well as a survey of different Robust MPC methodologies. The second part consists ofa discussion of the recently developed Tube-Based Robust MPC framework and its extension to output-feedback and tracking control problems. Guidelines on how to synthesize Tube-Based Robust ModelPredictive Controllers are given, and a software framework is developed allowing for the controllers tobe implemented both explicitly as a lookup table (using multiparametric programming) and implicitlyby using fast on-line optimization algorithms. The reviewed Tube-Based Robust MPC controllers aretested on illustrative benchmark problems and issues concerning their computational complexity arediscussed. The last part of this thesis presents the novel contribution of “Interpolated Tube MPC”, anapproach that combines interpolation techniques with the basic ideas behind Tube-Based Robust MPC.Important properties of this new type of controller are proven in a rigorous theoretical analysis. Finally, theapplicability of Interpolated Tube MPC is tested in a case study, which shows the superior computationalperformance of the controller compared to standard Tube-Based Robust MPC.

Keywords: Model Predictive Control, Constrained Robust Control, Reference Tracking, Output-Feedback,Tube-Based Robust MPC, Explicit MPC, Interpolated Tube MPC

ZusammenfassungDiese Diplomarbeit beschäftigt sich mit den theoretischen Grundlagen der Robusten ModellprädiktivenRegelung (MPC) und deren Anwendung auf Trajektorienfolgeregelungen. Neben einer Einführung inMPC für beschränkte linear Systeme wird im ersten Teil dieser Arbeit zudem eine umfassende Litera-turübersicht zu verschiedenen Robust MPC Methoden gegeben. Der zweite Teil diskutiert das kürzlichentwickelte „Tube-Based Robust MPC“, sowie dessen Erweiterung auf die Regelung von Systemen mit Aus-gangsrückführung (“output-feedback”) und auf Führungsgrößenfolgeregelungen (“tracking”). Es werdenLeitlinien zur Synthese von Reglern dieser Art vorgestellt und die zur Realisierung nötigen Algorithmenimplementiert. Die Realisierung der Regler kann zum einen implizit geschehen, d.h. unter der “on-line”Verwendung mathematischer Optimierungsalgorithmen, zum anderen explizit als “lookup-table” (mitHilfe von “multiparametric programming”). Die Anwendbarkeit von Tube-Based Robust MPC wird anHand von Simulationsbeispielen untersucht, weiterhin werden Fragen bezüglich der Komplexität derImplementierung diskutiert. Der letzte Teil dieser Arbeit präsentiert mit “Interpolated Tube MPC” neueForschungsergebnisse, bei denen Interpolationstechniken mit den Grundideen hinter Tube-Based RobustMPC kombiniert sind. In einer umfassenden theoretische Analyse werden wichtige Eigenschaften diesesneuen Reglertyps gezeigt. Zum Abschluss wird die Anwendbarkeit der Regelung an einem Simulationsbei-spiel untersucht und die überlegene Rechengeschwindigkeit von Interpolated Tube MPC im Vergleich zuregulärem Tube-Based Robust MPC demonstriert.

Schlagwörter: Modellprädiktive Regelung, Beschränkte Robuste Regelung, Trajektorienfolgeregelung,Regelung mit Ausgangsrückführung, Tube-Based Robust MPC, Explicit MPC, Interpolated Tube MPC

v

Contents

Abstract vi

List of Symbols xi

1. Introduction 11.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. The Basic Idea Behind Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. MPC – A Brief History and Current Developments . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Model Predictive Control for Constrained Linear Systems 52.1. The Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1. The Model Predictive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2. Stability of the Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3. Choosing the Terminal Set and Terminal Cost Function . . . . . . . . . . . . . . . . . 92.1.4. Solving the Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2. Robustness Issues and Output-Feedback Model Predictive Control . . . . . . . . . . . . . . . 112.3. Reference Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1. Nominal Set Point Tracking Model Predictive Control . . . . . . . . . . . . . . . . . . 122.3.2. Offset Problems in the Presence of Uncertainty . . . . . . . . . . . . . . . . . . . . . . 152.3.3. Reference Governors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4. Explicit MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1. Obtaining an Explicit Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2. Issues with Explicit MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3. Explicit MPC in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Robust Model Predictive Control 213.1. Inherent Robustness in Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2. Modeling Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1. Parametric and Polytopic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2. Structured Feedback Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3. Bounded Additive Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.4. Stochastic Formulations of Model Predictive Control . . . . . . . . . . . . . . . . . . . 25

3.3. Min-Max Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1. Open-Loop vs. Closed-Loop Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2. Enumeration Techniques in Min-Max MPC Synthesis . . . . . . . . . . . . . . . . . . . 28

3.4. LMI-Based Approaches to Min-Max MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1. Kothare’s Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.2. Variants and Extensions of Kothare’s Controller . . . . . . . . . . . . . . . . . . . . . . 313.4.3. Other LMI-based Robust MPC Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5. Towards Tractable Robust Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . 373.5.1. The Closed-Loop Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5.2. Interpolation-Based Robust MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5.3. Separating Performance Optimization from Robustness Issues . . . . . . . . . . . . . 39

vii

3.6. Extensions of Robust Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6.1. Output-Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6.2. Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6.3. Offset-Free Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4. Tube-Based Robust Model Predictive Control 474.1. Robust Positively Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2. Tube-Based Robust MPC, State-Feedback Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.2. The State-Feedback Tube-Based Robust Model Predictive Controller . . . . . . . . . 524.2.3. Tube-Based Robust MPC for Parametric and Polytopic Uncertainty . . . . . . . . . . . 554.2.4. Case Study: The Double Integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3. Tube-Based Robust MPC, Output-Feedback Case . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2. The Output-Feedback Tube-Based Robust Model Predictive Controller . . . . . . . . 664.3.3. Case Study: Output-Feedback Double Integrator Example . . . . . . . . . . . . . . . . 694.3.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4. Tube-Based Robust MPC for Tracking Piece-Wise Constant References . . . . . . . . . . . . . 734.4.1. Tube-Based Robust MPC for Tracking, State-Feedback Case . . . . . . . . . . . . . . . 744.4.2. Tube-Based Robust MPC for Tracking, Output-Feedback Case . . . . . . . . . . . . . . 824.4.3. Offset-Free Tube-Based Robust MPC for Tracking . . . . . . . . . . . . . . . . . . . . . 85

4.5. Design Guidelines for Tube-Based Robust MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5.1. The Terminal Weighting Matrix P and the Terminal Set X f . . . . . . . . . . . . . . . 884.5.2. The Disturbance Rejection Controller K . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5.3. Approximate Computation of mRPI Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 934.5.4. The Offset Weighting Matrix T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.6. Computational Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6.1. Problem Setup and Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6.2. Observations and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5. Interpolated Tube MPC 1055.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2. The Interpolated Terminal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.1. Controller Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.2.2. The Maximal Positively Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2.3. Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3. The Interpolated Tube Model Predictive Controller . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.1. The Optimization Problem and the Controller . . . . . . . . . . . . . . . . . . . . . . . 1115.3.2. Properties of the Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3.3. Choosing the Terminal Controller Gains Kp . . . . . . . . . . . . . . . . . . . . . . . . . 1175.3.4. Extensions to Output-Feedback and Tracking MPC . . . . . . . . . . . . . . . . . . . . 118

5.4. Possible Ways of Reducing the Complexity of Interpolated Tube MPC . . . . . . . . . . . . . 1185.4.1. Reducing the Number of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.4.2. Reducing the Number of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.5. Case Study: Output-Feedback Interpolated Tube MPC . . . . . . . . . . . . . . . . . . . . . . . 1205.5.1. Problem Setup and Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5.2. Comparison of the Regions of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5.3. Computational and Performance Benchmark . . . . . . . . . . . . . . . . . . . . . . . . 122

viii Contents

6. Conclusion and Outlook 127

A. Appendix 131A.1. Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.2. Definition of the Matrices Mθ and Nθ in Lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . 132

List of Figures 133

List of Tables 135

Bibliography 137

Contents ix

List of SymbolsAbbreviations

ARE Algebraic Riccati Equation

BMI Bilinear Matrix Inequality

CLF Control Lyapunov Function

ERPC Efficient Robust Predictive Control

FIR Finite Impulse Response

GERPC Generalized Efficient Predictive Control

PI Positively Invariant

ISS Input-to-State Stability

KKT Karush-Kuhn-Tucker (optimality conditions)

LFT Linear Fractional Transformation

LMI Linear Matrix Inequality

LP Linear Program

LPV Linear Parameter-Varying

LQG Linear Quadratic Gaussian

LQR Linear Quadratic Regulation

MPC Model Predictive Control

MPI Maximal Positively Invariant

mpQP multiparametric Quadratic Program

MRPI Maximal Robust Positively Invariant

mRPI minmal Robust Positively Invariant

PWA Piece-Wise Affine

PWQ Piece-Wise Quadratic

QP Quadratic Program

RHC Receding Horizon Control

RPI Robust Positively Invariant

SDP Semidefinite Program

SOCP Second Order Cone Program

Functions

||x ||2Q ||x ||2Q := x TQx

Convh(·) Convex Hull

d(z,Ω) distance of a point z from a set Ω,d(z,Ω) := inf

||z− x || | x ∈ Ω

matrix inequality (with respect to positivedefiniteness)

hΩ(a) support function of a set Ω evaluated at a

int(·) interior of a set

J(x,u) cost of a state trajectory x driven by u

κN (·) implicit model predictive control law

κip(·) interpolated feedback controller

κipN (·) implicit Interpolated Tube MPC law

κ∗N (·) implicit Tube-Based Robust MPC law

l(·) stage cost function

λmax(·) maximum eigenvalue of a p.s.d matrix

λmin(·) minimum eigenvalue of a p.s.d matrix

µ feedback control policy

Pontryagin set difference

⊕ Minkowski set addition

Φ(i; x ,u) state of the system at time i controlled by uwhen the initial state at time 0 is x

Φ(i; x ,u,ψ) state of the system at time i controlled by uwhen the initial state at time 0 is x and therealization of the generic uncertainty is ψ

Φ(i; x ,u,w) state of the system at time i controlled by uwhen the initial state at time 0 is x and thestate disturbance sequence is w

PN (x) optimization problem with time horizon Nfor initial state x

P0N (x) conventional optimal control problem for

initial state x (Tube-Based Robust MPC)

PclN (x) closed-loop Min-Max MPC optimization

problem with horizon N

PipN (x) Interpolated Tube MPC optimization prob-

lem for horizon N

PolN (x) open-loop Min-Max MPC optimization prob-

lem with horizon N

P∗N (x) modified optimal control problem for initialstate x in Tube-Based Robust MPC

Pre(Ω) predecessor set of a set Ω

Projx(Ω) projection of the set Ω on the x-space

ρ(·) spectral radius of a matrix

σ(·) maximum singular value of a matrix

tr(·) trace of a matrix

vert(Ω) set of vertices of a set Ω

Vf (·) terminal cost function

V∞ infinite horizon cost function

V∞ unconstrained infinite horizon cost function

VN (·) cost function for an optimal control problemof horizon N

Vo(·) offset cost function

xi

Matrices

0 zero matrix/vector of appropriate dimension

A system matrix

A nominal system matrix, A := 1L

∑Lj=1 A j

AE augmented system matrix

AK closed-loop system matrix, AK := A+ BK

AL closed-loop observer system matrix,AL := A− LC

B input matrix

B nominal input matrix, B := 1L

∑Lj=1 B j

Bd virtual disturbance input matrix

C output matrix

Cd virtual disturbance output matrix

D feedthrough matrix

I identity matrix of appropriate dimension

K disturbance rejection controller

K∞ unconstrained infinite horizon optimal con-troller (“Kalman-Gain”)

Kp pth terminal controller gain

L observer feedback gain matrix

Ld observer disturbance-feedback gain matrix

Lx observer state-feedback gain matrix

Mθ steady state parametrization matrix,zs = (xs, us) = Mθθ

Mθ set point parametrization matrix, ys = Nθθ

P terminal weighting matrix

P∞ unique pos. definite solution to the ARE

Pp infinite horizon cost matrix correspondingto the controller gain Kp

Q state weighting matrix

R control weighting matrix

T offset weighting matrix

V⊥ a matrix such that V T V⊥ = 0 andthat [V, V⊥] is a non-singular square matrix

Sets

Bnp the n-dimensional p-norm unit ball

Ctrl(Ω) one-step controllable set of Ω

∆c bound on the artificial disturbance δc

∆e bound on the artificial disturbance δe

E RPI set bounding the error x − x betweenactual and nominal system state

Ec RPI set for the system e+c = AK ec +δc

Ee RPI set for the system e+e = ALee +δe

F∞ minimal robust positively invariant set

MN set of admissible control policies

; empty set

Ωε ε-approximation of the set Ω

Ωet invariant set for tracking

Ω∞ maximal positively invariant set

ΩE∞ MPI set for the augmented system

( x E)+ := AE x E

Ωp∞ MPI set for the system x+ = (A+ BKp) x

Pl l th polyhedral partition of the explicit solu-tion of a multiparametric program

Ψ generic disturbance set

ψ generic disturbance realization

T “tube” of trajectories: T (t) = x∗0(x(t))⊕EU control constraint set

U tightened control constraint set

UN set of admissible control sequences u

Us set of admissible steady state inputs us

V bound on the output disturbance v

W bound on the state disturbance w

w bound on the virtual additive disturbance

X state constraint set

X tightened state constraint set

XE augmented state constraint set

X f terminal constraint set for the nominal sys-tem in Tube-Based Robust MPC

XN region of attraction

XN region of attraction of nominal states

XN region of attraction of state estimates

X pN region of attraction of Tube-Based Robust

MPC with terminal controller Kp

X f terminal constraint set

Xx pl exploration region for a multiparametric pro-gram

Xs set of admissible steady states xs

xN set of slack state variables

Ys set of trackable set points

Z tightened constraint set in the (x , u)-space:Z := X× U

Variables

d virtual integrating disturbance

δc artificial disturbance, δc := L(Cec + v )

δe artificial disturbance, δe := w− Lv

d estimate of the virtual integrating distur-bance d

e error between actual and predicted statee := x − x

ec error between observer state and nominalsystem state, ec := x − x

xii List of Symbols

ee state estimation error, ee := x − x

ε relaxation factor in the computation of anapproximated mRPI set

k∗ determinedness index

λ contraction factor in the computation of K

N prediction horizon

Ncom overall number of scalar constraints in theoptimization problem

Nreg number of regions of the explicit solution

Nv ar overall number of scalar variables in the op-timization problem

ν number of terminal controllers in Interpo-lated Tube MPC

ρ tradeoff parameter for the computation of K

θ vector parametrizing all admissible steadystates

θ vector parametrizing all artificial admissiblesteady states

u control input

u control input to the nominal system

u control sequence, u :=

u0, u1, . . . , uN−1

u predicted nominal control sequence,u :=

u0, u1, . . . , uN−1

v output disturbance

v output disturbance sequence,v :=

v0, v0, . . . , vN−1

w state disturbance

w state disturbance sequence,w :=

w0, w1, . . . , wN−1

w virtual additive disturbance

x system state

x state of the nominal system

x E augmented state, x E :=

x T ( x1)T . . . ( xν−1)TT

x+ successor state of the nominal system

( xs, us) artificial steady state, xs = Axs + Bus

x state trajectory, x :=

x0, x1, . . . , xN−1

x predicted nominal state trajectory,x :=

x0, x1, . . . , xN−1

x state estimate

x+ successor state estimate

x+ successor state

(xs, us) steady state, xs = Axs + Bus

x p pth state slack variable

y system output

y output estimate

ym measured output variables

yre f output reference

ys output set point

yt tracked output variables

zs steady state and input, zs := (xs, us)

xiii

1 Introduction

1.1 Motivation

Over the past decades, model-based optimal control has become one of the most commonly encounteredcontrol methodologies for multivariable control problems, both in theory and in practical applications.Since it is more or less impossible to obtain perfect models for real-world plants, and because of the factthat there will always be some exogenous disturbances acting on the plant, the presence of uncertainty isa characteristic of virtually any control problem. Controller synthesis methods that constructively dealwith prior information about uncertainty (such as bounds, stochastic distributions, etc.) are referred to asRobust Control methods. Linear Robust Control, in particular methods such as H2- or H∞-control, havebeen successfully brought to maturity and today find widespread use in practice. However, these methodsusually do not take constraints on the states and/or control inputs of the system directly into account.Frequently, controllers are therefore synthesized for an idealized, unconstrained problem while additionalmeasures are added a posteriori to ensure constraint satisfaction in an ad-hoc way. Clearly, this does notonly make analysis very difficult, but also yields potentially conservative controllers. Moreover, in mostapplications, the control task is to not only stabilize the system, but to control it in such a way that itsoutput tracks a given reference value (a set point) or reference trajectory. Furthermore, the number andquality of available measurements in real-world applications is generally limited, such that erroneousmeasurements of the system output are often times the only source of information that can be used forcontrolling the system. This task is usually referred to as output-feedback control.

One of the few (if not the only) control methodologies that is able to handle hard constraints on thesystem in a non-conservative way is Model Predictive Control (MPC). Model Predictive Control is acontrol strategy based on solving on-line, at each sampling instant, a mathematical optimization problembased on a dynamic model of the plant to be controlled. In this optimization problem, the predictedevolution of the system is optimized with respect to some cost function, and only the first element ofthe predicted optimal control sequence is applied to the plant. This is then repeated at all subsequentsampling instances. Over time, MPC has become the preferred control method in the process industry,where system dynamics and sampling times are relatively slow. Advances in computer technology andalso in MPC theory today have made MPC an interesting and viable option also for fast sampled systems.If both uncertainty and hard constraints on the system are treated in an integrative way, the resultingapproaches are referred to as “Robust MPC”, a term that subsumes all those flavors of MPC that directlytake uncertainties into account. Although conventional (non-robust) MPC has quite a long history inapplications, the computational challenges pertaining to Robust MPC so far have prevented the use ofRobust Model Predictive Controllers for all but very simple (or very slow) systems.

The purpose of this thesis is to give an overview of the wide and constantly expanding field of RobustMPC, and to present, discuss and eventually extend the recently proposed framework of Tube-BasedRobust Model Predictive Control. Tube-Based Robust MPC is a very interesting variant of Robust MPC,for a number of different reasons. One reason is that it is fairly easy to develop reference tracking andoutput-feedback controllers using the Tube-Based Robust MPC ideas. Maybe the most important reasonis that, due to its rather low computational complexity as compared to other Robust MPC methods,Tube-Based Robust MPC seems very attractive for practical applications.

1

After some general information on Model Predictive Control in the following section, the basic theoryof Model Predictive Control for linear systems will be presented in chapter 2 in a condensed form. Thischapter also addresses some additional extensions and answers questions that go beyond standard MPC.Chapter 3 then provides an overview over the most important Robust MPC approaches, before Tube-BasedRobust Model Predictive Control as the main part this thesis is discussed in detail in chapter 4. Finally,chapter 5 presents with “Interpolated Tube MPC” the novel contribution of this thesis. Interpolated TubeMPC is an extension of Tube-Based Robust MPC that allows for the design of controllers with reducedcomputational complexity.

1.2 The Basic Idea Behind Model Predictive Control

Model Predictive Control is, at the most basic level, a method of controlling dynamic systems using thetools of mathematical optimization. The common feature of all Model Predictive Control approaches isto solve on-line, at each sampling instant, a finite horizon optimal control problem based on a dynamicmodel of the plant, where the current state is the initial state. Only the first element of the computedsequence of predicted optimal control actions is then applied to the plant. At the next sampling instant,the prediction horizon is shifted, and the finite horizon optimal control problem is solved again for newlyobtained state measurements. This idea is not new, already in Lee and Markus (1967) one can find thefollowing statement: “One technique for obtaining a feedback controller synthesis from knowledge ofopen-loop controllers is to measure the current control process state and then compute very rapidly forthe open-loop control function. The first portion of this function is then used during a short time interval,after which a new measurement of the process state is made and a new open-loop control function iscomputed for this new measurement. The procedure is then repeated.” The technique described by Leeand Markus (1967) is commonly referred to as “Receding Horizon Control” (RHC), and is today usedmore or less synonymously to the term Model Predictive Control.

Figure 1.1.: Receding Horizon Control (Bemporad and Morari (1999))

2 1. Introduction

Figure 1.1, which has been inspired by Bemporad and Morari (1999), illustrates the concept of RecedingHorizon Control for a SISO system. At time t, the (open-loop) optimal control problem is solved for theinitial state x(t) with a prediction horizon of length N . From the sequence of predicted optimal controlinputs u∗(t) :=

u∗0(t), . . . , u∗N−1(t)

, the first element u∗0(t) is applied to the plant until the next samplinginstant. At time t+1, the prediction horizon is shifted by one, new measurements of the state x(t+1)are obtained, and the optimal control problem is solved again for the new data. If the predictions wereaccurate and there was no uncertainty present, then of course the first N−1 elements of u∗(t+1), wouldcoincide with the last N−1 elements of u∗(t). But since in reality there will always be some degree ofuncertainty, u∗0(t+1) will generally differ from u∗1(t). Thus, repeatedly shifting the horizon and using newmeasurements of the state of the system provides some degree of robustness against modeling errors andperturbations. Receding Horizon Control therefore introduces feedback into the closed-loop system.

1.3 MPC – A Brief History and Current Developments

The first practical applications of Model Predictive Control, at the time mainly in the process industry,date back already about 35 years (Garcia et al. (1989); Qin and Badgwell (2003); Camacho and Bordons(2004)). A variety of different versions of MPC emerged from the concepts of the first generation MPCmethods like IDCOM (Richalet et al. (1978)) and DMC (Cutler and Ramaker (1980)). Lacking compre-hensive theory, the use of MPC in industry until the late 1980’s was often times merely an ad-hoc solution,usually without formal guarantees on stability and feasibility of the solution. This began to change inthe early 1990’s, when an increasing number of researchers in academia became interested in the theoryof MPC. Since then, there has been a vast and ever increasing number of contributions to the field: InMorari (1994) it is reported that a simple database search for “predictive control” generated 128 refer-ences for the years 1991–1993 alone. Bemporad and Morari (1999) already reported 2.802 hits for theyears 1991–1999. Today, “predictive control” generates more than 22.200 results for the years 1991–2010.

The reason for the popularity of MPC in both industry and academia is simple: Model Predictive Controlis one of the few (if not the only) control methodologies that can guarantee optimality (with respectto some performance measure) while ensuring the satisfaction of hard constraints on system states andinputs 1. One of the main limitations of MPC has always been its substantial computational complexityin comparison to classical controller types. After all, a mathematical optimization problem has to besolved on-line at each sampling instant. Thus, the practical application of MPC in the past has beenrestricted to “slow” dynamical systems. This also explains the rather isolated success of MPC in the processindustry, where time constants are usually relatively large, constraint satisfaction is essential, and the costof expensive computer technology is of minor significance.

During the last decade, the situation has however changed, and Model Predictive Control has become in-creasingly interesting for a wider range of applications. There are three important factors that contributedand still contribute to this development. The first one is the considerable progress in computer technologythat allows the development of increasingly fast, cost-effective, miniaturized, and energy-efficient proces-sors. The second important factor is the development of more powerful and more reliable optimizationalgorithms, that continuously widen the spectrum of possible applications of MPC methods. Finally, as athird factor, the theoretical advances in MPC itself must of course not be forgotten. After a consensushad been reached within the control community on what kind of “ingredients” were necessary to ensurestability and feasibility of MPC, a process which was more or less completed with the seminal surveypaper Mayne et al. (2000), researchers have now turned themselves to the development of extensions tostandard nominal Model Predictive Control.

1 Clearly, simply saturating the control action of unconstrained optimal controllers (i.e. LQR) is NOT an optimal controltechnique and may exhibit arbitrarily poor performance or even result in loss of stability.

1.3. MPC – A Brief History and Current Developments 3

In order to enable the application of Model Predictive Control to commonly encountered practical controlproblems, one of the main requirements is to be able to design controllers that are robust with respect touncertainties. It is well known that nominal Model Predictive Controllers inherently provide some degreeof robustness (de Nicolao et al. (1996); Santos and Biegler (1999)), yet this robustness is very hard toquantify and may even, except for linear systems subject to convex constraints, be arbitrarily small, as waspointed out in Teel (2004); Grimm et al. (2004). Because of the use of on-line optimization, Bemporadand Morari (1999) consider robustness analysis of MPC control loops generally far more difficult thantheir synthesis. This insight, together with the motivation coming from the success of Linear RobustControl theory (Green and Limebeer (1994); Zhou et al. (1996)), led to considerable research activity in“Robust MPC” (Bemporad and Morari (1999); Chisci et al. (2001); Cuzzola et al. (2002); Kothare et al.(1995); Mayne et al. (2005); Scokaert and Mayne (1998)). Robust MPC is, as Linear Robust Controlis, a constructive technique that takes uncertainties (either in the model, or caused by exogenous distur-bances) directly into account already during the design process of the controller. This field has recentlyseen a number of enticing contributions, and will be the main theme of this thesis in the chapters 3, 4 and 5.

Another very interesting line of work in the field of linear MPC proposes to use parametric program-ming to precompute the solution of the optimal control problem (which is traditionally solved on-linefor a measured initial state) for all initial states off-line, and to store the resulting piecewise-affinecontrol law in a lookup table (Bemporad et al. (2002); Alessio and Bemporad (2009)). This method,commonly referred to as “Explicit MPC”, seems a promising alternative to conventional on-line opti-mization, at least for applications to fast systems of lower complexity (in terms of state dimension andnumber of constraints). As one recent example, Mariéthoz et al. (2009) report FPGA implementationsof Explicit MPC that achieve sampling frequencies up to 2.5Mhz. The basic ideas of Explicit MPC willbe presented in section 2.4. Moreover, Explicit Robust Model Predictive Controllers will also be imple-mented the context of Tube-Based Robust MPC and Interpolated Tube MPC in chapter 4 and 5, respectively.

Other contributions include extensions of MPC to the output-feedback case, i.e. when only incompleteinformation about the system state is available, and to tracking problems (Mayne et al. (2000)), bothof which will be addressed in this thesis. Current research on Model Predictive Control includes thedevelopment of suboptimal linear MPC algorithms (Zeilinger et al. (2008); Canale et al. (2009)) and thenonlinear Robust MPC (Lazar et al. (2008); Limon et al. (2009)). Due to the vast amount of availableliterature this thesis can not aim at giving an exhaustive overview over the developments and trends inthe field of Model Predictive Control. The following chapters are therefore restricted to (Robust) ModelPredictive Control for discrete-time constrained linear systems.

4 1. Introduction

2 Model Predictive Control for ConstrainedLinear Systems

The purpose of the following chapter is to give a brief introduction to nominal Model Predictive Controlof constrained discrete-time linear systems. The varieties of MPC are of course far greater than thislimited view can provide, they include the design of controllers for nonlinear as well as for time-varyingsystems in both discrete and continuous time. At least the discrete-time formulation is however not reallya restriction: Since the employed Receding Horizon Control strategy inherently introduces a discretizationof time, it is only plausible to formulate the MPC problem in discrete time, and most researchers interestedin implementable algorithms do so.

The theory of nonlinear MPC1 is, although important progress has been made (Findeisen et al. (2007)),not yet as well developed as the one of linear MPC. Nonlinear MPC naturally involves more complexoptimization problems than linear MPC does, a fact that immediately exacerbates the problem of develop-ing sufficiently fast optimization algorithms for on-line implementation. So far there is also no methodcomparable to Explicit MPC for linear models (see section 2.4) in sight, only suboptimal approximativetechniques based on linear multiparametric programming have been developed. Because of the namedissues, nonlinear MPC will not be addressed any further in this thesis. The use of MPC for unconstrainedlinear systems is more than questionable, since LQR/LQG control is considered to solve this problemwell (Bitmead et al. (1990)). In fact, is is easy to show that if the LQR infinite horizon cost is used asa terminal cost in the unconstrained MPC formulation, the LQR controller itself is recovered for anyprediction horizon. Clearly, implementing this kind of controller would be an absurd thing to do.

Over time, numerous different approaches to the question of how to ensure stability of the closed-loop sys-tem when using a Model Predictive Controller have been pursued in the literature, all of which have someimportant features in common. The following review will only encompass those ideas that in the process ofa continuous refinement have been condensed out of these approaches and have become widely acceptedin the Model Predictive Control community. For details, the reader is referred to the excellent survey paperMayne et al. (2000) and the recent book Rawlings and Mayne (2009), on which much of the content andnotation of this introductory section is based. In order to allow for an overall coherent exposition, this basicnotation will also be adopted (and, if necessary, extended) throughout the following chapters of this thesis.

As its name suggests, Model Predictive Control is based on an underlying model of the process to becontrolled. In the context of MPC for constrained linear systems one usually considers a discrete-timelinear system of the form

x(t + 1) = Ax(t) + Bu(t)y(t) = C x(t),

(2.1)

where x(t) ∈ Rn, u(t) ∈ Rm and y(t) ∈ Rp are the system state, the applied control action, and thesystem output at time t, respectively. The matrices A ∈ Rn×n, B ∈ Rn×m, and C ∈ Rp×n are the systemmatrix, the input matrix and the output matrix, respectively.

1 nonlinear MPC refers to MPC of nonlinear models, as MPC is, even for linear systems, inherently nonlinear

5

System (2.1) is subject to the following constraints on state and input:

x(t) ∈ X, u(t) ∈ U, (2.2)

where the control constraint set U ⊂ Rm is convex and compact (closed and bounded), and the stateconstraint set X ⊂ Rn is convex and closed. Both U and X are assumed to contain the origin in theirinterior, i.e. 0 ∈ int(U) and 0 ∈ int(X).

2.1 The Regulation Problem

For now, the attention will be restricted to the regulation problem only. In the regulation problem, theobjective is to optimally (with respect to some performance measure) steer the state x(t) of system (2.1)to the origin while satisfying the constraints (2.2) at all times. In order for this to be an achievable goal,the following (reasonable) assumption is necessary and will be adopted throughout this thesis:

Assumption 2.1:The pair (A, B) is controllable.

2.1.1 The Model Predictive Controller

As with all Model Predictive Controllers, a Receding Horizon Control strategy as described in section 1.2 isemployed to control the system (2.1). Hence, at each time step, a finite horizon optimal control problemneeds to be solved. The cost function VN (·) of this optimal control problem is defined by

VN (x(t),u(t)) :=t+N−1∑

i=t

l(x i, ui) + Vf (x t+N ) (2.3)

where N is the prediction horizon, u(t) :=

ut , ut+1, . . . , ut+N−1

denotes the sequence of predictedcontrol inputs and x(t) :=

x t , x t+1, . . . , x t+N

denotes the predicted state trajectory whose elementssatisfy x i+1= Ax i+Bui. For clarity it makes sense to distinguish between the notations x(t) and x t+i asfollows: The argument in parentheses denotes actual values, whereas the subscript denotes predictedvalues. Thus, x(t) denotes the actual state of the system at time t, whereas x t+i denotes the predictedstate of the system at time t + i, given the information about the system at time t. Because the currentstate is measured and the first element of the predicted control sequence is really applied to the system,the actual values of the respective elements x t and ut of u(t) and x(t) are given by x t= x(t) and ut=u(t).

The stage cost function l(·, ·) in (2.3) is a positive definite function of both state x and control input u,satisfying l(0,0) = 02, and is usually chosen as

l(x i, ui) := ||x i||2Q + ||ui||

2R (2.4)

where ||x ||2Q := x TQx and ||u||2R := uT Ru denote the squared weighted euclidean norms with the positivedefinite state weighting matrix Q 0 and control weighting matrix R 0, respectively. Similarly, theterminal cost function Vf (·) is also a positive definite function of the state and satisfies Vf (0) = 0. Forreasons that will become clear in the following sections, the terminal cost is usually chosen of the form

Vf (x t+N ) := ||x t+N ||2P (2.5)

2 in the following 0 will denote the scalar zero, whereas 0 will denote the zero matrix or the zero vector of appropriatedimension. Similarly, I will denote the identity matrix of appropriate dimension

6 2. Model Predictive Control for Constrained Linear Systems

with a terminal weighting matrix P0. In addition to the externally specified constraints (2.2), a terminalconstraint of the form

x t+N ∈ X f (2.6)

is imposed on the predicted terminal state x t+N .

Remark 2.1 (Polytopic norms in the cost function):It is not necessary to use a cost function based on quadratic norms in the formulation of the optimizationproblem. The use of polytopic norms is beneficial from a computational point of view as it yields a LinearProgram (LP), for which very efficient and reliable solvers exist. However, this type of cost may result in aninferior closed-loop behavior compared to the use of a quadratic cost (Rao and Rawlings (2000)).

Since the system and its constraints are assumed to be time-invariant, it is possible to simplify notation byrewriting the system dynamics (2.1) as

x+ = Ax + Bu

y = C x ,(2.7)

where x+ denotes the successor state of x . The cost function VN (·) only depends on the value of thecurrent state and not on the current time t, therefore one may write

VN (x ,u) :=N−1∑

i=0

l(x i, ui) + Vf (xN ), (2.8)

where u :=

u0, u1, . . . , uN−1

and x := x0, x1, . . . , xN−1, with x0 := x . This formulation is equivalentto (2.3). Let Φ(i; x ,u) denote the solution of (2.7) at time i controlled by u when the initial state attime 0 is x (by convention, Φ(0; x ,u) = x). Furthermore, for a given state x , denote by UN (x) the set ofadmissible control sequences u, i.e.

UN (x) =¦

u | ui ∈ U, Φ(i; x ,u) ∈ X for i=0, 1, . . . , N−1, Φ(N ; x ,u) ∈ X f

©

. (2.9)

Let XN denote the domain of the value function V ∗N (·), i.e. the set of initial states x for which the the setof admissible control sequences UN (x) is non-empty:

XN =

x | UN (x) 6= ;

. (2.10)

Usually, one refers to XN as the region of attraction of the Model Predictive Controller. At each time t, itis assumed that the current state x of the system is known. The sequence of optimal predicted controlinputs u∗(x) for a given state x is obtained by minimizing the cost function (2.8). Denote by PN (x) thefollowing finite horizon constrained optimal control problem:

V ∗N (x) =minu

VN (x ,u) | u ∈ UN (x)

(2.11)

u∗(x) = arg minu

VN (x ,u) | u ∈ UN (x)

. (2.12)

At each sampling instant, PN (x) is solved on-line, and the first element u∗0(x) of the predicted optimalcontrol sequence u∗(x) is applied to the system. The repeated execution of measuring the state, computingthe optimal control input and applying it to the plant can be regarded as an implicit time-invariant ModelPredictive Control law κN (·) of the form

κN (x) := u∗0(x). (2.13)

The dynamics of the closed-loop system can then be expressed as

x+ = Ax + BκN (x)y = C x .

(2.14)

2.1. The Regulation Problem 7

2.1.2 Stability of the Closed-Loop System

In order to discuss the parameter choices that are necessary to guarantee stability and feasibility of theModel Predictive Controller, the notion of a positively invariant set is required. Invariant sets play animportant role in Control Theory and are used extensively especially in Model Predictive Control (Rakovic(2009)). For a detailed treatment of the theory and application of set invariance in controls, the reader ispointed to the very good survey paper Blanchini (1999) and the recent book Blanchini and Miani (2008).

Definition 2.1 (Positively invariant set, Blanchini (1999)):A set Ω is said to be positively invariant (PI) for the autonomous system x(t + 1) = f (x(t)) if, forall x(0) ∈ Ω, then the solution x(t) ∈ Ω for all t > 0.

Corollary 2.1 (Positive invariance for closed-loop linear systems):Let κ(·) : Rn 7→ Rm be a state-feedback controller (not necessarily a linear one). A set Ω⊆ Rn is positivelyinvariant for the closed-loop system (2.14) if, for all x ∈ Ω, then Ax + Bκ(x) ∈ Ω.

Stability of MPC is usually established using Lyapunov arguments, where the optimal cost V ∗N (·) is used asa Control Lyapunov Function (CLF). The additional assumptions on the terminal set X f and the terminalcost function Vf (·) that are necessary to ensure stability and feasibility of the closed-loop system (2.14)can be summarized as follows:

Assumption 2.2 (Mayne et al. (2000)):

1. All states inside the terminal set X f satisfy the state constraints, X f is closed, and it contains theorigin, i.e. 0 ∈ X f ⊂ X

2. The control constraints are satisfied inside the terminal set, i.e. κ f (x) ∈ U, ∀x ∈ X f ,where κ f (·) : Rn 7→ Rm is a local state-feedback controller

3. The terminal set X f is positively invariant under κ f (·), i.e. Ax + Bκ f (x) ∈ X f , ∀x ∈ X f

4. The terminal cost function Vf (·) is a local Control-Lyapunov function (CLF), i.e.Vf (Ax + Bκ f (x))≤ Vf (x)− l(x ,κ f (x)), ∀x ∈ X f

Assumption 2.2 is straightforward: items 1 and 2 ensure feasibility of the origin, of all states withinthe terminal set X f , and of all control actions generated by the terminal feedback controller κ f (·)acting on any state within X f . Item 3 ensures persistent feasibility of the states and control inputsbeyond the actual prediction horizon N while item 4 ensures stability by requiring that the terminalcost along the trajectory of the closed-loop system controlled by terminal controller κ f (·) is non-increasing.

The following theorem states the main stability result for nominal MPC for constrained linear systems:

Theorem 2.1 (MPC stability, Rawlings and Mayne (2009)):Suppose that the cost function is of the form (2.8) and that Assumption 2.2 holds. Then, ifXN is bounded,the origin is exponentially stable with a region of attraction XN for the system x+= Ax+BκN (x). If XN isunbounded, the origin is exponentially stable with a region of attraction that is any sublevel set of V ∗N (·).

Although the policy of this thesis is to generally refrain from stating proofs for results drawn from externalreferences (and instead point the reader to the corresponding references), an exception will be made forthe above theorem. The reason for doing this is that Theorem 2.1 is a fundamental basis for everythingthat will be presented in the later chapters of this thesis. Furthermore, the proof serves as a template formore or less all MPC stability proofs, and its basic idea can be found throughout the MPC literature andother proofs within this thesis. The proof of Theorem 2.1 is given in Appendix A.1.


2.1.3 Choosing the Terminal Set and Terminal Cost Function

Theorem 2.1 merely requires the terminal cost function Vf (·) to be any local CLF for the system (2.7).Clearly, if Vf (·) was chosen as the infinite horizon cost function V ∗∞(·) (obtained by taking the limitN →∞ in (2.8)), then, by the principle of optimality (Bertsekas (2007)), V ∗N (·) = V ∗∞(·). Regardless of theprediction horizon N , infinite horizon optimal control would be recovered. However, due to the presenceof the constraints, V ∗∞(·) is unknown. A common move in MPC therefore is to choose Vf (·) as V∞(·), thecost of the unconstrained infinite horizon optimal control problem:

V ∗∞(x) =minu

∞∑

i=0

l(x i, ui) =minu

∞∑

i=0

||x i||2Q + ||ui||

2R (2.15)

The solution of (2.15) is the well known solution of the classic discrete-time LQR problem, i.e.

V ∗∞(x) = ||x ||P∞= x T P∞x , (2.16)

with P∞ being the unique positive definite solution to the Discrete-Time Algebraic Riccati Equation (ARE)

P∞ =Q+ AT (P∞− P∞B(R+ BT P∞B)−1BT P∞)A. (2.17)

The optimal linear feedback controller that minimizes (2.15) (often times referred to as “Kalman-Gain”)is given as K∞ =−(R+ BT P∞B)−1BT P∞ A. If one chooses Vf (·) = V ∗∞(·) or, equivalently, P = P∞ in (2.5),Assumption 2.2 requires the terminal set X f to be positive invariant under κ f (x) = K∞x , and that stateand control constraints be satisfied, i.e. X f ⊂ X and κ f (x) ∈ U for all x ∈ X f . A set that satisfies theabove is called a constraint admissible positively invariant set.

Definition 2.2 (Constraint admissible positively invariant set, Kerrigan (2000)):Consider the autonomous system x+= (A+BK)x subject to the constraints x∈X and K x∈U. A positivelyinvariant set Ω for this system is constraint admissible if Ω⊆ X and KΩ⊆ U.

The essential role of the terminal set X f is to permit the replacement of the actual infinite horizoncost V ∗∞(·) by the infinite horizon cost V ∗∞(·) of the unconstrained system (Mayne et al. (2000)). In orderto obtain a region of attraction XN of the Model Predictive Controller as large as possible, X f is usuallychosen as the maximal positively invariant set for the closed-loop system x+= (A+ BK∞)x .

Definition 2.3 (Maximal positively invariant set, Kerrigan (2000)):A constraint admissible positively invariant set Ω for the autonomous system x+= (A+ BK)x subject tothe constraints x ∈ X and K x ∈ U is said to be the maximal positive invariant (MPI) set Ω∞ if 0 ∈ Ω∞and Ω∞ contains every constraint admissible invariant set that contains the origin.

Note that there is quite a variety of different definitions for invariant sets in the literature. All of thesedefinitions differ slightly from each other, but basically describe the same concepts. One well-knowndefinition is for example that of the maximal output admissible set (Gilbert and Tan (1991)), which isessentially the same as the definition of the maximal positively invariant set used in this thesis.

For the previously assumed case that the state constraint set X contains an open region around the origin,Gilbert and Tan (1991); Kolmanovsky and Gilbert (1998) show that if A+ BK∞ is Hurwitz3 (which canalways be achieved if Assumption 2.1 holds true), the MPI set Ω∞ exists and contains a nonempty regionaround the origin. Clearly, if the terminal set X f is chosen as the MPI set Ω∞ it follows from Definition 2.3that Vf (x) = V ∗∞(x) for all x ∈ X f .

3 a quadratic matrix is Hurwitz when all its eigenvalues lie strictly inside the unit disk

2.1. The Regulation Problem 9

The following Lemma states two interesting results about the optimality of the solution obtained from theconstrained finite horizon optimal control problem PN (x).

Lemma 2.1 (Optimality of the solution):Let Vf (·) = V∞(·) in the cost function (2.8), and let Assumption 2.2 be satisfied. Then,

1. V ∗N (x) = V ∗∞(x) for all x∈X f

2. V ∗N (x) = V ∗∞(x) for all x ∈ XN for which the terminal constraint xN ∈ X f is inactive in problem PN (x)

Proof. Both items 1 and 2 of Lemma 2.1follow directly from the principle of optimality (Bertsekas (2007))and from the fact that Vf (x) = V ∗∞(x) for all x ∈ X f .

Remark 2.2 (Synthesizing controllers that provide “real” optimality):Item 1 in Lemma 2.1 states that the unconstrained LQR controller is recovered for all initial states x ∈ X f .This is because for all states within the terminal set X f the unconstrained LQR controller is persistentlyfeasible and hence optimal. Item 2 states the less obvious fact that in case the terminal constraint is inactive,the optimal cost V ∗N (x) of the finite horizon optimal control problem PN (x) is equal to the optimal cost V ∗∞(x)of the constrained infinite horizon optimal control problem. An interesting approach that somewhat reverselyexploits this fact is taken by the authors of Sznaier and Damborg (1987); Scokaert and Rawlings (1998). Theypropose to employ a variable prediction horizon N in the following fashion: At each time step, problem PN (x)is solved without explicitly invoking the terminal constraint xN ∈ X f for increasing prediction horizons N ,starting from some small initial horizon N0. The prediction horizon is then increased until the predictedterminal state satisfies xN ∈ X f . A similar approach was proposed in the context of switched linear systemsby Balandat et al. (2010).

2.1.4 Solving the Optimization Problem

In order to implement a Model Predictive Controller in practice, the optimization problem PN (x) frompage 7 must be solved on-line at each sampling instant4. If stage and terminal cost in the cost function (2.8)are chosen as (2.4) and (2.5), respectively, the objective function VN (x ,u) of PN (x) is quadratic. The stateand control weighting matrices Q and R are given and the terminal weighting matrix P is easily obtainedby solving the ARE (2.17) of the associated unconstrained LQR problem off-line using standard algorithms.

So far there have been no assumptions made on the nature of the constraint sets X, U and X f . In casethese sets are polytopic (i.e. they can be represented by the intersection of a finite set of closed halfspaces(Ziegler (1995))), then the optimization problem PN (x) becomes comparably easy to solve. Therefore,the common assumption made in the literature is the following:

Assumption 2.3 (Nature of the constraint sets):The constraint sets X, U and X f in problem PN (x) are polytopic.

Remark 2.3 (Implied polytopic shape):Note that it in Assumption 2.3 it would actually be sufficient to only assume that X and U are polytopic. Thisis because linearity of the system then implies that the maximal positively invariant set, which will usually bethe choice for X f , is also polytopic (Gilbert and Tan (1991); Kolmanovsky and Gilbert (1998)).

4 There exist MPC algorithms that skip measurements and apply the computed optimal control input with some delay, henceallowing for more than one sampling interval for solving PN (x). This however significantly complicates analysis in thepresence of uncertainty. Therefore, these and other variants of MPC are not considered here for simplicity


Under Assumption 2.3, the optimization problem PN (x) can be posed as

V ∗N (x) = minu0,...,uN−1

x0,...,xN

x TN P xN +

N−1∑

i=0

x Ti Qx i + uT

i Rui

s.t. x i+1 = Ax i + Bui i = 0, . . . , N−1

x0 = x

Hx x i ≤ kx i = 0, . . . , N−1

Huui ≤ ku i = 0, . . . , N−1

H f xN ≤ k f ,

(2.18)

where X = x | Hx x ≤ kx, U = u | Huu ≤ ku and X f = x | H f x ≤ k f are the “H -representations”(Ziegler (1995)) of the respective polyhedral sets. Clearly, (2.18) is a Quadratic Programming prob-lem (QP), which can be solved fast, efficiently and reliably using modern optimization algorithms (Boydand Vandenberghe (2004); Nocedal and Wright (2006); Dostál (2009)).

Obtaining the Terminal Set X fThe only remaining question now concerns the computation of the terminal set X f . As outlined in Gilbertand Tan (1991); Blanchini (1999); Blanchini and Miani (2008), there exist efficient algorithms for thecomputation of (polytopic) maximal positively invariant sets for polytopic constraint sets. Hence, withthe Kalman-gain K∞ obtained from the unconstrained LQR problem, the terminal set X f can easily becomputed off-line as the maximal positively invariant set Ω∞ of the closed-loop system x+= (A+ BK∞)x .An algorithm for the computation of Ω∞ as well as some computational issues pertaining to it will bediscussed in section 4.5.1 in the context of Tube-Based Robust MPC.

2.2 Robustness Issues and Output-Feedback Model Predictive Control

An important question that arises in the assessment of any control strategy is how well the controllerdeals with uncertainty. In Model Predictive Control this question is: What happens if, due to the effectsof uncertainty, the predicted evolution of the nominal system is different from its actual behavior? Atbest, the only thing that will happen when the nominal controller is used for the uncertain system isa degradation in performance. But if the uncertainty is “large”, or if the closed-loop system has smallrobustness margins, the controlled uncertain system may also become unstable. The Robust Control issueis generally much harder to come by for constrained systems, as the control objective is to not only ensurerobust stability but also robust constraint satisfaction.

The causes of uncertainty are manifold: there may be modeling errors, the state of the system may not beexactly known, or exogenous disturbances may affect the system. Evidently, all of the above applies, atleast to some degree, to virtually any practical application. Hence, it is clear that controllers need to berobust with respect to these uncertainties. Chapter 3 therefore introduces the Robust MPC framework,whose objective is to synthesize robust controllers based on some underlying model of the uncertainty.

Output-Feedback Model Predictive ControlOne particular kind of uncertainty which is prevalent in more or less any real-world situation are measure-ment errors. The previous sections dealt with the design of Model Predictive Controllers based on the exactknowledge of the current state x of the system. In applications however, exact measurements of all systemstates are generally not available. Thus, it is necessary to develop extensions of the classic “state-feedbackMPC” to “output-feedback MPC”, which only use the available (possibly inaccurate) measurements of thesystem’s output y .

2.2. Robustness Issues and Output-Feedback Model Predictive Control 11

It is standard practice in control engineering to combine a state-feedback controller with an observer thatestimates the system state x from the available measurements y (Skogestad and Postlethwaite (2005)).If the observer is chosen as a Kalman-Filter (Kalman (1960)) this approach is usually referred to as“LQG-Control”. In virtue of the separation principle (Luenberger (1971)), closed-loop stability can thenbe ensured for the composite (linear) system. However, due to the nonlinear control law, the separationprinciple in general does not hold for systems controlled by Model Predictive Controllers (Teel and Praly(1995)). As a result things become more involved and additional caution is needed in the design ofoutput-feedback MPC. Albeit numerous reasons for why it is actually not a great idea, nominal ModelPredictive Controllers in loop with separately designed observers to date are still widely used in industry(Rawlings and Mayne (2009)).

A variety of different ideas for a more theoretical approach to output-feedback MPC have been proposedin the literature (Bemporad and Garulli (2000); Löfberg (2002); Chisci and Zappa (2002); Findeisen et al.(2003)). A brief overview over some important contributions is given in section 3.6.1 in the context ofRobust MPC. Moreover, the recently proposed and very elegant method of “output-feedback Tube-BasedRobust Model Predictive Control” will be presented in more detail in chapter 4.

2.3 Reference Tracking

So far the exposition has been concerned solely with the MPC regulation problem, the goal of whichis to steer the system state to the origin. But what application engineers are really interested in isto optimally track a given output reference trajectory yre f (t), while ensuring that state and controlconstraints are satisfied at all times. Obtaining general theoretical results on stability, feasibility, androbustness for constrained tracking of arbitrary, time-varying references is however extremely hard.Instead, the tracking problem is often confined to the problem of optimally tracking arbitrary, but constantreference signals yre f (t) = ys. This is commonly referred to as “set point tracking”. The following sectionswill give a short introduction on how constrained set point tracking controllers can be realized usingModel Predictive Control methods.

2.3.1 Nominal Set Point Tracking Model Predictive Control

Set Point CharacterizationFor the purpose of this section, assume that there is no uncertainty present. In this case, for the output y(t)of system (2.1) to be able to track a constant reference signal ys (set point) without exhibiting any offset,there must exist a feasible steady state5 (xs, us) ∈ X×U satisfying

xs = Axs + Bus (2.19)

ys = C xs. (2.20)

Hence, for any set point ys, there must exist a steady state input us such that

C(I− A)−1B us = ys. (2.21)

In order to hold for arbitrary set points ys ∈Rp, the mapping in (2.21) needs to be surjective, i.e. thematrix C(I− A)−1B ∈ Rp×m must have full row rank. The following Lemma states an equivalent condition:

5 with some abuse of notation the pair (xs, us) of actual steady-state xs and associated constant control input us will in thefollowing simply be referred to as “the steady state”


Lemma 2.2 (Pannocchia and Rawlings (2003)):Consider the linear discrete-time system (2.1). If and only if

rank

I− A −BC 0

= n+ p, (2.22)

then there exists an offset-free steady state (xs, us) for any constant set point ys.

Remark 2.4 (Pannocchia and Rawlings (2003); Pannocchia and Kerrigan (2005)):Note that condition (2.22) in Lemma 2.2 implies that the number of controlled variables cannot exceed eitherthe number of control inputs or the number of states, i.e. p ≤minm, n.

If (2.22) holds true then the steady state (xs, us) is not necessarily unique for a given set point ys. Ifthis is the case, a common approach (Muske and Rawlings (1993); Scokaert and Rawlings (1999)) isto determine artificially a unique steady state (x∗s , u∗s ) by solving the following quadratic optimizationproblem:

(x∗s , u∗s ) = argminxs ,us

(us − ud)T Rs(us − ud)

s.t. xs = Axs + Bus

ys = C xs

xs ∈ Xus ∈ U,

(2.23)

where ud is a desired steady state input and Rs0 is a positive definite weighting matrix penalizing thedeviation of us from ud . The equality constraints in (2.23) hereby ensure that the obtained pair (x∗s , u∗s ) isindeed a steady state for the system (2.1). Obviously, if ud is an admissible steady state input, then u∗s = ud .

If, on the other hand, the system does not provide enough degrees of freedom to track the desiredreference ys without offset, one can determine a feasible steady state (x∗s , u∗s ) such that the output trackingerror is minimized in the least-squares sense (Muske and Rawlings (1993); Muske (1997)) by solving thefollowing quadratic program:

(x∗s , u∗s ) = arg minxs ,us

(ys − C xs)TQs(ys − C xs)

s.t. xs = Axs + Bus

xs ∈ X,

us ∈ U,

(2.24)

where Qs 0 is a positive definite weighting matrix penalizing the output tracking error. The actualoutput that is achieved at steady state then is ys = C x∗s . For the purpose of the remainder of this section itwill be assumed that the condition in Lemma 2.2 is satisfied, i.e. that for any target set point ys thereexists feasible offset-free steady state (xs, us).

Nominal Model Predictive Control for TrackingGiven an offset-free steady state pair (xs(ys), us(ys)) for the desired set point ys, a Tracking ModelPredictive Controller (Rawlings and Mayne (2009)) is realized by solving on-line the modified optimalcontrol problem PN (x , ys)

V ∗N (x , ys) =minu

VN (x , ys,u) | u ∈ UN (x , ys)

(2.25)

u∗(x , ys) = arg minu

VN (x , ys,u) | u ∈ UN (x , ys)

, (2.26)

2.3. Reference Tracking 13

in which the cost function VN (·) and the set of admissible control sequences UN , which now both dependon the value of the target set point ys, are defined by

VN (x , ys,u) :=N−1∑

i=0

l(x i − xs(ys) , ui − us(ys)) + Vf (xN − xs(ys)) (2.27)

UN (x , ys) =¦

u | ui ∈ U, Φ(i; x ,u) ∈ X for i=0, 1, . . . , N−1, Φ(N ; x ,u) ∈ X f (ys)©

. (2.28)

Terminal cost function Vf (·) and stage cost function l(·, ·) are again chosen as (2.5) and (2.4), therespective cost functions used in the regulation problem. The reference-dependent terminal set X f (ys)can, since the system is linear, be chosen as a shifted version of the terminal set X f from the regulationproblem (which is centered at the origin), i.e.6

X f (ys) =

xs(ys)

⊕X f ⊂ X, (2.29)

where the steady state xs(ys) must satisfy the additional constraint that all states within the shiftedterminal set X f (ys) be contained in X. This additional requirement limits the set points that can betracked to the set

Ys :=¦

ys | xs(ys)⊕X f ∈ X, us(ys) ∈ U©

. (2.30)

For a given admissible set point ys ∈ Ys the region of attraction XN of the tracking controller is

XN (ys) :=

x | UN (x , ys) 6= ;

. (2.31)

Remark 2.5 (Choice of the terminal set):Note that it is not possible to simply use a terminal set of the form

xs(ys)⊕X f

∩X, as this set, in general,is not positively invariant and therefore does not ensure persistent feasibility of the closed-loop system beyondthe prediction horizon. One possible way to enlarge the set of trackable set points Ys is to choose a terminalset of the form xs(ys)⊕αX f with α ∈ [0,1], which is positively invariant (because of linearity). However,this choice at the same time yields a controller with a smaller region of attraction XN . Alternatively, anappropriate positively invariant terminal set could be recomputed on-line for each new set point ys. Theon-line computation of positively invariant sets is however computationally prohibitive in general.

Employing the Receding Horizon Control approach and applying only the first element u∗0(x , ys) of thepredicted sequence u∗(x , ys) of optimal control inputs to the system, the implicit Model Predictive Controllaw is given by

κN (x , ys) := u∗0(x , ys). (2.32)

Stability of the steady state (and hence of the set point ys) can be established as follows:

Corollary 2.2 (Stability of nominal Tracking MPC, Rawlings and Mayne (2009)):Suppose that the rank condition (2.22) holds, that ys∈Ys is an admissible constant reference set point,and that (xs(ys), us(ys)) is an associated steady state of the system satisfying both state and controlconstraints. Furthermore, suppose that the cost function is of the form (2.27) and that Assumption 2.2holds with the terminal set X f (ys) as in (2.29). Then, the steady state xs(ys) is exponentially stable witha region of attraction XN (ys) =

x | UN (x , ys) 6= ;

for the closed-loop system x+ = Ax + BκN (x , ys).

6 in (2.29) the so-called Minkowski set addition, denoted by ⊕, is used. Its formal definition is postponed to Definition 4.3to allow for a more coherent presentation of chapter 4. Here it can simply be seen as shifting the set X f by xs(ys)


As indicated in Remark 2.5, Tracking MPC is generally more involved than just shifting the system to thedesired steady state. In particular, for this approach to be feasible it is necessary that (2.29) is satisfied.This requirement, however, may lead to a potentially small region of admissible steady states. For theregulation problem, one generally wants to use a terminal set X f as large as possible, such that theregion of attraction XN of the controller is also large. If the size of X f is primarily limited by the stateconstraints X (meaning that the set U of feasible control actions is comparatively large), then (2.29)will be satisfied only for steady states xs close to to origin. Conversely, in order to enlarge the region ofadmissible steady states it is necessary to use a smaller terminal set X f . Hence, the method of shifting theoriginal system to a desired steady state to achieve tracking is inherently a tradeoff.

An Improved Approach to Tracking Model Predictive ControlAn elegant method for Tracking Model Predictive Control is developed in Limon et al. (2005, 2008a);Ferramosca et al. (2009a). The proposed controller features an additional artificial steady state to whichthe system is driven. An additional offset cost penalizing the deviation of the artificial steady statefrom the steady state corresponding to the output reference is introduced in the cost function. Sincethe constraints do not depend on the desired output reference, the controller ensures feasibility for allreference values and drives the system to the closest admissible steady state. The details of this approachwill be discussed in the context of Tube-Based Robust MPC in chapter 4 of this thesis. A further extensiondeveloped in Ferramosca et al. (2009b, 2010) addresses the problems of tracking target sets, i.e. whenthe desired output reference is required to lie in a specific set in the output space, where the exact valuesof the outputs are not important.

2.3.2 Offset Problems in the Presence of Uncertainty

In the previous sections, it was assumed that a perfect model of the system was available and that therewere no external disturbances present. In this nominal case, if there exists a feasible steady state for thegiven output set point ys, offset-free tracking is achieved, i.e. limt→∞ y(t) = ys. In real-world applications,however, perfect models do not exist and the system will always be subject to some exogenous disturbance.It is well known that if there is a non-vanishing disturbance (with zero mean) present, standard ModelPredictive Control methods as discussed in the previous sections generally exhibits offset, i.e. there isa mismatch between measured and predicted outputs. This is true for both tracking controllers andregulators (clearly, ys,reg = xs,reg = us,reg = 0).

The prevailing approach in the literature to overcome this deficiency is to augment the system statewith fictitious integrating disturbances (Maeder et al. (2009); Muske and Badgwell (2002); Pannocchiaand Kerrigan (2005); Pannocchia (2004); Pannocchia and Rawlings (2003)). By doing so it is possible,under proper conditions, to achieve offset-free MPC given that the external disturbance is asymptoticallyconstant. This idea will be revisited in more detail in section 3.6.3. In the context of Tube-Based RobustMPC, section 4.4.3 furthermore presents a method that enables offset-free control without augmentingthe system state. This is achieved by determining the offset value from the measured output and a stateestimate, and by scaling the reference input appropriately such that this offset is cancelled.

2.3.3 Reference Governors

As another way of addressing the constrained tracking control problem, so-called “reference governors”(also: “command governors”) have been proposed (Bemporad and Mosca (1994b,a, 1995); Gilbert et al.(1995); Bemporad et al. (1997)). The basic concept of reference governors is to separate the issue ofconstraint satisfaction from the issue of designing a stable closed-loop system. A reference governoris an auxiliary nonlinear device that operates between the external reference command and the input

2.3. Reference Tracking 15

Figure 2.1.: Reference governor block diagram (Gilbert and Kolmanovsky (2002))

to the primal compensated control system, as depicted in Figure 2.1. The primal plant compensation,which may be performed using a wide range of different controller synthesis methods, yields a stableclosed-loop system that performs satisfactorily in the absence of constraints. Whenever it is necessary, thereference governor modifies the reference input yre f (t), generating a virtual reference input y∗re f (t) so asto avoid constraint violation of the pre-compensated system. The necessary modification of yre f (t) can beperformed in different ways: While Bemporad and Mosca (1994a); Bemporad et al. (1997); Casavolaand Mosca (1996); Chisci and Zappa (2003) and related concepts employ forms of Model PredictiveControl, the approach advocated by Gilbert et al. (1994, 1995) uses a nonlinear low-pass filter. TheMPC-based approach generates the virtual reference input y∗re f (t) by solving on-line a mathematicaloptimization problem. The nonlinear filter, on the other hand, attenuates, if necessary, the externalreference input yre f (t) appropriately.

In addition to the nominal case, robust reference governors have been proposed for uncertain systemsin Casavola and Mosca (1996); Gilbert and Kolmanovsky (1999a); Casavola et al. (2000b). Theseapproaches take uncertainties in the model and external disturbances directly into account. Moreover, theideas behind reference governors have also been used for the purpose of controlling nonlinear systems(Bemporad (1998b); Angeli and Mosca (1999); Gilbert and Kolmanovsky (1999b, 2002)). Referencegovernors are an interesting approach to guaranteeing constraint satisfaction, in particular as they leavethe basic stability properties of the pre-compensated system untouched. This allows for a large flexibilityin the design of controllers for the pre-stabilization of the plant. However, the resulting overall controlleris clearly not “optimal” in the sense a Model Predictive Controller would be. Among other drawbacks,this usually results in smaller regions of attraction (Casavola et al. (2000a)). Nevertheless, referencegovernors are a useful tool in constrained control.

2.4 Explicit MPC

Section 2.1.4 argues that modern optimization algorithms are able to solve the quadratic optimizationproblem that arises in linear MPC fast and efficiently. However, the computational effort may still exceedthe available computing resources, especially when the objective is to control fast7 processes with limitedhardware resources. A very interesting approach to overcome this problem was proposed in the seminalpaper by Bemporad et al. (2002), where the authors advocated the explicit solution of the optimizationproblem (2.18). The proposed approach is based on multiparametric programming, for which the currentstate x of the system is regarded as a parameter vector in the optimization problem. This way, the on-linecomputation effort can be reduced to a simple function evaluation (of a piece-wise affine optimizerfunction defined over polytopic regions in the state space). The following sections discuss only thebasic ideas and properties of Explicit MPC, for further reading consult Bemporad et al. (2002); Tøndelet al. (2003b); Alessio and Bemporad (2009); Borrelli et al. (2010). Explicit controllers will also beimplemented in the context of Tube-Based Robust MPC and Interpolated Tube MPC in chapter 4 and 5.

7 the definition of “fast” is of course constantly changing with the rapid progress in computer technology


2.4.1 Obtaining an Explicit Control Law

In section 2.1.4 it has been shown how the on-line optimization problem PN (x) of a nominal ModelPredictive Controller is readily formulated as the quadratic programming problem (2.18). By substitutingx i = Ai x+

∑i−1j=0 AjBui−1− j and performing a (linear) change of variables, the optimization problem (2.18)

can be rewritten in the new variable z as

V ∗N (x) =minz

zT Hz

s.t. Gz ≤W + Sx ,(2.33)

where H 0, G, W and S are suitably defined matrices depending on the system dynamics and theconstraints (Bemporad et al. (2002)). Note that in (2.33) the parameter vector x appears only on theright hand side of the constraints. This renders the problem a multiparametric Quadratic Program (mpQP),for which an explicit solution can be obtained. The properties of this solution are summarized in thefollowing important theorem:

Theorem 2.2 (Properties of the explicit solution, Bemporad et al. (2002); Alessio and Bemporad (2009)):Consider a multiparametric Quadratic Program of the form (2.33) with H 0. The setXN of parameters xfor which the problem is feasible is a polyhedral set, the value function V ∗N : XN 7→ R is continuous,convex, and piecewise quadratic (PWQ), and the optimizer function z∗ :XN 7→ RN×m is piecewise affine(PWA) and continuous.

Because of the linearity of the change of variables, the properties of the optimizer z∗ from Theorem 2.2are inherited by the optimal control sequence u∗(x) and hence also hold for the Model Predictive Controllaw κN (x) := u∗0(x). In contrast to the on-line solution, this control law can be stated explicitly as

κN (x) =

F1 x + g1 if H1 x ≤ k1...

......

FNregx + gNreg

if HNregx ≤ kNreg

(2.34)

where κN ,l(x) = Fl x+gl is the affine control law defined on the l th polyhedral regionPl :=

x | Hl x ≤ kl

of the solution. The domain XN of the value function V ∗N (·) is partitioned into Nreg different polyhedral

regions, i.e.⋃Nreg

l=1 Pl = XN . In case the constraint set X is unbounded, it is futhermore necessary toexplicitly specify a so-called exploration region Xx pl , since in this case the set XN may also be unbounded.Specifying an exploration region (that should contain the desired operation region of the controller)is however recommendable in any case, since otherwise the number of regions Nreg over which thecontroller is defined can grow very large (see section 2.4.2).

A number of different algorithms for the computation of the polyhedral regions Pl and the associatedaffine state-feedback control laws κN ,l(·) have been developed (Alessio and Bemporad (2009); Bemporadet al. (2002, 2001); Spjøtvold et al. (2006); Tøndel et al. (2003b)). A common feature of these algorithmsis that they are based on the first-order Karush-Kuhn-Tucker (KKT) optimality conditions (Boyd andVandenberghe (2004)) for (2.33), from which they identify “sets of active constraints” that characterizeso-called “critical regions” in the x-space. The region of attraction XN is then successively explored byobtaining the critical regions for all possible sets of active constraints. The algorithms differ mainly in theexploration strategy they use for covering the region of attraction with critical regions. The computation ofthe polyhedral partition and the piece-wise affine control law can be performed off-line, which reduces theon-line computation to a simple evaluation of (2.34). This allows the implementation of MPC controllersalso for fast dynamic systems with high sampling frequencies (Alessio and Bemporad (2009)).

2.4. Explicit MPC 17

Remark 2.6 (Other benefits of Explicit MPC):An additional, more subtle benefit of Explicit MPC is that guarantees on the solution (such as an upper boundon the computation time, etc.) are comparably easy to derive, since all computations involving mathematicaloptimization algorithms are performed off-line. If instead on-line optimization is carried out “in the loop”,it is generally very hard to obtain formal guarantees that are not excessively conservative, since everythingdepends on the (potentially very complex, often also closed-source8) optimization algorithm. This benefitof Explicit MPC is especially useful in safety-critical applications such as aerospace engineering, which aresubject to rigorous software certification processes.

Naturally, the approaches and algorithms of Explicit MPC can be applied to any optimization problemthat can be brought in the form (2.33). This includes other types of Model Predictive Controllers (e.g. thetracking controller from section 2.3), but more generally any optimization problem that depends affinelyon a parameter, and that needs to be solved repeatedly at high speeds.

2.4.2 Issues with Explicit MPC

Complexity of the SolutionAlthough Explicit MPC seems a compelling alternative to on-line optimization at the first glance, there arealso some drawbacks associated with it. A major problem is the growth of the number of regions Nreg intowhich the domain XN is partitioned (Alessio and Bemporad (2009); Borrelli et al. (2010)). Although Nreggrows only moderately with the state dimension (there is no “curse of dimensionality”), it depends, in theworst case, exponentially on the number of constraints in (2.33) (Bemporad et al. (2002)). The numberof constraints again primarily depends on NX, NU, and NX f

, the numbers of inequalities9 defining thepolytopic sets X, U, and X f , respectively. From that it is obvious that the constraint sets should be chosenas simple as possible. Furthermore, since the number of constraints increases linearly with the predictionhorizon N , exact Explicit MPC is limited to comparably short prediction horizons.

Different ideas have been presented in the literature that aim at reducing the complexity of ExplicitModel Predictive Controllers. Geyer et al. (2008) propose to merge those regions P for which theaffine gain F x + g is the same, so that the exact solution of the problem is expressed with a minimalnumber of partitions. Other approaches introduce sub-optimality in order to reduce the complexity ofthe explicit solution (Grieder and Morari (2003); Bemporad and Filippi (2003); Rossiter and Grieder(2004); Rossiter et al. (2005); Pannocchia et al. (2007); Christophersen et al. (2007)). Yet another wayis the use of semi-explicit methods as in Zeilinger et al. (2008), which pre-process off-line as much aspossible of the MPC optimization problem without characterizing all possible outcomes, but rather leavesome optimization operations for on-line computation. These topics as well as some further ideas are welldiscussed in the survey paper Alessio and Bemporad (2009).

Controller ImplementationThe on-line evaluation of the piecewise state-feedback control law (2.34) requires the determination ofthe polyhedron in which the measured state x of the system lies. This information is crucial in order todecide which “piece” of the piece-wise affine control law to apply to the plant. This procedure has beenreferred to as the “point location problem” in the literature. The easiest and most straightforward wayto implement an algorithm would be to store the polyhedral regions Pl and perform an on-line searchthrough them until the region Pl∗ which contains the current state x of the system is identified. Theoptimal control action is then simply obtained as u∗ = Fl∗ x + gl∗ . However, this approach is, in general,not very efficient in terms of evaluation time. Since the evaluation time is crucial for fast-samplingsystems, researchers have developed alternate methods, which are more efficient especially in case the

8 optimization software is a competitive business, companies will usually try everything to keep their code secret9 NX, NU, and NX f

appear as the number of rows of Hx , Hu, and H f in (2.18), respectively


solution is characterized by a large number of regions Nreg . These advanced approaches are, for example,based on the the use of binary search trees (Tøndel et al. (2003a)), PWA descriptor functions (Borrelliet al. (2001a); Baotic et al. (2008)), reachability analysis (Spjotvold et al. (2006)) or minimal volumebounding boxes (Christophersen (2007)). Research on the point location problem is ongoing, and efficientmethods to evaluate the explicit control law (2.34) will be a key prerequisite for the successful applicationof Explicit MPC in the future.

Other AspectsOne of the general shortcomings of Explicit MPC is that its underlying approach is hard to generalizeto nonlinear MPC. Although general nonlinear optimization algorithms are undoubtedly considerablyharder to implement in real-time than, for example, Quadratic Programming, the basic MPC ideas alsoapply to nonlinear problems (with appropriate modifications to the cost function and the terminal set).Exact multiparametric programming, however, is generally restricted to optimization problems with linear(mpLP) or quadratic (mpQP) cost functions subject to linear constraints. Therefore, approximative explicitsolution to nonlinear MPC problems have been proposed (Johansen (2002, 2004); Grancharova andJohansen (2006)). Another drawback of Explicit MPC is that it is not possible to change parameters of theoptimization problem (i.e. weighting matrices and constraints) on-line. If any parameter (except x ofcourse, which is regarded as the parameter vector in the multiparametric program) in (2.33) needs tobe adjusted, the explicit solution needs to be recomputed. This is not the case when the optimizationproblem is solved on-line10. On-line optimization based controllers therefore provide more flexibility bythe possibility of a quicker re-parametrization in applications.

2.4.3 Explicit MPC in Practice

When Should Explicit MPC be Used?It is very hard to make a general statement about when and for what kind of problems the use of ExplicitMPC is beneficial, since the complexity of the solution depends not only on the number of variables andconstraints in the optimization problem but also on the internal structure of the very problem itself. Thiswill become clear in the case studies of chapter 4, where different Explicit Tube-Based Robust ModelPredictive Controllers are implemented and compared to controllers based on on-line optimization. Inaddition, research on advanced, on-line optimization based MPC approaches that use specially tailoredsolvers that exploit the structure of the optimization problem is ongoing and reported results, for examplethose in Ferreau et al. (2008); Milman and Davison (2008); Wang and Boyd (2008); Biegler and Zavala(2009); Richter et al. (2009, 2010), are promising. Those specialized solvers are fast enough to competewith Explicit MPC, in particular for problems of higher complexity. Finally, the semi-explicit variants ofMPC mentioned in the previous section seem a suitable way of combining the advantages of both on-lineand off-line implementations. Explicit MPC to date can not yet be considered a mature technology. Furtherresearch effort is necessary to yield a practically applicable framework that is suitable for a noteworthynumber of real-world applications. Nevertheless, several successful applications of Explicit MPC havealready been reported in the literature by various researchers.

Successful Applications of Explicit MPCThe potential speed of Explicit MPC has been impressively demonstrated in a number of applications,most of which were in automotive and power systems control. Alessio and Bemporad (2009) characterizethe applications most suitable for Explicit MPC as fast-sampling problems (with a sampling time in therange of 1-50ms) of relatively small size (involving 1-2 manipulated inputs and 5-10 parameters). In thecontext of automotive control, Borrelli et al. (2001b, 2006) report the successful application of ExplicitMPC (for hybrid systems) to traction control. Recently, there have also been increasing research efforts to10 this does not mean that on-line optimization based MPC has “hot-plug” capabilities, since simply changing parameters of a

running controller induces other problems that have to be accounted for, i.e. stability and/or feasibility issues

2.4. Explicit MPC 19

implement Explicit MPC in hardware, in order to make it suitable for applications with very high samplingfrequencies that appear, for example, in mechatronics, the control of MEMS, automotive control, or powerelectronics. In this context, Johansen et al. (2007) showed that Explicit MPC implementations on anapplication specified integrated circuit (ASIC), which use about 20,000 gates, allow sampling frequenciesin the Mhz range. The authors also identified that the main problem of this kind of implementation isthat the memory requirements increase rapidly with the problem dimensions. In a more recent article,Mariéthoz et al. (2009) report an FPGA implementation of Explicit MPC for a buck DC-DC converter thatachieves sampling frequencies of up to 2.5Mhz.

Software for Explicit MPCAlgorithms for multiparametric programming and other tasks pertaining to the design of Explicit ModelPredictive Controllers are implemented in the free Multiparametric Toolbox (MPT) for MATLAB (Kvas-nica et al. (2004)). The Hybrid Toolbox (Bemporad (2004)), also free and developed for MATLAB,provides similar and some extended features. Note that the commercial Model Predictive ControlToolbox for MATLAB (Bemporad et al. (2010)) to date does not support the synthesis of Explicit ModelPredictive Controllers (as of version 3.2).


3 Robust Model Predictive ControlAlthough nominal Model Predictive Control as discussed in the previous chapter provides strong theo-retical results about nominal stability and feasibility, it does not consider the question of what happenswhen the predicted system evolution differs from the actual system behavior. It has become a widelyaccepted consensus that for practical applications, Model Predictive Controllers need not only guaranteeperformance under nominal operating conditions, but also provide satisfactory robustness against un-certainty. Possible causes for uncertainty are modeling errors, unknown or neglected system dynamicsor exogenous disturbances. The following chapter addresses the problem of how to account for thesedifferent kinds of uncertainty in the design of Model Predictive Controllers. It distills from the extensive lit-erature on Robust MPC the most influential ideas and approaches that have been proposed for this purpose.

The most straightforward approach to deal with uncertainty is of course to simply ignore it. The implica-tions of this ad-hoc “treatment” of uncertainty are outlined in section 3.1, where the inherent robustnessproperties of nominal MPC are examined. In order to be able to synthesize Robust Model PredictiveControllers, i.e. to incorporate prior knowledge about the uncertainty already during the design process,suitable models for this uncertainty are necessary. This is the topic of section 3.2.

Robust Model Predictive Control as a design tool had its first appearance in Campo and Morari (1987).This seminal contribution motivated to minimize a worst-case objective function, i.e. an objective functionthat corresponds to the worst possible realization of the uncertainty. This method is usually referredto as “Min-Max Model Predictive Control” and has become a widely adopted approach in the RobustMPC literature. The basic idea of Min-Max MPC is discussed in section 3.3, while actual Min-Maxcontroller implementations based on Linear Matrix Inequality (LMI) optimization techniques are reviewedin section 3.4. Section 3.5 deals with controllers that reduce the high on-line computational complexityfrom which many LMI-based Robust Model Predictive Controllers suffer. Finally, some extensions ofRobust MPC (e.g. to output-feedback and tracking control, and to explicit controller implementations)are presented in section 3.6. “Tube-Based Robust MPC” and the newly proposed “Interpolated Tube MPC”approach, though also Robust MPC methods, will be addressed separately in chapter 4 and 5, respectively.

3.1 Inherent Robustness in Model Predictive Control

As has been remarked, employing a Receding Horizon Control strategy introduces feedback into thesystem. It is well known that feedback control is superior to open-loop control, since it provide somedegree of robustness against perturbations (even if the controller is not specifically designed for thistask). It is natural to assume that similar qualities also hold for Model Predictive Controllers. However,the presence of constraints and the implicit form of the control law make robustness analysis of MPCcontrol loops a very difficult task (Bemporad and Morari (1999)). As a result, only a few approaches foranalyzing robustness of nominal MPC have appeared in the literature. The following section summarizessome important contributions that are concerned with robustness analysis of Model Predictive Control.

The authors of de Nicolao et al. (1996) are mostly concerned with nonlinear systems, but they alsostate that for linear MPC without state or control constraints, the robustness margins of the infinitehorizon LQR controller can be approximated indefinitely when using a suitably long prediction horizon N .Although this was found for terminal constrained MPC (the term “terminal constrained MPC” is usedfor MPC methods where X f =0), it can readily be extended to the case when using a (robust) positively

21

invariant terminal set with appropriate cost function, as has been discussed in chapter 2. However,in light of Remark 2.2, this finding for the linear case is only of theoretical value. Since the authorsassume neither state nor control constraints, one will always use a standard LQR controller which ismuch easier to implement and has superior robustness margins. Another result on robustness analysisof unconstrained MPC for general nonlinear systems can be found in Scokaert et al. (1997). In thiscontribution, the authors show that nonlinear Model Predictive Control with a quadratic cost providesnominal exponential stability. Under the additional assumption of Lipschitz continuity of the Model Pre-dictive Control law, they obtain an asymptotic stability result for systems subject to decaying perturbations.

A framework for robustness analysis of input constrained linear MPC was introduced in Primbs and Nevistic(2000). Sufficient LMI conditions for stability are developed, based on checking whether the finite horizoncost is decreasing along the trajectories for all possible uncertainties. However, this framework can not beeasily extended to the case when the system is subject to state constraints. In Teel (2004) and Grimm et al.(2004), it was found that it is possible for nonlinear systems controlled by a nominally stabilizing ModelPredictive Controller to have absolutely no robustness. Fortunately, it is also shown that this phenomenondoes not occur when the objective is to control a linear systems subject to convex constraints. In this case,the value function can be shown to be Lipschitz continuous, which is a sufficient condition for robustness.Under this assumption of Lipschitz continuity of the value function, it is futhermore possible to show thatfor system subject to bounded disturbances, there exists a set in which the perturbed closed-loop systemis Input-to-State stable (ISS) (Sontag (2008); Limon et al. (2009); Rawlings and Mayne (2009)).

Albeit the above results, which state that nominal Model Predictive Controllers for linear systems subject toconvex constraints provide some degree of robustness, the lack of analysis tools illustrates the difficultiesin quantifying the inherent robustness of MPC even in the linear case. This is why the number ofconstructive methods for synthesizing Robust Model Predictive Controllers is much greater than thenumber of robustness analysis tools for nominal MPC. The remainder of this thesis will therefore onlyaddress the synthesis of Robust MPC.

3.2 Modeling Uncertainty

From the previous section, it is clear that simply ignoring uncertainty in the formulation of ModelPredictive Control problems is generally a bad idea (though not as bad as it may be for nonlinear systems,compare Grimm et al. (2004) in this context). In order to be able to account for uncertainty already duringthe design process of the controller, it is necessary to have adequate means of modeling uncertainty. Thefollowing sections present some of the uncertainty models that are most prevalent in the MPC literature,including parametric and polytopic uncertainty, structured feedback uncertainty and bounded additivedisturbances. Section 3.2.4 furthermore comments on the increasing use of stochastic disturbance modelsin MPC. Modeling uncertainty through uncertain coefficients in impulse-response or step-response modelson the other hand will not be discussed. Although this framework has been used in the earlier works onRobust MPC, e.g. by Campo and Morari (1987); Zheng and Morari (1993); Bemporad and Mosca (1998),it seems to have been more or less discarded in recent publications that almost exclusively use a statespace approach. The Robust Model Predictive Controllers reviewed and proposed in the course of thisthesis utilize all of the uncertainty models presented in the following, except the stochastic models.

3.2.1 Parametric and Polytopic Uncertainty

Linear Parameter-Varying SystemsLinear Parameter-Varying (LPV) systems (Kothare et al. (1996)) are systems of the form

x(t + 1) = A(θ)x(t) + B(θ)u(t), (3.1)

22 3. Robust Model Predictive Control

where the matrices A(·) and B(·) are known functions of the parameter θ ∈Θ, whereΘ⊂ Rnθ is a compactset. At any time t, the parameter θ(t) may take on any value in Θ. Thus, the system (3.1) is time-varyingin general. LPV systems naturally arise when modeling dynamic system for which parameters are notexactly known, or whose parameters may vary with time or operating point of the system. Moreover, forcomplex systems one usually has to trade model exactness for simplicity and tractability and simplify thesystem dynamics even if a more detailed model is possible. The resulting modeling error can be accountedfor by using linear parameter-varying system models.

Polytopic SystemsA way of modeling uncertainty that is closely related to LPV systems is so-called polytopic uncertainty(Bemporad and Morari (1999)). Polytopic uncertainty is described by the time-varying system

x(t+1) = A(t)x(t) + B(t)u(t), (3.2)

where

[A(t) B(t)] ∈ Ω = Convh

[A1 B1], [A2 B2], . . . , [AL BL]

, (3.3)

with Convh(·) denoting the convex hull, i.e.

[A(t) B(t)] =L∑

i=1

αi[Ai Bi], withL∑

i=1

αi = 1, αi ≥ 0. (3.4)

Polytopic uncertainty models are often used when a number of “extreme” system dynamics are known,which then represent the vertices [Ai Bi] in (3.3). These extreme system dynamics may have beenobtained by identifying an unknown system through a sufficiently large number of measurements, or bymodeling a system under different extreme operating conditions. Another interesting possible source forthis kind of uncertainty model is remarked in Kothare et al. (1996); Bemporad and Morari (1999), wherethe authors point out that polytopic uncertainty can also be used as a conservative approach to modelnonlinear systems of the form x(t+1) = f (x(k), u(k)), when the Jacobian J f (x , u) =

∂ f∂ x

, ∂ f∂ u

is knownto lie within a polytopic set Ω.

Relationship Between LPV and Polytopic SystemsThere exists a direct connection between LPV and polytopic system descriptions: If the set Θ is polytopic,and if A(·) and B(·) in (3.1) are affine functions of θ , i.e.

A(θ) = Aθ ,0+nθ∑

i=1

θiAθ ,i, B(θ) = Bθ ,0+nθ∑

i=1

θiBθ ,i, (3.5)

then it is straightforward to represent a LPV system as a polytopic system. In this case, the vertex pairs[Ai Bi] in (3.3) are the mappings [A(θ 1) B(θ 1)], [A(θ 2) B(θ 2)], . . . , [A(θ L) B(θ L)], with θ i being anenumeration of the L vertices of Θ. To see this, note that under an affine mapping polytopes are againmapped into polytopes, and that the vertices of the original polytopes are mapped into the vertices of theimage polytopes (Ziegler (1995)).

3.2.2 Structured Feedback Uncertainty

Another popular framework in the Robust MPC literature to model uncertainty is so-called structuredfeedback uncertainty (Bemporad and Morari (1999)). The structured uncertainty description is based onLinear Fractional Transformations (LFT), which are a well-known framework in Linear Robust Controlused to model various kinds of uncertainty in dynamic systems (Zhou et al. (1996)). The basic idea

3.2. Modeling Uncertainty 23

behind the structured feedback uncertainty model is to split the uncertain system into two parts. Whilethe first part, an LTI system, incorporates everything that is known about the system, the second part,a feedback loop, contains all of the uncertainty that appears in the model. To be specific, consider thefollowing linear time-invariant system:

x(t+1) = Ax(t) + Bu(t) + Bww(t)z(t) = Cz x(t) + Dzu(t)y(t) = C x(t),

(3.6)

where the auxiliary variables z and w are introduced in addition to the system state x , the control input u,and the system output y . The feedback interconnection of this system is illustrated in Figure 3.1.

Figure 3.1.: LFT feedback interconnection (Zhou et al. (1996))

The operator ∆ is hereby of the following block-diagonal structure:

∆=

∆1∆2 . . .

∆r

(3.7)

where the blocks ∆i on the diagonal are either time-varying matrices that satisfy1 σ(∆i(t))≤ 1, or stableLTI systems with l2-gain less than 1, i.e. satisfying

t∑

l=0

wTi (l)wi(l)≤

t∑

l=0

zTi (l) zi(l), ∀t ≥ 0 (3.8)

for all i=1, 2, . . . , r (Bemporad and Morari (1999)).

There is a large number of possible causes for uncertainty, most of which can be modeled using theframework of structured feedback uncertainty. These causes include unknown parameters, unknown orneglected dynamics and nonlinearities (Zhou et al. (1996); Packard and Doyle (1993)). The authors ofKothare et al. (1996) furthermore point out that the framework of structured feedback uncertainty canalso be used to describe LPV systems as in section 3.2.1. The set Ω in (3.3) is then characterized as:

Ω =

[A+ Bw∆Cz B+ Bw∆Dz] | ∆ satisfies (3.7) with σ(∆i(t))≤ 1

. (3.9)

1 here, σ(·) denotes the maximum singular value of a matrix


3.2.3 Bounded Additive Disturbances

Another framework for describing uncertainty is that of bounded additive disturbances. Consider to thisend the perturbed linear time-invariant system

x+ = Ax + Bu+w

y = C x + v ,(3.10)

where w ∈W ⊂ Rn and v ∈ V ⊂ Rp are unknown, but bounded disturbances. W and V are given convexand compact sets and the system model itself is assumed to be accurate (i.e. the matrices A, B and Cin (3.10) are exact). The state disturbance w directly affects the state evolution and represents not onlyexternal disturbances, but also includes parameter uncertainties as well as unmodeled dynamics. Theoutput disturbance v accounts for measurement errors and uncertainties in the system output matrix.The output disturbance is of importance in the context of output-feedback MPC, where the state x is notknown exactly, but must be estimated from the output measurements y .

The sets W and V are usually assumed to be polytopic, but also other representations (e.g. ellipsoids)have been considered in the literature. The are two main reasons why polytopic bounds on the disturbanceare by far the most prevalent today. Firstly, polytopes can be used to approximate arbitrary convex sets,and they are also the straightforward way to express interval-bounded disturbances. Secondly, polytopicbounds result in linear constraints in the optimization problem. When the cost is linear or quadratic, thisleads to linear and quadratic optimization problems, respectively, which are a lot easier to solve than theSecond Order Cone Programs (SOCP) or, more generally, Semidefinite Programs (SDP) that result fromquadratic bounds on the disturbance.

In case the system (3.10) is subject to non-vanishing additive disturbances (i.e. limt→∞w 6= 0 orlimt→∞ v 6= 0), it is clearly impossible to show stability of the origin (which has been the notion of stabilityused for nominal MPC in chapter 2). Nevertheless, under appropriate assumptions, it is possible to showasymptotic stability of a positively invariant set, which plays the role of the origin in the uncertain case(Rawlings and Mayne (2009)). Bounded additive disturbances will be the kind of uncertainty consideredin “Tube-Based Robust MPC” and “Interpolated Tube MPC” in chapter 4 and 5, respectively.

3.2.4 Stochastic Formulations of Model Predictive Control

Recently, there has been considerable research activity in the field of Stochastic MPC, i.e. Model Pre-dictive Control for systems that are subject to uncertainties modeled in a stochastic framework. Thisis a very natural approach to modeling uncertainty, since noise and disturbances are often times ofstochastic nature. Stochastic uncertainty models in MPC also immediately arise when extending existingcontrol methodologies that use stochastic uncertainty models, such as LQG control, to constrained systems.

In one of the earlier publications on Model Predictive Control dealing with stochastic uncertainty, Clarkeet al. (1987) proposed the by now classic “Generalized Predictive Control” algorithm, which is based onfinite impulse response (FIR) plant models. The authors of Li et al. (2002) deal with probabilisticallyconstrained systems, while those of Cannon et al. (2007) investigate multiplicative stochastic uncertainty.In de la Peña and Alamo (2005), stochastic programming is applied to an MPC problem with a linearcost function, and Hokayem et al. (2009) proposes Stochastic MPC for input constrained systems withquadratic cost. The above references constitute only a small part of the work published on StochasticModel Predictive Control. For a more comprehensive overview of the literature, the reader is referred tothe survey paper Kouvaritakis et al. (2004).

3.2. Modeling Uncertainty 25

Judging from the results that have been established so far, Stochastic MPC seems a promising approachto robust optimal control of constrained systems. However, the developed theory is not yet as matureas for example Robust MPC for models affected by bounded uncertainties is. Therefore, and due to thelimitations on the amount of material that this thesis can cover, Stochastic MPC will not be investigatedany further in the following chapters.

3.3 Min-Max Model Predictive Control

The idea behind Min-Max Model Predictive Control (also: Minimax MPC) is to optimize robust perfor-mance. That is, instead of minimizing nominal performance as in the optimization problem PN (x) fromsection 2.1.1, the controller is designed to minimize the worst-case performance achievable under anyadmissible uncertainty. Min-Max MPC was introduced in the seminal paper Campo and Morari (1987)and since then has become a very popular way of formulating Robust MPC problems. Most Robust MPCmethods that have been developed following up on the initial ideas of Campo and Morari (1987) areessentially (obvious or non-obvious) based on the minimization of the worst-case performance.

In order to reduce the notational burden when making some conceptual statements in the following,the “generic uncertainty” set Ψ is introduced for the purpose this section. This loosely defined set shallcontain everything that is uncertain about a specific dynamic system. It may represent model uncertainty(e.g. the extreme plants in case of polytopic uncertainty, or the operator ∆ in case of structured feedbackuncertainty) as well as external disturbances (information about the bounding sets W and V in caseof additive disturbances) and combinations thereof. The set Ψ is assumed to be bounded and closed.Let ψ∈Ψ denote a specific realization of the generic uncertainty. Furthermore, denote by Φ(i; x ,u,ψ)state of the system at time i controlled by u, the sequence of predicted optimal control inputs, when theinitial state at time 0 is x and the realized uncertainty sequence is ψ.

3.3.1 Open-Loop vs. Closed-Loop Predictions

Open-Loop Min-Max MPCIn open-loop Min-Max MPC (Rawlings and Mayne (2009)), the optimization problem Pol

N (x) solved on-lineat each time step is

V ∗N (x) =minu

¦

VN (x ,u) | u ∈ UN (x ,Ψ)©

(3.11)

u∗(x) = argminu

¦

VN (x ,u) | u ∈ UN (x ,Ψ)©

, (3.12)

where VN (x ,u) denotes the worst-case cost of the perturbed system:

VN (x ,u) :=maxψ

VN (x ,u) | ψ ∈Ψ

. (3.13)

The set of admissible control sequences

UN (x ,Ψ) =¦

u | ui∈U, Φ(i; x ,u,ψ) ∈ X, Φ(N ; x ,u,ψ) ∈ X f for i=0, 1, . . . , N−1, ∀ψ ∈Ψ©

(3.14)

is smaller than UN (x) from (2.9), the set of admissible control sequences for nominal MPC. Althoughthe cost function VN (·) in (3.13) does not explicitly depend on the uncertainty ψ, it of course implicitlydoes through the perturbed state trajectory. Note that the more general “Inf-Sup” formulation, which alsoappears in the literature, in this case is equivalent to the “Min-Max” formulation. This is because stateand control constraints as well as uncertainties are assumed to be bounded by closed sets, and thereforeminimum and maximum, respectively, are attained (if the problem is feasible).


Remark 3.1 (Game-theoretic interpretation of Min-Max MPC):From a game-theoretic point of view, the optimization problem Pol

N (x) can be interpreted as a two playerzero sum finite horizon dynamic game (Lall and Glover (1994); Chen et al. (1997)). Due to the order ofminimization and maximization, the player of the generic uncertainty ψ has an advantage over the player ofthe control sequence u. Maximization of the cost over all uncertainty realizations corresponds to a maliciousplayer of the uncertainty. Because this maximization is performed over the whole prediction horizon N , theuncertainty player can plan out his whole move before the player of the control sequence may react and playhis controls such as to counteract the effects of the disturbance.

In problem PolN (x), the performance corresponding to the worst-case realization of the generic uncer-

tainty is minimized over the open-loop control actions of the control sequence u. This is clearly a veryconservative approach, since it does not take into account the additional information obtained throughfuture measurements of the state (see Remark 3.1). Without this feedback information, the trajectoriescorresponding to the different uncertainty realizations quickly diverge, and may differ substantially fromeach other. As a result, the region of attraction XN for open-loop Min-Max MPC is often small or evenempty for reasonable N (Mayne et al. (2000)). To mitigate this problem, closed-loop Min-Max MPC (also:feedback Min-Max MPC) has been proposed in Lee and Yu (1997); Rossiter et al. (1998); Scokaert andMayne (1998); Bemporad (1998a); Lee and Kouvaritakis (1999).

Closed-Loop Min-Max MPCInstead of optimizing over a nominal control sequence u=

u0, u1, . . . , uN−1

as performed in open-loopMin-Max MPC, closed-loop Min-Max MPC (Rawlings and Mayne (2009)) aims to find a control policy

µ :=

µ0,µ1(·), . . . ,µN−1(·)

, (3.15)

which is a sequence of control laws µi(·) : X 7→ U (note that if the state x is assumed to be known thenthere is no uncertainty associated with the first control move, and hence the first element in µ is a controlaction: µ0 = u0). The corresponding closed-loop optimization problem Pcl

N (x) solved at each time step is

V ∗N (x) =minµ

¦

VN (x ,µ) | µ ∈MN (x ,Ψ)©

(3.16)

µ∗(x) = arg minµ

¦

VN (x ,µ) | µ ∈MN (x ,Ψ)©

, (3.17)

where the worst-case cost VN (x ,µ) of the perturbed system is

VN (x ,µ) =maxψ

VN (x ,µ) | ψ ∈Ψ

(3.18)

and the set of admissible control policies is

MN (x ,Ψ) =n

µ | µi(Φ(i; x ,µ,ψ)) ∈ U, Φ(i; x ,µ,ψ) ∈ X, for i=0, 1, . . . , N−1,

Φ(N ; x ,µ,ψ) ∈ X f , ∀ψ ∈Ψo

.(3.19)

Problem PclN (x) can, in principle, be solved via dynamic programming (Bertsekas (2007)). However, the

exact problem is computationally intractable, since as the optimization is carried out over a sequence ofcontrol laws instead of control actions, the decision variable µ is infinite dimensional in general. Hence,in order to use dynamic programming, a discretization of the state space (“gridding”) is necessary, as isdiscussed in Lee and Yu (1997). Due to the exponential growth of the number of grid points with the statedimension (the so-called “curse of dimensionality”), this approach is however only feasible for problemsin low dimensions. In general, the universal form of Problem Pcl

N (x) is much too complex, and one hasto explore alternative formulations that sacrifice optimality for implementability by parametrizing thecontrol laws µi(·) by a finite number of decision variables. The remainder of this chapter will focus onreviewing corresponding approaches that have been discussed in the literature.

3.3. Min-Max Model Predictive Control 27

3.3.2 Enumeration Techniques in Min-Max MPC Synthesis

The original open-loop Min-Max formulation of Model Predictive Control introduced by Campo andMorari (1987) assumes a system model with an uncertain impulse response description and is based onan enumerative scheme. The authors show that the set of possible future states (under all admissibledisturbance realizations) defines a convex set, and hence that only the extreme points of this set need tobe checked to find the worst-case cost. It is furthermore shown that the resulting (potentially very large)overall Min-Max optimization problem can be recast as a Linear Program.

Enumeration techniques of a similar kind are also used in the closed-loop Min-Max MPC formulations ofLee and Yu (1997); Casavola et al. (2000a) and Schuurmans and Rossiter (2000), where linear models withpolytopic uncertainty description are considered. For systems subject to bounded additive disturbances arelated approach is developed in Scokaert and Mayne (1998). The main problem with all these methods,regardless whether they are based on open-loop or closed-loop predictions, is the complexity of theassociated optimization problems. Due to the enumeration of all possible uncertainty realizations, theseoptimization problems are generally of exponential complexity, and hence only applicable to problemswith short prediction horizons and a small number of uncertain parameters. Therefore, it has becomewidely accepted in the MPC community that enumerative methods are not suitable for all but the simplestpractical applications. The research focus today lies on alternative, possibly suboptimal methods that usesmart ways to bound the worst-case infinite horizon cost of the controlled uncertain system. The mostpopular ones among these methods use Linear Matrix Inequality constraints in the formulation of thecorresponding on-line optimization problem.

3.4 LMI-Based Approaches to Min-Max MPC

Some Basics on LMIsDuring the last two decades, Linear Matrix Inequalities (LMIs) have gained increasing importance inSystems and Control Theory, as a large number of different problem types can be formulated as convexoptimization problems subject to LMI constraints. The standard form of an LMI (Boyd et al. (1994)) is

F(x) := F0+m∑

i=1

x i Fi 0, (3.20)

where x ∈ Rm is the variable and the symmetric matrices Fi = F Ti ∈R

n×n for i = 0, 1, . . . , m are given.Expressions that contain matrices (e.g controller gains) as variables are another form of LMI that frequentlyappears in problems in System and Control Theory (Boyd et al. (1994)). The success of the practicalapplication of LMIs in Control Theory is mainly due to the development of powerful interior-pointalgorithms (Nesterov and Nemirovskii (1994)), which allow to solve the Semidefinite Programing (SDP)problems that arise LMI constrained optimization problems very efficiently (in polynomial time). Today,there exists a large number of fast algorithms and powerful software packages that facilitate an easy andefficient implementation of convex optimization problems involving LMI constraints (Sturm (1999); Tohet al. (1999); Löfberg (2004); Grant and Boyd (2010)). For the purpose of this thesis, it is not necessaryto present any further details on the theory of LMIs here. The point that should be emphasized is that LMIoptimization problems are tractable, and solvers as well as computer hardware today are fast enough toallow the implementation of LMI-based online-optimization in certain control systems.

3.4.1 Kothare’s Controller

A very important contribution that sparked increasing interest in Robust MPC and that had a strongimpact on its theoretical development was Kothare et al. (1994), which was later extended in Kothare


et al. (1996). In these seminal papers, the authors used the LMI framework to formulate a Min-Max ModelPredictive Control problem. The controller, which for brevity will be referred to as “Kothare’s controller” inthe following, was proposed for uncertain systems with both polytopic and structured feedback uncertainty.The following section reviews this controller for systems characterized by polytopic uncertainty (seesection 3.2.1). The controller for systems characterized by structured feedback uncertainty is very similar.

In Kothare’s controller, a time-invariant linear feedback controller F is recomputed at each samplinginstant. In the words of section 3.3.1, the stationary control policy µ(x) = F x is employed for theprediction model. The performance measure associated with this controller is, at each time step, chosenas an upper bound γ on the infinite-horizon worst-case cost. The controller gain F is the only degreeof freedom in the controller synthesis, hence it must be ensured that infinite horizon robust constraintsatisfaction is guaranteed for the perturbed closed-loop system controlled by u= F x .

Consider the discrete-time linear system

x+ = Ax + Bu

y = C x(3.21)

with polytopic uncertainty defined by

[A B] ∈ Ω = Convh

[A1 B1], [A2 B2], . . . , [AL BL]

. (3.22)

The system matrices A and B in (3.21) are time-varying in the sense that, at each time step, their valuesmay take on any value [A B] ∈ Ω. The worst-case infinite horizon cost is

V∞(x ,u) :=maxΩ

∞∑

j=0

x Tj Qx j + uT

j R u j, (3.23)

where Q0 and R0. The maximization over the set Ω in (3.23) means that, at each j, the matrix pair[A( j) B( j)] ∈ Ω is such that the infinite horizon cost V∞(x ,u) :=

∑∞j=0 x T

j Qx j + uTj Ru j is maximized.

Worst-Case Cost for the Unconstrained SystemIf the state and control constraints on the system are temporarily ignored, the problem at each time stepreduces to finding an upper bound on the worst-case infinite horizon cost (3.23) with x0 := x as initialstate. If the system (3.21) is quadratically stabilizable, then the stabilization of the uncertain systemis equivalent to the simultaneous quadratic stabilization of its system vertices [Al Bl] (Geromel et al.(1991)). Hence, the quadratic Control Lyapunov Function V (x) = x T P x , with P 0, is an upper boundon this worst-case cost if there exists a linear, time-invariant state-feedback controller F such that, for allpossible [A B] ∈ Ω, it holds that

(x+)T P(x+)− x T P x ≤−x TQx − uT R u. (3.24)

This can be seen by adding up left- and right-hand side of (3.24) from 0 to∞ (Kothare et al. (1996)). Byusing an epigraph formulation, the minimization of x T P x can be performed by

γ∗ =minγ,P

γ

s.t. x T P x ≤ γ.(3.25)

Performing the change of variables W := γP−1 and applying a Schur complement, the quadratic constraintx T P x ≤ γ in (3.25) is equivalent to the LMI

1 x T

x W

0. (3.26)

3.4. LMI-Based Approaches to Min-Max MPC 29

Substituting the control law u= F x and defining Y := FW (Geromel et al. (1991); Boyd et al. (1994)),the requirement (3.24) can be expressed in the set of LMIs

W (AlW + Bl FY )T W Y T

(AlW + Bl FY ) W 0 0W 0 γQ−1 0Y 0 0 γR−1

0 for l = 1, . . . , L. (3.27)

Although the system matrices [A B] may take on any value in the set Ω, the convexity of Ω implies thatit is sufficient to check only its vertices. Using the LMI constraints (3.27), an upper bound γ∗ on theworst-case infinite horizon cost of the unconstrained system given the current state x is given by

γ∗ =minγ,W,Y

γ

s.t. (3.26), (3.27).(3.28)

Incorporating ConstraintsAt each time step, Kothare’s controller computes a new (constant) feedback gain F . Since V (x) = x T P x ,where P = γW−1, is a Lyapunov Function for the (unconstrained) uncertain system, the set

E(P, γ∗) :=¦

x T P x ≤ γ∗©

(3.29)

defines an invariant ellipsoid (i.e. a positively invariant set in form of an ellipsoid) for the uncertainclosed-loop system (Kothare et al. (1996)). Invariant sets for uncertain systems are also referred to asrobust invariant sets (see section 4.1 for details). A (potentially conservative) way to ensure persistentfeasibility is therefore to require that both state and control constraints of the closed-loop system besatisfied for all states contained within the ellipsoidal set E. Then, due to the robust invariance of E, thecontrol law u= F x will keep the system state within E while satisfying the constraints for all times.

The types of constraints that are addressed in the original framework of Kothare’s controller are Euclideannorm bounds and component-wise peak bounds on both input and state. For the purpose of this exposition,consider instead the polytopic constraints x ∈ X and u ∈ U as they were introduced in (2.2). SufficientLMI conditions for those constraints to be satisfied can be obtained by requiring constraint satisfaction ofstate x and control u= F x for all states contained within the ellipsoid E(P, γ∗). Assume that X and U aregiven in their normalizedH -representations as X =

x ∈ Rn | Hx x ≤ 1

and U =

u ∈ Rn | Huu ≤ 1

,respectively. For an ellipsoidal set E(P,γ) to be contained in X it must hold that

Hx x ≤ 1, ∀ x ∈ E(P,γ). (3.30)

This requirement can be reformulated as the sufficient LMI condition

1 (Hx)Ti W T

W (Hx)i W

0, for i = 1, . . . , I , (3.31)

where I is the number of facets of X, the index (·)i denotes the i th row, and W = γP−1 as definedpreviously (Löfberg (2003b)). Equivalently, the requirement that the control constraint u ∈ U be satisfiedwithin an ellipsoid E(P,γ) can be expressed as

1 (HuF)Tj W T

W (HuF) j W

0, for j = 1, . . . , J . (3.32)


The Overall Optimization ProblemIncorporating the two additional LMI constraints (3.31) and (3.32) into (3.25) yields the overall opti-mization problem of Kothare’s controller, which needs to be solved on-line at each time step:

γ∗ =minγ,W,Y

γ

s.t. (3.26), (3.27)

(3.31), (3.32).

(3.33)

The optimization (3.33) at each time step determines a robust invariant ellipsoid (which is persistentlyfeasible in terms of state and control constraints) of the form (3.29) by minimizing a worst-case upperbound on the infinite horizon cost. Problem (3.33) is a Semidefinite Program (SDP), which can be solvedfairly efficiently using available optimization software (see references in the beginning of this section).If the control move is chosen as u = F ∗x , with F ∗ = Y ∗(W ∗)−1 being the in the sense of (3.33) optimalfeedback gain, and if this procedure is repeated at each subsequent time step in a Receding Horizonfashion, robust feasibility and robust stability of the uncertain system follow by construction.

Remark 3.2 (Relationship to “classic” MPC):Note that Kothare’s controller is essentially a Model Predictive Controller that uses a prediction horizonof N=0. The invariant ellipsoid E(P,γ∗) corresponds to a terminal set that is recomputed at each time step.

Properties of the ControllerBesides the main benefit of Kothare’s controller, which is the fact that robust feasibility and robust stabilityof the uncertain system follow by construction, there are also some drawbacks associated with it. Themost obvious one is certainly the parametrization of the control sequence by the linear state-feedbacklaw u= F x , which is clearly conservative. This becomes apparent when considering Remark 3.2 in lightof the fact that the performance and the size of the region of attraction of MPC generally increases withthe prediction horizon. Moreover, the treatment of constraints is potentially conservative as constraintsatisfaction is required for all states within the invariant ellipsoid E(P,γ∗) and not only for those thatare reachable from the current initial state. Although it was remarked in Kothare et al. (1996) that thissufficient condition has been found to be not overly conservative, it nevertheless is a general disadvantageof the controller2. Löfberg (2003b) furthermore rightly points out that treating the constraints by usingellipsoidal arguments cannot encompass asymmetric constraints in a non-conservative way. Finally, aninherent shortcoming of using polytopic uncertainty models is that non-vanishing, additive disturbancescan not be accounted for.

Kothare’s controller is an important theoretical contribution to Robust MPC because of its catalytic effecton the research on related approaches that employed similar methods. The ideas of the controller live onin many variants and extensions that have been proposed since its first appearance in the literature. Themost important ones among these extensions will be discussed in the following sections.

3.4.2 Variants and Extensions of Kothare’s Controller

Using Multiple Lyapunov FunctionsAn extension to Kothare’s controller that employs multiple Lyapunov functions was proposed in Cuzzolaet al. (2002). While the main characteristics of the original controller from Kothare et al. (1996) remainuntouched, the approach, instead of a single Lyapunov function V (x) = x T P x , uses L different Lyapunovfunctions, each one corresponding to one of the L vertices of the uncertainty polytope Ω. This results in areduced conservativeness, however at the expense of a higher on-line computational workload. Note thata corrected version of the paper’s initially incorrect proofs have later been provided in Mao (2003).2 the same reasoning of course also applies to the determination of positively invariant terminal sets for nominal Model

Predictive Controllers. However, in this case the longer prediction horizon mitigates this issue


Shifting the LMI Optimization Off-LineIn order to reduce the on-line optimization effort to a minimum, Wan and Kothare (2003b) presentan efficient off-line formulation of Kothare’s controller. The basic idea in their approach is to solve theoptimization problem (3.33) for a sequence of initial states off-line, and to store the obtained sequenceof robust positively invariant ellipsoids E(W−1) =

x ∈ Rn | x T W−1 x ≤ 1

and the associated optimalcontroller gains in a lookup table. The on-line effort for a measured state x can thereby be reduced to asimple bisection search over the matrices W−1 that define the invariant ellipsoids. This bisection searchdetermines the smallest invariant ellipsoid that contains the current state x of the system, i.e. for which itholds that ||x ||2W−1<1. The associated optimal controller gain F ∗ is then used to compute the optimalcontrol input u∗=F ∗x . Since only a finite number of initial states can be considered in the off-line part ofthe algorithm, the resulting controller generally exhibits inferior performance as compared to the originalon-line implementation of Kothare’s controller, where the optimization problem is solved exactly at eachtime step. The simulations presented in Wan and Kothare (2003b) however suggest that the performancedegradation is marginal if the number of initial states considered off-line is high enough. Besides the verysame drawbacks as the original controller, the main issue with this off-line approach is the question of howto choose the initial conditions for which to compute the invariant ellipsoids and associated controllergains off-line. On the other hand, the reduction in on-line complexity is significant, which makes thiscontroller attractive also for fast systems.

Ding et al. (2007) propose two controller types that feature only slight modifications in comparison tothe one presented in Wan and Kothare (2003b). One of them minimizes the nominal infinite horizoncost instead of the worst-case infinite horizon cost, thereby achieving improvements in feasibility; theother one uses a recursive approach to compute the robust positively invariant ellipsoids, which allowsboth optimality and feasibility to be improved. The authors of Angeli et al. (2002, 2008) also computerecursively a sequence of ellipsoidal sets

E0, E1, . . . , EI

(with Ei ⊆ X) such that for any Ei and anystate x ∈ Ei there exists a feasible control u∈U that drives the successor state x+ into the set Ei−1 forany admissible disturbance. In other words, any set Ei is an inner approximation of the one-step robustcontrollable set of Ei−1. The starting set E0 is chosen as a robust positively invariant set for the closed-loopsystem stabilized by a linear feedback controller. It should be pointed out that, in contrast to Wan andKothare (2003b); Ding et al. (2007), the computation of the ellipsoidal sets Ei does not depend onthe (stage) cost used during the on-line evaluation. Hence, the proposed approach also allows for theimplementation of time-optimal controllers. Simulations have shown that this controller type comparesfavorably with the other two techniques in terms of achievable region of attraction (Angeli et al. (2008)).

Quasi-Min-Max MPCImprovements in performance are possible if additional information about the uncertainty can be obtainedon-line. Lu and Arkun (2000b) discuss an MPC algorithm for LPV systems of the form (3.1), where theyassume that the current value of the parameter θ can be measured on-line in real-time. They refer to thisapproach as “Quasi-Min-Max MPC” since, as the current system matrices A(θ) and B(θ) are known, thereis no uncertainty associated to the choice of the first control input to the system. The basic ingredients ofhow to bound the worst-case infinite horizon optimal cost are the same as in Kothare’s controller. Theresulting Model Predictive Controller therefore effectively uses a prediction horizon of N=1. Anothernovel idea of this approach is the use of a parameter-dependent scheduling feedback controller of theform F :=

∑Ll=1 θl Fl , where Fl is a stabilizing controller computed on-line for the l th vertex θl of Θ.

Slightly different algorithmic variants of the controller are considered in Lu and Arkun (2000b). Caoand Xue (2004) comment on some flaws in the respective proofs. In the related paper Lu and Arkun(2000a), it was assumed that upper and lower bounds on the rate of change of the parameters (theelements of the parameter vector θ) are available, by which the possible range of values of the systemmatrices A(θ) and B(θ) at the next time step can be limited. By exploiting this additional informationsuperior performance as compared to the approach in Lu and Arkun (2000b) can be achieved.


3.4.3 Other LMI-based Robust MPC Approaches

Min-Max MPC for Increased Prediction HorizonsAs has been pointed out, the main drawback of Kothare’s controller is the simple parametrization of thecontrol sequence by the linear state-feedback law u = F x . A method that extends Kothare’s originalapproach by employing a prediction horizon N≥1, but that still uses the key ideas from section 3.4.1to compute a terminal constraint set on-line, was proposed independently by Casavola et al. (2000a)and Schuurmans and Rossiter (2000). The on-line optimization problem solved in this approach isan open-loop constrained robust optimal control problem of horizon N , where the terminal cost is anupper bound on the infinite horizon worst-case cost which is obtained on-line, at each time step, byKothare’s method (3.33). Because of the uncertainty, the predicted state evolution is set-valued. Startingfrom the initial state set X 0(x) = x (which is a singleton), the sets of predicted states are given byX k(x) = Convh

(Al + Bl F)z, ∀ z ∈ vert(X k−1(x)), l = 1, . . . , L3. Because of linearity of the system

and convexity of the set Ω that describes the uncertainty, it is sufficient to consider only the verticesof the sets of predicted states. By allowing N additional free control moves, a significant reduction inconservativeness as compared to Kothare’s controller can be achieved. However, there are two mainproblems associated with this method: Firstly, because of the enumeration of the vertices, its computationalcomplexity depends, in the worst-case, exponentially on the prediction horizon N (compare section 3.3.2).Secondly, the use of open-loop predictions results in a significant “spread” of the predicted trajectoriesand hence in large sets X k(x) and a small region of attraction. Therefore, an implementation of thisapproach is feasible only for short prediction horizons N .

Min-Max MPC using a Time-varying Terminal Constraint SetAs indicated above, the main problem with approaches that recompute the terminal set on-line andthat at the same time employ prediction horizons N ≥ 0 is their computational complexity. In order toachieve local optimality in some region around the origin, it is general practice in MPC to use the maximalpositively invariant set for the closed-loop system controlled by the optimal unconstrained controller as theterminal set. The same is true for Robust MPC, while in this case the maximal robust positively invariantset (see section 4.1) can be employed as a fixed terminal set. On the one hand, this significantly reducesthe computational complexity, since the controller gain F need not be recomputed at each time step (note,however, that the ellipsoidal terminal constraint still induces a quadratic constraint, which renders theoptimization problem a SOCP). On the other hand, since the maximal robust positively invariant set forthe unconstrained optimal controller may be very small, this may result in a small region of attraction forshort prediction horizons.

In order to increase the region of attraction, while at the same maintaining the named computationalbenefits, Wan and Kothare (2003a) propose a Robust Model Predictive Controller with a time-varyingterminal constraint set. The idea of this approach is to compute two different ellipsoidal terminalsets off-line. One of these sets, E(W−1

0 ), is the maximal robust positively invariant set for the closed-loop nominal system controlled by the optimal unconstrained controller. The other one, E(W−1

1 ), is asufficiently large constraint admissible robust positively invariant set for which the associated controller isobtained through an LMI-constrained convex optimization problem similar to (3.33). On-line, for a givenprediction horizon N , a time-varying terminal set X f (α) = E(W−1(α)), where W (α) = αW1+ (1−α)W0,is computed at each time step by finding the smallest α ∈ [0,1] such that X f (α) contains the set ofpredicted states X N (x) of the finite horizon optimal control problem. If the optimal value α∗ of theparameter α is found to be α∗=0, then the prediction horizon is reduced by one at the next time step.This control strategy will eventually use a prediction horizon of N = 0 and therefore asymptoticallyrecovers the optimal unconstrained LQR controller. The ideas of Wan and Kothare (2003a) were laterrefined by Pluymers et al. (2005d); Wan et al. (2006), who propose a slightly different on-line algorithm.

3 here, vert(Ω) denotes the set of vertices of a set Ω


What makes these methods attractive is that by using a time-varying terminal constraint set, it is possibleto obtain comparably large regions of attraction also for small prediction horizons. Since the complexityof the on-line optimization is caused mainly by the uncertainty propagation over the control horizon andthe enforcement of the terminal constraint over all propagated terminal states, this allows to significantlyreduce on-line computation. It should however be noted that the resulting optimization problem is still aSOCP and hence its solution is more involved than for example that of a QP.

Control Parametrizations of Higher ComplexityAn more general control parametrization is

u j =

F j x j + c j for j = 0, 1, . . . , N−1FN x j for j ≥ N , (3.34)

which is used, for example, by Casavola et al. (2002) and Jia et al. (2005). This general parametrizationallows for a great variety of controller structures. Setting F j=0 for all j<N in (3.34) corresponds to anopen-loop parametrization of the control law. For a prediction horizon of N=1, this results in the con-trollers proposed in Lu and Arkun (2000a,b). Setting N=0 on the other hand yields Kothare’s controller.By choosing a constant feedback component F j=F for all j the complexity of the parametrization can bereduced. The problem with the parametrization (3.34) in its general form is that the constraints on thefeedback matrices F j in the resulting optimization problem are non-convex in general (Jia et al. (2005);Goulart et al. (2006)). Different ways to address this problem and reformulate it as a convex optimizationproblem have been proposed in the literature.

For Linear Parameter-Varying systems, under the additional assumptions that the parameter vector θ ismeasurable on-line, and that bounds on its rate of change are available, Casavola et al. (2002) advocatethe use of a parameter-dependent state-feedback controller

F0 :=L∑

l=1

θ l F l , (3.35)

where F l denotes a stabilizing controller computed on-line for the lth vertex θ l of Θ. The control moveapplied to the system is

u∗0 = F ∗0 x + c∗0, (3.36)

where the optimal state-feedback controllers (F l)∗ for l=1, . . . , L and the optimal control input c∗0 areobtained through a (possibly very large) convex optimization problem subject to LMI constraints. Althoughthe very general form of the control law has the potential for a significant performance improvement ascompared to Kothare’s Controller or Quasi-Min-Max MPC, the resulting optimization problem is generallyof very high complexity and numerically intractable for large prediction horizons N . Nevertheless, thecontroller is interesting from a theoretical perspective.

Another contribution that circumnavigates the problem of non-convexity of the optimization problemresulting from the general control parametrization (3.34) is Jia et al. (2005). This approach, whichassumes ellipsoidal constraints on state and input, proposes a three step procedure that approximates theoriginal problem with a set of convex optimization problems. The idea is to 1) decompose the overallproblem into a set of N single-stage robust MPC problems; 2) compute ellipsoidal approximations tothe sets of reachable states using the control law obtained in the first step; 3) optimize the open-loopcontrol sequence in the affine feedback control law over the entire control horizon in order to reduceconservativeness. This sequential approach is obviously rather unwieldy and therefore does not seem veryuseful for practical control applications. In addition, its computational complexity is very high and henceits application is limited to systems with slow sampling frequencies.


Disturbance Parametrization of the Control LawAs mentioned in the previous paragraph, the main drawback of treating the feedback gains in (3.34) asindependent variables in the MPC on-line optimization problem is that the set of admissible decisionvariables is non-convex in general. Most implementable approaches therefore rely on solving a modifiedproblem which is convex, but which in general yields a more conservative solution. A promising ideathat uses an alternative control parametrization is presented in the series of papers Goulart et al. (2006,2008, 2009), which extend earlier contributions from van Hessem and Bosgra (2002); Löfberg (2003a);Ben-Tal et al. (2004). In this line of work, conservativeness in comparison to other approaches is reducedby parametrizing the control law as an affine function of the sequence of past disturbances as

ui =i−1∑

j=0

Mi, jw j + ci, (3.37)

where Mi, j ∈ Rm×n and ci ∈ Rm. The values of the past disturbances can easily be obtained from predictedand measured actual states as wi= x i+1−Ax i−Bui. The disturbances wi are hereby either the actual addi-tive disturbances of the uncertainty model from section 3.2.3, or they characterize the state error resultingfrom the difference between nominal and actual system matrices A and B in case the polytopic uncertaintymodel from section 3.2.1 is used. In Goulart et al. (2006), the disturbance-feedback parametrization (3.37)is shown to be equivalent to the state-feedback parametrization of the general form ui =

∑ij=0 Fi, j x j + ci

and, more importantly, to be convex4. Hence, the computation can be performed much more efficientlythan for the general state-feedback parametrization (3.34). Moreover, the inherent convexity of theresulting optimization problem eliminates the need to introduce conservative approximations, as is thecase for example in the approach of Jia et al. (2005).

An extension of the results from Goulart et al. (2006) has been developed in Goulart et al. (2009), which,by imposing additional (convex) constraints on the class of robust control policies, allows to formulate aRobust MPC law that, in addition to robust feasibility and stability, guarantees the closed-loop system tohave a bounded l2-gain. Specifically, the approach minimizes a parameter γ such that for a given initialstate x(0) of the system there exists a non-negative scalar β(x(0)) so that the following property holds:

∞∑

k=0

||x(k)||22+ ||u(k)||22 ≤ β(x(0)) + γ

2∞∑

k=0

||w(k)||22. (3.38)

The disturbance w(k) in (3.38) may take on any value in a compact and convex set W . By employing thedisturbance-feedback parametrization (3.37), it is shown that this problem can also be posed as a singleconvex optimization problem.

Reducing Computational ComplexityPark and Jeong (2004) also consider input constrained LPV systems with measurable parameter vector θand bounded rates of parameter variations, but in contrast to Casavola et al. (2002) they reformulate therobust control problem in a structured feedback uncertainty framework (see section 3.2.2). An open-loopparametrization of the control sequence is employed, which results in a significant reduction of theon-line complexity compared to Casavola et al. (2002). Although this reduced complexity is traded forcontrol performance (because of the open-loop formulation), the provided simulation results suggestthat the loss in performance as compared to Casavola et al. (2002) is minor. The proposed approach canalso be regarded as a generalization of Kothare et al. (1996) and Lu and Arkun (2000a) to predictionhorizons N≥1. Since the controller internally uses a structured feedback uncertainty formulation, theapproach can easily be extended to uncertain systems described by structured feedback uncertainty.

4 note that the values of the ci in the two parametrizations ui =∑i−1

j=0 Mi, jw j + ci and ui =∑i

j=0 Fi, j x j + ci are different


Structured feedback uncertainty is also the type of model uncertainty addressed in Casavola et al. (2004).The constraints that are considered in this approach are ellipsoidal constraints on the control input. Thecontrol sequence is parametrized as

u j =

F x j + c j for j = 0,1, . . . , N−1F x j for j ≥ N , (3.39)

which can be regarded as the special case of a single time-invariant feedback gain in the parametriza-tion (3.34) (time-invariant here refers to the time invariance during the prediction – the gain F is stillrecomputed at each time step). The distinctive difference of this approach compared to previous ones isthat large parts of the computation are shifted off-line, which is achieved through an extensive use of theS-Procedure (Boyd et al. (1994)). This results in a much more tractable on-line optimization problem. Infact, the complexity of the controller5 can be shown to increase only linearly with the prediction horizon.The additional conservativeness induced by the use of the S-Procedure is found to result only in a modestperformance degradation, at least for the examples considered in Casavola et al. (2004).

Using Ellipsoids not Centered at the OriginIn Smith (2004), the author pursues yet another LMI-type Robust MPC approach based on closed-looppredictions. A control law of the form u j = F(x j − z j) + c j is employed, where F is a local feedbackcontroller, the z j are the centers of a sequence of ellipsoids E j that are guaranteed to contain the predictedstates x j, and the c j are feedforward components. The feedforward component in the control law is usedto drive the centers of the ellipsoids E j, E j+1, . . . , E j+N to a specified target steady state xs (possibly theorigin), while the local feedback control gain F guarantees that, for each j, the state x j is containedin an ellipsoid E j that is no larger6 than the original ellipsoid E0. Hence, the approach is actually aTracking MPC approach. The variables in the on-line optimization problem are F , z j and c j, whereasthe original ellipsoid E0 is fixed and chosen a priori (the ellipsoids E j+1, . . . , E j+N are determined from E0and z1, . . . , z j+N ). In addition to bounded additive disturbances on the system state, model uncertainty inthe form of LFTs is taken into account. The proposed approach is conceptually similar to “Tube-BasedRobust MPC” and the newly proposed “Interpolated Tube MPC”, which will be treated in much greaterdetail in chapter 4 and 5, respectively. In both cases, a sequence of sets (a so-called “tube”) that boundsthe deviation of the actual system state from a nominal trajectory plays an important role. The maindifference is that Tube-Based Robust MPC uses polyhedral sets and a fixed feedback gain F , whereas theapproach in Smith (2004) uses ellipsoidal sets and considers F as an optimization variable. As a result,the latter requires the solution to a rather complex LMI optimization problem, which is practical only forsystems of rather low complexity.

Stability Constrained Robust MPCAn alternative to the common approach to ensure stability of Model Predictive Control, which is to chooseterminal cost and terminal constraint set appropriately, is to explicitly invoke a Lyapunov type stabilityconstraint on the predicted state that forces the cost function to decrease along the trajectory. Thismethod is commonly referred to as “Stability-Constrained MPC” and has been introduced by Cheng andKrogh (2001). In Cheng and Jia (2004), an extension of this method to the uncertain case is developed(uncertain in the sense that the system is uncertain, whereas the state is assumed to be known exactly).The proposed controller features an additional tuning parameter β , by which the desired contraction rate,i.e. the rate of the decrease in cost along the trajectory can be adjusted. The on-line computation involvesthe solution to a convex optimization problem subject to LMI constraints. Because the feasibility of theexplicit robust stability constraints does not depend on the parameters in the objective function, the costfunction can be chosen as any generic convex function of x and u for any finite prediction horizon without

5 the authors refer to this controller as the “NB-frozen MPC algorithm”, where “NB” stands for “norm-bound”6 since the centers z j of the ellipsoids change one cannot say that the E j need be contained in E0


jeopardizing robust stability. The control performance, however, is indeed affected by the choice of thecost function. It seems as if this rather unconventional controller type did not meet with much responseamong other researchers, supposably because its region of attraction is hard to determine explicitly andfurthermore strongly depends on the tuning parameter β .

3.5 Towards Tractable Robust Model Predictive Control

The previous sections presented a number of Robust Model Predictive Control approaches that werebased on LMI optimization techniques. Besides the noteworthy exception of Wan and Kothare (2003b)and related contributions that followed, all of these approaches perform the LMI optimization on-line.Though it has been claimed that modern optimization techniques and computer hardware render theseapproaches feasible for on-line implementation, their computational complexity may still be too high toallow for the control of fast dynamical systems with high sampling frequencies. LMI-based Robust MPCmethods have therefore been advocated mainly by researchers who are especially interested in processcontrol applications, where the system’s time constants are usually large. This luxury of sufficiently slowdynamics is however not given in all control applications. Many recent contributions in the field RobustMPC therefore aim at developing algorithms that involve simpler on-line optimization problems so thatRobust MPC becomes a feasible alternative to conventional controllers also for fast dynamical systems.The following sections will discuss some of these contributions.

3.5.1 The Closed-Loop Paradigm

Under the name “Efficient Robust Predictive Control” (ERPC), Kouvaritakis et al. (2000) propose a ModelPredictive Controller for systems with polytopic uncertainty that also uses a control parametrizationof the form (3.39), but for which the feedback controller F is pre-computed off-line as an optimal (inwhatever sense is appropriate for the particular application) robustly stabilizing feedback controller forthe unconstrained uncertain system. This parametrization has also been referred to as the “Closed-LoopParadigm”. Flexibility in the design is provided in the sense that any known technique for computinga robustly stabilizing F can be employed. The additional free control moves c j are used to enhancethe performance of the controller and to enlarge its region of attraction. For all those initial statesfor which the “optimal” state-feedback controller F is constraint admissible (i.e. for which the closed-loop system yields a feasible worst-case trajectory and sequence of control inputs), the elements c j arezero. Hence, the c j can be regarded as perturbations to the control action of the optimal unconstrainedcontroller that ensure constraint satisfaction. This insight is the rationale behind choosing a cost functionof the form VN (x) = f T f , where f := [cT

0 . . . cTN−1]

T . Note that although this cost function does notexplicitly depend on the current state, it implicitly does via the constraints that are imposed on theoptimization problem. The feasibility constraint used in Kouvaritakis et al. (2000) requires the augmentedstate z := [x T f T]T to be contained in the maximal volume constraint admissible invariant ellipsoidEmax(F) =

¦

z ∈ Rn | zT W−1z z ≤ 1, F x + c j ∈ U, ∀ j=0, . . . , N−1

©

computed in the augmented statespace. This essentially means that in the framework of this controller the membership of the state toan invariant set is invoked at the current time step instead of at the end of the prediction horizon. Themaximal volume ellipsoid Emax(F) can easily be computed off-line using LMI optimization (Boyd et al.(1994)). The on-line optimization problem then is

f ∗ =minf

f T f

s.t. zT W−1z z ≤ 1,

(3.40)

which can be shown to be a univariate problem and hence is very easy to solve. This computationalsimplicity is the main benefit of the proposed approach. It should however be noted that the treatment

3.5. Towards Tractable Robust Model Predictive Control 37

of the constraints is based on ellipsoidal arguments, which provides limited flexibility especially forasymmetric constraints and furthermore also introduces some sub-optimality into the problem.

The ideas of Kouvaritakis et al. (2000) were built upon in Kouvaritakis et al. (2002) to develop anominal Model Predictive Controller of very low complexity. In this approach, a simple Newton-Raphsoniteration is used to solve the univariate optimization problem on-line. Simulation results show that theproposed controller is about 10 times faster than a standard nominal Model Predictive Controller thatuses (warm-started7) Quadratic Programming. Because of the use of an ellipsoidal terminal set, thecontrol performance has however been found to be slightly worse than the one of standard MPC based onQuadratic Programming. In order to reduce this conservativeness, an extension to the basic controller typeis proposed which also explores the region outside the ellipsoidal terminal set (and inside the maximalpositively invariant set). This extended controller is only marginally more complex than the basic onebut achieves a control performance almost indistinguishable from that of standard QP-based MPC. Otherwork that uses similar ideas proposes “Generalized Efficient Predictive Control” (GERPC) (Imsland et al.(2005); Cannon and Kouvaritakis (2005)), which allows to obtain larger invariant ellipsoids, and “TripleMode MPC” (Rossiter et al. (2000); Cannon et al. (2001); Imsland et al. (2006)), which introduces anadditional mode in the prediction also in order to enlarge the region of attraction of the controller.

3.5.2 Interpolation-Based Robust MPC

Besides the obvious requirements of robust stability and robust feasibility, there are three additional maincriteria which a Model Predictive Controller should satisfy: 1) a good asymptotic control performance (atbest close to that of infinite horizon optimal control); 2) a large region of attraction; and 3) an on-lineoptimization problem of low computational complexity. It is easy to see that these three criteria arecontrary requirements: A good asymptotic control performance requires an aggressive terminal controller,which usually results in a small terminal set and hence a small region of attraction. In order to increase theregion of attraction for a given terminal controller, an increased prediction horizon is necessary, which inturn increases the computational complexity of the optimization problem. Enlarging the terminal set by de-tuning the terminal controller on the other hand immediately worsens the asymptotic control performance.

For a given availability of computational resources, the tradeoff between a large region of attraction andan “optimal” asymptotic control behavior can be accounted for by using a nonlinear terminal controller.A general nonlinear controller however makes it extremely difficult to compute the associated infinitehorizon cost and the associated (robust) positively invariant set. An important insight is that a nonlinearcontrol behavior can also be obtained by interpolating between a number of given linear controllers, whichthemselves can be designed to meet different objectives. This simplifies the analysis significantly, makinginterpolation between different controllers interesting for MPC applications. The common feature ofInterpolation-Based MPC approaches (Bacic et al. (2003); Rossiter et al. (2004); Pluymers et al. (2005c))is to use an on-line decomposition of the current state x , where each component of this decompositionbelongs to a separate invariant set. The controllers corresponding to the different invariant sets arethen applied to the respective state component separately in order to calculate the overall input value.Given a set of n j robustly stabilizing feedback controllers

K1, K2, . . . , Kn j

, and the set of correspondingrobust positively invariant sets

S1, S2, . . . , Sn j

, the decomposition of the current state x that is performedon-line at each time step is

x =n j∑

j=1

x j, with

∑n jj=1λ j = 1, λ j ≥ 0

x j ∈ S jx j = λ j x j

(3.41)

7 warm-starting an optimization algorithm means to provide initial guesses for the optimizer in order to reduce theevaluation time. In the context of MPC this means using the predicted optimal values from the previous iteration


Note that the invariant sets S j need not necessarily be ellipsoidal, hence the above formulation also allowsfor the use of other types of invariant sets, in particular polyhedral ones (Pluymers et al. (2005b)). Theinfinite horizon cost of the trajectories of the closed-loop system controlled by the control law

u=n j∑

j=1

K j x j (3.42)

is given by the quadratic function V ( x) = x T P x , where x := [ x T1 . . . x T

n j]T, and where P can be computed

off-line by solving an SDP. The optimization problem solved on-line at each time step is

V ∗ =minx j ,λ j

x T P x

s.t. (3.41).(3.43)

The decomposition of the state and the interpolated control law is based on essentially the same idea as theones from section 3.4.3 that use a time-varying terminal constraint set. The difference is that the terminalconstraint sets in Interpolation-based MPC are computed off-line, which significantly reduces the on-linecomplexity of the controller. While the use of ellipsoidal invariant sets S j renders (3.43) an SOCP (Bacicet al. (2003)), the use of polyhedral invariant sets results in a QP (Rossiter et al. (2004); Pluymers et al.(2005c)). The latter method has a number of advantages over the former and exhibits some additionalfavorable properties. Because it uses polyhedral invariant sets, it is possible to treat non-symmetrical stateand input constraints in a non-conservative way, which overcomes a major restriction of other RobustMPC approaches that are based on ellipsoidal invariant sets. Furthermore, due to the generally larger sizeof the polyhedral invariant sets, the region of attraction is larger as the ones of other controller types.Finally, since the optimization problem is a QP, it can be solved much more efficiently than the usual SDPs.

An interesting extension of Interpolation-based MPC to non-zero prediction horizons and systems subjectto bounded additive disturbances has been provided in Sui and Ong (2006). This approach will alsobecome the starting point for the development of the novel “Interpolated Tube MPC” in chapter 5. Itsproperties will be discussed in more detail in section 5.2. Additional information on other Model PredictiveControl methods that use interpolation can be found in the survey paper Rossiter and Ding (2010). Besidesgiving a comprehensive literature review on Interpolation-Based MPC, this paper also contains someinteresting novel results.

3.5.3 Separating Performance Optimization from Robustness Issues

The common feature of all Min-Max Model Predictive Controllers is that they optimize robust performance,i.e. the worst-case performance under all possible realizations of the uncertainty. The benefit of thisapproach is that it automatically guarantees robust stability (of course only if robust feasibility holdsand if terminal constraint set and terminal cost are chosen appropriately). However, besides the obviousfact that performance for “small” disturbances is potentially bad, the main drawback is that the resultingmin-max optimization problem is generally hard to solve. Hence, many of the approaches presented inthe previous sections are limited to comparably slow dynamical systems. An alternative way to ensurerobust stability is to optimize the cost associated to the evolution of the nominal system, while boundingthe deviation of the actual from the nominal system state by robust positively invariant sets. In orderto ensure that the original constraints of the system are satisfied, the nominal evolution of the systemmust be optimized for appropriately tightened constraints. By doing so, the computational complexityof the on-line optimization can be reduced significantly. In fact, if polyhedral invariant sets are used forbounding the state deviation, the optimization problem that needs to be solved on-line can be cast as asimple Quadratic Program, just as in nominal MPC. This makes these methods very attractive for Robust

3.5. Towards Tractable Robust Model Predictive Control 39

Control of systems with fast dynamics. One of the most interesting approaches along this line of work is“Tube-Based Robust MPC” (Langson et al. (2004); Mayne et al. (2005)), which will be discussed in detailin chapter 4. The novel “Interpolated Tube Model Predictive Controller” developed in chapter 5 as anextension to Tube-Based Robust MPC incorporates the same ideas.

3.6 Extensions of Robust Model Predictive Control

The previous sections presented an extensive overview over the most important Robust Model PredictiveControl methods that have been developed in the literature. Most of these approaches, however, onlyaddress the regulation problem and make the implicit assumption that an exact measurement of the currentstate of the system is available. Additional references that are concerned with output-feedback controland the explicit implementation of Robust MPC are presented in section 3.6.1 and 3.6.2, respectively. Inorder to contain the length of the exposition, this is performed without going into much detail. Finally,as the last part of this chapter, section 3.6.3 shows how offset-free tracking controllers may be realizedwithin the Robust Model Predictive Control framework.

3.6.1 Output-Feedback

A straightforward extension of the off-line Robust MPC scheme from Wan and Kothare (2003b) is pre-sented in Wan and Kothare (2002). The authors use a simple Luenberger observer to estimate the currentstate of the system. The on-line selection of the appropriate ellipsoid E∗ among the ones that have beencomputed off-line is performed using the current state estimate x . The actual control input is determinedas u∗ = F ∗ x , where F ∗ is the control gain matrix corresponding to the ellipsoid E∗. As simply combining alinear state estimator with a Robust Model Predictive Controller does neither guarantee robust stabilitynor robust feasibility, some additional requirements on the controller gains need to be satisfied. Theserequirements are, however, not explicitly incorporated as constraints in the optimization problem, butare checked a posteriori after the gains have been determined from the very same optimization problemas in Wan and Kothare (2003b). The drawback of this approach is that it is not obvious how the designparameters (i.e. the robust invariant ellipsoids) should be chosen in the first place. A clear benefit, on theother hand, is that the on-line evaluation of the control law only involves a simple bisection search, whichmakes this controller applicable also to fast sampling systems.

A related approach is discussed in Cheng and Jia (2006). Instead of checking necessary conditions aposteriori, additional constraints that guarantee robust stability and robust feasibility of the compositesystem of actual system state and observer state are instead explicitly invoked on the optimization problem.This controller is a generalization of the Stability Constrained Robust Model Predictive Controller fromsection 3.4.3 to the output-feedback case. If the cost function is chosen as a linear or quadratic function,the on-line computation involves the minimization of a linear objective function subject to LMI constraints.The additional robust stability constraints that are necessary because of the observer dynamics result inan optimization problem that, although of the same type, is more complex the the one occurring in theoriginal state-feedback controller from Cheng and Jia (2004).

Löfberg (2003b) proposes joint state estimation and control for a control parametrization of theform (3.34) by formulating an optimization problem involving a Bilinear Matrix Inequality (BMI)constraint. This non-convex constraint is then conservatively approximated by a convex LMI con-straint, and an algorithm to approximately solve the joint problem using Semidefinite Programming isdeveloped. Despite this relaxation, even the approximate algorithm is of considerable complexity andposes high requirements on the computational resources. Therefore, this approach can be considered tobe more of conceptual nature than actually applicable to practical problems.


The controller discussed in Jia et al. (2005) has been extended to the output-feedback case in Jia andKrogh (2005). The modified scheme uses two states estimators in the following way: While a standard Lu-enberger observer provides a state estimate x for the state-feedback part of the control law u j = F j x j + c j,a set-membership state estimator is employed to bound the current states of the physical system. Theset-membership estimator, which is not incorporated into the MPC optimization formulation but runs inparallel, takes into account previous predictions and is based on the recursive computation of ellipsoidalconstraint sets. The resulting on-line computation effort amounts to solving two rather complex SDPs.

For systems with unstructured feedback uncertainty8 Løvaas et al. (2008) propose a robust mixed objectiveMPC design. This design employes a linear state estimator and incorporates a closed-loop stability testbased on LMI optimization to determine a fixed state-feedback gain off-line. The parametrization of thecontrol law is similar to the “closed-loop” formulation from section 3.5.1. A quadratic upper bound on thenominal cost function is minimized on-line at each time step, while the parameters of the cost function aredetermined off-line from a convex parametrization of a class of cost functions that all lead to a sufficientlysmall closed-loop l2-gain.

In addition to the ones mentioned above, many other approaches for output-feedback Robust MPC havebeen proposed in the literature. A very interesting one among these is “output-feedback Tube-BasedRobust MPC”, which will be discussed in detail in chapter 4. Moreover, the newly proposed “InterpolatedTube MPC” from chapter 5 can also easily be extended to the output-feedback case.

3.6.2 Explicit Solutions

Motivated by the successful application to nominal MPC, the off-line computation of explicit solutions(see section 2.4) to Robust MPC problems has recently received increasing attention in the literature.Most of these “Explicit Robust MPC” methods that have been proposed are concerned with Min-MaxMPC problems. However, due to the universal nature of multiparametric programming, more or lessany Robust MPC method that is based on the solution of a parametric linear or quadratic program canbe implemented in an explicit form. In particular, this includes all variants of Tube-Based Robust MPCdiscussed in chapter 4 as well as the novel Interpolated Tube MPC framework presented in chapter 5.Although this thesis mainly addresses Model Predictive Control algorithms for quadratic cost functions,this section, in addition, also reviews some Explicit Min-Max MPC methods for linear cost functions.This is because explicit solutions to this problem type are much easier to obtain than for problems withquadratic cost and hence there are many more contributions in the literature addressing this task.

Explicit Min-Max MPC for Linear Cost FunctionsBemporad et al. (2003) employ dynamic programming to obtain the explicit solution of a closed-loop Min-Max MPC problems with linear cost function. A sequence of N multiparametric Linear Programs (mpLP)is solved off-line, where the solution of the last mpLP constitutes the optimal control law. Unfortunately,this approach is not easily extendable to problems with quadratic cost, since in this case the optimalityregions in the intermediate steps of the dynamic programming recursions would not be polyhedral regionsanymore. A different approach to the same problem was taken in Kerrigan and Maciejowski (2004),where only a single (but larger) mpLP is solved off-line.

In Diehl and Björnberg (2004), the authors employ robust dynamic programming to solve Min-Max MPCproblems, where they consider linearly constrained polytopic systems with piece-wise affine cost functions.Their method can therefore treat the same problem class as Bemporad et al. (2003). The difference is

8 unstructured feedback uncertainty can essentially be seen as a special case of structured feedback uncertainty (seesection 3.2.2), in which the operator ∆ is degenerate and consists of only a single block

3.6. Extensions of Robust Model Predictive Control 41

that it uses a dual9 instead of a primal approach to perform the robust dynamic programming recursion.An interesting aspect of this work is the joint representation of the feasible set and the cost-to-go in asingle polyhedron. In order to reduce the high computational complexity of the approach, a modifiedversion that uses approximate robust dynamic programming was developed in Björnberg and Diehl (2006).

For Linear Parameter-Varying systems for which the time-varying parameter can be measured on-line,Besselmann et al. (2008) propose an Explicit MPC approach for linear cost functions which is also basedon solving a sequence of mpLPs. By taking the additional information on the value of the parameter intoaccount the conservativeness compared to standard LPV approaches is reduced.

Explicit Min-Max MPC for Quadratic Cost FunctionsAs has been indicated, the computation of explicit solutions for Min-Max MPC problems with quadraticcost function is significantly harder than for problems with linear cost function. Kakalis et al. (2002)and Sakizlis et al. (2004) therefore move away from the Min-Max philosophy and take an approachthat minimizes the cost of the predicted nominal trajectory, while addressing the uncertainty only in theconstraints of the optimization. Unfortunately, no rigorous stability analysis is provided for the closed-loopversion of the proposed Model Predictive Controller.

In Ramirez and Camacho (2006), the authors were able to prove that the solution of a Min-Max MPCproblem with quadratic cost function is piece-wise affine and continuous (and hence of the same structureas that of a nominal MPC problem). These results hold true also for a control parametrization of theform u j = F x j + c j from (3.39). For this parametrization, it has been shown that the associated Min-MaxMPC problem can be transformed into an mpQP problem (de la Peña et al. (2007)). This is achievedthrough a smart reformulation of the optimization problem. This reformulation is an alternative and morecompact proof of the properties of the solution than the one provided in Ramirez and Camacho (2006).Note that although the used control parametrization contains a feedback part, the problem solved in de laPeña et al. (2007) is essentially an open-loop Min-Max MPC problem. Explicit solutions of closed-loopMin-Max MPC with quadratic performance measure today still seem to be an unresolved problem.

Other Work on Explicit Robust MPCThe authors of de la Peña et al. (2006) apply the general approximate multiparametric programmingalgorithm developed in Bemporad and Filippi (2006) to the on-line optimization problem of Kothare’scontroller. The resulting explicit control law, which is sub-optimal with respect to the original on-lineimplementation of Kothare’s controller, is shown to be robustly stabilizing and persistently feasible. Sincethe algorithm from Bemporad and Filippi (2006) is very versatile and able to determine approximatemultiparametric solutions of general convex nonlinear programming problems, it can be applied to a wideclass of Robust MPC methods. Further research will be necessary to identify those Robust MPC approachesthat are suitable for an application of the proposed approximate multiparametric programming algorithm.

In Mayne et al. (2006a), a constrained H∞-control problem is considered. The disturbance is negativelycosted in the objective function, which is minimized over control policies and maximized over disturbancesequences so that the solution yields a feedback controller. It is shown that, under certain conditions, thecorresponding value function is piece-wise quadratic and the optimal control policy piece-wise affine. Thiscontrol problem requires the solution of a parametric program in which the constraints are polyhedral andthe cost is piece-wise quadratic (rather than quadratic). To this end, the authors also present an algorithmfor parametric piece-wise quadratic programming. The results of this contribution are interesting from aconceptual point of view, whereas their implementation for control purposes is discussed only briefly inthe original reference.

9 based on the Lagrangian dual problem


Baric et al. (2008) discuss the “Max-Min MPC” framework, in which the current values of disturbancesand uncertainties are considered known, while the available information about their future realizations islimited to the knowledge that they lie within some set. This additional information about the currentvalues of the uncertainty allows to synthesize a control strategy with reduced conservativeness. Forcost functions that are based on polyhedral norms, the solution can be computed explicitly by solving asequence of mpLPs, similar to the approach in Bemporad et al. (2003).

3.6.3 Offset-Free Tracking

As has been pointed out in section 2.3.2, one of the major problems of MPC is that Model PredictiveControllers, without additional modifications, generally exhibit offset. This holds true not only for nominalMPC but also for Robust MPC methods. The standard approach to overcome this deficiency and to obtainoffset-free control is to augment the system model with a disturbance model, which is used to estimate andpredict the mismatch between measured and predicted outputs (Muske and Badgwell (2002); Pannocchiaand Rawlings (2003); Pannocchia (2004); Pannocchia and Kerrigan (2005); Maeder et al. (2009)). Thefollowing section discusses the basic ingredients of the approach taken in Maeder et al. (2009), whichitself in large parts is a refinement of previous material. For a more detailed exposition the reader isreferred to the original reference.

Disturbance Model and Controller Design

System DescriptionConsider the following general uncertain discrete-time time-invariant system (Pannocchia and Bemporad(2007); Maeder et al. (2009))

x+m = f (xm, u, w)ym = g(xm, w)yt = H ym,

(3.44)

where xm ∈ Rn, u ∈ Rm, and ym ∈ Rp. The tracked variables yt ∈ Rr are a linear combination of themeasured output ym (the matrix H is assumed to have full row rank), and w ∈ Rn is a vector of boundedexogenous disturbances. State xm and control u are subject to constraints xm ∈ X and u ∈ U. In addition,consider the following linear model of the “real” plant (3.44):

x+ = Ax + Bu+ Bd d

d+ = d

y = C x + Cd d.(3.45)

The virtual integrating disturbance d ∈ Rnd incorporates the effects of the exogenous disturbance w andthe sources of mismatch between the LTI model (x+, y) = (Ax + Bu , C x) and the real plant (3.44).

The objective is to design a Model Predictive Controller based on the system model (3.45) in order tohave the output yt of the actual system (3.44) track a reference signal yr(t). Moreover, for offset-freeMPC it is required that if limt→∞ yr(t) = yr,∞ and limt→∞w(t) = w∞, where yr,∞ and w∞ are a constantreference input and disturbance value, respectively, the steady state tracking error e∞ := yt,∞− yr,∞ iszero, i.e. it holds that limt→∞ yt(t)− yr,∞ = 0.


Control StrategyIn the following, an observer will be employed to estimate both state x and virtual disturbance d of (3.45)of (3.45). For this approach to be feasible, the augmented system model must be observable.

Proposition 3.1 (Observability of (3.45), Pannocchia and Rawlings (2003)):The augmented system (3.45) is observable if and only if the pair (C , A) is observable and

rank

A− I BdC Cd

= n+ nd (3.46)

Remark 3.3 (Number of virtual disturbances, Maeder et al. (2009)):Note that condition (3.46) in Proposition 3.1 can only be satisfied if nd ≤ p, i.e. if the number of virtualdisturbances d is smaller than the number of available measurements y .

The state and disturbance estimator for system (3.45) is

x+

d+

=

A Bd0 I

xd

+

B0

u+

LxLd

(C x + Cd d − ym), (3.47)

where the observer feedback gain matrices Lx and Ld are chosen such that the estimator is stable. Maederet al. (2009) show that if this is the case, the observer disturbance-feedback gain matrix Ld has rank nd .

Remark 3.3 states that the number of integrating disturbances must be chosen smaller or equal than thenumber of available measurements, i.e. nd ≤ p. On the other hand, the Internal Model Principle (Francisand Wonham (1976)) suggests that there must be at least as many virtual disturbances as tracked outputs,i.e. nd ≥ r. In Maeder et al. (2009); Pannocchia and Rawlings (2003), it is shown that if nd = p, i.e. whenthe number of virtual integrating disturbances equals the number of measured outputs, offset-free ModelPredictive Control can always be achieved. For simplicity, this choice will be assumed in the following.

Assumption 3.1 (Number of virtual disturbances):Assume that nd = p, i.e. that the number of virtual integrating disturbances d equals the number ofmeasured outputs ym.

Proposition 3.2 (Observer steady state, Maeder et al. (2009)):Suppose that the observer system (3.47) is stable, and that Assumption 3.1 is satisfied. Then, the observersteady state ( x∞, u∞, d∞) satisfies

A− I BC 0

x∞u∞

=

−Bd d∞ym,∞− Cd d∞

, (3.48)

where ym,∞ denotes the steady state output measurement of (3.44), u∞ denotes the steady state input,and x∞ and d∞ denote the steady state estimates of state x and virtual disturbance d, respectively.

Denote by yr,∞ the steady state output reference, i.e. limt→∞ yr(t) = yr,∞. Offset-free tracking requiresthat yr,∞ = yt,∞ = H ym,∞, where yt,∞ and ym,∞ denote the tracked and measured variables at steadystate, respectively. Using condition (3.48), this requirement can be expressed as

A− I BHC 0

x∞u∞

=

−Bd d∞yr,∞−HCd d∞

. (3.49)

Remark 3.4 (Number of controlled inputs):Note that for a pair (x∞, u∞) to exist for any yr,∞ and d∞, the matrix on the left hand side of (3.49) musthave full row rank. This implies that r ≤ m, i.e. that the number of tracked outputs yr must not exceed thedegrees of freedom in the control input u.


For a given output reference value yr and a given disturbance estimate d, define the set of admissiblesteady states (xs, us) in the (x , u)-space as

Z∞(yr , d) :=

¨

(xs, us) ∈ X×U

A− I BHC 0

xsus

=

−Bd dyr −HCd d

«

. (3.50)

Furthermore, for a given pair (x , d), define the set of admissible control sequences UN as

UN (x , d) =

u | ui ∈ U, Φ(i; x ,u, d) ∈ X for i=0,1, . . . , N

, (3.51)

where Φ(i; x ,u, d) denotes the predicted state of the system x t+1 = Ax t + But + Bd dt at time i controlledby u = u0, u1, . . . , uN−1 when the initial state at time 0 is x and the virtual disturbance is dt = d. Thecost function VN (·) for the optimization problem of the offset-free Model Predictive Controller is

VN ( x , d, yr ;u) :=N−1∑

i=0

||x i − xs(yr , d)||2Q + ||ui − us(yr , d)||2R+ ||xN − xs(yr , d)||2P (3.52)

and depends not only on the current state estimate x and the control sequence u, but also on the currentdisturbance estimate d and the reference input yr . The matrices Q, R, and P satisfy the usual assumptions,namely that they are positive definite and that P is chosen as the solution of the ARE (2.17).

Given the current the reference input yr and the current state and disturbance estimates x and d, theoffset-free MPC optimal control problem PN ( x , d, yr) solved on-line at each time step is

V ∗N ( x , d, yr) =minu

¦

VN ( x , d, yr ;u) | u ∈ UN ( x , d), (xs(yr , d), us(yr , d)) ∈ Zs

©

(3.53)

u∗( x , d, yr) = argminu

¦

VN ( x , d, yr ;u) | u ∈ UN ( x , d), (xs(yr , d), us(yr , d)) ∈ Zs

©

. (3.54)

Remark 3.5 (Uniqueness of the steady state):The set Zs is a singleton if there exists a unique steady state for any yr and d. This is the case if the matrix onthe left hand side of (3.49) is invertible. Otherwise, it is customary to determine a suitable pair (xs, us) bysolving an optimization problem similar to (2.23) in section 2.3.1.

Applying only the first element u∗0( x , d, yr) of the optimal sequence u∗( x , d, yr) to the system, the implicitReceding Horizon Control law is

κN ( x , d, yr) := u∗0( x , d, yr). (3.55)

Applying this control law to (3.44) yields the closed-loop dynamics of the composite system:

x+m = f (xm,κN ( x , d, yr), w)

x+ = (A+ Lx C) x + (Bd + Lx Cd)d + BκN ( x , d, yr)− Lx ym

d+ = Ld C x + (I+ Ld Cd)d − Ld ym

(3.56)

Theorem 3.1 (Offset-free tracking control, Maeder et al. (2009); Pannocchia and Rawlings (2003)):Consider that nd = p, i.e. the number of virtual disturbances equals the number of measured outputs.Assume that the output reference yr(t) is asymptotically constant, i.e. that limt→∞ yr(t) = yr,∞. Fur-thermore, assume that optimization problem PN ( x , d, yr) is feasible for all times, that the constraints inPN ( x , d, yr) are inactive for t →∞, and that the closed-loop system (3.56) converges asymptotically tox∞, d∞, ym,∞, i.e. x(t)→ x∞, d(t)→ d∞, and ym(t)→ ym,∞ as t →∞. Then, offset-free tracking isachieved, i.e. yt(t) = H ym(t)→ yr,∞ as t →∞.


Comments and Extensions

Note that the presented offset-free controller does not automatically ensure robust stability and robustconstraint satisfaction. Theorem 3.1 rather establishes offset-free tracking of asymptotically constantreference inputs, provided that robust stability and robust constraint satisfaction holds for the controller.This is clearly a very restrictive assumption. With additional assumptions on the state update function f (·)in (3.44), however, it is possible to use Theorem 3.1 in order to prove local Lyapunov stability of theclosed-loop system (3.56). In order to ensure robust stability and robust constraint satisfaction, one willgenerally apply a Robust MPC method to the augmented system (3.45).

Assumption 3.1 is restrictive in the sense that if r ≤ p, that is if the number of tracked outputs is smallerthan the number of measured outputs, it is possible to achieve offset-free control also when nd < p.The reason why one would choose nd < p is that the choice nd = p may introduce a large number ofadditional disturbance states, which directly increases the complexity of the MPC problem PN ( x , d, yr).Maeder et al. (2009) show how to design the observer such that offset-free tracking is achieved also forchoices r ≤ nd < p. For brevity, the necessary extensions will not be discussed here.

Other Approaches to Offset-Free Robust MPCIn Pannocchia and Kerrigan (2005), the overall offset-free Model Predictive Control problem is split intotwo separate parts: the design of a dynamic unconstrained linear offset-free controller, and the design ofa Robust Model Predictive Controller. The Robust Model Predictive Controller is in this framework usedto enlarge the region of attraction of the unconstrained offset-free controller. However, in order to yieldan easily tractable optimization problem, the set-point calculation is performed explicitly, which in turndoes not permit that state and control constraints are included. This inevitably yields a smaller region ofattraction of the proposed controller.

Another interesting idea for offset-free MPC was proposed by Pannocchia and Bemporad (2007), whoemploy a “dynamic” observer designed to minimize the effect of unmeasured disturbances and modelmismatch on the output prediction error. An interesting feature of this approach is that the disturbancemodel and observer for the augmented are designed simultaneously by solving an appropriately formulatedH∞-control problem, which can be cast as a convex optimization problem subject to LMI constraints.A scalar tuning parameter is used to adjust the tradeoff between good disturbance rejection propertiesand the resiliency to output and process noise. It is shown that in the context of offset-free control, theobtained observer is equivalent to choosing an integrating disturbance model and a Luenberger observerfor the augmented system. The proposed approach therefore significantly reduces the number of designparameters by eliminating the need to manually choose the disturbance model and the observer gains Lxand Ld , which is necessary in the approach from the previous section.


4 Tube-Based Robust Model Predictive ControlAs it has been argued in section 3.3.1, the use of open-loop predictions in Robust Model Predictive Controlis overly conservative in general. Not taking into account the influence of feedback on the predictionsalso often times results in controllers with very small regions of attraction. Instead, improved Robust MPCapproaches usually employ closed-loop predictions to contain the spread of predicted trajectories resultingfrom the influence of uncertainty. This however leads to Model Predictive Controllers of comparably highcomputational complexity, which is a major drawback of many of the Min-Max Robust MPC methods thathave been presented in chapter 3 of this thesis.

A very promising approach that is able to alleviate the on-line computational burden is so-called “Tube-Based Robust Model Predictive Control”. Initially proposed in Langson et al. (2004), Tube-Based RobustMPC builds upon the theory of invariant sets, and some of its core ideas can be traced back to the earlyworks of Witsenhausen (1968); Bertsekas and Rhodes (1971a,b); Glover and Schweppe (1971). Thebasic concept of Tube-Based Robust MPC is to solve a nominal Model Predictive Control problem forsuitably tightened constraints, while bounding the error between nominal1 and uncertain system stateby a robust positively invariant set. By doing so, the uncertain system can be guaranteed to evolve in a“tube” of trajectories around the predicted nominal trajectory, making it possible to control the nominalsystem in such a way that the original constraints are satisfied by the uncertain system at all times. Thecommon ingredient of all Tube-Based Robust MPC approaches is a control law of the form u= u+K(x−x),where x and u are state and control input of the nominal system. The linear time-invariant feedbackcontroller K , also referred to as the disturbance rejection controller, is computed off-line and ensures thatthe deviation x− x of the actual system state x from the nominal system state x is bounded. This boundcan be quantified in form of an invariant set E . An appropriate adjustment of the original constraint sets Xand U and the choice of a suitable terminal set X f , all of which depend on size and shape of E , thenallows the on-line computation to be reduced to the implementation of an only slightly modified standardModel Predictive Controller for the nominal system. This controller only involves the solution of a stan-dard Quadratic Program with a complexity comparable to that of conventional MPC (Mayne et al. (2005)).

Tube-Based Robust MPC can essentially be seen as a way of separating the problem of constrained optimalcontrol from the problem of ensuring robustness in the presence of uncertainty, an idea that was brieflyaddressed already in section 3.5.3 of the previous chapter. The underlying concept of “tubes” allowsfor an easy extension of basic Tube-Based Robust MPC to other problem types, i.e. to output-feedback(Mayne et al. (2006b)) and tracking problems (Alvarado et al. (2007b)) as well as to combinationsthereof (Alvarado et al. (2007a)). What makes Tube-Based Robust MPC especially interesting for practicalapplications is its computational simplicity: Because the optimization problem that needs to be solvedon-line is a standard Quadratic Program, it can be solved fast and efficiently using standard mathematicaloptimization algorithms. Hence, real-time optimization may be performed in a comparatively simple way,making Tube-Based Robust MPC interesting also for fast dynamical systems. In fact, since the on-lineoptimization problem can be formulated as a multiparametric program, it can, in principle, be solvedexplicitly (see section 2.4). An efficient implementation in hardware then enables the application ofTube-Based Robust Model Predictive Controllers also to systems with extremely high sampling frequencies.

The purpose of the following chapter is to provide a detailed exposition of the Tube-Based Robust ModelPredictive Control framework, which will also be useful in the development of “Interpolated Tube MPC”,

1 the nominal system in this context is the system that describes the plant dynamics when no uncertainty is present

47

the novel contribution of this thesis that is presented in chapter 5. Section 4.1 introduces the concept of arobust positively invariant set, which is fundamental to the Tube-Based Robust MPC approach. The finitehorizon optimal control problem that is solved on-line and the main results for state-feedback Tube-BasedRobust MPC are stated in section 4.2. Extensions of this basic framework to the output-feedback case andto the case of tracking piece-wise constant references are presented in section 4.3 and 4.4, respectively.Furthermore, for each of the different controller types a case study is provided for illustration. Thequestion of how to choose the different design parameters of the controllers is addressed in section 4.5.Finally, section 4.6 provides a computational benchmark of the on-line evaluation speed of the differentTube-Based Robust Model Predictive Controllers for their implicit as well as their explicit implementations.

The material presented in the following was composed from a large number of different references onTube-Based Robust MPC, including Langson et al. (2004); Mayne et al. (2005); Limon et al. (2005);Mayne et al. (2006b); Alvarado (2007); Alvarado et al. (2007a,b); Limon et al. (2008b,a); Mayne et al.(2009); Rawlings and Mayne (2009) and Limon et al. (2010).

4.1 Robust Positively Invariant Sets

Robust positively invariant sets play an important role not only in Tube-Based Robust MPC, but also invarious other Robust MPC approaches. Definition 2.1 introduced the notion of a positively invariant set.When there are exogenous disturbances present, i.e. when the system dynamics are described by

x(t+1) = f (x(t), u(t), w(t)), (4.1)

where w(t) ∈W and W is a compact set, this definition can be generalized as follows:

Definition 4.1 (Robust positively invariant set, Blanchini (1999)):A set Ω is said to be robust positively invariant (RPI) for the autonomous system

x(t+1) = f (x(t), w(t)) (4.2)

if for all x(0) ∈ Ω and all w(t) ∈W the solution x(t) ∈ Ω for all t>0.

Note that the term “robust invariance” is used somewhat loosely in Definition 4.1. Some authors refer torobust positively invariant sets as they are used in this thesis as disturbance invariant sets (e.g. Kolmanovskyand Gilbert (1998)). This is because robust invariance may be regarded as being more general, since italso encompasses systems where the uncertainty manifests itself not in exogenous disturbances but inan uncertain model description. That being said, the following parts of this thesis will always refer toDefinition 4.1 when speaking of robust invariant sets.

Definition 4.2 (Minimal robust positively invariant set, Rakovic et al. (2005)):A RPI set Ω is said to be the minimal robust positively invariant (mRPI) set F∞ for the system (4.2) if itis contained in every closed RPI set of (4.2).

For the special case of perturbed discrete-time linear systems, where

x(t+1) = f (x(t), w(t)) = Ax(t) +w(t), (4.3)

Kolmanovsky and Gilbert (1998) show that if the system matrix A is Hurwitz, the mRPI set F∞ exists,is unique, compact and contains the origin. The following sections, which discuss the properties ofTube-Based Robust MPC, will make use of robust positively invariant sets in a more conceptual sense.Section 4.5.3 later addresses the computation of RPI sets, putting an emphasis on the special requirementsin the context of Tube-Based Robust MPC.

48 4. Tube-Based Robust Model Predictive Control

4.2 Tube-Based Robust MPC, State-Feedback Case

This section introduces the basic state-feedback version of Tube-Based Robust Model Predictive Control,for which it is assumed that, at each sampling instant, an exact measurement of the state x of the systemis available. Furthermore, the control task in this section will be confined to regulating the state of thesystem to the origin. The developed framework will in later sections be extended to the more realisticoutput-feedback setting, where only limited information about x is available, and to the task of robustlytracking a non-zero target set point.

4.2.1 Preliminaries

Problem StatementThe problem considered in the following is that of regulating to the origin the state of a constrained,discrete-time linear system perturbed by a bounded, additive disturbance. The system dynamics are

x+ = Ax + Bu+w, (4.4)

where x ∈ Rn, u ∈ Rm and w ∈ Rn are the current state, control input, and disturbance, respectively, andwhere x+ denotes the successor state of the system at the next time step. The system is subject to thefollowing (hard) constraints on state and control input:

x ∈ X, u ∈ U, (4.5)

where X⊂ Rn is closed, U⊂ Rm is compact, and both sets are polyhedral and contain the origin in theirrespective interior. The additive disturbance w is unknown but bounded, that is

w ∈W , (4.6)

where W ⊂ Rn is assumed to be compact and to contain the origin. For the problem to be well-posed, thefollowing standard assumption on the system matrices is required:

Assumption 4.1 (Controllability):The pair (A, B) is controllable.

For brevity, a short-hand notation similar to the one in section 2.1.1 will be used in the following. Letu :=

u0, u1, . . . , uN−1

and w :=

w0, w1, . . . , wN−1

denote the control sequence and the disturbancesequence, respectively. Let Φ(i; x ,u,w) denote the solution of (4.4) at time i controlled by u when theinitial state at time 0 is x (by convention, Φ(0; x ,u,w) := x). Furthermore, let Φ(i; x , u) denote thesolution of the nominal system

x+ = Ax + Bu (4.7)

at time i controlled by the predicted nominal control sequence u :=

u0, u1, . . . , uN−1

when the initial stateat time 0 is x ( x0 := x). Denote the predicted nominal state trajectory by x :=

x0, x1, . . . , xN−1

.

As pointed out in section 2.3.2, the perturbation through a non-vanishing additive disturbance generallyprohibits stability of the origin to be established. Nevertheless, under certain conditions, it is possibleto achieve robust stability of a set of states, which may then be regarded as some kind of “origin” of theuncertain system. In Mayne et al. (2005), the authors obtain the strong result of robust exponentialstability of a robust positively invariant set E . This contribution may be regarded as the actual “birth”of Tube-Based Robust MPC and had a strong echo among the scientific community, triggering manyrecent contributions along the same line of work (see the ones listed in the beginning of this chapter andreferences therein).

4.2. Tube-Based Robust MPC, State-Feedback Case 49

Control StrategyFollowing the ideas in Lee and Kouvaritakis (2000), the employed control law consists of two separatecomponents: the first component is a is a feedforward control input computed for the nominal system,whereas the second component is a linear feedback controller that acts on the error2 e := x− x betweenactual state x and predicted nominal state x:

u= u+ Ke = u+ K(x− x). (4.8)

Assumption 4.2 (Stabilizing disturbance rejection controller):The linear disturbance rejection controller K ∈ Rm×n in (4.8) is chosen such that AK := A+ BK is Hurwitz.

Note that Assumption 4.2 can always be satisfied if Assumption 4.1 holds true. For what follows, thedefinition of the Minkowski set addition, denoted by the ⊕-operator, and the Pontryagin set difference,denoted by the -operator, will prove useful for expressing operations on sets.

Definition 4.3 (Minowski set addition, Schneider (1993)):Given two setsA ⊆ Rn andB ⊆ Rn, the Minkowsi set addition (also: Minkowski sum) is defined by

A ⊕B :=

x + y | x ∈A , y ∈B

. (4.9)

Definition 4.4 (Pontryagin set difference, Kolmanovsky and Gilbert (1998)):Given two setsA ⊆ Rn andB ⊆ Rn, the Pontryagin set difference is defined by

A B :=

z ∈ Rn | z+ y ∈A , ∀ y ∈B

. (4.10)

Remark 4.1 (Relationship between Minkowski sum and Pontryagin difference, Borrelli et al. (2010)):It is important to realize that the Pontryagin difference (also referred to as Set Erosion in Blanchini and Miani(2008)) is not the complement of the Minkowski sum, i.e. it does in general not hold that (A B)⊕B =A .Rather, it holds that (A B)⊕B ⊆A .

Using the newly defined Minkowski set addition, the following Proposition, which is of fundamentalimportance for Tube-Based Robust Model Predictive Control, is easily formulated.

Proposition 4.1 (Proximity of actual and nominal system state, Mayne et al. (2005)):Suppose that Assumption 4.2 is satisfied and that E ∈ Rn is a robust positively invariant set for theperturbed system x+ = AK x + w. If x ∈ x ⊕ E and u = u+ K(x − x), then x+ ∈ x+ ⊕ E for alladmissible disturbance sequences w ∈W , where x+ = Ax + Bu+w and x+ = Ax + Bu.

Proposition 4.1 states that if the control law (4.8) is employed, it will keep the states x(i) = Φ(i; x ,u,w) ofthe the uncertain system (4.4) “close” to the predicted states x(i) = Φ(i; x , u) of the nominal system (4.7)for all admissible disturbance sequences w, i.e.

x(0) ∈ x0 ⊕ E =⇒ x(i) ∈ x i ⊕ E ∀w(i) ∈W ,∀ i ≥ 0, (4.11)

where x(0) and x0 are the initial states of (4.4) and (4.7), respectively. Proposition 4.1 therefore suggeststhat if the optimal control problem for the nominal system (4.7) is solved for the tightened constraints

X := XE , U := U KE , (4.12)

then the use of the control law (4.8) will ensure persistent constraint satisfaction for the controlleduncertain system (4.4). For the problem to be well-posed, the following additional assumption is required:

2 clearly, e=0 in the nominal case, i.e. when W = ;


Assumption 4.3 (Existence of tightened constraint sets X and U):There exists a feedback controller gain K ∈ Rm×n such that the tightened constraint sets X and U existand contain the origin.

For Assumption 4.3 to hold it is required that the set W by which the disturbance w is bounded issufficiently “small”. This is of course an implicit requirement in any robust control problem, since oneclearly cannot ask for robustness guarantees in the presence of arbitrarily large disturbances. In order tominimize the cross-section of the robust positively invariant set E , it is desirable to choose the feedbackgain K “large”. However, this immediately results in an also “large” mapping KE , and therefore in a“small” tightened constraint set U. Hence, there is a tradeoff in choosing K between good disturbancerejection properties (K large) on the one hand, and the size of the tightened constraint set U (U large forsmall K) on the other hand. This tradeoff will be discussed in more detail in section 4.5.2, where also aconstructive way to determine a suitable K is presented.

Remark 4.2 (Freedom in choosing the nominal initial state x0):Note that there is no need for the initial state x0 of the nominal system to equal the current state x of theactual system. Clearly, the requirement x ∈ x0 ⊕ E in Proposition 4.1, i.e. that x0 is sufficiently “close”to x , is trivially fulfilled if x0 = x . However, as will be identified in the following, there may be better choicesfor x0 than this most obvious one.

The Cost FunctionDefine the cost function VN : Rn×Nm 7→ R+ for a trajectory of the nominal system (4.7) as

VN ( x , u) :=N−1∑

i=0

l

x i, ui

+ Vf

xN

(4.13)

with stage cost

l

x i, ui

= || x i||2Q + ||ui||

2R (4.14)

and terminal cost

Vf

xN

= || xN ||2P , (4.15)

with positive definite weighting matrices Q0, R0, and P0.

Remark 4.3 (Cost functions based on polytopic norms, Mayne et al. (2005)):It can be shown that the presented results, with some minor modifications, are valid also for the case when l(·)and Vf (·) are defined using polytopic vector norms, e.g. l1- and l∞-norms. However, in view of Remark 2.1,the use of the resulting linear cost functions in MPC may sometimes be problematic. Therefore polytopic vectornorms in the cost function will not be considered any further in the following.

The terminal cost function Vf (·) and the terminal constraint set X f are chosen to satisfy the usualassumptions for MPC stability as in Assumption 2.2, namely:

Assumption 4.4 (Terminal constraint set):The terminal constraint set X f is a constraint admissible, positively invariant set for the closed-loop systemx+ = Ax + Bκ f ( x), i.e. X f ⊂ X, and for all x ∈ X f it holds that Ax + Bκ f ( x) ∈ X f and κ f ( x) ∈ U.

Assumption 4.5 (Terminal cost function):The terminal cost function Vf (·) is a local control Lyapunov function for the system x+ = Ax + Bκ f ( x)in X f , i.e. it satisfies Vf (Ax + Bκ f ( x)) + l( x ,κ f ( x))≤ Vf ( x) ∀ x ∈ X f .


Note that requiring X f to be contained in the tightened state constraint set X rather than in the originalstate constraint set X ensures, in virtue of Proposition 4.1, that the terminal state x(N) of the uncertaintrajectory is contained in X. Also note that the disturbance rejection controller K need not be the sameas the infinite horizon controller κ f (·), which is usually chosen as the optimal unconstrained infinitehorizon controller K∞

3. In fact, since the purpose of K is not to yield an optimal cost but to provide gooddisturbance rejection properties, it can be optimized with respect to this objective. See section 4.5.2 fordetails and a constructive method to compute K . For the purpose of this section, assume that K is a givenstabilizing disturbance rejection controller.

For a given initial state x0 of the nominal system (4.7), let UN ( x0) denote the set of admissible nominalcontrol sequences u, i.e.

UN ( x0) =¦

u | ui ∈ U, Φ(i; x0, u) ∈ X, for i=0,1, . . . , N−1, Φ(N ; x0, u) ∈ X f

©

. (4.16)

The Conventional Optimal Control ProblemAssume for now that x0= x , i.e. that initial state of the nominal system and actual system state coincide.Define the following conventional optimal control problem P0

N (x):

V 0N (x) =min

u

VN (x , u) | u ∈ UN (x)

(4.17)

u0(x) = arg minu

VN (x , u) | u ∈ UN (x)

. (4.18)

The domain XN of the value function V 0N (·) can be expressed as

XN =

x | UN (x) 6= ;

. (4.19)

The solution of P0N (x) yields the predicted optimal control sequence u0(x) :=

u00(x), u0

1(x), . . . , u0N−1(x)

and the predicted optimal state trajectory x0(x) :=

x00(x), x0

1(x), . . . , x0N−1(x)

, where x00(x) := x and

x0i (x) := Φ(i; x , u0(x)). Under the implicit conventional state-feedback Model Predictive Control law

κ0N (x) := u0

0(x) (4.20)

persistent feasibility and satisfaction of the tightened constraints (4.12) is guaranteed for the closed-loopnominal system

x+ = Ax + Bκ0N ( x) (4.21)

for all initial states x = x0 ∈ XN . From Assumptions 4.4 and 4.5 and from Theorem 2.1 it furthermorefollows that the origin of the controlled nominal system (4.21) is exponentially stable with a region ofattraction XN .

4.2.2 The State-Feedback Tube-Based Robust Model Predictive Controller

As indicated in Remark 4.2, it is not necessary to choose the initial state x0 of the nominal systemequal to the current state x of the actual system, as has been the case for the conventional optimalcontrol problem P0

N (x). On the contrary: because of the influence of the disturbance w on the uncertainsystem (4.4), it is not necessarily true that V 0

N (Ax + Bκ0N ( x) + w) ≤ V 0

N (x), i.e. that the cost V 0N (·)

decreases along the actual, uncertain state trajectory for all initial states x ∈ XN \E (Rawlings andMayne (2009)). In other words: it is not possible to establish robust exponential stability of the set E3 that the form of (4.15) already implies that the terminal controller is assumed linear. The terminal weighting matrix P

therefore is the positive definite matrix that characterizes the unconstrained infinite horizon for the respective controller.


if the control (4.20) computed from the conventional optimal control problem P0N (x) is applied to the

uncertain system (4.4). This is why in Mayne et al. (2005) a novel optimization problem is introducedthat incorporates the nominal initial state x0 as an additional decision variable in the optimal controlproblem. This is possible since the states x of the nominal system (4.7) have no immediate physicalmeaning for the actual system (4.4). In particular, x may well differ from x . Hence, x0 can be seen as anadditional parameter in the optimal control problem of the virtual nominal system which can be chosenfreely (as long as it satisfies Proposition 4.1). By doing so, it is possible to synthesize a controller forwhich robust exponential stability of the set E can be established.

The Modified Optimal Control ProblemMotivated by the above discussion about the choice of the initial state of the nominal system, Mayne et al.(2005) introduce the modified optimal control problem P∗N (x):

V ∗N (x) =minx0,u

VN ( x0, u) | u ∈ UN ( x0), x0∈x⊕(−E )

(4.22)

( x∗0(x), u∗(x)) = argmin

x0,u

VN ( x0, u) | u ∈ UN ( x0), x0∈x⊕(−E )

, (4.23)

which in its core ingredients is the same as the conventional optimal control problem P0N (x), but which in

addition also includes the initial state x0 of the nominal syste as a decision variable. This new decisionvariable is subject to the constraint

x0 ∈ x ⊕ (−E ). (4.24)

Following previous notation, let u∗(x):=

u∗0(x), u∗1(x), . . . , u∗N−1(x)

denote the predicted optimal nomi-nal control sequence and let x∗(x):=

x∗0(x), x∗1(x), . . . , x∗N−1(x)

denote the associated predicted optimalnominal state trajectory obtained from P∗N (x), where x∗i (x) := Φ(i; x∗0(x), u

∗(x)). The domain XN of thevalue function V ∗N (·) of the modified problem P∗N (x) is

XN =

x | ∃ x0 such that x0∈x⊕(−E ), UN ( x0) 6= ;

. (4.25)

Inspecting (4.25) and (4.19), it it easy to see that XN = XN⊕E . If, motivated by Proposition 4.1, thecontrol law (4.8) is applied to the system uncertain when its current state is x , the implicit modifiedstate-feedback Model Predictive Control law is given by

κ∗N (x) := u∗0(x) + K(x − x∗0(x)), (4.26)

where u∗0(x) and x∗0(x) are obtained by solving the modified optimal control problem P∗N (x) on-line. Notethat the constraint sets X, U and hence also E , X and U are all polytopic (and thus convex) and cantherefore be expressed by a finite set of linear inequalities. Moreover, as the objective function of P∗N (x) isquadratic, the optimization problem P∗N (x) is a Quadratic Program (Mayne et al. (2005)).

Since x0 is now a parameter in the optimization problem (4.22) (and thus generally different from x),the applied control move κ∗N (x) will also, in contrast to conventional MPC, generally differ from the firstelement of u∗(x). The additional feedback component of the control action generated by K(x− x∗0(x)) issuch that it drives the state x of the actual system back towards the predicted state x of the nominal system.Hence, it counteracts the influence of the disturbance sequence w and keeps the system trajectory xinside the sequence of sets T (0),T (1),T (2), . . . , where T (t) = x∗0(x(t))⊕E . This tube of trajectories,illustrated in Figure 4.1, is where the name “Tube-Based Robust Model Predictive Control” originates.


Figure 4.1.: The “tube” of trajectories in Tube-Based Robust MPC

Properties of the ControllerFrom the definition of the modified optimal control problem P∗N (x) one can deduce the following:

Proposition 4.2 (Properties of region of attraction, value function and optimizer, Mayne et al. (2005)):Let XN and XN be the domains of the value functions V ∗N (x) and V 0

N (x), respectively. Then,

1. XN = XN ⊕E ⊆ X.

2. V ∗N (x) = V 0N ( x

∗0(x)) and u∗(x) = u0( x∗0(x)) for all x ∈ XN .

3. For all x ∈ E it holds that V ∗N (x) = 0, x∗0(x) = 0, u∗(x) = 0,0, . . . ,0, x∗(x) = 0,0, . . . ,0 andthat the control action is κ∗N (x) = K x .

Item 1 of Proposition 4.2 states that the Tube-Based Robust Model Predictive Controller obtained fromthe modified problem P∗N (x) yields an enlarged region of attraction XN as compared to XN , the region ofattraction of the controller from the conventional problem P0

N (x). Furthermore, item 3 states that theControl Lyapunov Function V ∗N (·) is zero inside the set E , a fact which is useful to establish Theorem 4.1,the main theorem given at the end of this section. In order to be able to formally address the stability ofsets in the following, introduce d(z,Ω) := inf

||z− x || | x ∈ Ω4 as a measure of the distance of a point z

from a set Ω. Clearly, d(z,Ω) = 0 for all z ∈ Ω.

Definition 4.5 (Robust exponential stability, Mayne et al. (2006b)):A set Ω is robustly exponentially stable (Lyapunov stable and exponentially attractive) for the systemx+= f (x ,κ(x), w), w ∈W , with a region of attraction X if there exists a c > 0 and a γ ∈ (0,1) such thatany solution x(·) of x+= f (x ,κ(x), w) with initial state x(0) ∈ X satisfies d(x(i),Ω)≤ cγid(x(0),Ω) forall i ≥ 0 and all admissible disturbance sequences w.4 here || · || denotes any norm


Theorem 4.1 (Robust exponential stability of E , Mayne et al. (2005)):Suppose that Assumptions 4.1–4.5 hold true and that E is a robust positively invariant set for the perturbedclosed-loop system x+= AK x +w, where w ∈W . Furthermore, let κ∗N (x) be the implicit modified ModelPredictive Control law obtained from solving P∗N (x) on-line at each sampling instant. Then the set E isrobustly exponentially stable for the controlled system

x+ = Ax + Bκ∗N (x) +w, (4.27)

where w ∈W . The region of attraction is XN .

Theorem 4.1 states a strong result for the Tube-Based Robust Model Predictive Controller described in theprevious sections: although the system is subject to non-vanishing time-varying disturbances, exponentialstability of the robust positively invariant set E can be guaranteed. This means that for any admissibleinitial state x ∈ XN \ E , the controller is able to drive the system state into E at a guaranteed rate ofconvergence, even if one alleges that the disturbance w has a malicious intent.

Complexity of the Optimization ProblemThe number of scalar variables in the modified optimal control problem P∗N (x) is Nv ar= Nm+ (N+1)n,and the number of scalar constraints is Ncon= N (n+NX+NU) + NX f

+ NE , where NX, NU, NX fand NE are

the number of inequalities defining the polytopic sets X, U, X f , and E , respectively. In addition to theobvious dependency on the prediction horizon N and the state and control input dimensions n and m, thecomplexity of P∗N (x) is strongly affected also by the complexity of the sets X, U, X f , and E .

4.2.3 Tube-Based Robust MPC for Parametric and Polytopic Uncertainty

It is possible to also treat parametric and polytopic uncertainty within the Tube-Based Robust MPCframework presented in the previous section. This can be achieved by introducing a “virtual additivedisturbance” w, whose bounding set W is determined by the model uncertainty of the original system.The polytopic uncertainty framework from section 3.2.1 assumes the (time-varying) system dynamics

x+ = Ax + Bu, (4.28)

in which the matrices A and B at any time can take on any value in the set Ω:

[A B] ∈ Ω = Convh

[A1 B1], [A2 B2], . . . , [AL BL]

. (4.29)

The following quadratic stabilizability assumption is needed:

Assumption 4.6 (Quadratic stabilizability, Rawlings and Mayne (2009)):The unconstrained uncertain system x+= Ax + Bu is quadratically stabilizable, i.e. there exists a positivedefinite function Vf (x) = x T Pf x , a linear feedback control law u= K x and a positive constant c such that

Vf ((A+ BK)x)− Vf (x)≤−c||x ||2 (4.30)

for all x ∈ Rn and all [A B] ∈ Ω. The origin is globally exponentially stable for the uncertain closed-loopsystem x+= (A+BK)x .

The set Ω, being the convex hull of a finite set of matrices, is convex. Therefore (4.30) holds true forall x ∈Rn and all [A B] ∈ Ω. The controller K and the associated matrix Pf in Vf (·) can be obtainedoff-line using standard LMI techniques (Boyd et al. (1994)). In fact, Kothare’s controller from section 3.4.1


performs the computation of such a K on-line at each time step, while additional LMI constraints incorpo-rated into the optimization problem ensure persistent feasibility. Because in the Tube-Based Robust ModelPredictive Control framework the question of robust feasibility is addressed by choosing appropriatelytightened constraints, these additional constraints are not necessary. Some of the ideas from Kothare’scontroller can however be adopted to find a “good” choice of the disturbance rejection controller K , aswill be outlined in section 4.5.2.

Define the nominal system as

z+ = Az+ Bv , (4.31)

with nominal system and input matrices

A :=1

L

L∑

j=1

A j, B :=1

L

L∑

j=1

B j. (4.32)

Because of the convexity of Ω the origin is globally exponentially stable for x+ = Ax + Bu. Introducingthe virtual disturbance w (Rawlings and Mayne (2009)), one may rewrite (4.28) as

x+ = Ax + Bu+ w, (4.33)

where w := (A−A)x + (B−B)u lies in the set W defined by

W :=¦

(A−A)x + (B−B)u | [A B] ∈ Ω, (x , u) ∈ X×U©

. (4.34)

Since X, U and Ω are all polytopic, so is the set W . Note that system (4.33) is of the same form as (4.4),the system with bounded additive disturbances addressed in the previous section. Consequently, thedesign process for a Tube-Based Robust Model Predictive Controller for the virtual nominal system (4.33)is the same as the one in section 4.2.2. Note however that the virtual disturbance w in contrast to the“real” exogenous disturbance w now implicitly depends on the value of x and u. Namely, from (4.34)it follows that w → 0 uniformly in [A B] ∈ Ω as (x , u)→ 0. Under some additional assumptions, it istherefore possible to prove robust asymptotic stability of the origin, as opposed to robust stability of theset E (Rawlings and Mayne (2009)).

Remark 4.4 (Mixed uncertainty):It has been shown how to translate polytopic model uncertainties to the framework of a additive, boundeddisturbances. This allows one to employ the methods of Tube-Based Robust MPC to robustly control systemswith polytopic uncertainty. More generally, essentially all kinds of disturbances and model uncertaintiescan be treated in the Tube-Based Robust MPC framework, as long as they can be represented by a bounded,additive disturbance. Hence, robust controller synthesis is possible also for systems subject to combinations ofpolytopic model uncertainty and an additive bounded disturbance by defining a corresponding virtual systemwith appropriately chosen bounds on w.

In virtue of Remark 4.4 only systems subject to additive disturbances of the form (4.4) will be addressedin the following. All subsequent extensions of the Tube-Based Robust MPC framework can also readily beapplied to systems subject to multiple kinds of uncertainty.


4.2.4 Case Study: The Double Integrator

The constrained discrete-time double integrator is by far the most frequently appearing illustrative ex-ample in the Model Predictive Control literature. In order to facilitate comparability with other MPCapproaches, this standard example will also be used in the case studies of this thesis. Within the chapteron Tube-Based Robust MPC, it will be used in a number of different versions, thereby accounting forthe different features of the presented approaches. An overview over the different case study examplesas well as the chosen design parameters of the respective controllers can be found in Table 4.1 on page 101.

The case study presented in this particular section is taken from Mayne et al. (2005). A state-feedbackTube-Based Robust Model Predictive Controller for a discrete-time constrained double integrator subjectto bounded additive disturbances is synthesized and its main properties are discussed. Both implicit andexplicit controller implementations are developed and compared with respect to their computationalcomplexity and on-line evaluation speed5. A more detailed computational benchmark, which in additionalso compares the two controllers with other controller variants developed in the following sections, willbe presented in section 4.6 at the end of this chapter.

Controller SynthesisConsider an LTI system subject to bounded additive uncertainty, with system dynamics described by

x+ =

1 10 1

x +

0.51

u+w. (4.35)

The state constraints are X =

x | x2 ≤ 2

, the control constraint is U =

u | |u| ≤ 1

and thedisturbance bound is W =

w | ||w||∞ ≤ 0.1

. The weighting matrices in the stage cost (4.14) are givenas Q = I2 and R = 0.01, respectively. The terminal weighting matrix P is chosen as P∞, characterizingthe optimal infinite horizon cost of the unconstrained control problem. In contrast to the examplein Mayne et al. (2005), the disturbance rejection controller K is not equal to the LQR gain K∞, butinstead optimized in order to obtain a smaller invariant set E . The method used for this computation isoutlined in section 4.5.2. Note that in this particular case, due to the small control weight R, the LQRgain K∞ = [−0.66 − 1.33] already provides good disturbance rejection properties. As a result, K∞and the optimized K = [−0.69 − 1.31] are almost identical. Hence, the approximated minimal robustpositively invariant set E obtained using the optimized K is only slightly smaller than the one obtainedusing K∞ in the original example in Mayne et al. (2005). The mRPI set E has been computed using thealgorithm that will be presented in section 4.5.3 with an error bound of ε = 10−2. This means that thealgorithm guarantees that6 E ⊆ E∞⊕ 0.01 ·B2

∞, where E∞ denotes the exact mRPI set.

Figure 4.2 shows the region of attraction of the Tube-Based Robust Model Predictive Controller fordifferent prediction horizons N . The rather limited admissible control action (small U and U) in thisexample leads to a controller whose region of attraction grows only slowly with the prediction horizon.For the following simulation the prediction horizon was chosen as N=9 in order to obtain a sufficientlylarge region of attraction.

The closed-loop system was simulated for the initial condition x(0) = [−5 − 2]T and a random, uniformlydistributed, time-varying disturbance w ∈ W for a simulation horizon Nsim = 15. The trajectory of thecontrolled system and the applied control inputs are depicted in Figure 4.3 and Figure 4.4, respectively.The regions shown in red are regions of infeasibility, i.e. they are the complement of X and U in thestate and input space, respectively. Note that in this example, the constraint set X is unbounded. The

5 the performance is of course the same for implicit and explicit implementation, as both represent the same controller6 in the following Bn

p will denote an n-dimensional p-norm unit ball


Figure 4.2.: Regions of attraction of the controller for different prediction horizons

terminal constraint set X f is shown in green, and the cross-section of the “tube” of trajectories is shown inyellow. The solid line corresponds to the actual state trajectory of the system, whereas the dash-dotted linecorresponds to the trajectory of the optimal initial states x∗0(x) of the nominal system. Furthermore, theboundaries of the region of attraction XN of actual states and of the region of attraction XN of nominalstates are also shown in Figure 4.3.

Simulation ResultsFrom Figure 4.3 and Figure 4.4 it is evident that state and control constraints are satisfied over the wholesimulation horizon. Moreover, is easy to see that the controller is non-conservative in the sense thatthe control input saturates for 0 ≤ t ≤ 3 and the (virtual) initial states of the nominal system at t = 4and t = 5 are chosen such that the tube just touches the facet x2 = 2 of the constraint set X. This clearlyis a highly nonlinear controller behavior and cannot be achieved by conventional Linear Robust Controlmethods. Figure 4.4 also shows the set KE ∈ R (between the two dashed lines), which characterizes abound on the disturbance rejection component K(x− x∗0) of the control action for t →∞. For the last 7control moves the tube is centered at the origin, which is explained by to Proposition 4.2.

In order to allow the application of the Tube-Based Robust Model Predictive Controller to “fast” dynamicalsystems, the on-line optimization problem must be efficiently solvable. If equality constraints of theform x i+1 = Ax i + Bui are used in the formulation of P∗N (x), the optimization problem for the predictionhorizon N=9 is characterized by Nv ar=29 scalar variables and Ncon=65 scalar constraints. For this andall following examples, the interior-point algorithm of the QPC-solver (Wills (2010)) was used to computethe solution of P∗N (x) on-line. Without special efforts directed to code efficiency7, the computation timefor solving the optimization problem P∗N (x) was found to be in the range of 2-10ms on a 2.5 GHz IntelCore2 Duo processor on Microsoft Windows Vista (32 bit). See section 4.6 for further details and abenchmark of the computational complexity of all the different controllers presented in this chapter.

7 such as storing the optimizers at each time step in order to use them to warm-start the solver at the next iteration


Figure 4.3.: Trajectories for the state-feedback Tube-Based Robust MPC Double Integrator example

Figure 4.4.: Control inputs for the state-feedback Tube-Based Robust MPC Double Integrator example


Remark 4.5 (Reducing the number of optimization variables, Borrelli et al. (2010)):Although the number of optimization variables may be reduced by substituting x i = Ai x +

∑i−1j=0 AjBui−1−j, it

is important to be aware that doing this may result in an even worse computational performance than incase equality constraints of the form x i+1 = Ax i + Bui are used. Clearly, this does not seem very intuitive atfirst. But this paradoxon is relatively easy to explain: The use of equality constraints yields highly structuredoptimization problems, whose properties (in particular sparsity of the constraint matrices) can be exploitedby advanced numerical algorithms. If equality constraints are eliminated this structure is lost, and as a resultthe solver may become less efficient even if the overall number of constraints is reduced.

A Property of the Optimization ProblemAn interesting observation one can make in Figure 4.3 is that the actual states of the system always(except for a small region around the origin) lie on the boundary of the tube T (t) = x∗0(x(t))⊕E . This isdue to an important general property of the quadratic optimization problem in combination with the factthat, in this specific example, the set E is symmetric about the origin.

Proposition 4.3 (An important property of the optimization problem):Consider the optimization problem P∗N (x) from (4.23). For this problem, the optimizer x∗0(x) is unique,x∗0(x) = 0 for all x ∈ E and x∗0(x) lies on the boundary of the set x ⊕ (−E ) for all x /∈ E .

Proof. Because VN (·) is quadratic, V ∗N (·)≥ 0 and V ∗N (0) = 0, the level sets of the objective function VN (·)are ellipsoids centered at the origin. Therefore, the minimum is achieved either at x∗0 = 0 (if feasible), oron the boundary of the feasible set. Note that the only constraint that is put on x0 in the optimizationproblem P∗N (x) is x0 ∈ x ⊕ (−E ) or, equivalently, x ∈ x0 ⊕ E . Hence, if the current state x is notcontained within E , the minimum is obtained for some x0 on the boundary of the set x ⊕ (−E ). Theuniqueness of the optimizer is a basic property of convex Quadratic Programming.

The proof of Proposition 4.3 is illustrated in Figure 4.5. The dotted lines correspond to sections of theellipsoidal level sets of the objective function projected on the x-space. For this illustration, the set Efrom the case study was used, which is symmetric about the origin (E =−E). Hence, with the propertyof the optimization problem from Proposition 4.3, it follows that for all x /∈ E , the system state lies on theboundary of the tube T = x∗0(x)⊕E , which is the observation that has been made in Figure 4.3.

Figure 4.5.: Actual state and optimal initial state of the nominal system


The Explicit SolutionSince the optimization problem P∗N (x) is a Quadratic Program, it is, in principle, possible to solve itexplicitly using the methods presented in section 2.4. The explicit solution of P∗N (x) is a piece-wise affinefeedback control law defined over a polyhedral partition of the seat of feasible states x , and was computedusing the mpQP solver implemented in the MPT toolbox (Kvasnica et al. (2004)). For an explorationregion Xx pl =

x | −10≤ x1≤5, −5≤ x2≤2

, the resulting polyhedral partition of the set XN ∩Xx plconsists of Nreg=366 polytopic regions. Together with the piece-wise quadratic value function V ∗N (x),this partition is depicted in Figure 4.6. In addition, Figure 4.7 on page 62 shows the associated piece-wiseaffine optimizer function determining the optimal nominal control input u∗0(x). Note that the range ofvalues of u∗0(x) is determined by the tightened constraint set U and not by the original constraint set U.

Figure 4.6.: Explicit state-feedback Tube-Based Robust MPC: PWQ value function V ∗N (x) over polyhedralpartition of the set XN ∩Xx pl

Clearly, the number of regions is relatively high even for the comparatively simple optimization problemat hand. This is reminiscent of the remarks in section 2.4.2. For the given example, it takes, on average,about 10ms to obtain the optimizer u∗0(x) from the PWA optimizer function for a given x , when usingnon-optimized Matlab-code. Compared with the 2-12ms needed to solve the optimization problemon-line, it is apparent that for this specific example this rather crude implementation of Explicit MPC is,on average, actually slower than the on-line Model Predictive Controller. The reason for this can be foundin the comparatively large number of regions of the polyhedral partition of the set XN ∩Xx pl . However,an efficient implementation of the evaluation of the explicit control law, as remarked in section 2.4.2, willmost likely yield a different picture. The efficient on-line evaluation of explicit control laws by itself is acomplex problem, and research in this field is still ongoing. Therefore, this topic will not be addressedany further in this thesis.


Figure 4.7.: Explicit state-feedback Tube-Based Robust MPC: PWA optimizer function u∗0(x)

4.2.5 Discussion

Tube-Based Robust MPC provides the strong theoretical result of robust exponential stability of a robustpositively invariant set (Theorem 4.1), even in the presence of non-vanishing additive disturbances.Furthermore, by separating the task of constrained optimal control from the task of robust constraint sat-isfaction, Tube-Based Robust Model Predictive Controllers are potentially less conservative than Min-MaxRobust MPC approaches that optimize worst-case performance.

The main benefit of Tube-Based Robust MPC, however, is its relative simplicity – at least from a com-putational point of view. As the computation of the approximated mRPI set E is performed off-line,its complexity is not an issue for the on-line implementation of the controller. Therefore, as remarkedin section 4.2.2, the on-line optimization problem P∗N (x) is a simple Quadratic Programming problem.Including the initial state x0 ∈ Rn of the nominal system as an additional decision variable does not affectthe problem structure and increases the problem size only moderately. Hence, the resulting optimizationproblem P∗N (x) is usually only slightly more complex than that of a nominal Model Predictive Controllerfor a comparable problem and can therefore readily be solved using available optimization algorithms.Moreover, it is also possible to compute an explicit solution using the ideas reported in section 2.4.

It should however be pointed out that in certain cases (especially in higher-dimensional state spaces),the number of facets of the approximated mRPI set E may grow large if a small approximation error εis desired (see section 4.5.3 for details on this issue). Since each additional facet of E introducesadditional constraints in P∗N (x), this may possibly result in an optimization problem of considerably highercomplexity. However, in comparison to the SDP or SOCP problems that commonly arise in the LMI-basedMin-Max MPC approaches of section 3.4, Tube-Based Robust Model Predictive Controllers generallyinvolve significantly less complex optimization problems. On the other hand, the questions of how toappropriately choose the design parameters K , X f , Vf (·) and N , and of how to compute the approximatedmRPI set E during the synthesis of the controller are nontrivial and need further investigation. Findinganswers to these questions is however postponed to section 4.5. Before that, extensions of the Tube-BasedRobust Model Predictive Control framework to the output-feedback and to the tracking case will beintroduced in section 4.3 and 4.2, respectively.


4.3 Tube-Based Robust MPC, Output-Feedback Case

As has been discussed in chapter 2, the widespread assumption that full state feedback is available isusually not met in applications. In most real-world problems it will not be possible to obtain exactmeasurements of all n states. There will rather be a noisy measurement y(t) (and its past values) of thesystem output available, based on which the current system state x(t) needs to be estimated. This stateestimation problem can be seen as the dual to the control problem.

For linear state-feedback control problems, the separation principle (Luenberger (1971)) suggests thata stable observer can be designed independently from the controller, yielding stability of the compositesystem of observer and controller. This is, however, generally not true for nonlinear systems (see forexample Atassi and Khalil (1999)). Since the closed-loop dynamics of a plant controlled by a (constrained)Model Predictive Controller are clearly nonlinear (even if the plant itself is linear), it is necessaryto address observer design and controller design simultaneously. The following section does this byintroducing output-feedback Tube-Based Robust MPC, which was first proposed in Mayne et al. (2006b).To circumnavigate the issue of separated state estimation and control mentioned above, this approachdirectly takes into account the estimation error by regarding it as an additional disturbance to the plant.The ideas of standard state-feedback Tube-Based Robust MPC from section 4.2 are then used to againestablish robust exponential stability of a robust positively invariant set. The state estimation task ishereby performed by a simple Luenberger observer.

4.3.1 Preliminaries

Consider the discrete-time linear time-invariant system (4.4) with additional output behavior:

x+ = Ax + Bu+w

y = C x + v ,(4.36)

where x ∈ Rn and u ∈ Rm are the current state and control input, respectively. The successor state x+ isaffected by an unknown, but bounded additive state disturbance w ∈ W ⊂ Rn. In addition, the currentmeasured output y ∈ Rp is also affected by an unknown, but bounded additive output disturbance v ∈V ⊂ Rp. Both W and V are assumed to be compact, convex, and to contain the origin in their respectiveinterior. A, B and C are system matrices of appropriate dimension. As in section 4.2, the system is subjectto the following constraints on state and control input

x ∈ X, u ∈ U, (4.37)

where X⊆ Rn is polyhedral, U⊆ Rm is polytopic, and both X and U contain the origin in their respectiveinterior. Since in output-feedback MPC one needs to estimate the system state x , a necessary assumptionon the system matrices is the following:

Assumption 4.7 (Controllability and Observability):The pair (A, B) is controllable and the pair (A, C) is observable.

Following the ideas of Shamma (2000), the authors of Mayne et al. (2006b) propose to estimate thestate x of system (4.36) using a simple Luenberger observer described by the estimator dynamics

x+ = Ax + Bu+ L(y − y)y = C x ,

(4.38)

4.3. Tube-Based Robust MPC, Output-Feedback Case 63

where x ∈ Rn is the current observer state (i.e. the current state estimate), x+ is the successor state of theobserver system, y ∈ Rp is the current observer output and L ∈ Rn×p is the observer feedback gain matrix.Defining the state estimation error ee := x − x , equation (4.38) can be expressed as:

x+ = Ax + Bu+ LCee + Lv . (4.39)

The state estimation error ee satisfies

e+e = ALee + (w− Lv ), (4.40)

where L is chosen such that AL := A− LC is Hurwitz (this is always possible because of Assumption 4.7).As in state-feedback Tube-Based Robust MPC, the underlying strategy is to control a nominal system

x+ = Ax + Bu (4.41)

in such a way that the actual system (4.36) is guaranteed to satisfy state and control constraints for allpossible realizations of the disturbance sequences w and v :=

v0, v0, . . . , vN−1

. The difficulty herebyis that the additional uncertainty induced through the state estimation error ee also has to be accounted for.

Define ec := x − x as the error between observer state and state of the nominal system. Analogous to thestate-feedback case, the proposed control law

u= u+ Kec = u+ K( x − x) (4.42)

consists of a predicted nominal control action u and a feedback component K( x− x), where the disturbancerejection controller K is chosen such that AK = A+ BK is Hurwitz (again, this is always possible becauseof Assumption 4.7). As the current value of the system state x is unknown, one has to rely on the currentstate estimate x to compute the feedback component of the control input. Applying the control (4.42) tothe system, the observer state x and the error ec between observer state and state of the nominal systemsatisfy the following difference equations:

x+ = Ax + Bu+ BKec + L(Cee + v ) (4.43)

e+c = AK ec + L(Cee + v ). (4.44)

From the definitions of ec and ee the actual state x of the uncertain system can be expressed as:

x = x + ee = x + ec + ee = x + e. (4.45)

Provided that the errors ec and ee can be bounded, one may choose the predicted nominal control sequenceu :=

u0(x), u1(x), . . .

and thus the predicted nominal state trajectory x :=

x0(x), x1(x), . . .

in such away that the actual control sequence u := u0, u1, . . .

and the actual state trajectory x :=

x0, x(1), . . . satisfy the original constraints (4.37) for all times.

Remark 4.6 (Relation to state-feedback Tube-Based Robust MPC):In the framework of output-feedback Tube-Based Robust MPC, the additional uncertainty that stems from thestate estimation error ee is treated in a similar fashion as the external disturbance w that affects the actualstate x in the state-feedback case. Due to this additional uncertainty, the minimal robust positively invariantset will be larger. Consequently, the tightened constraint sets will be smaller. Hence, the incomplete stateinformation inevitably limits the possible control performance as compared to the state-feedback controller.


If one defines an artificial disturbance δe := w− Lv , equation (4.40) can be rewritten as

e+e = ALee +δe, (4.46)

where

δe ∈∆e :=W ⊕ (−LV ). (4.47)

Section 4.1 states that, since the observer gain L is chosen such that AL is Hurwitz, an approximatedminimal robust positively invariant set Ee for the perturbed artificial system (4.46) exists and can becomputed using the algorithm presented in section 4.5.3. This motivates the following:

Proposition 4.4 (Proximity of state and state estimate, Mayne et al. (2006b)):Suppose that Ee is a RPI set for the uncertain system (4.46). If the initial system and observer states x(0)and x(0), respectively, satisfy ee(0) = x(0)− x(0) ∈ Ee, then x(i) ∈ x(i) ⊕ Ee for all i ≥ 0 and for alladmissible realizations of the disturbance sequences w and v.

Remark 4.7 (Steady state assumption on the state estimation error, Mayne et al. (2006b)):The assumption that ee(0) = x(0)− x(0) ∈ Ee made in Proposition 4.4 is essentially a steady state assumptionon the state estimation error. Although this assumption is reasonable during the regular operation of theplant, it may, in general, not be met when the observer is first initialized. To overcome this restriction, theauthors of Mayne et al. (2009) provide an extension of the presented results to the time-varying case, i.e.when this steady state assumption is violated. They show that, under some reasonable additional assumptions,the state estimation error ee lies in a time-varying compact set that converges to the minimal robust positivelyinvariant set Ee. The output-feedback Tube-Based Robust Model Predictive Controller is then modified toaccount for these time-varying bounds on the state estimation error.

Proceeding in a similar fashion as with the state estimation error dynamics (4.46), define a secondartificial disturbance δc := L(Cec + v ) and rewrite (4.44) as

e+c = AK ec +δc, (4.48)

where

δc ∈∆c := LCEe ⊕ LV . (4.49)

Since the disturbance rejection controller K is chosen so that AK is Hurwitz, and since the artificialdisturbance δc is bounded, it is possible to compute an approximated mRPI set Ec also for the error ecbetween state estimate and state of the nominal system.

Proposition 4.5 (Proximity of state estimate and nominal system state, Mayne et al. (2006b)):Suppose that Ec is a RPI set for the uncertain system (4.48). If the initial observer and nominal systemstates x(0) and x(0), respectively, satisfy ec(0) = x(0)− x(0) ∈ Ec, then ec(i) ∈ x(i) ⊕ Ec for all i ≥ 0and for all admissible disturbance sequences w and v.

Defining E as the Minkowski set addition of the two RPI sets Ee and Ec, i.e.

E := Ee ⊕Ec (4.50)

and combining Propositions 4.4 and 4.5, it is straightforward to show the following:

Proposition 4.6 (Proximity of actual state and nominal state, Mayne et al. (2006b)):Suppose x(0), x(0) and u =

u0, u1, . . .

are given. If the initial states of actual system, observer andnominal system satisfy ee(0) = x(0)− x(0) ∈ Ee and ec(0) = x(0)− x(0) ∈ Ec, then under the control lawu(i) = ui + Kec(i) the system state satisfies x(i) ∈ x(i)⊕Ee ⊆ x(i)⊕E for all i ≥ 0 and for all admissiblerealizations of the disturbance sequences w and v.


Analogous to the state-feedback case, Proposition 4.6 suggests that if the control problem for the nominalsystem (4.41) is solved for the tightened constraints

X := XE , U := U KEc, (4.51)

then the use of the feedback policy (4.42) will ensure persistent constraint satisfaction for the controlleduncertain system (4.36). For the problem to be well-posed, an additional assumption is needed:

Assumption 4.8 (Existence of tightened constraint sets):There exists a controller gain matrix K ∈ Rm×n and an observer gain matrix L ∈ Rn×p such that thetightened constraint sets X and U exist and contain the origin.

Combining the above, it is possible to show the following:

Theorem 4.2 (Persistent feasibility, Mayne et al. (2006b)):Suppose the initial states of actual system, observer system and nominal system all lie in X and satisfyee(0)= x(0)− x(0)∈Ee and ec(0)= x(0)− x(0)∈Ec, respectively. Suppose that, in addition, the initialstate x(0) of and the control input sequence u to the nominal system satisfy the tightened constraintsx(i) = Φ(i; x(0), u) ∈ X and ui ∈ U for all i ≥ 0. Then, the state x(i) of and the control inputs u(i) to thecontrolled uncertain system x(i+1) = Ax(i)+Bu(i)+w(i), where u(i) = ui+Kec(i) = ui+K( x(i)− x(i)),satisfy the original constraints x(i) ∈ X and u(i) ∈ U for all i ≥ 0 and all admissible realizations of thedisturbance sequences w and v.

4.3.2 The Output-Feedback Tube-Based Robust Model Predictive Controller

In the state-feedback Tube-Based Robust MPC framework from section 4.2.2, the control input sequence uto the nominal system is chosen in such a way that the actual system satisfies its original constraints onstate x and control input u. But since this section is concerned with output-feedback, the true currentsystem state x is not exactly known. This is why the computation of the predicted optimal control inputscan not be based on the actual system states, but must be based on the state estimates. In other words, thismay be regarded as controlling the observer system. Namely, the predicted nominal control sequence uyields a predicted nominal state trajectory x, which is the center of a predicted tube with cross-section Ecin which the trajectory x of the state estimates is guaranteed to lie. Since the state estimation error ee isbounded by the RPI set Ee, it is therefore possible to guarantee constraint satisfaction of the actual systemtrajectory x if the nominal control problem is solved for the tightened constraints (4.51).

Define the cost function VN : Rn×Nm 7→ R+ for a trajectory of the nominal system (4.41) as

VN ( x0, u) :=N−1∑

i=0

l( x i, ui) + Vf ( xN ), (4.52)

with stage cost function

l( x i, ui) = || x i||2Q + ||ui||

2R (4.53)

and terminal cost function

Vf ( xN ) = || xN ||2P , (4.54)

where the weighting matrices Q, R, and P are assumed positive definite. As in the state-feedback case,terminal cost function Vf (·) and terminal set X f ⊂ X are chosen to satisfy the usual Assumptions 4.4and 4.5, respectively. State x of and control input u to the nominal system (4.36) are subject to the


tightened constraints X and U from (4.51). In addition, initial state x0 and terminal state xN of thenominal system are required to satisfy

x0 ∈ x ⊕ (−Ec), xN ∈ X f , (4.55)

respectively, where x is the current state estimate. As in what has been called the “modified” problem insection 4.2.2, the initial state x0 of the nominal system is considered a decision variable in the optimizationproblem of the controller. For a given x0, the set of admissible nominal control sequences u is:

UN ( x0) =¦


©

. (4.56)

The optimal control problem PN ( x) that is solved on-line is:

V ∗N ( x) =minx0,u

VN ( x0, u) | u ∈ UN ( x0), x0∈ x⊕(−Ec)

(4.57)

( x∗0( x), u∗( x)) = argmin

x0,u

VN ( x0, u) | u ∈ UN ( x0), x0∈ x⊕(−Ec)

. (4.58)

It is easy to see that the optimization problem (4.57) has the same structure as (4.22), the optimizationproblem in state-feedback Tube-Based Robust MPC. In particular, PN ( x) is also a convex QuadraticProgram and hence can be solved efficiently using standard mathematical optimization algorithms.

Because the value function V ∗N (·) is now a function of the current state estimate and not the current state,the domain of the of the output-feedback Tube-Based Robust Model Predictive Controller is the set ofadmissible initial state estimates

XN =

x | ∃ x0 such that x0∈ x⊕(−Ec), UN ( x0) 6= ;

. (4.59)

Define XN :=

x | UN ( x) 6= ;

and XN :=


as the setof admissible nominal initial states and the set of admissible actual initial states, respectively. Then it iseasy to see that XN = XN ⊕Ec, and XN = XN ⊕Ee = XN ⊕E .

Let u∗0( x) denote the first element in the sequence u∗( x) obtained from the solution of PN ( x). Then, theimplicit output-feedback Model Predictive Control law κN (·) is

κN ( x) := u∗0( x) + K( x − x∗0( x)). (4.60)

Using the above ingredients, the main result of output-feedback Tube-Based Robust MPC can be stated:

Theorem 4.3 (Robust exponential stability of E , Mayne et al. (2006b)):Suppose that the set XN is bounded. Then, the set Ec×Ee is robustly exponentially stable for the compositesystem

x+

e+e

=

Ax + BκN ( x) +δcALee +δe

(4.61)

of state estimate x and estimation error ee. The region of attraction is XN×Ee. Any state x(0)= x(0)+ee(0)for which

x(0), ee(0)

∈ XN×Ee is robustly steered to E = Ee⊕Ec exponentially fast while satisfying thestate and control constraints at all times

The output-feedback Tube-Based Robust Model Predictive Controller is almost identical to the state-feedback Tube-Based Robust Model Predictive Controller discussed in section 4.2. The only difference,apart from the modified constraints in the optimization problem PN ( x), is that instead of the actual state x(which is unknown), the state estimate x is used to generate the feedback component K( x − x∗0( x)) inthe control law (4.60). Clearly, because of the additional uncertainty which is due to the state estimationerror ee, the cross-section of the tube T (t) = x∗0( x(t))⊕ E = x∗0( x(t))⊕ Ee ⊕ Ec is larger than in thestate-feedback case. Conversely, the terminal set X f is smaller. This is illustrated in Figure 4.8.


Figure 4.8.: The “tube” of trajectories in output-feedback Tube-Based Robust MPC

Number of Variables and Constraints in the Optimization ProblemThe number of scalar variables Nv ar = Nm + (N+1)n in in the optimization problem PN ( x) is thesame as in PN (x), the optimization problem for the state-feedback controller from section 4.2.2. WithNcon = N (n+NX+NU) + NX f

+ NEc, the number of scalar constraints is also comparable. Note, however,

that the set Ec is often times defined by a larger number of linear inequalities than the set E in thestate-feedback case.

Remark 4.8 (Additional complexity due to the observer):The use of an output-feedback Tube-Based Robust Model Predictive Controller requires that an observer is runin parallel with the on-line optimization, which means that additional computations need to be performedon-line at each time step. The complexity of the simple explicit operations associated with the state estimationis however negligible in comparison to that of solving the optimization problem.

Remark 4.9 (Optimizing over feedback policies on-line, Goulart and Kerrigan (2007)):The ideas of output-feedback Tube-Based Robust MPC are also used in Goulart and Kerrigan (2007), wherethe authors present an approach that is based on a fixed linear state observer combined with the on-lineoptimization over the class of feedback policies which are affine in the sequence of prior outputs. Similar as intheir earlier works (Goulart et al. (2006); Goulart (2006)), the resulting non-convex optimization problem isconvexified using an appropriate reparametrization. Since the feedback controllers are optimization variables,the approach provides a larger region of attraction than methods based on calculating control perturbationsto a static linear feedback law. However, the authors only considered the problem of finding a feasible controlpolicy at each time, without regard to optimality. In addition, the optimization problem is, though convex, ofmuch higher complexity than the one for output-feedback Tube-Based Robust MPC.


4.3.3 Case Study: Output-Feedback Double Integrator Example

Controller SynthesisThe example considered in this section is once again a constrained double integrator, this time taken fromMayne et al. (2006b). The system dynamics are described by

x+ =

1 10 1

x +

11

u+w

y =

1 1

x + v ,(4.62)

where the state and control constraints are X =

x | −50 ≤ x i ≤ 3, i = 1,2

and U =

u | |u| ≤ 3

,respectively, and the disturbance bounds are W =

w | ||w||∞ ≤ 0.1

and V =

v | |v | ≤ 0.05

. Theweighting matrices in the stage cost (4.53) are given by Q = I2 and R= 0.01. The terminal cost matrix Pwas chosen as P∞ and the prediction horizon was set to N=13. The disturbance rejection controller Kwas optimized using the approach from section 4.5.2, yielding K = [−0.7 − 1.0], and the observergain matrix L was chosen such that the eigenvalues of AL are one tenth of those of A+ BK∞, resultingin L = [1.00 0.96]T . Furthermore, the error bound for the computation of the approximated mRPI setswas again chosen as ε = 10−2 (see section 4.5.3). The closed-loop system was simulated for the initialstate estimate x(0) = [−3 − 8]T, a random initial condition x(0) ∈ x(0)⊕Ee and random, uniformly dis-tributed, time-varying state and output disturbances w ∈W and v ∈ V for a simulation horizon Nsim = 15.

The trajectory of the controlled system and the control inputs applied to the system are depicted in Fig-ure 4.9 and Figure 4.10, respectively. The bright yellow inner part of the cross-section of the tube oftrajectories is the RPI set Ec, which bounds the error ec between observer state and nominal systemstate. The darker outer rim is the additional enlargement due to the RPI set Ee, which bounds the stateestimation error ee. The solid line corresponds to the actual state trajectory of the system, whereasthe dash-dotted line corresponds to the trajectory of the initial states x∗0( x) of the nominal system. Forclearness, because the set Ee is comparatively small in this case, the trajectory of the state estimates xis not shown. Since the control action in output-feedback Tube-Based Robust MPC is determined fromthe current state estimate, the region of attraction of interest is XN . Hence, the boundary of XN is alsoshown in Figure 4.9, together with the boundary of XN , the region of attraction of nominal states.

Simulation ResultsFigure 4.9 and Figure 4.10 on page 70 show a similar situation as in the state-feedback case: Thesystem state is regulated to the set E , while state and control constraints are both satisfied over thewhole simulation horizon. From Figure 4.10 a non-conservative, highly nonlinear control behavior canbe verified for the output-feedback Tube-Based Robust Model Predictive Controller. In contrast to thesituation in Figure 4.3 from the case study in section 4.2.4, the actual system states generally do notlie on the boundary of the tube T (t) = x∗0( x(t))⊕ E . This is due to the fact that only estimates ofthe state are available, which requires additional conservativeness in the computation of the controlinput. Since u∗0 = 0 for t →∞ it follows from the control law (4.60) that u(t) ∈ KEc for t →∞. Thisset KEc ⊂ R, which asymptotically bounds the control action, is shown as the region between the twodashed lines in Figure 4.10.

Using the equality constraint formulation, the optimization problem PN ( x) is characterized by Nv ar=41scalar variables and Ncon=117 scalar constraints8. The computation time needed for solving P∗N ( x) wasfound to be in the range between 5ms and 12ms, using the same setup (machine and QP-solver) as insection 4.2.4. Further details on the computational complexity and speed of the controller are given inthe benchmark in section 4.6.8 note that due to the different constraints in this example, the number of variables and constraints can not directly be

compared to the number of variables and constraints in the example from section 4.2.4


Figure 4.9.: Trajectories for the output-feedback Tube-Based Robust MPC Double Integrator example

0 2 4 6 8 10 12 14

−3

−2

−1

0

1

2

3

Figure 4.10.: Control inputs for the output-feedback Tube-Based Robust MPC Double Integrator example


The Explicit SolutionAs already for the state-feedback Tube-Based Robust MPC, the optimization problem PN ( x) of output-feedback Tube-Based Robust MPC is also a Quadratic Program and hence can be solved explicitly usingmultiparametric programming. The exploration region for the output-feedback example of this case studyhas been chosen as Xx pl =

x | −14≤ x1 ≤ 3, −12≤ x2 ≤ 3

. The resulting polyhedral partition of theset XN ∩Xx pl consists of Nreg=142 polytopic regions, which, together with the associated piece-wisequadratic value function V ∗N ( x), are shown in Figure 4.11. The corresponding piece-wise affine optimizerfunction u∗0( x) is depicted in Figure 4.12. The on-line evaluation time of the explicit controller implemen-tation in this example varied between 5ms and 6ms. Although this is not really faster than the fasteston-line solutions of P∗N ( x), the variation of only 1ms strongly emphasizes a major benefit of the explicitimplementation: The necessary on-line computation time is much more predictable.

Figure 4.11.: Explicit output-feedback Tube-Based Robust MPC: PWQ value function V ∗N ( x) over polyhedralpartition of the set XN ∩Xx pl

Note that the region of attraction of the explicit controller in this example is smaller that the one ofthe controller that is based on on-line optimization. This may be the case if Xx pl ∩ X 6= X, i.e if theexploration region does not fully cover the set of feasible states. This condition is however only necessaryand not sufficient, since the region of attraction will usually also be limited by the available control action.Restricting the solution of the optimization problem to some exploration region Xx pl ⊂ X in the statespace is necessary if otherwise the complexity (in terms of the number of regions of the partition) of theresulting explicit controller would be prohibitively high.


Figure 4.12.: Explicit output-feedback Tube-Based Robust MPC: PWA optimizer function u∗0( x)

4.3.4 Discussion

The previous sections introduced output-feedback Tube-Based Robust Model Predictive Control, which al-lows robust stabilization of systems for which only incomplete state information, obtained from erroneousmeasurement data, is available. For the uncertainty model of an additional unknown, but bounded outputdisturbance v , the same basic approach as in state-feedback Tube-Based Robust MPC from section 4.2 canbe employed. Satisfaction of the original constraints X and U is guaranteed by solving a robust controlproblem for the observer system, while bounding the state estimation error ee by an invariant set. Becauseof the additional uncertainty that stems from the state estimation error, the tightened constraint sets Xand U for the nominal problem are smaller than in the state-feedback case. This is equivalent to theintuitive fact that the potential performance of the output-feedback Tube-Based Robust Model PredictiveController is inferior to that of state-feedback kind. Clearly, if X and U are small, and W and V are large,it may happen that the resulting sets X and U are empty9. This would mean that the region of attractionis empty, i.e. that the control problem is infeasible for any initial state.

By treating observer and controller dynamics in an integrated fashion, the problems mentioned in thebeginning of this section (such as that the separation principle loses its validity when a Receding HorizonController is used) can be overcome in a very elegant way. It is once more possible to establish the strongtheoretical result of robust exponential stability of a robust positively invariant set. Naturally, because ofthe limited information about the current system state, this set is larger than in the state-feedback case.As has been indicated in Remark 4.7, the basic time-invariant version of the controller that is discussedhere may not be directly applicable in case the observer has not been running for a sufficiently long timeto reach its steady state. An ad-hoc approach to avoiding this problem would be to first bring the systemto a reference state using a “safe mode” robust auxiliary controller of simpler structure, and to then switchto the output-feedback Tube-Based Robust Model Predictive Controller once the state estimation errormay be regarded sufficiently small. Alternatively, a direct implementation of the extended time-varying

9 in this case, Assumption 4.8 is not satisfied


output-feedback Tube-Based Robust Model Predictive Controller proposed in Mayne et al. (2009) ispossible. For brevity, this modified controller will however not be discussed in this thesis.

The optimization problem PN ( x) is a Quadratic Program also for the output-feedback Tube-Based RobustMPC, a fact that allows to efficiently compute solutions on-line using standard mathematical optimizationalgorithms. Alternatively, an explicit solution of PN ( x) can be obtained using the ideas from section 2.4. Inthis case the on-line computation reduces to evaluating a piece-wise affine optimizer function defined overa (possibly complex) polyhedral partition of the region of attraction, a feature which makes the presentedoutput-feedback Tube-Based Robust Model Predictive Controller attractive also for fast dynamical systemswith high sampling frequencies.

4.4 Tube-Based Robust MPC for Tracking Piece-Wise Constant References

The two Tube-Based Robust Model Predictive Controllers that have been presented in section 4.2 and 4.3are both restricted to the regulation problem only. Chapter 2 however motivates that for a practicalapplication of MPC it is necessary that controllers, in addition to providing robustness in the presence ofuncertainties are also able to handle non-zero target steady states. The goal of the following section istherefore to review a class of Tube-Based Robust Model Predictive Controllers that achieve exactly that.

The classic way of using MPC to track a constant output reference has been discussed in section 2.3. Asa first step in this task, a suitable steady state of the system must be determined. One way to do this isby using the optimization problem (2.24), which yields an admissible steady state such that the outputtracking error is minimized in the least square sense. The Tube-Based Robust Model Predictive Controllerfor Tracking (Limon et al. (2005); Alvarado (2007); Alvarado et al. (2007a,b); Limon et al. (2010))presented in the following moves this computation on-line by introducing an artificial steady state as anadditional decision variable into the optimization problem. To limit the resulting offset, an additionalterm that penalizes the deviation between desired and actual system output is added to the cost function.In section 2.3.1 it has been emphasized that the main problem with the common approach of simplyshifting the system to the desired steady state is that, in order to ensure feasibility, the terminal set X factually would need to be recomputed for each new steady state. The Tube-Based Robust Model PredictiveController for Tracking avoids this problem by computing an “invariant set for tracking” (Alvarado (2007))in an augmented state space. This invariant set for tracking incorporates an additional decision variableparametrizing all possible steady state values.

Section 4.4.1 first introduces the state-feedback version of Tube-Based Robust MPC for Tracking. Sec-tion 4.4.2 will then make use of the ideas from section 4.3 to extend the applicability of this controlleralso to the output-feedback case. Since both controller types in general do not guarantee zero steadystate offset for asymptotically constant disturbances (compare section 2.3.2), section 4.4.3 shows howoffset-free tracking can still be achieved through additional modifications. Illustrative case studies areagain presented for the different controller types.

4.4. Tube-Based Robust MPC for Tracking Piece-Wise Constant References 73

4.4.1 Tube-Based Robust MPC for Tracking, State-Feedback Case

Preliminaries

Problem StatementFor the purpose of the following sections, system (2.1) is formulated in a more general fashion, includinga feedthrough path acting on the system output. The uncertain linear time-invariant system is

x+ = Ax + Bu+w

y = C x + Du,(4.63)

where x ∈ Rn, u ∈ Rm and y ∈ Rp are the current state, control input, and measured output, respectively.The successor state x+ is affected by the unknown, but bounded disturbance w ∈ Rn. The system matricesA, B, C and D are of appropriate dimension. Once again, the floating assumption of controllability ofthe pair (A, B) (Assumption 4.1) is needed. The disturbance w is bounded by the compact and convexset W ⊂ Rn. The system is subject to the usual constraints on state and control input

x ∈ X, u ∈ U, (4.64)

where X⊆ Rn is polyhedral, U⊆ Rm is polytopic, and both X and U contain the origin in their interior.

The objective of the state-feedback Tube-Based Robust Model Predictive Controller for Tracking presentedin this section is to robustly stabilize the system and steer its output to a neighborhood of a desired setpoint ys, while satisfying the constraints (4.64) for all admissible disturbance realizations w.

Set Point ParametrizationExtending the ideas from section 2.3, it is clear that for a given set point ys any complying steady statezs := (xs, us) of the system (4.63) has to satisfy

A− In BC D

xsus

=

0n,1ys

, (4.65)

which can be rewritten as

Ezs = F ys (4.66)

with appropriately defined matrices E and F . As pointed out in section 2.3.1, there may exist more thanone admissible steady state zs for a given set point ys. In order to parametrize the set of feasible steadystates, the following Lemma is useful:

Lemma 4.1 (Alvarado (2007)):Assume that the pair (A, B) is stabilizable. Then, a pair (zs, ys) is a solution to (4.66) if and only if thereexists a vector θ ∈ Rnθ such that

zs = Mθθ

ys = Nθθ ,(4.67)

where Mθ ∈ R(n+m)×nθ and Nθ ∈ Rp×nθ are suitable matrices as defined in Appendix A.2.

Remark 4.10 (Uniqueness of the steady state, Alvarado (2007)):For a given admissible set point ys, the steady state zs = (xs, us) is unique if and only if the rank of the matrixE in (4.66) is equal to n+m. If the rank of E is less than n+m, then there exist infinitely many steadystates zs such that ys = C xs + Dus.


Consider now the nominal system

x+ = Ax + Bu

y = C x + Du,(4.68)

and, analogously to section 4.2, the control law

u= u+ K(x − x). (4.69)

Following the same ideas as in the regulation problem, let E be a robust positively invariant set for theperturbed system (4.63) controlled by (4.69). The tightened constraints for the nominal system then are

X := XE , U := U KE . (4.70)

Clearly, any admissible steady state zs = (xs, us) has to satisfy the constraints

zs = Mθθ = (xs, us) ∈ X× U=: Z. (4.71)

The set of admissible steady states Zs in the (x , u)-space, and the set of admissible set points Ys arecharacterized, respectively, by

Zs =

zs = Mθθ | Mθθ ∈ Z

(4.72)

Ys =

ys = Nθθ | Mθθ ∈ Z

. (4.73)

Consider the following Definition of the projection of a set:

Definition 4.6 (Projection of a set, Blanchini and Miani (2008)):The projection of a setA ⊆ Rn+k onto the x-space Rn is defined by

Projx(A ) :=

x ∈ Rn

∃ y ∈ Rk such that

xy

∈A

. (4.74)

Making use of Definition 4.6, the set of admissible steady states Xs and the set of admissible steadystate inputs Us can be obtained as Xs = Projx(Zs) and Us = Proju(Zs), that is as the projection of Zson the x- and u-space, respectively. Alternatively, if one defines the set of admissible parameters θas Θs =

θ | Mθθ ∈ Z

, then these sets are given by Xs = MxΘs and Us = MuΘs, respectively, whereMx = [In 0n,m]Mθ and Mu = [0m,n Im]Mθ .

Remark 4.11 (Tracking of target sets):The above parametrization of all admissible set points is unambiguous in the sense that every θ is associatedwith exactly one particular output value. Sometimes it may however be desirable to track a target set, i.e. torequire the system outputs to lie in a specific set in the output space, whereas the exact values of the outputsare not important. A straightforward extension of Tube-Based Robust MPC to this “set-tracking problem” canbe developed by using the ideas from Ferramosca et al. (2009b, 2010).

The State-Feedback Tube-Based Robust Model Predictive Controller for Tracking

From section 2.1.2, it is known that one common “ingredient” to ensure stability of nominal MPC is touse a positively invariant terminal constraint set for the finite horizon optimal control problem solvedon-line (Mayne et al. (2000)). The local stabilizing controller is hereby usually a linear feedback controllerof the form u = K x , where K is, for example, the LQR gain. In the framework of Tube-Based Robust


MPC, the terminal set is simply computed as a positively invariant set for the nominal system subject toappropriately tightened constraints. However, if one wants to formulate a general tracking problem forarbitrary set points, things get a little more involved.

The objective of the tracking controller discussed in this section is to keep state of and control inputto the uncertain system (4.63) within the neighborhood of some admissible steady state (us, xs) that iscompatible with the output set point requirement ys = C xs+Dus. After having reached this neighborhoodof the target steady state, the control input to the system will essentially be of the form u= us+K(x− xs),as opposed to u = K x in the regulation problem. Because the system is subject to state and controlconstraints, one cannot simply shift the invariant set computed for the regulation problem to other,non-zero steady states (see section 2.3.1 for details on this issue). A different terminal constraint set isneeded, and the invariant set for tracking defined below provides an appropriate extension.

Definition 4.7 (Invariant set for tracking, Alvarado et al. (2007b)):Let x e denote the extended state (x ,θ) ∈ Rn+nθ . Furthermore, let KΩ ∈ Rm×n be a control gain such thatA+ BKΩ is Hurwitz and define Kθ := [−KΩ Im]Mθ . Then, a set Ωe

t ⊂ Rn×nθ is an admissible invariant set

for tracking if for all (x ,θ) ∈ Ωet , then x ∈ X, KΩx + Kθθ ∈ U and

(A+ BKΩ)x + BKθθ , θ

∈ Ωet .

From Definition 4.7 it follows that for any initial state (x(0),θ) ∈ Ωet , the trajectory of the system

x+ = Ax + Bu controlled by u = KΩ(x − xs) + us, where (xs, us) = Mθθ , will satisfy x(i) ∈ Projx(Ωet)

for all i ≥ 0. The important benefit of using an invariant set for tracking Ωet as the terminal constraint

set in the optimization problem is that in this case, since Ωet is computed in the augmented state space

x e = (x ,θ) ∈ Rn+nθ , the terminal set need not be recomputed in case of a set point change. Notethat Definition 4.7 is based on the tightened constraint sets X and U. This will allow that the task ofconstrained optimal control and the task of robust constraint satisfaction can be treated independently inthe overall Tube-Based Robust Model Predictive Controller for Tracking.

In order to obtain an increased region of attraction, the Tube-Based Robust Model Predictive Controllerfor Tracking features an artificial steady state zs = ( xs, us) that is incorporated as a decision variable intothe optimization problem. In Lemma 4.1 it is stated that any admissible artificial steady state can beparametrized as zs = Mθ θ , where θ ∈Θs. Hence, for simplicity, the parameter vector θ will be considereda decision variable in the cost function

VN (x ,θ ; x0, u, θ) :=N−1∑

i=0

l

x i, xs, ui, us

+ Vf ( xN , xs) + Vo(θ ,θ). (4.75)

In addition to the modified stage cost

l

x i, xs, ui, us

= || x i − xs||2Q + ||ui − us||

2R (4.76)

and terminal cost

Vf

xN , xs

= || xN − xs||2P (4.77)

a so-called steady state offset cost

Vo(θ ; θ) = ||θ − θ ||2T (4.78)

that penalizes the deviation between the artificial steady state ( xs, us) = Mθ θ and the desired steady state(xs, us) = Mθθ is incorporated into the cost function VN (·). The matrix T 0 in (4.78) is called the steadystate offset weighting matrix.


Following the Tube-Based Robust MPC ideas from the previous sections, state x and control u of thenominal system (4.68) are subject to the tightened constraints (4.70). Furthermore, the initial state x0and the predicted terminal state xN of the nominal system and the steady state parameter θ have tosatisfy, respectively:

x0 ∈ x ⊕ (−E ), ( xN , θ) ∈ Ωet , (4.79)

where Ωet is an invariant set for tracking as in Definition 4.7.

For a given pair ( x0, θ), the set of admissible nominal control sequences u is:

UN ( x0, θ) =¦

u | ui ∈ U, Φ(i; x0, u) ∈ X, for i=0,1, . . . , N−1, (Φ(N ; x0, u), θ) ∈ Ωet

©

. (4.80)

The optimal control problem PN (x ,θ) that is solved on-line at each time step (Alvarado (2007); Alvaradoet al. (2007b); Limon et al. (2008b)) is

V ∗N (x ,θ) = minx0,u,θ

¦

VN (x ,θ ; x0, u, θ) | u ∈ UN ( x0, θ), x0 ∈ x ⊕ (−E )©

(4.81)

( x∗0(x ,θ), u∗(x ,θ), θ ∗(x ,θ)) = arg minx0,u,θ

¦

VN (x ,θ ; x0, u, θ) | u ∈ UN ( x0, θ), x0 ∈ x ⊕ (−E )©

, (4.82)

which is again easily identified as a Quadratic Programming problem.

Remark 4.12 (Feasibility of arbitrary target set points, Alvarado et al. (2007b)):It is very important to note that the feasible region of PN (x ,θ) only depends on the current state x , and noton the parameter θ by which the target steady state (xs, us) is parametrized. In other words, the controllercan ensure feasibility under any change of the desired set point ys, even in the case this set point is notadmissible, i.e. not contained in Ys. In either case, the system is steered to the admissible artificial steadystate ( x∗s , u∗s ) obtained from (4.81). By heavily penalizing the deviation θ − θ in the offset cost Vo(·) it isachieved that, at least for all ys ∈ Ys, the tracking error of the nominal system output can be made arbitrarilysmall10. Section 4.5.4 gives additional details and guidelines on how to choose the offset weighting matrix T .

Remark 4.13 (Similar ideas in the literature):The idea of computing the invariant set in an augmented state space and appropriately modifying the targetreference value has also appeared in Chisci and Zappa (2003). In this contribution, the controller is howeversplit into two modes (“dual-mode predictive tracking”) and not designed all of a piece as is the case inTube-Based Robust MPC for Tracking.

The set of admissible nominal initial states for problem PN (x ,θ) is

XN =¦

x | ∃ θ such that UN ( x , θ) 6= ;©

, (4.83)

and the set of admissible actual initial states, i.e. the domain of the value function VN (·), is

XN =¦

x | ∃ ( x0, θ) such that x0 ∈ x ⊕ (−E ), UN ( x0, θ) 6= ;©

. (4.84)

From the solution of PN (x ,θ) the sequence of predicted optimal control inputs u∗(x ,θ) is obtained. Ateach sampling instant, the first element u∗0(x ,θ) of this sequence is used as the feedforward part in theimplicit state-feedback Tube-Based Robust Model Predictive Control law for Tracking κN (·):

κN (x ,θ) := u∗0(x ,θ) + K(x − x∗0(x ,θ)). (4.85)10 this however does not solve the offset problems when asymptotically constant disturbances are present


Number of Variables and Constraints in the Optimization ProblemBecause of the additional optimization variable θ that parametrizes the artificial steady state, thecomplexity of PN (x ,θ) is slightly higher than the one of the optimization problems PN (x) and PN ( x)associated with the Tube-Based regulators from sections 4.4.1 and 4.4.2. The number of scalar variables inthe optimization problem PN (x ,θ) is Nv ar = Nm+ (N+1)n+ nθ and the number of scalar constraints isNcon = N (n+NX+NU) + NE + NΩe

t, where NΩe

tis the number of linear inequalities defining the polytopic

invariant set for tracking Ωet .

Properties of the Controller

In order to be able to establish robust stability and robust constraint satisfaction of the closed-loop system,a number of conditions on the various design parameters have to be assumed, some of which have alreadybeen stated (either explicitly or implicitly). The following assumption aggregates all those conditions:

Assumption 4.9 (Limon et al. (2008b); Alvarado (2007)):The matrices Q, R, T , P, K , KΩ and the sets Ωe

t and E fulfill:

1. Q 0 and R 0.

2. There exists a constant σ > 0 such that σT M Tx Mx , where Mx = [In 0n,m]Mθ .

3. The feedback gain matrix K is such that AK = A+ BK is Hurwitz.

4. KΩ and P are such that A+BKΩ is Hurwitz, P 0, and P−(A+BKΩ)T P(A+BKΩ) =Q+ K T

ΩRKΩ.

5. The set E ⊂ X is an admissible robust positively invariant set for the system x+= AK x +w, i.e. E issuch that AKE ⊕W ⊆ E and KE ⊂ U.

6. The set Ωet is an invariant set for tracking (as in Definition 4.7) for the nominal system (4.68) subject

to the tightened constraints (4.70) when using KΩ as the terminal controller.

Remark 4.14 (Comments on Assumption 4.9):Items 1 and 3 are usual assumptions. Item 2 is a rather technical assumption which is necessary to proveconvergence of the closed-loop system to the desired steady state (see Alvarado (2007) for details). Item 4requires the terminal controller KΩ to be stabilizing (note that, in general, KΩ 6= K) and P to be the positivedefinite matrix characterizing the infinite horizon cost of the nominal system (4.68) controlled by u= KΩx .Item 5 requires feasibility of the RPI set used to bound the error e between actual and nominal system state xand x . Finally, item 6 requires the terminal set to be an invariant set for tracking, which is a necessaryextension of merely requiring a standard invariant terminal set as in the regulation problem.

In contrast to the Tube-Based regulators from the previous sections, it is not possible to show robustexponential stability of an invariant set for the closed-loop system controlled by a Tube-Based RobustModel Predictive Controller for Tracking. This is because the modified cost function (4.75) with theadditional steady state offset cost (4.78) can not be guaranteed to be strictly decreasing along all possibletrajectories of the system. Nevertheless, if Assumption 4.9 is satisfied, it is possible to show robustasymptotic stability of an invariant set centered at a desired robustly reachable steady state xs:

Theorem 4.4 (Robust asymptotic stability, Alvarado (2007); Alvarado et al. (2007b)):Consider the uncertain system (4.63) subject to the constraints (4.64), and suppose that Assumption 4.9holds. Furthermore, let κN (x ,θ) be the implicit state-feedback Model Predictive Control law for Trackingresulting from the solution of the optimization problem PN (x ,θ) at each time step. Then, for anyinitial state x ∈ XN , and for any desired admissible steady state xs ∈ Xs, the state of the closed-loop


system x+ = Ax + BκN (x ,θ) + w converges asymptotically to the set xs ⊕ E while satisfying theconstraints (4.64) for all admissible realizations of the disturbance sequence w.

Corollary 4.1 (Robust output tracking):From Theorem 4.4 it directly follows that for any initial state x ∈ XN , and any admissible target setpoint ys ∈ Ys, the system output y converges asymptotically to the set ys⊕CE ⊕DKE for all admissiblerealizations of the disturbance sequence w.

Case Study: Tracking Control of the Double Integrator

Controller SynthesisThe case study presented in this section is a slightly modified version of the double integrator trackingexample from Limon et al. (2008b). In order to demonstrate the properties of the tracking controllerwhen additional degrees of freedom are available, the control input in this example is two-dimensional.The system dynamics are

x+ =

1 10 1

x +

0 0.51 0.5

u+w

y =

1 0

x ,(4.86)

with state and control constraints X =

x | ||x ||∞ ≤ 5

and U =

u | ||u||∞ ≤ 0.3

, respectively, anddisturbance bound W =

w | ||w||∞ ≤ 0.1

. The weighting matrices in the modified stage cost (4.76)are chosen as Q = I2 and R = 10 I2, the prediction horizon as N =10, and the terminal cost matrix asthe matrix P∞ characterizing the infinite horizon cost for the system controlled by the unconstrainedLQR controller. In order to minimize offset in case the desired steady state is admissible (offset isclearly unavoidable if the desired steady state is not admissible), the offset weighting matrix is chosenas T=1000P∞. The optimized disturbance rejection controller K can be found in Table 4.1 (together withall of the above parameters).

In order to illustrate the ability of the Tube-Based Robust Model Predictive Controller for Tracking todeal with on-line set point changes, the tracking task in this example is twofold: Starting from theinitial state x(0)=[−3 1.5]T, the first target output to be tracked is given by ys,A=−4. After ys,A hasbeen successfully tracked, the target output switches to ys,B = 4. Note that the rank of the matrix Ein (4.66) is rank(E) = 3= n+ p < n+m= 4. From Lemma 2.2 it therefore follows that for any constraintadmissible target output there exists an constraint admissible steady state. However, as pointed outin Remark (4.10), this steady state is not unique (this is because of the additional degree of freedomdue to the two-dimensional control input). The parameters θA and θB corresponding to the two targetoutputs are chosen such that xs,A = [−4 0]T ∈ Xs and xs,B = [4 − 0.5]T /∈ Xs. A simulation horizonof Nsim=18 for each of the transition intervals proved to be adequate. The state disturbance w was againa time-varying random variable uniformly distributed on W .

Simulation ResultsBesides the “tube” of trajectories T (t) = x∗0(x(t),θ(t)) ⊕ E (shown with yellow cross-section), Fig-ure 4.13 contains a number of other important sets. The one shown in blue is the projection of theinvariant set for tracking Ωe

t on the x-space, representing those states for which there exists a parametervalue θ such that the associated steady state is contained in Ωe

t . Hence, this set can be regarded asthe “terminal set in the x-space”. The set of admissible steady states Xs is shown in green, and theboundaries of the regions of attraction XN and XN are represented by solid and dashed lines, respectively.In x1-direction, the regions of attraction are restricted by the state constraints; their extension in x2-direction can be increased by choosing higher values for the prediction horizon N (though not indefinitely).


Figure 4.13.: Trajectories for state-feedback Tube-Based Robust MPC for Tracking example

Figure 4.14.: Control inputs for state-feedback Tube-Based Robust MPC for Tracking example


The simulation results from Figure 4.13 show that after about 15 time steps, the system starting fromthe initial state x(0) = [−3 1.5]T reaches the neighborhood of the first (admissible) target steady statexs,A = [−4 0]T. After that, the target steady state is changed to xs,B = [4 − 0.5]T. It is important to notethat xs,B is not contained in Xs, the set of feasible steady states. However, instead of becoming infeasible,the Tube-Based Robust Model Predictive Controller for Tracking steers the system to the neighborhood ofthe steady state x∗s,B = [4 − 0.134]T, which is the set point “closest” to xs,B, i.e. the artificial set pointobtained from the optimization problem (4.82). Strictly speaking, the first target steady state that is usedinternally by the controller is also not xs,A but rather x∗s,A. However, due to the large offset penalty (theoffset weighting matrix was chosen as T=1000P∞), the values of xs,A and x∗s,A are indistinguishable.

Figure 4.14 shows the “trajectory” of the actual control inputs applied to the system in the u-space. Theset of admissible steady state control inputs Us, which is one-dimensional in this case, is depicted as thegreen line. The black dashed lines represent the boundaries of the sets u∗s,A⊕KE and u∗s,B⊕KE , whichare the sets that are guaranteed to contain the control input generated for the uncertain system at steadystate for any admissible disturbance sequence W .

From the simulation results visualized in Figure 4.13 and Figure 4.14 it can be verified that in theneighborhood of the tracked steady states as well as throughout both transients, state and control con-straints are indeed both satisfied. Non-conservativeness of the controller becomes evident especially wheninspecting the control inputs in Figure 4.14, where frequently u ∈ ∂U, i.e the control input lies on theboundary of the set of feasible control inputs. The reason why the bounding set KE around the optimalnominal control inputs u∗s,A and u∗s,B at the respective steady states is significantly larger than the actualdeviations u∗− u∗s,A and u∗− u∗s,B, respectively, is that the disturbance w that acts on the system is a zeromean random variable. The deviations would be far larger if the disturbance w had a malicious intentand was trying to actively destabilize the system. However, also in this case the Tube-Based Robust ModelPredictive Controller for Tracking would be able to ensure robust asymptotic stability of the sets x∗s,A⊕Eand x∗s,B ⊕ E , respectively, and therefore yield control inputs to the system guaranteed to be containedwithin the sets u∗s,A⊕ KE and u∗s,A⊕ KE , respectively.

The on-line optimization problem PN (x ,θ) for the example is characterized by Nv ar=44 scalar variablesand Ncon=185 scalar constraints, provided the equality constraint formulation is used. For the giveninitial state, the solver time varied between 4ms and 15ms, using the standard benchmark setup fromsection 4.2.4. A benchmark comparing the different Tube-Based Robust Model Predictive Controllers andproviding additional details on the controller of this case study will be presented in section 4.6.

The (not so) Explicit SolutionThe optimization problem PN (x ,θ) is, with its 44 variables and 185 constraints, only slightly morecomplex than the ones of the two examples from sections 4.2.4 and 4.3.3. Nevertheless, the exactexplicit solution of PN (x ,θ) is extremely involved for this case study, making an explicit computationunreasonable. Whereas the piece-wise affine optimizer functions in the previous two examples weredefined over 366 and 142 polyhedral regions, respectively, the computation of the tracking example wasmanually aborted after the solver had identified more than 40.000 regions. The reasons for this extremelyhigh number of regions lie in the internal structure of the optimization problem and are hard to pinpointwithout profound knowledge of the algorithms used in the multiparametric programming solver.


4.4.2 Tube-Based Robust MPC for Tracking, Output-Feedback Case

The state-feedback Tube-Based Robust Model Predictive Controller for Tracking discussed in the previoussection assumed full state information to be available. Using the ideas from section 4.3, it is straightforwardto develop a Tube-Based Robust Model Predictive Controller for Tracking also for the output-feedbackcase, where, in addition to the state disturbance w ∈W , the system

x+ = Ax + Bu+w

y = C x + Du+ v(4.87)

is subject to an additional unknown, but bounded output disturbance v ∈ V . The disturbance bounds Wand V satisfy the usual assumptions, and the system is again subject to the usual constraints x ∈ Xand u ∈ U on state and control input. The controller structure of the output-feedback Tube-Based RobustModel Predictive Controller for Tracking is essentially the same as the one of the state-feedback Tube-Based Robust Model Predictive Controller for Tracking from section 4.4.1. The most important differenceis that the controlled system in the output-feedback case is not the actual system but the observer system(see section 4.3.2). The employed control law is therefore

u= u+ K( x − x), (4.88)

where x is the current state estimate, and x and u are the state of and control input to the nominalsystem (4.68), respectively. Because of the additional uncertainty due to the state estimation error ee, thetightened constraint sets

X := XE = X (Ec ⊕Ee), U := U KEc (4.89)

for the nominal system are smaller than in the state-feedback case. Consequently, this is true also for Xs,Us, and Ys, the sets of admissible steady states, steady state inputs, and target outputs, respectively.

Consider again the modified cost function (4.75) and the definition ofUN ( x0, θ) in (4.80), which is the setof admissible nominal control sequences u for a given pair ( x0, θ). The optimal control problem PN ( x ,θ)of the output-feedback Tube-Based Robust Model Predictive Controller for Tracking can then be stated asfollows (Alvarado (2007); Alvarado et al. (2007a)):

V ∗N ( x ,θ) = minx0,u,θ

¦

VN ( x ,θ ; x0, u, θ) | u ∈ UN ( x0, θ), x0∈ x⊕(−Ec)©

(4.90)

( x∗0( x ,θ), u∗( x ,θ), θ ∗( x ,θ)) = arg minx0,u,θ

¦

VN ( x ,θ ; x0, u, θ) | u ∈ UN ( x0, θ), x0∈ x⊕(−Ec)©

. (4.91)

The domain of the value function V ∗N (·) of the controller is the set of admissible initial state estimates

XN =¦

x | ∃ ( x0, θ) such that x0 ∈ x ⊕ (−Ec), UN ( x0, θ) 6= ;©

. (4.92)

Denote by XN and XN the set of admissible nominal states and the set of admissible actual states,respectively. The following relationship between the sets XN , XN , and XN holds:

XN ⊂ XN = XN ⊕Ec ⊂XN = XN ⊕Ee = XN ⊕E . (4.93)

Using the first element of the the optimizer u∗( x ,θ) obtained from (4.91), the implicit output-feedbackModel Predictive Control law for Tracking κN (·) is

κN ( x ,θ) := u∗0( x ,θ) + K( x − x∗0( x ,θ)). (4.94)


Number of Variables and Constraints in the Optimization ProblemThe number of scalar variables in the optimization problem PN ( x ,θ) is with Nv ar = Nm+ (N+1)n+ nθthe same as in PN (x ,θ). The number of scalar constraints is Ncon = N (n+NX+NU)+NEc

+NΩet, where NΩe

tis the number of linear inequalities defining the polytopic invariant set for tracking Ωe

t . The only causesfor a difference in the number of constraints is the possibly different complexity of the sets Ec, X and U.

Properties of the Controller

To account for the differences between the output-feedback and the state-feedback version of Tube-BasedRobust MPC for Tracking, Assumption 4.9 needs to be modified appropriately:

Assumption 4.10 (Alvarado et al. (2007a)):Assume that items 1 – 4 and item 6 in Assumption 4.9 are satisfied. Furthermore, assume that the sets Eeand Ec are robust positively invariant for the perturbed systems e+e = ALee + δe and e+c = AK ec + δc,respectively, and that E = Ee ⊕Ec ⊂ X and KEc ⊂ U.

With the above, one can now state the main theorem for output-feedback Tube-Based Robust ModelPredictive Control for tracking piece-wise constant references:

Theorem 4.5 (Robust asymptotic stability, Alvarado et al. (2007a)):Suppose that Assumption 4.10 holds and that ee(0) = x − x ⊂ Ee. Let κN ( x ,θ) be the implicit output-feedback Model Predictive Control law for Tracking based on the on-line solution of problem PN ( x ,θ)at each time step. Then, for any initial state estimate x ∈ XN , and for any desired admissible steadystate xs ∈ Xs, the state of the closed-loop system x+ = Ax + BκN ( x ,θ) +w converges asymptotically tothe set xs ⊕ E while satisfying the constraints x ∈ X and u ∈ U for all admissible realizations of thedisturbance sequences w and v.

An equivalent of Corollary 4.1 from section 4.4.1 is the following:

Corollary 4.2 (Robust output tracking):For any initial state estimate x ∈ XN that satisfies ee(0) = x − x ⊂ Ee and any admissible target setpoint ys ∈ Ys, the system output y converges asymptotically to the set ys⊕CE ⊕DKEc for all admissiblerealizations of the disturbance sequences w and v.

Case Study: Output-Feedback Tracking Control of the Double Integrator

Controller SynthesisThis case study considers again the output-feedback double integrator example from section 4.3.3, whosesystem dynamics are given by

x+ =

1 10 1

x +

11

u+w

y =

1 1

x + v .(4.95)

Constraints, weights, bounds on the disturbances and controller and observer feedback gain matrices Kand L are the same as in the original example, all of which are summarized in Table 4.1 at the end ofthis chapter. The prediction horizon is also left unchanged at N=13. In this example, the control inputis scalar (m = 1), and the rank of the matrix E in (4.66) is rank(E) = 3 = n+ p = n+m. Hence, byvirtue of Lemma 2.2 and Remark 4.10, there exists a unique steady state for every admissible targetset point. Consequently, the parameter θ is a scalar and so is the offset weighting matrix T , which in


this example is chosen as T = 1000 in order to minimize offset. For simulation, the initial state of thesystem is chosen randomly in the set x(0) ∈ x(0)⊕ Ee, where the initial state estimate is assumed tobe x(0) = [−3 − 8]T. The control objective is again to sequentially track two different output targetvalues, which in this case are given by ys,A = −5 ∈ Ys and ys,B = 4 /∈ Ys. For both parts of the problem,a simulation horizon of Nsim=15 is used. The state and output disturbances w and v are time-varyingrandom variables uniformly distributed on W and V , respectively.

Simulation ResultsFigure 4.15 and Figure 4.16 show the tube of trajectories in the x-space and the scalar control inputsover time. Because any steady state of (4.95) must satisfy x2=0, the set of admissible steady states Xs inFigure 4.15 degenerates to a line (shown in green). Solid and dashed black lines furthermore correspondto the boundaries of the regions of attraction XN and XN . The region of attraction XN of state estimatesis the important one this case since the output-feedback Tube-Based Robust Model Predictive Controllerfor Tracking controls the observer system rather than the actual system. Note that the boundary of XNdoes not touch the boundary of the set of feasible states, which previously has been the case for theregion of attraction XN in the state-feedback tracking case study. This is because of the incomplete stateinformation it must hold that XN ⊕Ee ⊆ X, where Ee is fully dimensional.

Figure 4.15.: Trajectories for output-feedback Tube-Based Robust MPC for Tracking example

After successfully tracking the first (admissible) steady state x∗s,A=[−5 0]T that corresponds to the targetset point ys,A=−5, the target set point is changed to ys,B=4. Since the constraints on the system do notallow the corresponding steady state xs=[4 0]T, the controller steers the system to the “closest” admissi-ble steady state x∗s,B=[2.13 0]T obtained from the optimization (4.91). The actual control input u, shownin Figure 4.16, is zero at both steady states except for the small control action generated by the feedbackcomponent K( x− x) that is necessary to compensate the effect of the disturbances. The region between the


0 5 10 15 20 25

−3

−2

−1

0

1

2

3

Figure 4.16.: Control inputs for output-feedback Tube-Based Robust MPC for Tracking example

two horizontal dashed lines in Figure 4.16 corresponds is the set 0⊕ KEc = u∗s,A⊕ KEc = u∗s,B⊕ KEc,in which the control inputs at steady state are guaranteed to be contained for all possible realizations ofthe disturbance sequences w and v. This set is obviously very conservative for the random disturbancesequences that used in this simulation. Once again. it is easy to see from Figure 4.15 and Figure 4.16 thatalso the output-feedback Tube-Based Robust Model Predictive Controllers for Tracking ensures that bothstate and control constraints are satisfied at all times, and that the resulting state trajectory and controlinputs are non-conservative.

The optimization problem PN ( x) for the given example involves Nv ar=42 scalar variables and Ncon=121scalar constraints (using the equality constraint formulation), the solver time has been found to be in therange between 8ms and 25ms. For further details on the computational complexity of this controller andalso the other Tube-Based Robust Model Predictive Controllers see the benchmark in section 4.6.

The Explicit SolutionIn contrast to the state-feedback Tube-Based Robust Model Predictive Controller from the case studyin section 4.3.3, the explicit solution of the controller for the output-feedback tracking example is ofmoderate complexity. For an exploration region Xx pl =

x | −12≤ x1≤3, −10≤ x2≤ 3

, the optimizerfunction for the problem PN ( x) is defined over a polyhedral partition consisting of Nreg=613 regions.Because of the additional optimization parameter θ , value function as well as optimizer function aredefined in the augmented (x ,θ)-space, which makes them hard to visualize. The on-line evaluation timesof the explicit control law in this example varied between 16ms and 18ms.

4.4.3 Offset-Free Tube-Based Robust MPC for Tracking

In section 3.6.3 it was shown how offset-free MPC for asymptotically constant disturbances can be achievedby augmenting the system state with virtual, integrating disturbances. This method can of course alsobe employed to achieve offset-free tracking when using Tube-Based Robust Model Predictive Controllers.The problem with this approach is however that it necessitates the solution of an optimal control problemfor an augmented composite system of higher dimension, which increases the computational complexity


of the on-line optimization problem. For the purpose of controlling fast systems, it is therefore desirableto find alternate ways of ensuring offset-free tracking for Tube-Based Robust Model Predictive Controllers.

Cancellation of the Tracking ErrorFor the (common) case that there is no feedthrough path, i.e. D = 0, Alvarado (2007) and Alvaradoet al. (2007a) propose a simple method that is able to cancel the output tracking error by appropriatelyadjusting the reference input. The main benefit of this approach is that it does not increase the complexityof the on-line optimization problem, as is the case for the method discussed in section 3.6.3. Because ofits simplicity and effectiveness, this idea will be described in the the following.

Let w∞ and v∞ be the values of the asymptotically constant disturbances w ∈W and v ∈ V . From (4.46)and (4.47) it is easy to see that the asymptotic value of the state estimation error ee then becomes

ee(∞) = (I− AL)−1(w∞− Lv∞), (4.96)

where AL := A− LC . From (4.48) and (4.49) it furthermore follows that the error ec between observerstate and state of the nominal system asymptotically takes on the value

ec(∞) = (I− AK)−1 L(Cee(∞) + v∞), (4.97)

while the output y(∞) of the system asysmptotically becomes

y(∞) = C x(∞) + v∞ = C( x(∞) + ee(∞) + ec(∞)) + v∞. (4.98)

Let ys denote the output set point to be tracked. If ys is robustly reachable, it follows from Proposition 4.2that ys = C x(∞). Using (4.97), the output (4.98) can be rewritten as

y(∞) = ys + (I+ C(I− AK)−1 L)(Cee(∞) + v∞), (4.99)

from which is clear that it exhibits an offset∆y := y− ys = F(Cee(∞) + v∞), where F := I+C(I − AK)−1 L.

Noting that Cee+v = C(x− x)+v = y−C x , and that both y (measured) and x (internal) are available, itis easy to see that a modified reference signal

ys(k) = ys − F(y(k)− C x(k)) (4.100)

leads to asymptotic cancellation of the output offset for asymptotically constant disturbances w and v .This is because limk→∞ ys(k) = ys − F(y(∞)− C x(∞)) = ys − F(Cee(∞)+v∞) = ys−∆y .

Offset-Free Tracking ExampleThe above ideas have been incorporated into the developed Tube-Based Robust MPC software framework.In the following simulation, the behavior of the resulting offset-free controller using the modified referencesignal (4.100) is compared to that of the standard output-feedback Tube-Based Robust Model PredictiveController for Tracking from section 4.4.2. The example system used is thereby the same as the one inthe output-feedback tracking case study from section 4.4.2. The simplified tracking task consists onlyof tacking the target set point ys = −5. For both controllers, the simulation horizon is Nsim= 15 andthe initial state estimate used in the simulation is x(0) = [−3 − 8]T. In order to produce visible offset,state and output disturbance is chosen as w = [0.1 0.1]T= const. and v =−0.05= const., respectively.Figure 4.17 shows the system output y for both controller types. For the first 5 time steps, the outputvalues are more or less identical. For larger t, however, one can see that the standard output-feedbackTube-Based Robust Model Predictive Controller for Tracking indeed exhibits offset. The controller thatuses the modified reference output ys instead achieves offset-free tracking.


Figure 4.17.: Comparison of standard and offset-free tracking controller

4.5 Design Guidelines for Tube-Based Robust MPC

Throughout this chapter, a number of different design parameters for the presented Tube-Based RobustModel Predictive Controllers have been introduced. These parameters include the weighting matrices Q,R and P in the cost function, the disturbance rejection controller K , the robust positively invariant set Eand the terminal set X f as well as the prediction horizon N . In the output-feedback case, one also has tochoose the observer gain matrix L. Moreover, when synthesizing Tube-Based Robust Model PredictiveControllers for Tracking, additional choices are those of the offset weighting matrix T and the invariantset for tracking Ωe

t . The purpose of the following section is to discuss the effect of these parameters onthe properties of the controller and to provide some guidelines on how to choose them appropriately.To this end, section 4.5.1 addresses the question of how to choose the terminal weighting matrix P andpresents an algorithm for the related computation of the terminal constraint set X f . Section 4.5.2 thendevelops a constructive approach for computing an “optimal’ disturbance rejection controller gain K . Theapproximate computation of a minimal robust positively invariant set is the topic of section 4.5.3. Finally,section 4.5.4 deals with appropriately choosing the offset weighting matrix T . Before that, the followingparagraph briefly addresses the less substantial questions pertaining to the weights in the cost functionand the prediction horizon of the optimization problem.

Weights in the Cost Function and Prediction HorizonThe most immediate question in the controller design concerns the choice of the weighting matrices in thecost function. As with all optimal control approaches, it is often times not so much a question of how tosolve the optimal control problem, but how to pose it in the first place. There exist no general rules on howthe weighting matrices Q and R should be chosen, at best some loose guidelines can be given. In order toavoid a philosophical debate about optimal control at this point, it will be assumed in the following thatboth Q and R are given. The design parameter easiest to understand is probably the prediction horizon N ,which is typically chosen as large as possible, It is well known that a larger prediction horizon results in alarger region of attraction and an improved closed loop performance. However, since the complexity ofthe on-line optimization problem grows with N (fortunately only linearly in Tube-Based Robust MPC),this immediately increases the required computation time. Hence, depending on the speed requirementson the controller, a tradeoff is necessary.

4.5. Design Guidelines for Tube-Based Robust MPC 87

4.5.1 The Terminal Weighting Matrix P and the Terminal Set X f

In light of the arguments of section 2.1.3 it is desirable to choose the terminal cost function Vf (·) in thefinite horizon optimal control problem of a Model Predictive Controller as the true infinite horizon costfunction. This is achieved if the following conditions on P and X f are satisfied (compare section 2.1.3):

• The terminal weighting matrix P is chosen as P∞, i.e. as the solution of the Algebraic RiccatiEquation (2.17) of the unconstrained LQR problem.

• The terminal constraint set X f is a constraint admissible positively invariant set under the controllaw κ f ( x) = K∞ x , where K∞ is the unconstrained LQR controller.

The computation of P∞ (and hence of K∞) is a standard problem in unconstrained optimal control, forwhich numerous efficient algorithms exist. In order to obtain a region of attraction as large as possible,the terminal set X f should be chosen as the maximal positively invariant set (see Definition 2.3). Thissection will therefore review how the MPI set (or a suitable approximation of it) may be computed.

Computation of the Maximal Positively Invariant SetThe system considered for this purpose is of the simple form

x+ = A∞ x , (4.101)

where A∞ := A+BK∞ is the system matrix of the nominal system controlled by the optimal unconstrainedinfinite horizon controller K∞. System (4.101) is subject to the constraints

x ∈ X, K∞ x ∈ U (4.102)

on nominal state and control input, respectively. As stated above, the objective of this section is tocompute the maximal positively invariant (MPI) set Ω∞ defined by

Ω∞(A∞, K∞, X, U) :=¦

x ∈ Rn | Ak∞x ∈ X, K∞Ak

∞x ∈ U, ∀ k ≥ 0©

. (4.103)

For a set Ω ∈ Rn, define the predecessor set Pre(Ω) as

Pre(Ω) :=

x ∈ X | A∞x ∈ Ω, K∞A∞x ∈ U

, (4.104)

i.e. as the set of states that, under the system dynamics (4.101), evolve into the target set Ω in onestep while satisfying the constraints (4.102). If the target set Ω and the constraint sets X and U arepolyhedral, i.e. when they can be expressed as Ω =

x ∈ Rn | HΩx ≤ kΩ

, X =

x ∈ Rn | Hx x ≤ kx

and U=

u ∈ Rm | Huu≤ ku

, then Pre(Ω) is simply given by the polyhedron

Pre(Ω) =

x ∈ Rn

HΩA∞Hx

HuK∞

x ≤

kΩkxku

. (4.105)

Remark 4.15 (Removing redundant inequalities, Borrelli et al. (2010)):The representation (4.105) of the predecessor set Pre(Ω) may not be minimal, i.e. it may contain redundantinequalities that can be removed without changing the shape of Pre(Ω). Removing these redundant inequalitiesis advisable in any algorithmic implementation in order to reduce the complexity of the solution. Computingthe minimal representation of a polyhedron P generally requires to solve one Linear Program for eachhalf-space defining the non-minimal representation of P .

The Pre-operator (4.104) can be used to formulate a geometric condition for the invariance of a set Ω:

Theorem 4.6 (Geometric condition for invariance, Borrelli et al. (2010)):A set Ω is positively invariant for the autonomous system (4.101) subject to the constraints (4.102) ifand only if Ω⊆ Pre(Ω) or, equivalently, if and only if Pre(Ω)∩Ω = Ω.


Theorem 4.6 motivates the following conceptual Algorithm for the computation of the MPI set Ω∞ (Gilbertand Tan (1991); Kerrigan (2000); Borrelli et al. (2010)):

Algorithm 1 Computation of the maximal positively invariant set Ω∞Input: X, U, A∞, K∞Output: Ω∞

k← 0Ω0← XΩ1← Pre(Ω0)∩Ω0while Ωk+1 6= Ωk do

k← k+ 1Ωk+1← Pre(Ωk)∩Ωk

end whilereturn Ω∞ = Ωk = Ωk+1

It is easy to see that the sequence

Ω0,Ω1, . . .

returned by Algorithm 1 is a sequence of constraintadmissible sets of decreasing size. Hence, together with the condition from Theorem 4.6, it follows thatthe algorithm returns the maximal positively invariant set if and only if Ωk = Ωk+1 for some k. In general,however, Algorithm 1 is not guaranteed to terminate in finite time. In this case the MPI set Ω∞ is said tobe not finitely determined. If the algorithm does terminate, then the returned k is called the determinednessindex and denoted by k∗. Gilbert and Tan (1991) state necessary conditions for a finite termination ofAlgorithm 1. From these conditions, it can be inferred that, at least in the context of Tube-Based RobustMPC for the regulation problem, the MPI set Ω∞ is guaranteed to be finitely determined. This is because inthis case A∞ is strictly stable, U is bounded, and both X and U contain the origin in their respective interior.

Assume now that the parameters X, U, A∞ and K∞ are given, and that X and U are polyhedral andpolytopic, respectively. Then, Algorithm 1 can be carried out very efficiently. Since the starting set Ω0 = Xis polyhedral, so is Pre(Ω0) and hence all subsequently computed Ωk, including, Ωk∗ = Ω∞. Thus, thecomputation of Pre(Ωk) for any k is straightforward and can be carried out by (4.105). The intersectionof two polytopes (as in Pre(Ωk)∩Ωk) is also trivially obtained by combining the sets of defining equalitiesof the two partner polytopes (Borrelli et al. (2010)). Moreover, checking whether Ωk+1=Ωk reduces to asimple comparison of matrices, given that Ωk+1 and Ωk are both available in their minimal representation(Kvasnica et al. (2004)). For this polytopic case, Algorithm 1 has been implemented in Matlab, using theMPT-toolbox (Kvasnica et al. (2004)) and the CDD-solver (Fukuda (2010)).

Obtaining a Finitely Determined Invariant Set for Tracking Ωet

For the Tube-Based Robust Model Predictive Controllers for Tracking from section 4.4, the terminal set isthe invariant set for tracking Ωe

t , which is computed in the in the augmented state space x e=(x ,θ)∈Rn+nθ .From Definition 4.7 it can easily be seen that the system matrix of the corresponding closed-loopsystem (x e)+=Ae x e is given by

Ae =

A+ BKΩ BKθ0 Inθ

. (4.106)

Defining the auxiliary set

X eγ =

x e ∈ Rn+nθ | (x , KΩx + Kθθ) ∈ Z, Mθθ ∈ γZ

, (4.107)

which for γ= 1 is the set of feasible x e (Limon et al. (2008b)), it follows from the geometric condition forinvariance in Theorem 4.6 that the maximal positively invariant set for tracking can be characterized as

Ωet,∞ =

¦

x e ∈ Rn+nθ | Ake x e ∈ X e

1 , ∀ k ≥ 0©

. (4.108)


Obviously, Ae is not strictly stable, but has nθ unitary eigenvalues. Hence, the set Ωet,∞ might not be

finitely determined (Gilbert and Tan (1991)). Fortunately, a finitely determined under-approximation

Ωet(γ) =

n

x e ∈ Rn+nθ | Ake x e ∈ X e

γ , ∀ k ≥ 0o

(4.109)

can always be obtained by choosing γ ∈]0,1[ (Gilbert and Tan (1991)). Since the set X eγ is polytopic,

the implementation of Algorithm 1 can, with some straightforward modifications to the Pre-operator,directly be used to compute Ωe

t(γ). Furthermore, since it holds that γΩet,∞⊂ Ω

et(γ)⊂ Ω

et,∞ (Limon et al.

(2008b)), the exact maximal positively invariant set for tracking Ωet,∞ can be approximated arbitrarily

close by choosing the relaxation parameter γ close to 1.

4.5.2 The Disturbance Rejection Controller K

The robustness properties of a Tube-Based Robust Model Predictive Controller are fundamentally de-termined by the disturbance rejection controller K , which directly affects size and shape of the robustpositively invariant set E that bounds the error e between actual state x and nominal state x . If thespectral radius ρ of AK =A+BK is small (i.e. when the eigenvalues of AK are all grouped close to theorigin), one can expect E to be smaller than when ρ(AK) is close to 1. Clearly, for a small RPI set E thecontroller has good disturbance rejection properties and the tightened constraint set X= XE is large.However, depending on the dynamics of the uncontrolled system, placing the eigenvalues of AK close tothe origin may require a large11 feedback gain K , which necessarily leads to a smaller tightened constraintset U. This in turn limits the available nominal control action and hence the possible performance of theTube-Based Robust Model Predictive Controller. This inevitable tradeoff between disturbance rejectionproperties and controller performance will be the topic of the following section.

The most straightforward way to obtain a disturbance rejection controller K is to simply choose it equalto K∞, the LQR gain for the unconstrained system. Although this is a rather naive approach, it in somecases yields surprisingly good results. There are however no guarantees that K∞ is a good choice, mostlybecause the tradeoff between disturbance rejection properties and controller performance mentionedabove is not taken into account explicitly. The following section therefore presents a constructive way todetermine an “optimal” disturbance rejection controller with respect to a different measure of optimalitythat does account for this tradeoff.

Remark 4.16 (Computing K as the LQR gain for modified weights):The tradeoff between disturbance rejection and control performance can also be addressed “manually” bycomputing K as the unconstrained LQR controller for modified weighting matrices Q and R. Choosing R“large” in this case will result in a less aggressive controller gain, whereas choosing Q “large” on the otherhand will result in a controller with good disturbance rejection properties.

Alvarado (2007) and Limon et al. (2008b) propose to synthesize the disturbance rejection controller K byminimizing the size of a constraint admissible ellipsoidal robust positively invariant set for the closed-loopsystem driven by u = K x . The reason for using an ellipsoidal set instead of a polyhedral one is that inthis case, it is possible to use LMI-based optimization techniques to determine an “optimal” controller.Consider to this end the ellipsoid

E(P) :=

x ∈ Rn | x T P x ≤ 1

, (4.110)

which is uniquely defined by the positive definite matrix P. Assume furthermore that the sets X and U aresymmetric polyhedra given in their normalizedH -representations X=

x∈Rn | | f Tx ,i x |≤1, i=1, . . . , I

and U=

u∈Rn | | f Tu, ju|≤1, j=1, . . . , J

, respectively12.

11 “large” in this sense means a large maximum singular value σ(K)12 here 2I and 2J are the number of facets of X and U, respectively


Remark 4.17 (Using symmetric polytopes):The above representations of X and U are those of symmetric polytopes. Because the linear control law u= K xand the ellipsoidal set E(P) are both inherently symmetric, the assumption of symmetric constraints is inthis case in fact non-restrictive. For the purpose of computing K , one can therefore always use the largestsymmetric polytope contained in X and U, respectively.

For a general ellipsoid E(P) to become constraint admissible and positively invariant, a number ofconstraints need to be imposed on both E(P) and the mapped ellipsoid E(P)+.

State ConstraintsEvidently, to satisfy the state constraints, the ellipsoid E(P) must be fully contained in X. Hence, it musthold that | f T

x ,i x | ≤ 1, i=1, . . . , I for all x ∈ E(P) or, equivalently,

maxi=1,...,I

| f Tx ,i x | ≤ 1, ∀ x ∈ E(P). (4.111)

In Boyd et al. (1994) it is shown that (4.111) can be expressed as f Tx ,i P

−1 fx ,i ≤ 1 for i=1, . . . , I . Applyinga Schur complement, this condition is equivalent to the set of LMI

1 f Tx ,i

fx ,i P

0, i = 1, . . . , I . (4.112)

Input ConstraintsInput constraints can be treated in a similar way as the state constraints in the previous paragraph. Theauthors of Limon et al. (2008b) propose that, in order to guarantee a sufficiently large tightened constraintset U, the input constraints be relaxed in the computation of K . One way to do this is by introducing anadditional relaxation parameter ρ ∈]0, 1] and requiring | f T

u, jK x | ≤ ρ, j=1, . . . , J for all x ∈ E(P). Then,for small ρ the control input is subject to tighter constraints, ensuring that the set U is large, whereasfor ρ close to 1 U is allowed to be small. Therefore the parameter ρ characterizes the tradeoff betweengood disturbance rejection properties (ρ large) and good nominal controller performance (ρ small).Analogous to the state constraints, the modified input constraints can be expressed by the set of LMIs

ρ2 f Tu, jK

K T fu, j P

0, j = 1, . . . , J . (4.113)

Invariance ConstraintThe LMI constraints (4.112) and (4.113) would ensure that the ellipsoid E(P) is constraint admissible,but not that it is robust positively invariant in the presence of an exogenous disturbance w. For theperturbed closed-loop system x+= AK x+w, where AK :=A+BK , the invariance condition can be stated as

(x+)T P(x+)≤ 1, ∀ x ∈ E(P), ∀w ∈W . (4.114)

Because of the convexity of E(P) and the linearity of the system dynamics, it is sufficient to checkcondition (4.114) only for the vertices w of the set W . Applying the S-Procedure (Boyd et al. (1994)), itcan be shown that condition (4.114) is satisfied if there exists a λ≥ 0 such that

((AK x + w)T P(AK x + w)− 1)−λ(x T P x − 1)≤ 0, ∀ w ∈ W , (4.115)

where W := vert(W ) denotes the set of vertices of W . The parameter λ plays the role of a contractionfactor, describing the decrease in size from E(P) to E(P)+. A small λ corresponds to a significant decrease


in size, whereas a λ close to 1 means that E(P) and E(P)+ are of approximately the same size. It ispossible to rewrite (4.115) as the following set of LMIs:

λ

P 0

0 −1

−

ATK PAK AT

K Pw

wT∗ PAK wT Pw− 1

0, ∀ w ∈ W . (4.116)

Note that (4.116) is not jointly convex in P and K . However, as already in the optimization problem ofKothare’s controller from section 3.4.1, a change of variables of the form

W = P−1, Y = KP−1 (4.117)

solves this problem (Boyd et al. (1994)) and results in the following set of LMI constraints which arejointly convex in the new variables W and Y :

λW 0 W T AT + Y T BT

0 1−λ wT

AW + BY w W

0, ∀ w ∈ W . (4.118)

Objective FunctionsThe underlying purpose of this section is to compute a disturbance rejection controller K such that theresulting robust positively invariant set E is as small as possible, while at the same time ensuring asufficiently large tightened constraint set U. Once the input constraints have been appropriately tightenedusing the relaxation parameter ρ in (4.113), the objective becomes to determine the controller gain Kand the corresponding positive definite matrix P in such a way that the associated invariant ellipsoid E(P)is as small as possible. It is well known that the volume of the ellipsoid E(P) is proportional to det(P−1),or, using the variable change (4.117), to det(W ). The determinant maximation problem is a convexoptimization problem that is well studied in the literature (Boyd and Vandenberghe (2004); Vandenbergheet al. (1998)) and that can be solved efficiently using Semidefinite Programming. Minimizing the volumeof E(P) using the new variables W and Y would on the other hand require the minimizaton of a concavefunction, which clearly is a non-convex problem and therefore intractable for all but the simplest systems.

A tractable alternative approach has therefore been proposed in Alvarado (2007) and Limon et al. (2008b),where the objective function is chosen as a scalar γ>0, which is minimized subject to the constraintthat E(P)⊆pγX. The optimal value of γ denotes the minimal factor by which the constraint set X canbe shrunk so that it still contains the ellipsoid E(P). Although this formulation of the optimization maynot yield the minimal volume ellipsoid, its benefit is that it also takes the shape of the constraint set Xinto account. Using (4.112), the constraint E(P)⊆pγX can readily be expressed as the set of LMIs

γ f Tx ,i

fx ,i P

0, i = 1, . . . , I , (4.119)

where, for admissibility of the solution, 0< γ≤ 1.

Another reasonable measure for the size of the ellipsoid E(P) is the trace13 of the matrix P, denoted tr(P).The smaller tr(P) (or, equivalently, the larger tr(W )), the larger the volume of the ellipsoid E(P).

13 the trace of a matrix P is given by tr(P) =∑

Pi,i , where Pi,i denotes the i th diagonal element of P. It furthermore holdsthat tr(P) =

∑

λi(P), where λi(P) denotes the i th eigenvalue of P


The Overall Optimization ProblemReformulating the LMI constraints (4.113) and (4.119) in the new variables W and Y yields the followingoptimization problem for finding an “optimal” disturbance rejection controller K = Y W−1:

minW,Y,γ

γ

s.t.

γ f Tx ,iW

T

fx ,iW W

0, i=1, . . . , I

ρ2 f Tu, jY

Y T fu, j W

0, j=1, . . . , J

(4.116).

(4.120)

Similarly, if the objective function is chosen as tr(W ), the problem becomes

maxW,Y

tr(W )

s.t.

1 f Tx ,iW

T

fx ,iW W

0, i=1, . . . , I

ρ2 f Tu, jY

Y T fu, j W

0, j=1, . . . , J

(4.116).

(4.121)

Both (4.120) and (4.121) are tractable Semidefinite Programming Problems in case the contractionparameter λ in the constraint (4.116) is fixed. However, it is not immediately clear how λ should bechosen. It would therefore be desirable to not simply fix λ to some value, but to incorporate it into theoptimization problem as an additional variable. The problem with this approach is however that theconstraint (4.116) then becomes a Bilinear Matrix Inequality (BMI), which is not jointly convex in Wand λ and hence results in an optimization problem that is by far harder to solve than one that onlyinvolves convex LMI constraints (VanAntwerp and Braatz (2000)). Algorithms for solving BMIs have beendeveloped (Beran et al. (1997); Fukuda and Kojima (2001); Goh et al. (1994); Hassibi et al. (1999);VanAntwerp et al. (1999); Yamada and Hara (1996)), their applicability is however limited to verysimple problems. Although the ongoing research in the field can be expected to produce algorithms withimproved efficiency, it is well-known that solving optimization problems subject to BMI constraints is, ingeneral, NP-hard.

Remark 4.18 (Choosing λ):In order to get an idea of how the design parameter λ should be chosen, it is useful to compute the contractionfactor λ∞ for the LQR gain K∞ and the associated matrix P∞. This can be done by simply plugging in thefixed values of AK = A+BK∞ and P = P∞ into (4.116) and finding the minimal value of λ for which (4.116)is satisfied. This is a simple SDP with λ being the only variable. In the optimization problems (4.120)and (4.121), respectively, λ should then be chosen such that 0< λ < λ∞.

4.5.3 Approximate Computation of mRPI Sets

Once the disturbance rejection controller gain K and, in case of output-feedback control, the observergain L has been chosen, the next step in the synthesis of Tube-Based Robust Model Predictive Controllersis to compute the minimal robust positively invariant sets for the resulting perturbed closed-loop sys-tem(s). Conceptually, there is no difference between a system of the form x+= AK x+w and the error


dynamics (4.46) and (4.48), which are subject to the artificial disturbances δe and δc. Without loss ofgenerality, this section therefore only considers systems of the form

x+ = Ax +w, (4.122)

where the unknown time-varying disturbance w is bounded by W . The following assumption will bemade throughout the remainder of this section:

Assumption 4.11 (Properties of W ):The set W is convex, compact and contains the origin in its interior.

Assumption 4.11 is certainly plausible for the bounds on an actual exogenous disturbance w affecting thesystem as in, say, (4.4). In the context of output-feedback control, it is easy to see that also the bound ∆eon the artificial disturbance δe in (4.47) is always fully dimensional14 whenever W , the first addend inthe Minkowski sum in (4.47), is fully dimensional. On the other hand, Assumption 4.11 may be violatedby the set ∆c in (4.49) since the observer gain matrix L is generally not invertible (not even square). Itshould be pointed out that it is possible to relax Assumption 4.11 and to extend the algorithm for theapproximate computation of mRPI sets that will be presented in the following also to sets W that containthe origin in their relative interior (Rakovic et al. (2004)). However, this extension of the algorithm isnontrivial and yields conditions that are hard to verify computationally. For simplicity of implementation,it will therefore be ensured in the following that the standard algorithm is always applicable. This can beachieved, if necessary, by artificially enlarging the set ∆c such that it contains the origin, i.e. by choos-ing∆c := LCEe⊕ LV ⊕ηBn

∞ with an η>0 sufficiently small, so that increase in the size of∆c is negligible.

In contrast to section 4.5.1, where, in order to maximize the region of attraction, the objective has been tocompute the maximal positively invariant set Ω∞, this section is instead concerned with the computationof the minimal robust positively invariant set F∞ (see Definitions 4.1 and 4.2). Unfortunately, an exactcomputation of the mRPI set F∞ is, in general, not possible. Instead, the computation of a positivelyinvariant outer approximation of F∞ will be discussed in the following. Kolmanovsky and Gilbert (1998)give a detailed exposition about robust positively invariant sets, their motivation is however to computethe maximal robust positively invariant (MRPI) set. Their work is therefore much like an extension of thecontents of section 4.5.1 to the uncertain case. It is not necessary to address this extension here, sincerequiring the invariant terminal set X f to be constraint admissible with respect to the tightened constraintsets X and U eliminates the need of computing a robust invariant terminal set for Tube-Based RobustModel Predictive Controllers.

Remark 4.19 (The mRPI set in time-optimal control):The minimal robust positively invariant set F∞ also appears in other applications than just Tube-BasedRobust MPC. For example, in Blanchini (1992); Mayne and Schroeder (1997b) it is a used as a target set intime-optimal control. The availability of efficient algorithms for the computation of approximated mRPI setsis therefore a requirement also for other purposes and not limited to the design of Tube-Based Robust ModelPredictive Controllers.

As mentioned above, an exact representation of the mRPI set F∞ is generally hard to obtain. An explicitcomputation is possible only under the restrictive assumption that the system dynamics are nilpotent15,as has been discussed in Lasserre (1993); Mayne and Schroeder (1997a). Several methods have thereforebeen proposed in the literature to compute approximations of the mRPI set (Gayek (1991); Blanchini(1999)). However, the problem with many of these approaches is that they generally do not yield invariantapproximations of the mRPI set. Since invariance of the approximated sets is crucial for their use in

14 a fully dimensional polytope has a non-empty interior15 a discrete-time linear system x+= Ax is nilpotent if Ak=0 for some positive integer k


Tube-Based Robust MPC, it is necessary to overcome this issue. To this end, the authors of Rakovic et al.(2005) propose an algorithm for the computation of an invariant outer approximation of the mRPI set. IncaseW is a polytope, their approach allows to specify a priori the accuracy of the obtained approximation.To measure this accuracy, consider the following definition:

Definition 4.8 (ε-approximations, Rakovic et al. (2005)):Denote by Bn the unit ball in Rn, i.e. Bn := x ∈ Rn | ||x || ≤ 1, where || · || is any norm. Given ascalar ε>0 and a set Ω⊂ Rn, the set Ωε ⊂ Rn is an outer ε-approximation of Ω if Ω⊆ Ωε ⊆ Ω⊕ εBn, andit is an inner ε-approximation of Ω if Ωε ⊆ Ω⊆ Ωε ⊕ εBn.

An Algorithm for Computing Approximations of mRPI Sets

If, as is the case in Tube-Based Robust MPC, the system matrix A in (4.122) is Hurwitz and the set Wbounding the additive disturbance w is a polytope, then for a specified error tolerance ε > 0 thealgorithm proposed in Rakovic et al. (2005) allows an efficient computation of an invariant polytopicouter ε-approximation of the mRPI set F∞. This algorithm will be reviewed in the following.

Preliminaries on RPI SetsFor all s ∈ N+, define the set Fs as

Fs :=s−1⊕

i=0

AiW , F0 := 0, (4.123)

where⊕b

i=aAi :=Aa⊕Aa+1⊕ · · · ⊕Ab. Note that because W is assumed convex and compact, so is Fs.

Theorem 4.7 (Relation between Fs and F∞, Kolmanovsky and Gilbert (1998)):Suppose A is Hurwitz. Then, there exists a compact set F∞ ⊂ Rn with the following properties:

1. 0 ∈ Fs ⊂F∞ for all s ∈ N+.

2. Fs→F∞ for s→∞, i.e. for any ε>0 there exists s ∈ N+ such that F∞ ⊂Fs ⊕ εBn.

3. F∞ is robust positively invariant.

From Theorem 4.7, it is evident that the exact mRPI set F∞ is given by

F∞ =∞⊕

i=0

AiW . (4.124)

Clearly, (4.124) can in general not be used to determine F∞, as this would entail computing an infinite(Minkowski) sum of sets. However, it is possible to obtain an invariant outer approximation of F∞ byproperly scaling an Fs for some finite s.

Theorem 4.8 (Obtaining an RPI set via scaling, Kouramas (2002)):Suppose that A is Hurwitz and that 0∈ int(W ). Then, there exists a finite integer s ∈ N+ and an associatedscalar α ∈ [0,1[ that satisfy

AsW ⊆ αW . (4.125)

If the pair (α, s) satisfies (4.125), then the scaled set

F (α, s) := (1−α)−1Fs (4.126)

is a convex and compact robust positively invariant set for system (4.122). Furthermore, it holds that0 ∈ int(F (α, s)) and F∞ ⊆F (α, s).


Although Theorem 4.8 suggests that a pair (α, s) can be used to obtain an RPI approximation of F∞, itdoes not provide any information about the accuracy of this approximation.

Theorem 4.9 (Error bound on F (α, s), Rakovic et al. (2005)):Suppose the pair (α, s), where α ∈ [0, 1[ and s ∈ N+, satisfies

ε ≥ α(1−α)−1 maxx∈Fs||x || = α(1−α)−1 min

γ

γ | Fs ⊆ γBn. (4.127)

Then, F∞ ⊆ F (α, s) ⊆ F∞ ⊕ εBn, i.e. in the words of Definition 4.8, F (α, s) is a robust positivelyinvariant outer ε-approximation of the minimal robust positively invariant set F∞.

Obtaining a Suitable Pair (α, s)Theorem 4.9 provides a relationship between a pre-specified error tolerance ε for the outer approx-imation F (α, s) of the sought-after mRPI set and the parameters α and s from condition (4.125) inTheorem 4.8. What is still needed is a way of evaluating the right hand side of (4.127). As will becomeclear in the following, it is possible, under the additional assumption that polytopic norms are used todefine the error bound, to do this without having to calculate the set Fs explicitly. In order to showthis, some additional results are needed first. Denote by α0(s) and s0(α) smallest values of α and s suchthat (4.125) holds for a given s and α, respectively, i.e.

α0(s) :=min

α ∈ R | AsW ⊆ αW

(4.128)

s0(α) :=min

s ∈ N+ | AsW ⊆ αW

. (4.129)

With the above definitions of α0(s) and s0(α) it is possible to show the following:

Theorem 4.10 (Limiting behavior of the RPI approximation, Rakovic et al. (2005)):Let 0 ∈ int(W ). Then,

1. F (α0(s), s)→F∞ as s→∞.

2. F (α, s0(α))→F∞ as α→ 0.

The algorithm presented in the following will also make use of the support function, which plays animportant role in set-theoretic methods in controls and optimization (Blanchini and Miani (2008); Boydand Vandenberghe (2004)).

Definition 4.9 (Support function, Boyd and Vandenberghe (2004)):The support function of a set W ⊂ Rn, evaluated at a vector a ∈ Rn, is defined as

hW (a) := supw∈W

aT w (4.130)

Remark 4.20 (Efficient computation of the support function for zonotopic sets, Rakovic et al. (2004)):It is a useful observation that if W is a zonotope, i.e. the image of a cube under an affine mapping (Girard(2005)), the computation of the support function (4.130) is trivial. In this case, W can be characterized byW =

Φx + c | ||x ||∞ ≤ η

, where Φ ∈ Rn×n and c ∈ Rn describe the affine mapping. Then,

hW (a) = supw∈W

aT w = max||x ||∞≤η

aTΦx + aT c = η||ΦT a||∞+ aT c, (4.131)

and hence the support function can be evaluated explicitly in a straightforward way.


Let W be given by itsH -representation W =

w ∈ Rn | Hww ≤ kw

, where Hw = [ fw,1 . . . fw,I]T and

kw = [kTw,1 . . . kT

w,I]T . Note that since 0 ∈ int(W ), the elements of kw satisfy kw,i > 0 for i=1, . . . , I . In

Kolmanovsky and Gilbert (1998) it is shown that

AsW ⊆ αW if and only if hW

(As)T fi

≤ αki, ∀ i=1, . . . , I . (4.132)

From (4.132) it is straightforward to see that α0(s) can be computed as

α0(s) = maxi=1,...,I

hW

(As)T fi

ki. (4.133)

Moreover, checking whether the Fs is contained in a polyhedral set P =

x ∈ Rn | Hp x ≤ kp

, withHp = [ fp,1 . . . fp,J]

T and kp = [kp,1 . . . kp,J]T, can be performed in a similar way (Rakovic et al. (2005)):

Fs ⊆P if and only ifs−1∑

k=1

hW

(Ak)T

fp, j

≤ αkp, j, ∀ j = 1, . . . , J . (4.134)

Motivated by (4.127), define M(s) :=minγ

γ ∈ R | Fs ⊆ γBn∞

, where Bn∞ denotes the∞-norm unit ball

in Rn. Applying (4.134), M(s) can be computed as

M(s) = maxj=1,...,n

(

s−1∑

k=0

hW

(Ak)Te j

,s−1∑

k=0

hW

−(Ak)Te j

)

, (4.135)

where e j denotes the j th standard basis vector in Rn (Rakovic et al. (2005)). A simple algebraic manipula-tion of (4.127) yields the following relationship between α, M(s) and ε:

α(1−α)−1Fs ⊆ εBn∞ if and only if α≤

ε

ε+M(s). (4.136)

The relationship (4.136) motivates Algorithm 2, which, for a given error bound ε > 0, is based onincreasing the value of s until it holds that α0(s)≤ ε(ε+M(s))−1.

Algorithm 2 Computation of a RPI outer ε-approximation to the mRPI set F∞, Rakovic et al. (2005)Input: A, W and ε > 0Output: F (α, s) such that F∞ ⊆F (α, s)⊆F∞⊕ εBn

∞s← 1α← α0(1)compute M(1)while α > ε

ε+M(s)do

s = s+ 1α← α0(s)compute M(s)

end whilereturn F (α, s) = (1−α)−1Fs


Properties of Algorithm 2 and Computational Issues

By virtue of Theorem 4.7, Algorithm 2 is guaranteed to terminate in finite time for any error bound ε>0.However, it is intuitive that the value of s may need to be very high in order to satisfy condition (4.136)for small error bounds ε≈ 0. A tradeoff between accuracy and simplicity of the returned approxima-tion F (α, s) of the exact minimal robust positively invariant set F∞ is therefore inevitable. An importantbenefit of the presented algorithm, in comparison to related approaches from the literature, is that itallows to specify the accuracy of the obtained approximation a priori.

There are two computationally expensive operations performed in Algorithm 2: One is the reoccur-ring evaluation of support functions hW(·) during the computation of α0(s) and M(s), the other onethe one-time evaluation of the partial sum (4.123) in order to compute Fs after a suitable pair (α, s)has been found. The computation of the support function hW(·) is, since W is assumed polyhedral, aLinear Program. Numerous efficient solvers even for very large-scale linear programming problems exist(Todd (2002)), therefore this computation can certainly be considered tractable. Moreover, if W is azonotope (which is in fact a very reasonable way to model disturbances), then the computation of hW(·)is actually trivial (see Remark 4.20). The computation of the set Fs has to be performed only once,at the very end of Algorithm 2. Nevertheless, a total of s−1 Minkowski sums need to be computedin (4.123). The Minkowski sum is a computationally expensive operation (Gritzmann and Sturmfels(1993); Borrelli et al. (2010)), such that this last step is what limits the applicability of Algorithm 2 inpractice. Simulations have shown that especially the computation of Fs is indeed very expensive and, inaddition, prone to numerical problems when involving complex shaped sets and high-dimensional systems.

Algorithm 2 has been implemented in Matlab, using the GLPK solver (The GNU project (2010)) to computethe support functions and relying on methods implemented in MPT-toolbox (Kvasnica et al. (2004))and the efficient vertex enumeration and convex hull algorithms of CDD (Fukuda (2010)) to computethe sequence of Minkowski sums in its last step. Other, possibly more efficient computational geometrysoftware16 could have also been used for the polytopic manipulations, but this was refrained from for thesake of simplicity.

Remark 4.21 (An alternate algorithm for computing mPRI sets):In Rakovic (2007), a modified algorithm was presented that makes use of the theoretic results on Minkowskialgebra and the Banach Contraction Principle from Artstein and Rakovic (2008). The main advantageof this modified algorithm, however, is mainly a more elegant theory rather than a significant increase incomputational efficiency. Nevertheless, this modified algorithm has also been implemented using the samesoftware framework as the implementation of Algorithm 2.

Remark 4.22 (Extension to systems with additional polytopic model uncertainty, Kouramas et al. (2005)):The ideas presented in this section can, with appropriate modifications, also be applied to systems that aresubject to both polytopic model uncertainty and bounded external disturbances (Kouramas et al. (2005)). Bynot lumping all uncertainty into a virtual disturbance w, which was the approach taken in section 4.2.3, thisyields a possibly tighter approximation of the minimal robust positively invariant set F∞.

Remark 4.23 (Obtaining polyhedral RPI sets from an ellipsoidal ones, Alessio et al. (2007)):It is noteworthy that there exist algorithms (Alessio et al. (2007)) that allow for the computation of polyhedralrobust positively invariant sets from ellipsoidal ones. Doing this can be useful for complex systems for whichthe computation of polyhedral RPI sets is intractable, whereas the computation of ellipsoidal RPI sets is not(see section 4.5.2 for the computation of ellipsoidal RPI sets). However, this approach will generally result insignificantly larger invariant sets, especially in case of asymmetric constraints and/or disturbance bounds.

16 such as polymake (Gawrilow and Joswig (2000))


4.5.4 The Offset Weighting Matrix T

For the Tube-Based Robust Model Predictive Controllers from section 4.4, the offset weighting matrix T isan additional design parameter that needs to be chosen during the synthesis of the controller. Importantproperties of the closed-loop system that are affected by this choice are the resulting offset, the speed ofconvergence to the neighborhood of the “optimal” steady state ( x∗s , u∗s ) and the local optimality property,i.e. the asymptotic control performance for states close to the “optimal” steady state.

Offset MinimizationAs already indicated in Remark 4.12, it is an important feature of the Tube-Based Robust Model PredictiveControllers for Tracking that the constraints of the respective optimization problems PN (x ,θ) and PN ( x ,θ)do not depend on the parameter θ . Hence, PN (x ,θ) and PN ( x ,θ) are feasible for any desired steady stateparametrized by θ , even if this steady state is not reachable. If this is the case then the system state isdriven to the neighborhood of the (admissible) artificial steady state ( x∗s , u∗s ) = Mθ θ

∗ obtained from

θ ∗ = arg minθ||θ − θ ||2T

s.t. Mθ θ = Projx(Ωet)× U.

(4.137)

The output y of the closed-loop system then tracks the artificial set point y∗s = Nθ θ∗. The offset weighting

matrix T also allows one to prioritize some outputs (by weighting more heavily the correspondingterms in T) to achieve a minimum offset on these outputs. One can therefore argue that, in a way,the Tube-Based Robust Model Predictive Controllers for Tracking have a form of the classical set-pointoptimizer (2.24) already “built-in” (Limon et al. (2008b)).

Speed of ConvergenceThe offset weighting matrix T determines how heavily the difference ∆θ := θ−θ is penalized in in thesteady state offset cost (4.78). The speed of convergence of θ ∗ to θ (and hence the speed of convergenceof the artificial steady state (x∗s , u∗s ) to the desired steady state (xs, us) and of the artificial set point y∗sto the desired set point ys) can therefore be adjusted by appropriately choosing T . Clearly, if T is large,then this convergence is fast, otherwise it will be slower (Limon et al. (2008b)). Since this simple scalingof T does not affect the possible prioritization of specific outputs discussed above, it can be consideredsimultaneously in the controller synthesis.

Local OptimalityIf the terminal set of a Model Predictive Controller for the regulation problem is chosen as a constraintadmissible, positively invariant set for the closed-loop system controlled by the optimal infinite horizoncontroller K∞, then this controller is locally infinite horizon optimal for the nominal system (see item 1of Lemma 2.1). The same holds true also for simple tracking controllers that are obtained by shiftingthe system to the desired steady state. By introducing the additional decision variable θ , this propertyhowever is lost in Tube-Based Robust MPC for Tracking. Fortunately, it can be shown that this loss ofoptimality can be made arbitrarily small by choosing T “large” enough (Alvarado (2007); Limon et al.(2008a)). Given the above properties, a reasonable way to design the steady state offset weightingmatrix T is to first pick its structure according to the offset minimization requirements, and to then scalethe matrix in order to achieve a quick transient with a small enough optimality loss.

Remark 4.24 (Alternative formulation of the steady state offset cost):It is not necessary for the steady state offset cost Vo(·) to be a quadratic function of the parameters θ and θ ,as has been assumed in (4.78). In Ferramosca et al. (2009a), it is shown that by choosing Vo(·) as a convex,positive definite and subdifferentiable function of desired and virtual output set point ys and ys, respectively,the local optimality property can be retained. For simplicity, this alternate formulation of the steady stateoffset cost has not been considered in section 4.4.


4.6 Computational Benchmark

The on-line computation effort is more or less comparable for all the different Tube-Based Robust ModelPredictive Controllers that have been presented in this chapter. Essentially, it amounts to finding thesolution of a convex Quadratic Program at each sampling instant. Two very different ways to accomplishthis task have been discussed and were implemented in software: The first one employs fast, specializedQuadratic Programming algorithms to solve the optimization problem on-line, whereas the second oneuses multiparametric programming to obtain an explicit solution, thereby effectively reducing the on-linecomputation to the evaluation of a piece-wise affine function defined over a polyhedral partition of theregion of attraction. In the following, these two implementations will for simplicity be referred to as the“on-line controller” and the “explicit controller”, respectively.

The case studies of the previous sections already indicate that the on-line evaluation speed of Tube-BasedRobust Model Predictive Controllers may be high enough to allow for their application also to fastdynamical systems with high sampling frequencies. However, since in each of the presented examples onlya single initial condition is used, a general statement about the computational performance of Tube-BasedRobust MPC is not possible at this point. The following section therefore develops a comprehensivecomputational benchmark of the controllers to investigate this question in more depth.

4.6.1 Problem Setup and Benchmark Results

Benchmark ScenarioIn order to compare the different Tube-Based Robust Model Predictive Controller types and their implicitand explicit implementations in terms of their computational complexity, the following benchmark sce-nario was developed: For a simulation horizon of Nsim=15, each of the four examples from the casestudies of this chapter was simulated for 100 different, randomly generated initial conditions scatteredover the respective region of attraction of the corresponding controller. The disturbance sequences wand v17 were generated by random, time-varying disturbances w and v , uniformly distributed on Wand V , respectively. For every example, the computation time for each of the 1500 single solutions ofthe optimization problem was determined. From this data, the minimal (tmin), maximal (tmax), andaverage (tav g) computation time for the different controllers was extracted and collected in Table 4.2.This was performed for both on-line and explicit controller implementations for identical initial conditions.Table 4.2 furthermore contains, for each example, the number of variables Nv ar and constraints Ncon inthe optimization problem and the number Nreg of regions over which the explicit solution is defined. Themachine used for the benchmark was a 2.5 Ghz Intel Core2 Duo running Matlab R2009a (32 bit) underMicrosoft Windows Vista (32 bit).

Table 4.1 was created as a reference and contains the weighting matrices and constraints of the casestudies examples as well as the design parameters that were used for the respective controllers.

On-Line Controller ImplementationAll on-line controllers that have been used in the various case studies of this thesis are based on the “qpip”interior-point algorithm18 of the QPC solver package (Wills (2010)). This Quadratic Programming solveris written in C and provides fast and efficient algorithms while at the same time allowing for a fairly easyinterfacing with Matlab, especially in combination with the optimization toolbox YALMIP (Löfberg (2004,2008)). QPC is free for academic use, the YALMIP-package is freely distributed for all purposes.

17 only in the output-feedback examples18 it has been found during simulations that the QPC solver’s “qpas” active-set algorithm is less reliable for this problem type


Tabl

e4.

1.:D

esig

nPa

ram

eter

suse

dfo

rthe

Case

Stud

ies

QR

TX

UW

VK

LN

Reg

ulat

ion,

stat

e-fb

I 20.

01-

x 2≤

2|u|≤

1||w|| ∞≤

0.1

-

−0.

69−

1.31

-9

Reg

ulat

ion,

outp

ut-f

bI 2

0.01

-−

50≤

x i≤

3|u|≤

3||w|| ∞≤

0.1|v|≤

0.05

−0.

7−

1.0

1.00

0.96

T13

Trac

king

,sta

te-f

bI 2

10I 2

1000

P ∞||x|| ∞≤

5||u|| ∞≤

0.3||w|| ∞≤

0.1

-

−0.

021−

0.69

−0.

28−

0.64

-10

Trac

king

,out

put-

fbI 2

0.01

1000

−50≤

x i≤

3|u|≤

3||w|| ∞≤

0.1|v|≤

0.05

−0.

7−

1.0

1.00

0.96

T13

Tabl

e4.

2.:C

ompu

tatio

nalB

ench

mar

kRe

sults

Nv

arN

con

Nre

gO

n-Li

neC

ontr

olle

raEx

plic

itC

ontr

olle

rto m

in/

ms

to avg/

ms

to ma

x/

ms

tx min/

ms

tx avg/

ms

tx ma

x/

ms

Reg

ulat

ion,

stat

e-fe

edba

ck29

6536

62.

32.

812

.210

.410

.614

.4R

egul

atio

n,ou

tput

-fee

dbac

k41

117

142

4.7

5.6

30.3

5.2

5.3

7.6

Trac

king

,sta

te-f

eedb

ack

4418

5>

40,0

00b

3.6

6.6

27.3

--

-Tr

acki

ng,o

utpu

t-fe

edba

ck42

121

613

4.2

5.5

48.0

16.6

16.9

20.9

aal

lon-

line

cont

rolle

rsar

eba

sed

onth

ein

teri

or-p

oint

algo

rith

mof

theQPC-s

olve

r(W

ills

(201

0))

bco

mpu

tati

onab

orte

dm

anua

lly,n

oex

plic

itso

luti

onob

tain

ed

4.6. Computational Benchmark 101

Explicit Controller ImplementationIn section 2.4 the concept of multiparametric programming has been introduced, which allows the com-putation of the explicit solution of a Quadratic Program (or Linear Program) as a function of a parameterthat enters the constraints of the optimization problem affinely. This computation is performed off-line,yielding a piece-wise affine optimizer function defined over a polyhedral partition of the feasible set inthe parameter space. In the context of Tube-Based Robust MPC, the parameter is the current systemstate x (or the current state estimate x for output-feedback), while in the context of Tube-Based RobustMPC for Tracking it is the current augmented state (x ,θ) (or ( x ,θ) for output-feedback). Since it isperformed off-line, the computation of the control law itself has no implication on the actual suitability ofthe resulting explicit controller for on-line implementation. The only task performed on-line is to identifyin what region of the partition the current parameter lies, and to evaluate the control law assigned tothis region. However, as has been indicated in section 2.4, also this task may become computationallyexpensive in case the number of regions is very high.

The computation of the piece-wise affine optimal control laws for the explicit controllers from the casestudies was performed using the mpQP algorithm of the freely distributed MPT-toolbox (Kvasnica et al.(2004)). For simplicity, a straightforward Matlab implementation was used to perform the on-lineevaluation of the resulting PWA optimizer functions. Note that the computation of the explicit solutionfor the state-feedback Tube-Based Robust Model Predictive Controller for Tracking from section 4.4.1was manually aborted after the solver had identified more than 40,000 regions. Apparently, the specificinternal structure of the associated optimization problem requires an extremely high number of regions inorder to characterize the explicit solution.

4.6.2 Observations and Conclusions

Complexity of the Explicit SolutionAn indicator for how the internal structure of the optimization problem determines the complexity of thecorresponding explicit solution is the output-feedback regulator example. In the associated optimizationproblem, the number of variables is about 40% higher, and the number of constraints is about 80%higher than in the optimization problem for the state-feedback example. Nevertheless, the number ofregions is only about 40% of the total number of regions of the state-feedback controller (the respectiveexploration regions Xx pl are comparable in size). This shows that the complexity of the explicit solutionis not only determined simply by the number of variables and constraints in the optimization problem,but depends on the internal problem structure in a complex and non-obvious way. Further research and amore detailed analysis of multiparametric programming will be necessary to understand this dependency.

On-Line Computation TimesFrom the benchmark results in Table 4.2, it can be observed that the average solver time of the on-line con-troller correlates strongly with the number of constraints Ncon in the optimization problem. It is interestingthat although the average solver time to

av g is comparatively low (in the sense that it is close to the minimalsolver time to

min), the maximal solver time tomax is up to 9 times the average solver time of the respective

controller. This shows that although the solver algorithm is generally very efficient (i.e. it requires onlya small number of iterations to obtain the optimal solution), situations do indeed occur in which thecomputation time is significantly higher (caused by a higher number of required iterations). Curiouslyenough, the computation times of the on-line implementation of the state-feedback Tube-Based RobustModel Predictive Controller for Tracking are completely inconspicuous compared to the ones of the otheron-line controllers. Knowing about the extremely high complexity of the associated explicit controller(with more than 40,000 regions), this is certainly surprising. Apparently, the employed interior-pointalgorithm has no problems whatsoever with the internal structure of this optimization problem.


A similar correlation as the one between the number of constraints and the average computation time ofthe on-line controller can be found between the number of regions Nreg over which the explicit controlleris defined and the average time t x

av g that is necessary to evaluate the PWA control law. In contrast to thoseof the on-line controller, the computation times of the explicit controller span only a rather small intervalt x

max− t xmin. The maximal ratio t x

max/t xav g = 7.6/5.3 = 1.43 of all explicit controllers is therefore much

smaller than the ratios tomax/to

av g found among the on-line controllers (here the maximal ratio is 8.73). Inother words, the time needed for determining the optimal control input given the current state (or stateestimate) is much more predictable for the explicit controller, though it is not necessarily shorter than theaverage time needed by the on-line controller.

Further StepsThe obtained benchmark results suggest that all of the reviewed Tube-Based Robust Model PredictiveControllers have at least the potential to be sufficiently fast to allow their application to “fast”-samplingdynamic systems (sampling frequencies of more than 100Hz are thinkable for the example systemsstudied in the benchmark). There are however still a number of open questions that need to be addressedthoroughly before the computational properties of the controllers can be fully assessed. Since thebenchmark computations were performed on a standard desktop computer, the large fluctuations in thesolver times may also be due to the resource allocation management of the operating system. Hence, thecontrollers should be implemented and tested on a real-time capable hard- and software architecture. Inaddition, developing a more efficient implementation for the evaluation of the explicit control law canbe expected to lower the on-line evaluation time of the explicit controllers significantly (note that in thebenchmark a simple Matlab code was used for this task). Another relation that needs to be understood ishow the computation speed scales with the available hardware resources, especially when using simplerprocessors like the ones found in embedded systems.

4.6. Computational Benchmark 103

5 Interpolated Tube MPCThe following chapter contains the novel contribution of this thesis, which constitutes an extension of theTube-Based Robust MPC approaches that have been discussed in chapter 4. In this extension, the ideasof Interpolation-Based Robust MPC from section 3.5.2 are combined with those of Tube-Based RobustMPC in order to synthesize a new type of Robust Model Predictive Controller. The advantage of this new“Interpolated Tube MPC” approach is a significantly larger region of attraction for a comparable on-linecomputational complexity of the controller. Equivalently, for a desired region of attraction the controllerfeatures a reduced on-line complexity. To the best of the author’s knowledge, the work presented in thefollowing is a novel contribution and so far has not been proposed in the literature.

After motivating the necessity for the new controller in section 5.1, the interpolated terminal controller isreviewed in section 5.2. Section 5.3 then discusses the overall Interpolated Tube Model Predictive Con-troller and its most important properties before section 5.4 provides an outlook on how the computationalcomplexity of the controller could be further reduced. Finally, the output-feedback Double Integrator casestudy from chapter 4 is revisited in section 5.5, where performance and computational complexity ofstandard Tube-Based Robust MPC and Interpolated Tube MPC are compared.

5.1 Motivation

In general, the region of attraction of any controller1 is defined as the set of initial states from which thecontrolled system can be stabilized. More specifically, the region of attraction XN of a Model PredictiveController is defined as the set of initial states from which the terminal constraint set X f can be reachedin N steps via an admissible state trajectory x ∈ X using an admissible control sequence u ∈ U. Thus,the size of XN directly depends on the size of the terminal set X f – a large terminal set implies a largeregion of attraction for a fixed prediction horizon N , while a smaller terminal set for the same N bringsabout a smaller region of attraction (provided that the size of XN is limited not only by the state con-straint set X but also by the control constraint set U). In order to achieve local optimality (and hencegood asymptotic control performance), Tube-Based Robust MPC usually employs the infinite horizonunconstrained LQR controller K∞ as the terminal controller. If the terminal set X f is chosen as a (usuallythe maximal) positively invariant set for the nominal closed-loop system x+= (A+BK∞) x subject to thetightened constraints X and U, then the infinite horizon cost V∞ for any state xN within X f is givenby V∞( xN ) = x T

N P∞ xN , where P∞ is the unique positive definite solution to the ARE (2.17). However,the choice of this optimal unconstrained controller is likely to result in a rather small terminal set X fand consequently in a small region of attraction of the Model Predictive Controller. Because of this,Tube-Based Robust MPC generally requires comparably large prediction horizons to obtain sufficientlylarge regions of attraction, which in turn increases the on-line complexity of the controller. In orderto reduce the necessary prediction horizon one could detune the terminal controller so that it yields alarger X f . This however comes at the cost of a loss in local optimality and hence an inferior closed-loopperformance. Therefore, the choice of the terminal controller in Tube-Based Robust MPC inherentlyconstitutes a tradeoff between the controller’s optimality properties and the size of its region of attraction.

1 strictly speaking, the region of attraction is defined for the closed-loop system and depends on both controller and systemdynamics. When thinking of it as comparing multiple controllers for the same system, the region of attraction can forsimplicity be regarded as belonging to the controller

105

One way to overcome this limitation is to use a nonlinear terminal controller that achieves good local con-trol performance while at the same time ensuring a sufficiently large terminal set X f . The use of a generalnon-linear controller, however, causes problems in both theoretical analysis and implementation. On theone hand, it is much more difficult to guarantee stability of the MPC control loop. On the other hand, thecomputation of both terminal cost function and terminal set is also more involved and generally results inan on-line optimization problem that is not a Quadratic Program anymore. Interpolation-Based RobustMPC approaches, as they have been proposed in Bacic et al. (2003); Rossiter et al. (2004); Pluymers et al.(2005c); Rossiter and Ding (2010) and discussed in section 3.5.2, take an alternative approach that allowsto retain the QP structure of the optimization problem by using interpolation between multiple predefinedlinear terminal controllers. Because this interpolation is performed by solving an optimization problemon-line at each time step, the resulting overall terminal controller is indeed nonlinear. Its correspondingmaximal positively invariant set can be shown to be the convex hull of the maximal positively invariantsets of the constituent linear feedback controllers. Hence, depending on the choice of the individualcontroller gains, the resulting terminal set can be significantly enlarged, which immediately leads to alarger region of attraction of the overall controller.

The novel Interpolated Tube Model Predictive Control approach proposed in this thesis uses theseinterpolation techniques in order to enlarge the terminal set and consequently the region of attraction ofTube-Based Robust Model Predictive Controllers. Its basic ideas are reminiscent of the work in Sui andOng (2006, 2007); Sui et al. (2008, 2009, 2010a,b), which is based on a similar interpolation method,but which instead uses the controller structure proposed in Chisci et al. (2001).

5.2 The Interpolated Terminal Controller

Before proceeding to the presentation of the overall Interpolated Tube MPC framework in section 5.3,the following sections first review structure and properties of the employed interpolated terminal con-troller. This controller is a modified version of the Interpolation-Based Model Predictive Controller fromsection 3.5.2 and, as the terminal controller, determines the terminal cost function that will be used tobound the infinite horizon cost in the overall on-line optimization problem.

5.2.1 Controller Structure

Assume for the purpose of this section that ν (different) stabilizing linear controllers Kp, p=0, . . . ,ν−1have already been designed for the nominal system

x+ = Ax + Bu, (5.1)

which is subject to the usual tightened constraints x ∈ X and u ∈ U on state and control input.

Assumption 5.1 (Properties of the controller gains Kp):The controller gains K0, . . . , Kν−1 are assumed to fulfill the following:

1. K0 is the infinite horizon optimal controller for the unconstrained infinite horizon LQR problem ofthe nominal system, i.e. K0 is given as K0=−(R+ BT P0B)−1BT P0 A, where P0 is the unique positivedefinite solution to the Algebraic Ricatti Equation P0 =Q+ AT (P0− P0B(R+ BT P0B)−1BT P0)A.

2. The controllers K1, . . . , Kν−1 are stabilizing linear state-feedback controllers, i.e. ρ(A+ BKp) < 1for p=1, . . . ,ν−1. Hence, for each Kp there exists a positive definite matrix Pp0 such that

ATKp

Pp AKp− Pp =−Q− K T

p RKp. (5.2)

With Pp satisfying (5.2), the infinite horizon cost for the trajectory of the unconstrained closed-loopsystem x+= (A+ BKp) x starting from an initial state x is given by V∞( x) = x T Pp x .

106 5. Interpolated Tube MPC

Following Pluymers et al. (2005a), an interpolated control law of the form

u= κip( x) =ν−1∑

p=0

Kp x p (5.3)

is used, where the current nominal state

x =ν−1∑

p=0

x p (5.4)

is decomposed into ν slack state variables x p which determine the proportions of the different feedbackgains in the overall controller (Sui et al. (2009)). Noting that x o can be expressed as x o = x −

∑ν−1p=1 x p,

the closed-loop nominal system under the control law (5.3) can be written as

x+ = AK0x +

ν−1∑

p=1

(AKp− AK0

) x p, (5.5)

where AKp:=A+BKp. For the purpose of the analysis of the controller, it is useful to define the auxiliary

systems ( x p)+ :=AKpx p for p=1, . . . ,ν−1. Stacking the nominal state x and the slack state variables

x1 . . . xν−1 yields the augmented closed-loop system (Sui et al. (2008))

x+

( x1)+

...

( xν−1)+

=

AK0AK1− AK0

. . . AKν−1− AK0

0 AK1. . . 0

......

. . ....

0 0 . . . AKν−1

·

xx1

...xν−1

(5.6)

which is subject to the constraints

x ∈ X, K0 x +ν−1∑

p=1

(Kp − K0) x p ∈ U. (5.7)

Remark 5.1 (Equivalence of the control constraint):It is easy to see2 that the second constraint in (5.7) is equivalent to the original control constraint u ∈ U.The benefit of the form in (5.7) is that it allows the control constraint to be invoked as an equivalent stateconstraint on the autonomous augmented system (5.6), for which a constraint admissible positively invariantset can thereupon be computed.

5.2.2 The Maximal Positively Invariant Set

In all Model Predictive Control approaches, the purpose of the terminal constraint set is to ensurepersistent feasibility of the closed-loop system stabilized by the terminal controller. Therefore, if theinterpolated controller (5.3) was to be used as the terminal controller in Interpolated Tube MPC, itis necessary to compute a constraint admissible positively invariant set for the closed-loop nominalsystem (5.5). Motivated by Remark 5.1, the idea in the following is to make a detour and use theaugmented closed-loop system (5.6) to ultimately determine a positively invariant set in the x-space. Tothis end, define the augmented state

x E :=

x T ( x1)T . . . ( xν−1)TT

(5.8)

2 by plugging in (5.4) and simplifying

5.2. The Interpolated Terminal Controller 107

and write the augmented closed-loop system as ( x E)+ := AE x E, where AE denotes the system matrixin (5.6). Moreover, denote by XE the constraint set for the augmented state x E. Since only the nominalstate x is constrained (by X), there are no additional constraints on the slack state variables x p. If, inaddition, Assumption 4.3 is satisfied3, this means that XE contains an open neighborhood of the origin.Furthermore, it is easy to see that the augmented system matrix AE is Hurwitz. This follows from itsblock diagonal structure and Assumption 5.1. Hence, the maximal positively invariant set ΩE

∞ for theaugmented system (5.6) exists and contains a nonempty region around the origin (Gilbert and Tan (1991);Kolmanovsky and Gilbert (1998)). Since the constraint sets X and U are assumed polytopic, and becausethe system (5.6) is linear, the set ΩE

∞ is also polytopic. Denote by X f the projection of ΩE∞ onto the

x-space, i.e. X f = Proj x(ΩE∞). It is easy to show that X f is a constraint admissible, positively invariant

set for the closed-loop nominal system (Proposition 5.1). Moreover, X f contains all maximal positivelyinvariant sets corresponding to the different linear terminal controllers K0, . . . , Kν−1 (Proposition 5.2).

Proposition 5.1 (Invariance of X f ):Let

K0, . . . , Kν−1

be a set of linear state-feedback controllers for the nominal system (5.1) satisfyingAssumption 5.1 and let ΩE

∞ be the maximal positively invariant set for the augmented system (5.6) subjectto the constraints (5.7). Then X f = Proj x(Ω

E∞) is a constraint admissible, positively invariant set for the

system (5.5) subject to the constraints x ∈ X and u ∈ U.

Proof. Since X f is the projection of ΩE∞ onto the x-space, it holds that X f ⊆ X. Moreover, the definition of

the projection operator in Definition 4.6 also implies that for any x ∈ X f there exists an x E ∈ ΩE∞. Because

of the invariance of ΩE∞ it holds that AE x E∈ ΩE

∞, and hence x+∈ X f . Furthermore, the control constraintin (5.7) is satisfied for all x E ∈ ΩE

∞ and hence for all x ∈ X f . Therefore, the set X f is a constraintadmissible, positively invariant set for the system (5.5) subject to the constraints x ∈ X and u ∈ U.

Proposition 5.2 (Size of the set X f ):Let

K0, . . . , Kν−1

be a set of linear state-feedback controllers for the nominal system (5.1) satisfyingAssumption 5.1 and let

Ω0∞, . . . ,Ων−1

∞

be the set of maximal positively invariant sets corresponding tothe respective closed-loop systems subject to the constraints x ∈ X and u ∈ U. Furthermore, let X f bethe projection of the maximal positively invariant set ΩE

∞ for the augmented system (5.6) subject to theconstraints (5.7) on the x-space. Then, it holds that X f ⊇ Convh(Ω0

∞, . . . ,Ων−1∞ ), i.e. the set X f contains

the convex hull of all sets Ω0∞, . . . ,Ων−1

∞ .

Proof. Consider the case when the decomposition of x =∑ν−1

p=0 x p is degenerate, i.e. when only one ofthe slack state variables x p is non-zero. Denote the index of this non-zero variable by p∗, and distinguishthe two cases p∗=0 (case 1) and p∗ 6=0 (case 2).

case 1: When p∗= 0, then clearly x= x0. The augmented system (5.6) reduces to x+= AK0x , as x p=0

for all p 6=0. The associated maximal positively invariant set is Ω0∞, hence X f =Proj x(Ω

E∞)=Ω

0∞.

case 2: When p∗ 6=0, then x p∗= x and the augmented system (5.6) can be reduced to

x+

x+

=

AK0AKp∗− AK0

0 AKp∗

·

xx

, (5.9)

since x p=0 for all p 6= p∗. It is straightforward to see that (5.9) is equivalent to x+= AKp∗x , for

which the associated maximal positively invariant set is Ωp∗∞ . Hence, X f =Proj x(Ω

E∞) = Ω

p∗∞ .

From the above it is clear that X f contains every Ωp∞ for p=0, . . . ,ν−1. Since the system (5.6) is linear and

the constraints (5.7) are polytopic (and thus convex), it furthermore holds that Convh(Ω0∞, . . . ,Ων−1

∞ )⊆ X f .

3 the tightened constraint sets X and U exist and contain the origin


Theorem 5.1 states that X f contains the convex hull of all ν maximal positively invariant sets Ωp∞ for the

respective closed-loop systems under the linear feedback controllers Kp. The size of X f is therefore atleast equal to the size of the largest of the sets Ωp

∞, which is potentially considerably larger than the sizeof Ω0

∞, the maximal positively invariant set for the closed-loop system under the optimal unconstrainedinfinite horizon controller K0. Hence, using an interpolated terminal controller κip(·) of the form (5.3)can significantly enlarge the terminal set X f and consequently the region of attraction XN of Tube-BasedRobust Model Predictive Controllers. It is this favorable property of the interpolated terminal controllerthat will be taken advantage of by the Interpolated Tube MPC framework in section 5.3.

5.2.3 Stability Properties

This section explores the stability properties of the interpolated terminal controller (5.3). Although thedecomposition of the state x is not recomputed at each sampling instant (as will later be the case forInterpolated Tube MPC in section 5.3), this analysis of the simplified controller will nevertheless be veryuseful in the later analysis of the overall Interpolated Tube Model Predictive Controller.

Theorem 5.1 (Stability of the interpolated terminal controller):Let

K0, . . . , Kν−1

be a set of linear state-feedback controllers for the nominal system (5.1) satisfyingAssumption 5.1 and suppose that x ∈ X f , i.e. that the initial state of system (5.1) is contained within theset X f . Furthermore, suppose that x E∈ ΩE

∞, where x E is given by (5.8), and that

x0, . . . , xν−1 is a setof slack state variables satisfying (5.4). Then, the origin of the closed-loop system x+= Ax + Bκip( x) isexponentially stable while state and control constraints x ∈ X and u ∈ U are satisfied for all times.

Proof. Since X f is a constraint admissible, positively invariant set for the constrained closed-loop systemx+= Ax + Bκip( x) (see Proposition 5.1), it suffices to show exponential stability of the origin for theunconstrained closed-loop system in order to prove Theorem 5.1. Consider as a candidate LyapunovFunction the cost function

V∞( x) =ν−1∑

p=0

|| x p||2Pp, (5.10)

where, under slight abuse of notation, the argument x is used for simplicity. Under the interpolatedterminal controller (5.3), the successor state of x is given by x+=

∑ν−1p=0 AKp

x p, and the successor slack

state variables are given by ( x p)+= AKpx p for all p=0, . . . ,ν−1. Hence, the following holds:

V∞( x+)− V∞( x) =

ν−1∑

p=0

||( x p)+||2Pp−ν−1∑

p=0

|| x p||2Pp=ν−1∑

p=0

||AKpx p||2

Pp− || x p||2Pp

=ν−1∑

p=0

( x p)T (ATKp

Pp AKp−Pp)( x

p) =−ν−1∑

p=0

|| x p||2Q+KTp RKp

(5.11)

where the last step in (5.11) follows from (5.2). Hence, there exist constants β>α>0 and γ>0 such that

V∞( x)≥ α|| x ||2 (5.12)

V∞( x+)≤ V∞( x)− γ|| x ||

2 (5.13)

V∞( x)≤ β || x ||2, (5.14)

where α, β and γ satisfy α ≥ minp

λmin(Pp)

, β ≤ maxp

λmax(Pp)

and γ ≥ minp

λmin(Q+ K Tp RKp)

,respectively4. Hence, the origin of the closed-loop system x+= Ax + Bκip( x) is exponentially stable witha region of attraction X f (Rawlings and Mayne (2009)).4 here λmin(M) and λmax(M) denote the minimum and maximum eigenvalue of a p.s.d. matrix M , respectively

5.2. The Interpolated Terminal Controller 109

Remark 5.2 (Determining the slack state variables on-line):In Theorem 5.1 it is assumed that the decomposition of the initial state is given and that the evolutionof the slack state variables is determined by ( x p)+ = AKp

x p for all p= 0, . . . ,ν−1. In order to increasethe performance of the controller and retain local optimality for states sufficiently close to the origin, ithas been proposed in the context of Interpolation-Based Model Predictive Control (see section 3.5.2) thatthe decomposition of the initial state be performed on-line at each time step (Pluymers et al. (2005c)). Inthis case, it can be shown that, as the system state approaches the maximal positively invariant set Ω0

∞,the interpolation controller recovers the optimal infinite horizon controller K0 and hence is locally optimal.This kind of on-line decomposition, in combination with a non-zero prediction horizon, is the basis of theInterpolated Tube MPC approach proposed in the following section.

5.3 The Interpolated Tube Model Predictive Controller

Consider the general Tube-Based Robust Model Predictive Control law (4.8), where u is the control inputto the nominal system, K is the disturbance rejection controller and x and x are the state of the actualand the nominal system, respectively. Using identical arguments as in section 4.2, the idea of InterpolatedTube MPC is to regulate the nominal system to the origin, while bounding the deviation between actualand nominal system state by an invariant set E . Constraint satisfaction of the actual system is ensured byinvoking the appropriately tightened state and control constraints (4.12) on the nominal system.

The only difference between Tube-Based Robust MPC and Interpolated Tube MPC is the way how thenominal system is controlled. While standard Tube-Based Robust MPC employs a single linear time-invariant terminal feedback controller, the novel contribution of this thesis is to use the interpolatedterminal controller from the previous section. This can be achieved by generalizing the terminal costfunction Vf (·) in the overall cost function VN (·) and by appropriately modifying the variables andconstraints of the associated optimization problem. Instead of a single quadratic term as in (4.15), theterminal cost in Interpolated Tube MPC is chosen as

Vf ( xN ) = minx0

N ,..., xνN

ν−1∑

p=0

|| x pN ||

2Pp

s.t. x E =

x T ( x1)T . . . ( xν−1)TT∈ ΩE

∞

xN =ν−1∑

p=0

x pN

(5.15)

where ΩE∞ is the maximal positively invariant set for the augmented system (5.6) subject to the con-

straints (5.7). Hence, computing the terminal cost Vf ( xN ) amounts to solving a Quadratic Program5

even for a given predicted terminal state xN . The obvious thing to do is therefore to lump together thecomputation of the terminal cost and the prediction of the state trajectory and the control sequence intoone big combined finite horizon optimal control problem.

5 note that (5.15) is a Quadratic Program because ΩE∞ is polytopic and hence the constraint x E ∈ ΩE

∞ can be expressed by afinite set of linear inequalities


5.3.1 The Optimization Problem and the Controller

Following the above arguments, consider the cost function

VN (x; x0, u, x) :=N−1∑

i=0

|| x i||2Q + ||ui||

2R+

ν∑

p=0

|| x pN ||

2Pp

, (5.16)

where xN :=

x0N , . . . , xν−1

N

denotes the set of slack state variables into which the terminal state xN isdecomposed, and where the weighting matrices Q, R and P0, . . . , Pν−1 satisfy Assumption 5.1. The set ofadmissible control sequences for a given nominal initial state x0 is given by

UN ( x0) =¦


©

, (5.17)

where X f = Proj x(ΩE∞). At each time step, given a measurement of the current state x of the system, the

following optimal control problem PipN (x) is solved on-line:

V ∗N (x) = minx0,u,xN

VN (x; x0, u, x)

u∈UN ( x0), x0∈x⊕(−E ), xN=ν−1∑

p=0

x pN , x E

N ∈ ΩE∞

(5.18)

( x∗0(x), u∗(x), x∗(x)) = arg min

x0,u,xN

VN (x; x0, u, x)

u∈UN ( x0), x0∈x⊕(−E ), xN=ν−1∑

p=0

x pN , x E

N ∈ ΩE∞

(5.19)

where x EN =

x TN ( x

1N )

T . . . ( xν−1N )

TTas defined in (5.8). The domain of the value function VN (·) is

XN =


. (5.20)

Applying only the first element of the the optimizer u∗(x) obtained from PipN (x) at each time step in a

Receding Horizon fashion, the implicit Interpolated Tube Model Predictive Control law κipN (·) becomes

κipN (x) := u∗0(x) + K(x − x∗0(x)). (5.21)

5.3.2 Properties of the Controller

As will be shown in the following, the proposed Interpolated Tube Model Predictive Control approachfeatures some interesting beneficial properties that make it attractive for applications. As to the controlperformance, these properties include the guaranteed robust exponential stability of a robust positivelyinvariant set, an increased region of attraction compared to Tube-Based Robust MPC, and local optimality6

for states sufficiently close to the origin. Another major benefit of Interpolated Tube MPC is its comparablylow computational complexity: the corresponding on-line optimization problem is a convex QuadraticProgram and can therefore be solved very efficiently, permitting an application of the proposed controlleralso to fast dynamical systems. In fact, for a given size of the region of attraction, it is possible to designInterpolated Tube Model Predictive Controllers that have a significantly lower on-line complexity thancomparable Tube-Based Robust Model Predictive Controllers.

6 local optimality in this chapter means a control performance identical to the one of Tube-Based Robust MPC

5.3. The Interpolated Tube Model Predictive Controller 111

Control Performance

The most important properties pertaining to the control performance of the newly proposed InterpolatedTube Model Predictive Controller are subsumed in the following main theorem of this section:

Theorem 5.2 (Performance of Interpolated Tube MPC):Let E be a robust positively invariant set for the perturbed closed-loop system x+ = (A+BK)x + w,where w ∈ W and where K is a disturbance rejection controller stabilizing the unconstrained nominalsystem x+= Ax+Bu. Assume that the associated tightened constraint sets X= XE and U= UKE arenonempty and contain the origin. Suppose that

K0, . . . , Kν−1

is a set of linear state-feedback controllersfor the nominal system x+= Ax + Bu satisfying Assumption 5.1. Furthermore, let Ω0

∞ be the maximalpositively invariant set for the system x+ = (A+ BK0) x subject to the constraints x ∈ X and u ∈ U,and let ΩE

∞ be the maximal positively invariant set for the augmented system (5.6) subject to theconstraints (5.7). Denote by κip

N (x) the implicit Interpolated Tube Model Predictive Control law (5.21),based on the solution of the optimization problem (5.18) at each time step. Then, the following holds:

1. Persistent Feasibility: For any initial state x(0) ∈ XN and any admissible disturbance sequence w,the resulting state trajectory x and the resulting sequence of control inputs u of the perturbedclosed-loop system x+ = Ax + Bκip

N (x) + w are persistently feasible, i.e. it holds that x(t) ∈ Xand u(t) ∈ U for all t ≥ 0 given that w(t) ∈W .

2. Robust Stability: The set E is robustly exponentially stable for the perturbed closed-loop systemx+= Ax + Bκip

N (x) +w with a region of attraction XN .

3. Increased Attractivity: The region of attractionXN satisfiesXN⊇Convh(X 0N , . . . ,X ν−1

N ), whereX pN

denotes the region of attraction of a Tube-Based Robust Model Predictive Controller that employs Kpas the terminal controller.

4. Local Optimality: For all states x ∈ X 0N , the Interpolated Tube Model Predictive Controller κip

N (x)yields the same closed-loop performance as a Tube-Based Robust Model Predicitve Controller κN (x)that employs the unconstrained infinite horizon optimal controller K0 as the terminal controller.

Proof.Persistent Feasibility: By the definition of the region of attraction XN , there exists a feasible sequenceof nominal control inputs u and a feasible decomposition of the predicted nominal terminal state xN forany x ∈ XN . Denote the optimizers obtained from Pip

N (x) by u∗ =

u∗0, . . . , u∗N−1

, x∗ =

x∗0, . . . , x∗N

andx∗N =

x0,∗N , . . . , xν−1,∗

N

, respectively. The successor state x+ at the next time step is

x+ = Ax + BκipN (x) +w = Ax + B(u∗0+ K(x − x∗0)) +w

= A(x − x∗0 + x∗0) + B(u∗0+ K(x − x∗0)) +w

= Ax∗0 + Bu∗0+ (A+ BK)(x − x∗0) +w

= x∗1 + AK(x − x∗0) +w

∈ x∗1 ⊕E ,

(5.22)

where the last step in (5.22) follows from the facts that x − x∗0 ∈ E (this is explicitly invoked as aconstraint in the optimization problem Pip

N (x)) and that E is a robust positively invariant set for thesystem x+= AK x +w for w∈W . Because x∗1 ∈ X and X⊕E ⊆ X, it holds that x+∈ X, i.e. the successorstate satisfies the state constraints. A feasible (but not necessarily optimal) choice for the initial state x+0of the nominal system at the next time step is x+0 = x∗1. A feasible (again not necessarily optimal) controlsequence for this initial state is given by u+=

u∗1, . . . , u∗N−1,κip( x∗N )

, where κip(·) is defined in (5.3). Thisholds true because of the invariance of the terminal set X f under the control law κip(·) (Proposition 5.1).


Hence, for any x ∈ XN and any w∈W , the Interpolated Tube Model Predictive Controller κipN (x) yields a

successor state x+∈ XN . Repeating this argument proves Persistent Feasibility (item 1).Robust Stability: Denote by V ∗N (x) the cost obtained from solving Pip

N (x) for the current state x ∈ XN .At the next time step, the cost VN (x+) for the feasible control sequence u+=

u∗1, . . . , u∗N−1,κip( x∗N )

andthe feasible initial state x+0 = x∗1 is

VN (x+) =

N∑

i=1

|| x∗i ||2Q + ||u

∗i ||

2R+

ν−1∑

p=0

||AKpx p,∗

N ||2

Pp

=N−1∑

i=0

|| x∗i ||2Q + ||u

∗i ||

2R+ || x

∗N ||

2Q + ||u

∗N ||

2R− || x

∗0||

2Q − ||u

∗0||

2R+

ν−1∑

p=0

||AKpx p,∗

N ||2

Pp

= V ∗N (x)−ν−1∑

p=0

|| x p,∗N ||

2Pp+ || x∗N ||

2Q + ||u

∗N ||

2R− || x

∗0||

2Q − ||u

∗0||

2R+

ν−1∑

p=0

||AKpx p,∗

N ||2

Pp

= V ∗N (x) +

ν−1∑

p=0

x p,∗N

2

Q

+

ν−1∑

p=0

Kp x p,∗N

2

R

− || x∗0||2Q − ||u

∗0||

2R+

ν−1∑

p=0

||AKpx p,∗

N ||2

Pp− || x p,∗

N ||2Pp

≤ V ∗N (x) +ν−1∑

p=0

|| x p,∗N ||

2Q +

ν−1∑

p=0

||Kp x p,∗N ||

2R− || x

∗0||

2Q − ||u

∗0||

2R+

ν−1∑

p=0

||AKpx p,∗

N ||2

Pp− || x p,∗

N ||2Pp

= V ∗N (x)− || x∗0||

2Q − ||u

∗0||

2R+

ν−1∑

p=0

||AKpx p,∗

N ||2

Pp− || x p,∗

N ||2Pp+ || x p,∗

N ||2Q + ||Kp x p,∗

N ||2R

= V ∗N (x)− || x∗0||

2Q − ||u

∗0||

2R+

ν−1∑

p=0

( x p,∗N )

T (ATKp

PpAKp− Pp +Q+ K T

P RKp)( xp,∗N )

= V ∗N (x)− || x∗0||

2Q − ||u

∗0||

2R

(5.23)

where in the third step of (5.23) x∗N and u∗N are expressed by their decompositions x∗N =∑ν−1

p=0 x p,∗N and

u∗N =∑ν−1

p=0 Kp x p,∗N , respectively. The fourth step of (5.23) follows from the Cauchy-Schwartz Inequality

(Horn and Johnson (1990)) and the last step follows from item 2 of Assumption 5.1. Since the controlsequence u+=

u∗1, . . . , u∗N−1,κip( x∗N )

and the initial state x+0 = x∗1 are feasible but not necessarily optimal,it holds that

V ∗N (x+)− V ∗N (x)≤−|| x

∗0||

2Q − ||u

∗0||

2R (5.24)

Note that V ∗N (x) = 0 for all x ∈ E as u∗ =

0, . . . ,0

, x∗ =

0, . . . ,0

and x∗N =

0, . . . ,0

are feasible forall x ∈ E (see Proposition 4.2). On the other hand, the constraint x − x∗0 ∈ E implies that x∗0 6= 0 ∀ x /∈ E .Hence, for all x ∈ XN \E there exist constants β > α > 0 (Rawlings and Mayne (2009)) such that

V ∗N (x)> α||x ||2 (5.25)

V ∗N (x+)≤ V ∗N (x)−α||x ||

2 (5.26)

V ∗N (x)< β ||x ||2. (5.27)

From (5.26) and (5.27) it follows that V ∗N (x+)≤ (1−α/β)V ∗N (x) and thus V ∗N (x(i))≤ γ

iV ∗N (x(0)) for allx ∈ XN \E , where γ := (1−α/β) ∈ [0, 1] and x(i) = Φ(i; x(0), u,w). Hence,

||x(i)||2 ≤ (1/α)V ∗N (x(i))≤ (1/α)γiV ∗N (x(0))≤ (β/α)γ

i||x(0)||2, (5.28)

or |x(i)| ≤ cδi for all x(0) ∈ XN \E , with c :=p

(β/α) and δ :=pγ. Robust exponential stability of theset E (item 2) is proven.


Increased Attractivity: In section 4.5.1 the “Pre”-operator on a set Ω was introduced (4.104). Definethe one-step controllable set Ctrl(Ω) of a target set Ω for the nominal system as

Ctrl(Ω) :=

x ∈ X | ∃ u ∈ U such that Ax + Bu ∈ Ω

. (5.29)

The one-step controllable set is the set of all states in the state space from which the target set Ω can bereached in one step by applying a constraint admissible control input. It is easy to see that the regionof attraction of nominal states of a Tube-Based Robust Model Predictive Controller is XN = CtrlN (X f ),where CtrlN (Ω):=Ctrl(Ctrl(. . . Ctrl(Ω))). Now let Ω = Convh(v1, . . . , vJ) be a polytope defined by theconvex hull of its J vertices v j, j=1, . . . , J . Because the linear system dynamics are linear and the sets Ω,X and U are convex, it holds that

Ctrl(Ω) = Convh(Ctrl(v1), . . . , Ctrl(vJ)). (5.30)

Therefore, for convex sets ΩA and ΩB,

Ctrl(Convh(ΩA,ΩB)) = Convh(Ctrl(ΩA), Ctrl(ΩB)), (5.31)

and hence XN (Convh(ΩA,ΩB)) = Convh(XN (ΩA),XN (ΩB)). Together with Theorem 5.2 it then followsthat XN (X f ) ⊇ Convh(X 0

N (X f ), . . . ,X ν−1N (X f )) which proves the Increased Attractivity property of the

Interpolated Tube Model Predictive Controller (item 3).Local Optimality: Consider the terminal cost

∑ν−1p=0 || x

pN ||

2Pp

in the cost function (5.16). Since K0 is theinfinite horizon unconstrained optimal controller it holds that Pp P0 for all p 6= 0. Hence,

VN (x) =N∑

i=1

|| x∗i ||2Q + ||u

∗i ||

2R+

ν−1∑

p=0

|| x pN ||

2Pp

≥N∑

i=1

|| x∗i ||2Q + ||u

∗i ||

2R+

ν−1∑

p=0

|| x pN ||

2P0

≥N∑

i=1

|| x∗i ||2Q + ||u

∗i ||

2R+

ν−1∑

p=0

x pN

2

P0

=N∑

i=1

|| x∗i ||2Q + ||u

∗i ||

2R+ || xN ||

2P0

,

(5.32)

where the second step in (5.32) follows from the Cauchy-Schwartz Inequality (Horn and Johnson (1990)).Therefore, if feasible, the optimal values of the slack state variables are x0,∗

N = x∗N and x p,∗N = 0 for all p 6=0.

This combination is feasible for all x∗N ∈ Ω0∞ and hence for all x∗0 ∈ X

0N . Note that in this case the cost

function (5.32) as well as the state and control constraints become the same as in Tube-Based Robust MPC.Consequently, κ∗N (x) = κ

ipN (x) for all x ∈ X 0

N , i.e. the Tube-Based Robust Model Predictive Controller isrecovered within the set X 0

N . This proves Local Optimality (item 4).

Remark 5.3 (Tube-Based Robust MPC as a special case of Interpolated Tube MPC):Note that the framework of Interpolated Tube MPC also covers standard Tube-Based Robust MPC as a specialcase. In Tube-Based Robust MPC only a single terminal controller is used (ν=1) so that x0

N = xn. All theresults of this chapter and in particular the proof of Theorem 5.2 remain valid.


Possible Performance Degradation of Interpolated Tube MPCOne issue with Interpolated Tube MPC is the potential loss of optimality7 for states far from the origin,caused by the modified terminal cost function and terminal constraint set of the controller. Item 4 ofTheorem 5.2 states that this loss of optimality may only occur outside of X 0

N , the region of attraction of aTube-Based Robust Model Predictive Controller with prediction horizon N that employs K0 as the terminalcontroller. It is again very hard to make general quantitative statements about how the two controllertypes perform in comparison, since this depends of a variety of factors and their interaction (such asthe length N of the prediction horizon, the number ν of terminal controllers, and the actual terminalcontroller gain values Kp). However, the simulations in the case study presented in section 5.5.3 suggestthat the performance loss is possibly very small, at least for the given examples. Nevertheless, a morerigorous analysis will be necessary in order to fully understand to what extent the control performance ofInterpolated Tube MPC may be inferior to that of Tube-Based Robust MPC. For lack of time and space thisanalysis will not be pursued any further in this thesis.

Remark 5.4 (Similarities to other controllers proposed in the literature):Similar ideas for Interpolation-Based Robust MPC have been proposed in the series of papers Sui and Ong(2006, 2007); Sui et al. (2008, 2009, 2010a,b). Despite some similarities, for example the formulation ofthe augmented system for the interpolated terminal controller, the Interpolated Tube MPC approach proposedin this thesis differs distinctly from above contributions (in the following referred to as “Sui’s Controller”) andfeatures some important advantages. The most apparent difference can be found in the cost function and thecontrol law: Sui’s Controller uses a control parametrization of the form ui = K x + ci for the first N−1 timesteps. The ci, which are computed on-line, can be regarded as perturbations to a linear control law u= K x(this is reminiscent of the “closed-loop paradigm” from section 3.5.1). Interpolated Tube MPC insteadassumes that the control inputs ui to the nominal system are solely determined through the optimizationproblem Pip

N (x). By separating the evolution of the (virtual) nominal system from the evolution of the actualsystem, analysis as well as synthesis of the controller is simplified. Specifically, Interpolated Tube MPC onlyinvolves the computation of a positively invariant set for the augmented system (5.6) subject to appropriatelytightened constraints, whereas Sui’s Controller requires the computation of a robust positively invariant set,a task which is significantly more involved. The main difference between the two controllers, however, isthat Interpolated Tube MPC allows the initial state x0 of the nominal system to differ from the actual systemstate x , which is not the case in Sui’s Controller. By doing so, it is possible to prove robust exponential stabilityof the closed-loop system, whereas Sui’s Controller only guarantees robust asymptotic stability. Additionally,the important local optimality property is treated more rigorously in this thesis, yielding the quantitativeresult that local optimality is recovered for all x ∈ X 0

N . Although it is mentioned in more than one of theabove references that Sui’s Controller will asymptotically yield a good control performance, the authors remainvague, in particular they do not identify the region of optimality.

Computational Complexity

One of the main advantages of Tube-Based Robust Model Predictive Control is its rather low computationalcomplexity as compared to other Robust MPC approaches. Bounding the error x− x between actual andnominal system state by a robust positively invariant set E allows the on-line optimization problem of thecontroller to be cast as a Quadratic Program, which can be solved very efficiently. As it turns out, thisimportant feature of Tube-Based Robust MPC can be retained for Interpolated Tube MPC.

Lemma 5.1 (Type of the optimization problem):The on-line optimization problem Pip

N (x) is a convex Quadratic Programming problem.

7 “optimality” in this context means the same performance as standard Tube-Based Robust MPC


Proof. By the assumptions on X, U and W the sets E , X, U and ΩE∞ are all polytopic (see section 5.2.2).

Hence, the optimization problem (5.18) can be written as

V ∗N (x) = minx0,u0,...,uN−1, x0

N ,..., xνN

N−1∑

i=0

|| x i||2Q + ||ui||

2R+

ν−1∑

p=0

|| x pN ||

2Pp

s.t. xk+1 = Axk + Buk for k=0, . . . , N−1

HX x j ≤ kX for j=0, . . . , N−1

HUu j ≤ kU for j=0, . . . , N−1

HΩE∞

x TN ( x

1N )

T. . . ( xν−1

N )TT≤ kΩE

∞

xN =ν−1∑

p=0

x pN

HE (x − x0)≤ kE

(5.33)

where X =

x ∈ Rn | HXx ≤ kX

, U =

u ∈ Rm | HUx ≤ kU

, E =

x ∈ Rn | HE x ≤ kE

andΩE∞ =

x ∈ Rνn | HΩE∞

x ≤ kΩE∞

are the H -representations of the polytopes X and U, E and ΩE∞,

respectively. Now it is easy to see that (5.33) and thus PipN (x) is a convex Quadratic Program.

Remark 5.5 (Redundant constraints):Note that the constraint Φ(N ; x0, u) = xN ∈ X f in (5.17) is redundant and therefore does not need to beaccounted for in (5.33). This is because since ΩE

∞ is convex (a polyhedron) and because X f = Proj x(ΩE∞), it

holds that xN ∈ X f for all x EN ∈ Ω

E∞.

Number of Variables and Constraints in the Optimization ProblemFor the same prediction horizon N , Interpolated Tube Model Predictive Control involves solving anoptimization problem with ν−1 additional scalar variables compared to Tube-Based Robust MPC, suchthat the overall number variables is Nv ar= Nm+ (N + ν)n+ nθ . The number of additional constraints isgenerally hard to determine a priori, since it depends on the complexity of the set ΩE

∞. With NΩ denotingthe number of linear inequalities defining the set ΩE

∞, the overall number of scalar constraints in PipN (x) is

Ncon= N (n+ NX+ NU) + n+ NE + NΩ.

Therefore, for the same prediction horizon, the number of variables and constraints in the InterpolatedTube MPC optimization problem is higher than in the one of Tube-Based Robust MPC. However, theincreased region of attraction XN of the Interpolated Tube Model Predictive Controller allows one tochoose much smaller prediction horizons while obtaining a region of attraction of comparable size. Overall,this can significantly reduce the number of variables and constraints and thus the complexity of the on-lineoptimization. This approach is particularly effective if the constraint polyhedra X and U are complex, i.e.defined by a large number of inequalities. It is however hard to make a general statement about how thecomputational complexities of Interpolated Tube MPC and Tube-Based Robust MPC compare for a similarregion of attraction XN , since this depends on the particular problem setting. For a specific example, acomparison of the complexity of the two controller types is provided in section 5.5, where InterpolatedTube MPC is applied to the output-feedback case study from chapter 4.


5.3.3 Choosing the Terminal Controller Gains Kp

The Number ν of Terminal ControllersOne of the first questions that arises when designing Interpolated Tube Model Predictive Controllers iswhat the optimal number ν of terminal controllers is. Although a single detuned controller in additionto the unconstrained LQR controller K0 is in most cases sufficient to obtain a large enough region ofattraction, this choice is sometimes problematic as it may result in a poor closed-loop performance. Theindisputable benefit of this choice on the other hand is that the resulting optimization problem is of lowcomplexity. An upper bound on ν is given by the lowest number of terminal controllers from which on,for a specified closed-loop performance level, an enlargement of the region of attraction by increasingthe prediction horizon N is computationally cheaper than by adding additional terminal controllers. Formost applications of interest, this number will not be very high. This is because in the employed RecedingHorizon approach, only the first input of the predicted optimal control sequence is actually applied to thesystem, after which the predicted optimal control sequence is recomputed at the next time step. Hence, anincreasingly more exact characterization of the terminal cost function has only little effect on the overallclosed-loop performance. It is therefore reasonable to first start with a small ν (there is no reason for notusing standard Tube-Based Robust MPC (ν=1) if its implementation is feasible), and then add additionalterminal controllers if necessary. See also section 5.4.1 for further information on this issue.

Unconstrained LQR OptimizationIf ν ≥ 2, then the logical next question is how to best choose the additional terminal controller gainsK1, . . . , Kν−1. Since Assumption 5.2 places no other requirements on these controller gains besides stabilityof the closed-loop system, this task offers a great deal of design freedom. One of the easiest ways todetermine the Kp is to use unconstrained LQR controllers designed for different weighting matrices Qpand Rp (a similar idea was pursued by Alvarez-Ramírez and Suárez (1996) in the context of globalstabilization of input constrained discrete-time linear systems). By “increasing” the control weight Rp (or,equivalently, by “decreasing” the state weight Qp), the resulting infinite horizon unconstrained optimalcontrollers can be made less aggressive and can therefore be expected to yield larger maximal positivelyinvariant sets for the closed-loop system. Although it in many cases yields usable results, the main problemwith this approach is that it completely ignores the constraints on the system.

LMI Optimization TechniquesAn alternative, more systematic way to determine the controller gains Kp is to use LMI optimizationtechniques. In this context, Sui et al. (2009) propose solving an LMI optimization problem that maximizesthe volume of the projection of a positively invariant ellipsoid in the x E-space onto the x-space, wherethe controller gains K1, . . . , Kν−1 are regarded as optimization variables (K0 is fixed as the unconstrainedinfinite horizon optimal controller). The resulting optimization problem is, in general, subject to non-convex Bilinear Matrix Inequality (BMI) constraints and hence very hard to solve. However, it can beshown that under some reasonable assumptions, it reduces to a convex optimization problem subjectto LMI constraints, which allows for an efficient solution using standard Semidefinite Programmingsolvers. Although this approach explicitly takes the constraints on the system into account, it relies onellipsoidal arguments and hence is only an approximative tool. For further details consult Sui et al. (2009).

Finally, yet another possibility is to again use LMI optimization techniques, while considering each of theterminal controllers Kp for itself. Specifically, a reasonable thing to do in case of symmetric constraintsis to compute the least aggressive terminal controller Kν−1 as the controller gain that maximizes apositively invariant ellipsoidal set subject to the constraints x ∈ X and Kν−1 x ∈ U. This optimizationproblem can be reformulated as a so-called determinant maximization problem, which can be solvedefficiently using Semidefinite Programming (Boyd and Vandenberghe (2004)). The remaining controllergains

K1, . . . , Kν−2

should then lie “between” the gains K0 and Kν−1. Possible ways to determine these


remaining gains are, for example, to use linear interpolation techniques, or to solve additional LMIoptimization problems with modified constraints.

Obtaining the Cost Matrices PpThe above discussion illustrates that choosing both the number ν and the actual gains K1, . . . , Kν−1 ofthe terminal controllers is a complex problem for itself. Although some initial guidelines can be given,further research efforts will be necessary in order to determine the most effective way to address thistask. In any case, however, once the terminal controller gains K1, . . . , Kν−1 have been determined, thematrices Pp 0 that characterize the (unconstrained) infinite horizon cost of the respective closed-loopsystems x+= (A+ BKp)x need to be computed. This can be performed by solving

(A+ BKp)T Pp(A+ BKp)− Pp +Q+ K T

p RKp = 0, (5.34)

a task for which standard SDP solvers can be employed. Note that since K0 is the (cost-)optimal infinitehorizon controller, it holds that Pp P0 for any Kp 6= K0.

5.3.4 Extensions to Output-Feedback and Tracking MPC

The extension of the Interpolated Tube MPC framework to the output-feedback case is straightforward. Infact, besides adjusting the tightened constraints X and U as discussed in section 4.3, there is no need tomake any other modifications to the controller introduced in section 5.2. The Interpolated Tube MPCcase study that will be presented in section 5.5 also deals with an output-feedback problem, revisiting theoutput-feedback double integrator example from section 4.3.3.

Unfortunately, the situation is much more involved for the tracking controllers from section 4.4. Inparticular, generalizing the “Invariant Set for Tracking” Ωe

t from Definition 4.7 to Interpolated Tube MPCseems hard, since the decomposition of the predicted terminal state xN is performed on-line. Hence, theterminal cost function Vf (·) can not simply be chosen as Vf

xN , xs

= || xN − xs||2P as in Tube-Based Robust

MPC for Tracking. One possible approach to this problem could be to perform an on-line decompositionnot only of the predicted terminal state xN , but also of the artificial steady state xs = Mx θ . A detailedexploration of this issue at this point would however go beyond the scope of this thesis. Further researchwill therefore be necessary in order to understand how interpolation techniques can be used to enlargethe region of attraction of Tube-Based Robust MPC for Tracking.

5.4 Possible Ways of Reducing the Complexity of Interpolated Tube MPC

As pointed out in section 5.3.2, Interpolated Tube Model Predictive Control only involves the solutionof a Quadratic Program and therefore outperforms many other Robust MPC approaches in terms ofcomputational complexity and thus applicability to practical control problems. In order to speed upcomputation and facilitate the application of Interpolated Tube Model Predictive Controllers also to fastsystems with high sampling frequencies, it is nevertheless desirable to further reduce the number ofvariables and constraints in the optimization problem. The following sections briefly address possibleways how this can be achieved.

5.4.1 Reducing the Number of Variables

There are essentially two possible ways to reduce the number of variables in the optimization prob-lem (5.33): One is to shorten the prediction horizon N , the other one is to reduce the number of terminalcontrollers ν . Both lead to a reduced size of the region of attraction XN (under the assumption that


the terminal controller gains remain unchanged). Hence, for a desired minimum size of the region ofattraction, the appropriate balance between N and ν has to be found. Decreasing N requires a larger νand possibly results in a performance degradation, decreasing ν on the other hand entails a smallerterminal set and therefore requires a larger N . In practice, simulations will be necessary in order to find agood tradeoff between N and ν . Another way of reducing complexity while maintaining the size of theregion of attraction XN is to use fewer terminal controllers and leave the prediction horizon untouched,but to detune the remaining controllers such that XN remains sufficiently large. This may however alsoresults in a loss of closed-loop performance, as the required terminal controllers are less aggressive andwill therefore generally yield a higher infinite horizon cost (“larger” matrices Pp).

5.4.2 Reducing the Number of Constraints

For a given prediction horizon N and a given number of terminal controllers ν , the number of constraintsin problem (5.33) is mainly affected by the number of inequalities defining the polyhedral sets E , X, Uand ΩE

∞. Hence, the use of constraint sets of lower complexity seems the most promising approach toreduce the overall complexity of the on-line optimization.

The Robust Positively Invariant Set EOne might speculate that replacing the robust positively invariant set E by a (polyhedral) subset0 ∈ E sub ⊆ E renders the results of Theorem 5.2 valid for a smaller region of attraction X sub

N ⊆ XN .This is however not the case. The problem with this approach lies in the proof of persistent feasibility ofthe resulting closed-loop system: Although in this case the successor state x+ of any feasible x can indeedbe shown to be constraint admissible, the fact that the set E sub is not a robust positively invariant set forthe perturbed closed-loop system does not guarantee the existence of a feasible initial condition x+0 atthe next time step. This is because for some x+∈ X there may not exist an x+0 ∈ X such that x+∈ x+0 ⊕E

sub.

Changing the set E is only reasonable if the alternate choice E al t is also a robust positively invari-ant set for the perturbed closed-loop system x+ = AK x + w. In section 4.5.3 it was stated that anapproximation of the exact mRPI set F∞ can be obtained by scaling the partial sum Fs in (4.123) appro-priately. Clearly, the number of vertices of E grows rapidly with the numbers of summands in (4.123).Choosing a small error bound ε, which generally leads to a high number of summands in (4.123), cantherefore result in an E of very high complexity. Hence, if permitted8 by the size of the constraintsets X and U, a straightforward way of reducing the number of constraints is to use a less strict errorbound ε for the computation of E . The drawback of doing this is that, as the resulting tightened constraintsets X= XE and U= UKE will be smaller, it will in turn lead to a degradation in control performance.

It seems as if the computation of a sufficiently tight, yet at the same time sufficiently simple robustpositively invariant set for the perturbed closed-loop system x+= AK x+w is the major obstacle in makingTube-Based or Interpolated Tube MPC easily applicable also to higher-dimensional systems. Evidently, atight approximation and a simple representation are contrary requirements. Nevertheless, it is reasonableto suspect that improved (possibly far more complex) algorithms will allow for the computation of lesscomplex robust invariant sets. The complexity of these potential improved algorithms is less an issue,since the computation of the RPI set is performed off-line and hence does not affect on-line performance.

The Tightened State and Control Constraint SetsUnlike with E , it is possible to replace the sets X and U by subsets 0 ∈ Xsub ⊆ X and 0 ∈ Usub ⊆ U,respectively, without jeopardizing the results of Theorem 5.2. However, the sets X and U will usually beof comparably low complexity anyway, so that replacing them by simpler subsets will not simplify the

8 in the sense that the resulting tightened constraint sets X and U exist and contain the origin

5.4. Possible Ways of Reducing the Complexity of Interpolated Tube MPC 119

optimization problem much. This is because the original constraint sets X and U themselves are usuallyof comparably low complexity. Furthermore, due to the nature of the Pontryagin Set Difference operation(see Definition 4.4), the number of defining inequalities of the sets X = XE and U = U KE can beshown to be less or equal than the number of defining inequalities of X and U (Borrelli et al. (2010)).Underapproximating the sets X and U does therefore not seem to be a very promising approach.

The Terminal Constraint SetAnother possibility to reduce the number of constraints in the optimization problem lies in replacingthe terminal set ΩE

∞ in the augmented state space by a polytopic subset 0 ∈ ΩE,sub∞ ⊆ ΩE

∞. Althoughthis immediately results in a smaller region of attraction of the overall Model Predictive Controller, itnevertheless is a reasonable thing to do in case the set ΩE

∞ is of high complexity and can be under-approximated sufficiently close by a much simpler polytope ΩE,sub

∞ .

5.5 Case Study: Output-Feedback Interpolated Tube MPC

Consider again the output-feedback double integrator example from section 4.3.3. In order to illustratethe benefits of the proposed Interpolated Tube MPC approach, the following sections will compare twodifferent output-feedback Interpolated Tube Model Predictive Controllers with the previously designedoutput-feedback Tube-Based Robust Model Predictive Controller from section 4.3.3. Revisiting the othercase studies will be omitted for brevity. Section 5.5.1 briefly recalls the example system and the controlproblem. Section 5.5.2 will then compare the regions of attraction of the different controllers. Finally,section 5.5.3 will provide a benchmark comparison of the controllers’ on-line computational complexityas well as their control performance.

5.5.1 Problem Setup and Controller Design

The system dynamics of the considered double integrator example are

x+ =

1 10 1

x +

11

u+w

y =

1 1

x + v ,(5.35)

with state and control constraints X =

x | −50 ≤ x i ≤ 3, i=1, 2

and U =

u | |u| ≤ 3

, respectively.The disturbances w and v are bounded by the sets W =

w | ||w||∞ ≤ 0.1

and V =

v | |v | ≤ 0.05

,respectively. The weighting matrices in the cost function are given by Q= I2 and R=0.01. The samedisturbance rejection controller and observer gains as in section 4.3.3 were used, namely K=[−0.7 −1.0]and L = [1.00 0.96]T. Note that all of the above parameters have been summarized in Table 4.1 onpage 101 in the previous chapter.

For comparison with the standard Tube-Based Robust Model Predictive Controller (Controller A) withprediction horizon NA= 13 from , two Interpolated Tube Model Predictive Controllers (Controllers Band C) were designed for system (5.35). Controller B uses νB=2 terminal controllers and a predictionhorizon of NB=6, while Controller C uses νC=3 terminal controllers and a prediction horizon of NC=4.The infinite horizon unconstrained optimal controller K0 was designed for the specified weightingmatrices Q = I2 and R = 0.01. Following the simple LQR approach from section 5.3.3, the terminalcontrollers of Controller B and C were computed as the infinite horizon unconstrained optimal controllersfor the modified input weights RB,1 = 10, RC ,1 = 1 and RC ,2 = 100, respectively. An overview of therespective prediction horizons and terminal controller gains for the three Interpolated Tube ModelPredictive Controllers is provided by Table 5.1 on page 123.


5.5.2 Comparison of the Regions of Attraction

The terminal sets XAf , XB

f and XCf and the corresponding regions of attraction9 X A

N , X BN and X C

N ofthe synthesized Interpolated Tube Model Predictive Controllers are depicted in Figure 5.1. In addition,Figure 5.1 also contains the regions of optimality X B

opt and X Copt for Controller B and C, respectively (the

region of optimality of Controller A is its entire region of attraction X AN ).

Figure 5.1.: Terminal Sets and Regions of Attraction for Interpolated Tube MPC Controllers A, B and C

It is easy to see from Figure 5.1 that the size of the terminal set X f depends strongly on the employedterminal controller(s). For example, the terminal set XB

f of Controller B is more than three times the sizeof XA

f , the terminal set of the standard Tube-Based Robust Model Predictive Controller from section 4.3.3.The terminal set XC

f is even larger, covering more than ten times the area of XAf . As a result, even for

the significantly reduced prediction horizons NB=6 and NC=4, the regions of attraction X BN and X C

N ofController B and C are comparable to that of Controller A in terms of size. In fact, it is possible for thisexample to design an Interpolated Tube Model Predictive Controller with a prediction horizon of N=1 oreven N=0, and still obtain a region of attraction as large as the one of the standard Tube-Based RobustModel Predictive Controller with prediction horizon N=13. This is remarkable, especially when takinginto account that it has been shown in Theorem 5.2 that local optimality (i.e a performance identical tothat of Tube-Based Robust MPC) is guaranteed within the set X 0

N (which is equal to Ω0∞ if N=0) for any

Interpolated Tube Model Predictive Controller.

The shapes of the respective terminal sets XAf , XB

f and XCf are fairly similar, this can be traced back to the

fact that all of the terminal controller gains were determined by unconstrained LQR optimization, wherefor simplicity only scaled versions of the original weighting matrices Q and R were used.

9 the regions of attraction are defined for the state estimates, since the actual value of the current state is unknown

5.5. Case Study: Output-Feedback Interpolated Tube MPC 121

Remark 5.6 (Zero prediction horizon):A prediction horizon of N=0 means that no prediction of the system’s future evolution is performed in theoptimization problem. Nevertheless, the on-line computation still involves solving a Quadratic Program. Thisis because the decomposition (5.4) of the initial state x0 of the nominal system (which for N=0 is at the sametime the terminal state) needs to be performed on-line such that the terminal cost in (5.33) is minimized.Interpolated Tube MPC for a prediction horizon of N=0 essentially degenerates to a controller similar to theone proposed in Pluymers et al. (2005c).

From the above discussion it is evident that the newly proposed Interpolated Tube MPC approach isindeed capable of considerably reducing the necessary prediction horizon for a given size of the regionof attraction of the controller. Equivalently, the region of attraction can be significantly enlarged for aspecified prediction horizon when an Interpolated Tube Model Predictive Controller is used. What is leftto investigate is how the modified controller structure affects the closed-loop control performance, andfurthermore how the on-line computational complexity of Interpolated Tube MPC compares to that ofstandard Tube-Based Robust MPC.

5.5.3 Computational and Performance Benchmark

In this section, the same benchmark scenario as in section 4.6.1 is used in order to illustrate the efficiencyof Interpolated Tube MPC. For a simulation horizon of Nsim=15, the closed-loop system for each of thethree controllers from Table 5.1 was simulated for the same 100 randomly generated initial conditionsscattered over X B

N (this choice is due to the fact that X BN is the smallest of the three regions of attraction,

see Figure 5.1). The disturbance sequences w and v were thereby generated by random, time-varyingdisturbances w and v , uniformly distributed on W and V , respectively. To ensure comparability of theclosed-loop robust performance, the sequences w and v were the same for all three controllers fromTable 5.1. For every controller, the on-line computation time for each of the 1500 single solutions ofthe optimization problem was determined. From this data, the minimal (tmin), maximal (tmax), andaverage (tav g) computation time was determined and is reported in Table 5.2. As in section 4.6.1, thiswas performed for both on-line and explicit controller implementations for the same initial conditionsand disturbance sequences. Table 5.2 also contains, for each of the three controllers, the number ofvariables Nv ar and constraints Ncon in the optimization problem and the number Nreg of regions overwhich the explicit solution is defined. Figures 5.2–5.4 on page 125 show the polyhedral partitions of thedifferent explicit solutions. The machine used for the benchmark was the same as in section 4.6.1, namelya 2.5 Ghz Intel Core2 Duo running Matlab R2009a (32 bit) under Microsoft Windows Vista (32 bit) Theon-line controller was again based on the interior-point algorithm of the QPC-solver (Wills (2010)), andthe evaluation of the explicit control law was performed in Matlab.

Remark 5.7 (Presentation of the simulation results):It will become evident in the following that the control performance of the three different controllers is almostidentical. Hence, it does not make much sense to present plots of the simulated state trajectory and controlinputs, as those would essentially be the same as the ones in Figure 4.9 and Figure 4.10 from the case studyin section 4.6.1. What is more interesting in this context is how the different controllers perform in terms ofcomputational complexity and on-line evaluation time.

Speed of On-Line ComputationThe on-line computation times for the three different controllers from section 5.5.1 are summarized inTable 5.2, where both on-line and explicit controller implementation have been considered. Note thatthe discrepancy between the computation times for Controller A reported in Table 5.2 and Table 4.2,respectively, is caused by the different set of possible initial conditions. Furthermore, the reason why thenumber of regions Nreg over which the explicit version of Controller A is defined differs between the twotables is that in this benchmark, the exploration region Xx pl for all three controllers was chosen as X B

N ,


Tabl

e5.

1.:P

aram

eter

soft

heIn

terp

olat

edTu

beM

odel

Pred

ictiv

eCo

ntro

llers

νN

Con

trol

ler

Gai

nsC

ontr

olle

rA

113

K0=[−

0.61

4−

0.99

6]C

ontr

olle

rB

26

K0=[−

0.61

4−

0.99

6]K

1=[−

0.20

5−

0.57

8]C

ontr

olle

rC

34

K0=[−

0.61

4−

0.99

6]K

1=[−

0.42

2−

0.82

2]K

2=[−

0.08

0−

0.36

9]

Tabl

e5.

2.:C

ompu

tatio

nalC

ompl

exity

ofIn

terp

olat

edTu

beM

PC

Nv

arN

con

Nre

gO

n-lin

eC

ontr

olle

raEx

plic

itC

ontr

olle

rm

ax

J−J A

J A

∅

J−J A

J A

to min/

ms

to avg/

ms

to ma

x/

ms

tx min/

ms

tx avg/

ms

tx ma

x/

ms

Con

trol

ler

Ab

4111

715

34.

55.

318

.75.

35.

410

.40

0C

ontr

olle

rB

2276

138

2.3

2.8

12.9

4.9

5.0

8.4

0.00

34%

0.00

02%

Con

trol

ler

C18

7998

2.2

2.6

8.5

4.0

4.0

6.5

0.21

98%

0.00

92%

aus

ing

the

inte

rior

-poi

ntal

gori

thm

ofth

eQPC-s

olve

r(W

ills

(201

0))

bth

edi

ffer

ence

sin

the

repo

rted

com

puta

tion

tim

esfo

rth

est

anda

rdTu

be-B

ased

cont

rolle

rbe

twee

nTa

ble

5.2

and

Tabl

e4.

2ar

edu

eto

the

diff

eren

tse

tof

init

ialc

ondi

tion

s;th

edi

ffer

ence

inth

enu

mbe

rof

regi

ons

over

whi

chC

ontr

olle

rA

isde

fined

isbe

caus

eof

the

diff

eren

tex

plor

atio

nre

gion


the region of attraction of Controller B. As to the number of variables and constraints in the on-lineoptimization problem, Table 5.2 shows that the complexity of the Interpolated Tube Model PredictiveControllers B and C is significantly lower than that of Controller A, the standard Tube-Based Robust ModelPredictive Controller. It however also stands out that although the prediction horizon of Controller C isonly two thirds of the one of Controller B, the reduction in the number of variables from Controller Bto Controller C is only minor, while the number of constraints actually grows. This is due to the factthat with each additional terminal controller, the dimension of the augmented system (5.6) and hencealso of the terminal set ΩE

∞ is increased by n, which generally results in maximal positively invariantsets of higher complexity. It is therefore necessary to find a tradeoff between the number ν of terminalcontrollers and the prediction horizon N .

Table 5.2 furthermore reports that the average computation time toav g for the on-line implementation

of Controller B is about 53% of that for Controller A. For Controller C, it is only about 49% of that forController A. The maximal and minimal computation times to

min and tomax have also been significantly

reduced for Controllers B and C. Moreover, it can be observed that the evaluation times t xmin, t x

av g and t xmax

for the explicit implementations of Controller B and C, respectively, have also been reduced in comparisonto those for Controller A. The effect is however not as pronounced as for the on-line implementation of thecontrollers. From these simulation results it is evident that, at least for the given example, InterpolatedTube MPC is clearly superior to standard Tube-Based Robust MPC in terms of computational efficiency. Itis conjecturable that this is not an exceptional case but rather a general feature of Interpolated Tube MPCand holds true for a wide range of systems.

Control PerformanceAlthough Theorem 5.2 guarantees local optimality for all three controllers from section 5.5.1, it is notobvious how the reduced prediction horizons of Controller A and B together with the modified terminalcost affect the performance of Interpolated Tube MPC during transients. Fortunately, it turns out that, atleast for the example considered in this case study, there is hardly any measurable difference between thecost of the closed-loop trajectories resulting from the three different controllers.

Consider to this end the cost J(x,u) :=∑Nsim

i=0 ||x(i)||2Q + ||u(i)||

2R of a state trajectory x driven by the

control sequence u, where Nsim denotes the simulation horizon. The last two columns of Table 5.2 containthe maximal and the average relative difference between J(x,u) and the cost JA(xA,uA) of the “optimaltrajectory” (the one resulting from Controller A) over all 100 initial conditions. Motivated by Figure 4.9,only the first Nsim=9 states and control inputs were considered in determining J(x,u). A maximal relativedifference in the trajectory cost of less than 0.01% for Controller B, and of about 0.2% for Controller Cshows that the performance loss is negligible. Although, strictly speaking, these results are valid only forthe specific example considered in this case study, it is reasonable to assume that for non-zero predictionhorizons and a reasonable choice of the terminal controllers Kp the control performance of InterpolatedTube MPC will generally be close to that of Tube-Based Robust MPC.

Remark 5.8 (Dependency of the result on the initial conditions):One might argue that since the initial conditions are scattered randomly over X B

N , in a number of instancesthey will be contained in the regions of optimality X B

opt and X Copt for Controller B and C, respectively. However,

of the 100 random initial conditions used in the benchmark, there were more than 30% contained neitherin X B

opt nor in X Copt and hence indeed affected the maximal relative difference between J(x,u) and JA(xA,uA).

The local optimality property of Interpolated Tube MPC can also be demonstated graphically. Consider tothis end the polyhedral partitions of the three different controllers depicted in Figure 5.2–5.4 on page 125,where Figure 5.2 shows the partion of the “optimal” Tube-Based Robust Model Predictive Controller A,which uses the unconstrained LQR controller as the only terminal controller. Close to the origin, it is easy


Figure 5.2.: Polyhedral Partition of the Explicit Solution for Controller A (153 regions)

Figure 5.3.: Polyhedral Partition of the Explicit Solution for Controller B (138 regions)

Figure 5.4.: Polyhedral Partition of the Explicit Solution for Controller C (98 regions)


to see that the three different partitions are more or less identical (it has been verified that also the PWAcontrol laws in these regions are almost indistinguishable). The farther away from the origin, however,the coarser the partitions of the Interpolated Tube Controllers B and C become. This illustrates the gradualloss of optimality for states outside the respective regions of optimality X B

opt and X Copt . In this context

it is hard to say why the partition of Controller B in Figure 5.3 is acually more refined close to the leftboundary of the feasible set. Further inverstigations into the interrelation between the different designparameters and properties of the Interpolated Tube Model Predictive Controller are therefore necessary.


6 Conclusion and OutlookCommon Misconceptions of Model Predictive ControlEven today, many researchers misconceive Model Predictive Control as “tinkering with some optimizationalgorithms” without any formal stability guarantees. As the previous chapters have shown, this is anantiquated belief. Modern MPC approaches are based on sound theoretical foundations, and especiallythe theory of nominal Model Predictive Control for constrained linear systems can be regarded as more orless mature. The most important hurdle that linear MPC still has to take lies therefore not in theory, but inimplementation. Undoubtedly, the on-line complexity of Model Predictive Controllers is disproportionatelyhigh in comparison to that of other classic control approaches. While this is less an issue for applicationsthat are characterized by slow dynamics (such as in process control, the traditional domain of MPC), itcertainly becomes a limitation when applying MPC to fast systems. One of the main problems thereby isnot so much the actual achieved speed itself, but that it is usually very hard to find reasonable bounds onthe computation times. Although worst-case bounds can be obtained, they are generally much too high.This makes it hard to give formal guarantees for stability of the closed-loop system, which is of particularimportance in safety-critical applications. However, the rapid progress in both software and hardwaretechnology will continue to push these boundaries and permit the application of Model Predictive Controlto an increasingly wider range of problems.

Robust MPC as a Universal Framework for Constrained Robust Optimal ControlIt is a widely accepted consensus that controllers need not only guarantee performance under nominaloperating conditions, but must also provide satisfactory robustness to uncertainty, which may arise inform of modeling errors or exogenous disturbances. Established Robust Control methods, however,generally do not take constraints into account explicitly, which renders their application to constrainedsystems problematic and at best very conservative. Robust MPC fills this existing void by providingconstructive methods for synthesizing non-conservative controllers that ensure both robust stability aswell as robust constraint satisfaction for the uncertain closed-loop system. Being based on optimal controlideas, the controller behavior can be adequately influenced by appropriately choosing the weights inthe cost function. Because of these features, Robust MPC can be regarded as the only real universalframework for robust optimal control of constrained systems. One of the most promising among the manyexisting Robust MPC methods is Tube-Based Robust MPC, which sparked notable interest only recently. Byseparating the problem of robustness from the problem of constrained optimal control, Tube-Based RobustMPC finds a balance between good closed-loop performance and sufficiently low on-line complexity.

Tube-Based Robust Model Predictive ControlTube-Based Robust MPC is an approach that is appealing for various reasons. First and foremost, its coretheoretical result of robust exponential stability of an invariant set is unusually strong compared to otherRobust MPC approaches for systems subject to bounded additive disturbances. Moreover, the structureof the controller allows to easily extend the framework also to output-feedback and tracking problems,which make it a very versatile tool in constrained robust optimal control. The main benefit of Tube-BasedRobust MPC, however, is its simple on-line computation, which only involves solving a Quadratic Program.Optimization algorithms for Quadratic Programming have been developed to a mature level, so that thiscomputation can be performed on-line fast and reliably. In addition, the structure of the optimizationproblem also permits the computation of an explicit control law, by which the on-line effort can be reducedto a simple function evaluation. The benchmark results provided in this thesis show that both on-line andexplicit implementation have the potential to be applied to fast-sampling systems.

127

As any other approach, Tube-Based Robust MPC nonetheless also has some shortcomings. Maybe themost important one pertains to the computation of the approximated minimal robust positively invariantset, the “cross-section” of the tube of trajectories. This computation may in some cases yield polytopicapproximations of considerable complexity and, at least in the current implementation, seems prone tonumerical problems. Hence, for a wider application of Tube-Based Robust MPC, improvements in thiscomputation are essential. Moreover, the off-line computation of tightened constraints limits the possibilityof re-parametrizing the controller on-line, since any change of constraints or disturbance bounds, strictlyspeaking, requires to recompute the tightened constraint sets and the terminal set. Another potentialrestriction of Tube-Based Robust MPC is that, in some cases, the region of attraction of the controller isgrowing only slowly with the prediction horizon, resulting in controllers of rather high complexity whenrequiring a large region of attraction. One way to address this issue is to find ways to enlarge the terminalset. This is where newly proposed Interpolated Tube MPC framework comes into play.

Interpolated Tube Model Predictive ControlThe Interpolated Tube Model Predictive Control framework as the main contribution of this thesis isessentially an extension of Tube-Based Robust MPC, and hence inherits all its beneficial properties (and,unfortunately, also its shortcomings). In particular, robust exponential stability of an invariant set androbust feasibility have been proven for the closed-loop system. By employing a nonlinear terminalcontroller based on interpolation techniques, Interpolated Tube MPC achieves a significantly larger regionof attraction for a comparable on-line complexity of the controller. Equivalently, for a desired region ofattraction it is possible to design controllers with reduced on-line complexity. Since the optimizationproblem is again a Quadratic Program, it is furthermore possible to implement Explicit Interpolated TubeModel Predictive Controllers. Moreover, it has been shown that despite the modified terminal controller,Interpolated Tube MPC retains local optimality, i.e. it achieves the same closed-loop performance as Tube-Based Robust MPC in a region around the origin. A straightforward extension of Interpolated Tube MPCto the output-feedback case, i.e. when the exact state of the system is unknown, is possible. Unfortunately,an extension of the proposed controller to facilitate tracking of piece-wise constant references by usingthe methods from Tube-Based Robust MPC for Tracking on the other hand does not seem obvious. Thesuperiority of Interpolated Tube MPC over Tube-Based Robust MPC regarding its on-line computationalcomplexity has been demonstrated in a computational benchmark.

Software ImplementationAnother major part of this work has been the implementation of the discussed Tube-Based Robustand Interpolated Tube Model Predictive Controllers in software. The essential task thereby was toformulate the respective on-line optimization problems of the controllers, given the parameters andconstraints. Advanced optimization toolboxes, which provide a higher-level interface to the actualsolvers engines, were employed for this purpose. Formulating the optimization problem however alsorequired the implementation of some auxiliary algorithms, e.g. for computing an approximate minimalrobust positively invariant set (Algorithm 2), the maximal positively invariant set (Algorithm 1) or anoptimized disturbance rejection controller (optimization problem (4.120)). The developed softwareframework was written in Matlab and interfaces various toolboxes and solvers (MPT-toolbox (Kvasnicaet al. (2004)), Yalmip (Löfberg (2004))), CDD (Fukuda (2010)) and QPC Wills (2010)). The sourcecode can be downloaded from http://www.fsr.tu-darmstadt.de. If properly referenced1, the authorencourages its use for further projects.

1 M. Balandat. Constrained Robust Optimal Trajectory Tracking: Model Predictive Control Approaches. Diploma thesis,Technische Universität Darmstadt, 2010.

128 6. Conclusion and Outlook

http://www.fsr.tu-darmstadt.de

Experimental Evaluation of Interpolated Tube MPCThe next step in the evaluation of the practical applicability of Interpolated Tube MPC would be theactual experimental test outside the deceptive world of simulations. Since one of the major benefitsof the proposed approach is its potential speed, the author suggests the application of the controllerto a plant with “fast” dynamics, i.e. a mechatronic system. Because of the (still) existing (off-line)computational issues mentioned above, suitable systems should be of rather low complexity (a statedimension of 2-5 seems reasonable). Furthermore, it is of importance that the system is not stiff, i.e.that its time constants do not differ greatly (in the order of several magnitudes) from each other. This isbecause otherwise, in order to suitably capture the behavior of the slow system dynamics, the predictionhorizon would have to be very high, provided that the sampling frequency is chosen sufficiently high toaccount for the fast system dynamics. If the system does have this property, it is recommendable that,whenever possible, the fast dynamics are controlled in a subordinate control loop. For this purpose eithersimpler controllers (e.g. PID-controllers) or again fast MPC can be employed. The resulting closed-loopdynamics of the subordinate control loop are then regarded as the plant by the top-level Interpolated TubeModel Predictive Controller, where possible uncertainties can be modeled as virtual bounded additivedisturbances.

The implementation of the controller can be realized either on a standard Desktop Computer in a“hardware in the loop” setting, or on an embedded system platform. In the latter case, because ofthe different hardware environment and the generally lower computational power, the code must becompletely rewritten with a focus on efficiency. This can be expected to yield significant improvementsin terms of computation speed especially for the explicit implementation of the controller, for which inthe previous benchmarks the evaluation of the piece-wise affine control law was performed in Matlab.Existing quadratic programming solvers can be compiled for the respective environments to enable afast on-line implementation. Implementing this kind of testbed would help to identify the central issuespertaining to an actual controller implementation that can not be adequately understood in simulations.

Promising Research DirectionsAs has been remarked, the extension of the Interpolated Tube MPC framework to the tracking case isnot immediately obvious. It does however not seem a hopeless endeavor either. Therefore, furtherresearch efforts should aim at developing Interpolated Tube Model Predictive Controllers for Tracking.Another task of equal importance is to find smart ways to determine both the number and the valuesof the additional terminal controller gains such that a good tradeoff between the size of the region ofattraction, the complexity of the controller and the closed-loop performance is achieved. For a moreefficient implementation of the controller, it would furthermore be enticing to investigate the practicalityof suboptimal explicit solutions with a reduced number of regions, or also of hybrid approaches, whichcombine coarse explicit solutions with on-line optimization.

Recently, there has been an increasing research effort to develop a better understanding of the theoryof set invariance (Rakovic and Baric (2009, 2010); Artstein and Rakovic (2008, 2010); Rakovic et al.(2007); Rakovic (2007)). From these results, an extension of Tube-Based Robust MPC has been developed.This “Homothetic Tube Model Predictive Control” approach (Rakovic et al. (2010)) uses the same ideas asTube-Based Robust MPC, but allows the cross-section of the “tube of trajectories” to vary with time. Thisallows a less conservative on-line constraint handling, performed only locally with respect to the actualprocess, while yielding only a modest increase in computational complexity as compared to standardTube-Based Robust MPC. The idea of tubes has also been addressed in the context of nonlinear systems(Mayne and Rakovic (2003); Rakovic et al. (2006); Mayne and Kerrigan (2007); Cannon et al. (2010a))and system subject to stochastic disturbances (Cannon et al. (2009, 2010b); Kouvaritakis et al. (2010)).It will be interesting to see how these approaches can contribute to the further development of RobustNonlinear MPC and Stochastic MPC.

129

The Long-Term Potential of Interpolated Tube MPCDue to its rather involved controller design, Interpolated Tube MPC can not be expected to become the“swiss army knife of controls”, as for example PID controllers are today. The approach is simply toocomplex and not easy enough to grasp for application engineers to be a “plug and play” tool for everydaycontrol problems in industry. But when the plant is subject to hard constraints on state and control input,and when both good performance and robustness in the presence of uncertainty are key requirements,Interpolated Tube MPC is a viable and promising control approach. The computational issues pertainingto the on-line optimization are largely solved and will further diminish considering the rapid progress inoptimization algorithms and computer technology.

130 6. Conclusion and Outlook

A Appendix

A.1 Proof of Theorem 2.1

As most other proofs of MPC stability in the literature, the proof of Theorem 2.1, the main stabilitytheorem for nominal Model Predictive Control for linear systems, is based on Lyapunov arguments.Consider to this end the following Lemma:

Lemma A.1 (Optimal cost decrease along the trajectory, Rawlings and Mayne (2009)):Suppose that Assumption 2.2 is satisfied. Then,

V ∗N (Ax + BκN (x))≤ V ∗N (x)− l(x ,κN (x)), ∀x ∈ XN . (A.1)

Proof. For any state x ∈ XN the optimal cost is V ∗N (x) = VN (x ,u∗(x)), where the sequence of predictedoptimal control inputs u∗(x) is

u∗(x) =

u∗0(x), u∗1(x), . . . , u∗N−1(x)

, (A.2)

so that, in virtue of the Receding Horizon approach, κN (x) = u∗0(x). Obviously, the sequence (A.2) isfeasible for PN (x) (by construction). The associated predicted optimal state trajectory is

x∗(x) =

x , x∗1(x), . . . , x∗N (x)

, (A.3)

where x∗1(x) = Ax + BκN (x) and x∗N (x) ∈ X f (also by construction). Due to the lack of uncertainty in thenominal case the successor state is x+ = x∗1(x). At the next time step, consider a (possibly non-optimal)control sequence u∗ given by

u∗ =

u∗1(x), u∗2(x), . . . , u∗N−1(x), u

, (A.4)

where the last element u ∈ U is still to be chosen. The control sequence u∗ is feasible, but not necessarilyoptimal for the problem PN (x

∗1(x)) if the last element of the associated predicted state trajectory

x∗ =

x∗1(x), x∗1(x), . . . , x∗N (x), Ax∗N (x) + Bu

(A.5)

satisfies Ax∗N (x)+ Bu ∈ X f . Provided now that for all x ∈ X f there exists a u ∈ U such that Vf (Ax+Bu)≤Vf (x)− l(x , u) and Ax+Bu ∈ X f , then

V ∗N (Ax + BκN (x))≤ V ∗N (x)− l(x ,κN (x)), ∀x ∈ XN , (A.6)

which was the claim in Lemma A.1. Clearly, under Assumption 2.2 this requirement on u is satisfied forall x ∈ X f if u is chosen as u= κ f (x).

The proof of Theorem 2.1 will in the following distinguish between the case when the set1 XN is boundedand the case when it is unbounded.

1 strictly speaking, XN can only be called “region of attraction” if stability of the closed-loop system for all x ∈ XN has beenproven. In case XN is unbounded technicalities in the proof only allow to show stability for any bounded x ∈ XN

131

Case 1: XN bounded (Rawlings and Mayne (2009))Proof. Consider first the case when the set XN is bounded. With quadratic stage cost function (2.4) andterminal cost function (2.5), respectively, and with Assumption 2.2 satisfied, there exist constants α > 0and β <∞ such that for all x ∈ XN the value function satisfies

V ∗N (x)≥ α||x ||2 (A.7)

V ∗N (x)≤ β ||x ||2 (A.8)

V ∗N (Ax + BκN (x))≤ V ∗N (x)−α||x ||2. (A.9)

Consider any initial state x(0) ∈ XN and denote by x(i) the solution of the system x+= Ax + BκN (x)with initial condition x(0). From the proof of Lemma A.1 it is clear that XN is positively invariant for thesystem x+ = Ax + BκN (x), hence the entire sequence

x(0), x(1), . . .

is contained in XN , provided thatthe initial state x(0) is. The states x(i), for all i ≥ 0, satisfy

V ∗N (x(i+ 1))≤ V ∗N (x(i))−α||x(i)||2 ≤ (1−α/β)V ∗N (x(i)), (A.10)

from which it follows that

V ∗N (x(i))≤ δi V ∗N (x(0)) (A.11)

for all i ≥ 0 with δ := (1−α/β) ∈ [0,1]. Consequently,

||x(i)||2 ≤ (1/α)V ∗N (x(i))≤ (1/α)δi V ∗N (x(0))≤ (β/α)δ

i ||x(0)||2 (A.12)

and, with γ := β/α,

||x(i)||2 ≤ γδi ||x(0)||2, ∀ x ∈ XN ∀ i ≥ 0. (A.13)

Because x(i) ∈ XN for all i ≥ 0 the origin is exponentially stable with a region of attraction XN .

Case 2: XN unbounded (Rawlings and Mayne (2009))Proof. Consider now the case when the set XN is unbounded. From (A.7) it follows that any sublevel setof V ∗N (·) is bounded. Moreover, from (A.9) it follows that any sublevel set is also positively invariant forthe system x+ = Ax + BκN (x). Hence, the origin is exponentially stable with a region of attraction equalto any sublevel set of V ∗N (·).

A.2 Definition of the Matrices Mθ and Nθ in Lemma 4.1

Lemma 4.1 states that if the system is stabilizable, then a pair (zs, ys) is a solution to the equationEzs = F ys if and only if there exists a vector θ ∈ Rnθ such that

zs = Mθθ

ys = Nθθ .(A.14)

Consider the minimal singular value decomposition of E, i.e. E = UΣV T , where U ∈ R(n+p)×r , Σ ∈ Rr×r

is a non-singular diagonal matrix and V ∈ R(n+m)×r such that U and V are unitary (U T U=Ir , V T V=Ir).Furthermore, denote by V⊥ a matrix such that V T V⊥=0 and the matrix [V, V⊥] is a non-singular squarematrix. Then, the matrices Mθ and Nθ are given by (Alvarado (2007))

Mθ =

(

VΣ−1U T FG V⊥

if r < n+m

VΣ−1U T FG if r = n+m(A.15)

Nθ =

(

G 0p,n+m−r

if r < n+m

G if r = n+m(A.16)

where

G =

¨

Ip if r = n+ p

(F T U⊥)⊥ if r < n+ p(A.17)

132 A. Appendix

List of Figures1.1. Receding Horizon Control (Bemporad and Morari (1999)) . . . . . . . . . . . . . . . . . . . . 2

2.1. Reference governor block diagram (Gilbert and Kolmanovsky (2002)) . . . . . . . . . . . . . 16

3.1. LFT feedback interconnection (Zhou et al. (1996)) . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1. The “tube” of trajectories in Tube-Based Robust MPC . . . . . . . . . . . . . . . . . . . . . . . 544.2. Regions of attraction of the controller for different prediction horizons . . . . . . . . . . . . 584.3. Trajectories for the state-feedback Tube-Based Robust MPC Double Integrator example . . 594.4. Control inputs for the state-feedback Tube-Based Robust MPC Double Integrator example 594.5. Actual state and optimal initial state of the nominal system . . . . . . . . . . . . . . . . . . . 604.6. Explicit state-feedback Tube-Based Robust MPC: PWQ value function V ∗N (x) over polyhedral

partition of the set XN ∩Xx pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.7. Explicit state-feedback Tube-Based Robust MPC: PWA optimizer function u∗0(x) . . . . . . . 624.8. The “tube” of trajectories in output-feedback Tube-Based Robust MPC . . . . . . . . . . . . . 684.9. Trajectories for the output-feedback Tube-Based Robust MPC Double Integrator example . 704.10.Control inputs for the output-feedback Tube-Based Robust MPC Double Integrator example 704.11.Explicit output-feedback Tube-Based Robust MPC: PWQ value function V ∗N ( x) over polyhe-

dral partition of the set XN ∩Xx pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.12.Explicit output-feedback Tube-Based Robust MPC: PWA optimizer function u∗0( x) . . . . . . 724.13.Trajectories for state-feedback Tube-Based Robust MPC for Tracking example . . . . . . . . 804.14.Control inputs for state-feedback Tube-Based Robust MPC for Tracking example . . . . . . 804.15.Trajectories for output-feedback Tube-Based Robust MPC for Tracking example . . . . . . . 844.16.Control inputs for output-feedback Tube-Based Robust MPC for Tracking example . . . . . 854.17.Comparison of standard and offset-free tracking controller . . . . . . . . . . . . . . . . . . . . 87

5.1. Terminal Sets and Regions of Attraction for Interpolated Tube MPC Controllers A, B and C 1215.2. Polyhedral Partition of the Explicit Solution for Controller A (153 regions) . . . . . . . . . . 1255.3. Polyhedral Partition of the Explicit Solution for Controller B (138 regions) . . . . . . . . . . 1255.4. Polyhedral Partition of the Explicit Solution for Controller C (98 regions) . . . . . . . . . . . 125

133

List of Tables4.1. Design Parameters used for the Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2. Computational Benchmark Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1. Parameters of the Interpolated Tube Model Predictive Controllers . . . . . . . . . . . . . . . 1235.2. Computational Complexity of Interpolated Tube MPC . . . . . . . . . . . . . . . . . . . . . . . 123

135

BibliographyA. Alessio and A. Bemporad. A Survey on Explicit Model Predictive Control, volume 384/2009 of Lecture

Notes in Control and Information Sciences, pages 345–369. Springer Verlag, Berlin, 2009. 4, 16, 17, 18,19

A. Alessio, M. Lazar, A. Bemporad, and W. Heemels. Squaring the circle: An algorithm for generatingpolyhedral invariant sets from ellipsoidal ones. Automatica, 43(12):2096–2103, Dec 2007. 98

I. Alvarado. Model Predictive Control for Tracking Constrained Linear Systems. PhD thesis, Universidad deSevilla, 2007. 48, 73, 74, 77, 78, 82, 86, 90, 92, 99, 132

I. Alvarado, D. Limon, T. Alamo, and E. Camacho. Output feedback Robust tube based MPC for tracking ofpiece-wise constant references. In Proceedings of the IEEE Conference on Decision and Control, volume 12,pages 2175–2180, New Orleans, USA, Dec 2007a. 47, 48, 73, 82, 83, 86

I. Alvarado, D. Limon, T. Alamo, M. Fiacchini, and E. Camacho. Robust tube based MPC for tracking ofpiece-wise constant references. In Proceedings of the IEEE Conference on Decision and Control, volume 12,pages 1820–1825, New Orleans, USA, Dec 2007b. 47, 48, 73, 76, 77, 78

J. Alvarez-Ramírez and R. Suárez. Global stabilization of discrete-time linear systems with boundedinputs. International Journal of Adaptive Control and Signal Processing, 10(4–5):409–416, 1996. 117

D. Angeli and E. Mosca. Command Governors for Constrained Nonlinear Systems. IEEE Transactions onAutomatic Control, 44(4):816–820, Apr 1999. 16

D. Angeli, A. Casavola, and E. Mosca. Ellipsoidal Low-Demanding MPC Schemes for Uncertain PolytopicDiscrete-Time Systems. In Proceedings of the 41st IEEE Conference on Decision and Control, volume 3,pages 2935–2940, Las Vegas, Nevada, USA, Dec 2002. 32

D. Angeli, A. Casavola, G. Franzè, and E. Mosca. An ellipsoidal off-line MPC scheme for uncertainpolytopic discrete-time systems. Automatica, 44(12):3113–3119, Dec 2008. 32

Z. Artstein and S. V. Rakovic. Feedback and invariance under uncertainty via set-iterates. Automatica, 44(2):520–525, Feb 2008. 98, 129

Z. Artstein and S. V. Rakovic. Set invariance under output feedback: a set-dynamics approach. InternationalJournal of Systems Science, 2010. Scheduled for publication. 129

A. Atassi and H. Khalil. A separation principle for the stabilization of a class of nonlinear systems.Automatic Control, IEEE Transactions on, 44(9):1672–1687, Sep 1999. 63

M. Bacic, M. Cannon, Y. Lee, and B. Kouvaritakis. General interpolation in MPC and its advantages. IEEETransactions on Automatic Control, 48(6):1092–1096, Jun 2003. 38, 39, 106

M. Balandat, W. Zhang, and A. Abate. On the Infinite Horizon Constrained Switched LQR Problem. In49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, Dec 2010. (accepted). 10

M. Baotic, F. Borrelli, A. Bemporad, and M. Morari. Efficient On-Line Computation of Constrained OptimalControl. SIAM Journal on Control and Optimization, 47(5):2470–2489, 2008. 19

137

M. Baric, S. V. Rakovic, T. Besselmann, and M. Morari. Max-Min Optimal Control of Constrained Discrete-Time Systems. In Proceedings of the 17th IFAC World Congress, pages 8803–8808, Seoul, South Korea,Jul 2008. 42

A. Bemporad. Reducing Conservativeness in Predictive Control of Constrained Systems with Disturbances.In Proceedings of the 37th IEEE Conference on Decision and Control, volume 2, pages 1384–1389, Tampa,Florida, USA, Dec 1998a. 27

A. Bemporad. Reference Governor for Constrained Nonlinear Systems. IEEE Transactions on AutomaticControl, 43(3):415–419, Mar 1998b. 16

A. Bemporad. Hybrid Toolbox - User’s Guide, 2004. http://www.ing.unitn.it/~bemporad/hybrid/toolbox. 20

A. Bemporad and C. Filippi. Suboptimal Explicit Receding Horizon Control via Approximate Multipara-metric Quadratic Programming. Journal of Optimization Theory and Applications, 117(1):9–38, Apr2003. 18

A. Bemporad and C. Filippi. An Algorithm for Approximate Multiparametric Convex Programming.Computational Optimization and Applications, 35(1):87–108, Sep 2006. 42

A. Bemporad and A. Garulli. Output-feedback predictive control of constrained linear systems viaset-membership state estimation. International Journal of Control, 73(8):655–665, 2000. 12

A. Bemporad and M. Morari. Robust Model Predictive Control: A Survey, volume 245/1999 of Lecture Notesin Control and Information Sciences, pages 207–226. Springer Verlag, Berlin, 1999. 2, 3, 4, 21, 23, 24,133

A. Bemporad and E. Mosca. Constraint Fulfilment in Control Systems via Predictive Reference Management.In Proceedings of the 33rd IEEE Conference on Decision and Control, volume 3, pages 3017–3022, LakeBuena Vista, Fl, USA, Dec 1994a. 15, 16

A. Bemporad and E. Mosca. Constraint fulfilment in feedback control via predictive reference management.In Proceedings of the Third IEEE Conference on Control Applications, pages 1909–1914, Glasgow, Scotland,UK, Aug 1994b. 15

A. Bemporad and E. Mosca. Nonlinear Predictive Reference Governor for Constrained Control Systems.In Proceedings of the 34th IEEE Conference on Decision and Control, volume 2, pages 1205–1210, NewOrleans, USA, Dec 1995. 15

A. Bemporad and E. Mosca. Fulfilling Hard Constraints in Uncertain Linear Systems by ReferenceManaging. Automatica, 34(4):451–461, 1998. 22

A. Bemporad, A. Casavola, and E. Mosca. Nonlinear Control of Constrained Linear Systems via PredictiveReference Management. IEEE Transactions on Automatic Control, 42(3):340–349, 1997. 15, 16

A. Bemporad, K. Fukuda, and F. D. Torrisi. Convexity recognition of the union of polyhedra. ComputationalGeometry, 18(3):141–154, Apr 2001. 17

A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos. The Explicit Linear Quadratic Regulator forConstrained Systems. Automatica, 38(1):3–20, Jan 2002. 4, 16, 17, 18

A. Bemporad, F. Borrelli, and M. Morari. Min-max control of constrained uncertain discrete-time linearsystems. IEEE Transactions on Automatic Control, 48(9):1600–1606, 2003. 41, 43

A. Bemporad, M. Morari, and N. L. Ricker. Model Predictive Control Toolbox 3.2 User’s Guide. TheMathworks, Mar 2010. 20

138 Bibliography

http://www.ing.unitn.it/~bemporad/hybrid/toolbox

http://www.ing.unitn.it/~bemporad/hybrid/toolbox

A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski. Adjustable robust solutions of uncertain linearprograms. Mathematical Programming, 99(2):351–376, Mar 2004. 35

E. Beran, L. Vandenberghe, and S. Boyd. A global BMI algorithm based on the generalized BendersDecomposition. In Proceedings of the European Control Conference, number 934, Bruxelles, Belgium,1997. 93

D. Bertsekas and I. Rhodes. On the Minimax Reachability of Target Sets and Target Tubes. Automatica, 7(2):233–247, Mar 1971a. 47

D. Bertsekas and I. Rhodes. Recursive State Estimation for a Set-Membership Description of Uncertainty.IEEE Transactions on Automatic Control, 16(2):117–128, Apr 1971b. 47

D. P. Bertsekas. Dynamic Programming and Optimal Control, volume 1. Athena Scientific, 3rd edition, Jan2007. 9, 10, 27

T. Besselmann, J. Löfberg, and M. Morari. Explicit Model Predictive Control for Linear Parameter-VaryingSystems. In IEEE Conference on Decision and Control, pages 3848–3853, Cancun, Mexico, Dec 2008. 42

L. Biegler and V. Zavala. Large-scale nonlinear programming using IPOPT: An integrating frameworkfor enterprise-wide dynamic optimization. Computers & Chemical Engineering, 33(3):575–582, 2009.Selected Papers from the 17th European Symposium on Computer Aided Process Engineering held inBucharest, Romania, May 2007. 19

R. Bitmead, M. Gevers, and V. Wertz. Adaptive Optimal Control: the Thinking Man’s GPC. Prentice Hall,Sydney, Australia, 1990. 5

J. Björnberg and M. Diehl. Approximate robust dynamic programming and robustly stable MPC. Automat-ica, 42(5):777–782, May 2006. 42

F. Blanchini. Minimum-Time Control for Uncertain Discrete-Time Linear Systems. In Proceedings of the31st IEEE Conference on Decision and Control, volume 3, pages 2629–2634, Tucson, AZ, USA, Dec 1992.94

F. Blanchini. Set invariance in control. Automatica, 35(11):1747–1767, 1999. 8, 11, 48, 94

F. Blanchini and S. Miani. Set-Theoretic Methods in Control. Systems & Control: Foundations & Applications.Birkhäuser, Boston, 2008. 8, 11, 50, 75, 96

F. Borrelli, M. Baotic, A. Bemporad, and M. Morari. Efficient on-line computation of constrained optimalcontrol. In Proceedings of the 40th IEEE Conference on Decision and Control, volume 2, pages 1187–1192,Orlando, FL, USA, Dec 2001a. 19

F. Borrelli, A. Bemporad, M. Fodor, and D. Hrovat. A Hybrid Approach to Traction Control, volume 2034 ofLecture Notes in Computer Science, pages 162–174. Springer Verlag, 2001b. 19

F. Borrelli, A. Bemporad, M. Fodor, and D. Hrovat. An MPC/hybrid system approach to traction control.IEEE Transactions on Control Systems Technology, 14(3):541–552, May 2006. 19

F. Borrelli, A. Bemporad, and M. Morari. Constrained Optimal Control and Predictive Control for Linear andHybrid Systems. Springer Verlag, 2010. DRAFT VERSION, YET UNPUBLISHED. 16, 18, 50, 60, 88, 89,98, 120

S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. 11, 17, 92, 96,117

Bibliography 139

S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and ControlTheory, volume 15 of Studies in Applied Mathematics. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, USA, 1994. 28, 30, 36, 37, 55, 91, 92

E. F. Camacho and C. A. Bordons. Model Predictive Control. Advanced Textbooks in Control and SignalProcessing. Springer Verlag, 2nd. edition, 2004. 3

P. J. Campo and M. Morari. Robust Model Predictive Control. In Proceedings of the 1987 American ControlConference, pages 1021–1026, 1987. 21, 22, 26, 28

M. Canale, L. Fagiano, and M. Milanese. Set Membership approximation theory for fast implementationof Model Predictive Control laws. Automatica, 45(1):45–54, Jan 2009. 4

M. Cannon and B. Kouvaritakis. Optimizing Prediction Dynamics for Robust MPC. IEEE Transactions onAutomatic Control, 50(11):1892–1897, Nov 2005. 38

M. Cannon, B. Kouvaritakis, and J. Rossiter. Efficient active set optimization in triple mode MPC. IEEETransactions on Automatic Control, 46(8):1307–1312, Aug 2001. 38

M. Cannon, P. Couchman, and B. Kouvaritakis. MPC for Stochastic Systems, volume 358/2007 of LectureNotes in Control and Information Sciences, pages 255–268. Springer Verlag, Berlin, 2007. 25

M. Cannon, B. Kouvaritakis, and D. Ng. Probabilistic tubes in linear stochastic model predictive control.Systems & Control Letters, 58(10-11):747–753, Nov 2009. 129

M. Cannon, J. Buerger, B. Kouvaritakis, and S. V. Rakovic. Robust tubes in nonlinear model predictivecontrol. In 8th IFAC Symposium on Nonlinear Control Systems, Sep 2010a. 129

M. Cannon, Q. Cheng, B. Kouvaritakis, and S. V. Rakovic. Stochastic tube MPC with state estimation. In19th International Symposium on Mathematical Theory of Networks and Systems, Jul 2010b. 129

Y.-Y. Cao and A. Xue. Comments on “Quasi-min-max MPC algorithm for LPV systems by Y. Lu and Y. Arkun,Automatica 36 (2000) 527-540”. Automatica, 40(7):1281–1282, 2004. 32

A. Casavola and E. Mosca. Reference Governor for Constrained Uncertain Linear Systems Subject toBounded Input Disturbances. In Proceedings of the 35th Conference on Decision and Control, volume 3,pages 3531–3536, Kobe, Japan, Dec 1996. 16

A. Casavola, M. Giannelli, and E. Mosca. Min-max predictive control strategies for input-saturatedpolytopic uncertain systems. Automatica, 36(1):125–133, Jan 2000a. 16, 28, 33

A. Casavola, E. Mosca, and D. Angeli. Robust Command Governors for Constrained Linear Systems. IEEETransactions on Automatic Control, 45(11):2071–2077, Nov 2000b. 16

A. Casavola, D. Famularo, and G. Franze. A Feedback Min–Max MPC Algorithm for LPV Systems Subjectto Bounded Rates of Change of Parameters. IEEE Transactions on Automatic Control, 47(7):1147–1153,Jul 2002. 34, 35

A. Casavola, D. Famularo, and G. Franzé. Robust constrained predictive control of uncertain norm-boundedlinear systems. Automatica, 40(11):1865–1876, Nov 2004. 36

H. Chen, C. Scherer, and F. Allgöwer. A Game Theoretic Approach to Nonlinear Robust Receding HorizonControl of Constrained Systems. In Proceedings of the American Control Conference, volume 5, pages3073–3077, Jun 1997. 27

X. Cheng and D. Jia. Robust Stability Constrained Model Predictive Control. In Proceedings of the 2004American Control Conference, volume 2, pages 1580–1585, Boston, MA, USA, Jul 2004. 36, 40

140 Bibliography

X. Cheng and D. Jia. Robust Stability Constrained Model Predictive Control with State Estimation. InProceedings of the American Control Conference, pages 1581–1586, Minneapolis, Minnesota, USA, Jun2006. 40

X. Cheng and B. Krogh. Stability-Constrained Model Predictive Control. IEEE Transactions on AutomaticControl, 46(11):1816–1820, Nov 2001. 36

L. Chisci and G. Zappa. Feasibility in predictive control of constrained linear systems: the output feedbackcase. International Journal of Robust and Nonlinear Control, 12:465–487, 2002. 12

L. Chisci and G. Zappa. Dual mode predictive tracking of piecewise constant references for constrainedlinear systems. International Journal of Control, 76(1):61–72, Jan 2003. 16, 77

L. Chisci, J. Rossiter, and G. Zappa. Systems with persistent disturbances: predictive control with restrictedconstraints. Automatica, 37(7):1019–1028, Jul 2001. 4, 106

F. Christophersen. Efficient Evaluation of Piecewise Control Laws Defined Over a Large Number of Polyhedra,volume 359/2007 of Lecture Notes in Control and Information Sciences, pages 150–165. Springer Verlag,Berlin, 2007. 19

F. Christophersen, M. Zeilinger, C. Jones, and M. Morari. Controller Complexity Reduction for PiecewiseAffine Aystems Through Safe Region Elimination. In Proceedings of the 46th IEEE Conference on Decisionand Control, pages 4773–4778, New Orleans, USA, Dec 2007. 18

D. Clarke, C. Mohtadi, and P. Tuffs. Generalized Predictive Control. Automatica, 23(2):137–160, Mar1987. 25

C. R. Cutler and B. L. Ramaker. Dynamic matrix control - a computer control algorithm. In Proceedings ofthe joint automatic control conference, 1980. 3

F. A. Cuzzola, J. C. Geromel, and M. Morari. An improved approach for constrained robust modelpredictive control. Automatica, 38(7):1183–1189, 2002. 4, 31

D. M. de la Peña and T. Alamo. Stochastic Programming Applied to Model Predictive Control. InProceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference,pages 1361–1366, Seville, Spain, Dec 2005. 25

D. M. de la Peña, A. Bemporad, and C. Filippi. Robust Explicit MPC based on Approximate multi-parametricConvex Programming. IEEE Transactions on Automatic Control, 51(8):1399–1403, Dec 2006. 42

D. M. de la Peña, T. Alamo, D. Ramirez, and E. Camacho. Min-max Model Predictive Control as a QuadraticProgram. IET Control Theory & Applications, 1(1):328–333, Jan 2007. 42

G. de Nicolao, L. Magni, and R. Scattolini. On the robustness of receding-horizon control with terminalconstraints. IEEE Transactions on Automatic Control, 41(3):451–453, Mar 1996. 4, 21

M. Diehl and J. Björnberg. Robust dynamic programming for min-max model predictive control ofconstrained uncertain systems. IEEE Transactions on Automatic Control, 49(12):2253–2257, Dec 2004.41

B. Ding, Y. Xi, M. T. Cychowski, and T. O’Mahony. Improving off-line approach to robust MPC based-onnominal performance cost. Automatica, 43(1):158–163, Jan 2007. 32

Z. Dostál. Optimal Quadratic Programming Algorithms. Springer Verlag, 2009. 11

A. Ferramosca, D. Limon, I. Alvarado, T. Alamo, and E. Camacho. MPC for tracking with optimalclosed-loop performance. Automatica, 45(8):1975—1978, Aug 2009a. 15, 99

Bibliography 141

A. Ferramosca, D. Limon, A. González, D. Odloak, and E. Camacho. MPC for tracking target sets. InProceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference,pages 8020–8025, Shanghai, China, Dec 2009b. 15, 75

A. Ferramosca, D. Limon, A. González, D. Odloak, and E. Camacho. MPC for tracking zone regions.Journal of Process Control, 20(4):506–516, Apr 2010. 15, 75

H. J. Ferreau, H. G. Bock, and M. Diehl. An online active set strategy to overcome the limitations ofexplicit MPC. International Journal of Robust and Nonlinear Control, 18(8):816–830, 2008. 19

R. Findeisen, L. Imsland, and B. A. Foss. State and Output Feedback Nonlinear Model Predictive Control:An Overview. European Journal of Control, 9(2-3):190–206, 2003. 12

R. Findeisen, F. Allgöwer, and L. T. Biegler. Assessment and Future Directions of Nonlinear Model PredictiveControl, volume 358 of Lecture Notes in Control and Information Sciences. Springer Verlag, Berlin, 2007.5

B. Francis and W. Wonham. The internal model principle of control theory. Automatica, 12(5):457–465,Sep 1976. 44

K. Fukuda. The cdd and cddplus Homepage, 2010. http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html. 89, 98, 128

M. Fukuda and M. Kojima. Branch-and-Cut Algorithms for the Bilinear Matrix Inequality EigenvalueProblem. Computational Optimization and Applications, 19(1):79–105, Apr 2001. 93

C. E. Garcia, D. M. Prett, and M. Morari. Model predictive control: Theory and practice – A survey.Automatica, 25(3):335–348, May 1989. 3

E. Gawrilow and M. Joswig. polymake: a Framework for Analyzing Convex Polytopes. In G. Kalai andG. M. Ziegler, editors, Polytopes — Combinatorics and Computation, volume 29 of DMV Sem., pages43–73. Birkhäuser, Basel, 2000. 98

J. Gayek. A survey of techniques for approximating reachable and controllable sets. In Proceedings of theIEEE Conference on Decision and Control, volume 2, pages 1724–1729, Brighton, U.K., Dec 1991. 94

J. C. Geromel, P. L. D. Peres, and J. Bernussou. On a Convex Parameter Space Method for Linear ControlDesign of Uncertain Systems. SIAM Journal on Control and Optimization, 29(2):381–402, Mar 1991.29, 30

T. Geyer, F. D. Torrisi, and M. Morari. Optimal complexity reduction of polyhedral piecewise affine systems.Automatica, 44(7):1728–1740, Jul 2008. 18

E. Gilbert and I. Kolmanovsky. Fast Reference governors for Systems with State and Control Constraintsand Disturbance Inputs. International Journal of Robust and Nonlinear Control, 9(15):1117–1141, Dec1999a. 16

E. Gilbert and I. Kolmanovsky. Set-Point Control of Nonlinear Systems with State and Control Constraints:A Lyapunov-Function, Reference-Governor Approach. In Proceedings of the 38th IEEE Conference onDecision and Control, volume 3, pages 2507–2512, Phoenix, Arizona, USA, Dec 1999b. 16

E. Gilbert and I. Kolmanovsky. Nonlinear tracking control in the presence of state and control constraints:a generalized reference governor. Automatica, 38(12):2063–2073, Dec 2002. 16, 133

E. Gilbert and K. Tan. Linear systems with state and control constraints: The theory and application ofmaximal output admissible sets. IEEE Transactions on Automatic Control, 36(9):1008–1020, Sep 1991.9, 10, 11, 89, 90, 108

142 Bibliography

http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html

http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html

E. Gilbert, I. Kolmanovsky, and K. T. Tan. Nonlinear Control of Discrete-time Linear Systems with Stateand Control Constraints: A Reference Governor with Global Convergence Properties. In Proceedings ofthe 33rd IEEE Conference on Decision and Control, volume 1, pages 144–149, Lake Buena Vista, Fl, USA,Dec 1994. 16

E. G. Gilbert, I. Kolmanovsky, and K. T. Tan. Discrete-time reference governors and the nonlinear controlof systems with state and control constraints. International Journal of Robust and Nonlinear Control, 5(5):487–504, 1995. 15, 16

A. Girard. Reachability of Uncertain Linear Systems Using Zonotopes. In M. Morari and L. Thiele, editors,HSCC (Hybrid Systems: Computation and Control), volume 3414/2005 of Lecture Notes in ComputerScience, pages 291–305, Zurich, Switzerland, Mar 2005. Springer Verlag. 96

J. Glover and F. Schweppe. Control of Linear Dynamic Systems with Set Constrained Disturbances. IEEETransactions on Automatic Control, 16(5):411–423, Oct 1971. 47

K.-C. Goh, M. Safonov, and G. Papavassilopoulos. A Global Optimization Approach for the BMI Problem.In Proceedings of the 33rd IEEE Conference on Decision and Control, volume 3, pages 2009–2014, LakeBuena Vista, Fl, USA, Dec 1994. 93

P. J. Goulart. Affine Feedback Policies for Robust Control with Constraints. PhD thesis, University ofCambridge, 2006. 68

P. J. Goulart and E. C. Kerrigan. Output feedback receding horizon control of constrained systems.International Journal of Control, 80(1):8–20, Jan 2007. 68

P. J. Goulart, E. C. Kerrigan, and J. M. Maciejowski. Optimization over state feedback policies for robustcontrol with constraints. Automatica, 42(4):523–533, Apr 2006. 34, 35, 68

P. J. Goulart, E. C. Kerrigan, and D. Ralph. Efficient robust optimization for robust control with constraints.Mathematical Programming, 114(1):115–147, Jul 2008. 35

P. J. Goulart, E. C. Kerrigan, and T. Alamo. Control of Constrained Discrete-Time Systems With Boundedl2-Gain. IEEE Transactions on Automatic Control, 54(5):1105–1111, May 2009. 35

A. Grancharova and T. Johansen. Explicit Approximate Approach to Feedback Min-Max Model PredictiveControl of Constrained Nonlinear Systems. In Proceedings of the 45th IEEE Conference on Decision andControl, pages 4848–4853, San Diego, CA, USA, Dec 2006. 19

M. Grant and S. Boyd. CVX: Matlab Software for Disciplined Convex Programming, version 1.21.http://cvxr.com/cvx, May 2010. 28

M. Green and D. J. Limebeer. Linear robust control. Information and System Sciences. Prentice Hall, 1994.4

P. Grieder and M. Morari. Complexity reduction of receding horizon control. In Proceedings of the 42ndIEEE Conference on Decision and Control, volume 3, pages 3179–3190, Maui, Hawaii, USA, Dec 2003. 18

G. Grimm, M. J. Messina, S. E. Tuna, and A. R. Teel. Examples when nonlinear model predictive control isnonrobust. Automatica, 40(10):1729–1738, Oct 2004. 4, 22

P. Gritzmann and B. Sturmfels. Minkowski Addition of Polytopes: Computational Complexity andApplications to Gröbner Bases. SIAM Journal on Discrete Mathematics, 6(2):246–269, May 1993. 98

A. Hassibi, J. How, and S. Boyd. A Path-following Method for Solving BMI Problems in Control. InProceedings of the American Control Conference, volume 2, pages 1385–1389, San Diego, CA, USA, Jun1999. 93

Bibliography 143

http://cvxr.com/cvx

P. Hokayem, D. Chatterjee, and J. Lygeros. On Stochastic Receding Horizon Control with Bounded ControlInputs. In IEEE Conference on Decision and Control, pages 6359–6364, Shanghai, China, Dec. 2009. 25

R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, New York, 1990. 113, 114

L. Imsland, N. Bar, and B. A. Foss. More efficient predictive control. Automatica, 41(8):1395–1403, Aug2005. 38

L. Imsland, J. Rossiter, B. Pluymers, and J. Suykens. Robust triple mode MPC. In Proceedings of the 2006American Control Conference, pages 869–874, Minneapolis, Minnesota, USA, Jun 2006. 38

D. Jia and B. Krogh. Min-max Feedback Model Predictive Control with State Estimation. In Proceedings ofthe American Control Conference, pages 262–267, Portland, OR, USA, Jun 2005. 41

D. Jia, B. Krogh, and O. Stursberg. LMI Approach to Robust Model Predictive Control . Journal ofOptimization Theory and Application, 127(2):347–365, Nov 2005. 34, 35, 41

T. Johansen. On Multi-parametric Nonlinear Programming and Explicit Nonlinear Model PredictiveControl. In Proceedings of the 41st IEEE Conference on Decision and Control, volume 3, pages 2768–2773,Las Vegas, Nevada, USA, Dec 2002. 19

T. A. Johansen. Approximate explicit receding horizon control of constrained nonlinear systems. Automat-ica, 40(2):293–300, Feb 2004. 19

T. A. Johansen, W. Jackson, R. Schreiber, and P. Tondel. Hardware Synthesis of Explicit Model PredictiveControllers. Control Systems Technology, IEEE Transactions on, 15(1):191–197, Jan 2007. 20

N. M. P. Kakalis, V. Dua, V. Sakizlis, J. D. Perkins, and E. N. Pistikopoulos. A Parametric OptimisationApproach for Robust MPC. In Proceedings of the 15th IFAC World Congress on Automatic Control,Barcelona, Spain, 2002. 42

R. E. Kalman. A New Approach to Linear Filtering and Prediction Problems. Transactions of the ASME –Journal of Basic Engineering, 82(Series D):35–45, 1960. 12

E. Kerrigan and J. Maciejowski. Feedback min-max Model Predictive Control Using a Single LinearProgram: Robust Stability and the Explicit Solution. International Journal of Robust and NonlinearControl, 14(4):395–413, Mar 2004. 41

E. C. Kerrigan. Robust Constraint Satisfaction: Invariant Sets and Predictive Control. PhD thesis, Universityof Cambridge, Cambridge, U.K., 2000. 9, 89

I. Kolmanovsky and E. Gilbert. Theory and Computation of Disturbance Invariant Sets for Discrete-TimeLinear Systems. Mathematical Problems in Engineering, 4(4):317–367, 1998. 9, 10, 48, 50, 94, 95, 97,108

M. V. Kothare, V. Balakrishnan, and M. Morari. Robust Constrained Model Predictive Control using LinearMatrix Inequalities. In Proceedings of the American Control Conference, volume 1, pages 440–444,Baltimore, Maryland, USA, Jun 1994. 28

M. V. Kothare, V. Balakrishnan, and M. Morari. Robust Constrained Model Predictive Control using LinearMatrix Inequalities. IFA-Report 95–02, California Institute of Technology, Mar 1995. 4

M. V. Kothare, V. Balakrishnan, and M. Morari. Robust Constrained Model Predictive Control using LinearMatrix Inequalities. Automatica, 32(10):1361–1379, Oct 1996. 22, 23, 24, 28, 29, 30, 31, 35

144 Bibliography

K. Kouramas, S. V. Rakovic, E. C. Kerrigan, J. Allwright, and D. Q. Mayne. On the Minimal RobustPositively Invariant Set for Linear Difference Inclusions. In Proceedings of the 44th IEEE Conference onDecision and Control, and the European Control Conference, pages 2296–2301, Sevilla, Spain, Dec 2005.98

K. I. Kouramas. Control of linear systems with state and control constraints. PhD thesis, Imperial College,London, U.K., 2002. 95

B. Kouvaritakis, J. A. Rossiter, and J. Schuurmans. Efficient Robust Predictive Control. IEEE Transactionson Automatic Control, 45(8):1545–1549, Aug 2000. 37, 38

B. Kouvaritakis, M. Cannon, and J. A. Rossiter. Who needs QP for linear MPC anyway? Automatica, 38(5):879–884, 2002. 38

B. Kouvaritakis, M. Cannon, and V. Tsachouridis. Recent developments in stochastic MPC and sustainabledevelopment. Annual Reviews in Control, 28(1):23–35, 2004. 25

B. Kouvaritakis, M. Cannon, S. V. Rakovic, and Q. Cheng. Explicit use of probabilistic distributions inlinear predictive control. Automatica, In Press, Corrected Proof, 2010. 129

M. Kvasnica, P. Grieder, M. Baotic, and M. Morari. Multi-Parametric Toolbox (MPT). In HSCC (HybridSystems: Computation and Control), Lecture Notes in Computer Science, pages 448–462, Philadelphia,USA, Mar. 2004. Springer Verlag. 20, 61, 89, 98, 102, 128

S. Lall and K. Glover. A game theoretic approach to moving horizon control, pages 131–144. Advances inModel-Based Predictive Control. Oxford University Press, Oxford, U.K., 1994. 27

W. Langson, I. Chryssochoos, S. V. Rakovic, and D. Mayne. Robust model predictive control using tubes.Automatica, 40(1):125–133, Jan 2004. 40, 47, 48

J. Lasserre. Reachable, controllable sets and stabilizing control of constrained linear systems. Automatica,29(2):531–536, Mar 1993. 94

M. Lazar, D. M. de la Peña, W. Heemels, and T. Alamo. On input-to-state stability of min-max nonlinearModel Predictive Control. Systems and Control Letters, (57):39–48, 2008. 4

E. B. Lee and L. Markus. Foundations of Optimal Control Theory. Wiley, New York, 1967. 2

J. H. Lee and Z. Yu. Worst-case Formulations of Model Predictive Control for Systems with BoundedParameters. Automatica, 33(5):763–781, 1997. 27, 28

Y. I. Lee and B. Kouvaritakis. Constrained receding horizon predictive control for systems with disturbances.International Journal of Control, 72(11):1027–1032, Aug 1999. 27

Y. I. Lee and B. Kouvaritakis. Robust receding horizon predictive control for systems with uncertaindynamics and input saturation. Automatica, 36(10):1497–1504, Oct 2000. 50

P. Li, M. Wendt, and G. Wozny. A probabilistically constrained model predictive controller. Automatica, 38(7):1171–1176, 2002. 25

D. Limon, I. Alvarado, T. Alamo, and E. Camacho. MPC for tracking of piece-wise constant references forconstrained linear systems. In Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, Jul2005. 15, 48, 73

D. Limon, I. Alvarado, T. Alamo, and E. Camacho. MPC for tracking piecewise constant references forconstrained linear systems. Automatica, 44(9):2382–2387, Sep 2008a. 15, 48, 99

Bibliography 145

D. Limon, I. Alvarado, T. Alamo, and E. Camacho. On the design of Robust tube-based MPC for tracking.In Proceedings of the 17th IFAC World Congress, pages 15333–15338, Seoul, South Korea, Jul 2008b. 48,77, 78, 79, 89, 90, 91, 92, 99

D. Limon, T. Alamo, D. Raimondo, D. M. de la Peña, J. Bravo, A. Ferramosca, and E. Camacho. Input-to-State Stability: A Unifying Framework for Robust Model Predictive Control, volume 384/2009 of LectureNotes in Control and Information Sciences, pages 1–26. Springer Verlag, Berlin, 2009. 4, 22

D. Limon, I. Alvarado, T. Alamo, and E. Camacho. Robust tube-based MPC for tracking of constrainedlinear systems with additive disturbances. Journal of Process Control, 20(3):248–260, Mar 2010. 48, 73

J. Löfberg. Towards joint state estimation and control in minimax MPC. In Proceedings of the 15th IFACWorld Congress on Automatic Control, Barcelona, Spain, 2002. 12

J. Löfberg. Approximations of closed-loop minimax MPC. In Proceedings of the 42nd IEEE Conference onDecision and Control, volume 2, pages 1438–1442, Maui, Hawaii, USA, Dec 2003a. 35

J. Löfberg. Minmax approaches to robust model predictive control. PhD thesis, Linköping University, 2003b.30, 31, 40

J. Löfberg. YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In Proceedings of the CACSDConference, Taipei, Taiwan, 2004. 28, 100, 128

J. Löfberg. Modeling and solving uncertain optimization problems in YALMIP. In Proceedings of the 17thIFAC World Congress, Seoul, South Korea, 2008. 100

C. Løvaas, M. M. Seron, and G. C. Goodwin. Robust Output-Feedback Model Predictive Control forSystems with Unstructured Uncertainty. Automatica, 44(8):1933–1943, Aug 2008. 41

Y. Lu and Y. Arkun. A quasi-min-max MPC algorithm for linear parameter varying systems with boundedrate of change of parameters. In Proceedings of the American Control Conference, volume 5, pages3234–3238, Chicago, IL, USA, Jun 2000a. 32, 34, 35

Y. Lu and Y. Arkun. Quasi-Min-Max MPC algorithms for LPV systems. Automatica, 36(4):527–540, 2000b.32, 34

D. Luenberger. An introduction to observers. IEEE Transactions on Automatic Control, 16(6):596–602, Dec1971. 12, 63

U. Maeder, F. Borrelli, and M. Morari. Linear offset-free Model Predictive Control. Automatica, 45(10):2214–2222, Oct 2009. 15, 43, 44, 45, 46

W.-J. Mao. Robust stabilization of uncertain time-varying discrete systems and comments on “an improvedapproach for constrained robust model predictive control”. Automatica, 39(6):1109–1112, Jun 2003.31

S. Mariéthoz, U. Mäder, and M. Morari. High-speed FPGA implementation of observers and explicit modelpredictive controllers. In IEEE IECON, Industrial Electronics Conference, Porto, Portugal, Nov. 2009. 4, 20

D. Mayne and W. Schroeder. Robust Time-Optimal Control of Constrained Linear Systems. Automatica, 33(12):2103–2118, Dec 1997a. 94

D. Mayne, J. B. Rawlings, C. Rao, and P. Scokaert. Constrained model predictive control: Stability andoptimality. Automatica, 36(6):789–814, Jun 2000. 3, 4, 5, 8, 9, 27, 75

D. Mayne, S. V. Rakovic, R. Vinter, and E. Kerrigan. Characterization of the solution to a constrainedH-infinity optimal control problem. Automatica, 43(3):371–382, Mar 2006a. 42

146 Bibliography

D. Mayne, S. V. Rakovic, R. Findeisen, and F. Allgöwer. Robust Output Feedback Model Predictive Controlof Constrained Linear Systems: Time Varying Case. Automatica, 45:2082–2087, 2009. 48, 65, 73

D. Q. Mayne and E. C. Kerrigan. Tube-based robust nonlinear model predictive control. In Proceedings ofthe 7th IFAC Symposium - NOLCOS2007, Preatoria, South Africa, Aug 2007. 129

D. Q. Mayne and S. V. Rakovic. Model predictive control of constrained piecewise affine discrete-timesystems. International Journal of Robust and Nonlinear Control, 13(3-4):261–279, 2003. 129

D. Q. Mayne and W. R. Schroeder. Robust Time-Optimal Control of Constrained Linear Systems. Automatica,33(12):2103–2118, Dec 1997b. 94

D. Q. Mayne, M. M. Seron, and S. V. Rakovic. Robust Model Predictive Control of Constrained LinearSystems with Bounded Disturbances. Automatica, 41(2):219–224, Feb 2005. 4, 40, 47, 48, 49, 50, 51,53, 54, 55, 57

D. Q. Mayne, S. V. Rakovic, R. Findeisen, and F. Allgöwer. Robust Output Feedback Model PredictiveControl of Constrained Linear Systems. Automatica, 42(7):1217–1222, Jul 2006b. 47, 48, 54, 63, 65,66, 67, 69

R. Milman and E. Davison. A Fast MPC Algorithm Using Nonfeasible Active Set Methods. Journal ofOptimization Theory and Applications, 139(3):591–616, Dec 2008. 19

M. Morari. Model predictive control: Multivariable control technique of choice in the 1990s?, pages 22–37.Advances in Model-Based Predictive Control. Oxford University Press, New York, Jun 1994. 3

K. R. Muske. Steady-State Target Optimization in Linear Model Predictive Control. In Proceedings of theAmerican Control Conference, volume 6, pages 3597–3601, Albuquerque, New Mexico, USA, Jun 1997.13

K. R. Muske and T. A. Badgwell. Disturbance modeling for offset-free linear model predictive control.Journal of Process Control, 12(5):617–632, Aug 2002. 15, 43

K. R. Muske and J. B. Rawlings. Model Predictive Control with Linear Models. AIChE Journal, 39(2):262–287, 1993. 13

Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming, volume 13of Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia,USA, 1994. 28

J. Nocedal and S. Wright. Numerical Optimization. Springer Series in Operations Research and FinancialEngineering. Springer Verlag, 2nd. edition, 2006. 11

A. Packard and J. Doyle. The Complex Structured Singular Value. Automatica, 29(1):71–109, 1993. 24

G. Pannocchia. Robust model predictive control with guaranteed setpoint tracking. Journal of ProcessControl, 14(8):927–937, Dec 2004. 15, 43

G. Pannocchia and A. Bemporad. Combined Design of Disturbance Model and Observer for Offset-FreeModel Predictive Control. IEEE Transactions on Automatic Control, 52(6):1048–1053, Jun 2007. 43, 46

G. Pannocchia and E. Kerrigan. Offset-free Receding Horizon Control of Constrained Linear Systems.AIChE Journal, 51(12):3134–3146, Dec 2005. 13, 15, 43, 46

G. Pannocchia and J. B. Rawlings. Disturbance Models for Offset-Free Model-Predictive Control. AIChEJournal, 49(2):426–437, 2003. 13, 15, 43, 44, 45

Bibliography 147

G. Pannocchia, J. B. Rawlings, and S. J. Wright. Fast, large-scale model predictive control by partialenumeration. Automatica, 43(5):852–860, May 2007. 18

P. Park and S. C. Jeong. Constrained RHC for LPV systems with bounded rates of parameter variations.Automatica, 40(5):865–872, May 2004. 35

B. Pluymers, L. Roobrouck, J. Buijs, J. Suykens, and B. D. Moor. Constrained linear MPC with time-varyingterminal cost using convex combinations. Automatica, 41(5):831–837, May 2005a. 107

B. Pluymers, J. Rossiter, J. Suykens, and B. De Moor. The efficient computation of polyhedral invariant setsfor linear systems with polytopic uncertainty. In Proceedings of the 2005 American Control Conference,volume 2, pages 804–809, Portland, OR, USA, Jun 2005b. 39

B. Pluymers, J. Rossiter, J. Suykens, and B. D. Moor. Interpolation Based MPC for LPV Systems usingPolyhedral Invariant Sets. In Proceedings of the American Control Conference, pages 810–815, Portland,USA, Jun 2005c. 38, 39, 106, 110, 122

B. Pluymers, J. Suykens, and B. D. Moor. Min-max feedback MPC using a time-varying terminal constraintset and comments on “Efficient robust constrained model predictive control with a time-varying terminalconstraint set”. Systems & Control Letters, 54(12):1143–1148, Dec 2005d. 33

J. A. Primbs and V. Nevistic. A Framework for Robustness Analysis of Constrained Finite Receding HorizonControl. IEEE Transactions on Automatic Control, 45(10):1828–1838, Oct 2000. 22

S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control EngineeringPractice, 11(7):733–764, Jul 2003. 3

S. V. Rakovic. Minkowski algebra and Banach Contraction Principle in set invariance for linear discretetime systems. In December, editor, Proceedings of the 46th IEEE Conference on Decision and Control,pages 2169–2174, New Orleans, USA, Dec 2007. 98, 129

S. V. Rakovic. Set Theoretic Methods in Model Predictive Control, volume 384/2009 of Lecture Notes inControl and Information Sciences, pages 41–54. Springer Verlag, Berlin, 2009. 8

S. V. Rakovic and M. Baric. Local Control Lyapunov Functions for Constrained Linear Discrete-TimeSystems: The Minkowski Algebra Approach. IEEE Transactions on Automatic Control, 54(11):2686–2692,Nov 2009. 129

S. V. Rakovic and M. Baric. Parameterized Robust Control Invariant Sets for Linear Systems: TheoreticalAdvances and Computational Remarks. IEEE Transactions on Automatic Control (to appear), 2010. 129

S. V. Rakovic, E. C. Kerrigan, K. Kouramas, and D. Q. Mayne. Invariant approximations of robustlypositively invariant sets for constrained linear discrete-time systems subject to bounded disturbances.Technical Report CUED/F-INFENG/TR.473, Dept. Eng., University of Cambridge, Cambridge, U.K., Jan2004. 94, 96

S. V. Rakovic, E. Kerrigan, K. Kouramas, and D. Mayne. Invariant Approximations of the Minimal RobustPositively Invariant Set. IEEE Transactions on Automatic Control, 50(3):406–420, Mar 2005. 48, 95, 96,97

S. V. Rakovic, A. Teel, D. Mayne, and A. Astolfi. Simple Robust Control Invariant Tubes for Some Classesof Nonlinear Discrete Time Systems. In Proceedings of the 45th IEEE Conference on Decision and Control,pages 6397–6402, San Diego, CA, USA, Dec 2006. 129

S. V. Rakovic, E. Kerrigan, D. Mayne, and K. Kouramas. Optimized robust control invariance for lineardiscrete-time systems: Theoretical foundation. Automatica, 43(5):831–841, May 2007. 129

148 Bibliography

S. V. Rakovic, B. Kouvaritakis, R. Findeisen, and M. Cannon. Simple Homothetic Tube Model PredictiveControl. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks andSystems, Budapest, Hungary, Jul 2010. 129

D. Ramirez and E. Camacho. Piecewise affinity of min-max MPC with bounded additive uncertainties anda quadratic criterion. Automatica, 42(2):295–302, Feb 2006. 42

C. V. Rao and J. B. Rawlings. Linear programming and model predictive control. Journal of Process Control,10(2-3):283–289, Apr 2000. 7

J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing, 2009.5, 8, 12, 13, 14, 22, 25, 26, 27, 48, 52, 55, 56, 109, 113, 131, 132

J. Richalet, A. Rault, J. Testud, and J. Papon. Model predictive heuristic control: Applications to industrialprocesses. Automatica, 14(5):413–428, Sep 1978. 3

S. Richter, C. Jones, and M. Morari. Real-Time Input-Constrained MPC Using Fast Gradient Methods. InProceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference,pages 7387–7393, Shanghai, China, Dec. 2009. 19

S. Richter, Sebastian, Mariéthoz, and M. Morari. High-Speed Online MPC Based on a Fast Gradient MethodApplied to Power Converter Control. In Proceedings of the American Control Conference, Baltimore, MD,USA, Jun 2010. 19

J. Rossiter and P. Grieder. Using Interpolation to Simplify Explicit Model Predictive Control. In Proceedingsof the 2004 American Control Conference, volume 1, pages 885–890, Boston, MA, USA, Jun 2004. 18

J. Rossiter, B. Kouvaritakis, and M. Cannon. Triple mode control in MPC. In Proceedings of the 2000American Control Conference, volume 6, pages 3753–3757, Chicago, IL, USA, Jun 2000. 38

J. A. Rossiter and Y. Ding. Interpolation methods in model predictive control: an overview. InternationalJournal of Control, 83(2):297–312, Feb 2010. 39, 106

J. A. Rossiter, B. Kouvaritakis, and M. J. Rice. A numerically robust state-space approach to stable-predictivecontrol strategies. Automatica, 34(1):65–73, Jan 1998. 27

J. A. Rossiter, B. Kouvaritakis, and M. Bacic. Interpolation based computationally efficient predictivecontrol. International Journal of Control, 77(3):290–301, Feb 2004. 38, 39, 106

J. A. Rossiter, B. Kouvaritakis, and M. Cannon. An algorithm for reducing complexity in parametricpredictive control. International Journal of Control, 78(18):1511–1520, Dec 2005. 18

V. Sakizlis, N. M. Kakalis, V. Dua, J. D. Perkins, and E. N. Pistikopoulos. Design of robust model-basedcontrollers via parametric programming. Automatica, 40(2):189–201, Feb 2004. 42

L. O. Santos and L. T. Biegler. A tool to analyze robust stability for model predictive controllers. Journal ofProcess Control, 9(3):233–246, 1999. 4

R. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 44 of Encyclopedia of mathematics andits applications. Cambridge University Press, New York, 1st edition, Feb 1993. 50

J. Schuurmans and J. Rossiter. Robust predictive control using tight set of predicted states. IEE ProceedingsControl Theory & Applications, 147(1):13–18, Jan 2000. 28, 33

P. Scokaert and D. Mayne. Min-max Feedback Model Predictive Control for Constrained Linear Systems.IEEE Transactions on Automatic Control, 43(8):1136–1142, Aug 1998. 4, 27, 28

Bibliography 149

P. Scokaert and J. B. Rawlings. Constrained Linear Quadratic Regulation. IEEE Transactions on AutomaticControl, 43(8):1163–1169, Aug 1998. 10

P. Scokaert, J. B. Rawlings, and E. Meadows. Discrete-time stability with perturbations: Application tomodel predictive control. Automatica, 33(3):463–470, 1997. 22

P. O. M. Scokaert and J. B. Rawlings. Feasibility issues in linear model predictive control. AIChE Journal,45(8):1649–1659, 1999. 13

J. Shamma. Anti-windup via constrained regulation with observers. Systems and Control Letters, 40:261–268, Jul 2000. 63

S. Skogestad and I. Postlethwaite. Multivariable Feedback Control: Analysis and Design. Wiley, 2nd. edition,2005. 12

R. Smith. Robust Model Predictive Control of Constrained Linear Systems. In Proceedings of the AmericanControl Conference, volume 1, pages 245–250, Jun 2004. 36

E. D. Sontag. Input to State Stability: Basic Concepts and Results, volume 1932/2008 of Lecture Notes inMathematics, pages 163–220. Springer Verlag, 2008. 22

J. Spjøtvold, E. C. Kerrigan, C. N. Jones, P. Tøndel, and T. A. Johansen. On the facet-to-facet property ofsolutions to convex parametric quadratic programs. Automatica, 42(12):2209–2214, Dec 2006. 17

J. Spjotvold, S. Rakovic, P. Tondel, and T. Johansen. Utilizing Reachability Analysis in Point LocationProblems. In Proceedings of the 45th IEEE Conference on Decision and Control, pages 4568–4569, SanDiego, CA, USA, Dec 2006. 19

J. F. Sturm. Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. OptimizationMethods and Software, 11(1):625–653, 1999. 28

D. Sui and C. Ong. Multi-mode model predictive controller for constrained linear systems with boundeddisturbances. In Proceedings of the 2006 American Control Conference, pages 172–176, Minneapolis,Minnesota, USA, Jun 2006. 39, 106, 115

D. Sui and C. Ong. Interpolated Model Predictive Controller for Linear Systems with Bounded Disturbances.In Proceedings of the 2007 American Control Conference, pages 4458–4463, New York City, USA, Jul2007. 106, 115

D. Sui, L. Feng, C. J. Ong, and M. Hovd. Robust Explicit Time Optimal Controllers for Linear Systems viaDecomposition Principle. In Proceedings of the 17th IFAC World Congress, pages 10196–10200, Seoul,South Korea, Jul 2008. 106, 107, 115

D. Sui, L. Feng, M. Hovd, and C. J. Ong. Decomposition principle in model predictive control for linearsystems with bounded disturbances. Automatica, 45(8):1917–1922, Aug 2009. 106, 107, 115, 117

D. Sui, L. Feng, and M. Hovd. Robust output feedback time optimal decomposed controllers for linearsystems via moving horizon estimation. Nonlinear Analysis: Hybrid Systems, 4(2):334–344, May 2010a.IFAC World Congress 2008. 106, 115

D. Sui, L. Feng, C. J. Ong, and M. Hovd. Robust explicit model predictive control for linear systems viainterpolation techniques. International Journal of Robust and Nonlinear Control, 20(10):1166–1175,2010b. 106, 115

M. Sznaier and M. J. Damborg. Suboptimal control of linear systems with state and control inequalityconstraints. In Proceedings of the IEEE Conference on Decision and Control, volume 26, pages 761–762,Dec 1987. 10

150 Bibliography

A. Teel and L. Praly. Tools for Semiglobal Stabilization by Partial State and Output Feedback. SIAM Journalon Control and Optimization, 33(5):1443–1488, 1995. 12

A. R. Teel. Discrete Time Receding Horizon Optimal Control: Is the Stability Robust?, volume 301/2004 ofLecture Notes in Control and Information Sciences, pages 3–27. Springer Verlag, Berlin, 2004. 4, 22

The GNU project. GLPK (GNU Linear Programming Kit) v.4.42 Reference Manual, Jan 2010. 98

M. J. Todd. The many facets of linear programming. Mathematical Programming, 91(3):417–436, Feb2002. 98

K. Toh, M. Todd, and R. Tutuncu. SDPT3 – a Matlab software package for semidefinite programming.Optimization Methods and Softwar, 11:545–581, 1999. 28

P. Tøndel, T. A. Johansen, and A. Bemporad. Evaluation of piecewise affine control via binary search tree.Automatica, 39(5):945–950, May 2003a. 19

P. Tøndel, T. A. Johansen, and A. Bemporad. An algorithm for multi-parametric quadratic programmingand explicit MPC solutions. Automatica, 39(3):489–497, Mar 2003b. 16, 17

D. van Hessem and O. Bosgra. A conic reformulation of Model Predictive Control including bounded andstochastic disturbances under state and input constraints. In Proceedings of the 41st IEEE Conference onDecision and Control, volume 4, pages 4643–4648, Las Vegas, Nevada, USA, Dec 2002. 35

J. G. VanAntwerp and R. D. Braatz. A tutorial on linear and bilinear matrix inequalities. Journal of ProcessControl, 10(4):363–385, Aug 2000. 93

J. G. VanAntwerp, R. D. Braatz, and N. V. Sahinidis. Globally optimal robust process control. Journal ofProcess Control, 9(5):375–383, 1999. 93

L. Vandenberghe, S. Boyd, and S.-P. Wu. Determinant Maximization with Linear Matrix InequalityConstraints. SIAM Journal on Matrix Analysis and Applications, 19(2):499–533, 1998. 92

Z. Wan and M. V. Kothare. Robust output feedback model predictive control using off-line linear matrixinequalities. Journal of Process Control, 12(7):763–774, Oct 2002. 40

Z. Wan and M. V. Kothare. Efficient robust constrained model predictive control with a time varyingterminal constraint set. Systems & Control Letters, 48(5):375–383, Apr 2003a. 33

Z. Wan and M. V. Kothare. An efficient off-line formulation of robust model predictive control using linearmatrix inequalities. Automatica, 39(5):837–846, 2003b. 32, 37, 40

Z. Wan, B. Pluymers, M. Kothare, and B. de Moor. Comments on: “Efficient robust constrained modelpredictive control with a time varying terminal constraint set” by Wan and Kothare. Systems & ControlLetters, 55(7):618–621, Jul 2006. 33

Y. Wang and S. Boyd. Fast Model Predictive Control Using Online Optimization. In Proceedings of the 17thIFAC World Congress, pages 6974–6997, Seoul, South Korea, Jul 2008. 19

A. Wills. QPC – Quadratic Programming in C, Apr 2010. http://sigpromu.org/quadprog/. 58, 100,101, 122, 123, 128

H. Witsenhausen. A minimax control problem for sampled linear systems. IEEE Transactions on AutomaticControl, 13(1):5–21, Feb 1968. 47

Y. Yamada and S. Hara. Global Optimization forH∞ Control with Block-diagonal Constant Scaling. InProceedings of the 35th IEEE Conference on Decision and Control, volume 2, pages 1325–1330, Kobe,Japan, Dec 1996. 93

Bibliography 151

http://sigpromu.org/quadprog/

M. Zeilinger, C. Jones, and M. Morari. Real-time suboptimal Model Predictive Control using a combinationof Explicit MPC and Online Optimization. In Proceedings of the 47th IEEE Conference on Decision andControl, pages 4718–4723, Cancun, Mexico, Dec 2008. 4, 18

Z. Q. Zheng and M. Morari. Robust Stability of Constrained Model Predictive Control. In Proceedings ofthe American Control Conference, pages 379–383, San Francisco, USA, Jun 1993. 22

K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, 1996. 4, 23, 24, 133

G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer Verlag, Berlin,1995. 10, 11, 23

152 Bibliography

constrained robust optimal trajectory tracking:...

Documents