Predictive Control Algorithms for Nonlinear Systems Mircea PhD.pdf · 2013-10-10 · Predictive...

Predictive Control Algorithms forNonlinear Systems

DOCTORAL THESIS

for receiving the doctoral degree from the“Gh. Asachi” Technical University of Iaşi, România

The Defense will take place on15 September 2009

by

Mircea Lazăr

born at Iaşi, România

Promoter: Prof. Dr. Mihail Voicu

Corresponding Member of the Romanian Academy

Defense Committee:

Prof. Dr. Vasile-Ion Manta, Chair

Prof. Dr. Mihail Voicu, PromoterCorresponding Member of the Romanian Academy

Prof. Dr. Ioan DumitracheCorresponding Member of the Romanian Academy

Prof. Dr. Vladimir Răsvan

Prof. Dr. Octavian Păstrăvanu

soţiei meleto my wife

Contents

Acknowledgements 4

Summary 7

1 Introduction 111.1 Model predictive control . . . . . . . . . . . . . . . . . . . . . 111.2 Open problems in stability and robustness of MPC . . . . . . 16

1.2.1 Stability of MPC . . . . . . . . . . . . . . . . . . . . . 161.2.2 Robust MPC schemes . . . . . . . . . . . . . . . . . . 18

1.3 Summary of publications . . . . . . . . . . . . . . . . . . . . . 211.4 Basic mathematical notation and definitions . . . . . . . . . . 22

2 Lyapunov Functions Subtleties for Discrete-time Systems 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Stability and input-to-state stability . . . . . . . . . . 272.2.2 Lyapunov functions . . . . . . . . . . . . . . . . . . . . 29

2.3 Illuminating examples . . . . . . . . . . . . . . . . . . . . . . 302.4 ISS tests based on discontinuous USL functions . . . . . . . . 362.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Predictive control of hybrid systems: Input-to-state stabilityresults for suboptimal solutions 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 MPC scheme set-up . . . . . . . . . . . . . . . . . . . . . . . 473.4 Input-to-state stability results . . . . . . . . . . . . . . . . . . 493.5 Asymptotic stability results . . . . . . . . . . . . . . . . . . . 543.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 On Input-to-State Stability of Min-max Nonlinear ModelPredictive Control 574.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 594.2 Input-to-state stability . . . . . . . . . . . . . . . . . . . . . . 594.3 Min-max nonlinear MPC: Problem set-up . . . . . . . . . . . 64

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4.4 ISpS results for min-max nonlinear MPC . . . . . . . . . . . . 664.5 Main result: ISS dual-mode min-max MPC . . . . . . . . . . 684.6 Illustrative example: A nonlinear double integrator . . . . . . 734.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Design of the terminal cost:H∞ and min-max MPC 775.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.1 Input-to-state stability . . . . . . . . . . . . . . . . . . 795.2.2 Input-to-state stability conditions for min-max robust

MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1 Existing solutions . . . . . . . . . . . . . . . . . . . . . 825.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4.1 LMI-based-solution . . . . . . . . . . . . . . . . . . . . 835.4.2 Relation to LMI-based H∞ control design . . . . . . . 84

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Self-optimizing robust nonlinear MPC 896.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Preliminary definitions and results . . . . . . . . . . . . . . . 90

6.2.1 ISS definitions and results . . . . . . . . . . . . . . . . 916.2.2 Inherent ISS through continuous and convex control

Lyapunov functions . . . . . . . . . . . . . . . . . . . . 926.3 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . 936.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4.1 Optimized ISS through convex CLFs . . . . . . . . . . 946.4.2 Self-optimizing robust nonlinear MPC . . . . . . . . . 966.4.3 Decentralized formulation . . . . . . . . . . . . . . . . 986.4.4 Implementation issues . . . . . . . . . . . . . . . . . . 101

6.5 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . 1026.5.1 Example 1: control of a nonlinear system . . . . . . . 1026.5.2 Example 2: control of a DC-DC converter . . . . . . . 1036.5.3 Example 3: control of networked nonlinear systems . . 110

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Conclusions 1137.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1.1 Stability theory for discrete-time systems . . . . . . . 113

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7.1.2 Input-to-State stability theory for discrete-time dis-continuous Systems . . . . . . . . . . . . . . . . . . . . 114

7.1.3 Stabilizing nonlinear model predictive control . . . . . 1157.1.4 Robust nonlinear model predictive control . . . . . . . 1157.1.5 Low complexity nonlinear MPC . . . . . . . . . . . . . 116

7.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography 119

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Acknowledgements

This thesis presents the results of the research carried out during the periodSeptember 2001 - September 2002 and September 2006 - September 2009,under the supervision of Prof. Mihail Voicu, thesis promotor, and in closecollaboration with Prof. Octavian Păstrăvanu.

The completion of the research that led to the results published in thisthesis would have not been possible without the constant support, patienceand advice received from Prof. Voicu and Prof. Păstrăvanu and as such, mygratitude goes to them.

I am very grateful to Prof. Ioan Dumitrache and Prof. Vladimir Răsvanfor kindly agreeing to participate in the committee of this thesis and at thedefence ceremony. Also, I am very grateful to Prof. Vasile-Ion Manta foragreeing to chair the defense committee of this thesis.

I would like to thank Prof. Paul van den Bosch for his helpful advice andencouragement. He has always been there for me when I needed his opinionand he supported me through my career as a researcher.

This thesis is largely based on a collection of articles published in inter-national peer reviewed conferences and journals. As most of the articles arejoint work with several collaborators, I would like to express my gratitudeto all the co-authors.

First and foremost I am very grateful to Prof. Maurice Heemels, withoutwhom the research gathered in this thesis would have not been possible. Hisconstant dedication, supervision and professionalism will always be a sourceof inspiration for me.

I would like to thank Prof. Andrew (Andy) R. Teel for his contributionsto the research presented in Chapter 2 and for sharing his knowledge. Iam also grateful to Dr. David Muñoz de la Peña, Dr. Teodoro Alamo, Dr.Davide M. Raimondo, Prof. Lalo Magni, Dr. Daniel Limon, Prof. EduardoF. Camacho, Dr. Bas J.P. Roset, Prof. Henk Nijmeijer for their contributionto our joint works.

A special thanks goes to my colleague and friend, Dr. Andrej Jokić, whohas provided me with constant support and has had important contributionsin several research matters. Working together with him will always be a niceexperience.

Special thanks also go to Prof. Alberto Bemporad, my mentor and guidein the MPC world and to Dr. Stefano Di Cairano, with whom I enjoyed very

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much working together and having fun at the conferences.I am very grateful to Prof. Ilya V. Kolmanovsky for his constant support

and encouragement and for sharing his knowledge.I am eternally indebted to my wife Raluca, my parents Roxana and

Corneliu, my parents-in-law, Paulina and Traian, my grandparents on thefather side, Eleonora and Ilie, and my grandparents on the mother side,Magdalena and Florin, for all their support and love.

This thesis is dedicated to my wife.

Mircea LazarEindhoven, June, 2009

Summary

This thesis considers the stabilization and the robust stabilization of discrete-time systems using model predictive control.

Model predictive control (MPC) (also referred to as receding horizon con-trol) is a control strategy that offers attractive solutions, already successfullyimplemented in industry, for the regulation of constrained linear or nonli-near systems. In this thesis, the MPC controller design methodology will beemployed for the regulation of constrained discrete-time systems. One of thereasons for the success of MPC algorithms is their ability to handle hard con-straints on states/outputs and inputs. Stability and robustness are probablythe most studied properties of MPC controllers, as they are indispensable topractical implementation. A complete theory on (robust) stability of MPChas been developed for linear and continuous nonlinear systems. However,these results do not carry over to discrete-time discontinuous systems easily.These challenges will be taken up in this thesis with the purpose of highligh-ting certain subtleties that arise in stabilization and robust stabilization viamodel predictive control.

As a starting point, in Chapter 2 of this thesis we consider stability ana-lysis of discrete-time discontinuous systems using Lyapunov functions. Wedemonstrate via simple examples that the classical second method of Lya-punov is precarious for discrete-time discontinuous system dynamics. Also,we indicate that a particular type of Lyapunov condition, slightly stron-ger than the classical one, is required to establish stability of discrete-timediscontinuous systems. Furthermore, we examine the robustness of the sta-bility property when it was attained via a discontinuous Lyapunov function.This is often the case for discrete-time systems in closed-loop with modelpredictive controllers. In contrast to existing results based on smooth Lya-punov functions, we develop several robust stability tests, in terms of theinput-to-state stability (ISS) property, that explicitly employ an availablediscontinuous Lyapunov function.

The subtleties exposed in Chapter 2 are employed in Chapter 3 to developa novel model predictive control scheme that achieves input-to-state stabi-lization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability is guaranteed when an optimal solution of the MPC op-timization problem is attained. Special attention is paid to the effect thatsub-optimal solutions have on ISS of the closed-loop system. This issue is

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of interest as firstly, the infimum of MPC optimization problems does nothave to be attained and secondly, numerical solvers usually provide onlysub-optimal solutions. An explicit relation is established between the devi-ation of the predictive control law from the optimum (called the optimalitymargin) and the resulting deterioration of the ISS property of the closed-loop system. By imposing stronger conditions on the sub-optimal solutions,ISS can even be attained in this case. Revealing this explicit relation is animportant result, as it provides an a priori bound on the evolution of theclosed-loop system state and leads to conditions that guarantee ISS even inthe presence of unaccounted sub-optimal solutions.

Discrete-time nonlinear systems that are affected, possibly simultaneous-ly, by parametric uncertainties and other disturbance inputs are consideredin Chapter 4. The min-max model predictive control methodology is employ-ed to obtain a controller that robustly steers the state of the system towardsa desired equilibrium. The aim is to provide a priori sufficient conditions forrobust stability of the resulting closed-loop system using the input-to-statestability framework. First, we show that only input-to-state practical stabi-lity can be ensured in general for closed-loop min-max MPC systems; andwe provide explicit bounds on the evolution of the closed-loop system sta-te. Then, we derive new conditions for guaranteeing ISS of min-max MPCclosed-loop systems, using a dual-mode approach.

The results developed in Chapter 4 hinge on the fact that a suitableterminal cost that satisfies the developed sufficient conditions for ISS mustbe a priori available. This problem is addressed in Chapter 5, which presentsa novel method for designing the terminal cost and the auxiliary controllaw (ACL) for robust MPC of uncertain linear systems, such that ISS is apriori guaranteed for the closed-loop system. The method is based on thesolution of a set of linear matrix inequalities (LMIs). An explicit relationis established between the proposed method and H∞ control design. Thisrelation shows that the LMI-based optimal solution of the H∞ synthesisproblem solves the terminal cost and ACL problem in min-max MPC, for aparticular choice of the stage cost. This result, which was somehow missingin the MPC literature, is of general interest as it connects well known linearcontrol problems to robust MPC design.

In Chapter 6 we start from the observation that the goal of existing designmethods for synthesizing control laws that achieve ISS is to a priori guaran-tee a predetermined closed-loop ISS gain. Consequently, the ISS property,with a predetermined, constant ISS gain, is in this way enforced for all statespace trajectories of the closed-loop system and at all time instances. As theexisting approaches, which are also employed in the design of MPC schemes

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that achieve ISS, can lead to overly conservative solutions along particulartrajectories, it is of high interest to develop a control (MPC) design methodwith the explicit goal of adapting the closed-loop ISS gain depending of theevolution of the state trajectory. Motivated by this, in Chapter 6 we proposea novel novel method for synthesizing robust MPC schemes with this fea-ture. The method employs convex control Lyapunov functions (CLFs) anddisturbance bounds to embed standard ISS conditions using a finite numberof inequalities. This leads to a finite dimensional optimization problem thathas to be solved on-line, in a receding horizon fashion. The proposed ine-qualities govern the evolution of the closed-loop state trajectory through thesublevel sets of the CLF. The unique feature of the proposed robust MPCscheme is to allow for the simultaneous on-line (i) computation of a controlaction that achieves ISS and (ii) minimization of the closed-loop ISS gaindepending of an actual state trajectory. As a result, the developed nonlinearMPC scheme is self-optimizing in terms of disturbance attenuation. Fromthe computational point of view, following a particular design recipe, theself-optimizing robust MPC algorithm can be implemented as a single linearprogram for discrete-time nonlinear systems that are affine in the control va-riable and the disturbance input. This renders the developed MPC schemesapplicable to fast nonlinear systems, which is demonstrated by controllinga Buck-Boost DC-DC converter that requires sampling times less than amillisecond. Furthermore, we demonstrate that the freedom to optimize theclosed-loop ISS gain on-line makes self-optimizing robust MPC suitable fordecentralized control of networks of nonlinear systems.

In conclusion, this thesis contains a series of significant advances in thesynthesis of model predictive controllers for discrete-time, possibly disconti-nuous systems that guarantees stable and robust closed-loop systems. Thelatter properties are indispensable for any application of these control algo-rithms in practice. In the set-ups of the MPC algorithms, a clear focus wasalso on keeping the on-line computational burden low via simpler stabilizingconstraints. The example on the control of DC-DC converters showed thatthe application to (very) fast systems comes within reach. This opens up acompletely new range of applications, next to the traditional process con-trol for typically slow systems. Therefore, the developed theory representsa fertile ground for future practical applications and it opens many roadsfor future research in model predictive control and stability of discrete-timesystems as well.

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Motto:

“Imagination is more important thanknowledge.”

- Albert Einstein

1

Introduction

1.1 Model predictive control1.2 Open problems in

stability and robustnessof MPC

1.3 Summary of publications1.4 Basic mathematical

notation and definitions

This thesis deals with the synthesis of stabilizing and robust controllersfor constrained discrete-time discontinuous nonlinear systems. An appealingsolution to the control of these systems is provided by the model predicti-ve control methodology, due to its capability to a priori take into accountconstraints when computing the control action. Also, since the principlesof model predictive control do not depend on the type of model applied forprediction, this methodology can be employed to formulate controller de-sign set-ups for general dynamical systems. However, the properties of suchcontrol schemes and the feasibility of their implementation have to be recon-sidered in the discontinuous context. In this thesis we focus in particular onstability and robustness. As such, in this chapter we will present a generalintroduction to the principles of MPC and then focus on open problems re-lated to stability and robustness that will be tackled in the remainder of thethesis.

1.1 Model predictive control

Model predictive control (MPC) (also referred to as receding horizon con-trol) is a control strategy that offers attractive solutions for the regulationof constrained linear or nonlinear systems and, more recently, also for theregulation of discontinuous and hybrid systems. Within a relatively short ti-me, MPC has reached a certain maturity due to the continuously increasinginterest shown for this distinctive part of control theory. This is illustratedby its successful implementation in industry and by many excellent articlesand books as well. See, for example, (Garcia et al., 1989; Mayne et al., 2000;Qin and Badgwell, 2003; Findeisen et al., 2003; Camacho and Bordons, 2004)and the references therein.

12 Introduction

The initial MPC algorithms utilized only linear input/output models. Inthis framework, several solutions have been proposed both in the industrialworld and in the academic world: IDCOM - Identification and command(later MAC - Model algorithmic control) at ADERSA (Richalet et al., 1978)and DMC - Dynamic matrix control at Shell (Cutler and Ramaker, 1980),which use step and impulse response models, (the adaptive control branch)MUSMAR - Multistep multivariable adaptive regulator (Mosca et al., 1984)- the first MPC formulation that is based on state-space linear models, andEPSAC - Extend predictive self-adaptive control (De Keyser and van Cau-wenberghe, 1985). Generalized frameworks for setting up MPC algorithmsbased on input/output models were also developed later on, from which themost significant ones are GPC - Generalized predictive control (Clarke et al.,1987) and UPC - Unified predictive control (Soeterboek, 1992). The nextstep of the academic community was to extend the MPC algorithms based onstate-space models to continuous (smooth) nonlinear systems, which includesthe following approaches: nonlinear MPC with zero state terminal equalityconstraint (Keerthi and Gilbert, 1988), dual-mode nonlinear MPC (Mic-halska and Mayne, 1993) and quasi-infinite horizon nonlinear MPC (Chenand Allgöwer, 1996). More recent general set-ups for synthesizing stabili-zing MPC algorithms for smooth nonlinear systems can be found in (Magniet al., 2001; Grimm et al., 2005). The first MPC approach for the con-trol of discontinuous and hybrid systems has been reported in the seminalwork (Bemporad and Morari, 1999), which was followed by many other re-searcher, see, for example, (Kerrigan and Mayne, 2002; Grieder et al., 2005;Lazar et al., 2006; Baotic et al., 2006) and the references therein.

One of the reasons for the fruitful achievements of MPC algorithms con-sists in the intuitive way of addressing the control problem. In comparisonwith conventional control, which often uses a pre-computed state or outputfeedback control law, predictive control uses a discrete-time1 model of thesystem to obtain an estimate (prediction) of its future behavior. This isdone by applying a set of input sequences to a model, with the measuredstate/ouput as initial condition, while taking into account constraints. Anoptimization problem built around a performance oriented cost function isthen solved to choose an optimal sequence of controls from all feasible se-quences. The feedback control law is then obtained in a receding horizonmanner by applying to the system only the first element of the computed

1Although continuous-time models can also be employed in theory of MPC, see (Mayneet al., 2000), most MPC algorithms and theory consider discrete-time models, as this yieldsa tractable optimization problem.

1.1. Model predictive control 13

sequence of optimal controls, and repeating the whole procedure at the nextdiscrete-time step. Summarizing the above discussion, one can conclude thatMPC is built around the following key principles:

• The explicit use of a process model for calculating predictions of thefuture plant behavior over a finite horizon in time;

• The optimization of an objective function subject to constraints, whichyields an finite optimal sequence of controls;

• The receding horizon strategy, according to which only the first ele-ment of the optimal sequence of controls is applied on-line and theoptimization problem is solved again at the next time instant with themeasured state as initial condition.

The MPC methodology involves solving on-line an open-loop finite horizonoptimal control problem subject to input, state and/or output constraints.

A graphical illustration of this concept is depicted in Figure 1.1.At each discrete-time instant k, the measured variables and the process

model (linear, nonlinear or hybrid) are used to (predict) calculate the futu-re behavior of the controlled plant over a specified time horizon, which isusually called the prediction horizon and is denoted by N . This is achie-ved by considering a future control scenario as the input sequence appliedto the process model, which must be calculated such that certain desiredconstraints and objectives are fulfilled. To do that, a cost function is mini-mized subject to constraints, yielding an optimal sequence of controls over aspecified time horizon, which is usually called control horizon and is denotedby Nu. According to the receding horizon control strategy, only the firstelement of the computed optimal sequence of controls is then applied to theplant and this sequence of steps is repeated at the next discrete-time instant,for the updated state.

The MPC methodology can be summarized formally as the followingconstrained optimization problem:

Problem 1.1.1 Let N ≥ 1 be given and let X ⊆ Rn and U ⊆ Rm be setsthat implement state and input constraints, respectively, and contain theorigin in their interior. The prediction model is x(k + 1) = g(x(k), u(k)),k ≥ 0, with g : Rn × Rm → Rn a nonlinear, possibly discontinuous functionwith g(0, 0) = 0. Let F : Rn → R+ with F (0) = 0 and L : Rn × Rm → R+

with L(0, 0) = 0 be known mappings. At every discrete-time instant k ≥ 0

14 Introduction

P a s t F u tu re /P re d ic t io n s

In it ia l S ta te x 0

D e s ire d e q u il ib r iu m p o in t x r

C lo s e d - lo o p s ta te x k

P re d ic te d s ta te kx

S ta te c o n s tra in t

C lo s e d - lo o p in p u t u k

O p e n - lo o p in p u t ku

In p u t c o n s t ra in t

k C o n tro l h o r iz o n

P re d ic tio n h o r iz o n

k + N u k + N

Figure 1.1: A graphical illustration of Model Predictive Control.

let x(k) ∈ X be the measured state, let x(0|k) , x(k) and minimize the costfunction

J(x(k),u(k)) , F (x(N |k)) +N−1∑i=0

L(x(i|k), u(i|k)),

over all input sequences u(k) , (u(0|k), . . . , u(N − 1|k)) subject to the con-straints:

x(i + 1|k) , g(x(i|k), u(i|k)), i = 0, . . . , N − 1,

x(i|k) ∈ X, for all i = 1, . . . , N,

u(i|k) ∈ U, for all i = 0, . . . , N − 1.

In Problem 1.1.1, F (·), L(·, ·) and N denote the terminal cost, the stage costand the prediction horizon, respectively. The term x(i|k) denotes the predic-ted state at future discrete-time instant i ∈ [0, N ], obtained at discrete-time

1.1. Model predictive control 15

instant k ≥ 0 by applying the input sequence u(i|k)i=0,...,N−1 to a modelof the system, i.e. x(k + 1) = g(x(k), u(k)), with the measured state x(k)as initial condition, i.e. x(0|k) = x(k). The control actions in the sequenceu(i|k)i=0,...,N−1 constitute the optimization variables. Suppose that theabove MPC optimization problem is solvable and let u(i|k)∗i=0,...,N−1 de-note an optimal solution. The MPC control action is obtained as follows:

uMPC(x(k)) , u(0|k)∗; k ≥ 0.

Although the key principles of MPC are independent of the type of system,e.g. linear, nonlinear or hybrid, the computational complexity of the MPCconstrained optimization problem, as well as the stability issues, stronglydepend on the type of model used for prediction. For instance, assumingthat the MPC cost is defined using quadratic forms (Hahn, 1967) and theconstraint sets are polyhedra,

• Problem 1.1.1 is a quadratic programming problem if the model islinear;

• Problem 1.1.1 is a nonlinear optimization problem if the model is non-linear;

• Problem 1.1.1 is a mixed integer quadratic programming problem (Bem-porad and Morari, 1999) if the model is piecewise affine.

Therefore, depending on the utilized prediction model and MPC cost func-tion, different tools are required for solving the MPC optimization problem.

One of the most studied research problems regarding MPC, which is alsoaddressed in this thesis, consists in how to guarantee stability of a system inclosed-loop with an MPC controller, e.g. obtained by solving Problem 1.1.1,as this is not automatically guaranteed and is the primal condition that anycontroller should satisfy. For linear and continuous nonlinear systems, manysolutions to this problem have been developed, see the survey (Mayne et al.,2000) for a comprehensive and well documented overview. The most popularapproach is the so-called terminal cost and constraint set method, which re-quires that the terminal predicted state, i.e. x(N |k), is constrained inside aterminal set that contains the origin (the equilibrium) in its interior. Then,under the assumption that the system dynamics and the MPC value func-tion corresponding to Problem 1.1.1 are continuous, sufficient stabilizationconditions, in terms of properties that a terminal cost F (·) and a terminalconstraint set (usually denoted by XT ) must satisfy, can be found in (Mayneet al., 2000).

16 Introduction

This concludes the general introduction to MPC and this chapter con-tinues with a discussion of several relevant open problems in the theory ofmodel predictive control.

1.2 Open problems in stability and robustness ofMPC

Typically, stability and robustness results for discrete-time systems are ob-taining by mutatis mutandis reproducing the classical results available forcontinuous-time systems, see, for example, (Kalman and Bertram, 1960a,b;Freeman, 1965; Willems, 1970; LaSalle, 1976; Vidyasagar, 1993; Khalil, 2002;Jiang and Wang, 2001; Kellett and Teel, 2004). In general, less attention ispayed to relaxations of the sufficient conditions for Lyapunov stability thatmight be allowed by the discrete-time setting and their implications in termsof robustness, e.g., in the form of input-to-state stability (ISS) (Jiang andWang, 2001). One particularly relevant point is whether global or local(i.e. on a neighborhood of the equilibrium) continuity of the system dyna-mics and/or of the candidate Lyapunov function is still required to establishasymptotic stability in the Lyapunov sense. This issue is of paramount im-portance to MPC closed-loop systems, as it is well known, especially sincethe seminal work on hybrid systems (Bemporad and Morari, 1999), see also(Borrelli, 2003), that MPC candidate Lyapunov functions and closed-loopsystems are discontinuous in general. This is due to the fact that MPCusually generates a discontinuous control law, even for continuous systemdynamics, which was shown for the first time in (Meadows et al., 1995).

1.2.1 Stability of MPC

The stability results within the MPC framework follow closely the abovementioned general stability results for discrete-time systems, but with asharp focus on removing continuity assumptions, as summarized next. Theusual approach to ensure stability in MPC is to consider the value functionof the MPC cost as a candidate Lyapunov function. Then, if the system dy-namics is continuous, the classical Lyapunov stability theory (Kalman andBertram, 1960b) can be used to prove that the MPC control law is stabili-zing, which was done in (Keerthi and Gilbert, 1988). The requirement thatthe system dynamics must be continuous is (partially) removed in (Alamirand Bornard, 1994; Meadows et al., 1995), where terminal equality cons-traint MPC is considered. In (Alamir and Bornard, 1994), continuity of the

1.2. Open problems in stability and robustness of MPC 17

system dynamics on a neighborhood of the origin is still used to prove Lya-punov stability, but not for attractivity. Although continuity of the systemis still assumed in (Meadows et al., 1995), where it is shown that MPC cangenerate discontinuous state-feedbacks, the Lyapunov stability proof (Theo-rem 2 in (Meadows et al., 1995)) does not use the continuity property. Lateron, an exponential stability result is given in (Scokaert et al., 1997) and anasymptotic stability theorem is presented in (Scokaert et al., 1999), wheresub-optimal MPC is considered. The theorems of (Scokaert et al., 1997,1999) explicitly point out that both the system dynamics and the candidateLyapunov function only need to be continuous at the equilibrium. Stabili-ty of sub-optimal MPC is proven in (Scokaert et al., 1999) under the usualassumptions (existence of class K bounds on the candidate Lyapunov func-tion V and its forward difference) plus the extra requirement that the MPCoptimal sequence of controls is upper bounded in norm by a K function ofthe norm of the state. A recent overview on stability of receding horizoncontrol in discrete-time can be found in (Goodwin et al., 2005). Althoughcontinuity of the system dynamics and local continuity of V are assumed in(Goodwin et al., 2005), the stability proof (Theorem 4.3.2 in (Goodwin et al.,2005)) only uses continuity of V at the equilibrium, as done in (Meadowset al., 1995). The interested reader can find a general stability theorem fordiscrete-time MPC that unifies most of the above results in (Lazar et al.,2007a).

Apart from removing the continuity assumption on the system dynamicsand MPC cost function, all these results employ the additional assumptionthat the (global) optimum of the MPC optimization problem is always at-tained, which is usually referred to as the “optimality assumption” in MPC.Recently, in (Spjøtvold et al., 2007) it was shown that, similarly to the con-tinuity assumption, the optimality assumption is not a realistic one as well.This is because in the presence of a discontinuous value function corres-ponding to the cost of the optimization problem, which is usually the casewith MPC cost functions, although the global optimum may exists, it is notnecessarily attainable.

This rises the following open problem in stability of MPC: (i) what canbe said about stability of classical terminal cost and constraint set MPCschemes (Mayne et al., 2000) in the presence of discontinuous dynamics,value functions and/or sub-optimal solutions?

Notice that although in (Scokaert et al., 1999) stability results are ob-tained for sub-optimal MPC schemes, this is attained via additional modi-fications to the classical MPC set-up (Mayne et al., 2000). More precisely,an explicit nonlinear and nonconvex constraint that involves the MPC cost

18 Introduction

function is added to the MPC set-up, which significantly hampers implemen-tation. In contrast to (Scokaert et al., 1999), our aim is to obtain stabilityresults for sub-optimal MPC solutions without bringing any modificationsto the original terminal cost and constraint set MPC set-up. A solution tothis open problem is provided in Chapter 4 of this thesis by making use ofthe general stability results presented in Chapter 3.

An equally relevant and disturbing issue was raised in (Grimm et al.,2004), where it was shown for the first time that MPC closed-loop systemsthat are asymptotically stable have zero robustness. That is, in the presenceof arbitrarily small perturbations, the asymptotic stability property is lost.The fragility of the stability of MPC closed-loop systems is in fact relatedto the absence of a continuous Lyapunov function. As the usual candidatefor a Lyapunov function in MPC is the value function corresponding to thecost J(·, ·), normally a discontinuous function, the following open problemarises: (ii) what can be said about inherent robustness of asymptoticallystable discrete-time systems, when either the system dynamics or the Lya-punov function employed to establish stability, or both, are discontinuous?Notice that while it is well known that smooth Lyapunov functions implyinherent robustness, even in the sense of ISS, to the best of the author’sknowledge, there are no robustness test that rely exclusively on a discon-tinuous Lyapunov function. As such tests are crucial for MPC closed-loopsystems, several possible solutions are presented in Chapter 3 of this thesis.

1.2.2 Robust MPC schemes

Next, we continue the discussion on stability and robustness of MPC bypresenting a short summary of methods for designing MPC schemes withan a priori guarantee of robustness in the sense of input-to-state stability(Sontag, 1989, 1990; Jiang and Wang, 2001).

There are several ways for designing robust MPC controllers for pertur-bed nonlinear systems. One way is to rely on the inherent robustness proper-ties of nominally stabilizing nonlinear MPC algorithms, e.g. as it was done in(Scokaert et al., 1997; Magni et al., 1998; Limon et al., 2002b; Grimm et al.,2003). Another approach is to incorporate knowledge about the disturbancesin the MPC problem formulation via open-loop worst case scenarios. Thisincludes MPC algorithms based on tightened constraints, e.g., as the one of(Limon et al., 2002a), and MPC algorithms, based on open-loop min-maxoptimization problems, see, for example, the survey (Mayne et al., 2000).

As it was the case with the nominal stability results discussed in thischapter, ISS results for tightened constraints terminal cost and constraint set

1.2. Open problems in stability and robustness of MPC 19

MPC rely on the same basic assumptions: continuity of the system dynamics(Grimm et al., 2003) or even Lipschitz continuity (Limon et al., 2002a) and,optimality of the MPC solution. This gives rise to an open problem similar tothe one raised for nominal stability, i.e.: (iii) what can be said about input-to-state stability of tightened constraints robust MPC schemes in the presenceof discontinuous dynamics, value functions and/or sub-optimal solutions? Apossible solution to this problem is presented in Chapter 4 of this thesis.

To incorporate feedback to disturbances, the closed-loop or feedback min-max MPC (or shortly, min-max MPC) problem set-up was introduced in (Leeand Yu, 1997) and further developed in (Mayne, 2001; Magni et al., 2003;Limon et al., 2006; Magni et al., 2006). The open-loop approach is computa-tionally somewhat easier than the feedback approach, but the set of feasiblestates corresponding to the feedback min-max MPC optimization problemis usually much larger and the disturbance rejection is improved. Sufficientconditions for robust asymptotic stability of closed-loop (feedback) min-maxMPC systems were presented in (Mayne, 2001) under the assumption thatthe (additive) disturbance input converges to zero as the state converges tothe origin.

Recently, input-to-state stability (ISS) (Sontag, 1989, 1990; Jiang andWang, 2001) results for min-max nonlinear MPC were presented in (Limonet al., 2006) and (Magni et al., 2006). In (Limon et al., 2006) it was shownthat, in general, only input-to-state practical stability (ISpS) (Jiang, 1993;Jiang et al., 1994, 1996) can be a priori ensured for min-max nonlinear MPC.ISpS is a weaker property than ISS, as ISpS does not imply asymptoticstability for zero disturbance inputs. The reason for the absence of ISS ingeneral is that the effect of a non-zero disturbance input is taken into accountby the min-max MPC controller, even if the disturbance input vanishes inreality. Still, in the case when the disturbance input converges to zero, it isdesirable that asymptotic stability is recovered for the controlled system.

The first open problem related min-max MPC is (iv) under what con-ditions/modifications can ISS, rather than ISpS, can be a priori guaranteedfor min-max MPC closed-loop systems?

In (Magni et al., 2006), an H∞ (Chen and Scherer, 2006a) strategy wasused to modify the classical min-max MPC cost function (Mayne et al., 2000)such that ISS is guaranteed for the closed-loop min-max MPC system. Fur-thermore, in (Magni et al., 2006) it was proven that a local upper boundon the min-max MPC value function, rather than a global one, is sufficientfor ISS. However, this method requires the modification of the stage cost byintroducing a negative term which consists of a disturbance norm. In thisway, the corresponding min-max optimization problem becomes non-convex

20 Introduction

in the disturbance, which is a significant drawback regarding implementa-tion. As such, our goal is to provide a solution to this problem withoutincorporating additional terms in the standard min-max MPC cost, which isstill possible by employing a dual-mode approach, as presented in Chapter 5of this thesis.

The second open problem in min-max MPC is (v) how to compute aterminal cost and auxiliary control law that satisfy the sufficient conditionsfor input-to-state stability? While a solution to the computational of theterminal cost exists in the nominal case, i.e. it amounts to take the terminalcost equal to a local control Lyapunov function, for the robust case, it wouldamount to the computation of ISS control Lyapunov functions, which is stillan open problem. In Chapter 6 of this thesis we present a possible solutionfor solving this problem for quadratic candidate ISS CLFs. Furthermore,we demonstrate that the solution of the H∞ synthesis problem solves thecorresponding terminal cost min-max MPC problem for a particular choiceof the terminal cost.

The problems raised so far with respect to existing techniques for desig-ning robust MPC schemes still do not offer a solution to the ultimate openproblem in robust MPC: (vi-a) how to provide feedback to the disturbancesactively, on-line, as a function of the closed-loop trajectory, rather than ina worst case manner, i.e. imposing a fixed ISS gain for all possible trajecto-ries; and (vi-b) how to render the corresponding robust MPC optimizationproblems computationally efficient?

A novel and innovative solution to this problem is presented in Chap-ter 7 of this thesis, which introduces the concept of “self-optimizing” robustMPC, in the sense that this MPC scheme provides the means to optimize theclosed-loop ISS gain on-line, as a function of the state trajectory. Furthermo-re, in terms of computational complexity, for a fairly wide class of nonlinearsystems it is shown that the corresponding self-optimizing robust MPC op-timization problem can be formulated as a single linear program, which is amajor step in complexity reduction compared with standard min-max MPC.

A case study on the control of DC-DC converters that includes prelimi-nary real-time computational results is included to illustrate the potential ofthe developed theory for practical applications. As the sampling period ofthe considered DC-DC converter is well below one millisecond, this indicatesthat the proposed self-optimizing robust MPC scheme is implementable for(very) fast systems, which opens up a whole new range of industrial appli-cations in electrical, mechatronic and automotive systems.

The following summarizing formal statement concludes the section onopen problems. This thesis focuses mainly on novel ways to design MPC

1.3. Summary of publications 21

controllers with a robust stability guarantee. Special attention is paid todiscontinuous nonlinear system dynamics, sub-optimal solutions, low com-putational complexity and improved disturbance rejection.

1.3 Summary of publications

This thesis is mostly based on published or submitted articles. A completelist of the publications that support this thesis is presented in this section,as follows.

Chapter 2 contains results presented in:

• (Lazar et al., 2007b): M. Lazar, W.P.M.H. Heemels, A.R. Teel. Subtletiesin robust stability of discrete-time PWA systems. In proceedings of the 26thAmerican Control Conference 2007, New York, USA.

• (Lazar et al., 2009c): M. Lazar, W.P.M.H. Heemels, A.R. Teel. Lyapunovfunctions, stability and input-to-state stability subtleties for discrete-timediscontinuous systems. IEEE Transactions on Automatic Control, accepted,scheduled to appear in the September, 2009 issue.

The results presented in Chapter 3 are published in:

• (Lazar and Heemels, 2008c): M. Lazar, W.P.M.H. Heemels. Predictive con-trol of hybrid systems: Stability results for sub-optimal solutions. 17th IFACWorld Congress, Seoul, Korea, 2009.

• (Lazar and Heemels, 2009): M. Lazar, W.P.M.H. Heemels. Predictive controlof hybrid systems: Input-to-state stability results for sub-optimal solutions.Automatica, Vol. 45, No. 1, pp. 180-185, 2009.

Chapter 4 is based on:

• (Lazar et al., 2008a): M. Lazar, D. Muñoz de la Peña, W.P.M.H. Heemels andT. Alamo. On input-to-state stability of min-max nonlinear model predictivecontrol. Systems & Control Letters, Vol. 57, pp. 39-48, 2008.

• (Raimondo et al., 2009): D.M. Raimondo, D. Limon, M. Lazar, L. Magni,E.F. Camacho. Min-max model predictive control of nonlinear systems: Aunifying overview on stability. Survey paper (discussants: J. Maciejowskiand J.A. Rossiter). European Journal of Control, Vol. 15, No. 1, pp. 1-17.

The results of Chapter 5 are presented in:

• (Lazar et al., 2009b): M. Lazar, W.P.M.H. Heemels , D. Muñoz de la Peñaand T. Alamo. Further results on “Robust MPC using Linear Matrix Inequa-lities”. L. Magni et al., Eds., Assessment and Future Directions of NonlinearModel Predictive Control, Lecture Notes in Control and Information Scien-ces, vol. 384, pages 89-98, Springer-Verlag.

22 Introduction

Chapter 6 contains results presented in:

• (Lazar and Heemels, 2008b): M. Lazar, W.P.M.H. Heemels. Optimizedinput-to-state stabilization of discrete-time nonlinear systems with boundedinputs. In Proceedings of the 27th American Control Conference, Seattle,USA, 2008.

• (Lazar et al., 2008b): M. Lazar, B.J.P. Roset, W.P.M.H. Heemels, H. Nijmeij-er and P.P.J. van den Bosch. Input-to-state stabilizing sub-optimal nonlinearMPC algorithms with an application to DC-DC converters. InternationalJournal of Robust and Nonlinear Control, Invited paper for the Special Issueon Nonlinear MPC of Fast Systems, Vol. 18, Issue 8, pages 890-904, 2008.

• (Lazar et al., 2009a): M. Lazar, W.P.M.H. Heemels, A. Jokic. Self-optimizingRobust Nonlinear Model Predictive Control. L. Magni et al., Eds., Assess-ment and Future Directions of Nonlinear Model Predictive Control, LectureNotes in Control and Information Sciences, vol. 384, pages 27-40, Springer-Verlag.

1.4 Basic mathematical notation and definitions

In this section, some basic mathematical notation and standard definitionsare recalled to make the manuscript self-contained.

Sets and operations with sets:

• R, R+, Z and Z+ denote the field of real numbers, the set of non-negative reals, the set of integers and the set of non-negative integers,respectively;

• Z≥c1 and Z(c1,c2] denote the sets k ∈ Z+ | k ≥ c1 and k ∈ Z+ | c1 <k ≤ c2, respectively, for some c1, c2 ∈ Z+;

• For a set S ⊆ Rn, SN denotes the N -dimensional Cartesian productS× . . .× S, for some N ∈ Z≥1;

• For a set P ⊆ Rn, ∂P denotes the boundary of P, int(P) denotesthe interior of P, cl(P) denotes the closure of P, card(P) denotes thenumber of elements of P and Co(P) denotes the convex hull of P;

• For any real λ ≥ 0 and set P ⊆ Rn, the set λP is defined as

λP , x ∈ Rn | x = λy for some y ∈ P;

1.4. Basic mathematical notation and definitions 23

• For two arbitrary sets P1 ⊆ Rn and P2 ⊆ Rn, P1 ∪ P2 denotes theirunion, P1 ∩ P2 denotes their intersection, P1 \ P2 denotes their setdifference, P1 ⊂ P2 (or P1 ( P2) denotes “P1 is subset of, but notequal to, P2”, P1 ⊆ P2 denotes “P1 is subset of, or equal to P2”;

• For two arbitrary sets P1 ⊆ Rn and P2 ⊆ Rn,

P1 ∼ P2 , x ∈ Rn | x + P2 ⊆ P1

denotes their Pontryagin difference and

P1 ⊕ P2 , x + y | x ∈ P1, y ∈ P2

denotes their Minkowski sum;

• A convex and compact set in Rn that contains the origin in its interioris called a C-set;

• A polyhedron (or a polyhedral set) in Rn is a set obtained as theintersection of a finite number of open and/or closed half-spaces;

• A piecewise polyhedral set is a set obtained as the union of a finitenumber of polyhedral sets.

Vectors, matrices and norms:

• For a real number a ∈ R, |a| denotes its absolute value and dae denotesthe smallest integer larger than a;

• For a sequence zjj∈Z+ with zj ∈ Rl, z[k] denotes the truncation ofzjj∈Z+ at time k ∈ Z+, i.e. z[k] = zjj∈Z[0,k]

, and z[k1,k2] denotesthe truncation of zjj∈Z+ at times k1 ∈ Z≥1 and k2 ∈ Z≥k1 , i.e.z[k1,k2] = zjj∈Z[k1,k2]

;

• The Hölder p-norm of a vector x ∈ Rn is defined as:

‖x‖p ,

(|x1|p + . . . + |xn|p)

1p , p ∈ Z[1,∞)

maxi=1,...,n |xi|, p = ∞,

where xi, i = 1, . . . , n is the i-th component of x, ‖x‖2 is also called theEuclidean norm and ‖x‖∞ is also called the infinity (or the maximum)norm;

24 Introduction

• Let ‖ · ‖ denote an arbitrary Hölder p-norm. For a sequence zjj∈Z+

with zj ∈ Rn,‖zjj∈Z+‖ , sup‖zj‖ | j ∈ Z+;

• In denotes the identity matrix of dimension n× n;

• For some matrices L1, . . . , Ln, diag([L1, . . . , Ln]) denotes a diagonalmatrix of appropriate dimensions with the matrices L1, . . . , Ln on themain diagonal;

• For a matrix Z ∈ Rm×n and p ∈ Z≥1 or p = ∞

‖Z‖p , supx 6=0

‖Zx‖p

‖x‖p,

denotes its induced matrix norm. It is well known, see, for example,(Golub and Van Loan, 1989), that ‖Z‖∞ = max1≤i≤m

∑nj=1 |Zij|,

where Zij is the ij-th entry of Z;

• For a matrix Z ∈ Rm×n, Z> denotes its transpose and Z−1 denotes itsinverse (if it exists);

• For a matrix Z ∈ Rn×n, Z > 0 denotes “Z is positive definite”, i.e. forall x ∈ Rn \ 0 it holds that x>Zx > 0, and Z = Z>;

• For a matrix Z ∈ Rm×n with full-column rank, Z−L , (Z>Z)−1Z>

denotes the Moore-Penrose inverse of Z, which satisfies Z−LZ = In;

• For a positive definite and symmetric matrix Z, Z12 denotes its Cho-

lesky factor, which satisfies (Z12 )>Z

12 = Z

12 (Z

12 )> = Z;

• For a positive definite matrix Z, λmin(Z) and λmax(Z) denote the smal-lest and the largest eigenvalue of Z, respectively.

2

Lyapunov Functions Subtleties forDiscrete-time Systems

2.1 Introduction2.2 Preliminaries2.3 Illuminating examples

2.4 ISS tests based ondiscontinuous USL functions

2.5 Conclusions

In this chapter we consider stability analysis of discrete-time discon-tinuous systems using Lyapunov functions. We demonstrate via simpleexamples that the classical second method of Lyapunov is precarious fordiscontinuous system dynamics. Also, we indicate that a particular type ofLyapunov condition, slightly stronger than the classical one, is required toestablish stability of discrete-time discontinuous systems. Furthermore, weexamine the robustness of the stability property when it was attained via adiscontinuous Lyapunov function, which is often the case for discrete-timesystems in closed-loop with MPC controllers. In contrast to existing resultsbased on smooth Lyapunov functions, we develop several input-to-state sta-bility tests that explicitly employ an available discontinuous Lyapunov func-tion.

2.1 Introduction

Discrete-time discontinuous systems, such as piecewise affine (PWA) sys-tems, are a powerful modeling class for the approximation of hybrid andnon-smooth nonlinear dynamics (Sontag, 1981; Heemels et al., 2001). Themodeling capability of discrete-time PWA systems has already been shown inseveral applications, including switched power converters (Leenaerts, 1996),direct torque control of three-phase induction motors (Geyer et al., 2005)and applications in automotive systems (Bemporad et al., 2003). Many nu-merically efficient tools for stability analysis and stabilizing controller syn-thesis for discrete-time PWA systems have already been developed, see, forexample, (Johansson, 1999; Mignone et al., 2000; Ferrari-Trecate et al., 2002;

26 Lyapunov Functions Subtleties for Discrete-time Systems

Feng, 2002; Daafouz et al., 2002) for static feedback methods and (Lazaret al., 2005; Grieder et al., 2005; Lazar et al., 2006; Baotic et al., 2006) formodel predictive control (MPC) techniques. Most of these methods ma-ke use of classical Lyapunov methods (Kalman and Bertram, 1960b). Thefirst contribution of this chapter is to illustrate the precariousness of thesecond method of Lyapunov, as presented in (Kalman and Bertram, 1960b),for discontinuous system dynamics. We illustrate via a simple example thatexistence of a Lyapunov function in the sense of Corollary 1.2 of (Kalman andBertram, 1960b) (and hence, a continuous function) does not even guaranteeglobal asymptotic stability (GAS) for discrete-time discontinuous systems.In the presence of discontinuity of the dynamics one needs to impose a classK∞ upper bound on the one-step rate of decrease of the Lyapunov functionin order to attain GAS.

The second contribution of this chapter concerns robustness of stabilityin terms of input-to-state stability (ISS) (Jiang and Wang, 2001). Firstly,we present a simple example inspired from (Kellett and Teel, 2004) (see also(Grimm et al., 2004) for a similar example in MPC) to illustrate that eventhe global exponential stability (GES) property is precarious for discrete-time discontinuous systems affected by arbitrary small perturbations. Thesevere lack of inherent robustness is related to the absence of a continuousLyapunov function. This example establishes that there exist GES discrete-time systems that admit a discontinuous Lyapunov function, but not a con-tinuous one. Notice that previous results on stability of discrete-time PWAsystems (Johansson, 1999; Mignone et al., 2000; Ferrari-Trecate et al., 2002;Feng, 2002; Daafouz et al., 2002) only indicated that continuous Lyapunovfunctions may be more difficult to find than discontinuous ones, while infact a continuous Lyapunov function might not even exist. As such, a validwarning regarding nominally stabilizing state-feedback synthesis methods fordiscrete-time discontinuous systems, including both static feedback appro-aches (Johansson, 1999; Mignone et al., 2000; Ferrari-Trecate et al., 2002;Feng, 2002; Daafouz et al., 2002) and MPC techniques (Lazar et al., 2005;Grieder et al., 2005; Lazar et al., 2006; Baotic et al., 2006) arises. Thesesynthesis methods lead to a stable, possibly discontinuous closed-loop sys-tem and often rely on discontinuous Lyapunov functions. For example, inMPC the most natural candidate Lyapunov function is the value functioncorresponding to the MPC cost, which is generally discontinuous when PWAsystems are used as prediction models (Lazar et al., 2006). Hence, these con-trollers may result in closed-loop systems that are GAS, but only admit adiscontinuous Lyapunov function. This means that such closed-loop systemsmay not be ISS to arbitrarily small perturbations, which are always present

2.2. Preliminaries 27

in practice.This brings us to the second contribution of this chapter: for discrete-

time systems for which only a discontinuous Lyapunov function is known,we propose several robustness tests that can establish ISS solely based onthe available discontinuous Lyapunov function.

2.2 Preliminaries

In this section we introduce some preliminary notions, definitions and results.Let R, R+, Z and Z+ denote the field of real numbers, the set of non-

negative reals, the set of integer numbers and the set of non-negative integers,respectively. For every subset Π of R+ we define ZΠ := k ∈ Z+ | k ∈ Π.Let ‖ · ‖ denote an arbitrary norm on Rn and let | · | denote the absolutevalue of a real number. For a sequence z := z(l)l∈Z+ with z(l) ∈ Rn,l ∈ Z+, let ‖z‖ := sup‖z(l)‖ | l ∈ Z+ and let z[k] := z(l)l∈Z[0,k]

. Fora set S ⊆ Rn, we denote by int(S) the interior, by ∂S the boundary andby cl(S) the closure of S. For two arbitrary sets S ⊆ Rn and P ⊆ Rn, letS ⊕ P := x + y | x ∈ S, y ∈ P denote their Minkowski sum. The distanceof a point x ∈ Rn from a set P is denoted by d(x,P) := infy∈P ‖x − y‖.For any µ ∈ R(0,∞) we define Bµ := x ∈ Rn | ‖x‖ ≤ µ. A polyhedron (ora polyhedral set) is a set obtained as the intersection of a finite number ofopen and/or closed half-spaces. The p-norm of a vector x ∈ Rn is definedas ‖x‖p := (|x1|p + . . . + |xn|p)

1p for p ∈ Z[1,∞) and ‖x‖∞ := maxi=1,...,n |xi|,

where xi, i = 1, . . . , n is the i-th component of x. For a matrix Z ∈ Rm×n let‖Z‖p := supx 6=0

‖Zx‖p

‖x‖p, p ∈ Z[1,∞), p = ∞ denote its induced matrix norm. A

function ϕ : R+ → R+ belongs to class K (ϕ ∈ K) if it is continuous, strictlyincreasing and ϕ(0) = 0. A function ϕ : R+ → R+ belongs to class K∞(ϕ ∈ K∞) if ϕ ∈ K and lims→∞ φ(s) = ∞. A function β : R+ × R+ → R+

belongs to class KL (β ∈ KL) if for each fixed k ∈ R+, β(·, k) ∈ K and foreach fixed s ∈ R+, β(s, ·) is decreasing and limk→∞ β(s, k) = 0.

2.2.1 Stability and input-to-state stability

To study robustness, we will employ the ISS framework (Sontag, 1990; Jiangand Wang, 2001). Consider the discrete-time perturbed nonlinear system:

ξ(k + 1) = g(ξ(k), z(k)), k ∈ Z+, (2.1)

where ξ : Z+ → Rn is the state trajectory, z : Z+ → Rdv is an unknowndisturbance input trajectory and g : Rn × Rdv → Rn is a nonlinear, possi-


bly discontinuous function. For simplicity, we assume that the origin is anequilibrium for (2.1), i.e. g(0, 0) = 0.

Definition 2.2.1 A set P ⊆ Rn with 0 ∈ int(P) is called a robustly positi-vely invariant (RPI) set with respect to V ⊆ Rdv for system (2.1) if for allx ∈ P it holds that g(x, v) ∈ P for all v ∈ V. A set P ⊆ Rn with 0 ∈ int(P)is called a positively invariant (PI) set for system (2.1) with zero input if forall x ∈ P it holds that g(x, 0) ∈ P. 2

Definition 2.2.2 Let X with 0 ∈ int(X) be a subset of Rn. We call system(2.1) with zero input (i.e. z(k) = 0 for all k ∈ Z+) asymptotically stablein X, or shortly AS(X), if there exists a KL-function β(·, ·) such that, foreach ξ(0) ∈ X it holds that ‖ξ(k)‖ ≤ β(‖ξ(0)‖, k), ∀k ∈ Z+. If the propertyholds with β(s, k) := θρks for some θ ∈ R(0,∞) and ρ ∈ R[0,1) we call system(2.1) with zero input exponentially stable in X (ES(X)). We call system (2.1)with zero input globally asymptotically (exponentially) stable if it is AS(Rn)(ES(Rn)). 2

Definition 2.2.3 Let X and V be subsets of Rn and Rdv , respectively, with0 ∈ int(X). We call system (2.1) input-to-state stable in X for inputs inV, or shortly ISS(X,V), if there exist a KL-function β(·, ·) and a K-functionγ(·) such that, for each initial condition ξ(0) ∈ X and all z = z(l)l∈Z+

with z(l) ∈ V for all l ∈ Z+, it holds that the corresponding state trajectoryof (2.1) with initial state ξ(0) and input trajectory z satisfies ‖ξ(k)‖ ≤β(‖ξ(0)‖, k)+γ(‖z[k−1]‖) for all k ∈ Z[1,∞). The system (2.1) is globally ISSif it is ISS(Rn, Rdv). 2

Throughout this chapter we will employ the following sufficient conditionsfor analyzing ISS.

Theorem 2.2.4 (Jiang and Wang, 2001; Lazar et al., 2008a) Let α1, α2, α3 ∈K∞, σ ∈ K and let V be a subset of Rdv . Let X with 0 ∈ int(X) be a RPI setwith respect to V for system (2.1) and let V : X → R+ be a function withV (0) = 0. Consider the following inequalities:

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), (2.2a)V (g(x, v))− V (x) ≤ −α3(‖x‖) + σ(‖v‖). (2.2b)

If inequalities (2.2) hold for all x ∈ X and all v ∈ V, then system (2.1) isISS(X,V). If inequalities (2.2) hold for all x ∈ Rn and all v ∈ Rdv , thensystem (2.1) is globally ISS. If X with 0 ∈ int(X) is a PI set for system


(2.1) with zero input and inequalities (2.2) hold for all x ∈ X (x ∈ Rn) andv ∈ V = 0, then system (2.1) with zero input is AS(X) (GAS).

A function V (·) that satisfies the hypothesis of Theorem 2.2.4 is called anISS Lyapunov function. Note the following aspects regarding Theorem 2.2.4.(i) The hypothesis of Theorem 2.2.4 allows that both g(·, ·) and V (·) arediscontinuous. The hypothesis only requires continuity at the point x = 0,and not necessarily on a neighborhood of x = 0. (ii) If the inequalities (2.2)are satisfied for α1(s) = asλ, α2(s) = bsλ, α3(s) = csλ, for some a, b, c, λ > 0,then the hypothesis of Theorem 2.2.4 implies exponential stability of system(2.1) with zero input.

2.2.2 Lyapunov functions

As an extension of classical Lyapunov functions (see Corollary 1.2 and Co-rollary 1.3 of (Kalman and Bertram, 1960b)), which are assumed to be con-tinuous and only required to have a negative one step forward difference, wewill introduce the following known types of Lyapunov functions for the zeroinput system corresponding to (2.1), i.e. ξ(k + 1) = g(ξ(k), 0), k ∈ Z+. LetX ⊆ Rn be a positively invariant set for ξ(k+1) = g(ξ(k), 0) with 0 ∈ int(X),let α1, α2, α3 ∈ K∞, let V : Rn → R+ denote a possibly discontinuous func-tion with V (0) = 0, and consider the inequalities:

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), ∀x ∈ X, (2.3a)V (g(x, 0))− V (x) ≤ 0, ∀x ∈ X, (2.3b)V (g(x, 0))− V (x) < 0, ∀x ∈ X \ 0, (2.3c)V (g(x, 0))− V (x) ≤ −α3(‖x‖), ∀x ∈ X. (2.3d)

Definition 2.2.5 A function V (·) that satisfies (2.3a) and (2.3b) is called aLyapunov function. A function V (·) that satisfies (2.3a) and (2.3c) is calleda strict Lyapunov (SL) function. A function V (·) that satisfies (2.3a) and(2.3d) is called a uniformly strict Lyapunov (USL) function. 2

For continuous V (·) and discrete-time continuous system dynamics it isknown that SL functions and USL functions are equivalent and both im-ply asymptotic stability and inherent robustness (ISS, under certain conditi-ons); see, for example, (Kellett and Teel, 2004). In the following section wewill investigate whether these properties still hold when either the systemdynamics or the Lyapunov function is discontinuous, or both.


Notice that a USL function can also be defined by replacing (2.3d) withthe intermediate property

V (g(x, 0))− V (x) ≤ −δ(x), ∀x ∈ X, (2.4)

where δ : Rn → R+ is a continuous and positive definite function. However,it can be shown that given such a USL function one can always find a newUSL function that satisfies (2.3d), using ideas from (Nesic and Teel, 2001).Also, in the case when g(·, 0) and V (·) are continuous it can be proven thatSL functions and USL functions that satisfy (2.4) are equivalent.

2.3 Illuminating examples

Consider the following generic discrete-time PWA systems, which form oneof the simplest class of discontinuous systems and will serve as a support forsetting up the examples:

ξ(k + 1) = G(ξ(k)) := Ajξ(k) + fj if ξ(k) ∈ Ωj , (2.5a)

ξ(k + 1) = g(ξ(k), z(k)) := Aj ξ(k) + fj + z(k) if ξ(k) ∈ Ωj , (2.5b)

with z(k) ∈ Bµ for some small µ ∈ R(0,∞), k ∈ Z+, and where Aj ∈ Rn×n,fj ∈ Rn for all j ∈ S (a finite set of indexes) and Ωj ⊆ Rn | j ∈ S defines apartition of X, meaning that ∪j∈SΩj = X and Ωi ∩Ωj = ∅, with the sets Ωj

not necessarily closed. Firstly, we present a simple one-dimensional exampleof a discontinuous system that admits a continuous SL function but it is notGAS.

Example 1: Consider the discrete-time system (2.5a) with j ∈ S :=1, 2, A1 = f1 = 0, A2 = 0.5, f2 = 0.5 and the partition given byΩ1 = x ∈ R | x ≤ 1, Ω2 = x ∈ R | x > 1. One can easilycheck that limk→∞ ξ(k) = 1 for any ξ(0) = x ∈ R(1,∞) = Ω2 and thus,this system is not GAS. Consider the function V (x) := |x|. Clearly, forx ∈ Ω1 \0 we have V (G(x))−V (x) = −V (x) < 0 and, for x ∈ Ω2 we haveV (G(x))−V (x) = 0.5|x+1|−|x| < |x|−|x| = 0. Hence, V (x) is a continuousSL function. However, V (x) is not a USL function, as for any α3 ∈ K∞ itholds that limx↓1 (V (G(x))− V (x)) = limx↓1(0.5|x+1|−x) = 0 > −α3(1). 2

As illustrated above, the system of Example 1 admits a continuous SLfunction but the trajectories do not converge to the origin globally. Thisindicates that SL functions (even continuous ones) which are not USL func-tions do not guarantee GAS for discrete-time discontinuous systems. Hence,

2.3. Illuminating examples 31

Figure 2.1: The function G(·) for the system of Example 2.

one must strive for a USL function to guarantee GAS of a discrete-time dis-continuous system. For a proof that (discontinuous) USL functions implyGAS see, for example, Chapter 4 in this thesis. The interested reader is alsoreferred to (Nesic et al., 1999) for a proof that a GAS discrete-time systemalways admits a possibly discontinuous USL function.

Example 2: Consider now the discrete-time system (2.5a) with j ∈S := 1, 2, A1 = A2 = 0, f1 = 0, f2 = 1 and the partition is given byΩ1 = x ∈ R | x ≤ 1, Ω2 = x ∈ R | x > 1. Figure 2.1 shows thevalues of the function G(x). One can easily observe that any trajectoryξ(k) at time k ∈ Z+ of system (2.5a) starting from an initial conditionξ(0) = x ∈ R satisfies |ξ(k)| ≤ |ξ(0)| (even |ξ(k)| < |ξ(0)| when ξ(0) = x 6= 0)and converges exponentially to the origin. Actually, any trajectory ξ(k)reaches the origin in 2 discrete-time steps or less. Furthermore, it can beproven that V (x) :=

∑∞i=0 ξ(i)2 is a USL function, where ξ(i) denotes the

trajectory of system (2.5a) at time i ∈ Z+, obtained from initial conditionξ(0) = x ∈ R. Indeed, since V (x) =

∑∞i=0 ξ(i)2 = ξ(0)2 + ξ(1)2 for any

ξ(0) = x ∈ R, it holds that V (G(x))− V (x) ≤ −α3(|x|) for all x ∈ R, whereα3(s) := s2. An explicit expression for V (·) is:

V (x) =∞∑i=0

ξ(i)2 = ξ(0)2 + ξ(1)2 =

x2 + 1 if x > 1x2 if x ≤ 1,

which shows that V (·) is discontinuous at x = 1.Next consider the case when z(k) = µ ∈ R(0,∞) for all k ∈ Z+ in (2.5b).

Then, the origin of the perturbed system (2.5b) corresponding to the no-minal system (2.5a) is not ISS, as x = 1 + µ is an equilibrium of (2.5b) towhich all trajectories with initial conditions ξ(0) = x ∈ R(1,∞) = Ω2 conver-ge. Hence, no matter how small µ ∈ R(0,∞) is taken, the system (2.5b) is


not ISS(R, Bµ). 2

The following conclusions can be drawn from Example 2:(i) GES discrete-time discontinuous systems are not necessarily ISS, even

to arbitrarily small inputs;(ii) existence of a discontinuous USL function does not guarantee ISS,

even to arbitrarily small inputs.This indicates that additional conditions must be imposed on USL func-

tions to attain ISS. For example, continuity of the USL function is known toguarantee inherent ISS (Lazar et al., 2009a), but this condition is too restric-tive for discrete-time discontinuous systems such as PWA systems. Thus, inthe next section we will propose ISS tests that can deal with discontinuousUSL functions.

Remark 2.3.1 The GES discrete-time system of Example 2 also admits acontinuous SL function, i.e. V (x) := |x|, which satisfies V (G(x))−V (x) < 0for all x 6= 0. However, as it was the case in Example 1, V (x) = |x| is not aUSL function, as for any α3 ∈ K∞ it holds that limx↓1 (V (G(x))− V (x)) =limx↓1(1−x) = 0 > −α3(1). Hence, the existence of a continuous SL functiondoes not necessarily guarantee any robustness for discontinuous systems. 2

The next example shows a constrained 2D PWA system that is expo-nentially stable but it has no robustness. Such constrained PWA systemsarise inherently in explicit model predictive control of linear or PWA systems(Grieder et al., 2005; Lazar et al., 2006; Baotic et al., 2006), as the dynamicsthat describe the closed-loop system. Therefore, this makes the followingexample especially relevant for MPC closed-loop systems.

Example 3: Consider the discontinuous nominal and perturbed PWAsystems (3.3) with v(k) ∈ Bµ = v ∈ R2 | ‖v‖ ≤ µ for some µ ∈ R(0,∞),j ∈ S := 1, . . . , 9, k ∈ Z+, and where

Aj =[1 00 1

]for j 6= 7; A7 =

[0.35 0.6062

0.0048 −0.0072

]; f1 = −f2 =

[0.50

];

f3 = f4 = f5 = f6 =[

0−1

]; f7 =

[00

]; f8 =

[0.4−0.1

]; f9 =

[−0.4−0.1

].

The system state takes values in the set X := ∪j∈SΩj , where the regi-ons Ωj are polyhedra (the exact representations are omitted due to spa-ce limitations), as shown in Figure 2.2. The state trajectories1 of sys-

1Note that the regions Ω1 and Ω2 are such that for all x ∈ ∂Ω1 ∩ ∂Ω2 the dynamicsx(k + 1) = A2x(k) + f2 is active, i.e. ∂Ω1 ∩ ∂Ω2 ⊆ Ω2.


Figure 2.2: A constrained 2D PWA system with no robustness: nominal(square,circle-dotted lines) and perturbed (star-solid line).

tem (2.5a) obtained for x(0) = [0.2 3.6]> ∈ Ω2 (square dotted line) andx(0) = [0.2 3.601]> ∈ Ω1 (circle dotted line) are plotted in Figure 2.2. 2

Theorem 2.3.2 The following statements hold:(i) The function V (x) := ‖x(10)‖∞+

∑9i=0 ‖Qx(i)‖∞, where Q = 0.04I2

and x(i) is the solution of system (2.5a) obtained at time i ∈ Z[0,10] frominitial condition x(0) := x ∈ X, is a discontinuous USL function for system(2.5a);

(ii) The PWA system (2.5a) is exponentially stable in X;(iii) For any µ ∈ R(0,∞) the PWA system (2.5b) is not ISS in X for inputs

in Bµ.

Proof: (i) The following properties hold for the PWA system (2.5a) ofExample 3, as it can be seen by inspection of the dynamics: (P1) ‖x(k +1)‖∞ ≤ ‖x(k)‖∞ for all x(k) ∈ X, k ∈ Z+; (P2) For any initial state x(0) ∈ Xthe state trajectory satisfies x(k) ∈ Ω7 for all k ∈ Z≥10; (P3) ‖A7‖∞ <1; (P4) Ω7 is a Positively Invariant (PI) set for the dynamics x(k + 1) =


A7x(k) + f7; (P5) X is a PI set for the PWA system (2.5a).First, we prove that V (x) = ‖x(10)‖∞ +

∑9i=0 ‖Qx(i)‖∞ satisfies ine-

quality (2.2a). For any τ ∈ (0, 0.04) it holds that ‖Qx‖∞ ≥ τ‖x‖∞. There-fore, α1(‖x‖∞) ≤ V (x) is satisfied for all x ∈ X with α1(‖x‖∞) := τ‖x‖∞.For any state trajectory x(i)i∈Z[0,10]

there exists a set of indices ji ∈ S,i ∈ Z[0,10], such that x(i) ∈ Ωji (note that by property (P2) j9 = j10 = 7 forany x(0) ∈ X). Then, using the triangle inequality, for any x ∈ X (note thatx(0) := x) we obtain that

V (x) = ‖x(10)‖∞ +9∑

i=0

‖Qx(i)‖∞ ≤ ‖Qx(0)‖∞ + ‖QAj0x(0)‖∞

+ ‖Qfj0‖∞ + ‖QAj1Aj0x(0)‖∞+ ‖QAj1fj0‖∞ + ‖Qfj1‖∞ + . . .+‖Aj9 . . . Aj0x(0)‖∞ + ‖Aj9 . . . Aj1fj0‖∞ + . . . + ‖fj9‖∞.

Note that, by property (P4), for all x(0) = x ∈ Ω7 we have that x(i) ∈ Ω7

for all i ∈ Z+ and hence, x(i + 1) = A7x(i) for all i ∈ Z+, as f7 = [0 0]>.Otherwise, if x(0) = x ∈ X \ Ω7, since 0 ∈ int(Ω7) and Ω7 is bounded, thereexists a positive number ζ > 0 such that

‖Qfj0‖∞ + (‖QAj1fj0‖∞ + ‖Qfj1‖∞) + . . .

+ (‖Aj9 . . . Aj1fj0‖∞ + . . . + ‖fj9‖∞) ≤ ζ‖x(0)‖∞.

Then, using x(0) = x and the inequality ‖Qx‖∞ ≤ ‖Q‖∞‖x‖∞ it followsthat V (x) ≤ α2(‖x‖∞) for all x ∈ X with α2(‖x‖∞) := θ‖x‖∞, where

θ := ‖Q‖∞ +9∑

i=1

‖Q i−1∏p=0

Ajp‖∞

+ ‖9∏

p=0

Ajp‖∞ + ζ.

Finally, for any x ∈ X ∩Ωj and any j ∈ S, by properties (P2), (P3) it holdsthat

V (Ajx + fj)− V (x) = −‖Qx‖∞ + (‖A7x(10)‖∞ − ‖x(10)‖∞ + ‖Qx(10)‖∞)≤ −‖Qx‖∞ ≤ −τ‖x‖∞ =: −α3(‖x‖∞).

In the above inequality we used the fact that

‖A7x‖∞ − ‖x‖∞ ≤ (‖A7‖∞ − 1)‖x‖∞ = −0.0438‖x‖∞≤ −0.04‖x‖∞ = −‖Q‖∞‖x‖∞ ≤ −‖Qx‖∞,


for all x ∈ Rn. Therefore, the function V (x) = ‖x(10)‖∞ +∑9

i=0 ‖Qx(i)‖∞is a USL function for system (2.5a) of Example 3. One can easily check thatV (x) is discontinuous, for example, at x = [0.2 3.6]> ∈ Ω2.

(ii) By property (P5), X is a PI set for the PWA system (2.5a) ofExample 3 and hence, a valid domain of attraction. Therefore, exponen-tial stability of the origin follows directly from the result of part (i), due tothe special form of the K-functions α1(·), α2(·) and α3(·) established in theproof of part (i).

(iii) To illustrate the non-robustness phenomenon for the perturbed PWAsystem (2.5b) of Example 3, we constructed an additive disturbance v(k),which at times k = 0, 2, 4, . . . is equal to [0 ε]> and at times k = 1, 3, 5, . . .is equal to [0 − ε]>, where ε > 0 can be taken arbitrarily small. Thesystem trajectory (see Figure 2.2 for a plot - red line) with initial statex(0) = [0.2 3.6]> ∈ ∂Ω2 ∩ ∂Ω1 is given by x(k) = [0.2 3.6]>, if k = 0, 2, 4, . . .and x(k + 1) = [−0.3 3.6 + ε]>, if k = 1, 3, 5, . . .. This is a limit cycle withperiod 2 and ‖x(k)‖∞ ≥ 3.6 for all k ∈ Z+. Then, for any β ∈ KL andγ ∈ K we can take ε > 0 arbitrarily small and k∗ ∈ Z+ large enough suchthat

β(k, ‖x(0)‖∞) + γ(‖w[k−1]‖∞) < 3.6 ≤ ‖x(k)‖∞, ∀k ≥ k∗.

Therefore, for any ε > 0, the PWA system (2.5b) of Example 3 is not ISSfor initial conditions in X and inputs in Bε.

Notice that, by taking any finite polyhedral partition of R2 \X, definingthe dynamics in each polyhedral region of this partition to be x(k + 1) =[ 0 00 0 ]x(k) +

[0.1−0.1

], k ∈ Z+, and adding these affine subsystems to the PWA

system (2.5a), one obtains a 2D PWA system that is GES, but it has norobustness to arbitrarily small disturbances.

Remark 2.3.3 While the disturbance signal used in Example 1 does nothave a particular structure, a specific disturbance signal was employed inExample 3 to destroy ISS. However, in practice there is often still somestructure in the disturbances (for example, time delays in embedded systemsor cyclic sensor/encoder errors), which makes such a situation not highlyunlikely to happen.

Remark 2.3.4 By Theorem 14 of (Kellett and Teel, 2004), Example 2implies that there exist GES discrete-time systems that do not admit acontinuous USL function. However, as shown above, the PWA system ofExample 2 does admit a discontinuous USL function, which is in conformi-


ty with the converse stability result for discrete-time discontinuous systemspresented in (Nesic et al., 1999). 2

2.4 ISS tests based on discontinuous USL functions

In this section we consider piecewise continuous (PWC) nonlinear systemsof the form

ξ(k + 1) = G(ξ(k)) := Gj(ξ(k)) if ξ(k) ∈ Ωj , k ∈ Z+, (2.6)

where each Gj : Rn → Rn, j ∈ S, is assumed to be a continuous function.PWA systems are obtained as a particular case by setting Gj(x) = Ajx+fj .Consider also a perturbed version of the above system, by including additivedisturbances, i.e.

ξ(k+1) = g(ξ(k), z(k)) := Gj(ξ(k))+z(k) if ξ(k) ∈ Ωj , k ∈ Z+. (2.7)

Furthermore, we consider discontinuous USL functions V : Rn → R+, withV (0) = 0,

V (x) := Vi(x) if x ∈ Γi, i ∈ J , (2.8)

where for each i ∈ J , Vi : Rn → R+ is a continuous function that satisfies

|Vi(x)− Vi(y)| ≤ σi(‖x− y‖), ∀x, y ∈ cl(Γi), (2.9)

for some σi ∈ K. Examples of functions that satisfy this property includeuniformly continuous functions on compact sets and Lipschitz continuousfunctions. This captures a wide range of frequently used Lyapunov functionsfor PWA systems, such as piecewise quadratic (PWQ), PWA or piecewisepolyhedral functions (i.e. functions defined using the infinity norm or the1-norm), including the value functions that arise in model predictive controlof PWA systems.

In (2.6) and (2.8), Ωj | j ∈ S and Γi | i ∈ J with S := 1, . . . , sand J := 1, . . . ,M finite sets of indices, denote partitions of Rn. Moreprecisely, we assume that ∪j∈SΩj = Rn, Ωi ∩Ωj = ∅ for i 6= j, (i, j) ∈ S × Sand int(Ωi) 6= ∅ for all i ∈ S and likewise for the regions Γi, i ∈ J . Weassume that a discontinuous USL function of the form (2.8) is available forsystem (2.6). We have seen from Example 2 in the previous section that thisdoes not necessarily guarantee anything in terms of ISS. However, the goalis now to develop tests for ISS of system (2.7) based on the discontinuousUSL function (2.8).

2.4. ISS tests based on discontinuous USL functions 37

The first result is based on examining the trajectory of the PWC system(2.6) with respect to the set of states at which V (·) may be discontinuous.Let µ ∈ R(0,∞) and let P ⊆ Rn with 0 ∈ int(P) be a RPI set for system (2.7)with respect to Bµ, i.e. R1(P) ⊕ Bµ ⊆ P, where R1(P) := G(x) | x ∈ Pis the one-step reachable set for system (2.6) from states in P. Let XD ⊂ Pdenote the set of all states in P at which V (·) is not continuous. If one canverify that any state trajectory ξ(k)k∈Z+ of (2.6) is a distance µ ∈ R(0,∞)

away from the set XD for all ξ(0) = x ∈ P and all k ∈ Z[1,∞), then it can beproven that ISS(P,Bµ) is achieved, as formulated in the following result.

Theorem 2.4.1 Suppose that the PWC system (2.6) admits a (disconti-nuous2) USL function of the form (2.8) and consequently, (2.6) is GAS.Furthermore, suppose that there exists a µ ∈ R(0,∞) and a set P ⊆ Rn with0 ∈ int(P) such that

d(x, XD) > µ for all x ∈ R1(P) (2.10)

and P is a RPI set3 for system (2.7) with respect to Bµ. Then, the PWCsystem (2.7) is ISS(P,Bµ).

Proof: First, we will prove that there exists a K-function σ(·) (inde-pendent of x) such that for all x and for any two points y, y ∈ G(x)⊕ Bµ itholds that |V (y)− V (y)| ≤ σ(‖y − y‖).

By (2.9), for each i ∈ J and any two points y, y ∈ cl(Γi) there exists a K-function σi(·) such that |Vi(y)− Vi(y)| ≤ σi(‖y − y‖). The inequality (2.10)implies that V (·) is continuous on the set G(x)⊕Bµ for any x ∈ P. For anytwo points y, y ∈ G(x)⊕Bµ consider the line segment L(y, y) := y+α(y−y) |0 ≤ α ≤ 1 between y and y. We will construct a set of points z0, . . . , zM ⊂L(y, y) with M ≤ M on this line segment such that: (i) z0 = y; zM = y and(ii) (zp−1, zp) ∈ cl(Γip−1)× cl(Γip−1) for some ip−1 ∈ J , for all p = 1, . . . ,M .To construct this set, take i0 ∈ J such that z0 = y ∈ cl(Γi0), α0 = 0 andα1 := maxα ∈ [0, 1] | y + α(y − y) ∈ cl(Γi0). Note that due to closednessof cl(Γi0) the maximum is attained and z1 := y + α1(y − y) ∈ cl(Γi0). Inaddition, for all α ∈ (α1, 1] it holds that y+α(y−y) 6∈ cl(Γi0). If α1 = 1 (andthus y ∈ cl(Γi0)) the construction is complete. If α1 6= 1, then there is ani1 ∈ J \ i0 with z1 ∈ cl(Γi1). Take α2 := maxα ∈ [α1, 1] | y + α(y − y) ∈cl(Γi1) and observe that z2 := y +α2(y− y) ∈ cl(Γi1) and for all α ∈ (α2, 1]we have that y + α(y − y) 6∈ cl(Γi0) ∪ cl(Γi1). If α2 = 1 the construction

2Note that the result also holds for continuous USL functions, as then XD = ∅.3Observe that P = Rn is a possible choice of a RPI set with respect to Bµ for any

µ ∈ R(0,∞).


is complete. Otherwise, continue the construction. This construction willterminate in at most M steps as the number of regions cl(Γi), i = 1, . . . ,M ,is finite and y lies in at least one of them. At termination, we arrived at theset of points z0, . . . , zM with the mentioned properties. Due to continuityof V (·) in the region G(x) ⊕ Bµ, for zp ∈ ∂Γip−1 ∩ ∂Γip , p = 1, . . . ,M , wehave that V (zp) = Vip−1(zp) = Vip(zp). Then, for any y, y ∈ G(x) ⊕ Bµ, itfollows that

|V (y)− V (y)| =

∣∣∣∣∣∣M∑

p=1

(V (zp−1)− V (zp))

∣∣∣∣∣∣ ≤M∑

p=1

|V (zp−1)− V (zp)|

=M∑

p=1

|Vip(zp−1)− Vip(zp)|

≤M∑

p=1

σip(‖zp−1 − zp‖) ≤M∑

p=1

σip(‖y − y‖).

Letting σ(s) := M maxi∈J σi(s) ∈ K, one obtains |V (y)−V (y)| ≤ σ(‖y−y‖)for any y, y ∈ G(x)⊕ Bµ.

Since for any v ∈ Bµ it holds that g(x, v) = G(x) + v ∈ G(x) ⊕ Bµ, itfollows that:

V (g(x, v))− V (G(x)) ≤ σ(‖v‖), ∀x ∈ P, ∀v ∈ Bµ. (2.11)

As by the hypothesis V (·) is a USL function for the PWC system (2.6), wehave that α1(‖x‖) ≤ V (x) ≤ α2(‖x‖) and

V (G(x))− V (x) ≤ −α3(‖x‖), ∀x ∈ P, (2.12)

for some α1, α2, α3 ∈ K∞. Adding (2.11) and (2.12) yields:

V (g(x, v))− V (x) ≤ −α3(‖x‖) + σ(‖v‖), ∀x ∈ P, ∀v ∈ Bµ.

Hence, V (·) is an ISS Lyapunov function for the PWC system (2.7). Thestatement then follows from Theorem 2.2.4.

The constant µ can be chosen as follows:

0 < µ ≤ µ∗ := minj∈S

inf

y∈Ωj∩P, y∈XD

‖Gj(y)− y‖

. (2.13)

If the set XD is the union of a finite number of polyhedra, the sets Ωj , j ∈ Sand P are polyhedra, each Gj , j ∈ S is an affine function and the infinity

2.4. ISS tests based on discontinuous USL functions 39

norm (or the 1-norm) is used in (2.13), a solution to the optimization problemin (2.13) can be obtained by solving a finite number of linear programmingproblems (quadratic programming problems if the 2-norm is used). If theoptimization problem in (2.13) yields a strictly positive µ∗, then µ∗ ∈ R(0,∞)

can be considered as a measure of the (worst case) inherent robustness ofsystem (2.6). The sufficient condition (2.10) can be relaxed, as shown bythe next result, in the sense that the trajectory ξ(k)k∈Z+ of system (2.6)is now allowed to intersect the set XD.

Proposition 2.4.2 Let P ⊆ Rn with 0 ∈ int(P) be a RPI set for system(2.7) with respect to Bµ for some µ ∈ R(0,∞). Suppose that the PWC system(2.6) admits a function of the form (2.8) that satisfies (2.3a) for all x ∈ P.Furthermore, suppose that there exists α3 ∈ K∞ such that

maxi∈I

Vi(G(x))− V (x) ≤ −α3(‖x‖), ∀x ∈ P. (2.14)

Then, the PWC system (2.7) is ISS(P,Bµ).

The above result is based on a stronger, more conservative extension of thestabilization conditions from (Johansson, 1999; Mignone et al., 2000; Ferrari-Trecate et al., 2002; Feng, 2002; Daafouz et al., 2002), as it requires that theLyapunov function is decreasing irrespective of which dynamics might beactive at the next step. The proof of Proposition 2.4.2 follows from theproof of the less conservative result formulated next in Theorem 2.4.3.

The sufficient condition (2.14) can be significantly relaxed, as follows.Consider the set Z := x ∈ P | G(x)⊕Bµ∩XD 6= ∅ and define for x ∈ Z

M(x) := i ∈ J | G(x) 6∈ Γi, G(x)⊕ Bµ ∩ Γi 6= ∅.

Theorem 2.4.3 Suppose that the PWC system (2.6) admits a (disconti-nuous) USL function of the form (2.8). Furthermore, suppose that thereexists a µ ∈ R(0,∞), a K∞-function α3(·) and a set P ⊆ Rn with 0 ∈ int(P)such that

maxi∈M(x)

Vi(G(x))− V (x) ≤ −α3(‖x‖), ∀x ∈ Z (2.15)

and P is a RPI set for system (2.7) with respect to Bµ. Then, the PWCsystem (2.7) is ISS(P,Bµ).

Proof: As done in the proof of Theorem 2.4.1, we will show thatV (·) satisfies the ISS inequalities (2.2). For any x ∈ P only the following


situations can occur: (A) G(x)⊕Bµ ∩XD = ∅ or (B) x ∈ Z. In case (A),as shown in the proof of Theorem 2.4.1, by continuity of V (·) on G(x)⊕Bµ,there exists a σ ∈ K (independent of x) as constructed in the proof ofTheorem 2.4.1 such that

V (g(x, v))− V (x) ≤ −α3(‖x‖) + σ(‖v‖), ∀v ∈ Bµ. (2.16)

In case (B), suppose that v ∈ Bµ is such that G(x) ∈ Γp and G(x)+v ∈ Γp forsome p ∈ J . In this case p 6∈ M(x). Then, since V (G(x)) = Vp(G(x)) andV (G(x) + v) = Vp(G(x) + v), by continuity of Vi(·), inequality (2.16) holdswith the same K-function σ(·) constructed in the proof of Theorem 2.4.1.

Otherwise, if v ∈ Bµ is such that G(x) ∈ Γp and G(x) + v ∈ Γi forsome p, i ∈ J , p 6= i, we have that V (G(x)) = Vp(G(x)), V (G(x) + v) =Vi(G(x) + v) and i ∈ M(x). Then, by continuity of Vi(·) and inequality(4.17) we obtain:

V (G(x) + v)− V (x) = Vi(G(x) + v)− V (x)= Vi(G(x))− V (x) + Vi(G(x) + v)− Vi(G(x))≤ max

i∈M(x)Vi(G(x))− V (x) + σi(‖v‖)

≤ −α3(‖x‖) + σ(‖v‖),

with σi(·) and σ(·) as defined in the proof of Theorem 2.4.1. Letting α3(s) :=min(α3(s), α3(s)) ∈ K∞, it follows that

V (g(x, v))− V (x) = V (G(x) + v)− V (x) ≤ −α3(x) + σ(‖v‖),∀x ∈ P, ∀v ∈ Bµ.

Therefore, V (·) is an ISS Lyapunov function for system (2.7). The statementthen follows from Theorem 2.2.4.

Observe that (2.13) amounts to an a posteriori check that must be perfor-med on a given USL function of the form (2.8). In contrast, condition (2.14)can be a priori specified when computing a USL function of the form (2.8),and it can be casted as a semidefinite programming problem for piecewisequadratic (PWQ) functions and PWA systems. On the same issue, conditi-on (4.17) involves the set XD and hence, amounts to an a posteriori checkthat must be performed on a given USL function of the form (2.8). Undercertain reasonable assumptions (e.g., XD is the union of a finite number ofpolyhedra, the regions Ωj , j ∈ S and Γi, i ∈ I are polyhedra, the systemis PWA, the USL function is convex) checking (4.17) amounts to solving afinite number of convex optimization problems.

2.5. Conclusions 41

Remark 2.4.4 The result of Theorem 2.4.3 also holds when condition (4.17)is replaced by

maxi∈M(x)

Vi(G(x))− V (G(x)) ≤ cα3(‖x‖), ∀x ∈ Z, (2.17)

for some c ∈ R[0,1), which might be easier to check than (4.17). 2

Remark 2.4.5 The tests developed in this section require that for eachi ∈ J , Vi is a continuous function that satisfies (2.9) and it is only defined on:(i) Γi for Theorem 2.4.1, (ii) P ⊆ Rn for Proposition 2.4.2 and (iii) cl(Γi)⊕Bµ for some µ ∈ R(0,∞) for Theorem 2.4.3. Some of these are additionalrequirements with respect to USL functions, which in principle, only requirethat each Vi is defined on Γi. An alternative to the tests presented in thissection is to directly check condition (2.2b), which for PWA dynamics andPWQ candidate ISS Lyapunov functions can lead to tractable optimizationproblems, as shown recently in (Lazar and Heemels, 2008a). 2

2.5 Conclusions

In this chapter we analyzed two types of Lyapunov functions in terms of theirsuitability for establishing stability and input-to-state stability of discrete-time discontinuous systems. Via examples we exposed certain subtleties thatarise in the classical Lyapunov methods when they are applied to discrete-time discontinuous systems, as follows:

• The existence of a continuous SL function does not necessarily implyGAS - Example 1;

• The existence of a continuous SL function or discontinuous USL func-tion does not necessarily imply ISS, even to arbitrarily small inputs -Example 2;

• GES does not necessarily imply the existence of a continuous USLfunction - Example 2 (see also (Kellett and Teel, 2004)).

These results, together with the fact that existence of a possibly disconti-nuous USL function is equivalent to GAS (Nesic et al., 1999) (see also Chap-ter 4 in this thesis), issue a strong warning regarding existing nominallystabilizing state-feedback synthesis methods for discrete-time discontinuoussystems, including both static feedback approaches (Johansson, 1999; Mig-none et al., 2000; Ferrari-Trecate et al., 2002; Feng, 2002; Daafouz et al.,


2002) and MPC techniques (Lazar et al., 2005; Grieder et al., 2005; Lazaret al., 2006; Baotic et al., 2006). This warning motivates the results on input-to-state stabilizing (sub-optimal) MPC of discontinuous systems presentedin the next chapter.

To render the many available procedures for obtaining Lyapunov func-tions, which typically yield discontinuous Lyapunov functions (e.g., valuefunctions in MPC or PWQ Lyapunov functions), applicable to discontinuoussystems, we presented several ISS tests based on discontinuous Lyapunovfunctions. These tests can be employed to establish ISS of nominally asymp-totically stable discrete-time PWC systems in the case when a discontinuousUSL function is available.

3

Predictive control of hybrid systems:Input-to-state stability results for

suboptimal solutions

3.1 Introduction3.2 Preliminaries3.3 MPC scheme set-up

3.4 Input-to-state stabilityresults

3.5 Asymptotic stability results3.6 Conclusion

This chapter presents a novel model predictive control (MPC) schemethat achieves input-to-state stabilization of constrained discontinuous nonli-near and hybrid systems. Input-to-state stability (ISS) is guaranteed whenan optimal solution of the MPC optimization problem is attained. Specialattention is paid to the effect that sub-optimal solutions have on ISS of theclosed-loop system. This issue is of interest as firstly, the infimum of MPCoptimization problems does not have to be attained and secondly, numericalsolvers usually provide only sub-optimal solutions. An explicit relation isestablished between the deviation of the predictive control law from the op-timum and the resulting deterioration of the ISS property of the closed-loopsystem. By imposing stronger conditions on the sub-optimal solutions, ISScan even be attained in this case.

3.1 Introduction

Discrete-time discontinuous systems form a powerful and general modelingclass for the approximation of hybrid and nonlinear phenomena, which alsoincludes the class of piecewise affine (PWA) systems (Heemels et al., 2001).The modeling capability of the latter class of systems has already been shownin several applications, including switched power converters, automotive sys-tems and systems biology. As a consequence, there is an increasing interestin developing synthesis techniques for robust control of discrete-time hybridsystems. The model predictive control (MPC) methodology (Mayne et al.,

44Predictive control of hybrid systems: Input-to-state stability results for


2000) has proven to be one of the most successful frameworks for this task,see, for example, (Bemporad and Morari, 1999; Kerrigan and Mayne, 2002;Lazar et al., 2006) and the references therein.

In this chapter we are interested in input-to-state stability (ISS) (Jiangand Wang, 2001) as a property to characterize robust stability of hybridsystems in closed-loop with MPC. More precisely, we consider systems thatare piecewise continuous and affected by additive disturbances. It is knownthat for such discontinuous systems most of the results obtained for smoothnonlinear MPC (Mayne et al., 2000; Limon et al., 2002a; Grimm et al.,2007) do not necessarily apply. The min-max MPC methodology (see, e.g.,(Lazar et al., 2008a) and the references therein) might be applicable, butits prohibitive computational complexity prevents implementation even forlinear systems. As such, computationally feasible input-to-state stabilizingpredictive controllers are widely unavailable.

In what follows we propose a tightened constraints MPC scheme for dis-continuous systems along with conditions for ISS of the resulting closed-loopsystem, assuming that optimal MPC control sequences are implemented.These results provide advances to the existing works on tightened constraintsMPC (Limon et al., 2002a; Grimm et al., 2007), where continuity of the sys-tem dynamics is assumed, towards discontinuous and hybrid systems. Gua-ranteeing robust stability and feasibility in the presence of discontinuities isdifficult and requires an innovative usage of tightened constraints, which isconceptually different from the approaches in (Limon et al., 2002a; Grimmet al., 2007). Therein tightened constraints are employed for robust feasibi-lity only. However, by carefully matching the new tightening approach withthe discontinuities in the system dynamics, we achieve both robust feasibili-ty and ISS for the optimal case. Another issue that is neglected in MPC ofhybrid systems is the effect of sub-optimal implementations. In particular,an important result was presented in (Spjøtvold et al., 2007), where it wasshown that in the case of optimal control of discontinuous PWA systems itis not uncommon that there does not exist a control law that attains theinfimum. Moreover, numerical solvers usually provide only sub-optimal so-lutions. As a consequence, for hybrid systems it is necessary to study ifand how stability results for optimal predictive control change in the caseof sub-optimal implementations, which forms one the main topics in thischapter.

To cope with MPC control sequences (obtained by solving MPC opti-mization problems) that are not optimal, but within a margin δ ≥ 0 fromthe optimum, we introduce the notion of ε-ISS as a particular case of theinput-to-state practical stability (ISpS) property (Jiang et al., 1996). Next,


we establish an analytic relation between the optimality margin δ of the so-lution of the MPC optimization problem and the ISS margin ε(δ). Whilethe ISS results presented in this chapter require the use of a specific robustMPC problem formulation (i.e. based on tightened constraints), we alsoshow that nominal asymptotic stability can be guaranteed for sub-optimalMPC of hybrid systems without any modification to the standard MPC set-up presented in (Mayne et al., 2000). Compared to classical sub-optimalMPC (Scokaert et al., 1999), where an explicit constraint on the MPC costfunction is employed, this provides a fundamentally different approach.

3.2 Preliminaries

First, we recall some basic definitions that will be employed in this chapterLet R, R+, Z and Z+ denote the field of real numbers, the set of non-

negative reals, the set of integer numbers and the set of non-negative integers,respectively. We use the notation Z≥c1 and Z(c1,c2] to denote the sets k ∈Z | k ≥ c1 and k ∈ Z | c1 < k ≤ c2, respectively, for some c1, c2 ∈ Z. Forx ∈ Rn let ‖x‖ denote an arbitrary norm and for Z ∈ Rm×n, let ‖Z‖ denotethe corresponding induced matrix norm. We will use both (z(0), z(1), . . .)and z(l)l∈Z+ with z(l) ∈ Rn, l ∈ Z+, to denote a sequence of real vectors.For a sequence z := z(l)l∈Z+ let ‖z‖ := sup‖z(l)‖ | l ∈ Z+ and letz[k] := z(l)l∈Z[0,k]

. For a set S ⊆ Rn, we denote by int(S) the interior ofS. For two arbitrary sets S ⊆ Rn and P ⊆ Rn, let S ∼ P := x ∈ Rn |x + P ⊆ S denote their Pontryagin difference. For any µ > 0 we define Bµ

as x ∈ Rn | ‖x‖ ≤ µ. A polyhedron (or a polyhedral set) is a set obtainedas the intersection of a finite number of open and/or closed half-spaces. Areal-valued function ϕ : R+ → R+ belongs to class K if it is continuous,strictly increasing and ϕ(0) = 0. A function β : R+ × R+ → R+ belongs toclass KL if for each fixed k ∈ R+, β(·, k) ∈ K and for each fixed s ∈ R+,β(s, ·) is decreasing and limk→∞ β(s, k) = 0.

Next, consider a discrete-time system of the form

x(k + 1) ∈ G(x(k), w(k)), k ∈ Z+, (3.1)

where x(k) ∈ Rn is the state, w(k) ∈ W ⊆ Rl is an unknown input atdiscrete-time instant k ∈ Z+ and G : Rn × Rl → 2(Rn) is an arbitrarynonlinear, possibly discontinuous, set-valued function. For simplicity of no-tation, we assume that the origin is an equilibrium in (6.1) for zero input,i.e. G(0, 0) = 0.



Definition 3.2.1 RPI We call a set P ⊆ Rn robustly positively invariant(RPI) for system (6.1) with respect to W if for all x ∈ P and all w ∈ W itholds that G(x,w) ⊆ P.

Definition 3.2.2 ε-ISS Let X with 0 ∈ int(X) and W be subsets of Rn andRl, respectively. For a given ε ∈ R+, the perturbed system (6.1) is calledε-ISS in X for inputs in W if there exist a KL-function β and a K-functionγ such that, for each x(0) ∈ X and all w = w(l)l∈Z+ with w(l) ∈ W for alll ∈ Z+, it holds that all state trajectories of (6.1) with initial state x(0) andinput sequence w satisfy

‖x(k)‖ ≤ β(‖x(0)‖, k) + γ(‖w[k−1]‖) + ε, ∀k ∈ Z≥1.

We call system (6.1) ISS in X for inputs in W if (6.1) is 0-ISS in X for inputsin W.

Definition 3.2.3 ε-AS For a given ε ∈ R+, the 0-input system (6.1), i.e.x(k + 1) ∈ G(x(k), 0), k ∈ Z+, is called ε-asymptotically stable (ε-AS) in Xif there exists a KL-function β such that, for each x(0) ∈ X it holds that allstate trajectories with initial state x(0) satisfy ‖x(k)‖ ≤ β(‖x(0)‖, k) + ε,∀k ∈ Z≥1. We call the 0-input system (6.1) AS in X if it is 0-AS in X.

We refer to ε by the term ISS (AS) margin.

Theorem 3.2.4 Let d1, d2 be non-negative reals, a, b, c, λ be positive realswith c ≤ b and α1(s) := asλ, α2(s) := bsλ, α3(s) := csλ and σ ∈ K.Furthermore, let X be a RPI set for system (6.1) with respect to W and letV : Rn → R+ be a function such that

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖) + d1, (3.2a)V (x+)− V (x) ≤ −α3(‖x‖) + σ(‖w‖) + d2 (3.2b)

for all x ∈ X, w ∈ W and all x+ ∈ G(x,w). Then the system (6.1) is ε-ISSin X for inputs in W with

β(s, k) := α−11 (3ρkα2(s)), γ(s) := α−1

1

(3σ(s)1− ρ

),

ε := α−11

(3(

d1 +d2

1− ρ

)), ρ := 1− c

b∈ [0, 1). (3.3)

If the inequalities (3.2) hold for d1 = d2 = 0, the system (6.1) is ISS in X forinputs in W.

3.3. MPC scheme set-up 47

The proof of Theorem 6.2.3 is similar in nature to the proof given in Chap-ter 4 of this thesis by replacing the difference equation by a difference inclu-sion as in (6.1) and is omitted here. We call a function V (·) that satisfiesthe hypothesis of Theorem 6.2.3 an ε-ISS function.

3.3 MPC scheme set-up

Consider the piecewise continuous (PWC) system

x(k + 1) = g(x(k), u(k), w(k)) := gj(x(k), u(k)) + w(k)if x(k) ∈ Ωj , k ∈ Z+, (3.4)

where each gj : Ωj × U → Rn, j ∈ S, is a continuous, possibly nonlinearfunction in x and S := 1, 2, . . . , s is a finite set of indices. We assume thatx and u are constrained in some sets X ⊆ Rn and U ⊆ Rm that contain theorigin in their interior. The collection Ωj ⊆ Rn | j ∈ S defines a partitionof X, meaning that ∪j∈SΩj = X and Ωi ∩ Ωj = ∅, with the sets Ωj notnecessarily closed. We also assume that w takes values in the set W := Bµ

with µ ∈ R>0 sufficiently small as determined later.

Assumption 3.3.1 For each fixed j ∈ S, gj(·, ·) satisfies a continuity con-dition in the first argument in the sense that there exists a K-function ηj(·)such that

‖gj(x, u)− gj(y, u)‖ ≤ ηj(‖x− y‖), ∀x, y ∈ Ωj , ∀u ∈ U,

and ∃j0 ∈ S such that 0 ∈ int(Ωj0) and gj0(0, 0) = 0.

As we allow g(·, ·, ·) to be discontinuous in x over the switching bounda-ries, discontinuous PWA systems are a sub-class of PWC systems as givenin (5.4). For a fixed N ∈ Z≥1, let (φ(1), . . . , φ(N)) denote a state sequencegenerated by the unperturbed system (5.4), i.e.

φ(i + 1) := gj(φ(i), u(i)) if φ(i) ∈ Ωj , (3.5)

for i = 0, . . . , N − 1, from initial condition φ(0) := x(k) and by applyingan input sequence u[N−1] = (u(0), . . . , u(N − 1)) ∈ UN := U × . . . × U.Let XT ⊆ X denote a set with 0 ∈ int(XT ). Define η(s) := maxj∈S ηj(s).As the maximum of a finite number of K-functions is a K-function, η ∈ K.



Let η[p](s) denote the p-times function composition with η[0](s) := s andη[k](s) = η(η[k−1](s)) for k ∈ Z≥1. For any µ > 0 and i ∈ Z≥1, define

Liµ :=

ζ ∈ Rn

∣∣∣∣‖ζ‖ ≤ i−1∑p=0

η[p](µ)

.

Define the set of admissible input sequences for x ∈ X as:

UN (x) := u[N−1] ∈ UN |φ(i) ∈ Xi, i = 1, . . . , N − 1,

φ(0) = x, φ(N) ∈ XT , (3.6)

where Xi := ∪j∈SΩj ∼ Liµ ⊂ X, ∀i = 1, . . . , N − 1. The purpose of

the above set of input sequences will be made clear in Lemma 3.4.3. Fora given N ∈ Z≥1, notice that µ > 0 has to be sufficiently small so that0 ∈ int(Ωj0 ∼ LN−1

µ ) 6= ∅. Let F : Rn → R+ and L : Rn × Rm → R+ withF (0) = L(0, 0) = 0 be arbitrary nonlinear mappings.

Problem 3.3.2 MPC optimization problem Let XT ⊆ X and N ∈ Z≥1

be given. At time k ∈ Z+ let x(k) ∈ X be given and infimize the costJ(x(k),u[N−1]) := F (φ(N)) +

∑N−1i=0 L(φ(i), u(i)) over all sequences u[N−1]

in UN (x(k)).

We call a state x ∈ X feasible if UN (x) 6= ∅. Problem 3.3.2 is said to befeasible for x ∈ X if UN (x) 6= ∅. Let Xf (N) ⊆ X denote the set of feasiblestates for Problem 3.3.2. Let V ∗(x) := infu[N−1]∈UN (x) J(x,u[N−1]). SinceJ(·, ·) is lower bounded by 0, the infimum exists. As such, V ∗(x) is welldefined for all x ∈ Xf (N). However, as shown in (Spjøtvold et al., 2007),the infimum is not necessarily attainable. Therefore, we will consider thefollowing set of sub-optimal control sequences. For any x ∈ Xf (N) andδ ≥ 0, we define

Πδ(x) := u[N−1] ∈ UN (x) | J(x,u[N−1]) ≤ V ∗(x) + δ

and πδ(x) := u(0) ∈ Rm | u[N−1] ∈ Πδ(x). We will refer to δ by theterm optimality margin. Note that δ = 0 and Πδ(x) 6= ∅ correspond tothe situation when the global optimum is attained in Problem 3.3.2. Anoptimality margin δ can be guaranteed a priori, for example, by using thesub-optimal mixed integer linear programming (MILP) solver proposed in(Spjøtvold et al., 2007) or by specifying a tolerance with respect to achievingthe optimum, which is a usual feature of most solvers.

3.4. Input-to-state stability results 49

Next, consider the following MPC closed-loop system corresponding to(5.4):

x(k + 1) ∈ Φδ(x(k), w(k)) :=g(x(k), u, w(k)) | u ∈ πδ(x(k)), k ∈ Z+. (3.7)

To simplify the exposition we will make use of the following commonly adop-ted assumptions in tightened constraints MPC (Limon et al., 2002a; Grimmet al., 2007). Let h : Rn → Rm denote a terminal control law and defineXU := x ∈ X | h(x) ∈ U.

Assumption 3.3.3 There exist K-functions αL(·), αF (s) := τsλ, α1(s) :=asλ and α2(s) := bsλ, τ, a, b, λ ∈ R>0, such that(i) L(x, u) ≥ α1(‖x‖), ∀x ∈ X, ∀u ∈ U;(ii) |L(x, u)− L(y, u)| ≤ αL(‖x− y‖), ∀x, y ∈ X, ∀u ∈ U;(iii) |F (x)− F (y)| ≤ αF (‖x− y‖), ∀x, y ∈ Ωj0 ∩ LN−1

µ ;(iv) V ∗(x) ≤ α2(‖x‖), ∀x ∈ Xf (N).

Assumption 3.3.4 There exist N ∈ Z≥1, θ > θ1 > 0, µ > 0 and a terminalcontrol law h(·) such that(i) αF (η[N−1](µ)) ≤ θ − θ1;(ii) Fθ := x ∈ Rn | F (x) ≤ θ ⊆ (Ωj0 ∼ LN−1

µ ) ∩ XU and gj0(x, h(x)) ∈ Fθ1

for all x ∈ Fθ;(iii) F (gj0(x, h(x)))− F (x) + L(x, h(x)) ≤ 0, ∀x ∈ Fθ.

Note that the hypotheses in Assumption 3.3.3-(i),(ii),(iii) usually hold bysuitable definitions of L(·, ·) and F (·). Also, it can be shown that the hy-pothesis of Assumption 3.3.3-(iv) may hold, even for discontinuous valuefunctions. For further details on how to satisfy Assumption 3.3.4-(i),(ii),(iii)we refer to (Lazar et al., 2006, 2007a).

3.4 Input-to-state stability results

The main result on ε-ISS of sub-optimal predictive control of hybrid systemsis stated next.

Theorem 3.4.1 Let δ ∈ R>0 be given, suppose that Assumption 3.3.1, As-sumption 3.3.3 and Assumption 3.3.4 hold for the nonlinear hybrid system(5.4) and Problem 3.3.2, and set XT = Fθ1 . Then: (i) If Problem 3.3.2 isfeasible at time k ∈ Z+ for state x(k) ∈ X, then Problem 3.3.2 is feasible at



time k + 1 for any state x(k + 1) ∈ Φδ(x(k), w(k)) and all w(k) ∈ Bµ. Mo-reover, XT ⊆ Xf (N); (ii) The closed-loop system x(k + 1) ∈ Φδ(x(k), w(k))

is ε(δ)-ISS in Xf (N) for inputs in Bµ with ISS margin ε(δ) :=(

3ba2 δ) 1

λ .

To prove Theorem 3.4.1 we will make use of the following technical lem-mas (see the appendix for their proofs).

Lemma 3.4.2 Let x ∈ Ωj ∼ Li+1µ for some j ∈ S, i ∈ Z+, and let y ∈ Rn.

If ‖y − x‖ ≤ η[i](µ), then y ∈ Ωj ∼ Liµ.

Proof: Consider y ∈ Rn with ‖y − x‖ ≤ η[i](µ). Let ζ ∈ Liµ and

define z := y − x + ζ. Then it holds that ‖z‖ ≤ ‖y − x‖ + ‖ζ‖ ≤ η[i](µ) +∑i−1p=0 η[p](µ) =

∑ip=0 η[p](µ) and thus, z ∈ Li+1

µ . Together with x ∈ Ωj ∼Li+1

µ this yields x + z ∈ Ωj . Hence, y + ζ = x + z ∈ Ωj . Since ζ ∈ Liµ was

arbitrary, we have y ∈ Ωj ∼ Liµ.

Lemma 3.4.3 Let (φ(1), . . . , φ(N)) be a state sequence of the unpertur-bed system (3.5), obtained from initial state φ(0) := x(k) ∈ X and byapplying an input sequence u[N−1] = (u(0), . . . , u(N − 1)) ∈ UN (x(k)). Let(j1, . . . , jN−1) ∈ SN−1 be the corresponding mode sequence in the sense thatφ(i) ∈ Ωji ∼ Li

µ ⊂ Ωji , i = 1, . . . , N − 1. Let (φ(1), . . . , φ(N)) be also a sta-te sequence of the unperturbed system (3.5), obtained from the initial stateφ(0) := x(k + 1) = φ(1) + w(k) for some w(k) ∈ Bµ and by applying theshifted input sequence u[N−1] := (u(1), . . . , u(N − 1), h(φ(N − 1))). Then,

(φ(i), φ(i + 1)) ∈ Ωji+1 × Ωji+1 , i = 0, N − 2, (3.8a)

‖φ(i)− φ(i + 1)‖ ≤ η[i](‖w(k)‖), i = 0, N − 1. (3.8b)

Proof: Property (3.8a) obviously holds for i = 0, since φ(0) = φ(1) +w(k), w(k) ∈ Bµ = L1

µ and φ(1) ∈ Ωj1 ∼ L1µ. Property (3.8b) holds for

i = 0 as ‖φ(0) − φ(1)‖ = ‖w(k)‖ = η[0](‖w(k)‖). We proceed by induction.Suppose that both (3.8a) and (3.8b) hold for 0 ≤ i− 1 < N − 2. Then, sinceφ(i− 1) ∈ Ωji and ‖φ(i− 1)− φ(i)‖ ≤ η[i−1](‖w(k)‖), it follows that:

‖φ(i)− φ(i + 1)‖ = ‖gji(φ(i− 1), u(i))− gji(φ(i), u(i))‖≤ ηji(‖φ(i− 1)− φ(i)‖) ≤ η(‖φ(i− 1)− φ(i)‖)≤ η(η[i−1](‖w(k)‖)) = η[i](‖w(k)‖), (3.9)

and thus, (3.8b) holds for i. Next, as η[i](‖w(k)‖) ≤ η[i](µ) ≤∑i

p=0 η[p](µ),it follows that φ(i)− φ(i + 1) ∈ Li+1

µ . Then, since φ(i + 1) ∈ Ωji+1 ∼ Li+1µ ,


we have that φ(i + 1) + (φ(i)− φ(i + 1)) = φ(i) ∈ Ωji+1 . Hence, (3.8a) holdsfor i. Thus, we proved that (3.8a) holds for i = 0, . . . , N−2 and (3.8b) holdsfor i = 0, . . . , N − 2. Then, (3.8a) and (3.8b) for i = N − 2 imply (3.8b) fori = N − 1 via the reasoning used in (3.9).

Proof: (Proof of Theorem 3.4.1)(i) We will show that u[N−1], as defined in Lemma 3.4.3, is feasible at timek + 1. Let (j1, . . . , jN−1) ∈ SN−1 be such that φ(i) ∈ Ωji ∼ Li

µ ⊂ Ωji ,i = 1, . . . , N − 1. Then, due to property (3.8b) and φ(i + 1) ∈ Ωji+1 ∼Li+1

µ , it follows from Lemma 3.4.2 that φ(i) ∈ Ωji+1 ∼ Liµ ⊂ Xi for i =

1, . . . , N − 2. From ‖φ(N − 1) − φ(N)‖ ≤ η[N−1](‖w(k)‖) ≤ η[N−1](µ) andAssumption 3.3.3-(iii) it follows that

F (φ(N − 1))− F (φ(N)) ≤ αF (η[N−1](µ)),

which implies F (φ(N−1)) ≤ θ1+αF (η[N−1](µ)) ≤ θ due to φ(N) ∈ XT = Fθ1

and αF (η[N−1](µ)) ≤ θ − θ1. Hence φ(N − 1) ∈ Fθ ⊂ XU ∩ (Ωj0 ∼ LN−1µ ) ⊂

XU ∩ XN−1, so that h(φ(N − 1)) ∈ U and φ(N) ∈ Fθ1 = XT . Thus, thesequence u[N−1] is feasible at time k + 1, which proves the first part of (i).Moreover, since gj0(x, h(x)) ∈ Fθ1 for all x ∈ Fθ and Fθ1 ⊂ Fθ it followsthat Fθ1 is a positively invariant set for system x(k + 1) = gj0(x, h(x(k))),k ∈ Z+. Then, as Fθ1 ⊂ Fθ ⊆ (Ωj0 ∼ LN−1

µ ) ∩ XU ⊂ Xi ∩ XU for alli = 1, . . . , N − 1 and XT = Fθ1 , the sequence (h(φ(0)), . . . , h(φ(N − 1))) isfeasible for Problem 3.3.2 for all φ(0) := x(k) ∈ Fθ1 , k ∈ Z+. Therefore,XT = Fθ1 ⊆ Xf (N), which concludes the proof of (i).

(ii) The result of part (i) implies that Xf (N) is an RPI set for the closed-loop system x(k + 1) ∈ Φδ(x(k), w(k)), k ∈ Z+. Moreover, 0 ∈ int(XT ) imp-lies that 0 ∈ int(Xf (N)). We will now prove that V ∗(·) is an ε-ISS functionfor the closed-loop system (3.7). Since for any x ∈ X and u[N−1] ∈ UN (x)it holds that J(x,u[N−1]) ≥ L(x, u(0)), from Assumption 3.3.3-(i) it followsthat V ∗(x) ≥ α1(‖x‖) for all x ∈ Xf (N), with α1(s) = asλ. Furthermore,by Assumption 3.3.3-(iv), for all x ∈ Xf (N) we have that V ∗(x) ≤ α2(‖x‖),α2(s) = bsλ. Hence, V ∗(·) satisfies inequality (3.2a) with d1 = 0 for allx ∈ Xf (N). Next, we prove that V ∗(·) satisfies inequality (3.2b). Letx(k + 1) ∈ Φδ(x(k), w(k)) for some arbitrary w(k) ∈ Bµ. Furthermore, forany u[N−1] ∈ UN (x(k)) let u[N−1] be defined as in Lemma 3.4.3. Using As-sumption 3.3.4-(iii), i.e. F (gj0(x, h(x))) − F (x) + L(x, h(x)) ≤ 0, ∀x ∈ Fθ,property (3.8a), Assumption 3.3.3-(ii),(iii) and φ(N − 1) ∈ XT , it follows



that:

V ∗(x(k + 1))− V ∗(x(k))≤ J(x(k + 1), u[N−1])− J(x(k),u[N−1]) + δ

= −L(φ(0), u(0)) + F (φ(N)) + δ

+[−F (φ(N − 1)) + F (φ(N − 1))

]− F (φ(N)) + L(φ(N − 1), h(φ(N − 1)))

+N−2∑i=0

[L(φ(i), u(i + 1))− L(φ(i + 1), u(i + 1))

]≤ −L(φ(0), u(0)) + F (φ(N))− F (φ(N − 1))

+ L(φ(N − 1), h(φ(N − 1)))

+ αF

(η[N−1](‖w(k)‖)

)+

N−2∑i=0

αL

(η[i](‖w(k)‖)

)+ δ

≤ −α3(‖x(k)‖) + σ(‖w(k)‖) + δ,

with σ(s) := αF (η[N−1](s)) +∑N−2

i=0 αL(η[i](s)) and α3(s) := α1(s) = asλ.Notice that σ ∈ K due to αF , αL, η ∈ K. The statement then follows fromTheorem 6.2.3. Moreover, from (3.3) it follows that the ε-ISS property of

Definition 4.2.4 holds with ε(δ) =(

3ba2 δ) 1

λ .Theorem 3.4.1 enables the proper selection of an optimality margin δ in

the numerical solver by choosing a desirable ISS margin ε(δ) and finding thecorresponding value of δ. Also, Theorem 3.4.1 recovers as a particular casethe following result for the optimal case (Lazar et al., 2007a), where onlyPWA systems were considered.

Corollary 3.4.4 Suppose that the hypothesis of Theorem 3.4.1 holds andthe global optimum is attained in Problem 3.3.2 for all k ∈ Z+. Then, theclosed-loop system x(k + 1) ∈ Φ0(x(k), w(k)) is ISS in Xf (N) for inputs inBµ.

Remark 3.4.5 The result of Corollary 3.4.4 recovers the result in (Limonet al., 2002a) as the following particular case: X = Ωj0 , S = j0 andgj0(·, ·) is Lipschitz continuous in X. In this case, the set of admissible in-put sequences UN (x) only plays a role in guaranteeing recursive feasibilityof Problem 3.3.2, while ISS can be established directly using the Lipschitzcontinuity property of the dynamics, see (Limon et al., 2002a) for details.See also (Grimm et al., 2007) where Lipschitz continuity of system dynamics


is relaxed to just continuity. Corollary 3.4.4 further relaxes the Lipschitzcontinuity requirement to discontinuous nonlinear dynamics, while the as-sumptions on the MPC cost, prediction horizon and disturbance bound µ > 0are not stronger than the ones employed in (Limon et al., 2002a).

Next, we present a modification to the set of δ sub-optimal MPC con-trollers that will guarantee ISS of the closed-loop system a priori, even fornon-zero optimality margins. For any x ∈ Xf (N) and δ ≥ 0 let Πδ(x) :=u[N−1] ∈ UN (x) | J(x,u[N−1]) ≤ V ∗(x)+δ‖x‖λ and πδ(x) := u(0) ∈ Rm |u[N−1] ∈ Πδ(x). The MPC closed-loop system corresponding to (5.4) isnow given by x(k +1) ∈ Φδ(x(k), w(k)) := g(x(k), u, w(k)) | u ∈ πδ(x(k)),k ∈ Z+. For the above set of δ sub-optimal MPC control actions it holds thatπδ(0) ≡ π0(0) for all δ > 0. Hence, compared to the absolute δ sub-optimalMPC control laws, now δ is a relative optimality margin that varies with thesize of the state norm. The closer the state gets to the origin, the better theapproximation of the optimal MPC control law has to be. This is a realisticassumption, as there exists a sufficiently small neighborhood of the originwhere all constraints in Problem 3.3.2 become inactive and there is no moreswitching in the predicted trajectory.

Theorem 3.4.6 Suppose that the hypotheses of Theorem 3.4.1 are satis-fied with the K-function α1(s) := asλ, a, λ ∈ R>0, as introduced in As-sumption 3.3.3. Let δ ∈ R>0 be given such that 0 < δ < a. Then: (i) IfProblem 3.3.2 is feasible at time k ∈ Z+ for state x(k) ∈ X, then Pro-blem 3.3.2 is feasible at time k + 1 for any state x(k + 1) ∈ Φδ(x(k), w(k))and all w(k) ∈ Bµ. Moreover, XT ⊆ Xf (N); (ii) The closed-loop systemx(k + 1) ∈ Φδ(x(k), w(k)) is ISS in Xf (N) for inputs in Bµ.

Proof: The proof of Theorem 3.4.6 readily follows by applying thereasoning used in the proof of Theorem 3.4.1. The modified set of sub-optimal control laws πδ(x) makes a difference only in the proof of statement(ii), where J(x(k), u[N−1]) ≤ V ∗(x(k)) + δ‖x(k)‖λ implies −V ∗(x(k)) ≤−J(x(k), u[N−1]) + δ‖x(k)‖λ and thus,

V ∗(x(k + 1))− V ∗(x(k)) ≤ . . .

≤ −α1(‖φ(0)‖) + σ(‖w(k)‖) + δ‖x(k)‖λ

= −α3(‖x(k)‖) + σ(‖w(k)‖),

with σ(s) := αF (η[N−1](s))+∑N−2

i=0 αL(η[i](s)) and α3(s) := (a−δ)sλ. Notethat α3 ∈ K as a− δ > 0.



Remark 3.4.7 In the particular case when system (5.4) is PWA, X, U,Ωj , j ∈ S are polyhedral sets and the MPC cost function is defined using1,∞-norms, Problem 3.3.2 can be formulated as a MILP problem, which isstandard in hybrid MPC (Bemporad and Morari, 1999). For methods tocompute a terminal cost and control law h(·) that satisfy Assumption 3.3.3,Assumption 3.3.4 and for illustrative examples we refer to (Lazar et al., 2006,2007a).

3.5 Asymptotic stability results

Sufficient conditions for asymptotic stability of discrete-time PWA systemsin closed-loop with MPC were presented in (Lazar et al., 2006), under thestanding assumption of global optimality for the MPC control law. As al-ready mentioned, it is important to analyze if and how the stability resultsof (Lazar et al., 2006) change in the case of sub-optimal implementations.

Consider the PWC nonlinear system

x(k + 1) = ξ(x(k), u(k)) :=gj(x(k), u(k))if x(k) ∈ Ωj , (3.10)

where the notation is similar to the one in Section 6.3. We still assumethat each gj(·, ·), j ∈ S, satisfies a continuity condition as was defined inAssumption 3.3.1. However, we do not require anymore that the origin liesin the interior of one of the regions Ωj in the state-space partition. The MPCproblem set-up remains the same as the one described by Problem 3.3.2, withthe only difference that the set of admissible input sequences for an initialcondition x ∈ X is now defined as (without any tightening):

UN (x) := u[N−1] ∈ UN |φ(i) ∈ X, i = 1, . . . , N − 1,

φ(0) = x, φ(N) ∈ XT . (3.11)

All the definitions introduced in Section 6.3 and Section 6.4 remain the same(e.g., Xf (N), V ∗(·), Πδ(·), πδ(·), Πδ(·), πδ(·), etc.) with the observation thatthe set of admissible input sequences defined in (5.6) is replaced everywherewith the set defined in (3.11). We will use Ξδ(x(k)) := ξ(x(k), u) | u ∈πδ(x(k)) and Ξδ(x(k)) := ξ(x(k), u) | u ∈ πδ(x(k)).

Theorem 3.5.1 Let δ ∈ R>0 be given and suppose that Assumption 3.3.3holds for system (6.16) and Problem 3.3.2. Take N ∈ Z≥1, XT with 0 ∈int(XT ) as a positively invariant set for system (6.16) in closed-loop with

3.6. Conclusion 55

u(k) = h(x(k)), k ∈ Z+. Furthermore, suppose F (ξ(x, h(x))) − F (x) +L(x, h(x)) ≤ 0 for all x ∈ XT . (i) If Problem 3.3.2 is feasible at time k ∈ Z+

for state x(k) ∈ X, then Problem 3.3.2 is feasible at time k + 1 for any statex(k + 1) ∈ Ξδ(x(k)). Moreover, XT ⊆ Xf (N); (ii) The closed-loop system

x(k + 1) ∈ Ξδ(x(k)) is ε-AS in Xf (N) with ε(δ) :=(

2ba2 δ) 1

λ ; (iii) Supposethat δ ∈ R>0 satisfies 0 < δ < a, where a ∈ R>0 is the gain of the K-functionα1(s) := asλ, introduced in Assumption 3.3.3. Then, the closed-loop systemx(k + 1) ∈ Ξδ(x(k)) is AS in Xf (N).

The proof of the above theorem can be obtained mutatis mutandis by combi-ning the reasoning used in the proof of Theorem III.2 in (Lazar et al., 2006)and Theorem 6.2.3 for the case when σ(s) ≡ 0.

Remark 3.5.2 The result of Theorem 3.5.1, statement (ii), establishes thatδ sub-optimal nonsmooth MPC is ε(δ)-AS without requiring any additionalassumption, other than the ones needed for AS of optimal smooth MPC(Mayne et al., 2000). Furthermore, the result of Theorem 3.5.1, statement(iii), introduces a slightly stronger condition, under which even AS can beguaranteed a priori for a specific class of sub-optimal predictive control laws.In contrast with the results in (Scokaert et al., 1999) this is achieved withoutintroducing additional stabilization constraints in the original MPC problemset-up.

3.6 Conclusion

In this chapter we have considered discontinuous hybrid systems in closed-loop with predictive control laws. We presented conditions for ε-ISS andε-AS of the resulting closed-loop systems. These conditions do not requirecontinuity of the system dynamics nor optimality of the predictive controllaw. The latter is especially important as firstly, the infimum in an MPCoptimization problem does not have to be attained and secondly, numeri-cal solvers usually provide only sub-optimal solutions. An explicit relationwas established between the deviation of the MPC control action from theoptimum and the resulting deterioration of the ISS (AS) property of theclosed-loop system. By imposing stronger conditions on the sub-optimalsolutions, ISS can even be attained in this case. The link between the op-timality margin of the MPC control action and the ISS (AS) margin of theclosed-loop system was further exploited to derive stronger conditions thatyield sub-optimal MPC controllers with an ISS (AS) guarantee, without ad-ding additional constraints to the MPC optimization problem.

4

On Input-to-State Stability of Min-maxNonlinear Model Predictive Control

4.1 Introduction4.2 Input-to-state stability4.3 Min-max nonlinear

MPC: Problem set-up4.4 ISpS results for min-max

nonlinear MPC

4.5 Main result: ISSdual-mode min-max MPC

4.6 Illustrative example: Anonlinear double integrator

4.7 Conclusions

In this chapter we consider discrete-time nonlinear systems that are affec-ted, possibly simultaneously, by parametric uncertainties and other distur-bance inputs. The min-max model predictive control (MPC) methodology isemployed to obtain a controller that robustly steers the state of the systemtowards a desired equilibrium. The aim is to provide a priori sufficient condi-tions for robust stability of the resulting closed-loop system using the input-to-state stability (ISS) framework. First, we show that only input-to-statepractical stability can be ensured in general for closed-loop min-max MPCsystems; and we provide explicit bounds on the evolution of the closed-loopsystem state. Then, we derive new conditions for guaranteeing ISS of min-max MPC closed-loop systems, using a dual-mode approach. An exampleillustrates the presented theory.

4.1 Introduction

One of the practically relevant problems in control theory is the robust regu-lation towards a desired equilibrium of discrete-time systems affected, possi-bly simultaneously, by time-varying parametric uncertainties and other dis-turbance inputs. In the case when hard constraints are imposed on stateand input variables, the robust model predictive control (MPC) methodo-logy provides a reliable solution for tackling this control problem, see, forexample, (Mayne et al., 2000) for an overview. The research related to ro-bust MPC is focused on solving efficiently the corresponding optimization

58On Input-to-State Stability of Min-max Nonlinear Model Predictive

Control

problems on one hand and guaranteeing (robust) stability of the controlledsystem, on the other hand. In this chapter we are interested in stabilityissues and therefore, we position our results only with respect to articles on(robust) stability of nonlinear MPC.

There are several ways for designing robust MPC controllers for pertur-bed nonlinear systems. One way is to rely on the inherent robustness proper-ties of nominally stabilizing nonlinear MPC algorithms, e.g. as it was done in(Scokaert et al., 1997; Magni et al., 1998; Limon et al., 2002b; Grimm et al.,2003). Another approach is to incorporate knowledge about the disturbancesin the MPC problem formulation via open-loop worst case scenarios. Thisincludes MPC algorithms based on tightened constraints, e.g., as the one of(Limon et al., 2002a), and MPC algorithms, based on open-loop min-maxoptimization problems, see, for example, the survey (Mayne et al., 2000). Toincorporate feedback to disturbances, the closed-loop or feedback min-maxMPC problem set-up was introduced in (Lee and Yu, 1997) and further de-veloped in (Mayne, 2001; Magni et al., 2003; Limon et al., 2006; Magni et al.,2006). The open-loop approach is computationally somewhat easier than thefeedback approach, but the set of feasible states corresponding to the feed-back min-max MPC optimization problem is usually much larger. Sufficientconditions for robust asymptotic stability of closed-loop (feedback) min-maxMPC systems were presented in (Mayne, 2001) under the assumption thatthe (additive) disturbance input converges to zero as the state converges tothe origin.

Recently, input-to-state stability (ISS) (Sontag, 1989, 1990; Jiang andWang, 2001) results for min-max nonlinear MPC were presented in (Limonet al., 2006) and (Magni et al., 2006). In (Limon et al., 2006) it was shownthat, in general, only input-to-state practical stability (ISpS) (Jiang, 1993;Jiang et al., 1994, 1996) can be a priori ensured for min-max nonlinear MPC.ISpS is a weaker property than ISS, as ISpS does not imply asymptoticstability for zero disturbance inputs. The reason for the absence of ISS ingeneral is that the effect of a non-zero disturbance input is taken into accountby the min-max MPC controller, even if the disturbance input vanishes inreality. Still, in the case when the disturbance input converges to zero, itis desirable that asymptotic stability is recovered for the controlled system.In (Magni et al., 2006), an H∞ (Chen and Scherer, 2006a) strategy wasused to modify the classical min-max MPC cost function (Mayne et al.,2000) such that ISS is guaranteed for the closed-loop min-max MPC system.Furthermore, in (Magni et al., 2006) it was proven that a local upper boundon the min-max MPC value function, rather than a global one, is sufficientfor ISS.

4.2. Input-to-state stability 59

In this chapter we propose a new approach for designing min-max MPCschemes for nonlinear systems with guaranteed ISS. In contrast with (Magniet al., 2006), our results apply to the classical min-max MPC problem set-up, which is also employed in (Mayne, 2001; Limon et al., 2006). First, wedevelop novel ISpS conditions for min-max nonlinear MPC that allow usto derive explicit bounds on the evolution of the MPC closed-loop systemstate. Furthermore, we prove that these conditions actually imply that thestate trajectory of the closed-loop system is ultimately bounded in a robustlypositively invariant set. Then, we use a dual-mode approach in combinationwith a new technique based on KL-estimates of stability, e.g., see (Khalil,2002), to derive a priori sufficient conditions for ISS of min-max nonlinearMPC. This result is important because it unifies the properties of (Limonet al., 2006) and (Mayne, 2001). More specifically, it can be used to designrobustly asymptotically stable min-max MPC closed-loop systems withouta priori assuming that the disturbance input converges to zero as the stateof the closed-loop system converges to the origin.

Section 6.4 and the sufficient conditions for ISS of dual-mode min-maxnonlinear MPC are given in Section 5.5. An illustrative example is workedout in Section 6.5. Conclusions are summarized in Section 6.6.

4.1.1 Preliminaries

Before introducing the notion of input-to-state (practical) stability we brieflyrecall some basic definitions.

Let R, R+, Z and Z+ denote the field of real numbers, the set of non-negative reals, the set of integer numbers and the set of non-negative integers,respectively. We use the notation Z≥c1 and Z(c1,c2] to denote the sets k ∈Z+ | k ≥ c1 and k ∈ Z+ | c1 < k ≤ c2, respectively, for some c1 ∈ Z+,c2 ∈ Z>c1 , and ZN to denote the N -times Cartesian product Z×Z× . . .×Z,for some N ∈ Z≥1. We use ‖ · ‖ to denote an arbitrary p-norm. With someabuse of notation we will use both (z0, z1, . . .) and zll∈Z+ with zl ∈ Rn,l ∈ R+, to denote a sequence. For a sequence z := zll∈Z+ let ‖z‖ :=sup‖zl‖ | l ∈ Z+ and let z[k] denote the truncation of z at time k ∈ Z+,i.e. z[k] = zll∈Z[0,k]

. For a set S ⊆ Rn, we denote by int(S) its interior.For any r > 0 define a ball of radius r as Br := x ∈ Rn | ‖x‖ ≤ r.

4.2 Input-to-state stability

In this section we present the ISS framework (Sontag, 1989, 1990; Jiang andWang, 2001) for discrete-time autonomous nonlinear systems, which will


Control

be employed in this chapter to study the behavior of perturbed nonlinearsystems in closed-loop with min-max MPC controllers.

Consider the discrete-time autonomous perturbed nonlinear system de-scribed by

xk+1 = G(xk, wk, vk), k ∈ Z+, (4.1)

where xk ∈ Rn, wk ∈ W ⊂ Rdw and vk ∈ V ⊂ Rdv are the state, un-known time-varying parametric uncertainties and other disturbance inputs(possibly additive), respectively, and, G : Rn × Rdw × Rdv → Rn is an arbi-trary nonlinear, possibly discontinuous, function. In what follows we assumethat W and V are bounded sets. Throughout the chapter let w := wl |l ∈ Z+, wl ∈ W and v := vl | l ∈ Z+, vl ∈ V denote some arbitrarysequences of disturbances.

Definition 4.2.1 RPI: A set P ⊆ Rn that contains the origin in its interioris called a robustly positively invariant (RPI) set for system (6.1) (withrespect to W and V) if for all x ∈ P it holds that G(x, w, v) ∈ P for allw ∈ W and all v ∈ V.

Definition 4.2.2 UB: System (6.1) is said to be ultimately bounded (UB)in a set P ⊂ Rn for initial conditions in X ⊆ Rn (with respect to W and V),if for all x0 ∈ X there exists an i(x0) ∈ Z+ such that for all w and all v thecorresponding state trajectory of (6.1) satisfies xk ∈ P for all k ∈ Z≥i(x0).

Definition 4.2.3 A real-valued scalar function ϕ : R+ → R+ belongs toclass K if it is continuous, strictly increasing and ϕ(0) = 0. It belongsto class K∞ if ϕ ∈ K and it is radially unbounded (i.e. ϕ(s) → ∞ ass →∞). A function β : R+×R+ → R+ belongs to class KL if for each fixedk ∈ R+, β(·, k) ∈ K and for each fixed s ∈ R+, β(s, ·) is non-increasing andlimk→∞ β(s, k) = 0.

Next, we introduce a regional version of global ISpS (Jiang, 1993; Jiang et al.,1994, 1996) and global ISS (Sontag, 1989, 1990; Jiang and Wang, 2001), res-pectively, for the discrete-time nonlinear system (6.1). This is useful whendealing with constrained nonlinear systems, such as NMPC closed-loop sys-tems, as it was observed in (Magni et al., 2006).

Definition 4.2.4 Regional ISpS (ISS): The system (6.1) is said to beISpS in X ⊆ Rn if there exist a KL-function β, a K-function γ and a numberd ∈ R+ such that, for each x0 ∈ X, all w and all v, it holds that thecorresponding state trajectory of (6.1) satisfies

‖xk‖ ≤ β(‖x0‖, k) + γ(‖v[k−1]‖) + d, ∀k ∈ Z≥1. (4.2)


If 0 ∈ int(X) and (4.2) holds for d = 0, the system (6.1) is said to be ISS inX.

In what follows we state a discrete-time version of the continuous-timeISpS sufficient conditions of Proposition 2.1 of (Jiang et al., 1996). Thisresult will be used throughout the chapter to prove ISpS and ISS for theparticular case of min-max nonlinear MPC.

Theorem 4.2.5 Let d1, d2 ∈ R+, let a, b, c, λ ∈ R>0 with c ≤ b and let1

α1(s) := asλ, α2(s) := bsλ, α3(s) := csλ and σ ∈ K. Furthermore, let X bea RPI set for system (6.1) and let V : X → R+ be a function such that

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖) + d1 (4.3a)V (G(x, w, v))− V (x) ≤ −α3(‖x‖) + σ(‖v‖) + d2 (4.3b)

for all x ∈ X, w ∈ W and all v ∈ V. Then it holds that:(i) The system (6.1) is ISpS in X and the ISpS property of Definition 4.2.4

holds for

β(s, k) := α−11 (3ρkα2(s)), γ(s) := α−1

1

(3σ(s)1− ρ

), d := α−1

1 (3ξ), (4.4)

where ξ := d1 + d21−ρ and ρ := 1− c

b ∈ [0, 1).(ii) If 0 ∈ int(X) and the inequalities (6.4) hold for d1 = d2 = 0, the

system (6.1) is ISS in X and the ISS property of Definition 4.2.4 (i.e. ford = 0) holds for

β(s, k) := α−11 (2ρkα2(s)), γ(s) := α−1

1

(2σ(s)1− ρ

), (4.5)

where ρ := 1− cb ∈ [0, 1).

Proof: (i) From V (x) ≤ α2(‖x‖) + d1 for all x ∈ X, we have that forany x ∈ X \ 0 it holds:

V (x)− α3(‖x‖) ≤ V (x)− α3(‖x‖)α2(‖x‖)

(V (x)− d1) = ρV (x) + (1− ρ)d1,

where ρ := 1 − cb ∈ [0, 1). In fact, the above inequality holds for all x ∈ X,

since V (0)−α3(0) = V (0) = ρV (0)+(1−ρ)V (0) ≤ ρV (0)+(1−ρ)d1. Then,inequality (4.3b) becomes

V (G(x,w, v)) ≤ ρV (x) + σ(‖v‖) + (1− ρ)d1 + d2, (4.6)1Note that α1, α2, α3 ∈ K∞.


Control

for all x ∈ X, w ∈ W and all v ∈ V. Due to robust positive invariance of X,inequality (6.8) yields repetitively

V (xk+1) ≤ ρk+1V (x0) +k∑

i=0

ρi[σ(‖vk−i‖) + (1− ρ)d1 + d2]

for all x0 ∈ X, w[k] = (w0, w1, . . . , wk) ∈ Wk+1, v[k] = (v0, v1, . . . , vk) ∈Vk+1, k ∈ Z+. Then, taking (4.3a) into account, using the property σ(‖vi‖) ≤σ(‖v[k]‖) for all i ≤ k and the identity

∑ki=0 ρi = 1−ρk+1

1−ρ , it holds that:

V (xk+1) ≤ ρk+1α2(‖x0‖) + ρk+1d1 +k∑

i=0

ρi [σ(‖vk−i‖) + (1− ρ)d1 + d2]

≤ ρk+1α2(‖x0‖) + ρk+1d1 +[σ(‖v[k]‖) + (1− ρ)d1 + d2

] k∑i=0

ρi

= ρk+1α2(‖x0‖) +1− ρk+1

1− ρσ(‖v[k]‖) + d1 +

1− ρk+1

1− ρd2

≤ ρk+1α2(‖x0‖) +1

1− ρσ(‖v[k]‖) + d1 +

11− ρ

d2,

for all x0 ∈ X, w[k] ∈ Wk+1, v[k] ∈ Vk+1, k ∈ Z+. Let ξ := d1 + d21−ρ .

Taking (4.3a) into account and letting α−11 denote the inverse of α1, we

obtain:

‖xk+1‖ ≤ α−11 (V (xk+1)) ≤ α−1

1

(ρk+1α2(‖x0‖) + ξ +

σ(‖v[k]‖)1− ρ

). (4.7)

Applying the inequality

α−11 (z + y + s) ≤ α−1

1 (3 max(z, y, s)) ≤ α−11 (3z) + α−1

1 (3y) + α−11 (3s),

(4.8)

we obtain from (4.7)

‖xk+1‖ ≤ α−11

(3ρk+1α2(‖x0‖)

)+ α−1

1

(3σ(‖v[k]‖)

1− ρ

)+ α−1

1 (3ξ),

for all x0 ∈ X, w[k] ∈ Wk+1, v[k] ∈ Vk+1, k ∈ Z+.We distinguish between two cases: ρ 6= 0 and ρ = 0. First, suppose

ρ ∈ (0, 1) and let β(s, k) := α−11 (3ρkα2(s)). For a fixed k ∈ Z+, we have


that β(·, k) ∈ K due to α2 ∈ K∞, α−11 ∈ K∞ and ρ ∈ (0, 1). For a fixed

s, it follows that β(s, ·) is non-increasing and limk→∞ β(s, k) = 0, due toρ ∈ (0, 1) and α−1

1 ∈ K∞. Thus, it follows that β ∈ KL.

Now let γ(s) := α−11

(3σ(s)1−ρ

). Since 1

1−ρ > 0, it follows that γ ∈ K due

to α−11 ∈ K∞ and σ ∈ K.Finally, let d := α−1

1 (3ξ). Since ρ ∈ (0, 1) and d1, d2 ≥ 0, we have thatξ ≥ 0 and thus, d ≥ 0.

Otherwise, if ρ = 0 we have from (4.7) that

‖xk‖ ≤ α−11 (3σ(‖v[k−1]‖)) + α−1

1 (3ξ)

≤ β(‖x0‖, k) + α−11 (3σ(‖v[k−1]‖)) + α−1

1 (3ξ)

for any β ∈ KL and k ∈ Z≥1.Hence, the perturbed system (6.1) is ISpS in X in the sense of Definiti-

on 4.2.4 and property (4.2) is satisfied with the functions given in (4.4).(ii) Following the proof of statement (i) it is straightforward to observe

that when the sufficient conditions (6.4) are satisfied for d1 = d2 = 0, thenISS is achieved, since d = α−1

1 (3ξ) = α−11 (0) = 0. From (4.7) and α−1

1 (z +y) ≤ α−1

1 (2 max(z, y)) ≤ α−11 (2z) + α−1

1 (2y), it can be easily shown that theISS property of Definition 4.2.4 actually holds with the functions given in(4.5).

Definition 4.2.6 A function V (·) that satisfies the hypothesis of Theo-rem 4.2.5 is called an ISpS (ISS) Lyapunov function.

Remark 4.2.7 The hypothesis of Theorem 4.2.5 part (i) does not requirecontinuity of G(·, ·, ·) or V (·), nor that G(0, 0, 0) = 0 or V (0) = 0. Thelatter makes the ISpS framework suitable for analyzing stability of nonlinearsystems in closed-loop with min-max MPC controllers, since in general, themin-max MPC value function is not zero at zero (see Section 6.4 for details).The hypothesis of Theorem 4.2.5 part (ii), which deals with ISS, also doesnot require continuity of G(·, ·, ·) or V (·). However, it implies G(0, w, 0) = 0for all w ∈ W and V (0) = 0, and continuity of G(·, w, ·) and V (·) at thepoint x = 0 only, for all w ∈ W.

Note that, due to the use of K∞-functions α1, α2, α3 of a special type(which is not restrictive for the commonly used cost functions in min-maxMPC, as shown in Section 6.4), Theorem 4.2.5 provides explicit bounds onthe evolution of the state.


Control

4.3 Min-max nonlinear MPC: Problem set-up

The results presented in this chapter can be applied to both open-loop andfeedback min-max MPC strategies. However, there seems to be a commonagreement that open-loop min-max formulations are conservative and unde-restimate the set of feasible input trajectories. For this reason, although wepresent both problem formulations, the stability results are proven only forfeedback min-max MPC set-ups. However, it is possible to prove, via a simi-lar reasoning and using the same hypotheses, that all the results developedin this chapter also hold for open-loop min-max MPC schemes.

Consider the discrete-time non-autonomous perturbed nonlinear system

xk+1 = g(xk, uk, wk, vk), k ∈ Z+, (4.9)

where xk ∈ Rn, uk ∈ Rm, wk ∈ W ⊂ Rdw and vk ∈ V ⊂ Rdv are the state,the control action, unknown time-varying parametric uncertainties and otherdisturbance inputs, respectively. The mapping g : Rn×Rm×Rdw×Rdv → Rn

is an arbitrary nonlinear, possibly discontinuous, function. Let X ⊆ Rn andU ⊆ Rm denote sets that contain the origin in their interior and representstate and input constraints for system (5.4). Furthermore, let XT ⊆ Xwith 0 ∈ int(XT ) denote a desired terminal set and let F : Rn → R+ withF (0) = 0 and L : Rn × Rm → R+ with L(0, 0) = 0 be arbitrary functions.The objective is to regulate the system towards the origin while minimizinga performance index defined by the functions F (·), L(·, ·) and with the setXT as terminal constraint.

For a fixed prediction horizon N ∈ Z≥1, open-loop min-max MPC eva-luates a single sequence of controls, i.e. uk := (u0|k, . . . , uN−1|k) ∈ UN . Letxk(xk,uk,wk,vk) := (x1|k, . . . , xN |k) denote the state sequence generated bysystem (5.4) from initial state x0|k := xk and by applying the input sequenceuk, where wk := (w0|k, . . . , wN−1|k) ∈ WN and vk := (v0|k, . . . , vN−1|k) ∈VN are the corresponding disturbance sequences and

xi|k := g(xi−1|k, ui−1|k, wi−1|k, vi−1|k), i = 1, . . . , N.

The open-loop min-max MPC class of admissible input sequences definedfor XT and state xk ∈ X is

UN (xk) :=uk ∈ UN | xk(xk,uk,wk,vk) ∈ XN , xN |k ∈ XT , ∀wk ∈ WN ,∀vk ∈ VN

.

Let the terminal set XT ⊆ X and N ∈ Z≥1 be given. At time k ∈ Z+ letxk ∈ X be given. The open-loop min-max MPC approach minimizes the cost

4.3. Min-max nonlinear MPC: Problem set-up 65

J(xk,uk) := maxwk∈WN ,vk∈VN

[F (xN |k) +

∑N−1i=0 L(xi|k, ui|k)

], with predic-

tion model (5.4), over all sequences uk in UN (xk).Feedback min-max MPC obtains a sequence of feedback control laws

that minimizes a worst case cost function, while assuring robust constrainthandling. In this chapter we employ the dynamic programming approach tofeedback min-max nonlinear MPC proposed in (Lee and Yu, 1997) for linearsystems and in (Mayne, 2001) for nonlinear systems.

In this approach, the feedback min-max optimal control input is obtainedas follows:

Vi(x) := minu∈U

maxw∈W,v∈V

[L(x, u) + Vi−1(g(x, u, w, v))

]such that g(x, u, w, v) ∈ Xf (i− 1),∀w ∈ W,∀v ∈ V

,

(4.10)

where the set Xf (i) contains all the states xi ∈ X which are such that (4.10) isfeasible, i = 1, . . . , N . The optimization problem is defined for i = 1, . . . , Nwith the boundary conditions

V0(x0) := F (x0),Xf (0) := XT .

(4.11)

Taking into account the definition of the min-max problem (4.10), Xf (i) isnow the set of all states that can be robustly controlled into the set XT ini ∈ Z≥1 steps.

The control law is applied to system (5.4) in a receding horizon manner.At each sampling time the problem is solved for the current state x and thevalue function VN (x) is obtained. The feedback min-max MPC control lawis defined as

u(x) := u∗N , (4.12)

where u∗N is the optimizer of problem (4.10) for i = N . For simplicityof exposition, in what follows we assume existence and uniqueness of u∗N ,and that the minimum and the maximum are well-defined in (4.10), for alli = 1, . . . , N . Notice that it is possible to show that the results developed inthis chapter also apply when the global optimum is not unique. Furthermore,following the reasoning employed in (Scokaert et al., 1999), ISpS results canalso be obtained for the sub-optimal case.

In the following sections the min-max MPC value function VN (x) will beused as a candidate ISpS Lyapunov function in order to establish ISpS ofthe nonlinear system (5.4) in closed-loop with the feedback min-max MPCcontrol (4.12). To simplify the notation, for the reminder of the chapter wewill use V (x) to denote VN (x).


Control

4.4 ISpS results for min-max nonlinear MPC

In this section we present sufficient conditions for ISpS of system (5.4) inclosed-loop with the feedback min-max MPC control (4.12) and we deriveexplicit bounds on the evolution of the closed-loop system state. Let h :Rn → Rm denote an arbitrary nonlinear function with h(0) = 0 and letXU := x ∈ X | h(x) ∈ U.

Assumption 4.4.1 There exist aL, aF , bF , λ ∈ R>0 with aL ≤ bF , e1, e2 ∈R+, a function h : Rn → Rm with h(0) = 0 and a K-function σ such that:(i) XT ⊆ XU and 0 ∈ int(XT );(ii) XT is a RPI set for system (5.4) in closed-loop with uk = h(xk), k ∈ Z+;(iii) L(x, u) ≥ aL‖x‖λ for all x ∈ X and all u ∈ U;(iv) aF ‖x‖λ ≤ F (x) ≤ bF ‖x‖λ + e1 for all x ∈ XT ;(v) F (g(x, h(x), w, v)) − F (x) ≤ −L(x, h(x)) + σ(‖v‖) + e2 for all x ∈ XT ,w ∈ W, and v ∈ V. 2

Note that Assumption 4.4.1 implies that F (·) is a local2 ISpS Lyapunovfunction. Then, from Theorem 4.2.5 it follows that system (5.4) in closed-loop with uk = h(xk), k ∈ Z+ is ISpS in XT , as formally stated below.

Proposition 4.4.2 Suppose that Assumption 4.4.1 holds. Then, system(5.4) in closed-loop with uk = h(xk), k ∈ Z+, is ISpS in XT . Moreover, ifAssumption 4.4.1 holds with e1 = e2 = 0, system (5.4) in closed-loop withuk = h(xk), k ∈ Z+, is ISS in XT .

Assumption 4.4.1 can be regarded as a generalization of the usual suf-ficient conditions for nominal stability of MPC, which imply that F (·) is alocal Lyapunov function, see, for example, the survey (Mayne et al., 2000).Techniques for computing a terminal cost and a function h(·) such thatAssumption 4.4.1 is satisfied for relevant subclasses of system (5.4) (i.e. per-turbed linear and piecewise affine systems) will be presented in the nextchapter. See also the illustrative nonlinear example in Section 6.5 of thischapter.

Theorem 4.4.3 Suppose that F (·), L(·, ·), XT and h(·) are such that As-sumption 4.4.1 holds for system (5.4). Furthermore, suppose that there existsa number θ ∈ R≥bF

such that V (x) ≤ θ‖x‖λ for all x ∈ Xf (N) \ XT . Then,

2ISS Lyapunov function when e1 = e2 = 0.

4.4. ISpS results for min-max nonlinear MPC 67

the perturbed nonlinear system (5.4) in closed-loop with the feedback min-max MPC control (4.12) is ISpS in Xf (N). Moreover, the property (4.2)holds with the following functions:

β(s, k) :=(

3θ

aL

) 1λ

ρks, γ(s) :=(

3δ

aL(1− ρ)

) 1λ

s, d :=(

3ξ

aL

) 1λ

, (4.13)

where ρ := ρ1λ ∈ (0, 1), ρ := 1 − aL

θ ∈ (0, 1), δ > 0 can be taken ar-bitrarily small, ξ := d1 + d2

1−ρ , d1 := e1 + N [maxv∈V σ(‖v‖) + e2] andd2 := maxv∈V σ(‖v‖) + e2.

Proof: The proof consists in showing that the min-max MPC valuefunction V (·) is an ISpS Lyapunov function, i.e. it satisfies the hypothesis ofTheorem 4.2.5. First, it is known (see (Mayne, 2001; Kerrigan and Macie-jowski, 2001)) that under Assumption 4.4.1-(i),(ii) the set Xf (N) is a RPIset for system (5.4) in closed-loop with the feedback min-max MPC control(4.12).

Second, we will obtain lower and upper bounding functions on the min-max MPC value function that satisfy (4.3a). From Assumption 4.4.1-(iii) itfollows that V (x) = VN (x) ≥ L(x, u(x)) ≥ aL‖x‖λ, for all x ∈ Xf (N), whereu(x) is the feedback min-max MPC control law defined in (4.12).

Next, letting x0 := x ∈ XT , by Assumption 4.4.1-(ii) (i.e. due to robustpositive invariance of XT ) one can apply Assumption 4.4.1-(v) repetitivelyfor the sequence of predicted states. Summing up the resulting inequalitiesit follows that for any w[N−1] ∈ WN and any v[N−1] ∈ VN

F (xN ) +N−1∑i=0

L(xi, h(xi)) ≤ F (x0) +N−1∑i=0

σ(‖vi‖) + Ne2,

where xi := g(xi−1, h(xi−1), wi−1, vi−1) for i = 1, . . . , N . Then, by optimali-ty and Assumption 4.4.1-(iv) we have that for all x ∈ XT ,

V (x) = VN (x) ≤ maxw∈W,v∈V

[F (xN ) +

N−1∑i=0

L(xi, h(xi))

]≤ F (x) + N [max

v∈Vσ(‖v‖) + e2] ≤ bF ‖x‖λ + d1,

where d1 := e1 + N [maxv∈V σ(‖v‖) + e2] > 0. As from the hypothesis ofTheorem 4.4.3 we also have that V (x) ≤ θ‖x‖λ for all x ∈ Xf (N)\XT (withbF ≤ θ) it follows that V (x) ≤ θ‖x‖λ + d1 for all x ∈ Xf (N). Hence, V (·)


Control

satisfies condition (4.3a) for all x ∈ Xf (N) with α1(s) := aLsλ, α2(s) := θsλ

and d1 = e1 + N [maxv∈V σ(‖v‖) + e2] > 0.Next, we show that V (·) satisfies condition (4.3b). By Assumption 4.4.1-

(v) and optimality, for all x ∈ XT = Xf (0) we have that:

V1(x)− V0(x) ≤ maxw∈W,v∈V

[L(x, h(x)) + F (g(x, h(x), w, v))]− F (x)

≤ maxv∈V

σ(‖v‖) + e2.

Then, it can be shown via induction that (see also (Limon et al., 2006)):

Vi+1(x)−Vi(x) ≤ maxv∈V

σ(‖v‖)+ e2, ∀x ∈ Xf (i), ∀i ∈ 0, . . . , N − 1. (4.14)

At time k ∈ Z+, for a given state xk ∈ X and a fixed prediction horizon Nthe min-max MPC control law u(xk) is calculated and then applied to system(5.4). The state evolves to xk+1 = g(xk, u(xk), wk, vk) ∈ Xf (N). Then, byAssumption 4.4.1-(v) and applying recursively (4.14) it follows that

VN (xk+1)− VN (xk) (4.15)= VN (xk+1)− max

w∈W,v∈V[L(xk, u(xk)) + VN−1(g(xk, u(xk), w, v))]

≤ VN (xk+1)− L(xk, u(xk))− VN−1(g(xk, u(xk), wk, vk))= VN (xk+1)− L(xk, u(xk))− VN−1(xk+1)≤ −L(xk, u(xk)) + max

v∈Vσ(‖v‖) + e2

≤ −aL‖xk‖λ + maxv∈V

σ(‖v‖) + e2

= −aL‖xk‖λ + d2, (4.16)

for all xk ∈ Xf (N), wk ∈ W, vk ∈ V and all k ∈ Z+, where d2 :=maxv∈V σ(‖v‖) + e2 > 0. Hence, the feedback min-max nonlinear MPCvalue function V (·) satisfies (4.3b) with α3(s) := aLsλ, any σ ∈ K andd2 = maxv∈V σ(‖v‖) + e2 > 0. The statements then follow from Theo-rem 4.2.5.

The functions β(·, ·), γ(·) and the constant d defined in (4.13) are obtainedby letting σ(s) := δsλ for some (any) δ > 0 and substituting the functionsα1(·), α2(·), α3(·), σ(·) and the constants d1, d2 obtained above in relation(4.4).

4.5 Main result: ISS dual-mode min-max MPC

As shown in the previous section, the hypothesis of Theorem 4.4.3 is sufficientfor ISpS, but not necessarily for ISS of system (5.4) in closed-loop with u(·),

4.5. Main result: ISS dual-mode min-max MPC 69

even when e1 = e2 = 0. This is due to the min-max MPC value functionV (·), which is only an ISpS Lyapunov function in general, and not an ISSLyapunov function. Therefore, it is unclear whether the min-max MPCcontrol law (4.12) results in an ISS closed-loop system.

In the case of persistent disturbances this is not necessarily a drawback,since ultimate boundedness in a RPI subset of Xf (N) is the most one canaim at, anyhow. It will be shown next that UB is indeed guaranteed underthe hypothesis of Theorem 4.4.3. However, in the case when the disturbanceinput vanishes after a certain time it is desirable to have an ISS closed-loopsystem.

In this section we present sufficient conditions for ISS of system (5.4)in closed-loop with a dual-mode min-max MPC strategy. The followingtechnical result will be employed to prove the main result for dual-modemin-max nonlinear MPC.

For any τ ∈ R(0,aL) define

Mτ :=

x ∈ Xf (N)∣∣∣∣‖x‖λ ≤ d2

aL − τ

and Mτ := Xf (N) \Mτ , (4.17)

where aL ∈ R>0 is from Assumption 4.4.1-(iii) and d2 = maxv∈V σ(‖v‖) +e2 > 0. Note that 0 ∈ int(Mτ ), as d2

aL−τ > 0 and 0 ∈ int(XT ) ⊆ int(Xf (N)).

Lemma 4.5.1 Suppose that F (·), L(·, ·), XT and h(·) are such that As-sumption 4.4.1 holds for system (5.4). Let τ ∈ R(0,aL) be such that Mτ 6= ∅and consider the closed-loop system (5.4)-(4.12). Then, for each x0 ∈ Mτ

there exists an i(x0) ∈ Z≥1 such that for all disturbances realizations w andv, it holds that xi(x0) ∈ Mτ .

Moreover, there exists a KL-function β such that for all x0 ∈ Mτ and alldisturbances realizations w and v, the corresponding trajectory of the closed-loop system (5.4)-(4.12) satisfies ‖xk‖ ≤ β(‖x0‖, k) as long as xk ∈ Mτ forall k ∈ Z[0,i), i ∈ Z≥1.

Proof: We prove the second statement of the lemma first. As shownin the proof of Theorem 4.4.3, the hypothesis implies that

aL‖x‖λ ≤ V (x) ≤ θ‖x‖λ + d1, ∀x ∈ Xf (N).

Let r > 0 be such that Br ⊆ Mτ . For all state trajectories xkk∈Z[0,i)∈ Mi

τ

(and thus, xk 6∈ Mτ for all k ∈ Z[0,i)) we have that ‖xk‖ ≥ r for all k ∈ Z[0,i).


Control

This yields:

V (xk) ≤ θ‖xk‖λ + d1

(‖xk‖

r

)λ

≤(

θ +d1

rλ

)‖xk‖λ, ∀xk ∈ Mτ , ∀k ∈ Z[0,i).

The hypothesis also implies (see (4.15)) that

V (xk+1)−V (xk) ≤ −aL‖xk‖λ+d2, ∀xk ∈ Xf (N), wk ∈ W, vk ∈ V, k ∈ Z+.

By the definitions in (4.17), for x ∈ Mτ it holds that−aL‖x‖λ+d2 ≤ −τ‖x‖λ,which yields:

V (xk+1)− V (xk) ≤ −τ‖xk‖λ, ∀xk ∈ Mτ , wk ∈ W, vk ∈ V, k ∈ Z[0,i).(4.18)

Then, following the steps of the proof of Theorem 4.2.5, it is straightforwardto show that the state trajectory satisfies for all k ∈ Z[0,i),

‖xk‖ ≤ β(‖x0‖, k); β(s, k) := α−11 (ρkα2(s)) =

(b

aL

) 1λ

s(ρ

1λ

)k, (4.19)

where α2(s) := bsλ, b := θ + d1

rλ , α1(s) := aLsλ and ρ := 1 − τb. Note that

ρ ∈ (0, 1) as 0 < τ < aL ≤ bF ≤ θ < θ + d1

rλ = b.Next, we prove that there exists an i ∈ Z≥1 such that xi ∈ Mτ . Assume

that there does not exist an i ∈ Z≥1 such that xi ∈ Mτ . Then, for all i ∈ Z+

we have that

‖xi‖ ≤ β(‖x0‖, i) =(

b

aL

) 1λ

‖x0‖(ρ

1λ

)i.

Since ρ1λ ∈ (0, 1), we have that limi→∞

(ρ

1λ

)i= 0. Hence, there exists an

i ∈ Z≥1 such that xi ∈ Br ⊆ Mτ and we reached a contradiction. Note that(4.19) is independent of w or v and thus, i can be taken to depend on x0

only.Before stating the main result, we make use of Lemma 4.5.1 to prove that theISpS sufficient conditions of Assumption 4.4.1 ensure ultimate boundednessof the min-max MPC closed-loop system. This property is achieved withrespect to a RPI sublevel set of the min-max MPC value function inducedby the set Mτ .

Theorem 4.5.2 Suppose that the hypothesis of Lemma 4.5.1 holds and let

Υ := maxx∈Mτ

V (x) + d2 and VΥ := x ∈ Xf (N) | V (x) ≤ Υ.

Then, the closed-loop system (5.4)-(4.12) is ultimately bounded in the setVΥ for initial conditions in Xf (N).

4.5. Main result: ISS dual-mode min-max MPC 71

Proof: By definition of Υ, x ∈ Mτ ⊆ Xf (N) implies that

V (x) ≤ maxx∈Mτ

V (x) ≤ maxx∈Mτ

V (x) + d2 = Υ.

Therefore, Mτ ⊆ VΥ. Suppose that x0 ∈ Xf (N) \ VΥ and thus, x0 ∈ Mτ .Then, by Lemma 4.5.1 it follows that there exists an i(x0) ∈ Z≥1 such thatxi(x0) ∈ Mτ ⊆ VΥ.

Next, we prove that VΥ is a RPI set for the closed-loop system (5.4)-(4.12). As shown in the proof of Lemma 4.5.1 (see (4.18)), for any x ∈ VΥ\Mτ

it holds that

V (g(x, u(x), w, v)) ≤ V (x)− τ‖x‖λ ≤ V (x) ≤ Υ,

for all w ∈ W and all v ∈ V. Now let x ∈ Mτ . By inequality (4.15) it holdsthat

V (g(x, u(x), w, v)) ≤ V (x)− aL‖x‖λ + d2 ≤ V (x) + d2 ≤ Υ.

Therefore, for any x ∈ VΥ, it holds that g(x, u(x), w, v) ∈ VΥ for all w ∈W and all v ∈ V, which implies that VΥ is a RPI set for the closed-loopsystem (5.4)-(4.12). Hence, the closed-loop system (5.4)-(4.12) is ultimatelybounded in VΥ.

In a worst case situation, i.e. when the disturbance input v ∈ V is toolarge and VΥ = Xf (N) the result of Theorem 4.5.2 diminishes to ultimateboundedness of Xf (N) itself.

To state the main result, let the dual-mode feedback min-max MPCcontrol law be defined as:

uDM(x) :=

u(x) if x ∈ Xf (N) \ XT

h(x) if x ∈ XT .(4.20)

Theorem 4.5.3 Suppose Assumption 4.4.1 holds with e1 = e2 = 0 forsystem (5.4) and there exists τ ∈ R(0,aL) such that Mτ ⊆ XT . Then, theperturbed nonlinear system (5.4) in closed-loop with the dual-mode feedbackmin-max MPC control uDM(·) is ISS in Xf (N).

Proof: In order to prove ISS, we consider two situations: in Case 1 weassume that x0 ∈ XT and in Case 2 we assume that x0 ∈ Xf (N) \ XT . InCase 1, F (·) satisfies the hypothesis of Proposition 4.4.2 with e1 = e2 = 0 andhence, the closed-loop system (5.4)-(4.20) is ISS. Then, using the reasoningemployed in the proof of Lemma 4.5.1, it can be shown that there exist a


Control

KL-function β(s, k) := α−11 (2ρkα2(s)), with α1(s) := aF sλ, α2(s) := bF sλ,

ρ := 1− aLbF

, and a K-function γ such that for all x0 ∈ XT the state trajectorysatisfies

‖xk‖ ≤ β(‖x0‖, k) + γ(‖v[k−1]‖), ∀k ∈ Z≥1. (4.21)

In Case 2, since Mτ ⊆ XT , by Lemma 4.5.1, for any x0 ∈ Xf (N), w andany v, there exists a p ∈ Z≥1 such that xk 6∈ XT for k ∈ Z[0,p) and xp ∈ XT .From Lemma 4.5.1 we also have that there exists a KL-function β(s, k) =α−1

1 (ρkα2(s)), with α1(s) = aLsλ, α2(s) = bsλ, ρ = 1− τb

such that the statetrajectory satisfies

‖xk‖ ≤ β(‖x0‖, k), ∀k ∈ Z≤p and xp ∈ XT .

Then, for all p ∈ Z≥1 and all k ∈ Z≥p+1 it holds that

‖xk‖ ≤ β(‖xp‖, k − p) + γ(‖v[k−p,k−1]‖)

≤ β(β(‖x0‖, p), k − p) + γ(‖v[k−p,k−1]‖) ≤ β(‖x0‖, k) + γ(‖v[k−1]‖),

where v[k−p,k−1] denotes the restriction of v to the interval [k− p, k− 1]. Inthe above inequalities we used

β(β(s, p), k − p) = α−11

(2ρk−pα2

((b

aL

) 1λ

s(ρ

1λ

)p))

≤(

2bF b

aLaF

) 1λ

s(ρ

1λ

)k:= β(s, k),

and ρ := max(ρ, ρ) ∈ (0, 1). Hence, β ∈ KL.Then, we have that

‖xk‖ ≤ β(‖x0‖, k) + γ(‖v[k−1]‖), ∀k ∈ Z≥1,

for all x0 ∈ Xf (N), w and all v, where β(s, k) := max(β(s, k), β(s, k), β(s, k)

).

Since β, β, β ∈ KL implies that β ∈ KL, and we have γ ∈ K, the state-ment then follows from Definition 4.2.4.

The interpretation of the condition Mτ ⊆ XT is that the min-max MPCcontroller steers the state of the system inside the terminal set XT for allw and all v. Then, ISS can be achieved by switching to the local feedbackcontrol law when the state enters the terminal set.

4.6. Illustrative example: A nonlinear double integrator 73

4.6 Illustrative example: A nonlinear double inte-grator

The following example will illustrate how one can verify the conditions forISS of min-max nonlinear MPC presented in this chapter. For examplesthat illustrate the benefits of using a min-max MPC scenario compared tousing a nominally stabilizing or inherently robust MPC approach we referthe interested reader to (Lee and Yu, 1997; Scokaert and Mayne, 1998; Magniet al., 2003; Wang and Rawlings, 2004) and the references therein.

Consider a perturbed discrete-time nonlinear double integrator obtainedfrom a continuous-time double integrator via a sample-and-hold device witha sampling period equal to one, as follows:

xk+1 = Axk + Buk + f(xk) + vk, k ∈ Z+, (4.22)

where A = [ 1 10 1 ], B = [ 0.5

1 ], f : R2 → R2, f(x) := 0.025 [ 11 ]x>x is a nonlinear

additive term and vk ∈ V := v ∈ R2 | ‖v‖∞ ≤ 0.03 for all k ∈ Z+ is anadditive disturbance input (we use ‖ · ‖∞ to denote the infinity norm). Thestate and the input are constrained at all times in the sets

X := x ∈ R2 | ‖x‖∞ ≤ 10 and U := u ∈ R | |u| ≤ 2.

The MPC cost function is defined using ∞-norms, i.e.

F (x) := ‖Px‖∞, L(x, u) := ‖Qx‖∞ + ‖Rx‖∞,

where P is a full-column rank matrix (to be determined), Q = 0.8I2 andR = 0.1. The stage cost satisfies Assumption 4.4.1-(iii) for λ = 1 and anyaL ∈ (0, 0.8).

We take the function h(·) as h(x) := Kx, where K ∈ R1×2 is the gainmatrix. To compute the terminal cost matrix P and the gain matrix Ksuch that Assumption 4.4.1-(v) holds, we first calculate P and K for thelinearization of system (4.22), i.e.:

xk+1 = Axk + Buk + vk, k ∈ Z+. (4.23)

To accommodate for the nonlinear term f(·), we employ a “larger” stagecost weight matrix for the state, i.e. Q = 2.4I2, instead of Q = 0.8I2, forwhich it holds that ‖Qx‖∞ ≥ ‖Qx‖∞ for all x ∈ R2. The terminal costF (x) = ‖Px‖∞ and local control law h(x) = Kx with the matrices

P =[12.1274 7.02670.4769 11.6072

], K =

[−0.5885 −1.4169

], (4.24)


Control

were computed (using a technique recently developed in (Lazar et al., 2006))such that the following inequality holds for the linear system (4.23), i.e.

‖P ((A + BK)x + v)‖∞ − ‖Px‖∞ ≤ −‖Qx‖∞ − ‖RKx‖∞ + σ(‖v‖), (4.25)

for all x ∈ R2 and all v ∈ R2, where σ(s) := ‖P‖∞s.The terminal cost satisfies Assumption 4.4.1-(iv) for λ = 1, bF = ‖P‖∞ =

19.1541, aF = 0.1 and e1 = 0. To obtain a suitable bound on ‖f(x)‖∞ weemploy the following tightened set of constraints for h(·) (see Figure 4.1 fora plot of XU):

XU := x ∈ X | ‖x‖∞ ≤ 1.72, |Kx| ≤ 2.

The terminal set XT , also plotted in Figure 4.1, is taken as the maximal RPIset contained in the set XU (and which is non-empty) for the linear system(4.23), in closed-loop with uk = h(xk), k ∈ Z+, and disturbances in the setv ∈ R2 | ‖v‖∞ ≤ 0.18. One can easily check that maxx∈XU ‖f(x)‖∞ < 0.15and thus, it follows that the terminal set XT chosen as specified above is aRPI set for the nonlinear system (4.22) in closed-loop with uk = h(xk),k ∈ Z+, and all disturbances v in V = v ∈ R2 | ‖v‖∞ ≤ 0.03.

Using the fact that (notice that below, in some cases, ‖ · ‖∞ denotes theinduced infinity matrix norm)

‖Qx‖∞ ≥ 2.3515‖x‖∞, ∀x ∈ R2, maxx∈XT

‖P0.025 [ 11 ]x>‖∞ = 1.5515,

inequality (4.25) and the triangle inequality, for all x ∈ XT and all v ∈ R2

we obtain:

‖P ((A + BK)x + v) + Pf(x)‖∞ − ‖Px‖∞≤ ‖P ((A + BK)x + v)‖∞ − ‖Px‖∞ + ‖Pf(x)‖∞≤ −‖Qx‖∞ − ‖RKx‖∞ + σ(‖v‖∞) + ‖Pf(x)‖∞

≤ −2.3515‖x‖∞ − ‖RKx‖∞ + σ(‖v‖∞) + maxx∈XT

(‖P0.025 [ 1

1 ]x>‖∞)‖x‖∞

≤ −2.3515‖x‖∞ − ‖RKx‖∞ + σ(‖v‖∞) + 1.5515‖x‖∞= −0.8‖x‖∞ − ‖RKx‖∞ + σ(‖v‖∞)= −‖Q‖∞‖x‖∞ − ‖RKx‖∞ + σ(‖v‖∞)≤ −‖Qx‖∞ − ‖RKx‖∞ + σ(‖v‖∞)= −L(x,Kx) + σ(‖v‖∞).

4.6. Illustrative example: A nonlinear double integrator 75

Figure 4.1: State trajectory for the nonlinear system (4.22) in closed-loopwith a dual-mode min-max MPC controller and an estimate of the set offeasible states Xf (4).

Hence, the terminal cost F (x) = ‖Px‖∞ and the control law h(x) = Kx,with the matrices P and K given in (4.24), satisfy Assumption 4.4.1-(v) forthe nonlinear system (4.22) with e2 = 0 and with σ(s) = ‖P‖∞s.

Consider now the set Mτ , which needs to be determined to establish ISSof the nonlinear system (4.22) in closed-loop with the dual-mode min-maxMPC control law (4.20). We can choose aL = 0.79 < 0.8, which ensures that‖Qx‖∞ ≥ aL‖x‖∞ for all x ∈ R2. Since d2 = maxv∈V σ(‖v‖∞) = 0.5746,it follows that a necessary condition to be satisfied is τ ∈ (0, 0.79) (withthe smallest set Mτ obtained for limτ→0

d2aL−τ = 0.7273). For τ = 0.0718,

which yields d2aL−τ = 0.8001, it holds that Mτ ⊂ XT , see Figure 4.1 for an

illustrative plot. Therefore, the closed-loop system (4.22)-(4.20) is ISS inXf (N), as guaranteed by Theorem 4.5.3.

As the feedback min-max MPC optimization problem was computatio-nally untractable for the nonlinear model (4.22), we have used an open-loopmin-max MPC problem set-up, as the one described in Section 6.3, to calcu-late the control input. The developed theory applies also for the open-loopmin-max MPC scheme, as pointed out in Section 6.3. Although the resultingopen-loop min-max optimization problem still has a very high computational


Control

1 2 3 4 5 6 7 8 9 10−0.04

−0.02

0

0.02

0.04

v 1−sol

id li

ne; v

2 − d

ashe

d lin

e

1 2 3 4 5 6 7 8 9 10

−2

−1

0

1

2

u

Samples

Figure 4.2: Dual-mode min-max nonlinear MPC control input and distur-bance input histories.

burden, we could obtain a solution using the fmincon Matlab solver. Theclosed-loop state trajectories for initial state x0 = [−7 − 4]> and predictionhorizon N = 4 are plotted in Figure 4.1. The dual-mode min-max MPCcontrol input and (randomly generated) disturbance input histories are plot-ted in Figure 4.2. The min-max MPC controller manages to drive the stateof the perturbed nonlinear system inside the terminal set, while satisfyingconstraints at all times.

4.7 Conclusions

In this chapter we have revisited the robust stability problem in min-maxnonlinear model predictive control. The input-to-state practical stability fra-mework has been employed to study robust stability of perturbed nonlinearsystems in closed-loop with min-max MPC controllers. New a priori condi-tions for ISpS were presented together with explicit bounds on the evolutionof the closed-loop system state. Moreover, it was proven that these con-ditions also ensure ultimate boundedness. Novel conditions that guaranteeISS of min-max nonlinear MPC closed-loop systems were derived using adual-mode approach. This result is useful as it provides a methodology fordesigning robustly asymptotically stable min-max MPC schemes without apriori assuming that the (additive) disturbance input converges to zero asthe closed-loop system state converges to the origin.

5

Design of the terminal cost:H∞ and min-max MPC

5.1 Introduction5.2 Preliminaries5.3 Problem formulation

5.4 Main results5.5 Conclusions

This chapter presents a novel method for designing the terminal costand the auxiliary control law (ACL) for robust MPC of uncertain linearsystems, such that ISS is a priori guaranteed for the closed-loop system.The method is based on the solution of a set of LMIs. An explicit relationis established between the proposed method and H∞ control design. Thisrelation shows that the LMI-based optimal solution of the H∞ synthesisproblem solves the terminal cost and ACL problem in min-max MPC, for aparticular choice of the stage cost. This result, which was somehow missingin the MPC literature, is of general interest as it connects well known linearcontrol problems to robust MPC design.

5.1 Introduction

Perhaps the most utilized method for designing stabilizing and robustly sta-bilizing model predictive controllers (MPC) is the terminal cost and con-straint set approach (Mayne et al., 2000). This technique, which appliesto both nominally stabilizing and min-max robust MPC schemes, relies onthe off-line computation of a suitable terminal cost along with an auxiliarycontrol law (ACL). For nominally stabilizing MPC with quadratic costs, theterminal cost can be calculated for linear dynamics by solving a discrete-time Riccati equation, with the optimal linear quadratic regulator (LQR) asthe ACL (Scokaert and Rawlings, 1998). In (Kothare et al., 1996) it wasshown that an alternative solution to the same problem, which also worksfor parametric uncertainties, can be obtained by solving a set of LMIs. Thedesign of min-max MPC schemes that are robust to additive disturbanceswas treated in (Magni et al., 2003), where it was proven that the terminal

78Design of the terminal cost:

H∞ and min-max MPC

cost can be obtained as a solution of a discrete-time H∞ Riccati equation,for an ACL that solves the corresponding H∞ control problem.

In this chapter we present an LMI-based solution for obtaining a ter-minal cost and an ACL, such that min-max MPC schemes (Magni et al.,2006; Lazar et al., 2008a) achieve input-to-state stability (ISS) (Jiang andWang, 2001) for linear systems affected by both parametric and additive dis-turbances. The proposed LMIs generalize the conditions in (Kothare et al.,1996) to allow for additive uncertainties as well. Moreover, we establish anexplicit relation between the developed solution and the LMI-based1 opti-mal solution of the discrete-time H∞ synthesis problem corresponding to aspecific performance output, related to the MPC cost. This result, whichwas somehow missing in the MPC literature, adds to the results of (Magniet al., 2003) and to the well known connection between design of nominal-ly stabilizing MPC schemes and the optimal solution of the LQR problem.Such results are of general interest as they connect well known linear controlproblems to MPC design.

5.2 Preliminaries

Let R, R+, Z and Z+ denote the field of real numbers, the set of non-negative reals, the set of integer numbers and the set of non-negative integers,respectively. We use the notation Z≥c1 and Z(c1,c2] to denote the sets k ∈Z+ | k ≥ c1 and k ∈ Z+ | c1 < k ≤ c2, respectively, for some c1, c2 ∈ Z+.For i ∈ Z+, let i = 1, N denote i = 1, . . . , N . For a set S ⊆ Rn, we denoteby int(S) the interior of S. A polyhedron (or a polyhedral set) in Rn isa set obtained as the intersection of a finite number of open and/or closedhalf-spaces. The Hölder p-norm of a vector x ∈ Rn is defined as ‖x‖p :=

(|[x]1|p + . . . + |[x]n|p)1p for p ∈ Z[1,∞) and ‖x‖∞ := maxi=1,...,n |[x]i|, where

[x]i, i = 1, . . . , n, is the i-th component of x and | · | is the absolute value.For a positive definite and symmetric matrix M , denoted by M 0, M

12

denotes its Cholesky factor, which satisfies (M12 )>M

12 = M

12 (M

12 )> = M

and, λmin(M) and λmax(M) denote the smallest and the largest eigenvalueof M , respectively. We will use 0 and I to denote a matrix with all elementszero and the identity matrix, respectively, of appropriate dimensions. Letz := z(l)l∈Z+ with z(l) ∈ Ro for all l ∈ Z+ denote an arbitrary sequence.Define ‖z‖ := sup‖z(l)‖ | l ∈ Z+, where ‖ · ‖ denotes an arbitrary p-

1A similar connection is established in (Magni et al., 2003), with the difference thatthe Riccati-based solution to the optimal H∞ synthesis problem is exploited, rather thanthe LMI-based solution; also, parametric uncertainties are not considered.


norm, and z[k] := z(l)l∈Z[0,k]. A function ϕ : R+ → R+ belongs to class

K if it is continuous, strictly increasing and ϕ(0) = 0. A function ϕ :R+ → R+ belongs to class K∞ if ϕ ∈ K and lims→∞ ϕ(s) = ∞. A functionβ : R+ ×R+ → R+ belongs to class KL if for each fixed k ∈ R+, β(·, k) ∈ Kand for each fixed s ∈ R+, β(s, ·) is decreasing and limk→∞ β(s, k) = 0.

5.2.1 Input-to-state stability

Consider the discrete-time nonlinear system

x(k + 1) = Φ(x(k), w(k), v(k)), k ∈ Z+, (5.1)

where x(k) ∈ Rn is the state and w(k) ∈ Rdw , v(k) ∈ Rdv are unknown dis-turbance inputs at the discrete-time instant k. The mapping Φ : Rn×Rdw ×Rdv → Rn is an arbitrary nonlinear function. We assume that Φ(0, w, 0) = 0for all w. Let W and V be subsets of Rdw and Rdv , respectively.

Definition 5.2.1 We call a set P ⊆ Rn robustly positively invariant (RPI)for system (6.1) with respect to (W, V) if for all x ∈ P it holds thatΦ(x,w, v) ∈ P for all (w, v) ∈ W× V.

Definition 5.2.2 Let X with 0 ∈ int(X) be a subset of Rn. We call system(6.1) ISS(X, W, V) if there exist a KL-function β(·, ·) and a K-function γ(·)such that, for each x(0) ∈ X, all w = w(l)l∈Z+ with w(l) ∈ W, ∀l ∈ Z+ andall v = v(l)l∈Z+ with v(l) ∈ V, ∀l ∈ Z+ it holds that the correspondingstate trajectory of (6.1) satisfies ‖x(k)‖ ≤ β(‖x(0)‖, k) + γ(‖v[k−1]‖), ∀k ∈Z≥1. We call the function γ(·) an ISS gain of system (6.1).

5.2.2 Input-to-state stability conditions for min-max robustMPC

In this subsection we briefly summarize some of the results presented in theprevious chapter, to prepare the problem formulation.

Consider the discrete-time constrained nonlinear system

x(k + 1) = φ(x(k), u(k), w(k), v(k)), k ∈ Z+, (5.2)

where x(k) ∈ X ⊆ Rn is the state, u(k) ∈ U ⊆ Rm is the control actionand w(k) ∈ W ⊂ Rdw , v(k) ∈ V ⊂ Rdv are unknown disturbance inputs atthe discrete-time instant k. φ : Rn × Rm × Rdw × Rdv → Rn is an arbitrarynonlinear function with φ(0, 0, w, 0) = 0 for all w ∈ W. We assume that0 ∈ int(X), 0 ∈ int(U) and W, V are bounded. Next, let F : Rn → R+ and



L : Rn×Rm → R+ with F (0) = L(0, 0) = 0 be arbitrary nonlinear functions.For N ∈ Z≥1 let u[N−1](k) := (u(k), u(k+1), . . . , u(k+N− 1)) ∈ UN = U×. . .×U denote a sequence of future inputs and, similarly, let w[N−1](k) ∈ WN ,v[N−1](k) ∈ VN denote some sequences of future disturbances. Consider theMPC cost

J(x(k), u[N−1](k), w[N−1](k),v[N−1](k))

:= F (x(k + N)) +N−1∑i=0

L(x(k + i), u(k + i)),

where x(k + i+1) := φ(x(k + i), u(k + i), w(k + i), v(k + i)) for i = 0, N − 1and x(k) := x(k). Let XT ⊆ X with 0 ∈ int(XT ) denote a target set anddefine the following set of feasible input sequences:

UN (x(k)) := u[N−1](k) ∈ UN | x(k + i) ∈ X, i = 1, N − 1, x(k + N) ∈ XT ,

x(k) := x(k), ∀w[N−1](k) ∈ WN ,∀v[N−1](k) ∈ VN.

Problem 5.2.3 Let XT ⊆ X and N ∈ Z≥1 be given. At time k ∈ Z+ letx(k) ∈ X be given and infimize

supw[N−1](k)∈WN ,v[N−1](k)∈VN

J(x(k), u[N−1](k), w[N−1](k), v[N−1](k))

over all input sequences u[N−1](k) ∈ UN (x(k)). 2

Assuming the infimum in Problem 6.4.2 exists and can be attained, theMPC control law is obtained as uMPC(x(k)) := u∗(k), where ∗ denotes theoptimum2.

Next, we summarize a priori sufficient conditions for guaranteeing robuststability of system (6.4) in closed-loop with u(k) = uMPC(x(k)), k ∈ Z+ thatwere presented in detail in the previous chapter. Let h : Rn → Rm denote anauxiliary control law (ACL) with h(0) = 0 and let XU := x ∈ X | h(x) ∈ U.

Assumption 5.2.4 There exist functions α1, α2, α3 ∈ K∞ and σ ∈ K suchthat:(i) XT ⊆ XU;(ii) XT is a RPI set for system (6.4) in closed-loop with u(k) = h(x(k)),k ∈ Z+;

2If the infimum does not exist, one has to resort to ISS results for sub-optimal solutions,see, e.g., the results presented in Chapter 3 of the thesis.

5.3. Problem formulation 81

(iii) L(x, u) ≥ α1(‖x‖) for all x ∈ X and all u ∈ U;(iv) α2(‖x‖) ≤ F (x) ≤ α3(‖x‖) for all x ∈ XT ;(v) F (φ(x, h(x), w, v)) − F (x) ≤ −L(x, h(x)) + σ(‖v‖), ∀x ∈ XT , ∀w ∈ W,∀v ∈ V.

In (Magni et al., 2006) and Chapter 3 of this thesis it was shown that As-sumption 5.2.4 is sufficient for guaranteeing ISS of the MPC closed-loop sys-tem corresponding to Problem 6.4.2. Notice that although in Problem 6.4.2we have presented the “open-loop” formulation of min-max MPC for simpli-city of exposition, Assumption 5.2.4 is also sufficient for guaranteeing ISSfor “feedback” min-max variants of Problem 6.4.2, see (Magni et al., 2006)and Chapter 3 for the details.

Remark 5.2.5 The sufficient ISS conditions of Assumption 5.2.4 are anextension for robust MPC of the well known terminal cost and constraintset stabilization conditions for nominal MPC, see A1-A4 in (Mayne et al.,2000). While the stabilization conditions for MPC (Mayne et al., 2000)require that the terminal cost is a local Lyapunov function for the systemin closed-loop with an ACL, Assumption 5.2.4 requires in a similar mannerthat the terminal cost is a local ISS Lyapunov function (Jiang and Wang,2001) for the system in closed-loop with an ACL. 2

5.3 Problem formulation

For a given stage cost L(·, ·), to employ Assumption 5.2.4 for setting-uprobust MPC schemes with an a priori ISS guarantee (or to compute statefeedback controllers that achieve local ISS), one needs systematic methodsfor computing a terminal cost F (·), a terminal set XT and an ACL h(·) thatsatisfy Assumption 5.2.4.

Once F (·) and h(·) are known, several methods are available for calcula-ting the maximal RPI set contained in XU for certain relevant subclasses ofsystem (6.4), in closed-loop with u(k) = h(x(k)), k ∈ Z+, see, for example,(Kolmanovsky and Gilbert, 1998; Alessio et al., 2007) and the references the-rein. As a consequence, therefore, we focus on solving the following problem.

Problem 5.3.1 Calculate F (·) and h(·) such that Assumption 5.2.4-(v)holds. 2

This problem comes down to computing an input-to-state stabilizing state-feedback given by h(·) along with an ISS Lyapunov function (i.e. F (·)) for



system (6.4) in closed-loop with the ACL. This is a non-trivial problem,which depends on the type of MPC cost, system class and on the type ofcandidate ISS Lyapunov function F (·). Furthermore, it would be desirablethat the MPC cost function is continuous and convex.

5.3.1 Existing solutions

Several solutions have been presented for the considered problem for par-ticular subclasses of system (6.4). Most methods consider quadratic costfunctions, F (x) := x>Px, P 0, L(x, u) = x>Qx + u>Ru, Q,R 0, andlinear state feedback ACLs given by h(x) := Kx.

(i) The nominal linear case: φ(x, u, 0, 0) := Ax + Bu, A ∈ Rn×n, B ∈Rn×m. In (Scokaert and Rawlings, 1998) it was proven that the solutions ofthe unconstrained infinite horizon linear quadratic regulation problem withweights Q,R satisfy Assumption 5.2.4-(v), i.e.

K = −(R + B>PB)−1B>PA

and

P = (A + BK)>P (A + BK) + K>RK + Q. (5.3)

Numerically, this method amounts to solving the discrete-time Riccati equa-tion (5.3).

(ii) The linear case with parametric disturbances: φ(x, u, w, 0) := A(w)x+B(w)u, A(w) ∈ Rn×n, B(w) ∈ Rn×m are affine functions of w ∈ W withW a compact polyhedron. In (Kothare et al., 1996) it was proven thatP = Z−1 and K = Y Z−1 satisfy Assumption 5.2.4-(v), where Z ∈ Rn×n

and Y ∈ Rm×n are solutions of the linear matrix inequalityZ (A(wi)Z + B(wi)Y )> (R

12 Y )> (Q

12 Z)>

(A(wi)Z + B(wi)Y ) Z 0 0R

12 Y 0 I 0

Q12 Z 0 0 I

0,

∀i = 1, E

with w1, . . . , wE the vertices of the polytope W. Numerically, this methodamounts to solving a semidefinite programming problem. This solution tri-vially applies also to the case (i) and, moreover, it was extended to piecewiseaffine discrete-time hybrid systems in (Lazar et al., 2006).

5.4. Main results 83

(iii) The nonlinear case with additive disturbances: φ(x, u, 0, v) = f(x)+g1(x)u + g2(x)v with suitably defined functions f(·), g1(·) and g2(·). Anonlinear ACL given by h(x) was constructed in (Magni et al., 2003) usinglinearization of the system, so that Assumption 5.2.4-(v) holds for all statesin a sufficiently small sublevel set of V (x) = x>Px, P 0. Numerically thismethod amounts to solving a discrete-time H∞ Riccati equation.

For the linear case with additive disturbances (i.e. f(x) = A, g1(x) = Band g1(x) = B1), it is worth to point out that an LMI-based design methodto obtain the terminal cost, for a given ACL, was presented in (Alamo et al.,2005).

5.4 Main results

In this section we derive a novel LMI-based solution to the problem of findinga suitable terminal cost and ACL that applies to linear systems affected byboth parametric and additive disturbances, i.e.

x(k+1) = φ(x(k), u(k), w(k), v(k)) := A(w(k))x(k)+B(w(k))u(k)+B1(w(k))v(k),(5.4)

where A(w) ∈ Rn×n, B(w) ∈ Rn×m, B1(w) ∈ Rn×dv are affine functions ofw. We will also consider quadratic cost functions, F (x) := x>Px, P 0,L(x, u) = x>Qx + u>Ru, Q,R 0, and linear state feedback ACLs givenby h(x) := Kx.

5.4.1 LMI-based-solution

Consider the linear matrix inequalities,Z 0 (A(wi)Z + B(wi)Y )> (R

12 Y )> (Q

12 Z)>

0 τI B1(wi)T 0 0(A(wi)Z + B(wi)Y ) B1(wi) Z 0 0

R12 Y 0 0 I 0

Q12 Z 0 0 0 I

0,

∀i = 1, E, (5.5)

where w1, . . . , wE are the vertices of the polytope W, Q ∈ Rn×n and R ∈Rm×m are known positive definite and symmetric matrices, and Z ∈ Rn×n,Y ∈ Rm×n and τ ∈ R>0 are the unknowns.

Theorem 5.4.1 Suppose that the LMIs (5.5) are feasible and let Z, Y and τbe a solution with Z 0, τ ∈ R>0. Then, the terminal cost F (x) = x>Px,the stage cost L(x, u) = x>Qx + u>Ru and the ACL h(x) = Kx with



P := Z−1 and K := Y Z−1 satisfy Assumption 5.2.4-(v) with σ(‖v‖) :=τ‖v‖2

2 = τv>v.

Proof: For brevity let ∆(wi) denote the matrix in the left-hand side of (5.5).Using W = Cow1, . . . , wE (where Co· denotes the convex hull) and thefact that A(w), B(w) and B1(w) are affine functions of w, it is trivial toobserve that if (5.5) holds for all vertices w1, . . . , wE of W, then ∆(w) 0holds for all w ∈ W.

Applying the Schur complement to ∆(w) 0 (pivoting after diag(Z, I, I))and letting M(w) := A(w)Z + B(w)Y yields the equivalent matrix inequa-lities:(

Z −M(w)>Z−1M(w)− Z>QZ − Y >RY −M(w)>Z−1B1(w)−B1(w)>Z−1M(w) τI −B1(w)>Z−1B1(w)

) 0

and Z 0. Letting Acl(w) := A(w) + B(w)K, substituting Z = P−1

and Y = KP−1, and performing a congruence transformation on the abovematrix inequality with diag(P, I) yields the equivalent matrix inequalities:(

P −Acl(w)>PAcl(w)−Q−K>RK −Acl(w)>PB1(w)−B1(w)>PAcl(w)) τI −B1(w)>PB1(w)

) 0

and P 0. Pre multiplying with ( xv )> and post multiplying with ( x

v ) theabove matrix inequality yields the equivalent inequality:

(Acl(w)x + B1(w)v)>P (Acl(w)x + B1(w)v)− x>Px

≤ −x>(Q + K>RK)x + τv>v,

for all x ∈ Rn and all v ∈ Rdv . Hence, Assumption 5.2.4-(v) holds withσ(‖v‖) = τ‖v‖2

2. 2

Remark 5.4.2 In (Lazar et al., 2008a) the authors established an explicitrelation between the gain τ ∈ R>0 of the function σ(·) and the ISS gain of thecorresponding closed-loop MPC system. Thus, since τ enters (5.5) linearly,one can minimize over τ subject to the LMIs (5.5), leading to a smaller ISSgain from v to x. 2

5.4.2 Relation to LMI-based H∞ control design

In this section we formalize the relation between the considered robust MPCdesign problem and H∞ design for linear systems. But first, we brieflyrecall the H∞ design procedure for the discrete-time linear system (5.4).


For simplicity, we remove the parametric disturbance w and consider onlyadditive disturbances v ∈ V. However, the results derived below that relateto the optimal H∞ gain also hold if parametric disturbances are considered,in the sense of an optimal H∞ gain for linear parameter varying systems.

Consider the system corresponding to (5.4) without parametric uncer-tainties, i.e.

x(k + 1) = Ax(k) + Bu(k) + B1v(k),z(k) = Cx(k) + Du(k) + D1v(k), (5.6)

where we added the performance output z ∈ Rdz . Using the results of(Kaminer et al., 1993), (Chen and Scherer, 2006b) it can be demonstratedthat system (5.6) in closed-loop with u(k) = h(x(k)) = Kx(k), k ∈ Z+, hasan H∞ gain less than √γ if and only if there exists a symmetric matrix Psuch that:

P 0 (A + BK)>P (C + DK)>

0 γI B>1 P D>

1

P (A + BK) PB1 P 0C + DK D1 0 I

0. (5.7)

Letting Z = P−1, Y = KP−1 and performing a congruence transformationusing diag(Z, I, Z, I) one obtains the equivalent LMI:

Z 0 (AZ + BY )> (CZ + DY )>

0 γI B>1 D>

1

AZ + BY B1 Z 0CZ + DY D1 0 I

0. (5.8)

Indeed, from the above inequalities, where V (x) := x>Px, one obtains thedissipation inequality:

V (x(k + 1))− V (x(k)) ≤ −‖z(k)‖22 + γ‖v‖2

2. (5.9)

Hence, we can infer that∑∞

i=0 ‖z(i)‖22 ≤ γ

∑∞i=0 ‖v(i)‖2

2 and conclude thatthe H∞ norm of the system is not greater than √γ. Minimizing γ subjectto the above LMI yields the optimal H∞ gain as the square root of theoptimum.

Remark 5.4.3 In (Kaminer et al., 1993), (Chen and Scherer, 2006b) anequivalent formulation of the matrix inequality (5.7) is used, i.e. with γI inthe south east corner of (5.7)-(5.8) instead of I, which leads to the adapted



dissipation inequality V (x(k+1))−V (x(k)) ≤ −γ−1‖z(k)‖22 +γ‖v‖2

2. Then,by minimizing over γ subject to the LMIs (5.8), one obtains the optimal H∞gain directly as the optimal solution, without having to take the square root.However, regardless of which LMI set-up is employed, the resulting optimalH∞ gain and corresponding controller (defined by the gain K) are the same,with a difference in the storage function V (x) = x>Px with a factor γ. 2

Theorem 5.4.4 Suppose that the LMIs (5.5) without parametric uncer-tainties and (5.8) with C =

(Q

12

0

), D =

(0

R12

)and D1 = 0 are feasible for

system (5.6). Then the following statements are equivalent:

1. Z, Y and τ are a solution of (5.5);

2. Z, Y and γ are a solution of (5.8) with C =(

Q12

0

), D =

(0

R12

)and

D1 = 0;

3. System (5.6) in closed-loop with u(k) = Kx(k) and K = Y Z−1 satisfiesthe dissipation inequality (5.9) with storage function V (x) = x>Pxand P = Z−1, and it has an H∞ norm less than √γ =

√τ ;

4. Assumption 5.2.4-(v) holds for F (x) = x>Px, L(x, u) = x>Qx+u>Ruand h(x) = Kx, with P = Z−1, K = Y Z−1 and σ(‖v‖) = τ‖v‖2

2 =γ‖v‖2

2.

The proof of Theorem 5.4.4 is trivially obtained by replacing C, D and D1

in (5.8) and (5.9), respectively, and using Theorem 5.4.1 and the results of(Kaminer et al., 1993), (Chen and Scherer, 2006b).

Theorem 5.4.4 establishes that the LMI-based solution for solving Pro-blem 5.3.1 proposed in this chapter guarantees an H∞ gain equal to thesquare root of the gain τ = γ of the σ(·) function for the system in closed-loop with the ACL. It also shows that the optimal H∞ control law obtainedby minimizing γ = τ subject to (5.8) (for a particular performance outputrelated to the MPC cost) solves the terminal cost and ACL problem in min-max robust MPC. These results establish an intimate connection betweenH∞ design and min-max MPC, in a similar way as LQR design is connectedto nominally stabilizing MPC. This connection is instrumental in improvingthe closed-loop ISS gain of min-max MPC closed-loop systems as follows: anoptimal gain τ = γ of the σ(·) function results in a smaller gain of the functi-on γ(·) of Definition 6.2.2 for the MPC closed-loop system, as demonstratedin the previous chapter.

5.5. Conclusions 87

5.5 Conclusions

In this chapter we proposed a novel LMI-based solution to the terminalcost and auxiliary control law problem in min-max robust MPC. The de-veloped conditions apply to a more general class of systems than previouslyconsidered, i.e. linear systems affected by both parametric and additive dis-turbances. Since LMIs can be solved efficiently, the proposed method iscomputationally attractive. Furthermore, we have established an intimateconnection between the proposed LMIs and the optimal H∞ control law.This result, which was somehow missing in the MPC literature, adds to thewell-known connection between design of nominally stabilizing MPC schemesand the optimal solution of the LQR problem. Such results are of generalinterest as they connect well known linear control problems to MPC design.

6

Self-optimizing robust nonlinear MPC

6.1 Introduction6.2 Preliminary definitions

and results6.3 Problem definition

6.4 Main results6.5 Illustrative examples6.6 Conclusions

This chapter presents a novel method for designing robust MPC schemesthat are self-optimizing in terms of disturbance attenuation. The methodemploys convex control Lyapunov functions and disturbance bounds to op-timize robustness of the closed-loop system on-line, at each sampling instant- a unique feature in MPC. Moreover, the proposed MPC algorithm is com-putationally efficient for nonlinear systems that are affine in the control inputand it allows for a decentralized implementation.

6.1 Introduction

Robustness of nonlinear model predictive controllers has been one of the mostrelevant and challenging problems within MPC, see, e.g., (Mayne et al., 2000;Lazar et al., 2007a; Magni and Scattolini, 2007; Mayne and Kerrigan, 2007;Raković, 2008). From a conceptual point of view, three main categories ofrobust nonlinear MPC schemes can be identified, each with its pros andcons: inherently robust, tightened constraints and min-max MPC schemes,respectively. In all these approaches, the input-to-state stability property(Sontag, 1989) has been employed as a theoretical tool for characterizingrobustness, or robust stability1.

The goal of the existing design methods for synthesizing control lawsthat achieve ISS (Sontag, 1999; Jiang and Wang, 2001; Kokotović and Ar-cak, 2001) is to a priori guarantee a predetermined closed-loop ISS gain.Consequently, the ISS property, with a predetermined, constant ISS gain, is

1Other characterizations of robustness used in MPC, such as ultimate boundedness orstability of a robustly positively invariant set, can be recovered as a particular case of ISSor shown to be related.

90 Self-optimizing robust nonlinear MPC

in this way enforced for all state space trajectories of the closed-loop systemand at all time instances. As the existing approaches, which are also em-ployed in the design of MPC schemes that achieve ISS, can lead to overlyconservative solutions along particular trajectories, it is of high interest todevelop a control (MPC) design method with the explicit goal of adaptingthe closed-loop ISS gain depending of the evolution of the state trajectory.

In this chapter we present a novel method for synthesizing robust MPCschemes with this feature. The method employs convex control Lyapunovfunctions (CLFs) and disturbance bounds to embed the standard ISS condi-tions of (Jiang and Wang, 2001) using a finite number of inequalities. Thisleads to a finite dimensional optimization problem that has to be solvedon-line, in a receding horizon fashion. The proposed inequalities govern theevolution of the closed-loop state trajectory through the sublevel sets of theCLF. The unique feature of the proposed robust MPC scheme is to allowfor the simultaneous on-line (i) computation of a control action that achie-ves ISS and (ii) minimization of the closed-loop ISS gain depending of anactual state trajectory. As a result, the developed nonlinear MPC schemeis self-optimizing in terms of disturbance attenuation. From the computati-onal point of view, following a particular design recipe, the self-optimizingrobust MPC algorithm can be implemented as a single linear program fordiscrete-time nonlinear systems that are affine in the control variable andthe disturbance input. Furthermore, we demonstrate that the freedom tooptimize the closed-loop ISS gain on-line makes self-optimizing robust MPCsuitable for decentralized control of networks of nonlinear systems.

6.2 Preliminary definitions and results

Let R, R+, Z and Z+ denote the field of real numbers, the set of non-negative reals, the set of integer numbers and the set of non-negative integers,respectively. We use the notation Z≥c1 and Z(c1,c2] to denote the sets k ∈Z+ | k ≥ c1 and k ∈ Z+ | c1 < k ≤ c2, respectively, for some c1, c2 ∈ Z+.For a set S ⊆ Rn, we denote by int(S) the interior of S. For two arbitrarysets S ⊆ Rn and P ⊆ Rn, let S ∼ P := x ∈ Rn | x + P ⊆ S denotetheir Pontryagin difference. A polyhedron (or a polyhedral set) in Rn is aset obtained as the intersection of a finite number of open and/or closedhalf-spaces. The Hölder p-norm of a vector x ∈ Rn is defined as ‖x‖p :=

(|[x]1|p + . . . + |[x]n|p)1p for p ∈ Z[1,∞) and ‖x‖∞ := maxi=1,...,n |[x]i|, where

[x]i, i = 1, . . . , n, is the i-th component of x and | · | is the absolute value. Fora matrix M ∈ Rm×n, let ‖M‖p := supx 6=0

‖Mx‖p

‖x‖pdenote its corresponding

6.2. Preliminary definitions and results 91

induced matrix norm. Then ‖M‖∞ = max1≤i≤m∑n

j=1 |[M ]ij |, where [M ]ijis the ij-th entry of M . Let z := z(l)l∈Z+ with z(l) ∈ Ro for all l ∈ Z+

denote an arbitrary sequence. Define ‖z‖ := sup‖z(l)‖ | l ∈ Z+, where‖ · ‖ denotes an arbitrary p-norm, and z[k] := z(l)l∈Z[0,k]

. A functionϕ : R+ → R+ belongs to class K if it is continuous, strictly increasing andϕ(0) = 0. A function ϕ : R+ → R+ belongs to class K∞ if ϕ ∈ K andlims→∞ ϕ(s) = ∞. A function β : R+×R+ → R+ belongs to class KL if foreach fixed k ∈ R+, β(·, k) ∈ K and for each fixed s ∈ R+, β(s, ·) is decreasingand limk→∞ β(s, k) = 0.

6.2.1 ISS definitions and results

Consider the discrete-time nonlinear system

x(k + 1) ∈ Φ(x(k), w(k)), k ∈ Z+, (6.1)

where x(k) ∈ Rn is the state and w(k) ∈ Rl is an unknown disturbanceinput at the discrete-time instant k. The mapping Φ : Rn × Rl → Rn is anarbitrary nonlinear set-valued function. We assume that Φ(0, 0) = 0. LetW be a subset of Rl.

Definition 6.2.1 We call a set P ⊆ Rn robustly positively invariant (RPI)for system (6.1) with respect to W if for all x ∈ P it holds that Φ(x,w) ⊆ Pfor all w ∈ W.

Definition 6.2.2 Let X with 0 ∈ int(X) and W be subsets of Rn and Rl,respectively. We call system (6.1) ISS(X, W) if there exist a KL-functionβ(·, ·) and a K-function γ(·) such that, for each x(0) ∈ X and all w =w(l)l∈Z+ with w(l) ∈ W for all l ∈ Z+, it holds that all corresponding statetrajectories of (6.1) satisfy ‖x(k)‖ ≤ β(‖x(0)‖, k) + γ(‖w[k−1]‖), ∀k ∈ Z≥1.We call the function γ(·) an ISS gain of system (6.1).

Theorem 6.2.3 Let W be a subset of Rl and let X ⊆ Rn be a RPI set for(6.1) with respect to W, with 0 ∈ int(X). Furthermore, let α1(s) := asδ,α2(s) := bsδ, α3(s) := csδ for some a, b, c, δ ∈ R>0, σ ∈ K and let V : Rn →R+ be a function such that:

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖), (6.2a)V (x+)− V (x) ≤ −α3(‖x‖) + σ(‖w‖) (6.2b)


for all x ∈ X, w ∈ W and all x+ ∈ Φ(x, w). Then the system (6.1) isISS(X,W) with

β(s, k) := α−11 (2ρkα2(s)), γ(s) := α−1

1

(2σ(s)1− ρ

), ρ := 1− c

b∈ [0, 1). (6.3)

If inequality (6.2b) holds for w = 0, then the 0-input system x(k + 1) ∈Φ(x(k), 0), k ∈ Z+, is asymptotically stable in X.

The proof of Theorem 6.2.3 is similar in nature to the proof given in (Jiangand Wang, 2001; Lazar et al., 2008a) by replacing the difference equationwith the difference inclusion as in (6.1).

6.2.2 Inherent ISS through continuous and convex controlLyapunov functions

Consider the discrete-time constrained nonlinear system

x(k + 1) = φ(x(k), u(k), w(k)) := f(x(k), u(k)) + g(x(k))w(k), k ∈ Z+,(6.4)

where x(k) ∈ X ⊆ Rn is the state, u(k) ∈ U ⊆ Rm is the control actionand w(k) ∈ W ⊂ Rl is an unknown disturbance input at the discrete-timeinstant k. φ : Rn × Rm × Rl → Rn, f : Rn × Rm → Rn and g : Rn → Rn×l

are arbitrary nonlinear functions with φ(0, 0, 0) = 0 and f(0, 0) = 0. Notethat we allow that g(0) 6= 0. We assume that 0 ∈ int(X), 0 ∈ int(U) andW is bounded. We also assume that φ(·, ·, ·) is bounded in X. Next, letα1, α2, α3 ∈ K∞ and let σ ∈ K.

Definition 6.2.4 A function V : Rn → R+ that satisfies (6.2a) for allx ∈ X is called a control Lyapunov function (CLF) for system x(k + 1) =φ(x(k), u(k), 0), k ∈ Z+, if for all x ∈ X, ∃u ∈ U such that V (φ(x, u, 0)) −V (x) ≤ −α3(‖x‖).

Problem 6.2.5 Let a CLF V (·) be given. At time k ∈ Z+ measure thestate x(k) and calculate a control action u(k) that satisfies:

u(k) ∈ U, φ(x(k), u(k), 0) ∈ X, (6.5a)V (φ(x(k), u(k), 0))− V (x(k)) + α3(‖x(k)‖) ≤ 0. (6.5b)

Let π0(x(k)) := u(k) ∈ Rm | (6.5) holds. Let

x(k + 1) ∈ φ0(x(k), π0(x(k))) := f(x(k), u) | u ∈ π0(x(k))

6.3. Problem definition 93

denote the difference inclusion corresponding to the 0-input system (6.4)in “closed-loop” with the set of feasible solutions obtained by solving Pro-blem 6.2.5 at each instant k ∈ Z+.

Theorem 6.2.6 Let α1, α2, α3 ∈ K∞ of the form specified in Theorem 6.2.3and a corresponding CLF V (·) be given. Suppose that Problem 6.2.5 isfeasible for all states x in X. Then: (i) The difference inclusion

x(k + 1) ∈ φ0(x(k), π0(x(k))), k ∈ Z+, (6.6)

is asymptotically stable in X; (ii) Consider a perturbed version of (6.6), i.e.

x(k + 1) ∈ φ0(x(k), π0(x(k))) + g(x(k))w(k), k ∈ Z+ (6.7)

and let X ⊆ X be a RPI set for (6.7) with respect to W. If X is compact,the CLF V (·) is convex and continuous2 on X and ∃M ∈ R>0 such that‖g(x)‖ ≤ M for all x ∈ X, then system (6.7) is ISS(X,W).

Proof: (i) Let x(k) ∈ X for some k ∈ Z+. Then, feasibility of Problem 6.2.5ensures that x(k + 1) ∈ φ0(x(k), π0(x(k))) ⊆ X due to constraint (6.5a).Hence, Problem 6.2.5 remains feasible and thus, X is a PI set for system(6.6). The result then follows directly from Theorem 6.2.3. (ii) By convexityand continuity of V (·) and compactness of X, V (·) is Lipschitz continuous onX (Wayne S.U., 1972). Hence, letting L ∈ R>0 denote a Lipschitz constant ofV (·) in X, one obtains |V (φ(x, u, w))−V (φ(x, u, 0))| = |V (f(x, u)+g(x)w)−V (f(x, u))| ≤ LM‖w‖ for all x ∈ X and all w. From this property, togetherwith inequality (6.5b) we have that inequality (6.2b) holds with σ(s) :=LMs ∈ K. Since X is an RPI set for (6.7) by the hypothesis, ISS(X,W) ofthe difference inclusion (6.7) follows from Theorem 6.2.3. 2

6.3 Problem definition

Theorem 6.2.6 establishes that all feasible solutions of Problem 6.2.5 arestabilizing feedback laws which, under additional assumptions even achieveISS. However, this inherent ISS property of a feedback law calculated bysolving Problem 6.2.5 relies on a fixed, possibly large gain of σ(·), whichdepends on V (·). This gain is explicitly related to the ISS gain of the closed-loop system via (6.3). To optimize disturbance attenuation for the closed-loop system, at each time instant k ∈ Z+ and for a given x(k) ∈ X, it

2Continuity of V (·) alone is sufficient, but it requires a somewhat more complex proof.


would be desirable to simultaneously compute a control action u(k) ∈ Uthat satisfies:

(i) V (φ(x(k), u(k), w(k)))− V (x(k)) + α3(‖x‖)− σ(‖w(k)‖) ≤ 0,

∀w(k) ∈ W (6.8)

and some function σ(s) := η(k)sδ and (ii) minimize η(k) (η(k), δ ∈ R>0,∀k ∈ Z+).

Remark 6.3.1 It is not possible to directly include (6.8) in Problem 6.2.5,as it leads to an infinite dimensional optimization problem. If W is a com-pact polyhedron, a possibility to resolve this issue would be to evaluate theinequality (6.8) only for w(k) taking values in the set of vertices of W. Howe-ver, this does not guarantee that (6.8) holds for all w(k) ∈ W due to the factthat the left-hand term in (6.8) is not necessarily a convex function of w(k),i.e. it contains the difference of two, possibly convex, functions of w(k). Thismakes the considered problem challenging and interesting. 2

6.4 Main results

In what follows we present a solution to the problem stated in Section 6.3.More specifically, we demonstrate that by considering continuous and convexCLFs and compact polyhedral sets X, U, W (that contain the origin in theirinterior) a solution to inequality (6.8) can be obtained via a finite set ofinequalities that only depend on the vertices of W. The standing assumptionthroughout the remainder of the chapter is that the considered system, i.e.(6.4), is affine in the disturbance input w.

6.4.1 Optimized ISS through convex CLFs

Let we, e = 1, ..., E, be the vertices of W. Next, consider a finite set ofsimplices S1, . . . , SM with each simplex Si equal to the convex hull of asubset of the vertices of W and the origin, and such that ∪M

i=1Si = W. Moreprecisely, Si = Co0, wei,1 , . . . , wei,l and wei,1 , . . . , wei,l ⊂ w1, . . . , wE(i.e. ei,1, . . . , ei,l ⊂ 1, . . . , E) with wei,1 , . . . , wei,l linearly independent.For each simplex Si we define the matrix Wi := [wei,1 . . . wei,l ] ∈ Rl×l,which is invertible. Let λe(k), k ∈ Z+, be optimization variables associatedwith each vertex we. Let α3 ∈ K∞, suppose that x(k) at time k ∈ Z+ isgiven and consider the following set of inequalities depending on u(k) and


λ1(k), . . . , λE(k):

V (φ(x(k), u(k), 0))− V (x(k)) + α3(‖x(k)‖) ≤ 0, (6.9a)

V (φ(x(k), u(k), we))− V (x(k)) + α3(‖x(k)‖)− λe(k) ≤ 0, ∀e = 1, E.(6.9b)

Theorem 6.4.1 Let V (·) be a convex CLF. If for α3 ∈ K∞ and x(k) attime k ∈ Z+ there exist u(k) and λe(k), e = 1, . . . , E, such that (6.9a) and(6.9b) hold, then (6.8) holds for the same u(k), with σ(s) := η(k)s and

η(k) := maxi=1,...,M

‖λi(k)W−1i ‖, (6.10)

where λi(k) := [λei,1(k) . . . λei,l(k)] ∈ R1×l.

Proof: Let α3 ∈ K∞ and x(k) be given and suppose (6.9b) holdsfor some λe(k), e = 1, . . . , E. Let w ∈ W =

⋃Mi=1 Si. Hence, there exists

an i such that w ∈ Si = Co0, wei,1 , . . . , wei,l, which means that thereexist non-negative numbers µ0, µ1, . . . , µl with

∑lj=0 µj = 1 such that w =∑l

j=1 µjwei,j + µ00 =

∑lj=1 µjw

ei,j . In matrix notation we have that w =Wi[µ1 . . . µl]> and thus [µ1 . . . µl]> = W−1

i w. Multiplying each inequalityin (6.9b) corresponding to the index ei,j and the inequality (6.9a) with µj ≥0, j = 0, 1, . . . , l, summing up and using

∑lj=0 µj = 1 yield:

µ0V (φ(x(k), u(k), 0)) +l∑

j=1

µjV (φ(x(k), u(k), wei,j ))

− V (x(k)) + α3(‖x(k)‖)−l∑

j=1

µjλei,j(k) ≤ 0.

Furthermore, using φ(x(k), u(k), wei,j ) = f(x(k), u(k)) + g(x(k))wei,j , con-vexity of V (·) and

∑lj=0 µj = 1 yields

V (φ(x(k), u(k),l∑

j=1

µjwei,j ))− V (x(k)) + α3(‖x(k)‖)−

l∑j=1

µjλei,j (k) ≤ 0,

or equivalently

V (φ(x(k), u(k), w))− V (x(k)) + α3(‖x(k)‖)− λi(k)[µ1 . . . µl]> ≤ 0.

Using that [µ1 . . . µl]> = W−1i w we obtain (6.8) with w(k) = w for σ(s) =

η(k)s and η(k) ≥ 0 as in (6.10).


6.4.2 Self-optimizing robust nonlinear MPC

For any x ∈ X let Wx := g(x)w | w ∈ W ⊂ Rn (note that 0 ∈ Wx) andassume that X ∼ Wx 6= ∅. Let λ := [λ1, . . . , λE ]> and let J(λ) : RE → R+

be a function that satisfies α4(‖λ‖) ≤ J(λ) ≤ α5(‖λ‖) for some α4, α5 ∈ K∞;for example, J(λ) := maxi=1,...,M ‖λiW

−1i ‖.

Problem 6.4.2 Let α3 ∈ K∞, J(·) and a CLF V (·) be given. At timek ∈ Z+ measure the state x(k) and minimize the cost J(λ1(k), . . . , λE(k))over u(k), λ1(k), . . . , λE(k), subject to the constraints

u(k) ∈ U, λe(k) ≥ 0, f(x(k), u(k)) ∈ X ∼ Wx(k), (6.11a)

V (φ(x(k), u(k), 0))− V (x(k)) + α3(‖x(k)‖) ≤ 0, (6.11b)

V (φ(x(k), u(k), we))− V (x(k)) + α3(‖x(k)‖)− λe(k) ≤ 0, ∀e = 1, E.(6.11c)

2

Let π(x(k)) := u(k) ∈ Rm | (6.11) holds and let

x(k + 1) ∈ φcl(x(k), π(x(k)), w(k)) := φ(x(k), u, w(k)) | u ∈ π(x(k))

denote the difference inclusion corresponding to system (6.4) in “closed-loop”with the set of feasible solutions obtained by solving Problem 6.4.2 at eachk ∈ Z+.

Theorem 6.4.3 Let α1, α2, α3 ∈ K∞ of the form specified in Theorem 6.2.3,a continuous and convex CLF V (·) and a cost J(·) be given. Suppose thatProblem 6.4.2 is feasible for all states x in X. Then the difference inclusion

x(k + 1) ∈ φcl(x(k), π(x(k)), w(k)), k ∈ Z+ (6.12)

is ISS(X, W).

Proof: Let x(k) ∈ X for some k ∈ Z+. Then, feasibility of Problem 6.4.2ensures that x(k+1) ∈ φcl(x(k), π(x(k)), w(k)) ⊆ X for all w(k) ∈ W, due tog(x(k))w(k) ∈ Wx(k) and constraint (6.11a). Hence, Problem 6.4.2 remainsfeasible and thus, X is a RPI set with respect to W for system (6.12). FromTheorem 6.4.1 we also have that V (·) satisfies (6.2b) with σ(s) := η(k)s andη(k) as in (6.10). Let

λ∗ := supx∈X,u∈U,e=1,...,E

V (φ(x, u, we))− V (x) + α3(‖x‖).


As V (·) is upper and lower bounded by K∞ functions, due to compactness ofX, U and boundedness of φ(·, ·, ·), λ∗ exists and is finite (the sup above is amax if φ(·, ·, ·) is continuous in x and u). Hence, inequality (6.11c) is alwayssatisfied for λe(k) = λ∗ for all e = 1, . . . , E, k ∈ Z+, and for all x ∈ X,u ∈ U. This in turn, via (6.10) ensures the existence of a η∗ ∈ R>0 such thatη(k) ≤ η∗ for all k ∈ Z+. Hence, we proved that inequality (6.8) holds forall x ∈ X and all w ∈ W. Then, since X is RPI, ISS(X,W) follows directlyfrom Theorem 6.2.3.

Remark 6.4.4 An alternative proof to Theorem 6.4.3 can be obtained bysimply applying the reasoning used in the proof of Theorem 6.2.6. Hence,inherent ISS can be established directly from constraint (6.11b). Also, no-tice that in the proof of Theorem 6.4.3 we used a worst case evaluation ofλe(k) to prove ISS. However, it is important to observe that compared toProblem 6.2.5, nothing is lost in terms of feasibility, while Problem 6.4.2,although it inherently guarantees a constant ISS gain, it provides freedom tooptimize the ISS gain of the closed-loop system, by minimizing the variablesλ1(k), . . . , λE(k) via the cost J(·). As such, in reality the gain η(k) of thefunction σ(·) can be much smaller for k ≥ k0, for some k0 ∈ Z+, dependingon the state trajectory x(k). 2

In Theorem 6.4.3 we assumed for simplicity that Problem 6.4.2 is feasiblefor all x ∈ X; in other words, feasibility implies ISS. Whenever Problem 6.4.2can be solved explicitly (see the implementation paragraph below), it is pos-sible to calculate the maximal RPI set for the closed-loop dynamics that iscontained within the explicit set of feasible solutions. Alternatively, we esta-blish next an easily verifiable sufficient condition under which any sublevelset of V (·) contained in X is a RPI subset of the set of feasible solutions ofProblem 6.4.2.

Lemma 6.4.5 Given a CLF V (·) that satisfies the hypothesis of Theo-rem 6.4.3, let V∆ := x ∈ Rn | V (x) ≤ ∆. Then, for any ∆ ∈ R>0

such that V∆ ⊆ X, if λ∗ ≤ (1−ρ)∆, with ρ as defined in (6.3), Problem 6.4.2is feasible for all x ∈ V∆ and remains feasible for all resulting closed-looptrajectories that start in V∆.

Proof: From the proof of Theorem 6.4.3 we know that inequalities(6.11c) are feasible for all x(k) ∈ X, u(k) ∈ U and e = 1, E by takingλ(k) = λ∗ for all k ∈ Z+. Thus, for any x(k) ∈ V∆ ⊆ X, ∆ ∈ R≥0, we have


that:

V (φ(x(k), u(k), w(k))) ≤ V (x(k))− α3(‖x(k)‖) + λ∗ ≤ ρV (x(k)) + λ∗

≤ ρ∆ + λ∗ ≤ ρ∆ + (1− ρ)∆ = ∆,

which yields φ(x(k), u(k), w(k)) ∈ V∆ ⊆ X. This in turn ensures feasibilityof (6.11a), while (6.11b) is feasible by definition of the CLF V (·), whichconcludes the proof.

Remark 6.4.6 The result of Theorem 6.4.3 holds for all inputs u(k) forwhich Problem 6.4.2 is feasible. To select on-line one particular controlinput from the set π(x(k)) and to improve closed-loop performance (in termsof settling time) it is useful to also penalize the state and the input. LetF : Rn → R+ and L : Rn × Rm → R+ with F (0) = L(0, 0) = 0 be arbitrarynonlinear functions. For N ∈ Z≥1 let u(k) := (u(k), u(k + 1), . . . , u(k +N− 1)) ∈ UN and JRHC(x(k), u(k)) := F (x(k+N))+

∑N−1i=0 L(x(k+i), u(k+

i)), where x(k + i + 1) := f(x(k + i), u(k + i)) for i = 0, N − 1 and x(k) :=x(k). Then one can add this cost to Problem 6.4.2, i.e. at time k ∈ Z+

measure the state x(k) and minimize JRHC(x(k), u(k))+J(λ1(k), . . . , λE(k))over u(k), λ1(k), . . . , λE(k), subject to constraints (6.11) and x(k + i) ∈ X,i = 2, N . Observe that the optimum needs not to be attained at eachsampling instant to achieve ISS, which is appealing for practical reasons butalso in the case of a possibly discontinuous value function. 2

Remark 6.4.7 Besides enhancing robustness, constraints (6.11b)-(6.11c)also ensure that Problem 6.4.2 recovers performance (in terms of settlingtime) when the state of the closed-loop system approaches the origin. Loo-sely speaking, when x(k) ≈ 0, solving Problem 6.4.2 will produce a con-trol action u(k) ≈ 0 (because of constraint (6.11b) and the fact that thecost JRHC(·) + J(·) is minimized). This yields V (φ(0, 0, we)) − λe(k) ≤ 0,e = 1, E, due to constraint (6.11c). Thus, solving Problem 6.4.2 with theabove cost will not optimize each variable λe(k) below the correspondingvalue V (φ(0, 0, we)), e = 1, E, when the state reaches the equilibrium. Thisproperty is desirable, since it is known from min-max MPC (Lazar et al.,2008a) that considering a worst case disturbance scenario leads to poor per-formance when the real disturbance is small or vanishes. 2

6.4.3 Decentralized formulation

In this paragraph we give a brief outline of how the proposed self-optimizingMPC algorithm can be implemented in a decentralized fashion. We consider


a connected directed graph G = (S, E) with a finite number of vertices S anda set of directed edges E ⊆ (i, j) ∈ S × S | i 6= j. A dynamical system isassigned to each vertex i ∈ S, with the dynamics governed by the followingequation:

xi(k + 1) = φi(xi(k), ui(k), vi(xNi(k)), wi(k)), k ∈ Z+. (6.13)

In (6.13), xi ∈ Xi ⊂ Rni , ui ∈ Ui ⊂ Rmi are the state and the control inputof the i-th system, and wi ∈ Wi ⊂ Rli is an exogenous disturbance input thatdirectly affects only the i-th system. With each directed edge (j, i) ∈ E weassociate a function vij : Rnj → Rni , which defines the interconnection signalvij(xj(k)), k ∈ Z+, between system j and system i, i.e. vij(·) characterizeshow the states of system j influence the dynamics of system i. The setNi := j | (j, i) ∈ E denotes the set of direct neighbors (observe that j ∈Ni 6⇒ i ∈ Nj) of the system i. For simplicity of notation we use xNi(k) andvi(xNi(k)) to denote xj(k)j∈Ni and vij(xj(k))j∈Ni , respectively. Bothφi(·, ·, ·, ·) and vij(·) are arbitrary nonlinear, possibly discontinuous functionsthat satisfy φi(0, 0, 0, 0) = 0, vij(0) = 0 for all (i, j) ∈ S ×Ni. For all i ∈ Swe assume that Xi, Ui and Wi are compact sets that contain the origin intheir interior.

Assumption 6.4.8 The value of all interconnection signals vij(xj(k)) isknown at all discrete-time instants k ∈ Z+ for any system i ∈ S.

From a technical point of view, Assumption 6.4.8 is satisfied, e.g., if allinterconnection signals vij(xj(k)) are directly measurable at all k ∈ Z+ or,if all directly neighboring systems j ∈ Ni are able to communicate theirlocal measured state xj(k) to system i ∈ S. Consider next the followingdecentralized version of Problem 6.4.2, where the notation and definitionsemployed so far are carried over mutatis mutandis.

Problem 6.4.9 For system i ∈ S let αi3 ∈ K∞, Ji(·) and a CLF Vi(·) be

given. At time k ∈ Z+ measure the local state xi(k) and the intercon-nection signals vi(xNi(k)) and minimize the cost Ji(λi

1(k), . . . , λiEi

(k)) overui(k), λi

1(k), . . . , λiEi

(k), subject to the constraints

ui(k) ∈ U, λie(k) ≥ 0, φi(xi(k), ui(k), vi(xNi(k)), 0) ∈ Xi ∼ Wxi(k),

(6.14a)

Vi(φi(xi(k), ui(k), vi(xNi(k)), 0))− Vi(xi(k)) + αi3(‖xi(k)‖) ≤ 0, (6.14b)

Vi(φi(xi(k), ui(k), vi(xNi(k)), wei ))− Vi(xi(k)) + αi

3(‖xi(k)‖)− λie(k) ≤ 0,

∀e = 1, Ei. (6.14c)


2

Let πi(xi(k), vi(xNi(k))) := ui(k) ∈ Rmi | (6.14) holds and let

xi(k + 1) ∈φcli (xi(k), πi(xi(k), vi(xNi(k)), vi(xNi(k)), wi(k)):= φi(xi(k), u, vi(xNi(k)), wi(k)) | u ∈ πi(xi(k), vi(xNi(k)))

denote the difference inclusion corresponding to system (6.13) in “closed-loop” with the set of feasible solutions obtained by solving Problem 6.4.9 ateach k ∈ Z+.

Theorem 6.4.10 Let, αi1, α

i2, α

i3 ∈ K∞ of the form specified in Theorem 6.2.3,

continuous and convex CLFs Vi(·) and costs Ji(·) be given for all systemsindexed by i ∈ S. Suppose Assumption 6.4.8 holds and Problem 6.4.9 isfeasible for each system i ∈ S and for all states xi in Xi and all corres-ponding vi(xNi). Then the interconnected dynamically coupled nonlinearsystem described by the collection of difference inclusions

xi(k + 1) ∈ φcli (xi(k), πi(xi(k), vi(xNi(k)), vi(xNi(k)), wi(k)), i ∈ S, k ∈ Z+

(6.15)is ISS(X1 × . . .× XS , W1 × . . .×WS).

The proof of the above theorem is obtained by a straightforward applicati-on of the centralized result presented in this chapter and properties of K∞functions. Its central argument is that each continuous and convex CLFVi(xi) is in fact Lipschitz continuous on Xi (Wayne S.U., 1972), which ma-kes

∑i∈S Vi(xi) =: V (xii∈S) a Lipschitz continuous CLF for the global

interconnected system. The result then follows similarly to the proof ofTheorem 6.2.6-(ii). Theorem 6.4.10 guarantees a constant ISS gain for theglobal closed-loop system, while the ISS gain of each closed-loop system i ∈ Scan still be optimized on-line.

Remark 6.4.11 Problem 6.4.9 defines a set of decoupled optimization pro-blems, implying that the computation of control actions can be performedin completely decentralized fashion, i.e. with no communication among con-trollers (if each vij(·) is measurable at all k ∈ Z+). Inequality (6.14b) canbe further significantly relaxed by replacing the zero on the righthand si-de with an optimization variable τi(k) and adding the coupling constraint∑

i∈S τi(k) ≤ 0 for all k ∈ Z+. Using the dual decomposition method, seee.g. (Bertsekas, 1999), it is then possible to devise a distributed control sche-me, which yields an optimized ISS-gain of the global interconnected system


in the sense that∑

i∈S Ji(·) is minimized. Further relaxations can be ob-tained by asking that the sum of τi(k) is non-positive over a finite horizon,rather than at each time step. 2

6.4.4 Implementation issues

In this section we briefly discuss the ingredients, which make it possible toimplement Problem 6.4.2 (or its corresponding decentralized version Pro-blem 6.4.9) as a single linear or quadratic program. Firstly, we considernonlinear systems of the form (6.4) that are affine in control. Then it ma-kes sense that there exist functions f1 : Rn → Rn with f1(0) = 0 andf2 : Rn → Rn×m such that:

x(k + 1) = φ(x(k), u(k), w(k)) := f1(x(k)) + f2(x(k))u(k) + g(x(k))w(k).(6.16)

Secondly, we restrict our attention to CLFs defined using the ∞-norm, i.e.V (x) := ‖Px‖∞, where P ∈ Rp×n is a matrix (to be determined) with full-column rank. We refer to (Lazar et al., 2006) for techniques to computeCLFs based on norms.

Then, the first step is to show that the ISS inequalities (6.11b)-(6.11c) canbe specified, without introducing conservatism, via a finite number of linearinequalities. Since by definition ‖x‖∞ = maxi∈Z[1,n]

|[x]i|, for a constraint‖x‖∞ ≤ c with c > 0 to be satisfied, it is necessary and sufficient to requirethat ±[x]i ≤ c for all i ∈ Z[1,n]. Therefore, as x(k) in (6.11) is the measuredstate, which is known at every k ∈ Z+, for (6.11b)-(6.11c) to be satisfied itis necessary and sufficient to require that:

± [P (f1(x(k)) + f2(x(k))u(k))]i − V (x(k)) + α3(‖x(k)‖) ≤ 0± [P (f1(x(k)) + f2(x(k))u(k) + g(x(k))we)]i − V (x(k)) + α3(‖x(k)‖)− λe(k) ≤ 0,

∀i ∈ Z[1,p], e = 1, E,

which yields 2p(E+1) linear inequalities in u(k), λ1(k), . . . , λE(k). If the setsX, U and Wx(k) are polyhedra, which is a reasonable assumption, then clearlythe inequalities in (6.11a) are also linear in u(k), λ1(k), . . . , λE(k). Thus,a solution to Problem 6.4.2, including minimization of the cost JRHC(·) +J(·) for any N ∈ Z≥1, can be obtained by solving a nonlinear optimizationproblem subject to linear constraints.

Following some straightforward manipulations, the optimization problemto be solved on-line can be further simplified as follows. If the model is(i) piecewise affine or (ii) affine and the cost functions JRHC(·) and J(·)


are defined using quadratic forms or infinity norms, then a solution to Pro-blem 6.4.2 (with the cost JRHC(·) + J(·)) can be obtained by solving (i) asingle mixed integer quadratic or linear program (MIQP - MILP), or (ii) asingle QP - LP, respectively, for any N ∈ Z≥1. Alternatively, for N = 1and quadratic or ∞-norm based costs, Problem 6.4.2 can be formulated asa single QP or LP for any discrete-time nonlinear model that is affine in thecontrol variable and the disturbance input.

6.5 Illustrative examples

6.5.1 Example 1: control of a nonlinear system

Consider the nonlinear system (6.16) where x(k) ∈ X = ξ ∈ R2 | ‖ξ‖∞ ≤5, u(k) ∈ U = ξ ∈ R | |ξ| ≤ 1 and w(k) ∈ W = ξ ∈ R2 | ‖ξ‖1 ≤ 0.2,k ∈ Z+. The dynamics are given by:

f1(x) =(

[x]1 + 0.7[x]2 + ([x]2)2

[x]2

),

f2(x) =(

0.245 + sin([x]2)0.7

), g(x) =

(1 00 1

).

The technique of (Lazar et al., 2006) was used to compute the weight P ∈R2×2 of the CLF V (x) = ‖Px‖∞ for α3(s) := 0.01s and the linearizationof (6.16) around the origin, in closed-loop with u(k) := Kx(k), K ∈ R2×1,yielding

P =[2.7429 0.71210.1989 4.0173

], K =

[−0.4379 −1.5508

].

To optimize robustness, 4 optimization variables λ1(k), . . . , λ4(k) were intro-duced, each one assigned to a vertex of the set W. The RHC cost was chosenas JRHC(x(k), u(k), λi(k)) = ‖Q1(f1(x(k))+f2(x(k))u(k))‖∞+‖Qx(k)‖∞+‖Ru(k)‖∞ +

∑4i=1 ‖λi(k)‖∞, where Q1 = 4I2, Q = 0.1I2 and R = 0.4. The

resulting linear program has 11 optimization variables and 42 constraints.During the simulations, the worst case computational time required by theCPU over 4000 runs was 0.02 seconds, which shows the potential for con-trolling fast nonlinear systems.

In the simulation scenario we tested the closed-loop system response forx(0) = [3, −1]> and for the following disturbance scenarios: w(k) = [0, 0]>

for k ∈ Z[0,40] (nominal stabilization), w(k) takes random values in W fork ∈ Z[41,80] (robustness to random inputs), w(k) = [0, 0.1]> for k ∈ Z[81,120]

(robustness to constant inputs) and w(k) = [0, 0]> for k ∈ Z[121,160] (toshow that asymptotic stability is recovered for zero inputs).

6.5. Illustrative examples 103

Figure 6.1: Evolution of the closed-loop system state (top figure: red andblue lines) and of the control input (bottom figure: blue line).

In Figure 6.4 the time history of the states and control input is depicted.The dashed horizontal lines give an approximation of the bounded region inwhich the system’s states remain despite disturbances, i.e. approximatelywithin the interval [−0.2, 0.2]. The dashed vertical lines delimit the timeintervals during which one of the four disturbance scenarios is active. Onecan observe that the feedback to disturbances is provided actively, resultingin good robust performance, while state and input constraints are satisfiedat all times. In Figure 6.7 the time history of the optimization variablesλ1(k), . . . , λ4(k) is presented. One can see that whenever the disturbanceis acting on the system, or when the state is far from the origin (in thefirst disturbance scenario), these variables act so as to optimize the decrea-se of V (·). Whenever the equilibrium is reached, the optimization varia-bles satisfy the constraint V (φ(0, 0, we)) ≤ λe(k), e = 1, . . . , 4, as explainedin Remark 6.4.7. In Figure 6.7 the values of V (φ(0, 0, we) for each vertex(0.5486 and 0.8432 for w1 = [0.2, 0]>, w3 = [−0.2, 0]> and w2 = [0, −0.2]>,w4 = [0, 0.2]>, respectively) are depicted with dashed horizontal lines.

6.5.2 Example 2: control of a DC-DC converter

In this section we illustrate the MPC scheme developed in this chapter byapplying it to control a Buck-Boost DC-DC converter power circuit. To


Figure 6.2: Evolution of the optimization variables λ1(k), . . . , λ4(k).

+ –

+

vo

–C ic

iL

L

ON/OFFvin

R

Figure 6.3: A schematic view of a Buck-Boost converter.

assess the enhancement of disturbance rejection, a comparison will be madewith the inherently robust MPC scheme obtained by removing the additionaloptimization variables denote with λ.

DC-DC converters are extensively used in power supplies for electronicequipment to control the energy flow between two DC systems. Buck-BoostDC-DC converters are currently used in a wide variety of relevant proces-ses, including electric and hybrid vehicles, solar plants, DC motor drives,switched-mode DC power supplies, and many more. In Figure 6.3 a schema-tic representation of an ideal Buck-Boost circuit (i.e. neglecting the parasitecomponents) is drawn.

The following discrete-time nonlinear averaged model of the converter,which was developed in (Lazar and De Keyser, 2004) by applying the theoryof (Kassakian et al., 1992), is used to obtain a prediction model:

xm(k+1) =[

[xm(k)]1 + TL [xm(k)]2 − T

L ([xm(k)]2 − Vin)um(k)−T

C [xm(k)]1 + TC [xm(k)]1um(k) + (1− T

RC )[xm(k)]2

], k ∈ Z+,

(6.17)


where xm(k) ∈ R2 and um(k) ∈ R are the state and the input, respectively.[xm]1 represents the current flowing through the inductor (iL), [xm]2 theoutput voltage (vo) and um represents the duty cycle (i.e. the fraction ofthe sampling period during which the transistor is kept ON). The samplingperiod is T = 0.65 milliseconds. The parameters of the circuit are theinductance L = 4.2mH, the capacitance C = 2200µF, the load resistanceR = 165Ω and the source input voltage vin, with nominal value Vin = 15V.

The control objective is twofold: at start-up, a desired value of the outputvoltage, i.e. xss

2 , should be reached as fast as possible and with minimumovershoot; after the output voltage reaches the desired value, it must keptclose to the operating point, i.e. within a range of ±3% around xss

2 (theindustrial operating margin for DC-DC converters) despite changes in theload R (within a 50% range around the nominal value) and disturbances.

Note that for a desired output voltage value xss2 one can obtain the steady

state duty cycle and inductor current as follows:

uss =xss

2

xss2 − Vin

, xss1 =

xss2

R(uss − 1). (6.18)

Furthermore, the following physical constraints must be fulfilled at all timesk ∈ Z+:

[xm(k)]1 ∈ [0.01, 5], [xm(k)]2 ∈ [−20, 0], um(k) ∈ [0.1, 0.9]. (6.19)

To implement the developed MPC scheme, we first perform the followingcoordinate transformation on (6.17):

[x(k)]1 = [xm(k)]1 − xss1 , [x(k)]2 = [xm(k)]2 − xss

2 , u(k) = um(k)− uss.(6.20)

We obtain the following system description

x(k + 1) =[

[x(k)]1 + α[x(k)]2 + (β − TL [x(k)]2)u(k)

(TC [x(k)]1 + γ)u(k) + (1− T

RC )[x(k)]2 + δ[x(k)]1

], (6.21)

where the constants α, β, γ and δ depend on the fixed steady state value xss2

as follows

α =T

L(1− xss

2

xss2 − Vin

), β =T

L(Vin − xss

2 ), γ =T

RCVinxss

2 (xss2 − Vin),

δ =T

C

(xss

2

xss2 − Vin

− 1)

.


Using (6.20) and (6.18), the constraints given in (6.19) can be converted to:

[x(k)]1 ∈ [bx1 , bx1 ], [x(k)]2 ∈ [bx2 , b

x2 ], u(k) ∈ [bu, bu], (6.22)

where

bx1 = 0.01− 1RVin

xss2 (xss

2 − Vin), bx2 = −20− xss2 , bu = 0.1− xss

2

xss2 − Vin

,

bx1 = 5− 1

RVinxss

2 (xss2 − Vin), b

x2 = −xss2 , b

u = 0.9− xss2

xss2 − Vin

.

The control objective can now be formulated as to robustly stabilize (6.21)around the equilibrium [0 0]> while fulfilling the constraints given in (6.22).

Next, to compute an ∞-norm based control Lyapunov function, we line-arize system (6.21) around the equilibrium [0 0]> (for zero input uk = 0 ∈[bu, b

u]). The linearized equations are:

∆x(k + 1) = A∆x(k) + B∆u(k), (6.23)

where ∆x(k) and ∆u(k) represent “small” deviations from the equilibrium[0 0]> and zero input uk = 0, respectively. The matrices A and B are givenby

A ,∂f

∂x

∣∣x=0,u=0

=[1 α

δ 1− TRC

], B ,

∂f

∂u

∣∣x=0,u=0

=[βγ

].

For the linear model corresponding to a steady state output voltage xss2 =

−4V (which yields uss = 0.2105 and xss1 = 0.0307A), by applying the method

of (Lazar et al., 2006) to find the matrix P and the feedback gain K satisfyingthe CLF condition for α3(s) = 0.001s, we have obtained the solution P =[

0.9197 −0.6895−0.5815 1.8109

]and K = [−0.4648 0.4125 ]. The MPC cost matrices have been

chosen as follows, to ensure a good performance: Q1 = [ 1 00 4 ], Q = [ 1 0

0 2 ] andR = 0.1 (notice that this is different from the load resistance, also denotedby R).

To test robustness, during the simulation we perturb the system with anadditive disturbance on the inductor current and we perform a load change.The disturbance is generated in the set W := w ∈ R2 | w = [w1 0]>, −0.1 ≤w1 ≤ 0. Therefore, to implement the self-optimizing robust MPC schemeit is sufficient to associate a single feedback optimization variable λ(k), cor-responding to the vertex w = [−0.1 0]>. The corresponding weight matrixfor λ(k) was taken equal to one.

To assess the real-time applicability of the developed theory for this typeof a very fast system with a sampling period well below one millisecond, we


0 0.01 0.02 0.03 0.04

−4

−2

0

Out

put v

olta

ge [V

]

Feedback ISS MPC scheme

0 0.01 0.02 0.03 0.04

−4

−2

0Inherent ISS MPC scheme

0 0.01 0.02 0.03 0.040

0.5

1

1.5

Indu

ctor

cur

rent

[A]

0 0.01 0.02 0.03 0.040

0.5

1

1.5

0 0.01 0.02 0.03 0.040.05

0.1

0.15

0.2

Dut

y cy

cle

Time (seconds)0 0.01 0.02 0.03 0.04

0.05

0.1

0.15

0.2

Time (seconds)

Figure 6.4: Start-up: State trajectories and MPC input histories - solidlines, desired steady state values and constraints - dotted lines.

we formulated the MPC optimization problems as Linear Programming (LP)problems. The LP problem corresponding to the inherently robust MPCscheme has 3 optimization variables and 14 constraints, while the LP problemcorresponding to the self-optimizing robust MPC scheme has 5 optimizationvariables and 20 constraints. Here we excluded the lower and upper boundson optimization variables, which are given directly as arguments of the LPsolver.

In one simulation, we tested first the start-up behavior (see Figure 6.4)and then, after reaching the desired operating point, we tested the distur-bance rejection (see Figure 6.7).

Note that, although the simulations were performed for the transformedsystem (6.21), we chose to plot all variables in the original coordinates cor-responding to system (6.17), which have more physical meaning.

During start-up, when no disturbance acts on the system and the valueof the load remains unchanged, the differences between the self-optimizingrobust MPC scheme and the inherently robust MPC scheme are very small,


0.04 0.06 0.08 0.1 0.12

−4.1

−4

−3.9

Ou

tpu

t vo

lta

ge

[V

]

Feedback ISS MPC scheme

0.04 0.06 0.08 0.1 0.12−4.2

−4

−3.8

−3.6

−3.4

−3.2

Inherent ISS MPC scheme

0.04 0.06 0.08 0.1 0.120

0.1

0.2

Ind

ucto

r cu

rre

nt [A

]

0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.04 0.06 0.08 0.1 0.12

0.2

0.25

0.3

Time (seconds)

Du

ty c

ycle

0.04 0.06 0.08 0.1 0.120.1

0.2

0.3

0.4

Time (seconds)

Figure 6.5: Disturbance rejection: State trajectories MPC input histories- solid lines, desired steady state values, constraints and industrial operatingmargins for DC-DC converters (±3% of the desired output voltage) - dottedlines.

as expected. Both schemes provide a very good start-up response.However, the difference in performance is significant in the second part

of the simulation, when the dynamics were simultaneously affected by anasymptotically decreasing (in norm) additive disturbance of the form w =[w1 0]> (see Figure 6.6 for a plot of w1 versus time) and a 50% drop of theload (i.e. R=82.5Ω) for k = 80, 81, . . . , 120. For k > 120 the disturbancewas set equal to zero and the load was set to its nominal value (i.e. R=165Ω)to show that the closed-loop system is ISS, i.e. that asymptotic stability isrecovered when the disturbance input vanishes. While the inherently robustMPC scheme does not manage to keep the output voltage within the desi-


0 0.02 0.04 0.06 0.08 0.1 0.12−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02Additive disturbance acting on the inductor current

0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.05

0.1

Optimization variable providing feedback to the disturbance

Time (seconds)

Figure 6.6: Time history of w1 and λ(k) - solid lines.

red operating range, the self-optimizing robust MPC scheme achieves verygood performance in spite of significant additive and parametric disturban-ces (changes in the load R). The time history of λ(k) is shown in Figure 6.6.One can observe in Figure 6.6 that when the state reaches the desired opera-ting point, λ(k) satisfies λ(k) ≥ ‖PV [−0.1 0]>‖ = 0.091, which means thatthe enhanced robustness is automatically deactivated when no the systemremain at the origin.

The LP problems equivalent to tMPC optimization problems were al-ways solved3 within the allowed sampling interval, with an worst case CPUtime over 20 runs of 0.6314 milliseconds. In total, 4000 LPs were alwayssolved within the allowed sampling interval for both algorithms. The verygood closed-loop performance obtained for N = 1 collaborated with thecomputational time estimate is encouraging for further development of thereal-time application of the presented theory to control DC-DC power con-verters, especially using faster platforms, such as Digital Signal Processors(DSP).

3The simulation platform was Matlab 7.0.4 (R14) (CDD Dual Simplex LP solver)running on a Linux Fedora Core 5 operating system powered by an Intel Pentium 4 witha 3.2 GHz CPU.


6.5.3 Example 3: control of networked nonlinear systems

Consider the nonlinear system (6.13) with S = 1, 2, N1 = 2, N2 = 1,X1 = X2 = ξ ∈ R2 | ‖ξ‖∞ ≤ 5, U1 = U2 = ξ ∈ R | |ξ| ≤ 2 andW1 = W2 = ξ ∈ R2 | ‖ξ‖1 ≤ 0.2. The dynamics are given by:

φ1(x1, u1, v1(xN1), w1) :=[1 0.70 1

]x1 +

[sin([x1]2)

0

]+[0.2450.7

]u1 +

[0

([x2]1)2

]+ w1,

(6.24a)

φ2(x2, u2, v2(xN2), w2) :=[1 0.50 1

]x2 +

[sin([x2]2)

0

]+[0.1250.5

]u2 +

[0

[x1]1

]+ w2.

(6.24b)

The technique of (Lazar et al., 2006) was used to compute the weightsP1, P2 ∈ R2×2 of the CLFs V1(x) = ‖P1x‖∞ and V2(x) = ‖P2x‖∞ forα1

3(s) = α23(s) := 0.01s and the linearizations of (6.24a), (6.24b), respec-

tively, around the origin, in closed-loop with u1(k) := K1x1(k), u2(k) :=K2x2(k), K1,K2 ∈ R2×1, yielding

P1 =[1.3204 0.62940.5629 2.0811

], K1 =

[−0.2071 −1.2731

],

P2 =[1.1356 0.56580.7675 2.1356

], K2 =

[−0.3077 −1.4701

].

Note that the control laws u1(k) = K1x(k) and u2(k) = K2x2(k) areonly employed off-line, to calculate the weight matrices P1, P2 and theyare never used for controlling the system. To optimize robustness, 4 op-timization variables λi

1(k), . . . , λi4(k) were introduced for each system, each

one assigned to a vertex of the set Wi, i = 1, 2, respectively. The follo-wing cost functions were employed in the optimization problem, as speci-fied in Remark 6.4.6: J i

RHC(xi(k), ui(k)) := ‖Qi1φi(xi, ui, vi(xNi), 0)‖∞ +

‖Qixi(k)‖∞ + ‖Riui(k)‖∞, Ji(λi1(k), . . . , λi

4(k)) := Γi∑4

j=1 |λij(k)|, where

i = 1, 2, Q11 = Q2

1 = 4I2, Q1 = Q2 = 0.1I2, R1 = R2 = 0.4, Γ1 = 1 andΓ2 = 0.1. For each system, the resulting linear program has 7 optimiza-tion variables and 42 constraints. During the simulations, the worst casecomputational time required by the CPU (Pentium 4, 3.2GHz, 1GB RAM)over 400 runs was 5 milliseconds, which shows the potential for controllingnetworks of fast nonlinear systems. In the simulation scenario we testedthe closed-loop system response for x1(0) = [3, −1]>, x2(0) = [1, −2]>

and for the following disturbance scenarios: w1(k) = w2(k) = [0, 0]> fork ∈ Z[0,40] (nominal stabilization), wi(k) takes random values in Wi, i = 1, 2,for k ∈ Z[41,80] (robustness to random inputs), w1(k) = w2(k) = [0, 0.1]> fork ∈ Z[81,120] (robustness to constant inputs) and w1(k) = w2(k) = [0, 0]> for


Figure 6.7: States, inputs and first optimization variable histories for eachsystem.

k ∈ Z[121,160] (to show that asymptotic stability is recovered for zero inputs).

In Figure 6.7 the time history of the states, control input and the op-timization variables λ1

1(k) and λ21(k), assigned to w1

1 = w12 = [0, 0.2]>, are

depicted for each system. In the state trajectories plots, the dashed horizon-tal lines give an approximation of the bounded region in which the system’sstates remain despite disturbances, i.e. approximately within the interval[−0.2, 0.2]. In the input trajectory plots the dashed line shows the inputconstraints. In all plots, the dashed vertical lines delimit the time inter-vals during which one of the four disturbance scenarios is active. One canobserve that the feedback to disturbances is provided actively, resulting ingood robust performance, while state and input constraints are satisfied atall times, despite the strong nonlinear coupling present. In the λ1 plot, one


can see that whenever the disturbance is acting on the system, or when thestate is far from the origin (in the first disturbance scenario), these variablesact to optimize the decrease of each Vi(·) and to counteract the influen-ce of the interconnecting signal. Whenever the equilibrium is reached, theoptimization variables satisfy the constraint Vi(φi(0, 0, we

i )) ≤ λie(k), e =

1, . . . , 4, as explained in Remark 6.4.7. In Figure 6.7, the λ1 plot, the valuesV1(φ1(0, 0, w1

1)) = 0.2641 and V2(φ2(0, 0, w12)) = 0.2271 are depicted with

dashed horizontal lines.

6.6 Conclusions

In this chapter we studied the design of robust MPC schemes with focuson adapting the closed-loop ISS gain on-line, in a receding horizon fashion.Exploiting convex CLFs and disturbance bounds, we were able to constructa finite dimensional optimization problem that allows for the simultaneouson-line (i) computation of a control action that achieves ISS, and (ii) mini-mization of the ISS gain of the resulting closed-loop system depending onthe actual state trajectory. As a consequence, the proposed robust nonli-near MPC algorithm is self-optimizing in terms of disturbance attenuation.Solutions for establishing recursive feasibility and for decentralized imple-mentation have also been briefly presented. Furthermore, we indicated adesign recipe that can be used to implement the developed self-optimizingMPC scheme as a single linear program, for nonlinear systems that are affinein the control variable and the disturbance input. This brings the applicationto (networks of) fast nonlinear systems within reach.

7

Conclusions

7.1 Contributions 7.2 Future research

A summary of the main contributions and a collection of several possibledirections for future research conclude this thesis.

7.1 Contributions

The major contributions are in the domains of

• Stability Theory for Discrete-time Discontinuous Systems;

• Input-to-State Stability Theory for Discrete-time Discontinuous Sys-tems;

• Stabilizing Nonlinear Model Predictive Control;

• Robust Nonlinear Model Predictive Control;

• Low Complexity Nonlinear Model Predictive Control.

We discuss the obtained results in more detail below.

7.1.1 Stability theory for discrete-time systems

The contributions of this thesis regarding stability of discrete-time systemsare presented in Chapter 2. The focus is on the assessment and generali-zation of the classical stability results (Kalman and Bertram, 1960b) in thecase when the system dynamics and/or the candidate Lyapunov functionis discontinuous. The most important observation is that, as opposed tothe continuous case, a uniformly strict Lyapunov function, rather than astrict Lyapunov function (see Chapter 2 for exact definitions), is needed forestablishing asymptotic stability in the Lyapunov sense. An example wasgiven that shows that the equilibrium of a discrete-time system that admits

114 Conclusions

a strict Lyapunov function, but not a uniformly strict one, is not necessarilyglobally attractive. While the uniform strictness condition is an additionalrequirement, compared to (Kalman and Bertram, 1960b), the fact that thesystems dynamics and the candidate Lyapunov function are allowed to bediscontinuous (only continuity at the equilibrium point is required, and noton a neighborhood of the equilibrium) is a significant relaxation. Notice thatglobally asymptotically stable discrete-time systems always enjoy a possiblydiscontinuous Lyapunov function, as shown in (Nesic et al., 1999), but notnecessarily a continuous Lyapunov function.

7.1.2 Input-to-State stability theory for discrete-time dis-continuous Systems

The subject of input-to-state stability theory for discrete-time systems ispresent throughout the thesis. Perhaps the most relevant contributions,which have an impact beyond the MPC context, can be found in Chapter 2,Chapter 4 and Chapter 6, as follows. In Chapter 6 we present a simpleway of establishing inherent robustness in the sense of ISS for possibly dis-continuous discrete-time systems that admit a continuous uniformly strictLyapunov function. Furthermore, in Chapter 2 we illustrate via an examplethat inherent robustness is no longer necessarily attained in the case of adiscontinuous Lyapunov function. Actually, in turns out that the sever phe-nomenon of zero robustness, i.e. loss of asymptotic stability in the presenceof arbitrarily small perturbations, is related to the absence of a continuousLyapunov function.

As most of the ISS results present in the literature assume continui-ty of the candidate (ISS) Lyapunov function, it is not clear how to esta-blish robustness from discontinuous (ISS) Lyapunov functions. In Chapter 2we presented several ISS tests based on discontinuous Lyapunov functions,which render the many available procedures for obtaining Lyapunov func-tions, which typically yield discontinuous Lyapunov functions, useful forestablishing robustness. These tests can be employed to establish ISS ofnominally asymptotically stable discrete-time discontinuous systems in thecase when a discontinuous USL function is available. Moreover, in Chap-ter 4 we have presented a general input-to-state (practical) stability theoremwhich allows for discontinuous system dynamics and candidate ISS Lyapunovfunctions. These results bring certain relevant relaxations with respect tothe original ISS work in discrete-time (Jiang and Wang, 2001) and prove tobe very useful in the context of optimization based control, such as MPCalgorithms.

7.1. Contributions 115

7.1.3 Stabilizing nonlinear model predictive control

While most of the existing results in the theory of MPC regarding closed-loop stability either assume optimality or employ non-trivial modificationsof the original terminal cost and constraint set MPC set-up (Mayne et al.,2000), in Chapter 3 of this thesis we attained stability results for sub-optimalMPC solutions. To cope with MPC control sequences (obtained by solvingMPC optimization problems) that are not optimal, but within a marginδ ≥ 0 from the optimum, we introduced the notion of ε-asymptotic stability(AS) as a particular case of regular AS. In this way we were able to showthat nominal asymptotic stability can be guaranteed for sub-optimal MPCwithout any modification to the standard terminal cost and constraint setMPC set-up presented in (Mayne et al., 2000). Compared to classical sub-optimal MPC (Scokaert et al., 1999), where an explicit constraint on theMPC cost function is employed, this result provides a fundamentally differentapproach to establishing closed-loop stability for sub-optimal MPC.

7.1.4 Robust nonlinear model predictive control

The contributions to robust nonlinear MPC form the richest core of thethesis and are present in all chapters, but Chapter 2. In particular, similarlyas discussed above about stability of sub-optimal MPC, in Chapter 3 wepresented input-to-state stability results that allow for sub-optimal MPCimplementations and discontinuous system dynamics. Within this contextwe introduced a novel approach in the framework of tightened constraintsrobust MPC, which recovers as a particular case existing set-ups. Notice thatallowing for sub-optimal solutions is of paramount importance as firstly, theinfimum in an MPC optimization problem does not have to be attained andsecondly, numerical solvers usually provide only sub-optimal solutions.

The main contributions to the framework of min-max MPC are presen-ted in Chapter 4 and Chapter 5. One of the drawbacks of min-max MPCwas, until now, the absence of an ISS guarantee for the closed-loop system.Due to the maximization of all possible realizations of the uncertain distur-bance, only input-to-state practical stability can be guaranteed in general.The effect of this drawback is rather disturbing, i.e. even if in reality thedisturbance vanishes, which should lead to recovering asymptotic stability(if ISS is established), the min-max MPC closed-loop system can only beguaranteed to be practically stable instead. A solution to solve this pro-blem was presented recently in (Magni et al., 2006), but at the cost of anon-trivial modification to the classical set-up, which unfortunately lead to

116 Conclusions

a non-convex maximization problem. In Chapter 4 novel conditions thatguarantee ISS of min-max nonlinear MPC closed-loop systems were derivedusing a dual-mode approach. This result is useful as it provides a metho-dology for designing robustly asymptotically stable min-max MPC schemeswithout a priori assuming that the (additive) disturbance input converges tozero as the closed-loop system state converges to the origin.

Another result that was missing in min-max MPC was a systematic pro-cedure for computing a terminal cost and an auxiliary control law that satisfythe developed sufficient conditions for ISS. In Chapter 5 we presented a fairlygeneral solution to this problem based on solving of a set of linear matrixinequalities (LMIs). An explicit relation was established between the propo-sed method and H∞ control design. This relation shows that the LMI-basedoptimal solution of the H∞ synthesis problem solves the terminal cost andACL problem in min-max MPC, for a particular choice of the stage cost.This result, which was somehow missing in the MPC literature, is of generalinterest as it connects well known linear control problems to robust MPCdesign.

One of the most important contributions of this thesis is the concept ofself-optimizing robust nonlinear MPC, which makes the subject of Chapter 6.The goal of the existing design methods for synthesizing control laws thatachieve ISS (Jiang and Wang, 2001) is to a priori guarantee a predeterminedclosed-loop ISS gain. Consequently, the ISS property, with a predetermined,constant ISS gain, is in this way enforced for all state space trajectories of theclosed-loop system and at all time instances. As such, it is obvious that theexisting approaches, which are also employed in the design of MPC schemesthat achieve ISS, can lead to overly conservative solutions along particulartrajectories. Therefore, it is of high interest to develop a control (MPC)design method with the explicit goal of adapting the closed-loop ISS gaindepending of the evolution of the state trajectory. A novel method for syn-thesizing robust MPC schemes with this feature was presented in Chapter 6of this thesis. Besides the benefits in performance obtained from “enhan-cing” robustness, which were illustrated by thorough examples, optimizedISS turned out to bring a viable solution to decentralized robust MPC.

7.1.5 Low complexity nonlinear MPC

In developing the theoretical results presented in this thesis we have alwayskept an eye toward obtaining a solution for implementation that has a lowcomputational complexity. Most of the existing methods in low complexityMPC start from a given, fixed optimization problem coming from classical

7.2. Future research 117

MPC design and provide solvers or explicit solutions for particular relevantclasses of systems. So far, these methods are restricted to linear systems, cer-tain types of uncertain systems (LPV systems) and hybrid systems (PWA,MLD systems). In contrast to these approaches concerned with numericaloptimization aspects, we focused on the design of the MPC algorithm, ai-ming at achieving a low complexity optimization problem for a large class ofsystems. In Chapter 6 we showed that for discrete-time nonlinear systemsthat are affine in the control and disturbance inputs, respectively, the self-optimizing robust MPC algorithm can be implemented by solving a singlelinear program. This was attained by considering infinity norms as controlLyapunov functions. The potential of this technique for real-life applicati-on to fast systems was illustrated in Chapter 6 by applying it to control aDC-DC converter (with a sampling period well below one millisecond). Thisopens up a complete new application domain, next to the traditional processcontrol for typically slow systems.

7.2 Future research

There are several interesting research directions possible on the basis of theresults presented in this thesis. In what follows we will briefly present somefuture lines of research that can be pursued.

In most of the algorithms developed in this thesis Lyapunov functioncandidates defined using infinity norms proved to be a fruitful alternativeto the classical quadratic Lyapunov functions. Although necessary and suf-ficient conditions for existence of infinity norm based Lyapunov functionsexist for linear systems (Molchanov, 1987), these conditions do not lead ingeneral to computationally tractable optimization problems. We have madeuse of some alternative, only sufficient conditions, which lead to tractableoptimization problems (although still, nonlinear, non-convex see, e.g.,(Lazaret al., 2006)) at the price of some conservativeness. As such, it would beof great interest to search for new, necessary and sufficient conditions forexistence of infinity norm based Lyapunov functions that can be implemen-ted in a systematic and tractable manner. Extensions to other classes thanlinear systems, such as linear polytopic difference inclusions and input affinenonlinear systems are also of interest.

In terms of robust MPC, in this thesis we have relaxed several stringentassumptions, such as continuity of the system dynamics, continuity of thevalue function corresponding to the MPC cost and most importantly, the as-sumption of optimality. What remains to be studied is an alternative to the

118 Conclusions

terminal cost and constraint set method for proving recursive feasibility. Inthe terminal set method, one needs to compute off-line an invariant set whichis usually difficult to obtain. Even then, this set is relatively “small”, in thesense that a long prediction may be required for reaching the set in N steps.As such, one either has to solve a computationally highly complex optimi-zation problem, due to a large N , or runs intro feasibility problems. That isthe reason why this method, although without rival in theory of MPC, is infact not applied in real-time MPC. An alternative way of proving recursivefeasibility, which does not employ finite time reachability to a predefinedneighborhood of the equilibrium, would really enhance the application ofrobust nonlinear MPC in real-time control.

Another relevant point for further research is related to providing feed-back to disturbances, i.e. to achieve optimal disturbance rejection, ratherthan just bounded trajectories for bounded disturbances. We have taken afirst step in this direction in Chapter 6 by providing a way to explicitly op-timize the ISS gain of the closed-loop system on line. This results however,only holds along a particular trajectory generated in closed-loop on-line. Itwould be interesting to extended this approach to a set of trajectories ori-ginating from a set of initial conditions of interest. Also, the developedself-optimizing MPC algorithm uses the fact that the nominal system ad-mits a continuous and convex control Lyapunov function. This condition isusually satisfied locally, i.e. if the nonlinear system is locally linearizable,but it may be conservative if imposed globally for general nonlinear systems.As such, it would be desirable to relax this conditions at the global level.

Decentralized control is another very active research direction, especiallysince the focus in the control systems community has shifted from complexhybrid and embedded systems to complex networked and embedded systemsin general and control of large scale networks of systems in particular. InChapter 6 we have pointed out a solution to decentralized input-to-statestabilization of networks of nonlinear systems with hard state and inputconstraints. It would be of interest to extended this solution to includesome coordination between different subsystems in the network and even toaccommodate a distributed implementation.

Bibliography

Alamir, M., Bornard, G., 1994. On the stability of receding horizon control ofnonlinear discrete-time systems. Systems and Control Letters 23, 291–296.

Alamo, T., Muńoz de la Peńa, D., Limon, D., Camacho, E. F., 2005. Con-strained min-max predictive control: Modifications of the objective func-tion leading to polynomial complexity. IEEE Transactions on AutomaticControl 50 (5), 710–714.

Alessio, A., Lazar, M., Bemporad, A., Heemels, W. P. M. H., 2007. Squa-ring the circle: An algorithm for obtaining polyhedral invariant sets fromellipsoidal ones. Automatica 43 (12), 2096–2103.

Baotic, M., Christophersen, F. J., Morari, M., 2006. Constrained optimalcontrol of hybrid systems with a linear performance index. IEEE Transac-tions on Automatic Control 51 (12), 1903–1919.

Bemporad, A., Borodani, P., Mannelli, M., 2003. Hybrid control of an auto-motive robotized gearbox for reduction of consumptions and emissions. In:Hybrid Systems: Computation and Control. Vol. 2623 of Lecture Notes inComputer Science. Springer Verlag, pp. 81–96.

Bemporad, A., Morari, M., 1999. Control of systems integrating logic, dyna-mics, and constraints. Automatica 35 (3), 407–427.

Bertsekas, D. P., 1999. Nonlinear programming, second edition. Athena Sci-entific.

Borrelli, F., 2003. Constrained optimal control of linear and hybrid systems.Vol. 290 of Lecture Notes in Control and Information Sciences. Springer.

Camacho, E. F., Bordons, C., 2004. Model Predictive Control. Springer-Verlag, London.

Chen, H., Allgöwer, F., 1996. A quasi-infinite horizon predictive control sche-me for constrained nonlinear systems. In: 16th Chinese Control Conferen-ce. Quindao, China, pp. 309–316.

Chen, H., Scherer, C. W., 2006a. Moving horizon H∞ control with per-formance adaptation for constrained linear systems. Automatica 42 (6),1033–1040.

120 BIBLIOGRAPHY

Chen, H., Scherer, C. W., 2006b. Moving horizon H∞ with performanceadaptation for constrained linear systems. Automatica 42, 1033–1040.

Clarke, D. W., Mohtadi, C., Tuffs, P. S., 1987. Generalized predictive control:I - the basic algorithm and II - Extensions and interpretations. Automatica23, 137–160.

Cutler, C. R., Ramaker, B. L., 1980. Dynamic matrix control - A computercontrol algorithm. In: Joint Automatic Control Conference. San Francisco,U.S.A.

Daafouz, J., Riedinger, P., Iung, C., 2002. Stability analysis and controlsynthesis for switched systems: A switched Lyapunov function approach.IEEE Transactions on Automatic Control 47 (11), 1883–1887.

De Keyser, R. M. C., van Cauwenberghe, A. R., 1985. Extended predictionself-adaptive control. In: IFAC Symposium on Identification and SystemParameter Estimation. York, U.K., pp. 1255–1260.

Feng, G., 2002. Stability analysis of piecewise discrete-time linear systems.IEEE Transactions on Automatic Control 47 (7), 1108–1112.

Ferrari-Trecate, G., Cuzzola, F. A., Mignone, D., Morari, M., 2002. Analysisof discrete-time piecewise affine and hybrid systems. Automatica 38 (12),2139–2146.

Findeisen, R., Imsland, L., Allgöwer, F., Foss, B. A., 2003. State and out-put feedback nonlinear model predictive control: An overview. EuropeanJournal of Control 9 (2–3), 190–206.

Freeman, H., 1965. Discrete-time systems. John Wiley & Sons, Inc.

Garcia, C. E., Prett, D. M., Morari, M., 1989. Model predictive control:theory and practice - a survey. Automatica 25 (3), 335–348.

Geyer, T., Papafotiou, G., Morari, M., 2005. Model Predictive Control inPower Electronics: A Hybrid Systems Approach. In: IEEE Conference onDecision and Control. Seville, Spain.

Golub, G. H., Van Loan, C. F., 1989. Matrix computations. The JohnsHopkins University Press.

Goodwin, G. C., Seron, M. M., De Dona, J. A., 2005. Constrained controland estimation An optimization approach. Communications and controlengineering. Springer.

BIBLIOGRAPHY 121

Grieder, P., Kvasnica, M., Baotic, M., Morari, M., 2005. Stabilizing lowcomplexity feedback control of constrained piecewise affine systems. Au-tomatica 41 (10), 1683–1694.

Grimm, G., Messina, M. J., Tuna, S., Teel, A. R., 2005. Model predictivecontrol: for want of a local control Lyapunov function, all is not lost. IEEETransactions on Automatic Control 50 (5), 546–558.

Grimm, G., Messina, M. J., Tuna, S., Teel, A. R., 2007. Nominally robustmodel predictive control with state constraints. IEEE Transactions on Au-tomatic Control 52 (10), 1856–1870.

Grimm, G., Messina, M. J., Tuna, S. E., Teel, A. R., 2003. Nominally robustmodel predictive control with state contraints. In: 42nd IEEE Conferenceon Decision and Control. Maui, Hawaii, pp. 1413–1418.

Grimm, G., Messina, M. J., Tuna, S. E., Teel, A. R., 2004. Examples whennonlinear model predictive control is nonrobust. Automatica 40 (10), 1729–1738.

Hahn, W., 1967. Stability of motion. Springer-Verlag.

Heemels, W. P. M. H., De Schutter, B., Bemporad, A., 2001. Equivalence ofhybrid dynamical models. Automatica 37 (7), 1085–1091.

Jiang, Z.-P., 1993. Quelques résultats de stabilisation robuste. Application ála commande (in french). Ph.D. thesis, École des Mines de Paris, France.

Jiang, Z.-P., Mareels, I., Wang, Y., 1996. A Lyapunov formulation of thenonlinear small-gain theorem for interconnected ISS systems. Automatica32 (8), 1211–1215.

Jiang, Z.-P., Teel, A. R., Praly, L., 1994. Small gain theorem for ISS systemsand applications. Journal of Mathematics of Control, Signals and Systems7, 95–120.

Jiang, Z.-P., Wang, Y., 2001. Input-to-state stability for discrete-time non-linear systems. Automatica 37, 857–869.

Johansson, M., 1999. Piecewise linear control systems. Ph.D. thesis, LundInstitute of Technology, Sweden.

Kalman, R. E., Bertram, J. E., 1960a. Control system analysis and design viathe second method of Lyapunov, I: Continuous-time systems. Transactionsof the ASME, Journal of Basic Engineering 82, 371–393.

122 BIBLIOGRAPHY

Kalman, R. E., Bertram, J. E., 1960b. Control system analysis and design viathe second method of Lyapunov, II: Discrete-time systems. Transactionsof the ASME, Journal of Basic Engineering 82, 394–400.

Kaminer, I., Khargonekar, P. P., Rotea, M. A., 1993. Mixed H2/H∞ controlfor discrete time systems via convex optimization. Automatica 29, 57–70.

Kassakian, J. G., Schlecht, M. F., Verghese, G. C., 1992. Principles of PowerElectronics. Adisson-Wesley, Publishing Company, Inc.

Keerthi, S. S., Gilbert, E. G., 1988. Optimal, infinite horizon feedback lawsfor a general class of constrained discrete time systems: Stability andmoving-horizon approximations. Journal of Optimization Theory and Ap-plications 57 (2), 265–293.

Kellett, C. M., Teel, A. R., 2004. Smooth Lyapunov functions and robustnessof stability for difference inclusions. Systems & Control Letters 52, 395–405.

Kerrigan, E. C., Maciejowski, J. M., 2001. Robust feasibility in model pre-dictive control: Necessary and sufficient conditions. In: 40th IEEE Confe-rence on Decision and Control. Orlando, Florida, pp. 728–733.

Kerrigan, E. C., Mayne, D. Q., 2002. Optimal control of constrained, piece-wise affine systems with bounded disturbances. In: 41st IEEE Conferenceon Decision and Control. Las Vegas, Nevada, pp. 1552–1557.

Khalil, H., 2002. Nonlinear Systems, Third Edition. Prentice Hall.

Kokotović, P., Arcak, M., 2001. Constructive nonlinear control: a historicalperspective. Automatica 37 (5), 637–662.

Kolmanovsky, I., Gilbert, E. G., 1998. Theory and computation of disturban-ce invariant sets for discrete-time linear systems. Mathematical Problemsin Engineering 4, 317–367.

Kothare, M. V., Balakrishnan, V., Morari, M., 1996. Robust constrained mo-del predictive control using linear matrix inequalities. Automatica 32 (10),1361–1379.

LaSalle, J. P., 1976. The stability of dynamical systems. In: SIAM (Ed.),Regional Conference Series in Applied Mathematics. No. 25. Philadelphia.

BIBLIOGRAPHY 123

Lazar, M., De Keyser, R., 2004. Nonlinear predictive control of a DC-to-DC converter. In: Symposium on Power Electronics, Electrical Drives,Automation & Motion - SPEEDAM. Capri, Italy.

Lazar, M., Heemels, W. P. M. H., 2008a. Global input-to-state stability andstabilization of discrete-time piece-wise affine systems. Nonlinear Analysis:Hybrid Systems 2, 721–734.

Lazar, M., Heemels, W. P. M. H., 2008b. Optimized input-to-state stabiliza-tion of discrete-time nonlinear systems with bounded inputs. In: AmericanControl Conference. Seattle, U.S.A.

Lazar, M., Heemels, W. P. M. H., 2008c. Predictive control of hybrid systems:Stability results for sub-optimal solutions. In: 17th IFAC World Congress.Seoul, Korea.

Lazar, M., Heemels, W. P. M. H., 2009. Predictive control of hybrid sys-tems: Input-to-state stability results for sub-optimal solutions. Automati-ca 45 (1), 180–185.

Lazar, M., Heemels, W. P. M. H., Bemporad, A., Weiland, S., 2007a.Discrete-time non-smooth nonlinear MPC: Stability and robustness. In:Assessment and Future Directions of Nonlinear Model Predictive Control.Vol. 358 of Lecture Notes in Control and Information Sciences. SpringerBerlin Heidelberg, pp. 93–103.

Lazar, M., Heemels, W. P. M. H., Jokic, A., 2009a. Self-optimizing robustnonlinear model predictive control. In: Assessment and Future Directi-ons of Nonlinear Model Predictive Control. Vol. 384 of Lecture Notes inControl and Information Sciences. Springer Berlin Heidelberg, pp. 27–40.

Lazar, M., Heemels, W. P. M. H., Muñoz de la Peña, D., Alamo, T., 2009b.Further results on “Robust MPC using Linear Matrix Inequalities”. In:Assessment and Future Directions of Nonlinear Model Predictive Control.Vol. 384 of Lecture Notes in Control and Information Sciences. SpringerBerlin Heidelberg, pp. 89–98.

Lazar, M., Heemels, W. P. M. H., Teel, A. R., 2007b. Subtleties in robuststability of discrete-time piecewise affine systems. In: American ControlConference. New York City, U.S.A., pp. 3464–3469.

Lazar, M., Heemels, W. P. M. H., Teel, A. R., 2009c. Lyapunov functions,stability and input-to-state stability subtleties for discrete-time disconti-

124 BIBLIOGRAPHY

nuous systems, accepted for publication in IEEE Transactions on Auto-matic Control.

Lazar, M., Heemels, W. P. M. H., Weiland, S., Bemporad, A., 2006. Stabi-lizing model predictive control of hybrid systems. IEEE Transactions onAutomatic Control 51 (11), 1813–1818.

Lazar, M., Heemels, W. P. M. H., Weiland, S., Bemporad, A., Pastravanu,O., 2005. Infinity norms as Lyapunov functions for model predictive con-trol of constrained PWA systems. In: Hybrid Systems: Computation andControl. Vol. 3414 of Lecture Notes in Computer Science. Springer Verlag,Zürich, Switzerland, pp. 417–432.

Lazar, M., Muñoz de la Peña, D., Heemels, W. P. M. H., Alamo, T., 2008a.On input-to-state stability of min-max nonlinear model predictive control.Systems & Control Letters 57, 39–48.

Lazar, M., Roset, B. J. P., Heemels, W. P. M. H., Nijmeijer, H., van denBosch, P. P. J., 2008b. Input-to-state stabilizing sub-optimal nonlinearMPC algorithms with an application to DC-DC converters. InternationalJournal of Robust and Nonlinear Control 18 (8), 890–904.

Lee, J. H., Yu, Z., 1997. Worst-case formulations of model predictive controlfor systems with bounded parameters. Automatica 33 (5), 763–781.

Leenaerts, D. M. W., 1996. Further extensions to Chua’s explicit piecewiselinear function descriptions. International Journal of Circuit Theory andApplications 24, 621–633.

Limon, D., Alamo, T., Camacho, E. F., 2002a. Input-to-state stable MPCfor constrained discrete-time nonlinear systems with bounded additive un-certainties. In: 41st IEEE Conference on Decision and Control. Las Vegas,Nevada, pp. 4619–4624.

Limon, D., Alamo, T., Camacho, E. F., 2002b. Stability analysis of systemswith bounded additive uncertainties based on invariant sets: Stability andfeasibility of MPC. In: American Control Conference. Anchorage, pp. 364–369.

Limon, D., Alamo, T., Salas, F., Camacho, E. F., 2006. Input to state sta-bility of min-max MPC controllers for nonlinear systems with boundeduncertainties. Automatica 42, 797–803.

BIBLIOGRAPHY 125

Magni, L., De Nicolao, G., Magnani, L., Scattolini, R., 2001. A stabilizingmodel-based predictive control algorithm for nonlinear systems. Automa-tica 37 (9), 1351–1362.

Magni, L., De Nicolao, G., Scattolini, R., 1998. Output feedback receding-horizon control of discrete-time nonlinear systems. In: 4th IFAC NOLCOS.Vol. 2. Oxford, UK, pp. 422–427.

Magni, L., De Nicolao, G., Scattolini, R., Allgöwer, F., 2003. Robust mo-del predictive control for nonlinear discrete-time systems. InternationalJournal of Robust and Nonlinear Control 13, 229–246.

Magni, L., Raimondo, D. M., Scattolini, R., 2006. Regional input-to-statestability for nonlinear model predictive control. IEEE Transactions on Au-tomatic Control 51 (9), 1548–1553.

Magni, L., Scattolini, R., 2007. Robustness and robust design of MPC fornonlinear discrete-time systems. In: Assessment and Future Directions ofNonlinear MPC. Vol. 358 of LNCIS. Springer Verlag, pp. 239–254.

Mayne, D. Q., 2001. Control of constrained dynamic systems. EuropeanJournal of Control 7, 87–99.

Mayne, D. Q., Kerrigan, E. C., 2007. Tube-based robust nonlinear model pre-dictive control. In: 7th IFAC Symposium on Nonlinear Control Systems.Pretoria, South Africa, pp. 110–115.

Mayne, D. Q., Rawlings, J. B., Rao, C. V., Scokaert, P. O. M., 2000. Con-strained model predictive control: Stability and optimality. Automatica36 (6), 789–814.

Meadows, E. S., Henson, M. A., Eaton, J. W., Rawlings, J. B., 1995. Rece-ding horizon control and discontinuous state feedback stabilization. Inter-national Journal of Control 62 (5), 1217–1229.

Michalska, H., Mayne, D. Q., 1993. Robust receding horizon control ofconstrained nonlinear systems. IEEE Transactions on Automatic Control38 (11), 1623–1633.

Mignone, D., Ferrari-Trecate, G., Morari, M., 2000. Stability and stabiliza-tion of piecewise affine and hybrid systems: An LMI approach. In: 39thIEEE Conference on Decision and Control. pp. 504–509.

126 BIBLIOGRAPHY

Molchanov, A. P., 1987. Lyapunov functions for nonlinear discrete-time con-trol systems. Avtomatika i Telemekhanika 6, 26–35.

Mosca, E., Zappa, G., Manfredi, C., 1984. Multistep horizon self-tuningcontrollers. In: 9th IFAC World Congress. Budapest, Hungary.

Nesic, D., Teel, A. R., 2001. Changing supply functions in input to statestable systems: the discrete-time case. IEEE Transactions on AutomaticControl 46 (6), 960–962.

Nesic, D., Teel, A. R., Kokotovic, P. V., 1999. Sufficient conditions for thestabilization of sampled-data nonlinear systems via discrete-time approxi-mations. Systems and Control Letters 38 (4–5), 259–270.

Qin, S. J., Badgwell, T. A., 2003. A survey of industrial model predictivecontrol technology. Control Engineering Practice 11, 733–764.

Raimondo, D. M., Limon, D., Lazar, M., Magni, L., Camacho, E. F., 2009.Min-max model predictive control of nonlinear systems: A unifying over-view on stability. European Journal of Control 15 (1), 1–17.

Raković, S. V., 2008. Set theoretic methods in model predictive control. In:3rd Int. Workshop on Assessment and Future Directions of NMPC. Pavia,Italy.

Richalet, J. A., Rault, A., Testud, J. L., Papon, J., 1978. Model predictiveheuristic control: applications to an industrial process. Automatica 14,413–428.

Scokaert, P. O. M., Mayne, D. Q., 1998. Min-max feedback model predictivecontrol for constrained linear systems. IEEE Transactions on AutomaticControl 43 (8), 1136–1142.

Scokaert, P. O. M., Mayne, D. Q., Rawlings, J. B., 1999. Suboptimal mo-del predictive control (feasibility implies stability). IEEE Transactions onAutomatic Control 44 (3), 648–654.

Scokaert, P. O. M., Rawlings, J. B., 1998. Constrained linear quadratic re-gulation. IEEE Transactions on Automatic Control 43 (8), 1163–1169.

Scokaert, P. O. M., Rawlings, J. B., Meadows, E. B., 1997. Discrete-timestability with perturbations: Application to model predictive control. Au-tomatica 33 (3), 463–470.

BIBLIOGRAPHY 127

Soeterboek, R., 1992. Predictive Control - A unified approach. EnglewoodCliffs, NJ: Prentice-Hall.

Sontag, E. D., 1981. Nonlinear regulation: the piecewise linear approach.IEEE Transactions on Automatic Control 26 (2), 346–357.

Sontag, E. D., 1989. Smooth stabilization implies coprime factorization.IEEE Transactions on Automatic Control AC–34, 435–443.

Sontag, E. D., 1990. Further facts about input to state stabilization. IEEETransactions on Automatic Control AC–35, 473–476.

Sontag, E. D., 1999. Stability and stabilization: Discontinuities and theeffect of disturbances. In: Nonlinear Analysis, Differential Equations, andControl. Clarke, F.H., Stern, R.J. (eds.), Kluwer, Dordrecht, pp. 551–598.

Spjøtvold, J., Kerrigan, E. C., Rakovic, S. V., Mayne, D. Q., Johansen,T. A., 2007. Inf-sup control of discontinuous piecewise affine systems. In:European Control Conference. Kos, Greece.

Vidyasagar, M., 1993. Nonlinear Systems Analysis (Second Edition).Prentice-Hall, Englewood Cliffs, NJ.

Wang, Y. J., Rawlings, J. B., 2004. A new robust model predictive controlmethod I: theory and computation. Journal of Process Control 14, 231–247.

Wayne S.U., Mathematical Dept., C. R., 1972. Every convex function islocally lipschitz. The American Mathematica Monthly 79 (10), 1121–1124.

Willems, J. L., 1970. Stability theory of dynamical systems. Thomas Nelsonand Sons, London, England.

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