Predictive Control Algorithms for Nonlinear Systems

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Predictive Control Algorithms for

Nonlinear Systems

DOCTORAL THESIS

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

The Defense will take place on 15 September 2009

by

Mircea Lazăr born at Iaşi, România

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Promoter: Prof. Dr. Mihail Voicu

Corresponding Member of the Romanian Academy

Defense Committee:

Prof. Dr. Vasile-Ion Manta, Chair Prof. Dr. Mihail Voicu, Promoter

Corresponding Member of the Romanian Academy Prof. Dr. Ioan Dumitrache

Corresponding Member of the Romanian Academy Prof. Dr. Vladimir Răsvan

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Acknowledgements

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

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

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

I would like to thank Prof. Paul van den Bosch for his helpful advice and encouragement. He has always been there for me when I needed his opinion and 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 are joint work with several collaborators, I would like to express my gratitude to all the co-authors.

First and foremost I am very grateful to Prof. Maurice Heemels, without whom the research gathered in this thesis would have not been possible. His constant dedication, supervision and professionalism will always be a source of inspiration for me.

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

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

Special thanks also go to Prof. Alberto Bemporad, my mentor and guide in 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 the father 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 Lazar Eindhoven, June, 2009

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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 successfully implemented in industry, for the regulation of constrained linear or nonli-near systems. In this thesis, the MPC controller design methodology will be employed for the regulation of constrained discrete-time systems. One of the reasons for the success of MPC algorithms is their ability to handle hard con-straints on states/outputs and inputs. Stability and robustness are probably the most studied properties of MPC controllers, as they are indispensable to practical implementation. A complete theory on (robust) stability of MPC has 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 via model 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. We demonstrate 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-time discontinuous 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 model predictive controllers. In contrast to existing results based on smooth Lya-punov functions, we develop several robust stability tests, in terms of the input-to-state stability (ISS) property, that explicitly employ an available discontinuous Lyapunov function.

The subtleties exposed in Chapter 2 are employed in Chapter 3 to develop a 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 that sub-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 not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the devi-ation of the predictive control law from the optimum (called theoptimality margin) 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 an important result, as it provides an a priori bound on the evolution of the closed-loop system state and leads to conditions that guarantee ISS even in the presence of unaccounted sub-optimal solutions.

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

The results developed in Chapter 4 hinge on the fact that a suitable terminal cost that satisfies the developed sufficient conditions for ISS must be a priori available. This problem is addressed in Chapter 5, which presents a novel method for designing the terminal cost and the auxiliary control law (ACL) for robust MPC of uncertain linear systems, such that ISS is a priori guaranteed for the closed-loop system. The method is based on the solution of a set of linear matrix inequalities (LMIs). An explicit relation is established between the proposed method and H∞ control design. This relation shows that the LMI-based optimal solution of the H∞ synthesis problem solves the terminal cost and ACL 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 general interest as it connects well known linear control problems to robust MPC design.

In Chapter 6 we start from the observation that the goal of existing design methods 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 state space trajectories of the closed-loop system and at all time instances. As the existing approaches, which are also employed in the design of MPC schemes

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Contents 9

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

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

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Motto

:

“Imagination is more important than

knowledge.”

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1 Introduction

1.1 Model predictive control 1.2 Open problems in

stability and robustness of MPC

1.3 Summary of publications 1.4 Basic mathematical

notation and definitions

This thesis deals with the synthesis of stabilizing and robust controllers for constrained discrete-time discontinuous nonlinear systems. An appealing solution to the control of these systems is provided by the model predicti-ve control methodology, due to its capability to a priori take into account constraints when computing the control action. Also, since the principles of model predictive control do not depend on the type of model applied for prediction, this methodology can be employed to formulate controller de-sign set-ups for general dynamical systems. However, the properties of such control schemes and the feasibility of their implementation have to be recon-sidered in the discontinuous context. In this thesis we focus in particular on stability and robustness. As such, in this chapter we will present a general introduction 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 the thesis.

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 regulation of constrained linear or nonlinear systems and, more recently, also for the regulation of discontinuous and hybrid systems. Within a relatively short ti-me, MPC has reached a certain maturity due to the continuously increasing interest shown for this distinctive part of control theory. This is illustrated by its successful implementation in industry and by many excellent articles and 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.

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The initial MPC algorithms utilized only linear input/output models. In this framework, several solutions have been proposed both in the industrial world 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, and EPSAC - Extend predictive self-adaptive control (De Keyser and van Cau-wenberghe, 1985). Generalized frameworks for setting up MPC algorithms based on input/output models were also developed later on, from which the most significant ones are GPC - Generalized predictive control (Clarke et al., 1987) and UPC - Unified predictive control (Soeterboek, 1992). The next step of the academic community was to extend the MPC algorithms based on state-space models to continuous (smooth) nonlinear systems, which includes the following approaches: nonlinear MPC with zero state terminal equality constraint (Keerthi and Gilbert, 1988), dual-mode nonlinear MPC (Mic-halska and Mayne, 1993) and quasi-infinite horizon nonlinear MPC (Chen and Allgöwer, 1996). More recent general set-ups for synthesizing stabili-zing MPC algorithms for smooth nonlinear systems can be found in (Magni et al., 2001; Grimm et al., 2005). The first MPC approach for the con-trol of discontinuous and hybrid systems has been reported in the seminal work (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 comparison with conventional control, which often uses a pre-computed state or output feedback control law, predictive control uses a discrete-time1 model of the system to obtain an estimate (prediction) of its future behavior. This is done by applying a set of input sequences to a model, with the measured state/ouput as initial condition, while taking into account constraints. An optimization problem built around a performance oriented cost function is then solved to choose an optimal sequence of controls from all feasible se-quences. The feedback control law is then obtained in a receding horizon manner by applying to the system only the first element of the computed

1

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

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1.1. Model predictive control 13

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

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

• The optimization of an objective function subject to constraints, which yields 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 the optimization problem is solved again at the next time instant with the measured state as initial condition.

The MPC methodology involves solving on-line an open-loop finite horizon optimal 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 is usually called the prediction horizon and is denoted by N. This is achie-ved by considering a future control scenario as the input sequence applied to the process model, which must be calculated such that certain desired constraints and objectives are fulfilled. To do that, a cost function is mini-mized subject to constraints, yielding an optimal sequence of controls over a specified time horizon, which is usually called control horizon and is denoted by Nu. According to the receding horizon control strategy, only the first element of the computed optimal sequence of controls is then applied to the plant 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 following constrained optimization problem:

Problem 1.1.1 Let N ≥ 1 be given and let _X ⊆_Rn _and

U ⊆Rm be sets

that implement state and input constraints, respectively, and contain the origin in their interior. The prediction model is x(k+ 1) = g(x(k), u(k)),

k≥0, withg:Rn×Rm →Rn a nonlinear, possibly discontinuous function

with g(0,0) = 0. LetF :Rn →R+ withF(0) = 0 and L:Rn×Rm → R+ with L(0,0) = 0 be known mappings. At every discrete-time instant k≥0

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

In itia l S ta te x0

D e s ire d e q u ilib riu m p o in t xr C lo s e d -lo o p s ta te xk P re d ic te d s ta tex _k S ta te c o n s tra in t C lo s e d -lo o p in p u t uk O p e n -lo o p in p u t u _k In p u t c o n s tra in t k C o n tro l h o riz o n P re d ic tio n h o riz o n k + Nu k + N

Figure 1.1: A graphical illustration of Model Predictive Control.

letx(k)∈Xbe the measured state, letx(0|k),x(k)and minimize the cost

function J(x(k),u(k)),F(x(N|k)) + N−1 X i=0 L(x(i|k), u(i|k)),

over all input sequencesu(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(·,·)andN denote the terminal cost, the stage cost and the prediction horizon, respectively. The termx(i|k)denotes the predic-ted state at future discrete-time instanti∈[0, N], obtained at discrete-time

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1.1. Model predictive control 15

instant k ≥0 by applying the input sequence{u(i|k)}_i=0,...,N−1 to a model of 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 sequence

{u(i|k)}_i=0,...,N−1 constitute the optimization variables. Suppose that the above 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 MPC constrained optimization problem, as well as the stability issues, strongly depend on the type of model used for prediction. For instance, assuming that the MPC cost is defined using quadratic forms (Hahn, 1967) and the constraint sets are polyhedra,

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

• Problem 1.1.1 is anonlinear optimization problemif the model is non-linear;

• Problem 1.1.1 is amixed 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 also addressed in this thesis, consists in how to guarantee stability of a system in closed-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 any controller should satisfy. For linear andcontinuousnonlinear systems, many solutions to this problem have been developed, see the survey (Mayne et al., 2000) for a comprehensive and well documented overview. The most popular approach is the so-calledterminal cost and constraint set method, which re-quires that the terminal predicted state, i.e. x(N|k), is constrained inside a terminal 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 stabilization conditions, in terms of properties that a terminal cost F(·) and a terminal constraint set (usually denoted byXT) must satisfy, can be found in (Mayne et al., 2000).

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

1.2 Open problems in stability and robustness of

MPC

Typically, stability and robustness results for discrete-time systems are ob-taining by mutatis mutandis reproducing the classical results available for continuous-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 is payed to relaxations of the sufficient conditions for Lyapunov stability that might be allowed by the discrete-time setting and their implications in terms of robustness, e.g., in the form of input-to-state stability (ISS) (Jiang and Wang, 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 establish asymptotic stability in the Lyapunov sense. This issue is of paramount im-portance to MPC closed-loop systems, as it is well known, especially since the seminal work on hybrid systems (Bemporad and Morari, 1999), see also (Borrelli, 2003), that MPC candidate Lyapunov functions and closed-loop systems are discontinuous in general. This is due to the fact that MPC usually generates a discontinuous control law, even for continuous system dynamics, 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 above mentioned general stability results for discrete-time systems, but with a sharp focus on removing continuity assumptions, as summarized next. The usual approach to ensure stability in MPC is to consider the value function of the MPC cost as a candidate Lyapunov function. Then, if the system dy-namics is continuous, the classical Lyapunov stability theory (Kalman and Bertram, 1960b) can be used to prove that the MPC control law is stabili-zing, which was done in (Keerthi and Gilbert, 1988). The requirement that the system dynamics must be continuous is (partially) removed in (Alamir and Bornard, 1994; Meadows et al., 1995), where terminal equality cons-traint MPC is considered. In (Alamir and Bornard, 1994), continuity of the

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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 system is still assumed in (Meadows et al., 1995), where it is shown that MPC can generate discontinuous state-feedbacks, the Lyapunov stability proof (Theo-rem 2 in (Meadows et al., 1995)) does not use the continuity property. Later on, an exponential stability result is given in (Scokaert et al., 1997) and an asymptotic stability theorem is presented in (Scokaert et al., 1999), where sub-optimal MPC is considered. The theorems of (Scokaert et al., 1997, 1999) explicitly point out that both the system dynamics and the candidate Lyapunov function only need to be continuous at the equilibrium. Stabili-ty of sub-optimal MPC is proven in (Scokaert et al., 1999) under the usual assumptions (existence of class K bounds on the candidate Lyapunov func-tion V and its forward difference) plus the extra requirement that the MPC optimal sequence of controls is upper bounded in norm by a K function of the norm of the state. A recent overview on stability of receding horizon control in discrete-time can be found in (Goodwin et al., 2005). Although continuity 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 (Meadows et al., 1995). The interested reader can find a general stability theorem for discrete-time MPC that unifies most of the above results in (Lazar et al., 2007a).

Apart from removing the continuity assumption on the system dynamics and MPC cost function, all these results employ the additional assumption that 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 case with MPC cost functions, although the global optimum may exists, it is not necessarily attainable.

This rises the following open problem in stability of MPC:(i) what can be said about stability of classical terminal cost and constraint set MPC schemes (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 atob-tained 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

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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 stability results for sub-optimal MPC solutions without bringing any modifications to the original terminal cost and constraint set MPC set-up. A solution to this open problem is provided in Chapter 4 of this thesis by making use of the 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 systems that are asymptotically stable have zero robustness. That is, in the presence of arbitrarily small perturbations, the asymptotic stability property is lost. The fragility of the stability of MPC closed-loop systems is in fact related to the absence of a continuous Lyapunov function. As the usual candidate for a Lyapunov function in MPC is the value function corresponding to the cost J(·,·), normally a discontinuous function, the following open problem arises: (ii) what can be said about inherent robustness of asymptotically stable 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 imply inherent robustness, even in the sense of ISS, to the best of the author’s knowledge, there are no robustness test that rely exclusively on a discon-tinuous Lyapunov function. As such tests are crucial for MPC closed-loop systems, 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 by presenting a short summary of methods for designing MPC schemes with an 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 disturbances in the MPC problem formulation via open-loop worst case scenarios. This includes MPC algorithms based on tightened constraints, e.g., as the one of (Limon et al., 2002a), and MPC algorithms, based on open-loop min-max optimization problems, see, for example, the survey (Mayne et al., 2000).

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

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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 to the 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 presence of discontinuous dynamics, value functions and/or sub-optimal solutions? A possible 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 (Lee and 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 feasible states corresponding to the feedback min-max MPC optimization problem is usually much larger and the disturbance rejection is improved. Sufficient conditions for robust asymptotic stability of closed-loop (feedback) min-max MPC systems were presented in (Mayne, 2001) under the assumption that the (additive) disturbance input converges to zero as the state converges to the origin.

Recently, input-to-state stability (ISS) (Sontag, 1989, 1990; Jiang and Wang, 2001) results for min-max nonlinear MPC were presented in (Limon et al., 2006) and (Magni et al., 2006). In (Limon et al., 2006) it was shown that, 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 asymptotic stability for zero disturbance inputs. The reason for the absence of ISS in general is that the effect of a non-zero disturbance input is taken into account by the min-max MPC controller, even if the disturbance input vanishes in reality. Still, in the case when the disturbance input converges to zero, it is desirable thatasymptotic 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 guaranteed for min-max MPC closed-loop systems?

In (Magni et al., 2006), anH∞ (Chen and Scherer, 2006a) strategy was used 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 bound on the min-max MPC value function, rather than a global one, is sufficient for ISS. However, this method requires the modification of the stage cost by introducing a negative term which consists of a disturbance norm. In this way, the corresponding min-max optimization problem becomes non-convex

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in the disturbance, which is a significant drawback regarding implementa-tion. As such, our goal is to provide a solution to this problem without incorporating additional terms in the standard min-max MPC cost, which is still possible by employing a dual-mode approach, as presented in Chapter 5 of this thesis.

The second open problem in min-max MPC is (v) how to compute a terminal cost and auxiliary control law that satisfy the sufficient conditions for input-to-state stability? While a solution to the computational of the terminal cost exists in the nominal case, i.e. it amounts to take the terminal cost equal to a local control Lyapunov function, for the robust case, it would amount to the computation of ISS control Lyapunov functions, which is still an open problem. In Chapter 6 of this thesis we present a possible solution for solving this problem for quadratic candidate ISS CLFs. Furthermore, we demonstrate that the solution of the H∞ synthesis problem solves the corresponding terminal cost min-max MPC problem for a particular choice of 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 open problem in robust MPC: (vi-a) how to provide feedback to the disturbances actively, on-line, as a function of the closed-loop trajectory, rather than in a 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 optimization problems 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” robust MPC, in the sense that this MPC scheme provides the means to optimize the closed-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 nonlinear systems it is shown that the corresponding self-optimizing robust MPC op-timization problem can be formulated as a single linear program, which is a major 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 of the developed theory for practical applications. As the sampling period of the considered DC-DC converter is well below one millisecond, this indicates that 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 on open problems. This thesis focuses mainly on novel ways to design MPC

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1.3. Summary of publications 21

controllers with a robust stability guarantee. Special attention is paid to discontinuous 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 complete list 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. Subtleties in robust stability of discrete-time PWA systems. In proceedings of the 26th

American Control Conference 2007, New York, USA.

• (Lazar et al., 2009c): M. Lazar, W.P.M.H. Heemels, A.R. Teel. Lyapunov functions, stability and input-to-state stability subtleties for discrete-time discontinuous 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 IFAC

World Congress, Seoul, Korea, 2009.

• (Lazar and Heemels, 2009): M. Lazar, W.P.M.H. Heemels. Predictive control of 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 and T. Alamo. On input-to-state stability of min-max nonlinear model predictive control. 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: A unifying overview on stability. Survey paper (discussants: J. Maciejowski and 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 Peña and T. Alamo. Further results on “Robust MPC using Linear Matrix Inequa-lities”. L. Magni et al., Eds.,Assessment and Future Directions of Nonlinear

Model Predictive Control, Lecture Notes in Control and Information

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Chapter 6 contains results presented in:

• (Lazar and Heemels, 2008b): M. Lazar, W.P.M.H. Heemels. Optimized input-to-state stabilization of discrete-time nonlinear systems with bounded inputs. 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 nonlinear MPC algorithms with an application to DC-DC converters. International

Journal of Robust and Nonlinear Control, Invited paper for the Special Issue

on 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-optimizing Robust Nonlinear Model Predictive Control. L. Magni et al., Eds.,

Assess-ment and Future Directions of Nonlinear Model Predictive Control, Lecture

Notes 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 definitions are 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-non-negative integers, respectively;

• Z≥c1 andZ(c1,c2] denote the sets{k∈Z+|k≥c1}and{k∈Z+|c1 <

k≤c2}, respectively, for somec1, c2∈Z+;

• For a set _S ⊆ _Rn_,

SN denotes the N-dimensional Cartesian product S×. . .×S, for someN ∈Z≥1;

• For a set P ⊆ _Rn_, _∂_P _{denotes the boundary of} _P_, _int(_P₎ _denotes the interior of P,cl(P) denotes the closure ofP,card(P) denotes the number of elements of P and Co(P) denotes the convex hull of P;

• For any real λ≥0 and setP ⊆_Rn_{, the set}_λ_P _{is defined as}

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1.4. Basic mathematical notation and definitions 23

• For two arbitrary sets P₁ ⊆ _Rn _and _P

2 ⊆ Rn, P1∪ P2 denotes their union, P1 ∩ P2 denotes their intersection, P1 \ P2 denotes their set difference, P1 ⊂ P2 (or P1 ( P2) denotes “P1 is subset of, but not equal to, P₂”,P₁⊆ P₂ denotes “P₁ is subset of, or equal toP₂”;

• 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 inRnthat contains the origin in its interior

is called a C-set;

• A polyhedron (or a polyhedral set) in Rn is a set obtained as the

intersection of a finite number of open and/or closed half-spaces;

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

Vectors, matrices and norms:

• For a real numbera∈R,|a|denotes its absolute value anddaedenotes

the smallest integer larger than a;

• For a sequence {zj}j∈Z+ with zj ∈ Rl, z[k] denotes the truncation of

{zj}j∈Z+ at time k ∈ Z+, i.e. z[k] = {zj}j∈Z[0,k], and z[k1,k2] denotes

the truncation of {zj}j∈Z+ at times k1 ∈ Z≥1 and k2 ∈ Z≥k1, i.e.

z[k1,k2]={zj}j∈Z[k₁,k_2];

• The Hölder p-norm of a vector x∈_Rn _{is defined as:}

kxkp , ( (|x1|p+. . .+|xn|p) 1 p_, _p∈_Z [1,∞) maxi=1,...,n|xi|, p=∞,

wherexi,i= 1, . . . , nis thei-th component ofx,kxk2is also called the Euclidean norm andkxk∞ is also called the infinity (or the maximum) norm;

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• Let k · k denote an arbitrary Hölderp-norm. For a sequence{zj}j∈Z+

with zj ∈Rn,

k{zj}j∈Z+k,sup{kzjk |j∈Z+};

• In denotes the identity matrix of dimensionn×n;

• For some matrices L1, . . . , Ln, diag([L1, . . . , Ln]) denotes a diagonal matrix of appropriate dimensions with the matrices L1, . . . , Lnon the main diagonal;

• For a matrix Z ∈_Rm×n _and _p_∈

Z≥1 or p=∞ kZkp,sup x6=0 kZxkp kxkp ,

denotes its induced matrix norm. It is well known, see, for example, (Golub and Van Loan, 1989), that kZk∞ = max1≤i≤mPnj=1|Z{ij}|, whereZ{ij} is theij-th entry of Z;

• For a matrixZ ∈Rm×n,Z> denotes its transpose andZ−1 denotes its

inverse (if it exists);

• For a matrix Z ∈Rn×n,Z >0 denotes “Z is positive definite”, i.e. for

all 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₎−1_Z> 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 1 2 =Z 1 2(Z 1 2)> =Z;

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

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2 Lyapunov Functions Subtleties for

Discrete-time Systems

2.1 Introduction 2.2 Preliminaries

2.3 Illuminating examples

2.4 ISS tests based on

discontinuous USL functions 2.5 Conclusions

In this chapter we consider stability analysis of discrete-time discon-tinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discontinuous system dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time systems in closed-loop with MPC controllers. In contrast to existing results based 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 and non-smooth nonlinear dynamics (Sontag, 1981; Heemels et al., 2001). The modeling capability of discrete-time PWA systems has already been shown in several 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, for example, (Johansson, 1999; Mignone et al., 2000; Ferrari-Trecate et al., 2002;

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Feng, 2002; Daafouz et al., 2002) for static feedback methods and (Lazar et al., 2005; Grieder et al., 2005; Lazar et al., 2006; Baotic et al., 2006) for model predictive control (MPC) techniques. Most of these methods ma-ke use of classical Lyapunov methods (Kalman and Bertram, 1960b). The first contribution of this chapter is to illustrate the precariousness of the second method of Lyapunov, as presented in (Kalman and Bertram, 1960b), for discontinuous system dynamics. We illustrate via a simple example that existence of a Lyapunov function in the sense of Corollary 1.2 of (Kalman and Bertram, 1960b) (and hence, a continuous function) does not even guarantee global asymptotic stability (GAS) for discrete-time discontinuous systems. In the presence of discontinuity of the dynamics one needs to impose a class

K∞ upper bound on the one-step rate of decrease of the Lyapunov function in order to attain GAS.

The second contribution of this chapter concerns robustness of stability in 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 even the global exponential stability (GES) property is precarious for discrete-time discontinuous systems affected by arbitrary small perturbations. The severe lack of inherent robustness is related to the absence of a continuous Lyapunov 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 PWA systems (Johansson, 1999; Mignone et al., 2000; Ferrari-Trecate et al., 2002; Feng, 2002; Daafouz et al., 2002) only indicated that continuous Lyapunov functions may be more difficult to find than discontinuous ones, while in fact a continuous Lyapunov function might not even exist. As such, a valid warning regarding nominally stabilizing state-feedback synthesis methods for discrete-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. These synthesis methods lead to a stable, possibly discontinuous closed-loop sys-tem and often rely on discontinuous Lyapunov functions. For example, in MPC the most natural candidate Lyapunov function is the value function corresponding to the MPC cost, which is generally discontinuous when PWA systems 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 a discontinuous Lyapunov function. This means that such closed-loop systems may not be ISS to arbitrarily small perturbations, which are always present

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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 on the 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-non-negative integers, respectively. For every subset Π of R+ we define ZΠ :={k∈Z+ |k ∈Π}. Let k · k denote an arbitrary norm on _Rn and let | · | denote the absolute value of a real number. For a sequence z := {z(l)}_l∈Z+ with z(l) ∈ Rn,

l ∈ _Z₊, let kzk := sup{kz(l)k | l ∈ _Z₊} and let z_[k] := {z(l)}_l∈_Z[0_,k_]. For a set S ⊆ _Rn_{, we denote by} _int(_S₎ _{the interior, by} _∂_S _{the boundary and} by cl(S) the closure of S. For two arbitrary sets S ⊆_Rn _and _{P ⊆}

Rn, let S ⊕ P :={x+y |x∈ S, y ∈ P} denote their Minkowski sum. The distance of a point x ∈ Rn from a set P is denoted by d(x,P) := infy∈Pkx−yk. For any µ∈_R_(0,_∞₎ we defineB_µ :={x ∈_Rn_{| k}_x_{k ≤}_µ_}_{. A polyhedron (or} a polyhedral set) is a set obtained as the intersection of a finite number of open and/or closed half-spaces. The p-norm of a vector x ∈ Rn is defined

askxk_p := (|x1|p+. . .+|xn|p)

1

p _for _p∈_Z

[1,∞) and kxk∞:= maxi=1,...,n|xi|, wherexi,i= 1, . . . , nis thei-th component ofx. For a matrixZ ∈Rm×nlet kZk_p := sup_x₆₌₀ kZxkp

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

function ϕ:R+→R+belongs to classK (ϕ∈ K) if it is continuous, strictly increasing and ϕ(0) = 0. A function ϕ : R+ → R+ belongs to class K∞ (ϕ∈ K∞) ifϕ∈ K and lims→∞φ(s) =∞. A functionβ :R+×R+ → R+

belongs to class KL (β ∈ KL) if for each fixed k∈_R₊,β(·, k) ∈ K and for each fixeds∈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; Jiang and 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 unknown

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

Definition 2.2.1 A set P ⊆Rn with0∈int(P) is called arobustly positi-vely invariant (RPI) set with respect to _V⊆ _Rdv _{for system (2.1) if for all} x∈ P it holds that g(x, v)∈ P for allv∈_V. A setP ⊆_Rn_with₀_∈_int(_P₎ is called apositively invariant (PI) setfor system (2.1) with zero input if for

allx∈ P it holds thatg(x,0)∈ P. 2

Definition 2.2.2 Let _Xwith0∈int(X) be a subset ofRn. We call system

(2.1) with zero input (i.e. z(k) = 0 for all k ∈ _Z₊) asymptotically stable in X, or shortly AS(X), if there exists a KL-function β(·,·) such that, for

eachξ(0)∈_X it holds thatkξ(k)k ≤β(kξ(0)k, k),∀k∈_Z₊. If the property holds withβ(s, k) :=θρk_s_{for some} _θ_∈

R(0,∞) and ρ∈R[0,1) we call system (2.1) with zero input exponentially stable inX(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 LetXandVbe subsets ofRn andRdv, respectively, with

0 ∈ int(X). We call system (2.1) input-to-state stable in X for inputs in V, or shortly ISS(X,V), if there exist a KL-functionβ(·,·) and aK-function γ(·) such that, for each initial condition ξ(0) ∈ X and all z = {z(l)}l∈Z+

withz(l)∈_Vfor alll∈_Z+, it holds that the corresponding state trajectory of (2.1) with initial state ξ(0) and input trajectory z satisfies kξ(k)k ≤ β(kξ(0)k, k) +γ(kz_[k−1]k)for allk∈Z[1,∞). The system (2.1) isglobally ISS

if it is ISS(Rn,Rdv). 2

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

Theorem 2.2.4 (Jiang and Wang, 2001; Lazar et al., 2008a)Letα1, α2, α3 ∈

K∞,σ∈ Kand letVbe a subset ofRdv. LetXwith0∈int(X) be a RPI set with respect to _V for system (2.1) and let V :X→ R+ be a function with

V(0) = 0. Consider the following inequalities:

α1(kxk)≤V(x)≤α2(kxk), (2.2a)

V(g(x, v))−V(x)≤ −α3(kxk) +σ(kvk). (2.2b)

If inequalities (2.2) hold for all x ∈ _X and all v ∈ _V, then system (2.1) is

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

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2.2. Preliminaries 29

(2.1) with zero input and inequalities (2.2) hold for allx∈_X (x∈_Rn_{) and}

v∈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 an

ISS 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(·) are discontinuous. The hypothesis only requires continuity at the point x= 0, and notnecessarily on a neighborhood ofx= 0. (ii)If the inequalities (2.2) are satisfied forα1(s) =asλ,α2(s) =bsλ,α3(s) =csλ, for somea, 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, we will introduce the following known types of Lyapunov functions for the zero input system corresponding to (2.1), i.e. ξ(k+ 1) =g(ξ(k),0),k∈_Z₊. Let

X⊆Rnbe a positively invariant set forξ(k+ 1) =g(ξ(k),0)with0∈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(kxk)≤V(x)≤α2(kxk), ∀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(kxk), ∀x∈X. (2.3d) Definition 2.2.5 A functionV(·)that satisfies (2.3a) and (2.3b) is called a

Lyapunov function. A functionV(·)that satisfies (2.3a) and (2.3c) is called a strict Lyapunov (SL) function. A function V(·) that satisfies (2.3a) and (2.3d) is called auniformly strict Lyapunov (USL) function. 2 For continuous V(·) and discrete-time continuous system dynamics it is known 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 we will investigate whether these properties still hold when either the system dynamics or the Lyapunov function is discontinuous, or both.

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

V(g(x,0))−V(x)≤ −δ(x), ∀x∈X, (2.4)

whereδ: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 new USL function that satisfies (2.3d), using ideas from (Nesic and Teel, 2001). Also, in the case wheng(·,0)andV(·) arecontinuous it can be proven that SL functions and USL functions that satisfy (2.4) are equivalent.

2.3 Illuminating examples

Consider the following generic discrete-time PWA systems, which form one of the simplest class of discontinuous systems and will serve as a support for setting 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)) :=Ajξ˜(k) +fj+z(k) if ξ˜(k)∈Ωj, (2.5b) withz(k) ∈ Bµ for some small µ ∈R(0,∞), k∈Z+, and where Aj ∈Rn×n, fj ∈Rnfor allj∈ S (a finite set of indexes) and{Ωj ⊆Rn|j ∈ S}defines a

partition of_X, meaning that∪_j∈SΩj =XandΩi∩Ωj =∅, with the setsΩj not necessarily closed. Firstly, we present a simple one-dimensional example of a discontinuous system that admits a continuous SL function but it is not GAS.

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 by

Ω1 = {x ∈ R | x ≤ 1}, Ω2 = {x ∈ R | x > 1}. One can easily

check 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, for

x∈Ω1\ {0}we have V(G(x))−V(x) =−V(x)<0and, forx∈Ω2 we have

V(G(x))−V(x) = 0.5|x+1|−|x|<|x|−|x|= 0. Hence,V(x)is a continuous SL function. However, V(x) is not a USL function, as for any α3 ∈ K∞ it holds thatlimx↓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 SL function but the trajectories do not converge to the origin globally. This indicates that SL functions (even continuous ones) which are not USL func-tions do not guarantee GAS for discrete-time discontinuous systems. Hence,

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2.3. Illuminating examples 31

Figure 2.1: The functionG(·) 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 (disdis-continuous) USL functions imply GAS see, for example, Chapter 4 in this thesis. The interested reader is also referred to (Nesic et al., 1999) for a proof that a GAS discrete-time system always 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 by

Ω1 = {x ∈ R | x ≤ 1}, Ω2 = {x ∈ R | x > 1}. Figure 2.1 shows the

values of the function G(x). One can easily observe that any trajectory

ξ(k) at time k ∈ Z+ of system (2.5a) starting from an initial condition

ξ(0) =x∈_Rsatisfies|ξ(k)| ≤ |ξ(0)|(even|ξ(k)|<|ξ(0)|whenξ(0) =x6= 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 be proven that V(x) := P∞

i=0ξ(i)2 is a USL function, where ξ(i) denotes the trajectory of system (2.5a) at time i ∈_Z₊, obtained from initial condition

ξ(0) = x ∈ R. Indeed, since V(x) = P∞i=0ξ(i)2 = ξ(0)2 +ξ(1)2 for any

ξ(0) =x∈_R, it holds thatV(G(x))−V(x)≤ −α3(|x|) for allx∈R, where α3(s) :=s2. An explicit expression for V(·)is:

V(x) = ∞ X i=0 ξ(i)2 =ξ(0)2+ξ(1)2= ( x2+ 1if x >1 x2 ifx≤1,

which shows that V(·) is discontinuous atx= 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) to which all trajectories with initial conditionsξ(0) =x∈R(1,∞)= Ω2 conver-ge. Hence, no matter how small µ ∈ _R_(0,∞) is taken, the system (2.5b) is

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not ISS(_R,B_µ). 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 to guarantee inherent ISS (Lazar et al., 2009a), but this condition is too restric-tive for discrete-time discontinuous systems such as PWA systems. Thus, in the next section we will propose ISS tests that can deal with discontinuous USL functions.

Remark 2.3.1 The GES discrete-time system of Example 2 also admits a

continuousSL function, i.e. V(x) :=|x|, which satisfiesV(G(x))−V(x)<0

for allx6= 0. However, as it was the case in Example 1, V(x) =|x|is not a USL function, as for anyα3 ∈ K∞ it holds thatlimx↓1(V(G(x))−V(x)) =

limx↓1(1−x) = 0>−α3(1). Hence, the existence of a continuous SL function does 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 systems arise 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 dynamics that describe the closed-loop system. Therefore, this makes the following example especially relevant for MPC closed-loop systems.

Example 3: Consider the discontinuous nominal and perturbed PWA systems (3.3) with v(k) ∈ Bµ = {v ∈ R2 | kvk ≤ µ} for some µ ∈ R(0,∞),

j∈ S :={1, . . . ,9},k∈Z+, and where Aj= 1 0 0 1 forj6= 7; A7= 0.35 0.6062 0.0048 −0.0072 ; f1=−f2= 0.5 0 ; f3=f4=f5=f6= 0 −1 ; f7= 0 0 ; 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-1_{Note that the regions} _Ω

1 andΩ2 are such that for allx∈∂Ω1∩∂Ω2 the dynamics

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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) and

x(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 functionV(x) :=kx(10)k∞+P_i=09 kQx(i)k∞, whereQ= 0.04I2

and x(i) is the solution of system (2.5a) obtained at time i ∈ _Z_[0,10] from initial 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 anyµ∈_R_(0,_∞₎the PWA system(2.5b)isnotISS in_Xfor inputs inB_µ.

Proof: (i) The following properties hold for the PWA system (2.5a) of Example 3, as it can be seen by inspection of the dynamics: (P1) kx(k+ 1)k∞≤ kx(k)k∞for allx(k)∈X,k∈Z+; (P2) For any initial statex(0)∈X

the state trajectory satisfies x(k) ∈ Ω7 for all k ∈ Z≥10; (P3) kA7k∞ <

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A7x(k) +f7; (P5)X is a PI set for the PWA system (2.5a).

First, we prove that V(x) = kx(10)k∞+P9_i=0kQx(i)k∞ satisfies ine-quality (2.2a). For anyτ ∈(0,0.04)it holds that kQxk∞ ≥τkxk∞. There-fore, α1(kxk∞) ≤V(x) is satisfied for all x ∈ Xwith α1(kxk∞) := τkxk∞. 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 for

anyx(0)∈_X). Then, using the triangle inequality, for anyx∈_X(note that

x(0) :=x) we obtain that V(x) =kx(10)k∞+ 9 X i=0 kQx(i)k∞≤ kQx(0)k∞+kQAj0x(0)k∞ +kQfj0k∞+kQAj1Aj0x(0)k∞ +kQAj1fj0k∞+kQfj1k∞+. . .+ kAj9. . . Aj0x(0)k∞+kAj9. . . Aj1fj0k∞+. . .+kfj9k∞.

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, ifx(0) =x∈_X\Ω7, since0∈int(Ω7) and Ω7 is bounded, there exists a positive numberζ >0such that

kQfj0k∞+ (kQAj1fj0k∞+kQfj1k∞) +. . .

+ (kAj9. . . Aj1fj0k∞+. . .+kfj9k∞)≤ζkx(0)k∞.

Then, using x(0) = x and the inequality kQxk∞ ≤ kQk∞kxk∞ it follows thatV(x)≤α2(kxk∞) for allx∈Xwithα2(kxk∞) :=θkxk∞, where

θ:=kQk∞+ 9 X i=1  kQ i−1 Y p=0 Ajpk∞  +k 9 Y p=0 Ajpk∞+ζ.

Finally, for anyx∈X∩Ωj and anyj∈ S, by properties (P2), (P3) it holds that

V(Ajx+fj)−V(x) =−kQxk∞+ (kA7x(10)k∞− kx(10)k∞+kQx(10)k∞)

≤ −kQxk∞≤ −τkxk∞=:−α3(kxk∞). In the above inequality we used the fact that

kA7xk∞− kxk∞≤(kA7k∞−1)kxk∞=−0.0438kxk∞

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for all x∈_Rn_{. Therefore, the function} _V₍_x_{) =}_k_x₍₁₀₎_k

∞+P9_i=0kQx(i)k∞ is a USL function for system (2.5a) of Example 3. One can easily check that

V(x) is discontinuous, for example, atx= [0.2 3.6]>∈Ω2.

(ii) By property (P5), X is a PI set for the PWA system (2.5a) of

Example 3 and hence, a valid domain of attraction. Therefore, exponen-tial stability of the origin follows directly from the result of part (i), due to the special form of the K-functionsα1(·),α2(·) and α3(·) established in the proof of part (i).

(iii) To illustrate the non-robustness phenomenon for the perturbed PWA system (2.5b) of Example 3, we constructed an additive disturbance v(k), which at times k= 0,2,4, . . . is equal to [0ε]> and at timesk = 1,3,5, . . .

is equal to [0 −ε]>, where ε > 0 can be taken arbitrarily small. The system trajectory (see Figure 2.2 for a plot - red line) with initial state

˜

x(0) = [0.2 3.6]>∈∂Ω2∩∂Ω1 is given byx˜(k) = [0.2 3.6]>, ifk= 0,2,4, . . . and x˜(k+ 1) = [−0.3 3.6 +ε]>, if k= 1,3,5, . . .. This is a limit cycle with period 2 and kx˜(k)k∞ ≥ 3.6 for all k ∈ Z+. Then, for any β ∈ KL and

γ ∈ K we can take ε > 0 arbitrarily small and k∗ ∈ Z+ large enough such that

β(k,kx˜(0)k∞) +γ(kw[k−1]k∞)<3.6≤ kx˜(k)k∞, ∀k≥k∗.

Therefore, for any ε > 0, the PWA system (2.5b) of Example 3 is not ISS for initial conditions in _Xand inputs in B_ε.

Notice that, by taking any finite polyhedral partition of_R2\_X, defining the dynamics in each polyhedral region of this partition to be x(k+ 1) = [0 0

0 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 no robustness to arbitrarily small disturbances.

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

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