Università degli Studi di Roma Tor Vergata. Facoltà di Ingegneria

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1 Università degli Studi di Roma Tor Vergata Facoltà di Ingegneria Corso di laurea in ingegneria dell Automazione Tesi di Laurea Magistrale On Quantization in Control Systems: Stabilization of Saturated Systems Subject to Quantization. Relatore Prof. Sergio Galeani Candidato Francesco Ferrante Corelatori Frédéric Gouaisbaut Sophie Tarbouriech Anno Accademico

2 On Quantization in Control Systems: Stabilization of Saturated Systems Subject to Quantization. Francesco Ferrante dell autore:

3 Ad Aldo e Luana, i miei genitori A Marta, mia sorella A Viola, il mio amore Coloro cui devo tutto.

4 ACKNOWLEDGEMENTS My gratitude goes to Luca Zaccarian, who allowed me to perform the activity which led to the present thesis. During this time, he constantly provided me valuable advices, being for me a mentor but mostly a friend. I am thankful to him since he supported my future career as PhD student, giving me the confidence. I would like to thank Sophie Tarbouriech and Frédéric Gouaisbaut, who provided me their experiences and their time during my activity at the LAAS-CNRS. I am very grateful to them, since they have introduced me to the research activity in automatic control and gave me the chance to go on, fulfilling my desire. I have learned a lot from their and I hope to learn much more in the future. I also thank all the members of the MAC group, who welcomed me in the best way, making me feel at home.

5 RINGRAZIAMENTI Arrivati alla fine di un percorso così lungo e così impegnativo, il capitolo dei ringraziamenti potrebbe superare di gran lunga la lunghezza di tutta la tesi! Per questo cercherò di riassumere, senza recar dispiacere a nessuno, o almeno lo spero. Voglio ringraziare per cominciare la mia famiglia, la quale mi ha permesso di giungere fino a questo punto, sopportandomi e supportandomi in ogni modo sempre ed incondizionatamente. Tra loro, ringrazio mio padre Aldo e mia madre Luana, i quali hanno permesso tutti ciò ed hanno sopportato il mio incessante brontolare. Un sentito grazie va a mia sorella Marta, del resto come non ringraziare espressamente colei che innumerevoli volte ha sentito parlare di esami durante le sue cene, ma che ha sopratutto sostenuto notti insonni permettendomi di scrivere tesi e tesine varie. Grazie anche a Viola, il mio amore, che con amore e pazienza mi è sempre stata accanto. Senza di lei sarebbe stato impossibile per me completare questo percorso, che rappresenta una delle tappe più importanti della mia vita. Spero che tutti i sacrifici da lei fatti in questo periodo, si trasformino in dei frutti che potremmo raccogliere insieme giorno dopo giorno. Voglio ringraziare anche mia suocera Miriam per l asilo ed i buoni pasti offertimi numerose volte durante questo periodo. Ringrazio anche la famiglia Baldini, la quale mi è stata accanto in ogni momento durante il mio primo periodo a Tolosa e continua

6 a farlo tutt oggi. Senza di loro tutto ciò sarebbe stato quasi impossibile. Un grazie va anche ad i miei amici: Emanuele Pellegrini il quale che ha sempre creduto in me. Giovanni Tancredi l amico di sempre, il quale non ha esitato a dispensare numerosi consigli di natura grafica durante la stesura di questa tesi. Raffaello Bonghi, l amico con cui ho condiviso gran parte di questa esperienza e con il quale ho superato numerosi esami, non dimenticherò mai il tempo trascorso insieme in una delle fasi più importanti della mia vita, l approccio con l automatica. Antonio De Palo, come dimenticare le giornate passate insieme a studiare con un entusiasmo senza fine che non dimenticherò mai. Federico Celletti compagno di studi ma sopratutto amico, ore e ore passate a studiare con una voglia insaziabile di capire, condividendo la passione per la teoria del controllo. Le numerose giornate trascorse a discutere hanno dato spunto a numerose ed interessanti riflessioni. Oltre gli amici, vorrei ringraziare anche i docenti che ho incontrato durante il mio percorso. Un grazie al prof. Lavino Ricciardi che ha dato inizio al meccanismo che mi ha condotto fino qui. Ringrazio anche il prof. Salvatore Monaco, il quale ha dato il via alla mia passione per l automatica, introducendomi alla teoria dei sistemi con curiosità e donandomi le solide basi che mi hanno permesso di giungere fin qui. Grazie anche al prof. Alessandro Savo, che ha saputo trasmettermi un metodo che tutt oggi caratterizza il mio modo di affrontare i problemi. Un ricordo anche alla memoria del prof. Alessandro De Carli. Ricorderò per sempre i suoi insegnamenti ed il suo saper essere ingegnere. Ringrazio infine, il prof. Sergio Galeani per la sua disponibilità sia come relatore, sia come professore sempre pronto a suggerire interessantissimi spunti ed esempi, ed il prof. Osvaldo Maria Grasselli per la sua cordialità e per il rigore e la passione con cui conduce il suo corso intriso di insegnamenti fondamentali.

7 Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? Paul Halmos, I want to be a Mathematician, Washington 1985

8 CONTENTS Introduction VII 1 Quantization in control systems Preliminary definitions Quantized systems Issues on discontinuity Stability and stabilization The works of R. Brockett and D. Liberzon Control by static quantized-state feedback Stabilization via dynamic quantizer The work of S.Tarbouriech and F.Gouaisbaut Quantization in Linear Saturated Systems Preliminary Definitions The Open-Loop System The Closed-Loop System Boundedness/Ultimate Boundedness via Lyapunov Like Analysis Existences of Limit Cycles: The Planar case I

9 2.4 Static State Feedback Control Design Problem statement and first steps toward the solution Models for the Saturation Nonlinearity Sector Conditions for the Uniform Quantizer Main results State Feedback with Quantized Input State Feedback with Quantized Measured State Optimization issues and numerical results Optimization Based Controller Synthesis Size criteria for ellipsoidal sets LMI Formulation Numerical Examples Quantized Input Case Quantized Measured State Case Conclusions and Outlooks Summary Critical Aspects Outlooks A Further Clarifications on the LMI Formulation 85 A.1 Quantized Input Case A.2 Quantized Measured State Case B Mathematical Review 89 B.1 Convex sets and Functions B.2 Convex Combination and Convex Hull B.3 Linear Matrix Inequalities B.4 S Procedure B.5 Ellipsoidal set contained in a symmetric polyhedron Bibliography 97 II

10 List of Figures 99 III

11 LIST OF SYMBOLS = equal to not equal to < (>) less (greater) than ( ) less (greater) than or equal to defined as : such that A\B max Subset of Set difference x A\B if x A and x / B exists for every belonging to summation maximum IV

12 min N Z Z Z l R n R n + R n m x x minimum strictly positive integer numbers integer numbers integer numbers multiple of l-component vector of integer numbers multiple of n-th dimensional Euclidean space subset of the n-th dimensional Euclidean space composed by vectors with strictly positive components Real n m matrices space the norm of a vector x the absolute value if x R, the vector component-wise absolute value if x R n B m the ball centered at the origin of radius m {x R n : x m} ( ) vector component-wise less (greater) than ( ) vector component-wise less (greater) than or equal to Co{x 1,..., x n } The convex hull of {x 1,..., x n } 0 the null scalar or the null matrix of appropriate dimension 1 m unitary vector belonging to R m, i.e. 1 m = [1,..., 1] } {{ } m I n n n identity matrix i.e } 0 0 {{ 1 } n V

13 diag{a 1 ;... ; A n } Denotes the block-diagonal matrix whose diagonal elements are A 1,..., A n λ min (P )(λ max (P )) the minimum (maximum) eigenvalue of a symmetric matrix P x, (A ) the transpose of a vector x (a matrix A) He(A) means A + A trace(a) trace of the matrix A P > 0 a symmetric positive definite matrix P 0 a symmetric positive semi-definite matrix A > B means that the matrix A B > 0 A B means that the matrix A B 0 f 2 f 1 it stands for symmetric blocks in the expression of a matrix the composition of two function LM I Linear Matrix Inequality ẋ f x the derivative of x respect to time the Jacobian matrix f : S 1 S 2 a function f mapping a set S 1 into a set S 2 sat( ) the saturation function VI

14 INTRODUCTION In the real realization of control systems, most of the time, some constraints are present on the variables that determine the evolution of the controlled system, these constraints arise from limited resources or technical limitations present in the real situations. One of the major limitations in real cases consists in a limit on the number of possible values for certain variables. This phenomena is commonly called quantization and the systems, whose variables are subjected to the quantization constraints, are commonly called quantized systems. For example, quantization arises when a plant is controlled by a digital system, which uses a finite-precision arithmetic to compute the control action, or when a controlled system is only a sub-system which exchanges information with a main system via a communication channel, with a limited set of symbols. Figure 1 depicts a typical situation wherein quantization is involved. The previous list of examples is only a small part of a wide class of real cases, since the presence of the quantization is massive in practice. It is straightforward find more and more examples that clarify the importance of a systematical study of the quantization phenomena. In the literature many works that deal with the quantization aspects in control system have been presented. For instance in [11] the effects induced by the quantization in digital control loop had been studied, VII

15 Figure 1: Quantization in control system showing the presence of chaotic behavior or limit-cycle in quantized system. Nevertheless, until the late of 80 s the quantization effect was considered as an undesirable effect and hence the presence of this phenomena was neglected in the control synthesis. Then, to estimate the effect induced by the quantization, the difference between the real signal and the quantized one was modeled as a noise (stochastic signal), with certain characteristics, and the behavior of the controlled system was studied in presence of that noise see [17]. In other words the quantization effect was been considered for a long time as an approximation and it was studied by the means of the tools provided by the information theory. However, since digital devices were becoming pervasive in control systems, new systematical analysis methods were necessary. D.F.Delchamps in 1998 [6] marked a watershed, proposing an alternative approach to deal with stability and stabilization in quantized control systems. This approach entails modeling the quantization phenomenon via a static nonlinear function, quantizer, which maps a real variable in a variable belonging to a given discrete set Q, that is: q: R Q The methodology proposed by D.F.Delchamps in [6] allows one to use the modern control theory tools in order to deal with quantized control systems. VIII

16 Such tools are based on the state-space description of the controlled plant, that is ẋ = f(x, u) where x: R R n, u: R R m, are respectively called state and control and f : R n+m R n. For example, describing the quantizer as a nonlinear map q( ), the effect induced by a quantizer on a given state feedback law v = k(x), can be considered, setting: u = q(k(x)) Accordingly to this approach, recently D. Liberzon and R. Brockett have been proposed a new framework, for studying quantized systems, based on the input to state stability concept proposed by E. Sontag in [19]. Moreover, in [16] D. Liberzon has highlighted that in many case, due the presence of the quantization, the Lyapunov asymptotic stability property may not be achieved. On the other hand, besides the stabilization problems, several aspects related to quantization effect in control systems has been studied. For instance, in control schemes whose control variables are subject to quantization, it is clear that the control properties could be influenced by the quantization phenomena. Regarding this aspect, for example, in [1] issues on reachability, for quantized linear discrete time, has been pointed. Let us take a step back to stabilization problem, in [3, 15, 16] D. Liberzon and R. Brockett have proposed new techniques aimed mainly at removing the limitation introduced by a quantizer in achieving the asymptotic stability. Such techniques deal with quantizers whose parameters can be varied on-line. Especially, in [3, 15, 16] a hybrid control scheme is proposed in order to obtain the global asymptotic stability for nonlinear systems, under precise conditions. However, these strategies can be adopted only in particular cases. Indeed, usually varying quantizer parameters is not allowed. In such situations, one have recourse to more classical control structure, hence disposing of synthesis conditions in these cases can be suitable. Concerning this aspect, in [22] some synthesis conditions have been provided in a quantized static state feedback control context and with respect to a restricted IX

17 class of system, that is linear saturated systems. Such class of systems can be represented by the following model: ẋ = Ax + B sat(u) This class of systems holds the interest of engineers and researchers as many real plants can be approximately modeled through a linear model, whereas saturation aries, since in real cases input variables cannot assume arbitrarily large values. Especially, in [22], via a slightly modification of the approach proposed by D. Liberzon, the authors have provided a synthesis procedure, which can be carried out through a solution of an associate LMI optimization problem, that is an optimization problem like: min c 1 x c n x n subject to A x 1 A x n A 1 n 0. A m 0 + x 1 A m x n A m n 0 where, x 1,..., x n R are the decision variables and A i 0, A i 1,..., A i n, i = 1,..., m are some given matrices. This latter feature represents the strong point of the exhibited methodology, since LMI optimizations problems can be efficiently solved by numerical procedures [2, 4]. On the other hand, as highlighted by the authors, putting an effort to simplify the problem, some conservativeness has been introduces. Such a conservativeness sometimes reveals some limitations in the applicability of the exposed method. In this thesis, we extend the work proposed in [22] reducing the above mentioned conservatism. To this end, a different approach to modeling the saturation nonlinearity is adopted. The research activity, which has led to this thesis, was conducted at: Laboratorie d Analyse et d Architecture des Systèmes (LAAS -CNRS) of Toulouse (France), under the supervision of Sophie Tarbouriech and Frédéric X

18 Gouaisbaut. For sake of clarity, we will expose as follow, which is the structure of the present thesis. Thesis Structure The thesis is composed of four chapters. Chapter 1 In this chapter, we will present an overview on the literature related to quantization in control systems. Firstly, we will provide some basic definitions and afterwards we will show how a quantizer can be represented and which are the related technical problems. In the end we will focus on the concept of stability for quantized system and we will show some stabilization techniques presented in the literature. In the end we will drops an hint on the works proposed in [22]. Chapter 2 In this chapter we will focus our attention on the effect induced by a quantizer in a control scheme involving a restricted class of controlled systems, that is the class of continuous-time LTI systems with saturated input. Especially in this chapter, we will firstly explain which are the models used in order to describe the controlled plant, afterwards we will illustrate which are the closed-loop systems analyzed in this work. Concerning this aspect, we will show how to cope with the difficulties due the presence of the saturation and the quantization. With respect to the saturation, a polytopic representation of a linear saturated system will be provided, instead with regards to quantization, sector conditions proposed in [22] will be shown. Through such tools some simplified models, of the closed-loop systems analyzed, will be proposed. In the end, we will expose the formal problem we want to solve. XI

19 Chapter 3 In this chapter, we will show some theoretical results aimed at solve the Problem 1 exposed in the Chapter 3, for both the case presented. These results are based on the simplified models shown in the previous chapter. Especially, these results represent sufficient conditions. Chapter 4 In this chapter, we will show how the conditions stated in the Chapter 3 can be effectively used in order to solve numerically the problem stated in Chapter 2. To this end, an optimization procedure will be presented in order to develop a suitable synthesis procedure, however this procedure is characterized by nonlinear constraint. Therefore, some additional constraints will be considered in order to obtain, in both cases, a LMI optimization problem. Finally, some numerical results will be shown. XII

20 CHAPTER 1 QUANTIZATION IN CONTROL SYSTEMS In this chapter we will present an overview on the literature related to quantization in control systems. Firstly, we will provide some basic definitions and afterwards we will show how a quantizer can be represented and which are the related technical problems. In the end we will focus on the concept of stability for quantized system and we will show some stabilization techniques presented in the literature. 1.1 Preliminary definitions In the sequel, we will deal with dynamical systems described by an ordinary differential equation with a specified initial condition, that is: ẋ = f(x, u) (1.1) x(t 0 ) = x 0 1

21 1.2 Quantized systems 2 where, considering only solutions that run forward in time, x: [t 0, + ) R n, u: [t 0, + ) R m, x 0 R n are said to be respectively, "state", "control" and "initial state" and ẋ denotes the time-derivative of x, i.e: ẋ(t) = dx(t) dt Moreover, without loss of generality, we will usually assume t 0 = Quantized systems By a quantizer we mean a function q that maps the Euclidean space R l to a discrete set Q, that is: R l Q q: x q(x) Depending on the model adopted for describing a particular situation, the set Q can be bounded or unbounded, but anyway we can state that a quantizer is a discontinuous map since it maps a connected space, R l, in an unconnected space Q [20]. The fact that the quantizer map is discontinuous can represent a non trivial problem in the field of quantized control systems. As stated previously, we are dealing with dynamical systems described by ordinary differential equations, hence the presence of a discontinuous element in the controller implies that the closed-loop system will be described by a discontinuous right-hand side ordinary differential equation. For instance, let us consider a static state-feedback control law for the system (1.1), obtained using a quantized state measure that is u = γ(q(x)), this controller provides the following closed-loop system ẋ = f(x, γ(q(x)))

22 1.2 Quantized systems 3 which is obviously described by a discontinuous right-hand side ordinary differential equation. Therefore in order to handle quantized systems we must face with the problem of discontinuous right-hand side differential equations Issues on discontinuity In the literature many examples are shown, wherein a discontinuous righthand side differential equation does not admit any solution in usual sense. For a recent survey we refer to [5]. Therefore in order to deal with systems described by a discontinuous right-hand side differential equation, we must extend the classical concept of solution for differential equations. Unfortunately there is not an unique way for doing this. Depending on the problem and on the objectives at hand, different notions have been proposed and each of them arises from a different point of view adopted in facing the right-hand side discontinuity. For example, in [8], a new concept of solution is introduced, looking at the values assumed by the vector field f in a neighborhood of each point rather than the single point. Formally a set valued map is associated to the vector field f. Specifically, for every x R n the vector field f is evaluated in the open ball B(x, δ) centered at x with radius δ and the analysis is carried out for smaller and smaller δ, that is when δ approaches zero. In [13] a similar approach is proposed. Another notion of solution is provides by the concept of Carathéodory solutions, shown in [8]. Especially, regarding at the differential equation (1.1), a Carathéodory solutions essentially is a classical solution, that satisfies the relation (1.1) for almost all t. In order to define formally what we mean with Carathéodory solutions it is suitable to introduce the concept of absolutely continuity Definition 1. [5] A function γ : [a, b] R is absolutely continuous if, for all ε (0, + ), there exists δ (0, + ), such that, for each finite collection {(a 1, b 1 ),..., (a n, b n )} (a, b)

23 1.2 Quantized systems 4 with it follows that n (b i a i ) < δ i=1 n γ(b i ) γ(a i ) < ε i=1 Clearly, every absolutely continuous function is continuous, but it can be shown by examples that the converse is not true [5]. Now we can provide the definition of Carathéodory solutions. Definition 2. A function x: I R n is a Carathéodory solution of the equation (1.1) on the interval I if is an absolutely continuous function that satisfies the following integral equation x(t) = x 0 + t t 0 f(x(τ))dτ t, t 0 I, t t 0 (1.2) The foregoing overview represents only an outline of the wide area concerning discontinuous right-hand side differential equations (see for example [5, 8, 13]). Looking at the above definitions, one may argue that every notion faces with the discontinuity in the right-hand side in different manners. Hence, choosing different notions of solution can provide different outcomes. Obviously, in choosing the notion of solution, one should be careful in obtaining a faithful model of the real problem, without adding any undesired phenomena. In fact, depending on the considered solution notion, different behaviors can be considered or excluded. For instance to consider Filippov s solutions allows us to analyze sliding mode motions, which are not solution in Carathéodory sense. However in this work, we will focus only on Carathéodory solutions. Therefore we will accordingly excluded particular solution like sliding mode motions between the quantization regions.

24 1.2 Quantized systems Stability and stabilization The most interesting topics in control area are problems related to stability and stabilization of an equilibrium point. In quantized system, the problem of stability plays a relevant role. Indeed, since almost all the controllers are feedback controller, they use the measured variables in order to compute the control action. Hence, when these variables are subject to quantization, it is clear that the closed-loop stability could be influenced by the quantization effect. Furthermore, without loss of generality, in the sequel we will consider the origin as equilibrium point. To deal with the problem of stability for quantized systems, first of all it should be noticed that in controlled systems involving a quantizer, sometimes the classical concept of Lyapunov asymptotic stability can be meaningless. Indeed in order to steer the state from a given initial condition to the origin, it is necessary to get an infinitely arbitrarily high precision close to the origin, but this requirement do not fit with the quantization constraint. For instance in a system controlled by a quantized input it is possible to drive the system only with a discrete set of values. However the impossibility in achieving the asymptotic stability objective it is not always so serious, indeed the real important task is often to keep the system trajectories bounded and sufficiently close to the origin. In order to handle this aspect it is suitable to introduce a weaker property than the asymptotic stability. At this end, in the literature several properties have been introduced. For example in [25] a notion of containability is provided in a particular case, whereas in [14] the concept of practical stability is introduced and studied. Furthermore, in [12] a more general property is introduced for nonlinear continuoustime systems. Such a property is called boundedness/ultimate boundedness property [12] and it is defined as follow. Boundedness and ultimate boundedness With respect to the system (1.1), the following definition holds.

25 1.3 The works of R. Brockett and D. Liberzon 6 Definition 3. [12] The solutions of (1.1) are: uniformly bounded if there exists a positive constant c and a (0, c) there is a β = β(a) > 0 such that: x 0 R n : x(t 0 ) a x(t) β t t 0 (1.3) globally uniformly bounded if (1.3) holds for arbitrary large a uniformly ultimately bounded with bound b, if there exist two positive constants, b, c and a (0, c) there exists a time T = T (a, b) 0 such that: x 0 R n : x(t 0 ) a x(t) b t t 0 + T (1.4) globally uniformly ultimately bounded if (1.4) holds for arbitrary large a Remark If the system (1.1) is autonomous, then the term uniformly can be omitted without ambiguity. 1.3 The works of R. Brockett and D. Liberzon The quantization phenomena and related problems discussed above can be englobed in a wider field called "Control with limited information" of which R. Brockett is one of the pioneers. Some representative references, concerning this research area, include [3, 6, 7, 23 25]. Many approaches can be adopted in order to cope the limited information control problem, for

26 1.3 The works of R. Brockett and D. Liberzon 7 instance in [15] a general methodology for handle this class of problems is proposed for state feedback case. Especially this approach can be summarized as follows. 1. Model the quantification effects via deterministic additive error signal e; 2. Design a nominal control law ignoring these errors, i.e. a control law acting on perfect information, which we assume by simplicity to be a static state feedback u = k(x); 3. ( certainty equivalence ) Apply the above control law to the imperfect/corrupted signals, resulting in u = k(x + e) and combine it with an estimation procedure aimed at reducing e to 0. Using this approach requires that the closed-loop system is robust, in some sense, with respect to e. Indeed one can show, by example, that even if the error converges to zero and the synthesized controller stabilizes the closedloop system when e 0, the state can blow up in finite time making the control strategy unable to stabilize the plant [15]. Hence the main problem is how to characterize the necessary closed-loop robustness with respect to e. In the works of D. Liberzon and R. Brockett this characterization is founding on the "Input to state stability"(iss) concept introduced by Eduardo Sontag [19]. Keeping in mind the approach provided by the control with limited information, we can cope with the problems of stability and stabilization for quantized systems Control by static quantized-state feedback In [16] D. Liberzon has shown that it is possible to obtain the boundedness and the ultimate boundedness properties for a general nonlinear system controlled by a static quantized-state feedback, under precise assumptions. Let us start to specify which the author means as quantizer in this work.

27 1.3 The works of R. Brockett and D. Liberzon 8 If the variable subject to quantization belongs to R l then the quantizer is a mapping q: R l Q where Q is a finite subset of R l. Therefore the quantization regions are the sets {z R l : q(z) = i}, i Q Furthermore the following assumption on the quantizer holds. Assumption 1. [16] There exist two positive real numbers M, such that: 1. If then 2. If then z M q(z) z z > M q(z) > M The above conditions define the so-called saturated quantizer. Condition 1 gives a bound on the quantization error when the quantizer does not saturate whereas condition 2 provides a way to detect the quantizer saturation. The numbers and M represent the quantizer sensitivity and the quantizer range. In Figure 1.1 the described quantizer is depicted. It should be noticed that the author does not provide a real description of a quantizer, but rather a set of properties which are commonly satisfied by a general quantizer. In this way, the results provided are not relate to a particular definition of the quantizer and they can be extended in every situation as long as Assumption 1 is verified. After the previous clarifications, assuming

28 1.3 The works of R. Brockett and D. Liberzon 9 Figure 1.1: Graphical description of quantizer defined by D. Liberzon that a general nonlinear system ẋ = f(x, u), x R n, u R m (1.5) and a state feedback law u = k(x) (1.6) that renders the origin globally asymptotically stable for the system (1.5) are given. Now suppose that the state x is quantized, hence the control law (1.6) becomes u = k(q(x)) = k(x + e) (1.7) where e q(x) x is the quantization error. One can straightforwardly argue that in general the origin of the obtained closed-loop system is not still globally asymptotically stable. Nevertheless one may expect in place of the global asymptotic stability property a weaker property like trajectories

29 1.3 The works of R. Brockett and D. Liberzon 10 boundedness/ultimately-boundedness. To this end, the author makes the following assumption Assumption 2. [16] There exists a class C 1 function V : R n R such that for some class K functions α 1, α 2, α 3, ρ and for all x, e R n we have and α 1 ( x ) V (x) α 2 ( x ) (1.8) x ρ( e ) V x f(x, k(x + e)) α 3( x ) (1.9) The relation (1.9) allows us to state that every solution that starts in the largest sub-level set of V (x), denoted with R 1, contained in the ball B 1 centered at origin and having radius M, converges toward the smallest sub-level set of V, denoted with R 2, containing the ball B 2 centered at origin and having radius ρ( ). Indeed since e and ρ( ) K, one can state that x ρ( ) x ρ( e ) the function V then is decreasing along the trajectories of the system (1.5), inside the set R 1 \R 2. The above discussion is resumed in Figure 1.2. In order to write down an explicit expression for the two sub-level sets, the author provides the following lemma. Lemma 1. [16] Considering Assumption 2 and assuming that we have α 1 (M) > α 2 ρ( ) Then the sets R 1 {x: V (x) α 1 (M)} (1.10) R 2 {x: V (x) α 2 ρ( )} (1.11) are invariant regions for the quantized closed-loop system. Moreover all so-

30 1.3 The works of R. Brockett and D. Liberzon 11 R 1 R 2 B 1 B 2 R 1 R 2 B 1 B 2 R 2 B 2 Figure 1.2: The sets B 1,B 2,R 1,R 2 lutions that start inside R 1 enter in R 2 in a finite time T such that: T α 1(M) α 2 ρ( ) α 3 ρ( ) According with Definition 3, the foregoing Lemma states that all trajectories starting in the set R 1 are bounded and ultimately bounded, indeed since x(t) R 1 t 0 then x(t) M whereas since x(t) R 2 t T then x α 1 1 α 2 ρ( ). Moreover the author highlights that the conditions stated in Assumption 2 are equivalent to input to state stability (ISS) for the system (1.5) with respect to the quantization error e, in other words in [15, 16] D. Liberzon shows that the ISS concept and the control with limited information problem are formally related. Moreover, the author states that for linear systems stabilized by a linear static state feedback, the above ISS requirement is

31 1.3 The works of R. Brockett and D. Liberzon 12 guaranteed. Indeed considering the linear system ẋ = Ax + Bu x R n, u R m (1.12) with the pair (A, B) stabilizable, then there exists a matrix K R m n such that the the matrix A + BK is Hurwitz. By Lyapunov theorem, there exist positive definite and symmetric matrices P, Q such that: (A + BK) P + P (A + BK) = Q Assuming, without loss of generality B 0, K 0, (trivial case i.e. A is Hurwitz is excluded). Denote with λ min ( ) and λ max ( ) respectively the minimal and the maximal eigenvalue of a symmetric matrix. Considering a linear static state feedback controller obtained using a quantized state measure, that is u = K q(x), one gets the following closed-loop system ẋ = Ax + BK q(x) or equivalently, highlighting the quantization error e ẋ = (A + BK)x + BKe (1.13) Since the matrix K stabilizes the plant (1.12) and since this plant is linear, then the closed-loop system (1.13) is ISS with respect to e, as it is shown in [12]. Furthermore in order to recover the sets R 1 and R 2 shown in Lemma 1, one may note that in this case we can set α 1 ( x ) λ min (P ) x 2 (1.14) α 2 ( x ) λ max (P ) x 2 (1.15) α 3 ( x ) 2 P BK ε x (1.16) ρ( x ) 2 P BK (1 + ε) λ min (Q) (1.17)

32 1.3 The works of R. Brockett and D. Liberzon 13 where ε is a small enough positive constant. To the end, under other technical assumptions which guarantee that R 2 R 1 (for more detail about this part we refer to [16]) the author shows that R 1 {x R n : x P x λ min (P )M 2 } and R 2 {x R n : x P x λ max (P ) 4λ } max(p ) 2 (1 + ε) 2 P BK 2 λ min (Q) Stabilization via dynamic quantizer In the above discussion quantizer range M and quantizer sensitivity were assumed to be constant. However in many situation these parameters can vary. In this case it is possible using other control policy in order to obtain an improvement on the results obtained in the previous section. In the work of Liberzon [16] it is proposed to use the following definition for the new quantizer q µ (z) µ q ( ) z µ (1.18) where µ > 0 and it is not fixed. This type of quantizer has a range equal to Mµ and provides a quantization error equal to µ and it is called Dynamic quantizer since the parameter µ can be adjusted on-line. Hence it is possible to consider µ as a zoom variable. Therefore, increasing µ the quantizer becomes coarse and vice versa decreasing µ one provides a more and more fine quantizer. In [16] D.Liberzon shows how it is possible to using the above quantizer features in order to improve the result stated in the previous section, mainly the author follows an approach based on the following idea: Update the values of µ accordingly with the values of a suitable Lyapunov function in order to maximize the available information on the system state. For the sake of simplicity, this update is doing by the author at discretetime instant. Hence using this strategy combined with a feedback control law, provides a hybrid system, since both discrete-time and continuous-time

33 1.3 The works of R. Brockett and D. Liberzon 14 dynamics are present. In [16] D. Liberzon develops an updating strategy aimed at improving the results provided from Lemma 1. Such a strategy is composed by two stages: 1. Zooming-out : At this stage the quantizer range is increase until the state of the system can be measured adequately. During this stage the system is in open loop. This stage takes place during the time interval [0, t 0 ); 2. Zooming-in : At this stage the system is controlled by a quantizedstate feedback and at the same time the quantizer range is decreased, improving the quantizer sensitivity, according to certain rule aimed at driving the state at the origin [3]. This stage takes place during the time interval [t 0, + ). The above procedure leads to the following closed-loop control law: 0, 0 t < t 0 u(t) = (1.19) K q µ (x(t)) t t 0 Furthermore, we will say that the system (1.5) is forward complete, if for every initial state x(0) the solutions of (1.5), are defined for all t 0. Using the control policy (1.19), the author shows that for the general nonlinear system (1.5) under the Assumption 2 and under certain technical assumptions, the system is globally asymptotically stable. Especially Liberzon provides the following result. Lemma 2. [16] Assuming that the unforced system ẋ = f(x, 0) is forward complete, and that α 1 2 α 1 (Mµ) > max{ρ( µ), χ(µ + 2 µ)} µ > 0 for some χ K then there exists a hybrid control policy (1.19), that makes the system (1.5) globally asymptotically stable.

34 1.4 The work of S.Tarbouriech and F.Gouaisbaut 15 Until now we have faced the problem of system stabilization by quantized state-feedback, that is problem in which the quantization phenomena affect the measured state. However, as mentioned at the beginning of this work, in many real problems the variables subject to quantization is the control variables, hence it is relevant to cope this class of problem. Concerning this problem, D. Liberzon in [16], has shown some interesting results centered on the concept of ISS with respect to the quantization error. These results are through and through similar to the ones provided in the case of quantized state-feedback. 1.4 The work of S.Tarbouriech and F.Gouaisbaut In [22], conditions for boundedness/ultimately boundedness are provided, for linear dynamical systems involving both input saturation and quantization control law. Especially in [22] a slightly version of the approach proposed by D.Liberzon in [16] is adopted in order to obtain synthesis conditions, in a quasi-linear matrix inequality context. However in [22] the authors adopt another definition for the quantizer with respect to that one provided by D.Liberzon in [16]. Basically in [22] the saturation nonlinearity and the quantization are disjointed, in such a way a larger class of systems can be handled. Especially two quantization cases are considered: input quantization case and state quantization case, both in a linear static state feedback context. Furthermore, [22] provides directly a convex optimization procedure that allows to computes in one shot the stabilizing controller and the sets R 1 and R 2 shown in [16]. However, as highlighted by the authors, the conditions provided in [22] are very conservative and then in several cases, this can represent a real limitation. On the other hand, as already mentioned, the techniques presented in [22] can provide a real outcome in engineering problems, since these ones are directly computable. Therefore, prompted by the result shown in [22], in the present work we intend to derive a set of conditions which may represent an alternative to those ones

35 1.4 The work of S.Tarbouriech and F.Gouaisbaut 16 proposed in [22], in order to reduce the conservatism presented in the considered approach. Hence in the next chapter, firstly we will introduce and analyze the class of the studied systems and afterwards we will explain in detail, how to assess the boundedness/ultimate boundedness property for the considered systems. At the end we will state formally the problem that we want cope with, and we will provide the first tools to solve it.

36 CHAPTER 2 QUANTIZATION IN LINEAR SATURATED SYSTEMS In this chapter we will focus our attention on the effect induced by a quantizer in a control scheme involving a restricted class of controlled systems, that is the class of continuous-time LTI systems with saturated input. This class of systems can be recognized in many real situations, thus it is interesting to study systematically the behavior of such a class of systems. Especially in this chapter, we will firstly explain which are the models used in order to describe the controlled plant, afterwards we will illustrate which are the closed-loop systems analyzed in this work. In the end, we will expose the formal problem we want to solve and we will also give some tools used in the sequel for solve such a problem. 2.1 Preliminary Definitions As already mentioned the quantizer definition is not unique, indeed a quantizer is mainly a function that maps a real variable in a discrete one and 17

37 2.1 Preliminary Definitions 18 clearly an unique way for doing this does not exist. Hence in order to cope with a specific problem, one should provide a precise definition of the quantizer, depending on which model is chosen for represent this one. According with [22], in this work for the uniform quantizer the following definition holds: Definition 4. [22] The uniform quantizer is the map q: R l Z l defined as follows: 0 if z(i) q i (z) k if k z(i) (k + 1) i = 1,..., l (2.1) k if (k + 1) z(i) k (2.2) where k N and is an arbitrary strictly positive real number. The i th component of the uniform quantizer defined in (2.1) is depicted in Figure 2.1.

38 2.1 Preliminary Definitions 19 q z z Figure 2.1: The i th component of the uniform quantizer Looking at the Definition 4 we can state that the uniform quantizer satisfies the following condition: q (i) (z) z (i) i {1,..., l} For this we will call quantization error bound. Furthermore, for the sequel providing the following definition is suitable. Definition 5. A polyhedron of R n S(R, µ 1, µ 2 ), with R R l n and µ 1, µ 2 R l +, is the set defined as: S(R, µ 1, µ 2 ) = {x R n : µ 2 Rx µ 1 } A symmetric polyhedron of R n S(Q, µ) with Q R l n and µ R l + is the set defined as: S( Q, µ) = {x R n : Qx µ}

39 2.1 Preliminary Definitions The Open-Loop System A physical plant P can be generally approximated by a linear dynamical system: ẋ = Ax + Bu (2.3) where: x: [0, + ) R n, u: [0, + ) R m, A R n n, B R n m (2.4) However in real systems the input variable cannot assume arbitrary large values. Hence in order to take into account the limitation on the input variable, some assumptions are needed. Especially we will assume that the control vector u takes values in the polyhedron S(I m, u min, u max ), that is u S(I m, u min, u max ) (2.5) Due to the magnitude constraints provided by relation (2.5) the input injected to the system can be modeled by the saturation function [21], i.e. u = sat(v) where sat( ) represents the function defined as follows: u max(i) if v (i) > u max(i) sat(v (i) ) v (i) if u min(i) v (i) u max(i) i = 1... m (2.6) u min(i) if v (i) < u min(i) The i th component of the saturation map defined in (2.6) is depicted in Figure 2.2. Therefore, since the saturation function is a part of the controlled system, the term Open-Loop system refers to the series connection between the system (2.3) and the static nonlinear system characterized by

40 2.1 Preliminary Definitions 21 sat v i u max u min u max v i u min Figure 2.2: The i th component of the saturation function the saturation nonlinearity defined by (2.6), i.e. u = sat(v) which provides: ẋ = Ax + B sat(v) (2.7) Such a system is illustrated in Figure 2.3. Furthermore for sake of simplicity, in the sequel we will assume u min = u max = ū where ū R m The Closed-Loop System Usually a generic plant is controlled by a closed-loop controller both in order to improving its performances and in order to stabilize it. Hence we want to investigate the behavior of the open-loop system (2.7) controlled involving the uniform quantizer defined in (2.1). The foregoing analysis can be per-

41 umax u min q z umax z 2.1 Preliminary Definitions 22 v u min sat v i v i u P x u K Figure 2.3: Open loop system formed looking at different control schemes and considering several variables subject to quantization. The most elementary case is to consider the presence of a quantizer in a linear static state feedback controller. Especially, we will consider two different control laws. State feedback with quantized input: In this case the control input computed by the controller is given by: v = Kx (2.8) where K R m n and we suppose that such an input is injected within the plant (2.3) via a uniform quantizer: v = q(kx) (2.9) Therefore controlling the system (2.3) via the state feedback with quantized input, one gets the following closed loop system ẋ = Ax + B sat(q(kx)) (2.10)

42 umax u min umax 2.1 Preliminary Definitions 23 In Figure 2.4 the foregoing control system is presented. v u min sat v i v i u P x u q( ) K Figure 2.4: State feedback controller with quantized input State Feedback With Quantized Measured State: In this other case we assume that the measured state is quantized. Hence using the available state measure in order to compute a linear state feedback as (2.8), one gets the following control law: v = K q(x) (2.11) which provides the following closed-loop system ẋ = Ax + B sat(k q(x)) (2.12) In Figure 2.5 the above described control system is depicted. Since we are interested in stability properties for the systems (2.10) and (2.12), it should be noticed that two limitations, which may affect asymptotic stability properties, exist. The first one is the effect introduced by the quantizer in the control systems (2.10) and (2.12) whereas the second one is the presence of the saturation. With respect to the quantization effect, as already touched

43 umax u min umax 2.1 Preliminary Definitions 24 v u min sat v i v i u P x u K q( ) Figure 2.5: State feedback controller with quantized measured state upon in the first chapter, the quantizer induces a lack of precision necessary to steer the state to the origin. One may note that there exists, for both systems (2.10) and (2.12), a subset S( ) of the state space containing the origin, wherein u = 0. Indeed with respect to the system (2.10), when x S(K, ) it follows that u = 0, instead regarding the system (2.12) the same phenomenon occurs when x S(I n, ). For this, one can set, respectively for the system (2.10) and for system (2.12): S( ) = S(K, ) S( ) = S(I n, ). Without ambiguity, we will denote, in both cases, S( ) as Linear Region, since when u = 0 both systems (2.10) and (2.12) behave as: ẋ = Ax Roughly speaking the properties of the controlled system (2.10) and (2.12), sufficiently close to the origin, are not influenced by the control action. We

44 2.1 Preliminary Definitions 25 briefly illustrate this aspect through the following example: Example Considering the closed-loop system (2.10), where: [ ] [ ] A =, B =, = 1, ū = In Figure 2.6 a closed-loop trajectory entering in the Linear Region is depicted. It is interesting to note that the considered trajectory, due the discontinuity introduced by the quantizer, experiences a corner point crossing the Linear Region Figure 2.6: A closed-loop trajectory ( ), initialized in ( ), enters in the Linear Region (- - -) Then it is easy to realize that the asymptotic stability property can be achieved, only if the matrix A is Hurwitz. With respect to the constraint introduced by the saturation, if A is not Hurwitz, implies that one cannot

45 2.2 Boundedness/Ultimate Boundedness via Lyapunov Like Analysis 26 ensure the global asymptotic stability of the origin [18]. Hence adding a quantizer, one may reasonably expect that such a limitation can be just enforced by the quantization effect. Therefore we can conclude that the asymptotic stability of the origin, for the closed-loop systems (2.10) and (2.12), can hold only if the matrix A is Hurwitz. Furthermore in such a case, also the global asymptotic stability of the origin could be proven [18],providing adequate conditions. However in many real situations the assumption on matrix A is not verified. Nevertheless, as we stated previously, if we are just interested in assuring that the system trajectories, obtained starting from a certain bounded set of initial states, are bounded and remain sufficiently close to the origin ultimately. Therefore this latest remark conduces us naturally to consider the local boundedness/ultimate boundedness property, for systems (2.10) and (2.12), in place of the asymptotic stability property. At this end as suggested in [12], we will use a Lyapunov like analysis exposed as follow. 2.2 Boundedness/Ultimate Boundedness via Lyapunov Like Analysis Consider a general nonlinear system ẋ = f(x) (2.13) Let V (x) be a continuous, smooth positive definite function and suppose that the sets: Ω c = {x: V (x) c} (2.14) Ω ε = {x: V (x) ε} (2.15) Λ = {x: ε V (x) c} (2.16) Are compact for some c > ε > 0 (see Figure 2.7).

46 2.2 Boundedness/Ultimate Boundedness via Lyapunov Like Analysis 27 Λ Ω ε Ω c Figure 2.7: The sets Ω c, Ω ε, Λ Suppose V (x) = V (x) x f(x) W 3(x), x Λ (2.17) where W 3 (x) is a continuous and positive definite function. Since V (x) is negative on the boundary of Ω c and Ω ε, which are level sets of V (x), then these ones are positively invariant sets for the system (2.13). By the virtue of compactness of Λ, the trajectories of the system (2.13) which are initialized in Λ are bounded. Furthermore setting, k = min x Λ W (x) then: V (x) k x Λ (2.18) Hence integrating the relation (2.18), along the trajectories of the system (2.13), on the interval [0, T ) with T > 0, one gets V (x(t )) V (x(0)) kt

47 2.3 Existences of Limit Cycles: The Planar case 28 then x(t) enters in the set Ω ε in the interval (0, c ε/k]. Hence, accordingly with Definition 3 provided in the first chapter, the trajectories of the system (2.13) are ultimately bounded in Ω ε. Looking at the above conditions, one may realize that this approach is a merely generalization of the Lyapunov stability theorem. Especially, roughly speaking the proposed method represents a way to relax the classical concept of asymptotic stability. Furthermore when ε approaches to zero, one recovers the Lyapunov theorem on asymptotic stability. Remark The above exposed method requires that V (x) is negative definite in the annular region between two level sets of V (x). It is clear that, since we are interested in keeping the state close at the origin, we want to find an inner level set which surrounding the origin as tight as possible. However, as previously mentioned, the quantizer induces a linear region which is a polytope. Therefore if the matrix A is not Hurwitz, then there exists a neighborhood of the origin B wherein V (x) cannot be definite negative. 2.3 Existences of Limit Cycles: The Planar case It is well known [11], that quantization phenomena can induce limit cycles and chaotic behavior in control loop. Usually in real controlled systems the presence of limit cycles is undesirable. Therefore it could be interesting to investigate on this phenomenon. Generally, it is not trivial to perform a formal analysis on the existence of a limit cycle. Indeed that analysis is nearly always carried out numerically. However in planar case one may take advantage of the Poincaré-Bendixson criterion, recalled here: Lemma 3 (Poincaré-Bendixson Criterion [12]). Let ẋ = f(x) (2.19)

48 2.3 Existences of Limit Cycles: The Planar case 29 be a second order system and let M be a closed bounded subset of the plane such that M contains no equilibrium points or contains only one equilibrium point such that the Jacobian matrix [ f(x)/ x] at this point has eigenvalues with positive real parts. Every trajectory starting in M stays in M for all future time. Then M contains a periodic orbit of (2.19). Hence, with respect to the closed-loop systems (2.10) and (2.12) in planar case, i.e x R 2, suppose that the boundedness/ultimate boundedness property holds for those systems and furthermore suppose that the origin is the unique equilibrium point contained in Ω ε. Then if the matrix A has eigenvalues with positive real parts by virtue of the Lemma 3, there exists a limit cycle inside the ultimate set Ω ε. Example Consider the system [ ] [ ] ẋ = x + sat(u) where ū = 30. In order to stabilize this system consider the following static state feedback controller: u = Kx (2.20) [ ] with K = 16 10, which provides the following saturated closed-loop system: [ ] [ ] ẋ = x + sat(kx) (2.21) Since the eigenvalues of the matrix A + BK are 3.8 and 0.6 from linear system Lyapunov s theory, we can conclude that there exists a matrix P = P > 0 such that: (A + BK) P + P (A + BK) < 0

49 2.3 Existences of Limit Cycles: The Planar case 30 Then the function V (x) = x P x is a Lyapunov function for the saturated linear system (2.21) inside the set S(Kx, ū), implying that the origin for the saturated linear system (2.21) is locally asymptotically stable [12]. Suppose now that the control action (2.20) is quantized via the uniform quantizer defined in Definition 2.1 and characterized by = 1. One gets the following closed-loop system: [ ] [ ] ẋ = x + sat(q(kx)) (2.22) Figure 2.8 illustrates that quantizing a stabilizing controller may be as the origin of limit cycles in the closed-loop system (2.22). Looking at this exam Figure 2.8: Example Limit cycle in state feedback with quantized input, closed-loop system. The initial condition is marked by a cross ple, one may argue intuitively that there is a link between the quantization error bound and the amplitude of the limit cycles induced by the quantizer. Indeed when = 0 no limit cycles arise, then we can informally guess that as little is as little will be the amplitude of the induced limit cycle.

50 2.3 Existences of Limit Cycles: The Planar case 31 The Figure 2.9 supports the foregoing intuition Figure 2.9: Example Two different amplitude limit cycles, in state feedback with quantized input closed-loop system, considering: = 1 ( ), = 0.5 (- - -).The initial condition is marked by a cross. Remark In order to carry out the foregoing analysis, one should verify that no equilibrium points, except the origin, belong to Ω ε. Generally, this analysis could be tricky, however it should be noticed that no equilibrium points can belong to the linear region, except the origin if the matrix A has eigenvalues with positive real parts. Hence, in order to find some equilibrium points inside Ω ε, only the set Θ Ω ε \S(K, ū) should be explored. Moreover, since Θ is bounded, then q(kx) can assume therein only a finite number of values. Therefore, as long as the shape of the set Ω ε is not too intricate, the equilibrium points researching can be performed by a simple iterative algorithm, without hazard in computational

51 2.4 Static State Feedback Control Design 32 overhead. Similar arguments could be used in the quantized measured state feedback case. The foregoing analysis does not aspire to provide a formal way to deal with limit cycles in quantized systems, but it drops just a hint on the problem. 2.4 Static State Feedback Control Design After the necessary clarifications about the direction we will adopt for facing the problem of boundedness/ultimate boundedness of the controlled systems (2.10) and (2.12), we can state a formal problem consistent with these limitations. Basically, in order to state a formal problem, one may follows two approaches: Synthesis approach or Analysis approach. With respect to the first one, the problem is to find a matrix K R n m and a function V (x), such that the closed-loop system (2.10) (or equivalently (2.12)), satisfies requirements stated in the Lyapunov like analysis (see Section 2.2). In the analysis approach, one assumes that K is known and then the unique problem is to find a function V (x) and the associated sets Ω c, Ω ε, Λ defined in (2.14) (2.15) (2.16). The last approach sometimes can be useful, especially in practical situations, for testing the effect of the saturation and quantization phenomena using a precomputed controller, which can be provided by linear techniques as LQR or pole placement. However, since the Lyapunov based analysis provides only a sufficient condition, a failure in the precomputed controller test it does not imply that the controller does not assure the desired boundedness properties. For that reason, the synthesis approach is nearly always preferable to the analysis one. Therefore in the sequel, we will put always an effort in adopting a synthesis point of view.

52 2.4 Static State Feedback Control Design Problem statement and first steps toward the solution The problem we intend to solve can be summarized as follows: Problem 1. Considering the systems (2.10) and (2.12), determine K and characterize two sets, S 0 and S u such that: x 0 S 0 the resulting trajectories for the closed-loop systems (2.10) and (2.12) are bounded. x 0 S 0 the resulting trajectories for the closed-loop systems (2.10) and (2.12) are ultimately bounded in the set S u which is contained in S 0. In order to solve the Problem 1, as already mentioned, we want to adopt the foregoing Lyapunov like analysis. However, the described procedure is not constructive. Clearly there is not an unique way for determine the candidate Lyapunov function. Therefore in order to provide some fruitful results, it is necessary to fix at least the structure of the function V (x). At this end, prompted by the results shown in [2, 21, 22], we will assume that V (x) is a positive definite quadratic function, that is: V = x P x (2.23) where: P R n n : P = P > 0 Obviously, on the other hand such a choice may revel to be very conservative. But on the other hand assuming that V (x) is a quadratic function provides many advantages. For instance, as shown in [2, 21, 22], using quadratic functions in Lyapunov s analysis leads to more numerical tractable conditions.

53 2.4 Static State Feedback Control Design Models for the Saturation Nonlinearity In order to cope with the Problem 1, facing saturation nonlinearity is necessary. However, it is well known that handling saturated systems can be very complicated [21]. Hence the first step, to provide a solution to Problem 1, is to render more tractable such a problem, that is to reduce the complexity due to the saturation. To this aim, a simpler representation of the saturation nonlinearity is needed. To this end, in [21] different representations are shown. Among, two of them lead to conditions which are more suitable in a synthesis framework. The first one is based on polytopic differential inclusions. It provides a local description of the saturated closed-loop system. Then a robust control approach is used to perform a Lyapunov analysis. The second representation consists in re-writing the saturated closed-loop system, replacing the saturation nonlinearity with a dead-zone nonlinearity. Then sector conditions can be used for analyze the closed-loop system behavior. For instance, in [22] the second approach is used in coping with the problem of boundedness and ultimate boundedness for the systems (2.10) and (2.12). Therefore in order to provide alternative conditions to those provided in [22], we will use an approach based on the polytopic representation. In such a context, several representation can be considered [21]. However in this work, we will focus just on the representation proposed in [9, 10], which we briefly explain as follows: Polytopic Model of Saturated Systems The basic idea consists in using an auxiliary variable h R m and represent the saturation function as a convex combination of the actual variable v ans h. In [21] some examples are proposed clarifying the described method. Formally the method derives from the following Lemma: Lemma 4. [9] Consider two vectors v R m and h R m. If h (i) ū (i), i {1... m}

54 2.4 Static State Feedback Control Design 35 then it follows that: sat(v) Co{Γ + j v + Γ j h, j = 1,..., 2m } where: Γ + j are diagonal matrices whose diagonal elements assume the value 1 or 0, j = 1,..., 2 m ; Γ j = I m Γ + j j = 1,..., 2 m. Let us consider the closed-loop systems (2.10) and (2.12) and define the following polyhedral set: S(H, ū) = {x R n : H x ū} From Lemma 4, if x S(H, ū), one gets: } sat(k q(x)) Co {Γ +j K q(x) + Γ j Hx, j = 1,..., 2m } (2.24) sat(q(kx)) Co {Γ +j q(kx) + Γ j Hx, j = 1,..., 2m As consequence if x(t) S(H, ū), t 0 then the closed-loop systems (2.10) and (2.12) can be respectively re-written as follows: ẋ = Ax + B ẋ = Ax + B 2 m j=1 2 m j=1 ) λ j (x) (Γ +j q(kx) + Γ j Hx ) λ j (x) (Γ +j K q(x) + Γ j Hx (2.25) where λ 1 ( )..., λ 2 m( ), for every x R n, are appropriate positive real num-

55 2.4 Static State Feedback Control Design 36 bers such that: 2 m j=1 λ j ( ) = 1 (2.26) Remark Every trajectories, of the closed-loop system (2.10) or (2.12), which are confined in the polyhedral set S(H, ū), are a trajectories of the polytopic model (2.25). However the converse is not true, indeed the equality (2.25) holds only for an appropriate choice of the coefficients λ j (x(t)). For this reason the polytopic representation is conservative Sector Conditions for the Uniform Quantizer Similarly to the saturation nonlinearity, the quantizer defined in Definition 4 is not easy to handle. Hence, in order to deal with the Problem 1, a simpler characterization of the uniform quantizer is needed. To this end, we will use the sector conditions provided in [22], which can be resumed in the following Lemma. Lemma 5. [22] The nonlinearity Ψ(v) q(v) v satisfies the following conditions: Ψ (v) T 2 v + 1 l T 2 v 0, v R l (2.27) Ψ (v) T 3 (Ψ(v) + v) 0, v R l (2.28) (Ψ(v) + v) T 4 (Ψ(v) v) 0, v R l (2.29) for any diagonal positive definite matrices T 1, T 2, T 3 R l l Proof. By using the definition of q(v) one can write: v (i) Ψ (i) (v)v (i), i {1,..., l} then equivalently, for every positive constant t i one can write: t i v (i) t i Ψ (i) (v)v (i), i {1,..., l}

56 2.4 Static State Feedback Control Design 37 Therefore, denoting with t i the elements of a generic diagonal positive matrix T 2 R l l, relation (2.27) then follows. By the same way, one obtains the relations (2.28) and (2.29). The i th component of Ψ( ) is illustrated in Figure 2.10 v v Figure 2.10: The i th component of function Ψ( ) Remark Using sector conditions, clearly introduces some conservativeness, since the sector analysis can be applied to every function satisfying the sector conditions. However, this approach leads to more numerical tractable conditions. Since Lemma 5 states conditions on the function Ψ( ), it is more suitable to re-write the polytopic models (2.25) for the closed-loop systems (2.10) and (2.12), highlighting such a function. At this aim, summing and subtracting the term: 2 m j=1 λ j Γ + j Kx at the right hand side of both the equations in (2.25), by virtue of the relation (2.26) and omitting the dependence by x for λ j, one gets respectively: ẋ = ẋ = 2 m j=1 2 m j=1 λ j ( A + B Γ + j K + B Γ j H ) x + λ j ( A + B Γ + j K + B Γ j H ) x + 2 m j=1 2 m j=1 λ j B Γ + j Ψ(Kx) (2.30) λ j B Γ + j KΨ(x) (2.31)

57 2.4 Static State Feedback Control Design 38 Summarizing, the foregoing representation together with Lemma 5, provides a simplified model for the closed-loop systems (2.10) and (2.12), although such a model is valid in a local sense and leads to a conservative analysis. In the next chapter a set of results, aimed at providing a solution to Problem 1, will be presented. Clearly these results will be founded on the polytopic models (2.30) and (2.31) and on the uniform quantizer description stated in Lemma 5.

58 CHAPTER 3 MAIN RESULTS In this chapter, we will show some theoretical results aimed at solve the Problem 1 exposed in the Chapter 2, for both the cases presented. These results represent sufficient conditions. 3.1 State Feedback with Quantized Input Consider the closed-loop system (2.10), the following result provides a solution to Problem 1. Proposition 1. If there exist two symmetric positive definite matrices Q, W R n n, three diagonal positive definite matrices S 1, S 2, S 3 R m m, two matrices Y, J R m n and two positive scalars τ 1, τ 2 such that: Z j < 0 j {1,..., 2 m } (3.1) Q W 0 (3.2) [ ] W J (i) 0 i {1,..., m} (3.3) J (i) ū 2 (i) 39

59 3.1 State Feedback with Quantized Input 40 where: He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q Z j S 2 Γ + j B + S 2 S1 1 Y Y 2S 2 S 2 S3 1 S 2 (τ Y 0 1 τ 2 ) m S2 1 Y 0 0 S 3 (3.4) then K = Y W 1 is solution to Problem 1, and the sets S 0 and S u are defined as follow: S 0 = E(P 1 ) {x R n : x P 1 x 1}, P 1 W 1, ( ) S u = E(P 1, η) {x R n : x P 1 x η 1 }, η λ min P2 P1 1, where P 2 W 1 QW 1. Proof. Consider the following Lyapunov function: V (x) = x P 1 x with P 1 = P 1 > 0. The time-derivative of V (x) along trajectories of (2.30), reads: V (x) = m j=1 2 m j=1 λ j x He [ P 1 ( A + B Γ + j K + B Γ j H)] x+ λ j x P 1 BΓ + j Ψ(Kx) (3.5) by virtue of the S-procedure and relations (2.27), (2.28) and (2.29), it follows that V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) L 1 (3.6)

60 3.1 State Feedback with Quantized Input 41 with: L 1 V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) + 2(Ψ (Kx) S 1 1 Kx+ + 1 m S 1 1 Kx ) 2Ψ (Kx) S 1 2 (Ψ(Kx) + Kx)+ 2(Ψ(Kx) + Kx) S 1 3 (Ψ(Kx) Kx) (3.7) For sake of simplicity it is suitable to alleviate the absolute value Kx. To this aim one can note that each component K (l) x can be written as: K (l) x = ±K (l) x, l {1,..., m}. Hence in order to represent the vector Kx, it is convenient to consider, 2 m diagonal matrices D i having the diagonal elements equal to ±1, in this manner we can claim that: Kx = 2 m i=1 α i D i Kx (3.8) where the coefficients α i are subject to the following constraints: α i 0 2 m i=1 α i = 1 (3.9) Therefore replacing the term Kx in (3.7) with the expression given in (3.8) and noting that: τ 1 τ 2 = 1 m1 m (τ 1 τ 2 ) m one can obtain, developing the right hand side of (3.6), the following inequality: 2 m V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) ξ α i L i ξ (3.10) i=1 where: ξ [x W 1 J HW Y KW Ψ (Kx)S ms 1 1 ]

61 3.1 State Feedback with Quantized Input 42 L i 2 m j=1 λ j He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q + Y S 1 3 Y 2 m j=1 λ j S 2 Γ + j B + S 2 S 1 1 Y Y 2S 2 S 2 S 1 3 S 2 D i Y 0 τ 1 τ 2 m S2 1 (3.11) By definition D id i = D 2 i = I m then since S 1 is a diagonal matrix D i S 2 1D i = S 2 1. Hence pre-and post-multiplying (3.11) by the matrix diag{i n ; I m ; D i }, one gets: 2 m j=1 λ j [ He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q + Y S 1 3 Y 2 m j=1 ] λ j S 2 Γ + j B + S 2 S 1 1 Y Y 2S 2 S 2 S 1 3 S 2 Y 0 τ 1 τ 2 m S2 1 Moreover, since 2 m j=1 λ j = 1, matrix (3.12) can be re-written as: (3.12) where: 2 m j=1 λ j ˆLj

62 3.1 State Feedback with Quantized Input 43 He ( AW + B(Γ + j Y + Γ j J)) + τ 1 W + τ 2 Q + Y S3 1 Y ˆL j S 2 Γ + j B + S 2 S1 1 Y Y 2S 2 S 2 S3 1 S 2 Y 0 τ 1 τ 2 m S2 1 (3.13) Finally, applying the Schur complement to ˆL j one gets the matrix (3.4). Hence the satisfaction of relation (3.1) implies that: L i < 0, i {1,..., 2 m } that is: Therefore: α i L i ξ < 0 ξ 0 ξ 2m i=1 ε 1 ; ε 2 ; ε 3 > 0: L 1 ξ diag{ε 1 I n ; ε 2 I m ; ε 3 I m }ξ and then: ε 1 ; ε 2 : L 1 ε 1 x x ε 2 Ψ (Kx)Ψ(Kx) Hence by the virtue of (3.10), it follows that: V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) ε 1 x x or equivalently, V (x) ε 1 x x, x S 0 \S (3.14) where S = E(P 2 ) {x R n : x P 2 x 1}. Furthermore, relation (3.2) implies that S is contained in S 0 and according to (3.3), the polyhedral

63 3.1 State Feedback with Quantized Input 44 region S(H, ū) = {x R n : H x ū} contains the set S 0. Consequently, by virtue of Lemma 4, x S 0 the relation (3.14) holds even along the solution of the closed-loop system (2.10). However, since S could not be a level set of the function V (x), the Lyapunov like analysis shown in Section 2.2, is not directly applicable. For this, one can determine the smallest level set of V (x) containing S. At this end, it suffices to solve the following optimization problem: max η subject to P 2 ηp 1 0, η 1 (3.15) whose solution provides 1 : η = λ min (P 2 P 1 1 ) Hence according to the procedure shown in Section 2.2 since the set S 0 is compact, it follows that the solutions of (2.10) initialized in S 0 are bounded and ultimately bounded in the set S u = E(P 1, η), concluding the proof. Looking at the proof of Proposition 1, one may argue that if the matrix A is Hurwitz, then a straightforwardly generalization of Proposition 1 can be derived. Such a generalization is provided by the following Corollary. 1 Since the condition (3.2) implies that P 2 > P 1 it follows that λ min (P 2 P 1 1 ) 1.

64 3.1 State Feedback with Quantized Input 45 Corollary 1. If there exist two symmetric positive definite matrices Q, W R n n, three diagonal positive definite matrices S 1, S 2, S 3 R m m, two matrices Y, J R m n and two positive scalars τ 1, τ 2 such that: Z j < 0, j {1,..., 2 m } (3.16) Q W 0, (3.17) [ ] W J (i) 0, i {1,..., m} (3.18) J (i) ū 2 (i) [ ] Q Y (j) 0, j {1,..., m} (3.19) Y (j) 2 AW + W A < 0 (3.20) where: He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q Z j S 2 Γ + j B + S 2 S1 1 Y Y 2S 2 S 2 S3 1 S 2 (τ Y 0 1 τ 2 ) m S2 1 Y 0 0 S 3 Then K = Y W 1 is a stabilizing gain, that is the origin is locally asymptotically stable for the system (2.10) and furthermore every solution of (2.10) initialized in S 0 = E(P 1 ) {x R n : x P 1 x 1}, P 1 W 1 converges to the origin. Proof. Considering the following Lyapunov function: V (x) = x P 1 x

65 3.2 State Feedback with Quantized Measured State 46 and the set S = E(P 2 ) {x R n : x P 2 x 1} where P 2 = W 1 QW 1. By virtue of Proposition 1, if relations (3.16), (3.17), (3.18) hold then S S 0 and: ϕ > 0: x S 0 \S V (x) ϕx x Furthermore relation (3.19) implies that S is contained inside the linear region S( ) of (2.10). Therefore, since in S( ) the time-derivative of V (x) along solutions of (2.10) reads: V (x) = A P 1 + P 1 A By the virtue of relation (3.20), there exists ε > 0 such that: V (x) εx x, x S( ) and then: V (x) min(ϕ, ε)x x, x S 0 Consequently, the origin for the system (2.10) is locally asymptotically stable and furthermore every solution initialized in S 0 converges asymptotically to the origin. 3.2 State Feedback with Quantized Measured State Consider the closed-loop system (2.12), the following result provides a solution to Problem 1. Proposition 2. If there exist two symmetric positive definite matrices Q, W R n n, three diagonal positive definite matrices S 1, S 2, S 3 R n n, two ma-

66 3.2 State Feedback with Quantized Measured State 47 trices Y, J R m n and two positive scalars τ 1, τ 2 such that: where: Z j < 0, j {1,..., 2 m } (3.21) Q W 0, (3.22) [ ] W J (i) 0, i {1,..., m} (3.23) J (i) ū 2 (i) He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q Z j Y Γ + j B + W (S1 1 S2 1 )W 2W ( ) S2 1 + S3 1 W τ W 0 1 τ 2 S2 2 n W 0 0 S 3 (3.24) then K = Y W 1 is solution to Problem 1, and the sets S 0 and S u are defined as follow: S 0 = E(P 1 ) {x R n : x P 1 x 1}, P 1 W 1, ( ) S u = E(P 1, η) {x R n : x P 1 x η 1 }, η λ min P2 P1 1, where P2 W 1 QW 1. Proof. Consider the following Lyapunov function: V (x) = x P 1 x with P 1 = P 1 > 0. The time-derivative of V (x) along trajectories of (2.31),

67 3.2 State Feedback with Quantized Measured State 48 reads: V (x) = m j=1 2 m j=1 λ j x He [ P 1 ( A + B Γ + j K + B Γ j H)] x λ j x P 1 BΓ + j KΨ(x) (3.25) by virtue of the S-procedure and relations (2.27), (2.28) and (2.29), it follows that with: V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) L 1 (3.26) L 1 V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) + 2(Ψ (x)s 1 1 x+ + 1 n S 1 1 x ) 2Ψ (x) S 1 2 (Ψ(x) + x)+ 2(Ψ(x) + x) S 1 3 (Ψ(x) x) (3.27) For sake of simplicity it is suitable to alleviate the absolute value x. For this aim one can note that each component z (l) can be written as: x (l) = ±x (l), l {1,..., n}. Hence in order to represent the vector x, it is convenient to consider, 2 n diagonal matrices D i having the diagonal elements equal to ±1, in this manner we can claim that: x = 2 n i=1 α i D i x (3.28) The coefficients α i are then subject to the following constraints: α i 0 2 n i=1 α i = 1 (3.29) Therefore replacing the term x in (3.27) with the expression given in (3.28) and noting that: τ 1 τ 2 = 1 n1 n (τ 1 τ 2 ) n

68 3.2 State Feedback with Quantized Measured State 49 one can obtain, developing the right hand side of (3.26), the following inequality: V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) ξ 2 n i=1 α i L i ξ (3.30) where: L i 2 m j=1 ξ [x W 1 Ψ (x)w 1 1 ns 1 1 ] J HW Y KW λ j [ He ( AW + B(Γ + j Y + Γ j J)) ] τ 1 W + τ 2 Q 2 m j=1 λ j Y Γ + j B + W (S1 1 S2 1 )W 2W ( ) S2 1 + S3 1 W D i W 0 τ 1 τ 2 n S2 2 (3.31) By definition D id i = D 2 i = I n then since S 1 is a diagonal matrix D i S 2 1D i = S 2 1. Hence pre- and-post multiplying (3.31) by the matrix diag{i n ; I n ; D i },

69 3.2 State Feedback with Quantized Measured State 50 one gets: 2 m λ j [ He ( AW + B(Γ + j Y + Γ j J)) j=1 ] τ 1 W + τ 2 Q 2 m j=1 λ j Y Γ + j B + W (S1 1 S2 1 )W 2W ( ) S2 1 + S3 1 W W 0 τ 1 τ 2 n S2 2 (3.32) Moreover, since 2 m j=1 λ j = 1, matrix (3.32) can be re-written as: where: 2 m j=1 λ j ˆLj He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q ˆL j Y Γ + j B + W (S1 1 S2 1 )W 2W ( ) S2 1 + S3 1 W τ W 0 1 τ 2 n S2 2 (3.33)

70 3.2 State Feedback with Quantized Measured State 51 Finally, applying the Schur complement to ˆL j one gets the matrix (3.24). Hence the satisfaction of relation (3.21) implies that: L i < 0 i {1,..., 2 n } that is: Therefore: 2n ξ i=1 α i L i ξ < 0 ξ 0 ε 1 ; ε 2 ; ε 3 > 0: L 1 ξ diag{ε 1 I n ; ε 2 I n ; ε 3 I n }ξ and then: ε 1 ; ε 2 : L 1 ε 1 x x ε 2 Ψ (x)ψ(x) Hence by the virtue of (3.30), it follows that: V (x) τ 1 (x P 1 x 1) τ 2 (1 x P 2 x) ε 1 x x or equivalently, V (x) ε 1 x x x S 0 \S (3.34) where S = E(P 2 ) {x R n : x P 2 x 1}. Furthermore, relation (3.22) implies that S is contained in S 0 and according to (3.23), the polyhedral region S(H, ū) = {x R n : H x ū} contains the set S 0. Consequently, by virtue of Lemma 4, x S 0 the relation (3.34) holds even along the solution of the closed-loop system (2.12). However, since S could not be a level set of the function V (x), the Lyapunov like analysis shown in Section 2.2, is not directly applicable. For this, it one can determine the smallest level set of V (x) containing S. At this end, it suffices to solve the

71 3.2 State Feedback with Quantized Measured State 52 following optimization problem: max η subject to P 2 ηp 1 0, η 1 (3.35) whose solution provides: η = λ min (P 2 P 1 1 ) Hence according to the procedure shown in Section 2.2 since the set S 0 is compact, it follows that the solutions of (2.12) initialized in S 0 are bounded and ultimately bounded in the set S u = E(P 1, η), concluding the proof. Even in this case, if A is Hurwitz, an extension of the Proposition 2 can be derived. Such an extension is provided by the following Corollary. Corollary 2. If there exist two symmetric positive definite matrices Q, W R n n, three diagonal positive definite matrices S 1, S 2, S 3 R n n, two matrices Y, J R m n and two positive scalars τ 1, τ 2 such that: Z j < 0, j {1,..., 2 m } (3.36) Q W 0 (3.37) [ ] W J (i) 0, i {1,..., m} (3.38) [ J (i) Q ū 2 (i) W (j) W (j) 2 ] 0, j {1,..., m} (3.39) AW + W A < 0 (3.40)

72 3.2 State Feedback with Quantized Measured State 53 where: He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q Z j Y Γ + j B + W (S1 1 S2 1 )W 2W ( ) S2 1 + S3 1 W τ W 0 1 τ 2 S2 2 n W 0 0 S 3 then K = Y W 1 is a stabilizing gain, that is the origin is asymptotically stable for the system (2.12) and furthermore every solution of (2.12) initialized in converges to the origin. S 0 = E(P 1 ) {x R n : x P 1 x 1}, P 1 W 1 Proof. Considering the following Lyapunov function: V (x) = x P 1 x and the set S = E(P 2 ) {x R n : x P 2 x 1} where P 2 = W 1 QW 1. By virtue of Proposition 2, if relations (3.36),(3.37),(3.38) hold then S S 0 and: ϕ > 0: x S 0 \S V (x) ϕx x Furthermore relation (3.39) implies that S is contained inside the linear region S( ) of (2.12). Therefore, since in S( ) the time-derivative of V (x)

73 3.2 State Feedback with Quantized Measured State 54 along solutions of (2.12) reads: V (x) = A P 1 + P 1 A By the virtue of relation (3.40), there exists ε > 0 such that: V (x) εx x x S( ) and then: V (x) min(ϕ, ε)x x x S 0 Consequently, the origin for the system (2.12) is locally asymptotically stable and furthermore every solution initialized in S 0 converges asymptotically to the origin.

74 CHAPTER 4 OPTIMIZATION ISSUES AND NUMERICAL RESULTS In this chapter, we will show how the conditions stated in the Chapter 3 can be effectively used in order to solve numerically Problem 1. To this end, an optimization procedure will be presented developing a suitable synthesis procedure. Some further constraints will be added to the constraints provided by Proposition 1 and Proposition 2, in order to obtain, in both cases, an LMI optimization problem. 4.1 Optimization Based Controller Synthesis Conditions provided by Proposition 1 and Proposition 2 are nonlinear in the variables W, Q, τ 1, τ 2, Y, S 1, S 2, S 3 defined in Proposition 1 and Proposition 2. On the other hand, looking at the Problem 1, it is clear that S 0 should be as large as possible, whereas S u should be as small as possible. Therefore, in order to get some numerical outcomes, the conditions provided by Proposition 1 and Proposition 2 can be involved in an optimization pro- 55

75 4.2 Size criteria for ellipsoidal sets 56 cedure aimed at provide the best solution to Problem 1. In other words the free variables W, Q, τ 1, τ 2, Y, S 1, S 2, S 3 introduced in Proposition 1 and Proposition 2 can be considered like decision variables of a suitable optimization problem. Such a problem is subject to the constraints stated in Proposition 1 and Proposition 2, which define a feasible set F for the related optimization problem. Clearly, in order to measure the quality of a given solution, an appropriate cost function Υ: F R must be defined. For this aim, several optimality criteria can be considered, depending on the size criterion chosen for measuring the sets S 0 and S u. Concerning this aspect, since in the proposed approach the considered sets are ellipsoids, many size criteria can be adopted. For example, as shown in [2], volume, minor axis and direction of interests could be a good measure of an ellipsoidal set, depending on the objective at the hand. Furthermore, one can reasonably expect that, regardless the size criteria chosen, since S u is the smallest level-set of V (x) = x P 1 x containing S, minimization of the set S u can be performed indirectly through the minimization of S. Remark Looking at the Proposition 1 and Proposition 2 proofs, it should be noticed that the set S actually could be removed, that is considering P 2 = ηp 1, η > 1. Hence the optimization problem could be designed in order to maximize η. However, in this way the optimization problem would be subject to a further constraint. Therefore, it could be harder finding a solution. Roughly speaking, the set S represents an additional degree of freedom which can be used in order to relax the optimization procedure. 4.2 Size criteria for ellipsoidal sets Size minimization (maximization) of an ellipsoidal set, as already mentioned, can be performed adopting different notions of measure. For example in [2]

76 4.2 Size criteria for ellipsoidal sets 57 the volume has been assumed as measure of an ellipsoidal set. Especially, consider a general ellipsoidal set E(P ) = {x R n : x P x 1} (4.1) where P = P > 0. Adopting a volume criterion leading, in volume maximization case, to the following optimization problem: min log det P 1 subject to P > 0 Such a problem is convex [2] and then it can be efficiently solved by numerical procedures. On the other hand, it should be noticed that sometimes adopting the foregoing volume criterion, it can lead to mediocre results. For instance, regarding to the set S 0, the desired task is clearly finding a set S 0, which covers as well as possible whole R n. However, as shown in [21], using a volume criterion can lead to an ellipsoidal set stretched in some directions, hence not really maximized as desired. In order to overcome this problem, in [21] two different optimality criteria has been considered. Trace minimization/maximization Maximization/minimization along specified directions. With respect to ellipsoidal set (4.1), first criterion consists in minimizing/maximizing trace(p 1 ). Basically, this approach is founded on the fact that λ i (P 1 ), for i = 1,..., n, are the semi-axes of E(P ), then since P > 0 and n trace(p 1 ) = λ i (P 1 ) minimizing/maximizing trace(p 1 ) amounts to minimize/maximize in one shot every λ i (P 1 ), obtaining a more homogeneous contraction/dilatation of E(P ). Concerning the second method, it consists in minimizing/maximizing i=1

77 4.2 Size criteria for ellipsoidal sets 58 the value of x P x in some specified directions: v i R n, i = 1,..., l For instance in minimization case, the foregoing criterion leads to the following optimization problem: min β subject to v ip v i β, β > 0, i = 1,..., l Analogous consideration holds for maximization case. However, we are interested in maximizing S 0 = E(P 1 ) and minimizing S = E(P 2 ) simultaneously. To this end, choosing a trace based criterion, one can minimize the cost function defined as: Υ = trace(p 1 ) + trace(p 1 2 ) (4.2) Indeed, since both P 1 and P 2 are positive definite, minimizing (4.2) provides an enlargement of S 0 and a contraction of S. On the other hand, the conditions stated in Proposition 1 and Proposition 2 are formulated on the variables Q and W which are related to P 1 and P 2 through: W = P 1 1 (4.3) Q = W P 2 W (4.4) Therefore, one should write down an optimization problem, which having as decision variables Q and W. For this aim, let us consider the following

78 4.2 Size criteria for ellipsoidal sets 59 optimization problem: O 1 : min trace(h 1 + H 2 ) (4.5) Q W 0 0 subject to W H W I n 0 (4.6) 0 0 I n H 1 where H 1, H 2 are two matrices such that: H 1 = H 1 > 0, H 2 = H 2 > 0 Solving such a problem is equivalent to minimize the function (4.2). In fact, by Schur complement, relation (4.6) implies: Hence, by relations (4.3) and (4.4), it follows that H 2 W Q 1 W (4.7) H 1 W 1 H 2 P 1 2 H 1 P 1 (4.8) By the positive definitiveness of P 1 and P 2 and by relation (4.8) one gets: trace(h 1 + H 2 ) trace(p 1 + P 1 2 ) 0 Therefore, the minimization of trace(h 1 + H 2 ) implies the minimization trace(p 1 + P 1 2 ). Clearly, the foregoing method can be extended in order to use a mixed criterion, that is a criterion based both on trace minimization criterion and on maximization along a specified direction criterion. For this

79 4.2 Size criteria for ellipsoidal sets 60 puropose, it suffices to consider the following optimization problem: O 2 : min trace(h 2 ) + µ (4.9) µ v i 0 0 subject to v i W Q W 0, i = 1,..., l; (4.10) 0 0 W H 2 µ > 0 (4.11) Indeed, by Schur complement and by (4.3) and (4.4), it follows that relation (4.10) implies: H 2 P 1 2 µ v ip 1 v i 0, i = 1..., l (4.12) Moreover, since H 2 > 0 and µ > 0 it follows that minimizing (4.9) is equivalent to minimize both trace(h 2 ) and µ. Hence by the virtue of the positive definiteness of P 1 and by relation (4.12), it follows that µ v ip 1 v i > 0, i = 1,..., l Thus minimizing µ is equivalent to minimize v ip 1 v i i = 1,..., l and the maximization of v ip 1 1 v i, i = 1,..., l follows. In order to solve Problem 1 optimizing the size of S 0 and S u both optimization problems O 1 and O 2, augmented with the conditions stated in Proposition 1 and Proposition 2, can be considered. At this stage, it is important to note that despite problems O 1 and O 2 are LMI optimization problems, since conditions (3.1) and (3.21) are nonlinear in the decision variables, augmenting problems O 1 and O 2 with such conditions does not

80 4.3 LMI Formulation 61 allows us to solve the augmented problem via convex optimization tools. Indeed, adding nonlinear constraints may lead to a non-convex feasible set F. Therefore, in order to obtain a numerical tractable problem, although adding more conservatism, some assumptions on the decisions variable are necessary. 4.3 LMI Formulation In order to obtain from conditions (3.1) and (3.21) an LMI optimization problem, we will consider further constraints on decision variables. As follow, we will show such constraints and the arising optimization problems, omitting for sake of exposition the relatively details, which are shown in Appendix A. State feedback with quantized input Concerning condition (3.1) in Proposition 1, let τ 3, τ 4 be two fixed real positive scalars, considering the following constraints: S 1 = S 2 (4.13) S 3 = 1 τ 4 I m (4.14) S 2 τ 3 I m (4.15)

81 4.3 LMI Formulation 62 hence condition (3.1) can be replaced by He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q S 1 Γ + j B (2 + τ 3 τ 4 )S 1 < 0 (τ Y 0 1 τ 2 ) τ m 3S 1 τ 4 Y 0 0 τ 4 I m for j = 1,..., 2 m (4.16) Therefore, Problem 1, in state feedback with quantized input case, depending on the optimization criteria chosen, leads to the following LMI optimization problems:

82 4.3 LMI Formulation 63 Optimization problem M 1 min trace(h 2 ) + µ subject to (3.2); (3.3); µ > 0; µ v i 0 0 v i W Q W 0, i = 1,..., l; 0 0 W H 2 He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q S 1 Γ + j B (2 + τ 3 τ 4 )S 1 < 0 (τ Y 0 1 τ 2 ) τ m 3S 1 τ 4 Y 0 0 τ 4 I m for j = 1,..., 2 m

83 4.3 LMI Formulation 64 Optimization problem M 2 min trace(h 1 + H 2 ) subject to (3.2); (3.3); Q W 0 0 W H W I 0, i = 1,..., l; n 0 0 I n H 1 He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q S 1 Γ + j B (2 + τ 3 τ 4 )S 1 < 0 (τ Y 0 1 τ 2 ) τ m 3S 1 τ 4 Y 0 0 τ 4 I m for j = 1,..., 2 m Remark Via Corollary 1, the asymptotic stabilization problem can be solved. In this case the real important task is to obtain a set S 0 as large as possible. Hence, the optimization scheme involving Corollary 1 could be modified aiming at optimize only the size of S 0. However, since the linear region S( ) containing E(P 2 ), allowing E(P 2 ) to be arbitrary large can leads to a small gain K. Indeed, when K approaches to zero the linear region S( ) covers whole the state space, then E(P 2 ) would be contained in S( ) for every P 2. For this, also in the asymptotic stability case, considering problem M 1 and M 2 is suitable.

84 4.3 LMI Formulation 65 State feedback with quantized measured state Concerning condition (3.21) in Proposition 2, let δ, τ 3, τ 4 be three fixed real positive scalars, considering the following constraints: hence condition (3.21) can be replaced by S 1 = S 2 (4.17) S 2 = 1 τ 3 I n (4.18) S 3 = 1 I n τ [ 4 ] (4.19) δi n I n W 0 (4.20) I n

85 4.3 LMI Formulation 66 He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q δy Γ + j B 2(τ 3 + τ 4 )W δ < 0 τ τ 3 W 0 1 τ 2 I n n τ 4 W 0 0 τ 4 I n j = 1,..., 2 m (4.21) Therefore, Problem 1, even in state feedback with quantized measured state case, depending on the optimization criteria chosen, to solve Problem 1, one

86 4.3 LMI Formulation 67 can consider the following LMI optimization problems: Optimization problem N 1 min trace(h 2 ) + µ subject to (3.2); (3.3); µ > 0; µ v i 0 0 v i W Q W 0, i = 1,..., l; 0 0 W H 2 He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q δy Γ + j B 2(τ 3 + τ 4 )W δ < 0 τ τ 3 W 0 1 τ 2 I n n τ 4 W 0 0 τ 4 I n for j = 1,..., 2 m

87 4.3 LMI Formulation 68 Optimization problem N 2 min trace(h 1 + H 2 ) subject to (3.2); (3.3); Q W 0 0 W H W I 0, i = 1,..., l; n 0 0 I n H 1 He ( AW + B(Γ + j Y + Γ j J)) + τ 1 W + τ 2 Q δy Γ + j B 2(τ 3 + τ 4 )W δ < 0 τ τ 3 W 0 1 τ 2 I n n τ 4 W 0 0 τ 4 I n for j = 1,..., 2 m Remark In the exposed methodology, some fixed scalars have been introduced. Such scalars can be used as tuning parameters in the considered optimization problems. However, fixing these scalars, there is no guarantee that the related problem remains feasible. Hence, getting an LMI optimization problem, via the above procedure, not only introduces additional conservatism but can also rend the synthesis procedure tricky, since the parameters tuning must be done in order to get a feasible problem. Also in this case, analogous considerations hold regarding to optimiza-

88 4.4 Numerical Examples 69 tion problem N 1 and N Numerical Examples In this section, some numerical examples will be shown Quantized Input Case Example Consider the system described by: [ ] [ ] ẋ = x + sat(v) 1 = u 0 = (4.22) Solving the related optimization problem M 2 leads to K = [ , ]. Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) obtained are shown in Figure 4.1. Furthermore, the circles depicted in Figure 4.1 are two unstable equilibrium points due to the saturation nonlinearity. Regarding optimization problem M 1, solving that problem in this case, leads to K = [ , ]. In Figure 4.2 sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) obtained are shown. Even in this case the circles represent two unstable equilibrium points due to the saturation nonlinearity.

89 4.4 Numerical Examples x x 1 Figure 4.1: Sets S 0 ( ), S u (- - -) S(H, ū) (-.-) obtained solving optimization problem M 2 and closed-loop equilibrium points ( ). Example 4.4.1

90 4.4 Numerical Examples x x 1 Figure 4.2: Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) obtained solving optimization problem M 1 and closed-loop equilibrium points ( ). Example 4.4.1

91 4.4 Numerical Examples 72 Example Consider the system described by: [ ] [ ] ẋ = x + sat(v) = 0.25 u 0 = 1 (4.23) In Figure 4.3 the sets S 0 ( ), S u (- - -), S(H, ū) (-.-) and the two closed-loop equilibrium points ( ) obtained solving the related optimization problem M 1, are shown. Solving instead optimization problem M 2, in this case leads x x 1 Figure 4.3: Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) and closed-loop equilibrium points ( ) obtained solving optimization problem M 1. Example to the sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) depicted in Figure 4.4. In

92 4.4 Numerical Examples 73 the same figure also the unstable closed-loop equilibrium points ( ) due the saturation are shown. To realize the conservatism of the proposed method, x x 1 Figure 4.4: Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) obtained solving problem M 2 and closed-loop equilibrium points ( ). Example in Figure 4.5 several closed-loop trajectories are shown.

93 4.4 Numerical Examples x x 1 Figure 4.5: Several closed-loop trajectories are shown. Trajectory obtained via Runge-Kutta method with sample time 10 4, the crosses represents the initial states. Example 4.4.2

94 4.4 Numerical Examples 75 Example To illustrate the asymptotic stabilization problem, consider the system described by: [ ] [ ] ẋ = x + sat(v) (4.24) u 0 = 2 In that case, we consider = 1. Therefore the controller uses only five values, i.e. u { 2, 1, 0, 1, 2}. Using Corollary 1 involved in optimization problem M 2, one gets: K = [ , ]. In Figure 4.6 set S 0 and S(H, ū) are shown. To showing the conservatism of the proposed method, x x 1 Figure 4.6: Sets S 0 ( ), and S(H, ū) (-.-). Example in Figure 4.7 several closed-loop trajectory are shown.

95 4.4 Numerical Examples x x 1 Figure 4.7: Sets S 0 ( ), S(H, ū) (-.-) and several trajectories, crosses represent the initial conditions. Trajectories obtained via Runge-Kutta method with sample time Example 4.4.3

96 4.4 Numerical Examples 77 Example To illustrate the limit cycles phenomenon in planar case, consider the system described by: [ ] [ ] ẋ = x + sat(v) u 0 = 5 (4.25) In that case, we consider = 1. Solving the related optimization M 1, one gets: K = [ , ]. In Figure 4.8 set S 0 and S(H, ū) are shown. According to Section 2.3, a limit cycle arises in the closed-loop x x 1 Figure 4.8: Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-). Example system (Figure 4.9).

97 4.4 Numerical Examples x x 1 (a) Limit cycle arises in the closed-loop system x x 1 (b) Zooming on the limit cycle Figure 4.9: Closed-Loop trajectory starting in ( ) converges to a limit cycle. Trajectory obtained via Runge-Kutta method with sample time Example 4.4.4

98 4.4 Numerical Examples Quantized Measured State Case Example To illustrate quantized measured state case, consider the system described by: [ ] [ ] ẋ = x + sat(v) u 0 = 2 In that case, we consider = (4.26) Solving the related optimization Proposition 2 involved in the optimization problem N 2, one gets: K = [ , ]. In Figure 4.10 set S 0 and S(H, ū) are shown x x 1 Figure 4.10: Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-). Example 4.4.5

99 CONCLUSIONS AND OUTLOOKS 4.5 Summary In this thesis, some conditions to assess the boundedness/ultimate boundedness property, for linear saturated systems subject to a uniform quantizer, have been developed via a Lyapunov like approach. Especially, we have faced with the quantization effect, in a static state feedback control scheme. Depending on the variable subject to quantization, two different cases have been analyzed. In the first case, we have assumed the control action subject to quantization, i.e. u = q(kx), and in the second one we have considered the measured state subject to quantization, i.e. u = K q(x). In both cases, in order to cope with the saturation nonlinearity, a polytopic model of the saturation system has been used, whereas sector conditions have been used to get a simpler characterization of the quantizer. Through the polytopic representation and the sector conditions, we have obtained two simplified models of the closed-loop systems, with respect to the analyzed cases. Afterwards, we have formulated a specific problem, Problem 1, which has been formulated looking at a synthesis approach. Especially solving this problem provides: stabilizing gain K and sets S 0 and S u such that x 0 S 0 every closed-loop trajectory is bounded and ultimately bounded in S u. To solve 80

100 this problem via a Lyapunov like analysis, quadratic Lyapunov functions have been assumed. Following this perspective, a set of quasi-lmi sufficient conditions has been provided. Furthermore, we have also highlighted that if matrix A is Hurwitz then local asymptotic stability can be proved. Such a highlighting was been done providing two sufficient conditions of local asymptotic stability also in a quasi-lmi context. In order to obtain, a set S 0 as large as possible and a set S u as small as possible the optimization problems M 1, M 2, N 1 and N 2 have been provided. However, in order to obtain numerical tractable conditions, further constraints have been added to the proposed optimization problems, providing an LMI optimization problem, although adding more conservatism to the problem. 4.6 Critical Aspects The proposed conditions provide a valuable alternative to such ones provided in [22], providing in some numerical case, the same results. However, also the proposed approach are affected by the conservatism reported by S.Tarbouriech and F.Gouaisabaut in [22]. Therefore, since representing in different ways the saturated closed-loop systems leads to comparable results, one realizes that the majority of conservatism arises for different reasons. Among which, we can include the sector conditions used for describing the uniform quantizer but mainly the conservatism introduced in considering only quadratic Lyapunov functions and in forcing the obtained conditions to be LMI conditions. On the other hand, dealing with LMI conditions provides undoubted advantages from a computational viewpoint. Another critical aspects is represented by the procedure to fix the tuning parameters necessary to solve the optimization problems M 1, M 2, N 1 and N 2. Indeed, sometimes fixing a set of parameters not affecting the feasibility of the related LMI optimization problem is tricky. 81

101 4.7 Outlooks In the present thesis, we have dealt with state feedback controllers. Therefore, in the analyzed cases, we have assumed to measure the system state, or at least we have assumed to have a partial state knowledge through quantized measures. However in many cases, knowing the state could be impossible or very expensive. In such cases, only some variables are measurable. These variables are generally called outputs and indicated by y. Such variables are usually nonlinear functions of the state, that is y = h(x) where y R p and h: R n R p. On the other hand, considering linear systems, it is logical to consider linear output, i.e. y = Cx where C R p n. For this reason, in order to analyze the effect of the quantization in output feedback control, one can extend the approach presented in this thesis, considering the class of saturated linear systems with linear output, that is: ẋ = Ax + B sat(u) y = Cx Especially, with respect to this case, two control schemes can be considered: Static Output Feedback, i.e. u = Ky 82

102 Dynamic Output Feedback, i.e.: ẋ c = A c x c + B c y u = C c x c + D c y where K, A c, B c, C c, D c are some opportune matrices. In such cases, a quantizer can affect the control variable u leading to two different quantized closed-loop systems, depending on the considered controller: Static Output Feedback Closed-Loop System ẋ = Ax + B sat(q(kcx)) Dynamic Output Feedback Closed-Loop System ẋ = Ax + B sat(q(c c x x + D c Cx)) ẋ c = A c x c + B c Cx In the static output feedback case, with a slightly modification, analogous consideration can be dealt, following the same steps discussed in the state feedback case with quantized input. With respect the second case, an observer based technique can be adopted, for instance via the Luenberger observer, leading to the following closed-loop system ẋ = Ax + B sat(q(k ˆx)) ˆx = Aˆx + B sat(q(k ˆx)) + LC(x ˆx) which can be handily analyzed via the standard change of variable: e x ˆx that provides: ẋ = Ax + B sat(q(k(x e))) ė = (A LC)e Therefore, in this case, analogous consideration can be dealt following the same steps discussed in the state feedback case with quantized input. On 83

103 the other hand, it should be noted that obtain LMI conditions in these cases may be complicated. Therefore this aspect has to be worked out, in order to obtain a synthesis numerically tractable technique. However, one may note that in dynamic output feedback case, using an observer adds an additional state variable to the closed-loop system. Such a variable is not influenced by the quantizer, hence involving also the state e in a boundedness analysis could lead to very conservative outcome. 84

104 APPENDIX A FURTHER CLARIFICATIONS ON THE LMI FORMULATION In this Appendix, we will provide the manipulations necessary to obtaining the LMI formulation of the problems M 1, M 2, N 1 and N 2 shown in Chapter 4. 85

105 A.1 Quantized Input Case In this case, using relations (4.13) and (4.14), one gets from the matrix on left-hand side of (3.4) He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q S 2 Γ + j B S 2 (2 + τ 4 S 2 ) < 0 (τ Y 0 1 τ 2 ) m S2 1 Y 0 0 S 3 (A.1) By relation (4.15) and assuming τ 1 τ 2 < 0 1, one can claim that: S 2 (2 + τ 4 S 2 ) S 2 (2 + τ 4 τ 3 ) Therefore, if (τ 1 τ 2 ) m S2 1 (τ 1 τ 2 ) m S 1τ 3 He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q S 1 Γ + j B (2 + τ 3 τ 4 )S 1 < 0 (τ Y 0 1 τ 2 ) τ m 3S 1 τ 4 Y 0 0 τ 4 I m j = 1,..., 2 m 1 Such an assumption is not restrictive, since if τ 1 τ 2 0 then relation (3.4) is not feasible 86

106 one gets the matrix in the left-hand side of (4.16). A.2 Quantized Measured State Case In this case, using relations (4.17), (4.18) and (4.19), from the matrix on left-hand side of (3.24) He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q Y Γ + j B 2W 2 (τ 2 + τ 3 ) < 0 τ 1 τ 2 W 0 I nτ3 2 n W 0 0 S 3 (A.2) By relation (4.20), it follows that W δ 1 I n. Hence, if He ( AW + B(Γ + j Y + Γ j J)) τ 1 W + τ 2 Q Y Γ + j B 2W δ 1 (τ 2 + τ 3 ) < 0 τ 1 τ 2 W 0 I nτ3 2 n W I n τ 4 (A.3) 87

107 then (A.2) holds. Finally multiplying both sides of (A.3) by diag{i n ; δi n ; I n ; τ 4 I n } one gets the matrix in the left-hand side of (4.21). 88

108 APPENDIX B MATHEMATICAL REVIEW Symmetric Matrices Properties Definition 6. A square matrix A is symmetric if A = A Symmetric matrices have notable properties, which are explained as follow. Theorem 1. Every symmetric matrix A has only real eigenvalues. Furthermore M : M 1 = M such that MAM is diagonal and having the eigenvalues of A as diagonal elements. Definition 7. A symmetric matrix A R n n is said to be: Positive definite, A > 0, if x Ax > 0, x R n : x 0; Positive semidefinite, A 0, if x Ax 0, x R n : x 0; Negative definite, A < 0, if A > 0; 89

109 Seminegative definite, A 0, if A 0 For symmetric positive semidefinite matrices, the following Lemma holds. Lemma 6 (Bellman 1970). Let A be a symmetric positive semidefinite matrix, there exists a matrix Q such that: A = Q 2 Q is called the square root of A, and is denoted by the symbol A 1/2. Furthermore, if A > 0 then A 1/2 > 0. Proposition 3. [4] A symmetric A R n n is positive definite if and only if λ i (A) > 0; i = 1,... n Proposition 4. [4] Let A be a partitioned with square diagonal blocks as A A 1n A =..... A n1... A nn then A > 0 implies A 11 > 0,..., A nn > 0 Lemma 7. Let A be a symmetric positive semidefinite matrix and let B be a nonsingular square matrix with adequate dimension. Then: BAB 0 Furthermore if A > 0 the foregoing inequality holds strictly. Dual relations hold if A is semidefinite/definite negative. 90

110 B.1 Convex sets and Functions Definition 8. [4] The set S in the vector space X is convex if λx 1 + (1 λ)x 2 S, x 1, x 2 S, λ (0, 1) Definition 9. A function F defined on S is convex if the domain S is convex and if F (λx 1 + (1 λ)x 2 ) λf (x 1 ) + (1 λ)f (x 2 ), x 1, x 2 S, λ (0, 1) B.2 Convex Combination and Convex Hull Definition 10. [4] A vector x X is convex combination of x 1,..., x l X if x = with λ i 0, l λ i x i i=1 l λ i = 1 i=1 Definition 11. [4] The convex hull Co(S) of any subset S X is the set of all convex combinations of points in S B.3 Linear Matrix Inequalities In stability analysis LMIs play an important role, for this reason it is convenient to spend some words about them. Let x be a vector belonging to R n defined as x = [x 1,..., x n ], we define a 91

111 system of m LMIs as: A x 1 A x n A 1 n 0. (B.1) A m 0 + x 1 A m x n A m n 0 where A i 0, A i 1,..., A i n, i = 1,..., m are real symmetric given matrices. The set F, such that x F satisfies system (B.1), is called Feasible set. In LMI context two problems can be considered: LMI feasibility problem: Test if there exists x = [x 1,..., x n ] that satisfies the given LMI system. LMI optimization problem: Minimize c 1 x 1 + +c n x n over all x = [x 1,..., x n ] that satisfy the given LMI system. Generally, a problem is said to be Feasible if F. Remark B.3.1. If should be noticed that the feasible set F for a generic LMI optimization problem is always a convex set. For this LMIs optimization problems assume an important role in optimization theory. An important result is now stated. Such a result is called Schur-Complement Lemma and it is very useful for dealing with LMIs. Lemma 8 (Schur Complement). Define X = [ Q S ] S R The following matrix inequality holds X > 0 if and only if: R > 0 Q S R 1 S > 0 (B.2) (B.3) 92

112 X > 0 if and only if: Q > 0 R S Q 1 S > 0 (B.4) (B.5) X 0 if and only if: R > 0 Q S R 1 S 0 (B.6) (B.7) X 0 if and only if: Q > 0 R S Q 1 S 0 (B.8) (B.9) The Schur complement can be used in order to linearize nonlinear matrix inequality. Let us illustrate it as follow: Example B.3.1. Consider the following matrix constraints: trace ( S(x) P 1 (x) S(x) ) < 1, P (x) > 0 (B.10) Introducing a slack matrix X such that: trace(x) < 1 93

113 By virtue of the linearity for trace operator, inequality (B.10) holds if: trace ( X S(x) P 1 (x)s(x) ) > 0 (B.11) therefore, if (X S(x) P 1 (x) S(x)) > 0 then (B.11) holds. Hence adding the positivity definite constraint on P, by the Schur Complement, relation (B.10) is equivalent to the following LMI: [ ] X S (x) > 0 Tr(X) < 1 (B.12) S(x) P (x) B.4 S Procedure The S procedure is a very powerful tool when dealing with LMIs in control system theory. Basically such a procedure is illustrated by the following Lemma. Lemma 9 (S procedure [2]). Let T 0,..., T p R n n be symmetric matrices. If there exist p non negative constants τ 1,..., τ p such that: T 0 p τ i T i > 0 i=1 (B.13) then ξ T 0 ξ > 0, ξ 0 : ξ T i ξ 0 i {1,..., p} Remark B.4.1. It should be noticed that (B.13) is an LMI in the variables τ 1,..., τ p. 94

114 B.5 Ellipsoidal set contained in a symmetric polyhedron Many times, simply conditions implying that a given polyhedron contains a given ellipsoid are necessary. To this end, Schur complements can be used providing a LMI constraint. This procedure is shown as follow. Let E(P ) be an ellipsoidal set defined as follow: E(P ) = {x R n : x P x 1} where P = P > 0 and let S( S, ρ) be a symmetric polyhedron defined as follow: S( S, ρ) = {x R n : Sx ρ} where ρ R m +. Imposing that E(P ) S( S, ρ) means: x R n : x P x 1 = Sx ρ (B.14) Hence, denoting with S (i), i = 1,... m, the i th row of S and with ρ (i), i = 1,..., m, the i th component of ρ, relation (B.14) is equivalent to x R n : x P x 1 = x S (i) S (i)x ρ 2 (i) 1, i = 1,..., m (B.15) Then E(P ) S( S, ρ) if: Relation (B.16) holds if x P x x S (i) S (i)x, i = 1,..., m (B.16) ρ 2 (i) P S (i) S (i) ρ 2 (i) 0, i = 1,..., m 95

115 Furthermore, by Schur complement, the foregoing relation is equivalent to: [ ] P S (i) 0, i = 1,..., m S (i) ρ 2 (i) 96

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119 LIST OF FIGURES 1 Quantization in control system VIII 1.1 Graphical description of quantizer defined by D. Liberzon The sets B 1,B 2,R 1,R The i th component of the uniform quantizer The i th component of the saturation function Open loop system State feedback controller with quantized input State feedback controller with quantized measured state A closed-loop trajectory ( ), initialized in ( ), enters in the Linear Region (- - -) The sets Ω c, Ω ε, Λ Example Limit cycle in state feedback with quantized input, closed-loop system. The initial condition is marked by a cross Example Two different amplitude limit cycles, in state feedback with quantized input closed-loop system, considering: = 1 ( ), = 0.5 (- - -).The initial condition is marked by a cross

120 2.10 The i th component of function Ψ( ) Sets S 0 ( ), S u (- - -) S(H, ū) (-.-) obtained solving optimization problem M 2 and closed-loop equilibrium points ( ). Example Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) obtained solving optimization problem M 1 and closed-loop equilibrium points ( ). Example Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) and closed-loop equilibrium points ( ) obtained solving optimization problem M 1. Example Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-) obtained solving problem M 2 and closed-loop equilibrium points ( ). Example Several closed-loop trajectories are shown. Trajectory obtained via Runge-Kutta method with sample time 10 4, the crosses represents the initial states. Example Sets S 0 ( ), and S(H, ū) (-.-). Example Sets S 0 ( ), S(H, ū) (-.-) and several trajectories, crosses represent the initial conditions. Trajectories obtained via Runge- Kutta method with sample time Example Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-). Example Closed-Loop trajectory starting in ( ) converges to a limit cycle. Trajectory obtained via Runge-Kutta method with sample time Example Sets S 0 ( ), S u (- - -) and S(H, ū) (-.-). Example