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

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