Symboic modes for noninear contro systems using approximate bisimuation Giordano Poa, Antoine Girard and Pauo Tabuada Abstract Contro systems are usuay modeed by differentia equations describing how physica phenomena can be infuenced by certain contro parameters or inputs. Athough these modes are very powerfu when deaing with physica phenomena, they are ess suitabe to describe software and hardware interfacing the physica word. This has spurred a recent interest in describing contro systems through symboic modes that are abstract descriptions of the continuous dynamics, where each symbo corresponds to an aggregate of continuous states in the continuous mode. Since these symboic modes are of the same nature of the modes used in computer science to describe software and hardware, they provided a unified anguage to study probems of contro in which software and hardware interact with the physica word. In this paper we show that every incrementay gobay asymptoticay stabe noninear contro system is approximatey equivaent (bisimiar) to symboic mode with a precision that can be chosen a priori. We aso show that for digita controed systems, in which inputs are piecewise constant, and under the stronger assumption of incrementa input to state stabiity, the symboic modes can be obtained, based on a suitabe quantization of the inputs. keywords: symboic modes, approximate bisimuation, digita contro systems, incrementa stabiity, noninear systems. I. INTRODUCTION The idea of using modes at different eves of abstraction has been very successfuy used in the forma methods community, with the purpose of mitigating the compexity of software verification. A centra notion when deaing with compexity reduction, is the one of bisimuation equivaence, introduced by Miner [1] and Park [] in the 80s. The key idea in the notion of bisimuation is to find and compute a symboic mode which shares with the origina mode most of the properties of interest. In fact, the use of symboic modes provides a unified framework for describing physica processes as we as software and hardware interacting with the physica word. Furthermore, deaing with symboic modes enabes one to everage the rich iterature on supervisory contro [3] and game theory [4], [5], that can be of hep when synthesizing controers. The probem of constructing symboic modes of contro systems is thus quite chaenging from a conceptua and technica point of view. After severa successfu resuts on the existence of finite bisimuations This work has been partiay supported by the Nationa Science Foundation CAREER award 0717188. G. Poa and P. Tabuada are with the Department of Eectrica Engineering, University of Caifornia at Los Angees, Los Angees, CA 90095-1594, USA poa,tabuada@ee.uca.edu. A. Girard is with Université Joseph Fourier, Laboratoire Jean Kuntzmann, B.P. 53, 38041, Grenobe, France, Antoine.Girard@imag.fr for discrete time contro systems [6], [7], a new twist in this research ine has been recenty given by the so caed approximate bisimuation, introduced in [8], that captures equivaence of systems in an approximate way. By reaxing the usua notion of bisimuation to approximate bisimuation, a arger cass of contro systems can be expected to admit symboic modes. In fact in [9], [10], a symboic mode is proposed, which is based on an approximate notion of simuation (one sided version of the notion of approximate bisimuation). More precisey [10] shows that, if a noninear contro system is asymptoticay stabiizabe it is possibe to construct a symboic mode that approximates the contro system, up to a given precision that is chosen a priori, as a design parameter. However, if a controer fais to exist for the symboic mode, nothing can be concuded regarding the existence of a controer for the origina mode. This drawback is a direct consequence of the one sided approximation notion used in [10]. This motivates the need to extend the resuts in [10] to bisimuation. The aim of this paper is to identify a cass of contro systems admitting symboic modes, that are approximatey bisimiar to the given contro system. The key idea in the resuts that we propose is to repace the assumption of asymptotic stabiizabiity of [10] with the stronger notion of asymptotic stabiity. By doing so, we show that every incrementay gobay asymptoticay stabe [11] noninear contro system admits a symboic mode that is an approximate bisimuation of the contro system, with a precision that is defined a priori, as a design parameter. Furthermore if the state space of the contro system is bounded, which is the case in many reaistic situations, the symboic mode is finite. Moreover, by focusing on digita contro systems, i.e. systems where contro signas are piecewise constant, a symboic mode can be obtained by quantizing the space of inputs. As a by product, our resuts aso shed some ight into the construction of finite abstractions in the context of quantized contro systems [1], [13]. Indeed, by performing a quantization on the input space, we can guarantee that the resuting symboic mode admits a attice structure. However, whie in this paper this attice structure is expoited for obtaining a finite abstraction of the contro system, in the context of quantized contro systems iterature, it has been used to obtain efficient motion panning agorithms. Simiar resuts have been recenty reported in [14]. This paper extends the work in [14] in two directions: (i) by enarging the cass of contro systems from inear to noninear and (ii) by enarging the cass of input signas from piecewise constant to measurabe. A fu version of this paper can be found in
[15]. II. CONTROL SYSTEMS AND STABILITY NOTIONS A. Notation The identity map on a set A is denoted by 1 A. If A B we denote by ı A : A B or simpy by ı the natura incusion map taking any a A to ı(a) = a B. Given a function f : A B, and a set C A, the symbo f C : C B denotes the restriction of f to C, so that f C (c) = f(c) for any c C. Given a, b R, we denote the cosed interva by [a, b] and the open interva by ]a, b[, i.e. [a, b] = {x R : a x b} and ]a, b[= {x R : a < x < b}. Given a vector x R n we denote by x i the i th eement of x and by x the infinity norm of x; we reca that x = max{ x 1, x,..., x n }, where x i denotes the absoute vaue of x i. Given a measurabe function f : R + 0 R, the (essentia) supremum of f is denoted by f. The cosed ba centered at x R n with radius ε is defined by B ε (x) = {y R n x y ε}. For any A R n and µ R set [A] µ = {a A a i = k i µ, k i Z, i = 1,..., n}. A continuous function γ : R + 0 R+ 0 beongs to cass K if it is stricty increasing and γ(0) = 0; γ beongs to cass K if γ K and γ(r) as r. A continuous function β : R + 0 R+ 0 R+ 0 beongs to cass KL if, for each fixed s, the map β(r, s) beongs to cass K with respect to r and, for each fixed r, the map β(r, s) is decreasing with respect to s and β(r, s) 0 as s. We identify a reation R A B with the map R : A B defined by b R(a) iff (a, b) R. Given a reation R A B, R 1 denotes the inverse reation defined by R 1 = {(b, a) B A : (a, b) R}. B. Contro Systems In this paper we consider the foowing cass of noninear contro systems: { ẋ = f(x, u) Σ : x R n, u U R m (1), where x is the state and u is the contro input. We suppose that contro input signas are the set, denoted by U, of a measurabe functions from intervas of the form ]a, b[ R to U with a < 0 and b > 0. Moreover we suppose that the function f : R n U R n is a continuous map satisfying the foowing Lipschitz assumption: for every compact set K R n, there exists a constant κ > 0 such that f(x, u) f(y, u) κ x y for a x, y K and a u U. An absoutey continuous curve x :]a, b[ R n is said to be a trajectory of Σ if there exists u U satisfying ẋ(t) = f(x(t), u(t)), for amost a t ]a, b[. Athough we have defined trajectories over open domains, we sha refer to trajectories x :[0, τ] R n defined on cosed domains [0, τ], τ R + with the understanding of the existence of a trajectory x :]a, b[ R n such that x = x [0,τ]. We wi aso write x(τ, x, u) to denote the point reached at time τ under the input u from initia condition x; this point is uniquey determined, since the assumptions on f ensure existence and uniqueness of trajectories. C. Stabiity notions The resuts presented in this paper wi assume certain stabiity assumptions on the contro systems. We briefy reca those notions in the foowing definitions: Definition 1: [11] A contro system Σ is said to be incrementay gobay asymptoticay stabe (δ GAS) if there exist a KL function β such that for any t R + 0, any x 1, x R n and any input signa u U the foowing condition is satisfied: x(t, x 1, u) x(t, x, u) β( x 1 x, t). () Definition : [11] A contro system Σ is said to be incrementay input to state stabe (δ ISS) if there exist a KL function β and a K function γ such that for any t R + 0, any x 1, x R n and any pair of input signas u 1, u U the foowing condition is satisfied: x(t, x 1, u 1 ) x(t, x, u ) β( x 1 x, t) (3) +γ( u 1 u ). By observing () and (3), it is not difficut to see that δ ISS impies δ GAS, whie the converse is not true in genera. In genera, inequaities () and (3) are difficut to check directy. An approach based on Lyapunov functions, can be of hep into checking these stabiity properties. Definition 3: Consider a contro system Σ and a smooth function V : R n R n R + 0. V is caed a δ GAS Lyapunov function for Σ if there exist K functions α 1, α and ρ such that: (i) for any (x 1, x ) R n R n α 1 ( x 1 x ) V (x 1, x ) α ( x 1 x ); (ii) for any u U and any x 1, x R n V x 1 f(x 1, u) + V x f(x, u) < ρ( x 1 x ). V is caed a δ ISS Lyapunov function for Σ, if there exist K functions α 1, α, ρ and σ such that condition (i) is satisfied and the foowing hods: (iii) for any u 1, u U and any x 1, x R n V f(x 1, u 1 ) + V f(x, u ) < ρ( x 1 x ) x 1 x +σ( u 1 u ). The foowing resut competey characterizes δ GAS and δ ISS in terms of existence of Lyapunov functions. Theorem 1: [11] Consider a contro system Σ. Then: If U is compact then Σ is δ GAS if and ony if it admits a δ GAS Lyapunov function; If U is cosed, convex, contains the origin and f(0, 0) = 0, then Σ is δ ISS if it admits a δ ISS Lyapunov function. Moreover if U is compact, existence of a δ ISS Lyapunov function is equivaent to δ ISS. III. APPROXIMATE BISIMULATIONS In this section we introduce a notion of approximate equivaence upon which a the resuts in this paper rey. We start by introducing the cass of transition systems that wi be used as abstract modes for contro systems.
Definition 4: A transition system is quintupe: consisting of: T = (Q, L,, O, H), A set of states Q; A set of abes L; A transition reation Q L Q; An output set O; An output function H : Q O. A transition system (Q, L,, O, H) is said to be: metric, if the output set O is equipped with a metric d : O O R + 0 ; countabe, if Q and L are countabe sets; finite, if Q and L are finite sets. We wi foow standard practice and denote an eement (p,, q) by p q. Transition systems capture dynamics through the transition reation. For any states p, q Q, p q simpy means that it is possibe to evove or jump from state p to state q under the action abeed by. We wi use transition systems as an abstract representation of contro systems. There are severa different ways in which we can transform contro systems into transition systems. We now describe one of these which has the property of capturing a the information contained in a contro system Σ. Given a contro system Σ define the transition system T (Σ) := (Q, L,, O, H), Q = R n ; L = U; u q p if there exists a trajectory x : [0, τ] R n of Σ satisfying x(τ, q, u) = p for some τ R + ; O = R n ; H = 1 R n. Note that T (Σ) is a metric transition system when we regard O = R n as being equipped with the metric d(p, q) = p q. We now introduce a notion of approximate equivaence. The notion of equivaence that we consider, is the one of bisimuation equivaence [1], []. Bisimuation reations are standard mechanisms to reate the properties of transition systems. Intuitivey, a bisimuation reation between a pair of transition systems T 1 and T is a reation between the corresponding state sets expaining how a sequence of transitions r 1 of T 1 can be transformed into a sequence of transitions r of T and vice versa. Whie typica bisimuation reations require that r 1 and r are observationay indistinguishabe, that is, H 1 (r 1 ) = H (r ), we sha reax this by requiring H 1 (r 1 ) to simpy be cose to H (r ) where coseness is measured with respect to the metric on the output set. The foowing notion has been introduced in [8] and in a sighty different formuation in [10]. Definition 5: Let T 1 = (Q 1, L 1, 1, O, H 1 ) and T = (Q, L,, O, H ) be metric transition systems with the same output set O and the same metric d and et ε R + 0. A reation R Q 1 Q is said to be an ε approximate bisimuation reation between T 1 and T, if for any (q 1, q ) R: (i) d(h 1 (q 1 ), H (q )) ε; (ii) q 1 1 1 p 1 impies the existence of q p such that (p 1, p ) R; (iii) q p impies the existence of q 1 1 1 p 1 such that (p 1, p ) R. Moreover T 1 is ε bisimiar to T if there exists a ε approximate bisimuation R between T 1 and T such that R(Q 1 ) = Q and R 1 (Q ) = Q 1. IV. APPROXIMATELY BISIMILAR SYMBOLIC MODELS In the foowing we wi work with a sub transition system of T (Σ) obtained by seecting those transitions from T (Σ) describing trajectories of duration τ for some chosen τ R +. This can be seen as a time discretization or samping process. Definition 6: Given a contro system Σ and a parameter τ R +, define the transition system: Q 1 = R n ; T τ (Σ) := (Q 1, L 1, 1, O 1, H 1 ), L 1 = {u U x(τ, x, u) is defined for a x R n }; u q 1 p if there exists a trajectory x : [0, τ] R n of Σ satisfying x(τ, q, u) = p; O 1 = R n ; H 1 = 1 R n. Note that the set of abes L 1 is composed by those contro inputs u U for which x(τ, x, u) is defined for a initia condition x R n. In the foowing we show existence of a countabe transition system that is approximatey bisimiar to T τ (Σ), provided that Σ satisfies some stabiity properties. For doing so we wi extract a countabe set of states and a countabe set of abes from T τ (Σ) in a way that the obtained transition system satisfies the required approximation. By simpe considerations on the infinity norm, for any given precision η R + we can approximate the state space Q 1 = R n of T τ (Σ) by means of the countabe set Q := [R n ] η so that for any x R n there exists q [R n ] η such that x q η/. The approximation of the set of abes L 1 of T τ (Σ) is more invoved and it requires the notion of reachabe sets. Given a contro system Σ, any time τ R + and any state x R n consider the set: R(τ, x) = {z R n : z = x(τ, x, u), u U}, of reachabe states at time τ from initia state x. Given a precision µ R +, we approximate the reachabe set R(τ, x) by: P µ (τ, x) = {y [R n ] µ : z R(τ, x) s.t. y z µ}. Since P µ (τ, x) [R n ] µ, it is countabe. Notice that for any y P µ (τ, x) there exists a (possiby infinite) set of abes L 1 so that d(y, x(τ, q, )) µ. In order to approximate the set of abes L 1 we consider for any y P µ (τ, x) ony one abe L 1, as representative of a abes
associated with p. The choice of representatives is defined by the function: ψ µ τ,x : P µ (τ, x) U, that associates to any y P µ (τ, x) one contro input u = ψ µ τ,x(y) U such that y x(τ, x, u) µ. Notice that the function ψ µ τ,x is not unique. We can now propose the foowing symboic mode. Given a contro system Σ, any τ R +, η R + and µ R + define the foowing transition system: T τ,η,µ (Σ) := (Q, L,, O, H ), (4) Q = [R n ] η ; L = q [R n ] η L (q), L (q) := { U : = ψ µ τ,q(p), p P µ (τ, q)}; (5) q p, if L (q) and p x(τ, q, ) η; O = R n ; H = ı : Q O. Parameters τ, η and µ in the transition system T τ,η,µ (Σ) can be thought of, respectivey, as a samping time, a state space and an input space quantization. Notice that the quantization µ is given on the space of reachabe states R(τ, q) rather than on the infinite dimensiona space U. The set appearing in (5) is countabe since it is the image through the function ψ τ,q µ of the countabe set P µ (τ, q). Hence, the set of abes L is the union of a countabe sequence of countabe sets and therefore it is countabe as we. Finay since aso the set of states Q is countabe, the transition system T τ,η,µ (Σ) is countabe. Furthermore if the state space of the contro system Σ is bounded, the corresponding transition system T τ,η,µ (Σ) is finite. We now have a the ingredients to state the main resut reating δ GAS to the existence of symboic modes. Theorem : Consider a contro system Σ and any desired precision ε R +. If Σ is δ GAS then for any τ R +, η R + and µ R + satisfying the foowing condition: β(ε, τ) + µ + η ε, (6) where β is a KL function satifying inequaity (), the transition system T τ (Σ) is ε bisimiar to T τ,η,µ (Σ). Before giving the proof of this resut we point out that if Σ is δ GAS, there aways exist parameters τ, η and µ satisfying condition (6). In fact, if Σ is δ GAS then there exists a sufficienty arge τ so that β(ε, τ) < ε; then by choosing sufficienty sma vaues of µ and η, condition (6) is fufied. As pointed out in Section II-C it is difficut in genera to find a KL function β satisfying inequaity (). However, once a δ GAS Lyapunov function V for Σ has been found, an expression for the function β can be derived on the basis of V. Proof: Consider the reation R Q 1 Q defined by (x, q) R if and ony if x q ε. By construction R(Q 1 ) = Q ; by geometrica considerations on the infinity norm, Q 1 q Q B η (q ) and therefore, since by (6), η < ε, we have that R 1 (Q ) = Q 1. We now show that R is an ε approximate bisimuation reation between T τ (Σ) and T τ,η,µ (Σ). Consider any (x, q) R. Condition (i) in Definition 5 is satisfied by the definition of R. Let us now show that condition (ii) in Definition 5 hods. Consider any u 1 L 1 and the transition x u1 1 y in T τ (Σ). Let v = x(τ, q, u 1 ); since R n w [R n ] µ B µ (w), there exists w [R n ] µ such that: v w µ. (7) Since v R(τ, q), it is cear that w P µ (τ, q) by definition of P µ (τ, q). Then, et L (q) be given by = ψ µ τ,q(w). By setting z = x(τ, q, ), it foows that: w z µ. (8) Since R n q Q B η (q ), there exists p Q such that: z p η. (9) Thus, q p and since Σ is δ GAS and by (7), (8), (9) and (6), the foowing chain of inequaities hods: y p = y v + v w + w z + z p y v + v w + w z + z p β( x q, τ) + v w + w z + z p β(ε, τ) + µ + µ + η ε. We now show that condition (iii) hods as we. Consider any L and the transition q p in T τ,η,µ (Σ). By definition of T τ,η,µ (Σ) there exists z Q 1 such that z = x(τ, q, ) and z p η. Choose u 1 = and consider now the transition x u1 1 y in T τ (Σ). Since Σ is δ GAS and by condition (6), the foowing chain of inequaities hods: y p = y z + z p y z + z p β( x q, τ) + z p β(ε, τ) + η ε. Thus (y, p) R, which competes the proof. This resut represents a substantia improvement over previousy known casses of contro systems admitting symboic modes, which incuded output controabe inear systems in discrete time [7] and stabe inear systems in discrete time [14]. Despite its conceptua importance, highighting stabiity as a sufficient condition for the existence of symboic modes, Theorem does not suggest how to construct such modes. In the next section we address this issue by identifying input quantizations, eading to the desired symboic modes. V. DIGITAL CONTROL SYSTEMS In this section we speciaize the resuts of the previous section to the case of digita contro systems, i.e. contro systems where contro signas are piecewise constant. In many man made systems, input signas are physicay impemented as piecewise constant signas. Our assumptions are then in consonance with rea physica constraints. Moreover, input quantization can be seen as a very powerfu compexity
reduction mechanism, simpifying severa contro synthesis probems [1], [13]. In the foowing we suppose that the input space U of the contro systems invoved, contains the origin and it is a hyper rectange, i.e. U = [a 1, b 1 ] [a, b ]... [a m, b m ], for some a i < b i, i = 1,,..., m. Given τ R +, we now consider the cass of constant inputs of duration τ, that is U τ = {u U : u(t) = u(0), t [0, τ]}. We denote by u the contro input u U τ for which u(t) = u, t [0, τ]. Let us denote by T Uτ (Σ) the sub transition system of T τ (Σ) where ony contro inputs in U τ are considered. More formay define: T Uτ (Σ) := (Q 1, L 1, 1, O 1, H 1 ), Q 1 = R n ; L 1 = U; q 1 p if there exists a trajectory x of Σ satisfying x(τ, q, ) = p; O 1 = R n ; H 1 = 1 R n. We now define a suitabe symboic mode associated with transition system T Uτ (Σ). Given a contro system Σ, any τ R +, η R + and µ R +, define the foowing transition system: T τ,η,µ (Σ) := (Q, L,, O, H ), (10) Q = [R n ] η ; L = [U] µ ; q p if p x(τ, q, ) η; O = R n ; H = ı : Q O. Notice that transition system in (10) differs from transition system in (4), (ony) in the way that we use for approximating contro inputs. In particuar, the choice of abes in transition system (10) is done without assuming the knowedge of reachabe set associated with the contro system, and therefore the construction of T τ,η,µ (Σ) is effective. We refer to [15] for a discussion on the computationa issues reated to the construction of the proposed symboic mode. We are now abe to give the foowing resut that reates δ ISS to the existence of symboic modes. Theorem 3: Consider a contro system Σ and any desired precision ε R +. If Σ is δ ISS then for any τ R +, η R +, and µ R + satisfying the foowing condition: β(ε, τ) + γ(µ) + η ε, (11) where β is a KL function and γ is a K function satisfying inequaity (3), the transition system T Uτ (Σ) is ε bisimiar to T τ,η,µ (Σ). Before giving the proof of this resut we point out that, anaogousy to conditions of Theorem, there aways exist parameters τ, η, and µ satisfying condition (11). As pointed out in Section II-C it is difficut to find in genera, a KL function β and a K function γ satisfying inequaity (3). However, once a δ ISS Lyapunov function V for Σ has been found, an expression for the functions β and γ can be derived on the basis of V ; we show this in the next section by means of an iustrative exampe. Proof: Consider the reation R Q 1 Q defined by (x, q) R if and ony if x q ε. By construction R(Q 1 ) = Q ; by geometrica considerations on the infinity norm, Q 1 q [R n ] η B η (q ) and therefore, since by (11), η < ε, we have that R 1 (Q ) = Q 1. We now show that R is an ε approximate bisimuation reation between T τ (Σ) and T τ,η,µ (Σ). Consider any (x, q) R. Condition (i) in Definition 5 is satisfied by the definition of R. Let us now show that condition (ii) in Definition 5 hods. Consider any u 1 U and the transition x u1 1 y. Consider an input L such that: u 1 µ, (1) and set z = x(τ, q, ). (Notice that such input L exists because the assumptions on U make [U] µ non empty.) Since Q 1 q [R n ] η B η (q ), there exists p Q such that: z p η, (13) and q p. Since Σ is δ ISS and by (11), (1) and (13), the foowing chain of inequaities hods: y p = y z + z p y z + z p β( x q, τ) + γ( u 1 ) + η β(ε, τ) + γ(µ) + η ε. Hence (y, p) R and condition (ii) in Definition 5 hods. We now show that condition (iii) hods as we. Consider any L and the transition q p in T τ,η,µ (Σ). By definition of T τ,η,µ (Σ): z p η, (14) where z = x(τ, q, ). Consider now the transition x 1 y in T Uτ (Σ). Since Σ is δ ISS and by (11) and (14), the foowing chain of inequaities hods: y p = y z + z p y z + z p β( x q, τ) + γ( ) + η β(ε, τ) + η ε. Thus (y, p) R, which competes the proof. VI. A SIMPLE EXAMPLE Consider a contro system: { ẋ = f(x, u) Σ : x R, u U R, (15) where U = [ 0.1, 0.1] and f : R U R is defined by: f((x 1, x ), u) = [ x 1 + x 7u (1 + u )x ]. We work in the compact set X = [ 1, 1] [ 1, 1]. The set X is invariant for the contro system Σ, i.e. x(t, x, u) X, for any x X, any u U, and any time t R + 0. Given a desired precision ε R +, the goa is to find suitabe parameters τ R +, η R + and µ R +, so that
transition system T τ,η,µ (Σ) as defined in (10) is ε bisimiar to transition system T Uτ. In order to find such parameters we can appy Theorem 3. We start by showing that the contro system Σ defined by (15) is δ ISS. Consider the function V : R R R + 0 defined by V (x, y) = 0.5 ((x 1 y 1 ) + (x y ) ), for any x = (x 1, x ), y = (y 1, y ) X. Function V satisfies condition (i) of Definition 3, by choosing α 1 (r) = 0.5 r and α (r) = r. Moreover it can be shown that: V V f(x, u) + x y f(y, v) x y + 14.8 u v. (16) Thus condition (iii) of Definition 3 is satisfied with ρ(r) = r and σ(r) = 14.8 r and therefore by Theorem 1, the contro system Σ is δ ISS. In order to appy Theorem 3, we need to find a KL function β and a K function γ satisfying inequaity (3). By inequaity (16), the definition of V and the comparison emma, the foowing inequaities hod for any x, y X, any u, v U and any time t R + 0 : x(t, x, u) x(t, y, v) V (t) ( ) e t V (0) + 9.6 e α dα u v 0 e t x y + 14.8 u v. (17) Define β(r, s) := e s r and γ(r) := 14.8 r for any r, s R. Functions β and γ are respectivey a KL function and a K function and by (17) they satisfy inequaity (3). We can now appy Theorem 3. Condition (11) becomes: e τ ε + 14.8 µ + η ε. (18) Set the precision ε = 0.5 and choose η = 1/3 and τ = 5; inequaity (18) impies µ 0.0017 and therefore we can choose µ = 0.001. The resuting transition system: T τ,η,µ (Σ) = (Q, L,, O, H ) is defined by: Q = { η, 0, η} { η, 0, η}; L = [U] 0.001 ; is depicted in Figure 1; O = X; H = ı : Q O. VII. DISCUSSION This paper extends the resuts of [9], [10], from approximate simuation to approximate bisimuation. The key idea was to repace the assumption of asymptotic stabiizabiity of [9], [10] by the stronger notion of asymptotic stabiity. Note that for contro systems with bounded inputs, which is the case in many reaistic situations, even if a feedback contro aw rendering the cosed oop system δ GAS were found, there is no guarantee that such feedback satisfies the input constraints. For the cass of digita contro systems, a symboic mode is proposed and based on a quantization of the contro input space. The construction of this symboic mode is effective. We refer to [15], for a discussion on the computationa issues arising in the construction of this symboic mode. η, η 0, η η, η η, 0 0, 0 η, 0 η, η 0, η η, η Fig. 1. Symboic mode T 5,1/3,10 3(Σ) associated with the contro system Σ, as defined in (15). An arrow from a state q to a state p means that there exists at east one abe in L so that x(5, q, ) is in B η (p). REFERENCES [1] R. Miner, Communication and Concurrency. Prentice Ha, 1989. [] D. Park, Concurrency and automata on infinite sequences, ser. 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