A Note on Input-to-State Stabilization for Nonlinear Sampled-Data Systems

Transcription

1 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY [0] J. C. Geromel, P. L. D. Peres, and J. Bernoussou, On a convex parameter space method for linear control design of uncertain systems, SIAM J. Control Optimiz., vol. 9, no., pp , 99. [] F. Blanchini, he gain scheduling and the robust state feedback stabilization problems, IEEE rans. Automat. Contr., vol. 45, pp , Nov [] M. M. Seron, G. A. De Doná, and G. C. Goodwin, Global analytical model predictive control under input constraints, in Proc. 39th IEEE Conf. Decision Control, Sydney, Australia, Dec [3]. Parisini and R. Zoppoli, A receding-horizon regulator for nonlinear systems and a neural approximation, Automatica, vol. 3, pp , 995. [4] B. D. O. Anderson and J. B. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM J. Optimiz., vol. 5, pp. 0 3, 98. [5] K. akaba, Robust servomechanism with preview action for polytopic uncertain systems, Int. J. Robust Nonlinear Control, vol. 0, pp. 0, 000. [6] H. O. Wang, K. anaka, and F. Griffin, An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE rans. Fuzzy Syst., vol. 4, pp. 4 3, Feb A Note on Input-to-State Stabilization for Nonlinear Sampled-Data Systems Dragan Nešić and Dina S. Laila Abstract We provide a framework for the design of stabilizing controllers via approximate discrete-time models for sampled-data nonlinear systems with disturbances. In particular, we present sufficient conditions under which a discrete-time controller that input-to-state stabilizes an approximate discrete-time model of a nonlinear plant with disturbances would also input-to-state stabilize (in an appropriate sense) the exact discrete-time plant model. Index erms Input-to-state stability, nonlinear, sampled-data. I. INRODUCION A stumbling block in controller design for nonlinear sampled-data control systems is the absence of a good model for the design. Indeed, even if the continuous-time plant model is known, we can not in general compute the exact discrete-time model of the plant since this requires an explicit analytic solution of a nonlinear differential equation. his has motivated research on controller design via approximate discrete-time models for sampled-data nonlinear systems [], [], [7]. A drawback of these early results was their limited applicability: they investigate a particular class of plant models, a particular approximate discrete-time plant model (usually Euler) and a particular controller. A more general framework for stabilization of disturbance-free sampled-data nonlinear systems via their approximate discrete-time models that is applicable to general plant models, controllers and approximate discrete-time models was first presented in [0], []. In this note, we generalize results in [] by i) considering sampled-data nonlinear systems with disturbances, and ii) providing a framework Manuscript received April 6, 00; revised November 9, 00 and January 8, 00. Recommended by Associate Editor Z. Lin. his work was supported by the Australian Research Council under the Large Grants Scheme. he authors are with the Department of Electrical and Electronic Engineering, he University of Melbourne, Parkville 300, Victoria, Australia ( d.nesic@ee.mu.oz.au; dsl@ee.mu.oz.au). Publisher Item Identifier 0.09/AC for the design of input-to-state stabilizing (ISS) controllers based on approximate discrete-time plant models (for more details on ISS see [6], [5], [3], and [4]). In particular, we provide sufficient conditions on the continuous-time plant model, the controller and the approximate discrete-time model, which guarantee that if the controller input-to-state stabilizes the approximate discrete-time plant model it would also input-to-state stabilize the exact discrete-time plant model. Our results apply to dynamic controllers and our approach benefits from the results on numerical integration schemes in [6], [3], and [4]. Related results were investigated in [8], on changes of supply rates for ISS discrete-time systems. II. PRELIMINARIES Sets of real and natural numbers (including 0) are denoted, respectively, as and. For a given function w: 0! n, we use the following notation: w [k] :=w(t); t [; (k +) ] where k and > 0, and w [k] is zero elsewhere (in other words w [k] is a piece of function w(t) in the kth sampling interval [; (k +) ]); and w(k) is the value of the function w() at t =, k. We denote the norms kw [k]k =sup [;(k+) ] jw( )j and kwk := sup 0 jw( )j and in the case when w() is a measurable function (in the Lebesgue sense) we use the essential supremum in the definitions. If kwk <, then we write w L. Consider a continuous-time nonlinear plant with disturbances _x(t) =f (x(t);u(t);w(t)) () where x n, u m and w p are respectively the state, control input and exogenous disturbance. It is assumed that f is locally Lipschitz and f (0; 0; 0) = 0. We will consider two cases: w() are measurable functions (in the Lebesgue sense), and w() are continuously differentiable functions. We will always make precise which case we consider. he control is taken to be a piecewise constant signal u(t) =u( )=:u(k); 8 t [; (k+) ), k, where > 0 is the sampling period. Also, we assume that some combination (output) or all of the states (x(k) :=x( )) are available at sampling instant, k. he exact discrete-time model for the plant (), which describes the plant behavior at sampling instants, is obtained by integrating the initial value problem _x(t) =f (x(t);u(k);w(t)) () with given w [k], u(k) and x 0 = x(k), over the sampling interval [; (k +) ]. If we denote by x(t) the solution of the initial value problem () at time t with given x 0 = x(k), u(k) and w [k], then the exact discrete-time model of () can be written as (k+) x(k +)=x(k) + f (x( );u(k);w())d =:F e (x(k);u(k);w [k]): (3) We refer to (3) as a functional difference equation since it depends on w [k]. We emphasize that e F is not known in most cases. Indeed, in order to compute F e we have to solve the initial value problem () analytically and this is usually impossible since f in () is nonlinear. Hence, we will use an approximate discrete-time model of the plant to design a controller. Different approximate discrete-time models can be obtained using different methods. For example, we may first assume that the disturbances w() are constant during sampling intervals, w(t) = w(k); 8 t [; (k + ) ] and then use a classical Runge Kutta numerical integration scheme (such as Euler) for the /0$ IEEE

2 54 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 00 initial value problem (). In this case, the approximate discrete-time model can be written as x(k +)=F a (x(k);u(k);w(k)): (4) We refer to the approximate model (4) as an ordinary difference equation since F a does not depend on w [k] but on w(k). For instance, the Euler approximate model is x(k+) = x(k)+f(x(k);u(k);w(k)). Recently, numerical integration schemes for systems with measurable disturbances were considered in [3] and [4]. Using these numerical integration techniques, we can obtain an approximate discrete-time model x(k +)=F a (x(k);u(k);w [k]) (5) which is in general a functional difference equation. For instance, the simplest such approximate discrete-time model, which is analogous to Euler model, has the following form x(k +)= x(k) + (k+) f (x(k);u(k);w(s))ds (see [3]). Since we will consider semiglobal ISS (see Definition II.), we will think of F e and F a as being defined globally for all small, even though the initial value problem () may exhibit finite escape times [, p. 6]. he sampling period is assumed to be a design parameter which can be arbitrarily assigned. Since we are dealing with a family of approximate discrete-time models F a, parameterized by, in order to achieve a certain objective we need in general to obtain a family of controllers, parameterized by. We consider a family of dynamic feedback controllers z(k +)=G (x(k);z(k)) u(k) =u (x(k);z(k)) (6) where z n. o shorten notation, we introduce ~x := (x z ), ~x n, where n ~x := n x + n z and F i (~x(k); ) := F i (x(k);u (x(k);z(k)); ) : (7) G (x(k);z(k)) he superscript i may be either e or a, where e stands for exact model, a for approximate model. We omit the superscript if we refer to a general model. he second argument of F i (~x; ) (third argument of F i )is either a vector w(k) or a piece of function w [k]. Similar to [0], we define the following. Definition II.: [Lyapunov semiglobally practically input-to-state stable (Lyapunov-SP-ISS)] he family of systems ~x(k + ) = F (~x(k);w [k]) is Lyapunov-SP-ISS if there exist functions ; ; 3 K and ~ K, and for any strictly positive-real numbers 3 ( ; ; ; ) there exist strictly positive-real numbers and L, such that for all 3 (0; ) there exists a function V : n! 0 such that for all ~x n with j~xj and all w L with kwk the following holds: (j~xj) V (~x) (j~xj) (8) [V (F (~x; w )) 0 V (~x)] 0 3(j~xj)+~(kw )+ k (9) and, moreover, for all x ;x ;z with (x z ) ; (x z ) [ ; ] and all 3 (0; ),wehavejv (x ;z) 0 V (x ;z)j L jx 0 x j. he function V is called an ISS-Lyapunov function for the family F. Remark II.: In the case when the family of parameterized closed-loop discrete-time nonlinear systems is an ordinary difference equation ~x(k +) = F (~x(k);w(k)), the condition (9) is replaced A function : is of class- if it is continuous, zero at zero and strictly increasing. It is of class- if it is of class- and is unbounded. A continuous function : is of class- if ( ) is of class- for each 0 and ( ) is decreasing to zero for each 0. by: for all 3 (0; ), all ~x n with j~xj and all w p with jwj we have [V (F (~x; w)) 0 V (~x)] 03(j~xj) +~(jwj) + (0) and the function V is called an ISS-Lyapunov function for the family F. he following definition is a semiglobal-practical version of the ISS property used in [3] and [5], and we use it in the case when we consider measurable disturbances w. Definition II.: [Semiglobal practical ISS ((SP-ISS)] he family of systems ~x(k +) = F (~x(k);w [k]) is SP-ISS if there exist KLand K such that for any strictly positive-real numbers ( ~x; w;) there exists 3 > 0 such that for all (0; 3 ), j~x(0)j ~x and w() with kwk w, the solutions of the system satisfy j~x(k)j (j~x(0)j ;)+(kwk )+, 8 k. he following semiglobal practical ISS-like property was used in [9], and we use it when the disturbances are continuously differentiable. Definition II.3: [Semiglobal practical derivative ISS (SP-DISS)] he family of systems ~x(k +) =F (~x(k); w [k]) is SP-DISS if there exist KLand Ksuch that for any strictly positive-real numbers ( ~x ; w ; _w ;) there exists 3 > 0 such that for all 3 (0; ), j~x(0)j ~x and all continuously differentiable w() such that kwk w, k _wk _w, the solutions of the family F satisfy j~x(k)j (j~x(0)j ;)+(kwk )+, 8 k. Note that a similar property to SP-ISS, called input-to-state practical stability (ISpS) was defined in [5] and [4] when considering nonparameterized systems. Definition II.4: u is said to be locally uniformly bounded if for any ~x > 0 there exist strictly positive numbers 3 and u such that for all (0; 3 ) and all j~xj ~x we have ju (~x)j u. In order to prove our main results, we need to guarantee that the mismatch between F e and F a is small in some sense. We define two consistency properties, which will be used to limit the mismatch. Similar definitions can be found in numerical analysis literature [6, Def. 3.4.], and recently in the context of sampled-data systems (see [, Def. ] and [0, Def. ]). In the sequel, we use the notation x = x(k), u = u(k), w = w(k), and w = w [k]. Definition II.5: (One-step weak consistency) he family F a is said to be one-step weakly consistent with F e if given any strictly positive real numbers ( x ; u ; w ; _w ), there exist a function K and 3 > 0 such that, for all (0; 3 ), all x n ;u m with jxj x, juj u and functions w() that are continuously differentiable and satisfy kw k w and k _w f k _w, wehave jf e 0 F a j( ). Definition II.6: (One-step strong consistency) he family F a is said to be one-step strongly consistent with F e if given any strictly positive real numbers ( x; u; w), there exist a function K and 3 > 0 such that, for all (0; 3 ), all x n ;u m ;w L with jxj x, juj u, kw k w,wehavejf e 0 F a j( ). Sufficient checkable conditions for one-step weak and strong consistency are given next (similar results for systems without disturbances are [0, Lemma ] and [, Lemma ]). Lemma II.: F a is one-step weakly consistent with F e if the following conditions hold: ) F a is one-step weakly consistent with F Euler (x; u; w) := x + f(x; u; w), and ) given any strictly positive-real numbers ( x; u; w; _w), there exist K, K, 3 > 0, such that, for all (0; 3 ), all x ;x with maxfjx j; jx jg x, all u m with juj u and all w ;w p with maxfjw j ; jw jg w, the following holds: jf (x ;u;w ) 0 f (x ;u;w )j (jx 0 x j)+ (jw 0 w j). Lemma II.: F a is one-step strongly consistent with F e if the following conditions hold: ) F a is one-step strongly consistent n

3 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY with F ~ Euler (x; u; w ) := x + (k+) f (x; u; w(s))ds, and ) given any strictly positive-real numbers ( x ; u ; w ), there exist K, 3 > 0, such that, for all (0; 3 ) and for all x ;x n with maxfjx j ; jx jg x, all u m with juj u and all w p with jwj w, the following holds: jf (x ;u;w) 0 f (x ;u;w)j (jx 0 x j). he proofs of Lemmas II. and II. are similar to the proofs of [0, Lemma ] and [, Lemma ]. Proof of Lemma II.: Let ( x ; u ; w ; _w ) be given. Using the numbers (R x; u; w; _w), where R x = x +, let the second condition of the lemma generate 3 > 0, K and K. Since f is locally Lipschitz, it is locally bounded and there exists a number M>0such that for all jxj R x, juj u, jwj w we have jf (x; u; w)j M. Let 3 := minf 3 ; =M g. It follows that for each jxj x, kw k w and all t [; (k +) ]; (0; 3 ), the solution x(t) of _x(t) =f (x(t);u;w(t)) x 0 = x(k) =x () satisfies jx(t)j R x and jx(t) 0 xj M (t 0 ) M and since w() is continuously differentiable by definition, we have jw(t) 0 w(k)j _w (t 0 ) _w, for all t [; (k +) ] and (0; 3 ). It then follows from condition ) of the lemma that, for all jxj x, juj u, kw k w, k _w f [k]k _w, and all (0; 3 ) (k+) [f (x( );u;w( )) 0 f (x; u; w)]d (k+) (jx( ) 0 xj)d + (k+) (jw( ) 0 wj)d (M)+ ( _w ) ~( ) () where ~(s) := (Ms)+ ( _ws) is a K function since and are K. Since F e = x + f(x; u; w) F + (k+) [f (x( );u; w( )) 0 f (x; u; w)]d (3) the result follows from () and the first condition of the lemma, which implies the existence of ~ K, such that F a 0 F Euler ~ ( ). Finally, by letting = ~ +~ we prove that F a is one-step weakly consistent with F e. he proof of Lemma II. is omitted since it follows closely the proof of Lemma II.. III. MAIN RESULS In this section, we state and prove our main results (heorems III. and III.). he results specify conditions on the approximate model, the controller and the plant, which guarantee that the family of controllers (G ;u ) that input-to-state stabilize F a would also input-to-state stabilize F e for sufficiently small. We emphasize that our results are given for general approximate discrete-time models F a (not only for the Euler approximation). We remark that under certain mild conditions on the plant and the controller, our results can be extended to include inter-sample behavior, to conclude SP-ISS results for the closed-loop sampled-data systems (see the results in []). Finally, an example is presented to illustrate our approach. heorem III.: Suppose that i) the family of approximate discrete-time models F a (~x; ) is Lyapunov-SP-ISS (where either (9) or (0) holds), ii) F a is one-step weakly consistent with F e, and iii) u is uniformly locally bounded. hen, the family of exact discrete-time models F e (~x; w ) is SP-DISS. heorem III.: Suppose that i) the family of approximate discrete-time models F a (~x; w ) is Lyapunov-SP-ISS (where (9) holds), ii) F a is one-step strongly consistent with F e, and iii) u is uniformly locally bounded. hen, the family of exact discrete-time models F e (~x; w ) is SP-ISS. he following lemmas are needed to complete proofs of both theorems. We prove only Lemma III. for the case of ordinary difference equations (i.e., when (0) holds) and then comment on the changes in the proof for the case of functional difference equations (i.e., when (9) holds) and the proof of Lemma III.. Lemma III.: If all conditions in heorem III. are satisfied, there exist ^ K such that for any strictly positive numbers (C ~x;c w;c _w;), there exists 3 > 0 such that for all (0; 3 ), we have j~xj C ~x ; kwk C w ; k _wk C _w maxfv (F e (~x; w ));V (~x)g ^(kwk )+ =) V (F e (~x; w )) 0 V (~x) 0 4 3(j~xj): (4) Lemma III.: If all conditions in heorem III. are satisfied, there exist ^ K such that for any strictly positive numbers (C ~x ;C w ;), there exists 3 > 0 such that for all (0; 3 ),wehave j~xj C ~x ; kwk C w maxfv (F e (~x; w ));V (~x)g ^(kwk )+ =) V (F e (~x; w )) 0 V (~x) 0 4 3(j~xj): (5) Proof of Lemma III.: First, we prove the following fact. Fact : Suppose that for any strictly positive numbers ( ; ; ) there exists 3 w > 0 such that for all (0; 3 w), j~xj and jwj we have that (0) holds. hen, for any strictly positive numbers ( ; ; 3 ; ) there exists 3 s > 0 such that for all (0; 3 s ), j~xj and continuously differentiable disturbances with kwk and k _wk 3 we have that V (F (~x; w)) 0 V (~x) 0 3 (j~xj) +~(kw k )+ : (6) Proof of Fact : Let ( ; ; 3 ; ) be given. Let be such that sup s[0; ] ~(s + ) 0 ~(s) ( =). Let :=, :=, := ( =) and let the numbers ; ; generate 3 w > 0 from the condition of Fact. Let s 3 := min w; 3 ( 3 ). Consider arbitrary (0;s 3 ), j~xj and any continuously differentiable disturbance with kwk and k _wk 3. From the mean value theorem and our choice of s 3, it follows that for all t [; (k+)], k we have that jwj = jw(k)j jw(t) 0 w()j + jw(t)j k _w f k (t0)+kw k 3 +kw k 3 s 3 +kw k + kw k. Finally, using our definitions of ; we can write [V (F (~x; w)) 0 V (~x)] 0 3 (j~xj) +~(jwj) + = 0 3(j~xj) +~(kw k )+~(jwj) 0 ~(kw k )+ 0 3(j~xj) +~(kw k )+~ + kw k 0 ~(kw k )+ 0 3(j~xj) +~(kw k )+ + (7) which completes the proof of the fact. Now we continue the proof of Lemma III..

4 56 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 00 Suppose that all conditions in heorem III. (where (0) holds) are satisfied. Using Fact it follows that all conditions in heorem III. (where (6) holds) are also satisfied. Let ^(s) := 0 (4~(s)). 3 Let arbitrary strictly positive numbers (C ~x;c w;c _w;) be given. Using (C ~x;c w ;C _w;), we define := (=) 0 (=); := min (=4) (=4); (=4) 3 0 ((=) ()) ; := 0 ((=) ()); and := 0 ( (C ~x)+~(c w)+ )+. Let the numbers ( ; ; ;) generate the numbers 3 > 0 and L > 0 from condition i) of heorem III. (where (6) holds). Let generate u > 0 and 3 > 0 from condition iii) of heorem III.. Let the quadruple (; u ;C w ;C _w) generate 3 3 > 0 and from condition ii) of heorem III.. Let strictly positive numbers 3 4 ; 3 5 ; 3 6 ; 3 7 be such that L( 3 4 ) (=4) 3 ( ); 3 5 ( 3 5 ) ; 3 6 ~(C w ) (=) ((=4)); and 3 7 ((=4) 3 (C ~x) +~(C w )+ + L( 3 7 )) (=). Finally, we take 3 = minf 3 ; 3 ; 3 3 ; 3 4 ; 3 5 ; 3 6 ; 3 7 ; g. In the calculations that follow, we consider arbitrary (0; 3 ), j~xj C ~x, kwk C w and k _wk C _w. From (8), (9), the definition of, and the fact that 3, wehave a jf a (~x; w)j 0 (V (F (~x; w))) 0 (V (~x) + ~(kwk )+ ) 0 ( (C ~x) +~(C w)+ ) < : (8) Using the condition ii) of heorem III., inequality (8) and our choice of and 3 (in particular the choice of 3 5 ), we can write jf e (~x; w )jjf a (~x; w)j + jf e (~x; w ) 0F a (~x; w)j 0 ( (C ~x) +~(C w)+ )+( ) 0 ( (C ~x) +~(C w )+ )+ =: (9) choice of and, and using (8) (3), we deduce that V (F e ) ^(C w )+(=) implies ^(C w )+ V (F e ) 0 V (~x) +V (~x) Hence, we can conclude that + V (F a ) 0 V (F a ) V (F a ) 0 V (~x) + jv (F e ) 0 V (F a )j + V (~x) ~(C w )+ + L ( )+V (~x) + V (~x): (4) V (F e ) ^(C w )+ =) V (~x) ^(C w ): (5) Again using the conditions i) and ii) of heorem III. and from the choice of 3 (in particular the choice of 3 4 ), the choice of and, and using (8) (5), we can write V (F e (~x; w )) 0 V (~x) V (F a (~x; w)) 0 V (~x) + jv (F e (~x; w )) 0 V (F a (~x; w))j 0 3 (j~xj) + ~(kwk )+ + L ( ) 0 4 3(j~xj) (j~xj) + ~(C w ) + 4 3( )+ 4 3( ) 0 4 3(j~xj) (V (~x)) + ~(C w ) Suppose that V (F e (~x; w )) ^(C w )+(=). From (8), the definition of and the choice of 3,wehave 0 3(j~xj) + 3( ) 0 jf e (~x; w )j 0 => (0) (j~xj): (6) and then using the condition ii) of heorem III. and our choice of 3 5, we have jf a (~x; w)j0jf e (~x; w ) 0F a (~x; w)j + jf e (~x; w )j 0( ) = : () From our choice of 3, 3 6,, and, and using the inequality (6), it follows that: () () + () 0 4 () 0 ~(C w) 0 which implies (jf a (~x; w)j) 0 ~(C w ) 0 V (F a (~x; w)) 0 ~(kwk ) 0 V (~x) (j~xj) () j~xj 0 () = : (3) Note that. From the conditions i) and ii) of heorem III. and from the choice of 3 (in particular, the choice of 3 4 and 3 7 ), the 4 Suppose now that V (F e (~x; w )) ^(C w)+(=) and V (~x) ^(C w )+. From our choice of 3 (in particular the choice of 3 7 ), it follows that: V (F e (~x; w )) 0 V (~x) ^(C w)+ 0 V (~x) (j~xj) (7) which shows that (4) is valid, and this completes the proof of Lemma III.. he proof of Lemma III. for the case of functional difference equations and the proof of Lemma III.3 follow the same steps, except that we do not need to use Fact since (9) holds. Also, in the case of functional difference equations of Lemma III. we use one-step weak consistency and in the case of Lemma III.3 we use one-step strong consistency. he next lemma is needed in proofs of heorems III. and III., and it was proved as a part of the proof of [, h. ]. Lemma III.3: Let W L and let ; ; 3 K. Let strictly positive-real numbers (d; D) be such that (D) > d and let 3 > 0 be such that for any (0; 3 ) there exists a function V : n! 0 such that for all (0; 3 ) and all ~x n we have (j~xj) V (~x) (j~xj) and, moreover, for all ~x n

5 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY with maxfv (F (~x; w ));V (~x)g d and j~xj D, all w W and all (0; 3 ), the following holds: ABLE I ROAS IN DISURBANCE FREE CASE V (F (~x; w )) 0 V (~x) 0 4 3(j~xj): hen, there exist a function KLsuch that for all (0; 3 ), j~x(0)j 0 (D) and w Wand all k the solutions of the family of discrete-time models ~x(k +) = F (~x(k);w [k]) exist and satisfy j~x(k)j (j~x(0)j ;)+ 0 (d). Proof of heorem III.: Let arbitrary strictly positive-real numbers ( ~x ; w ; _w ;) be given and let all conditions in heorem III. hold. Let ^ K come from Lemma III.. We define (C ~x ;C w ;C _w ;) as C w := w, C _w := _w, > 0 is such that sup s[0; ] [0 (^(s) +) 0 0 ^(s)], and the number C ~x := maxf 0 (^(w) +) +; 0 ( ~x)g. Using Lemma III., let (C ~x ;C w ;C _w ;) generate 3 > 0, such that (4) holds. Introduce D := C ~x and d := ^(kwk )+, and from the choice of (C ~x;c w;c _w;) we have that (D) >d. Let W be a set of continuously differentiable functions defined as follows: W := fw L jkwk C w ; k _wk C _w g. With these definitions of (D; d) and W, together with (8), we have that all conditions of Lemma III.3 hold. Hence, we can conclude that for all (0; 3 ), ~x(0) n, j~x(0)j ~x and w L with kwk w, k _wk _w and all k 0 we have that the solutions of F e (~x; w ) exist and satisfy j~x(k)j (j~x(0)j ;)+ 0 (d) (j~x(0)j ;)+ 0 (^(kwk )+) (j~x(0)j ;)+ 0 ^(kwk )+ =(j~x(0)j ;)+(kwk )+ (8) where (s) := 0 ^(s). his completes the proof of heorem III.. he proof of heorem III. is omitted since it follows closely the proof of heorem III.. We illustrate below our results via an example. Example III.: Consider the scalar continuous-time plant _x(t) = x 3 (t) + u(t) +w(t) and its approximate discrete-time model x(k +)= x(k) + (x 3 (k) +u(k)) + (k+) w(s)ds =: F a (x(k);u(k);w [k]), which can be obtained from numerical integration schemes described in [3]. he following three controllers: u (x) =0 x 3 0 x u (x) =0 x 3 0 x 0 x u 3 (x) =0 +x 0 p 0 4 x (9) can be shown to yield, respectively, the following three dissipation inequalities with V (x) =(=)x : [V (F a (x; u (x);w )) 0 V (x)] 0 x + kw k + kw k + x (30) [V (F a (x; u (x);w )) 0 V (x)] 0 x + kw k + + kw k x (3) [V (F a (x; u 3 (x);w )) 0 V (x)] 0 x + kw k + kw k : (3) From our choice of V (x) and (30) (3) it follows that the approximate discrete-time model with any of the controllers (9) is Lyapunov SP-ISS. Moreover, since the approximate discrete-time model is the ABLE II PERFORMANCE WIH A DISURBANCE same as F ~ Euler in the first condition of Lemma II., it follows that: F a is one-step strongly consistent with F e. Finally, all of the controllers in (9) are locally uniformly bounded (for u and u this is obvious and for u 3 this can be seen by using the aylor series expansion p 0 4 = 0 + O( )). herefore, for F a, V (x) and any controller in (9), we have that all conditions of heorem III. hold. Hence, we can conclude using heorem III. that each of controllers (9) semiglobally practically input-to-state stabilizes the exact discrete-time plant model. We applied the controllers (9) via a sampler and zero order hold to the continuous-time plant model and compared the performance of the three controllers via simulations in SIMULINK. Note that the controller u (x) may be obtained using a continuous-time design (obtain _V 0(=)x +(=)w for the continuous-time closed-loop) and controller discretization. In able I we estimated regions of attraction (ROA) of the closed-loop sampled-data system with controllers (9) for different sampling periods. he controller u gives the largest ROA for all tested sampling periods. In able II we summarize simulations for different sampling periods and fixed initial states with a sinusoidal disturbance of frequency rad/s. he values of amplitude of the sinusoidal disturbance recorded in able II are the largest values for which solutions of the sampled-data closed-loop system stay bounded. It is obvious that the controller u 3 is the most robust with respect to the test disturbance for all tested sampling periods. Similar observations were obtained for other initial states and disturbances that are not presented in able II. From ables I and II, we see that the performance of all controllers (9) becomes very similar for small sampling periods which can be expected since the dissipation inequalities in (30) (3) differ only in terms of order, which can be made arbitrarily small on compact sets by reducing. Difference in performance of controllers (9) is more pronounced for larger sampling periods (see ables I and II). REFERENCES [] D. Dochain and G. Bastin, Adaptive identification and control algorithms for nonlinear bacterial growth systems, Automatica, vol. 0, pp , 984. [] G. C. Goodwin, B. McInnis, and R. S. Long, Adaptive control algorithm for waste water treatment and ph neutralization, Optimiz. Control Appl. Meth., vol. 3, pp , 98. [3] L. Grüne and P. E. Kloeden, Higher order numerical schemes for affinely controlled nonlinear systems, Numer. Math., vol. 89, pp , 00. We used the following parameters in simulations: variable step size; ode-45; relative tolerance 0, absolute tolerance 0 ; max step size auto; and initial step size auto.

6 58 IEEE RANSACIONS ON AUOMAIC CONROL, VOL. 47, NO. 7, JULY 00 [4] R. Ferretti, Higher order approximations of linear control systems via Runge Kutta schemes, Computing, vol. 58, pp , 997. [5] Z. P. Jiang, A. R. eel, and L. Praly, Small gain theorem for ISS systems and applications, Math. Control, Signals, Syst., vol. 7, pp. 95 0, 994. [6] M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, Nonlinear and Adaptive Control Design. New York: Wiley, 995. [7] I. M. Y. Mareels, H. B. Penfold, and R. J. Evans, Controlling nonlinear time varying systems via Euler approximations, Automatica, vol. 8, pp , 99. [8] D. Nešić and A. R. eel, Changing supply functions in input-to-state stable systems: he discrete-time case, IEEE rans. Automat. Contr., vol. 46, pp , June 00. [9], Input-to-state stability for nonlinear time-varying systems via averaging, Math. Control, Signals, Syst., vol. 4, pp , 00. [0], Set stabilization of sampled-data nonlinear differential inclusions via their approximate discrete-time models, in Proc. 39th Conf. Decision Control, Sydney, Australia, 000, pp. 7. [] D. Nešić, A. R. eel, and P. Kokotović, Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, Syst. Control Lett., vol. 38, pp , 999. [] D. Nešić, A. R. eel, and E. Sontag, Formulas relating stability estimates of discrete-time and sampled-data nonlinear systems, Syst. Control Lett., vol. 38, pp , 999. [3] E. D. Sontag, he ISS philosophy as a unifying framework for stability-like behavior, in Nonlinear Control in the Year 000, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, Eds. Berlin, Germany: Springer-Verlag, 000, pp [4] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability with respect to compact sets, in Proc. IFAC Non-Linear Control Systems Design Symp. (NOLCOS 95), ahoe City, CA, June 995, pp [5] E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE rans. Automat. Contr., vol. 34, pp , Apr [6] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge, U.K.: Cambridge Univ. Press, 996. Sequential Versus Concurrent Languages of Labeled Conflict-Free Petri Nets Hsu-Chun Yen Abstract For structurally deterministic labeled conflict-free Petri nets (PNs), we show that two PNs have identical sequential languages if and only if their concurrent languages are identical as well, and whether a given labeled conflict-free PN is structurally deterministic or not can be checked in polynomial time. We also investigate a number of language-related problems in supervisory control theory for this class of PNs. As it turns out, the properties of controllability, observability, and normality in the sequential framework coincide with that in the concurrent framework. Index erms Concurrency, equivalence, Petri net (PN), supervisory control. I. INRODUCION Based upon the framework of Ramadge and Wonham [], the investigation of Petri net (PN) languages has emerged as a research area of increasing importance in supervisory control theory (see, e.g., [] [6]). Once modeled by a PN, one way to characterize the behavior of a system is to view the set of all executable transition sequences of the Manuscript received March 3, 000; revised May 9, 00. Recommended by Associate Editor R. S. Sreenivas. he author is with the Department of Electrical Engineering, National aiwan University, aipei, aiwan 06, Republic of China ( yen@cc.ee.ntu.edu.tw). Publisher Item Identifier 0.09/AC PN, which, according to the underlying transition firing semantics, can be classified as either a sequential language or a concurrent language (cf. [7]). he former associates a system s event to a single PN s transition, whereas in the latter, a system s event is captured by a set of concurrently fireable transitions. he contribution of this note is twofold. First, our work generalizes the equivalence theorem of structurally deterministic labeled marked graphs (reported in [8]) in the sense that for a wider class, namely, the class of structurally deterministic labeled conflict-free PNs, the equivalence theorem is shown to hold as well. (he equivalence theorem says that two PNs have identical concurrent languages if and only if their sequential languages are also identical). A polynomial time algorithm is also derived in this note to decide whether a labeled conflict-free PN is structurally deterministic or not. Our work also supplements the existing results concerning deterministic PN languages [9], in which the equivalence of two arbitrary, deterministically labeled, sequential PN languages has been shown to be decidable. Such a result, together with the equivalence theorem derived in this note, implies the decidability of equivalence for any two conflict-free, deterministically labeled, concurrent PN languages. he second contribution of our work lies in the applications to various problems in the theory of supervisory control in the framework of concurrent PN languages. As pointed out in [3], most research on supervisory control of labeled PNs was based upon the so-called no concurrency (NC) assumption. (See [4] for more). Here, we study three of the most important issues, namely, controllability, observability, and normality, in supervisory control of PNs with respect to both the sequential and the concurrent firing semantics. For structurally deterministic conflict-free controlled PNs (CPNs), it turns out that a PN being controllable (observable or normal) in the sequential framework coincides with that in the concurrent framework. As concurrent PN languages are less understood than their sequential counterparts, our result offers an alternative in reasoning about concurrent languages regarding controllability, observability and normality for the aforementioned class of PNs. Hopefully, our work will provide more insights into the theory of supervisory control for labeled PNs with the NC assumption lifted. II. PRELIMINARIES A PN is a three-tuple (P,, '), where P is a finite set of places, is a finite set of transitions, and ' is a flow function ' :(P)[ ( P )!f0; g. We assume that the reader is familiar with the basic definitions of PNs. (See [0] for more). We write 0! 0 to denote that 0 is reachable from through the firing of transition sequence ( 3 ). (We also write 0! to denote that can be fired from ). # (t) represents the number of occurrences of transition t in. (), the displacement of, is defined as () = 0 0, provided that 0! 0. r() =ftjt ;# (t) > 0g, denoting the set of transitions used in. p = ftj'(p; t) ;t g (resp., p = ftj'(t; p) ;t g) is the set of output (resp., input) transitions of place p. _0 0 is defined inductively as follows. Suppose 0 = t ;...;t n. Let 0 be. Ift i is in i0, let i be i0 with the leftmost occurrence of t i deleted; otherwise, let i = i0. Finally, let _0 0 = n. For example, if = t t t 3 t 4 t 5 and 0 = t 4 t 3 t, then _0 0 = t t 5. Given a computation 0! 0, a sequence 0 is said to be a rearrangement of if 8 t, # (t) =# (t) and 00! 0. Conflict-free PNs (cf-pn) [0]: A PN P =(P; ; ') is said to be conflict-free iff for every place p, either jp j, or 8 t p, t and p are on a self-loop (i.e., t (p \ p)). In words, a PN is conflict-free if every place which is an input of more than one transition is also /0$ IEEE