Automatica 35 (1999) 371—384

Towards a stability theory of general hybrid dynamical systems

Anthony N. Michel* ,BoHu

Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Received 29 September 1997; revised 18 May 1998; received in final form 27 August 1998

A general model for hybrid dynamical systems is presented which incorporates a concept of generalized time. For such systems, a stability theory is developed. Both Comparison ¹heorems and the Principal ¸yapunov Stability Results are included.

Abstract

In recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined on real time R>"[0, R). The motions of this class of systems are in general discontinuous. This class of systems may be finite or infinite dimensional. For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems. In the present paper we continue the work described above. In doing so, we first develop a general comparison theory for the class of hybrid dynamical systems (resp., discontinuous dynamical systems) considered herein, making use of stability preserving mappings. We then show how these results can be applied to establish some of the Principal Lyaponov Stability Theorems. For the latter, we also state and prove a converse theorem not considered previously. Finally, to demonstrate the applicability of our results, we consider specific examples throughout the paper. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Hybrid dynamical systems; stability analysis; Lyapunov stability; Lagrange stability

1. Introduction of the type encountered in science and applied mathe- matics (such as, e.g., ordinary differential equations, The demands of modern technology have resulted in ordinary difference equations, partial differential equa- the synthesis and implementation of systems of increas- tions, functional differential equations, Volterra integ- ing complexity. Such contemporary systems frequently rodifferential equations, and the like). Perhaps the most defy simple and tidy descriptions by classical equations important class of such contemporary systems are hybrid systems, whose descriptions involve mixtures of equa- tions of the type enumerated above; subsystems endowed with uncertainties which can best be represented by dif- ferential inclusions (e.g., differential inequalities rather * Corresponding author. Tel.: #001 219 6314395; fax: #001 219 than differential equations); ‘‘equation free’’ subsystems, 6314393; e-mail: [email protected]. such as discrete event systems (along with the necessary  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Guest Editors J.M. interface elements) for which there is no description by Schumacher, A.S. Morse, C. C. Pantelides, and S. Sastry. means of classical equations possible (e.g., subsystems Supported in part by an Alexander von Hamboldt Foundation representable by Petri nets, temporal logic units, etc.); Senior Research Award.  Supported in part by a Center of Applied Mathematics Fellowship, and the like (see, e.g., Grossman et al., 1993; Antsaklis University of Notre Dame. et al., 1993; Go¨ llu¨ and Varaiya, 1989; Brockett, 1993;

0005-1098/99/$—see front matter ( 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 1 6 5 - 4 372 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384

Branicky et al., 1994, Branicky, 1995; Pettersson, 1996 ¹"N"+0, 1, 2, 2, we speak of a discrete-time dynam- and Ye, 1996). ical system. Later on, when discussing hybrid dynamical In recent papers (Ye, 1996; Ye et al., 1995a; Ye et al., systems, we will generalize ¹. Every motion of a dynam- 3¹ 1995b; Ye et al., 1996a; Michel and Hou, 1997; Hou and ical system depends on initial data (t, a), where t is Michel, 1997; Ye et al., 1998), we proposed a general called initial time and a3ALX is called initial point, model for hybrid dynamical systems whose states are where X, the state space,isametric space with metric defined on arbitrary metric space and evolve along some d (i.e., (X, d) is a metric space), and A is an appropriate notion of generalized abstract time. For such systems we subset of X. For a given (t, a), we denote a motion,ifit introduced qualitative characterizations which include exists, by p(t, t , a), t3¹ , where ¹ "[t , t )5¹, and  R ? R ?   the usual notions of Lyapunov and Lagrange stability, where t may be finite or infinite. Thus, a motion is where convergence is defined relative to generalized time. a mapping p( ) , t , a):¹ PX with p(t , t , a)"a, and  R ?   Next, we showed that it is always possible to transform the family of motions which makes up the dynamical this class of hybrid dynamical systems into another class system is obtained by varying the initial point a over the ¹ of dynamical systems with equivalent qualitative proper- set A and the initial time t over , an appropriate ties, but is defined on real time. The motions of this class subset of ¹ (the set of initial times). If we denote such of systems are in general discontinuous. Furthermore, a family by S, then the dynamical system is signified by +¹ ¹ , ¹"¹ this class of systems may be finite or infinite dimensional, the quintuple , X, A, S,  . When , we simply and the motions may be generated by the solutions of write +¹, X, A, S,, and when all is clear from context, we classical equations of the type enumerated above, or by will simply speak of a dynamical system S (rather than +¹ ¹ , ¹" > ‘‘equation free’’ characterizations (e.g., Petri nets, discrete a dynamical system , X, A, S,  ). If R and all event systems, temporal logic elements, and the like). For p3S are continuous with respect to t, we speak of a the above discontinuous dynamical systems (and hence, continuous dynamical system. If the elements of S are for the above hybrid dynamical systems), we established not continuous, we speak of a discontinuous dynamical the Principal Lyapunov Stability Results (uniform sta- system (DDS). Most frequently, the system motions are bility, instability, uniform asymptotic stability (in the determined by means of the solutions of initial-value large), and exponential stability (in the large) of invariant problems. When such problems are defined on a finite- sets (resp., equilibria)) as well as Lagrange Stability Re- dimensional space (e.g., X"RL), they will generate sults (uniform boundedness and uniform ultimate boun- a finite-dimensional dynamical system. When X is not dedness of motions). For some of these results, we also finite-dimensional, the system on hand is said to be an established converse theorems. Finally, we demonstrated infinite-dimensional dynamical system. the applicability of these results by means of specific Examples of continuous-time, finite-dimensional classes of hybrid dynamical systems (including digital dynamical systems include systems determined by control systems and systems whose states are subjected ordinary differential equations and ordinary differential to impulses) (see Hou et al., 1997 and Ye et al., 1996b). inequalities; examples of discrete-time, finite-dimensional In the present paper we continue the work described dynamical systems include systems determined by ordi- above. In doing so, we first develop a general comparison nary difference equations and ordinary difference theory for the class of hybrid dynamical systems (resp., inequalities; and examples of infinite-dimensional dy- discontinuous dynamical systems) considered in these namical systems include systems determined by delay works, making use of stability preserving mappings (see differential equations, functional differential equations, Section 2). We then show how these results can be ap- Volterra integro-differential equations, partial differen- plied to establish some of the Principal Lyapunov Sta- tial equations, differential equations and inclusions on bility Theorems (see Section 3). For the latter, we state Banach space, and the like (Zubov 1964; Hahn 1967a; and prove a new converse theorem (see Section 4). To Yoshizawa 1996; Michel and Wang 1995 and Lakshmi- demonstrate the applicability of our results, we consider kantham and Leela 1969). specific examples throughout the paper. In the reminder of this section, we establish a proper 1.2. Hybrid dynamical systems context for what is to follow. Hybrid dynamical systems are capable of exhibiting 1.1. Dynamical systems simultaneously several kinds of dynamic behavior in different parts of the system (e.g., continuous-time dy- Dynamical systems are families of motions determined namics, discrete-time dynamics, logic commands, dis- by evolutionary processes (see, e.g., Zubov, 1964; Hahn, crete events, jump phenomena, and the like). Typical 1967a; Yoshizawa, 1996 and Michel and Wang, 1995). examples of such systems of varying degrees of com- The evolution of such processes takes place over time plexity include sampled-data control systems (Hou et al., which we denote by ¹. When ¹"R>"[0,R), we speak 1997), discrete event systems (see, e.g., Passino et al., 1995; of a continuous-time dynamical system and when Wang et al., 1994a; Wang et al., 1994b and Passino et al. A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 373

1994), switched systems (see, e.g., Peleties and DeCarlo, 1991 and Branicky, 1998), systems with impulse effects (e.g., Ye et al., 1996b), constrained robotic systems (see, e.g., Rui et al., 1997; Brogliato et al., 1997 and Brogliato, 1996), intelligent vehicle/highway systems Go¨ llu¨ and Varaiya, 1989), motion control systems (see, e.g., Brockett, 1993 and many other types of systems (see, e.g., the papers included in Grossman et al., 1993). Although some efforts have been made to provide a unified frame- work for describing such systems (see, e.g., Branicky et al., 1994; Branicky, 1995 and Petersson, 1996), most of the work reported in the literature to date seems to focus on specific classes of hybrid systems, utilizing ad hoc models. Recently, however, the present authors and their co- workers, have introduced a general model which appears to be suitable for the qualitative analysis of general hybrid dynamical systems Ye et al. (1995a) and Ye et al. (1998). This model incorporates a concept of generalized time. Fig. 1. ‘‘Graph’’ of time space T and embedding mapping for exa- mple 1.1.

Definition 1.1 (¹ime space). A metric space (¹, o)is called a time space if (i) ¹ is completely ordered with which is equipped with a metric o having the property that for any r "(t , q )3¹, r "(t , q )3¹, o(r , r )""t order relation ‘‘O’’; (ii) ¹ has a minimal element t 3¹,   *R   *R    * ! " ¹ i.e., for any t3¹ and tOt , it is true that t Ot; (iii) for t .Theset is completely ordered in such a way that * * O ( ¹ ¹ " any t , t , t 3¹ such that t Ot Ot , it is true that r r if and only if t t.Theset  is given by        + q 3  3 , o(t , t )"o(t , t )#o(t , t ); and (iv) ¹ is unbounded (k, I) R , k N . The motions determined by Eq. (1.1)       " 2 q 2 2 " q 3¹ from above, i.e., for any M'0, there exists a t3¹ such are of the form p(r) (x(t) , u( *R ) ) ,wherer (t, *R ) . o ' The state space for this HDS is given by X"RL;RK,and that (t, t *) M. ALX. We depict the ‘‘graph’’ of ¹ in Fig. 1. ) In the dynamical systems considered earlier (see Sub- 3 section 1.1), time is either R> or N. If we generalize this By characterizing motions p S as mappings that are by replacing time with the time space (¹, o) given in defined on equivalent but possibly different time spaces, Definition 1.1, we end up with a notion of hybrid dynam- the applicability of the preceding notion of HDS is +¹ ¹ , ical system +¹, X, A, S, ¹ , (HDS) which contains most broadened appreciably. In this context, , X, A, S,   ¹ of the specific classes of dynamical systems considered in is an HDS provided that denotes any one member of the literature as special cases. Before generalizing this a family of equivalent time spaces, where equivalence is ¹ ¹K notion of hybrid dynamical system even further, we con- defined as follows: and are equivalent time spaces if ¹P¹K sider a specific example. (Let A and B be sets and let f and there exists an isometric mapping h : such that the ¹ ¹K g map A into B, i.e., f : APB and g : APB. Throughout order relation in and are preserved under h.We ¹&¹K ¹ ¹K the paper, f3C[A, B] and g3C[A, B] will indicate that write to indicate that and are equivalent. ¹K L¹K ¹ L¹ f and g are continuous and continuously differentiable on Further, if  and  , the notation ¹ ¹ & ¹K ¹K ¹&¹K A to B, respectively.) ( , ) ( , ) indicates that with respect to h, and ¹ and ¹K are equivalent with respect to h , the   2 ¹ restriction of h to . (For a more formal and precise Example 1.1 (Nonlinear digital feedback control system). definition of the present notion of HDS, refer to Ye et al., We consider systems described by equations of the form 1995a and Ye et al., 1998.) xR (t)"f (x(t), u(q )), q 4t(q , I I I> (1.1) Example 1.2 (Motion control system, Brockett, 1993). An q " q\ q 3 u( I>) g(x( I>), u( I)), k N, engine-drive train system for an automobile with an where x3RL, u3RK, t3R>, f3C[RL;RK, RL], automatic transmission can be described by equations of g3C[RL;RK, RK] with f (0, 0)"0 and g(0, 0)"0 and the form E_+q , q , 2,04q 4q (2, is a fixed, unbounded,     R " closed set of discrete points. x(r) x(r), System (1.1) determines an HDS with time space given xR (r)"[!a(x (r))#u(r)]/[1#z([p])],   (1.3) by R " ' p(r) l(x, x, u), (l(x, x, u) 0), ¹"+ q 3 , # " (t, *R ) R , (1.2) z([p] 1) f(z([p]), x(r*N ), x(r*N )), 374 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384

3 where x, x R denote vehicle ground speed and engine ical systems encountered in the literature (refer, e.g., to rpm, respectively, u(r)3R denotes the external input as Sections 3.1 and 3.2 in Michel and Wang (1995). In the throttle position, the a( ) ) term describes the inability a similar manner as was done above, we can define 3 of the vehicle to produce torque at high rpms, z ZI uniform boundedness and uniform ultimate boundedness of represents the shift position of the transmission, where motions, and uniform asymptotic stability in the large, ; ; P ZI is some subset of N, and f : ZI R R ZI determines exponential stability in the large, and complete instability the shifting rule. The variable p3C[R>, R>] represents of invariant sets, for HDS’s of the type considered herein a clock or counter. The notation r*N denotes the most (refer to Ye, 1996; Ye et al., 1995a, Ye et al., 1996a; Michel recent time when p passes an integer. and Hou, 1997; Hou and Michel, 1997 and Ye et al., " ; ; As noted in Ye et al. (1998), X R R ZI is a state 1998). Note that in these definitions, convergence is rela- ¹" > ¹ "¹ space for Eq. (1.3), R with  is a time space tive to the generalized time defined earlier in Subsection and the motions of Eq. (1.3) are of the form 1.2 of the present section. 2 [x(r), x(r), z[p(r)]] . In Ye et al. (1998) it is also shown 5 that for different initial conditions r 0, p(r), the sets 1.4. Embedding of Hybrid Dynamical Systems into + 3  5 , determined by (r,[p]) R : r r, p(r) are also time Dynamical Systems Defined on R> spaces for system (1.3), and that these time spaces are all equivalent to R>. For such time spaces, we have Using the isometric mapping e : ¹PR> given by t"(r,[p])3R and the motions of (1.3) are of the form e(t)"o(t, t ), it is shown in Ye (1996), Ye et al. (1995a), " 2 * q(t) [x(r), x(r), z([p])] . For furter details concerning Ye et al. (1996), Michel and Hou (1997), Hou and Michel this example, refer to Ye et al. (1998). ) (1997) and Ye et al. (1998) that every hybrid dynamical +¹ ¹ , system , X, A, S,  can be embedded into a discon- 1.3. Some Qualitative Characterizations of + > I >, tinuous dynamical system (DDS), R , X, A, S, R , Hybrid Dynamical Systems >" ¹ L where R e( ), having the property that if M A, then (S, M) is invariant if and only if (SI , M) is invariant, +¹ ¹ , L For , X, A, S,  , a set M A is said to be invari- and (S, M) and (SI , M) possess identical stability proper- ant with respect to system S if a3M implies that ties. Rather than dwell on all the details (see Ye 1996; Ye p(t, a, t )3M for all t3¹ , all t 3¹ and all  ? R   et al., 1995a; Ye et al., 1995b; Ye et al., 1996a; Michel and ) 3 p( , a, t) S. We will state the above more compactly by Hou, 1997; Hou and Michel, 1997 and Ye et al., 1998), we saying that M is an invariant set of S,or(S, M) is invari- consider a specific example. "+ , ant. If in particular, M x , then x is called an equi- librium. Example 1.3. To embed the HDS determined by Eq. (1.1) The principal objective of the present paper is to study > "o into a DDS defined on R , let e(r) (r, r *). Assuming qualitative properties of invariant sets of HDS. In the " "" " r * (0, 0), then e(r) t . This mapping is depicted in following, we define some of these. Fig. 1. Let Let +¹, X, A, S, ¹ , be an HDS and let MLA  J " q 4 (q be an invariant set for S. We say that (S, M)isstable if for x(t) x(t), I t I>, e' 3¹ d"d e ' J " \ "q every 0 and t , there exists ( , t) 0 such x(t) x(t ), t I>, that d(p(t, a, t ), M)(e for all t3¹ and for all (1.4)  ? R uJ (t)"u(q ), q 4t(q , p( ) , a, t )3S, whenever d(a, M)(d. We say that (S, M)is I I I>  J " J \ \ "q uniformly stable if d"d(e). If (S, M) is stable and if for any u(t) g(x(t ), u(t )), t I> 3¹ g"g ' t  there exists an (t) 0 such that and let y(t)2"[xJ (t)2, uJ (t)2], P(y(t))"[ f (xJ (t), uJ (t))2,02]2 " e' limRd(p(t, a, t), M) 0 (i.e., for every 0, there and Q(y)"[xJ 2, g(xJ , uJ )2]2, where x, u, f, and g are defined 3¹ (e 3¹ exists a tC such that d(p(t, a, t), M) whenever t in Example 1.1. Then the HDS determined by Eq. (1.1) is O ) 3 (g and tC t) for all p( , a, t) S whenever d(a, M) , then embedded into the DDS defined on R> and determined (S, M) is said to be asymptotically stable. We call (S, M) by uniformly asymptotically stable if (S, M) is uniformly d' e' R " q 4 (q stable and if there exists a 0 and for every 0 there y(t) P(y(t)), I t I>, q"q e ' (e (1.5) exists a ( ) 0 such that d(p(t, a, t), M) for all y(t)"Q(y(t\)), t"q . ) t3+t3¹ : o(t, t )5q, and all p( ) , a, t )3S whenever I> ? R   d(a, M)(d. We call (S, M) exponentially stable if there Throughout the remainder of this paper we will assume a' e' 3¹ >" > ¹ "¹ exists 0, and for every 0 and t , there exists that R R (resp.,  ). To simplify our notation, d"d e ' (e \?M(R R) I ( ) 0 such that d(p(t, a, t), M) e for all we will henceforth drop the tilde from S, we will omit all t3¹ and for all p( ) , a, t )3S, whenever d(a, M)(d. reference to R>, and we will write +R>, X, A, S, in place ? R   Finally, we call (S, M) unstable if (S, M) is not stable. of +R>, X, A, SI , to denote the DDS under study. (All The above definitions constitute natural adaptations subsequent results can readily be modified to the more + > >, of corresponding concepts for the usual types of dynam- general case R , X, A, S, R .) A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 375

Summarizing, from the above it is clear that the quali- (ii) "» ; > _+ 3 "»  tative analysis of invariant sets of hybrid dynamical systems M (M R ) x X:x (x, t ) defined on abstract time space ¹ can be reduced to the for some x 3M and t3R>,, qualitative analysis of the same invariant sets of discontinu-   ous dynamical systems defined on R>. (2.2) (iii) the invariance of (S, M)isequivalent to the in- variance of (S, M), i.e., (S, M) is invariant if and only if 2. A General Comparison Theory for Discontinuous (S, M) is invariant; and Dynamical Systems (iv) the stability, uniform stability, asymptotic sta- bility, and uniform asymptotic stability of (S, M) and In the present section we will introduce the concept (S, M) are equivalent, respectively (i.e., (S, M) is stable if of stability preserving mapping between two dis- and only if (S, M) is stable; (S, M) is uniformly stable if + > , continuous dynamical systems R , X, A, S (with and only if (S, M) is uniformly stable; and so forth). + > , invariant set M) and R , X, A, S (with » invariant set M). Such mappings will serve as a basis The above definition states that the function from ; > P for developing a general comparison (stability) theory X R into X induces a mapping V : S S and that for discontinuous dynamical systems. For example, under V several stability properties of (S, M) and » ; > if a stability preserving mapping has been established (S, (M R )) are preserved. Our first result provides between S and S, and if the stability properties of the basis for a comparison theory for the stability anal- (S, M) are well understood, then it will be possible to ysis of the discontinuous (resp., hybrid) dynamical sys- deduce the stability properties of (S, M) from those of tems considered herein. (S, M). ¸ + > , " The initial results concerning stability preserving map- Theorem 2.1. et R , XG, AG, SG , i 1, 2 be two discon- L " pings were developed in Thomas (1964), Hahn (1967b) tinuous dynamical systems and let MG AG, i 1, 2, be » ; >P and subsequent results were reported in Passino et al. closed sets. Assume there exists :X R X which (1995), Wang et al. (1994a,b). In a recent book (Michel satisfies and Wang 1995), the role of stability preserving map- (i) V(S )LS where V(S ) is defined as in Eq. (2.1) and pings in the qualitative analysis of dynamical systems    M is defined as in Eq. (2.2); is explained in detail. All results in Michel and  (ii) there exist t , t 3K defined on R> such that Wang, (1995), Passino et al. (1995), Wang et al.   t 4 » 4t (1994a), Wang et al. (1994b), Thomas (1964) and Hahn (d(x, M)) d( (x, t), M) (d(x, M)) (2.3) (1967b) are concerned with continuous dynamical systems. 3 3 > In the present section we extend some of these previous for all x X, and t R , where d, d are the metrics t3 results to discontinuous dynamical systems. To the defined on X and X, respectively.( K means that t3 > > t " t authors’ knowledge, stability preserving mappings have C[R , R ], (0) 0, and (r) is monotonically not been utilized previously in the qualitative analysis increasing in r.) of discontinuous dynamical systems (resp., of hybrid ¹hen, dynamical systems). (a) the invariance of (S, M) implies the invariance of (S, M), 2.1. Stability Preserving Mappings (b) the stability, uniform stability, asymptotic stability, and uniform asymptotic stability of (S, M) imply the same The following concept will be essential. corresponding types of stability for (S, M); and t " @ ' ' (c) if in Eq. (2.3), (r) ar , a 0, b 0, then the ex- + > , + > , ponential stability of S M implies the exponential Definition 2.1. Let R , X, A, S and R , X, A, S ( , ) be two discontinuous dynamical systems with invariant stability for (S, M). sets M LA and MLA, respectively. We say that   3 » : X ;R>PX is a stability preserving mapping from Proof. (a) If (S, M) is invariant, then for any a M  ) 3 S to S (or more explicitly, from (S , M )to(S, M)) if and any p( , a, t) S, by assumption (i) and the defini-    ) " » satisfies the following conditions: tion of V(S) (see Eq. (2.1)), we have q( , b, t) » ) ) 3 "» 3 (i) (p( , a, t), ) S where b (a, t) M. (Discontinu- 3 ities of p S, at points in EN, will result in discontinuities " _+ ) 3 M S V(S) q( , b, t):q(t, b, t) of q S, at points in EO, where EN EO, since in general, "» "» discontinuities may disappear in the mapping process.) (p(t, a, t), t), with b (a, t) By the invariance of (S, M), we have that and ¹ "¹ , a3A, t 3R>,, (2.1) q(t, b, t )"»(p(t, a, t ), t)3M for all t3¹ "¹ . @ R ? R     @ R ? R 376 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384

Notice that M and M are closed sets and that inequal- the preceding results and of the useful observation given ity (2.3) holds. Therefore, p(t, a, t )3M for all t3¹ . in the following.   ? R Hence, (S, M) is invariant. e ' (b) If (S, M) is stable, then for every  0 and Remark 2.1. If inequalities (2.3) are replaced by the 3 > d "d ' t R , there exists a  (t) 0 such that weaker ones, d (q(t, b, t ), M )(e for all q( ) , b, t )3S and all       t 4 » t3¹ , whenever d (b, M )(d . We now prove that (d(p(t, a, t), M)) d( (p(t, a, t), t), M) @ R    e' 3 > (S, M) is stable. For every 0 and every t R , let 4t (d (p(t, a, t ), M )) (2.4) e "t e d"t\ d (d      ( ) and  ( ). If d(a, M) , then by 4t (t d "d ) 3 3 ; > (2.3), d(b, M) (d(a, M)) ( ) . It follows for every motion p( , a, t) S,(a, t) X R , that for all q( ) , b, t )3S we have d (q(t, b, t ), M )(e t 3¹ , then the conclusions of Theorem 2.1 still        ? R for all t3¹ . Now by Eq. (2.3), it follows that for all hold. @ R p( ) , a, t )3S and for all t3¹ "¹ , where   ? R @ R "» 4t\ » b (a, t), d(p(t, a, t), M)  (d( (p(t, a, t), t), Remark 2.2. If in Theorem 2.1, we replace hypothesis (i) "t\ 4t\ e "e " M))  (d(q(t, b, t), M))  ( ) whenever by the assumption V(S) S, then the following (d d(a, M) . Therefore (S, M) is stable. stronger results can be proved: The proof of the result for uniform stability follows (a) the invariance of (S , M ) and (S , M ) are equivalent along similar lines, choosing d independent of t .       (i.e., (S , M ) is invariant if and only if (S , M )is To prove asymptotic stability (resp., uniform asymp-     invariant); totic stability), it only remains to be shown that (S , M )   (b) » is a stability preserving mapping (see Definition is attractive (resp., uniformly attractive). 3 > 2.1); and If (S, M) is attractive, then for every t R , there t " @ ' ' " » g "g ' ) 3 (c) if in (2.3), G(r) aGr , aG 0, b 0, i 1, 2, then is exists an  (t) 0 such that for all q( , b, t) S, " (g exponential stability preserving, i.e., (S, M)is limRd(q(t, b, t), M) 0 whenever d(b, M) , e ' q"q e ' exponentially stable if and only if (S, M) is exponen- i.e., for every  0, there exists a ( , t, q) 0, " ) 3 (e tially stable. where q q( , b, t) S, such that d(q(t, b, t), M)  for all t3¹ whenever d (b, M )(g . By assump- The above results are of theoretical, rather than practical @ R>O    ) 3 "» tion (i), for every p( , a, t) S, let b (a, t). interest, since in applications it is rarely true that ) "» ) 3 " Then q( , b, t) (p( , a, t)) S. Now for every V(S) S. For this reason, and in the interests of e ' e "t e g "t\ g  0, choose the above  ( ). Let   ( ). economy, we will omit the proofs of these results. Then for every p( ) , a, t )3S ,ift3¹ "¹ ,   ? R>O @ R>O (g whenever d(a, M) , we have by Eq. (2.3) Remark 2.3. (a) Results which are in the spirit of 4t (t g "g that d(b, M) (d(a, M)) ( ) . Thus, Theorem 2.1 and Remark 2.2, can also be established d (q(t, a, t ), M )(e "t (e ) for all t3¹ . Again, for uniform asymptotic stability in the large, exponential       ? R>O by Eq. (2.3), it follows that d(p(t, a, t), M) stability in the large, instability, and complete instability 4t\ (e  (d(q(t, a, t), M)) . Therefore, (S, M)is of invariant sets and for uniform boundedness and attractive. uniform ultimate boundedness of solutions of dis- The proof of the result for uniform attractivity follows continuous, resp., hybrid dynamical systems of the g along similar lines, choosing  independent of t. type considered herein. Counterparts to all of these (c) Assume that (S, M) is exponentially stable. results for the case of continuous dynamical systems a ' By definition, there exits an  0, and for every are presented in Michel and Wang (1995, e ' d "d e '  0, there exists a  ( ) 0 such that for Chap. 3). ) 3 (e \?(R\R) all q( , b, t) S, d(q(t, b, t), M) e for all (b) There may be a temptation to view the notions of t3¹ , whenever d (b, M )(d . By assumption (i), stability preserving mapping and ¸yapunov function as @ R    ) 3 ) " for every p( , a, t) S, there exists q( , b, t) being identical concepts. This, however, is incorrect, as » ) ) 3 "» e ' +¹ ¹ , (p( , a, t), ) S where b (a, t). For every  0, can be seen by considering for ,X,A,S,  with e " e@ a "a " L "+ , » " 3 L choose  a , as above. Let  /b and X R and M 0 , the function (x) x R . This d "t\ d ) 3   ( ). Now for all p( , a, t) M and for all function is clearly a stability preserving mapping. How- t3¹ , whenever d (a, M )(d we have d (b, M ) ever, by any standards, it hardly qualifies as being ? R      4t (t d "d (d(a, M)) ( ) . Thus, it follows that a Lyapunov function. (e \?(R\R) t " @ d(q(t, b, t), M) e . Since (r) ar , we ob- ( e @ \?@(R\R)"e \?(R\R) tain d(p(t, a, t), M) ( /a) e e . 2.2. Applications ) Therefore, (S, M) is exponentially stable. To demonstrate the applicability of the results of In the next subsection where we address applications to the preceding subsection, we consider two specific specific classes of problems, we will make use of some of cases. A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 377

Example 2.1. We consider discontinuous dynamical sys- Assumption (2.6) implies that for e'0(e(1, which will tems determined by equations of the form be specified later) there exists d"d(e)'0 such that xR (t)"A x(t)#F (t,x(t)), q 4t(q , #F (t, x)#4e#x#, #G (t, x)#4e#x# (2.9) I I I I> (2.5) I I> x(t)"B x(t\)#G (t,x(t\)), t"q , k3N, I> I> I> for all x3B(d)"+x3RL:#x#(d, and t3R>, k3N. 3 L"L 3 >; L L 3 q q where AI, BI> R , FI, GI> C[R R , R ], and Combining Eqs. (2.8), (2.9), we obtain for t [ I, I>), " " 3 > FI(t,0) GI>(t,0) 0 for all t R and where _+q q q (q (2, yR (t)4(x(t)2x(t))\(2a#2e)#x(t)#4(a#1)y(t). (2.10) E , , 2 :   is a fixed, unbounded,  closed, discrete set, and Note that Eq. (2.10) is true even at points where x(t)"0. " d lim F (t, x)/#x#"0 and lim G (t, x)/#x#"0, (2.6) We now apply Theorem 2.1. We let X B( )to I I> "q V V derive local stability results. Now for t I>, we have > q "# q # "# q \ # q uniformly for t3R , k3N (# ) # denotes the Euclidean y ( I>) x ( I>) B I>x ( I>) GI>( I>, q\ # d ' norm). x( I>)) . By Eq. (2.6) we know that there exists  0 # #4# # # #(d such that FI(t, x) x whenever x . We show System (2.5) may be a consequence of a linearization d ' d (d \(?>)H that there exists a  0(  e ) such that process of the system of discontinuous differential equa- # #4d 3 q q # q #4d x(t)  for all t [ I, I>) whenever x( I)  for tions, 3 3 q q any k N. For otherwise, there must exist a t [ I, I>) R " q 4 (q # #"d # #(d 3 q x(t) fI(t, x(t)), I t I>, such that x(t) , while x(t)  for all t [ I, t). (2.7) # #4# # 3 q x(t)"g (t, x(t\)), t"q , k3N, Then FI(t, x(t)) x(t) for all t [ I, t). Since for I> I> t3[q , t ), x(t)"x(q )#R (A x(q)#F (q, x(q))) dq which I  I OI I I " about the point x% 0, where it is assumed that implies that f (t,0)"g (t,0)"0 and f , g 3C[R>;RL, RL] I I> I I> R and where we assume that the Jacobians # #4# q ## # ## q ### q q # q x(t) x( I)  ( AI x( ) FI( , x( )) )d * * " " * * " " OI [ fI(t, x)/ x] V AI and [ gI>(t, x)/ x] V BI> R 4# q ## a# # q # q are constant matrices. Assume that for every x( I) ( 1) x( ) d . 3 >; L  (t, x) R R , every solution of Eq. (2.5) can be ex- OI > tended to R . By the Gronwall inequality we obtain that # #4# q # (?>)(R\OI)4# q # (?>)H x(t) x( I) e x( I) e . It follows that Theorem 2.2. Assume that # #4d (?>)H(d x(t) e  which contradicts the previous a' # #"d (i) there exists a constant 0 such that for all assumption that x(t) . Thus, we know that when- 3 # #(a # ) # # q #4d k N, AI ( denotes the matrix norm induced ever x( I) , then it is true that # #4d # #4# # by the Euclidean vector norm); x(t) , FI(x(t)) x(t) and +q !q ,4j(R (ii) supIZ, I> I , and either #º #( ( 3 (iii) I q 1 for k N, where q is a constant, or #x(t)#4#x(q )#e(?>)H, (2.11) º Pº PR º I (iv) I as k , and is Schur stable, for t3[q , q ). We obtain º _ I(OI>\OI) 3 I I> where in (iii) and (iv) I BI>e , k N. OI> q\ " I(OI>\OI) q # I(O\OI) q q q " x( I>) e x( I) e FI( , x( )) d . Then the equilibrium xC 0 of system Eq. (2.5) is uni-  + , OI formly asymptotically stable (i.e., (S( ), 0 ) is uniformly (2.12) asymptotically stable, where S denotes the discon- ( ) d ' tinuous dynamical system determined by Eq. (2.5)). By Eq. (2.11) we know that there exists a  0 such that # #(d 3 q q # q #4d x(t) for t [ I, I>) whenever x( I) . There- # q #4d Proof. We only present the proof under the assumptions fore, whenever x( I)  is satisfied, we have (i), (ii), and (iii) of the theorem. (For the other cases the OI> OI> I(O\OI) q q q 4 ?He# q # q proof is similar.) e FI( , x( )) d e x( ) d "   Let S S( ) denote the dynamical system generated OI OI by the dynamical system described by Eq. (2.5), let OI> 2  4 ?He (?>)H# q # q »(x)"(x x) , and for any solution x(t) of Eq. (2.5), let  e e x( I) d "» 3 q q OI y(t) (x(t)). Then for t [ I, I>), R " 2 \ 2 R # R 2 4jee(?>)H#x(q )#. y(t) (x(t) x(t)) (x(t) x(t) x(t) x(t)) I " 2 \ 2 # 2 # 2 Since (x(t) x(t)) (x(t) (AI AI)x(t) x(t) FI(t, x(t)) # 2 # #"#º \ I(OI>\OI)#4 ?H FI(t, x(t)) x(t)). (2.8) BI> Ie qe 378 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384

# q\ #(d we have (since x( I>) ) where x t 3RL u t 3RK f3C RL;RK RL q "# q #"# I(OI>\OI) q ( ) , ( ) , [ , ], y( I>) x( I>) BI>e x( I) g3C[RL;RK, RK], f (0, 0)"0, and g(0, 0)"0. OI> Following Example 1.3, the dynamical system deter- #B e I(O\OI)F (q, x(q)) dq##e#x(q\ )# I> I I> mined by Eq. (2.15) is embedded into the dynamical OI system defined on R> and defined by Eq. (1.5). Lineariz- 4 # q ## (?>)Hje# q # q x( I) qe x( I) ing Eq. (1.5), we obtain the system #e ?H# q ##je (?>)H# q # AB (e x( I) e x( I) ) yR (t)" y(t)#FI (y(t)), q 4t(q , C00D I I> "[q(1#jee(?>)H)#e(e?H#jee(?>)H)]#x(q )#. (2.16) I I 0 y(t)" y(t\)#HI (y(t\)), t"q , k3N, e ' C D I> Therefore, there exists an  0 such that DC " #je (?>)H #e ?H#je (?>)H ( where q q(1 e ) (e e ) 1. (2.13) *f(x, u) *f(x, u) A" " , B" " , *x V S *u V S Clearly B(d(e )) is included in the region of attraction, and  *g(x, u) *g(x, u) we can pick d(e ) as large as possible in accordance with " " " "  C V S, D V S, Eqs. (2.9) and (2.13). *u *x Now consider the system and yR (t)4(a#1)y(t), q 4t(q , #FI (y)# #HI (y)# I I> (2.14) lim "0, lim "0. q 4 q 3 #y# #y# y( I>) qy( I), k N. W W Next, let The function »(x(t)) induces a mapping V from " " L I 0 AB S S( ) to S S( ) which satisfies V(S) S. º " q !q " I exp ( I> I) Since the equilibrium yC 0 of Eq. (2.14) is (uniformly) CDCD GC00D H + , asymptotically stable, i.e., (S, 0 ) is (uniformly) asymp- totically stable, it now follows from Theorem 2.1 that the OI>\OI " e (OI>\OI) e O dq B equilibrium x% 0 of system (2.5) is uniformly asymp- A B totically stable, i.e., (S , +0,) is uniformly asymptotically "  . (2.17)  OI>\OI stable. ) De (OI>\OI) C#DA e O dqBB  Remark 2.4. It can be shown that Theorem 2.2 still holds We are now in a position to apply Theorem 2.2 to obtain if either (iii) is replaced by the following result. #º #( (v) lim supI I 1, or, if (iv) is replaced by Corollary 2.1. For system (2.15), resp., (2.16), assume that ¹ º (vi) lim sup +max"j(º )",(1 and every subsequ- conditions (ii)-(iv) of heorem 2.2 are satisfied, where I is I I ¹ 2 2 2" 2 2 2 ence of +º , contains a subsequence which converges givenbyEq.(2.17). hen the equilibrium (x , u ) (0 ,0 ) I of system (2.15) is uniformly asymptotically stable. to a Schur stable matrix and the solutions PI of º2 º ! "! # ! #( IPI I PI I satisfy lim supI PI> PI 1, where I is the identical matrix. For sampled-data systems of the form (see Francis and Clearly, (v) and (vi) are less conservative than (iii) and Georgiou, 1988) (iv) in Theorem 2.2, respectively. We will not pursue the xR (t)"Ax(t)#Bu(q ), q 4t(q , I I I> (2.18) proofs of these statements. q " q # q 3 u( I>) Cu( I) Dx( I), k N which can be written in the form Example 2.2 (Sampled-data control systems). We consider R " # q q 4 (q once more the sampled-data control systems described x(t) Ax(t) Bu( I), I t I>, by equations of the form (see Example 1.1) OI\OI\ q " ! \ (OI\OI\) O q ) q u( I) (C e  e d B) u( I\) xR (t)"f(x(t), u(q )), q 4t(q ,  I I I> (2.15) q " q\ q 3 # \ (OI\OI\) q u( I>) g(x( I>), u( I)), k N, De x( I), A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 379 we know that when Theorem 3.1. ¸et +R>, X, A, S, be a DDS and let MLA be a closed set. Assume there exists a function » ; >P > I 0 : X R R which satisfies the following conditions: º " ) N 3 » ) N ) 3¼ I OI>\OI (i) for every motion p( , a, t) S, (p( , a, t), ) ! \ (OI>\OI) O q ) \ (OI>\OI) L "+ N"qN qN qN 4 C e  e d BDe with E4(N) EN where EN t , , , 2,:0  qN(qN(qN(2,    is the set of discontinuous points of ) N p( , a, t), which is assumed to be unbounded, discrete, and AB closed; and ;exp (q !q ) t t 3 GC00D I> I H (ii) there exist ,  K such that t 4» 4t OI>\OI (d(x, M)) (x, t) (d(x, M)) (3.1) e (OI>\OI)  e O dq ) B "  , (2.19) for all x3X and t3R>. DC Then the following statements are true: 3 » satisfies the hypotheses of Theorem 2.2, the following (a) If for any p S, (p(t, a, t), t) is nonincreasing for 5 N5 result holds. all t t 0, then (S, M) is invariant and uniformly stable. t 3 > (b) If there exists  K which is defined on R such Corollary 2.2. For system (2.18), assume that conditions 3 ¹ º that for all p S, (ii)—(iv) of heorem 2.2 are satisfied, where I is given by Eq. (2.19). ¹hen the equilibrium (x2, u2)2"(02,02)2 of »Q N 4!t N qN4 (qN (p(t, a, t), t) (d(p(t, a, t), M)), I t I>, system (2.18) is uniformly asymptotically stable. » N 4» \ \ "qN 3 (p(t, a, t), t) (p(t , a, t), t ), t I>, k N, (3.2) Note that if q !q "h for k3N, and if we let I> I then (S, M) is uniformly asymptotically stable. g(x, u)"x, then by Corollary 2.2 we can conclude that if (c) If in particular, in part (b) t (r)"a r@, a '0, e F#Fe O dq is Schur stable, then x "0 of a sampled- G G G  C b'0, i"1, 2, 3, then (S, M) is exponentially stable. data system of the form xR (t)"f (x(t), x(kh)), kh4t((k#1)h, k3N Proof. We apply Theorem 2.1. In the following we let _» y(t) (p(t, a, t), t) and consider dynamical systems is uniformly asymptotically stable. This result was re- determined by cently reported in Rui et al. (1997). Furthermore, for this yR (t)4g(y(t)), qN4t(qN , case, Corollary 2.2 reduces to a result given in Francis I I> (3.3) 4 \ "qN 3 and Georgiou (1988). y(t) y(t ), t I>, k N, where g3C(R>, R) with g(0)"0 and for the cases (a)—(c), we assume that g has the following forms: 3. The principal Lyapunov theorems , ,!t t\ _!t t3 (a) g(y) 0, (b) g(y)  °  (y) (y)( K) and (c) g(y),! a /a y_!ay. In the present section we employ the results of the   We denote by S the discontinuous system deter- preceding section (Theorem 2.1) to establish the principal ( ) mined by solutions of Eq. (3.3) with points of disconti- Lyapunov stability results for the class of DDS (resp., qN qN qN nuity at , , , 2. To simplify our discussion, we HDS) considered in the present paper. 5 N5 assume that the solutions of Eq. (3.3) exist for t t 0. " " " " "» ; > Let X X, A A, S S, X R, A (A R ), Definition 3.1. A motion p of a discontinuous dynamical S "S , and note that V(S )LS . We conclude by + > ,  ( )   system R , X, A, S is said to belong to class W (i.e., Theorem 2.1 that the invariance of (S , +0,) implies the 3¼ N ( ) p )ifp(t, a, t) is differentiable with respect to invariance of (S, M) and the uniform stability, uniform 3) ! t N EN and differentiable from the right at every asymptotic stability and exponential stability of (S , ) ( ) point in EN, where N is the domain of definition of p and +0,) imply the same corresponding types of stability for N R is of the form [t, ) and EN denotes an unbounded, (S, M). It remains to be shown that under the assump- discrete, closed set which is determined by the particular + , tions of the theorem, (S( ), 0 ) possesses the desired motion p being considered. respective properties. (a) If g(y),0, then from Eq. (3.1) we have that 4t\ » Without necessarily mentioning it, we will assume d(x, M)  ( (x, t)) and the invariance and uniform + , throughout this section that for a given DDS, S consists stability of (S( ), 0 ) follow. Also by Eq. (3.3), for arbi- only of motions that belong to class ¼ (i.e., SL¼). trary e'0, there exists a d"e such that whenever 380 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384

4 qN (d 4 4 qN 4e 0 y( ) , we always have 0 y(t) y( ) , which where implies that (S , +0,) is uniformly stable. ( ) 1 "! t 4 » qN qN qN _ » qN qN qN (b) For the present case, since g(y) (y) 0, D (p( I, a, ), I) [ (p( I>, a, ), I>) 3 qN qN R 4 4! t qN qN !qN we have for t [ I, I>), y(t) g(y(t)) (y( I>)). I> I Integrating from qN to qN\ , we obtain (since y(qN )4 !» qN qN qN I I> I> (p( I, a, ), I)], y(qN\ )) I> then (S, M) is uniformly asymptotically stable. qN ! qN 4 qN\ ! qN y( I>) y( I) y( I>) y( I) (c) If in addition to the assumptions in (a) and (b), t r "a r@ a ' b' i" and there exists 4!t y qN qN !qN G( ) G , G 0, 0, 1, 2, 3, ( ( I>))( I> I). ' h "h"O" a constant q 0 such that limF h( )/ 0, and if +qN !qN,(R Taking the sum of the above inequalities for supIZ, NZ1 I> I , then (S, M) is exponentially k"0, 1, 2, n!1, we obtain stable. y(qN)!y(qN)4!t(y(qN))(qN!qN). _» qN L  L L  Proof. Let y(t) (p(t, a, ), t). We apply Theorem 2.1 to Therefore, prove the present results. In doing so, we employ as a comparison system the dynamical system determined y(qN)!y(qN) y(qN) y(qN)4t\  L 4t\  . by inequalities of the form L A qN!qN B AqN!qNB L  L  y(t)4h(y(qN)), qN4t(qN , I I I> (3.5) For arbitrary e'0, there exist d"d(e)'0 and qN 4u qN qN qN 3 y( I>) (y( I), I, I>), k N. ¹"¹(e), such that t\(d/¹)(e. Now, whenever qN (d e !qN5¹ For the different cases (a)—(c), we let u assume the follow- y( ) ( ) and t  , there always exists an n such (qN qN!qN'¹ ing forms: that t L. We therefore have L  , which implies that y(t)4y(qN)4t\(d/¹)(e, i.e., (S , +0,) is uni- u qN qN qN " qN L ( ) (a) (y( I), I, I>) y( I); formly asymptotically stable. u qN qN qN " qN !t qN qN !qN (b) (y( I), I, I>) y( I) (y( I))( I> I) with (c) In the current case, Eq. (3.3) assumes the form t"t t\3   K; u qN qN qN " qN ! qN qN !qN yR (t)4!ay(t), qN4t(qN , (c) (y( I ), I, I>) y( I) ay( I)( I> I) with I I> " ° (3.4) a a/a. y(t)4y(t\), t"qN , k3N, I> Proceeding now in a similar manner as in the proof of ' + , where a 0. It is easily shown that (S( ), 0 ) is expo- Theorem 3.1, the present results follow. We omit the nentially stable. We omit the details. details. )

Results for uniform asymptotic stability in the large, Theorem 3.1 (and similar results), constitute natural ex- exponential stability in the large, uniform boundedness tensions to discontinuous dynamical systems of existing and uniform ultimate boundedness of motions, which are results for continuous dynamical systems. For a detailed in the spirit of the above results can also be established by coverage of the latter, refer to Michel and Wang (1995, the methodology employed in the proof of Theorem 3.1. Chap. 4). If in particular M"+0,, then the conditions in Theorem 3.1 require that » be positive definite and Theorem 3.2. ¸et +R>, X, A, S, be a DDS and let MLA decrescent, that »Q be less than or equal to zero, or less » ; >P > 3 q q " be a closed set. Assume there exists :X R R and than zero, for all t ( I>, I), k 0, 1, 2, 2, and that at t t 3 > t 4» ,  K defined on R such that (d(x, M)) (x, t) the points of discontinuity, jumps in » be in the ‘‘down- 4t 3 3 > (d(x, M)) for all x X and t R . ward’’ direction. ) qN 3 » ) qN ) Theorem 3.2, and other similar results appeared first in (a) Assume that for any p( , a, ) S, (p( , a, ), ) L Ye et al. (1995b) and in several subsequent publications has a set of discontinuities E4(N) EN, where "+qN qN 4qN(qN(2 , Ye, 1996; Ye et al. 1996a; Michel and Hou, 1997; Hou and EN , , 2,:0   is the set of points of ) qN Michel, 1997; Ye et al., 1998). Results which are in the discontinuities of p( , a, ) which is assumed to be an qN R spirit of Theorem 3.2 are less conservative than corre- unbounded, closed, and discrete subset of [ , ). If »(p(qN, a, qN), qN) is nonincreasing for k3N and if there sponding results that are in the spirit of Theorem 3.1. If in I  I M"+ , exists a function h3C[R>, R>], independent of p3S, such particular 0 , then results of the type given in » that h(0)"0 and »(p(t, a, qN), t)4h(»(p(qN, a, qN), qN)) for Theorem 3.2 require that satisfy certain kinds of  I  I q q k" t3(qN, qN ), k3N, then (S, M) is invariant and uniformly bounds over ( I, I>), 0, 1, 2, 2, and that at I I> q k" » stable. I, 0, 1, 2, 2, be nonincreasing, or strictly de- (b) If in addition to the assumptions in (a), there exists creasing, and so forth. Prior to the publication of Ye et al. t 3K such that (1995b), Ye (1996), Ye et al. (1996a), Michel and Hou  (1997), Hou and Michel (1997), Ye et al. (1998), stability » qN qN qN 4!t qN qN qN D (p( I, a, ), I) (d(p( I, a, ), I), M), studies of special classes of hybrid dynamical systems and A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 381

J " 4 ( # 3 3 " discontinuous dynamical systems were primarily in the p(t, a, t) p(k, a, k), k t k 1, k N, k N, t k. spirit of results of the type given in Theorem 3.1 (see, e.g., It is not difficult to show that the qualitative properties of Branicky et al. 1994; Branicky, 1995; Branicky, 1998, SI imply corresponding qualitative properties of S. (E. g., if Pavlidis, 1967; Barabanov and Starozhilov, 1988 and MLA,(SI , M) invariant implies (S, M) invariant, (SI , M) Bainov and Simeonov, 1989). Recently, however, works stable implies (S, M) stable, and so forth.) The converse to on HDS have appeared which are more along the lines of this statement is also true. Theorem 3.2 (see, e.g. Pettersson, 1996 and Pettersson In view of the above discussion, we can now apply and Lennartson, 1996b). Theorem 3.2 directly to system +R>, X, A, SI , to conclude The following example demonstrates that Theorem 3.2 the following result for the corresponding discrete-time is applicable to a much larger class of problems than is dynamical system +N, X, A, S,. Theorem 3.1. Theorem 3.3. ¸et +N, X, A, S, be a discrete-time dynam- Example 3.1. Consider discontinuous dynamical systems ical system and let MLA be a closed set. Assume there » ; P > t t 3 determined by equations of the form exists a function : X N R and ,  K defined > t 4» 4t on R such that (d(x, M)) (x, k) (d(x, M)) for xR (t)"A x(t), qN4t(qN , all x3X and k3N. I I I> (3.6) x(t)"B x(t\), t"qN , k3N. 3 5 » I> I> (a) If for all p S and k k, (p(k, a, k), k) is nonin- creasing, then (S, M) is invariant and uniformly stable. 3 Suppose that for each k N, AI has only eigenvalues with (b) If in addition to the above assumptions there exists positive real parts. Then, it is impossible to find t 3K defined on R> such that a Lyapunov function » which satisfies Eq. (3.2). There-  » fore, Theorem 3.1 is not applicable to the present D (p(k, a, k), k) example. _»(p(k#1, a, k ), k#1)!»(p(k, a, k ), k) Now assume that conditions (i)—(iii) of Theorem 2.2   hold for the present example. Let »(x)"#x#. Then since 4!t (d(p(k, a, k ), M), N N   » qN qN qN "# I(O I>\OI) qN qN #4 (p( I>, a, ), I>) BI>e p( I, a, ) q » qN qN qN then (S, M) is uniformly asymptotically stable. (p( I, a, ), I), we have (c) If in addition to the assumptions in (a) and (b) it is q!1 t @ D»(p(qN, a, qN), qN)4 »(p(qN, a, qN), qN) true that (r) is of the form a r , a '0, b'0, i"1, 2, 3, I  I qN !qN I  I G G G I> I then (S, M) is exponentially stable. 1!q 4! #p qN a qN # h "h j ( I, , ) , To obtain the above results, we let h( ) , qN" qN " # 3 I k, I> k 1 for all p S, and we make other where D» is as defined in Theorem 3.2. To apply The- necessary (obvious) modifications in applying t "t " orem 3.2, we let in conditions (a) and (c) (r) (r) r, Theorem 3.2. t " ! j " ?H (r) (1 q)/ r and h(x) e x. All conditions of The preceding stability results for discrete-time dynam- Theorem 3.2 are now satisfied and it therefore follows ical systems, as well as other stability results for discrete- that the trivial solution of Eq. (3.6) is exponentially time systems, not explicitly stated above, are well known stable. (see, e.g., [Michel and Wang, 1995, Chap. 4]).

The reduction in conservatism for results such as The- orem 3.2, when compared to results such as Theorem 3.1, 4. Converse Theorems is made possible by a judicious choice of the comparison system in the application of comparison results. In addi- In the case of continuous dynamical systems, tion to the reduction in conservatism, the choice of com- +R>, X, A, S,, necessary conditions for various types of parison system in Theorem 3.2 enables us also to deduce stability, instability, and boundedness have been estab- the stability properties of invariant sets for discrete-time lished for results involving scalar-valued Lyapunov dynamical systems. functions (i.e., for the Principal Lyapunov Stability To see this, we consider an arbitrary discrete-time Results). The same statement is also true for discrete-time dynamical system +N, X, A, S, (see, e.g., Michel and dynamical systems +N, X, A, S,. For a presentation of Wang, 1995). The motions p3S for this system are of the these results, which are called Converse ¹heorems, refer, form p( ) , a, k ):¹ PX where ¹ denotes the do- e.g., to Michel and Wang (1995).  ? I ? I ) main of definition of p( , a, k). We can associate with In Ye (1996), Ye et al. (1995b), Ye et al. (1996a), Michel +N, X, A, S, a unique continuous-time dynamical system, and Hou (1997), Hou and Michel (1997), Ye et al. (1998) with discontinuous motions (i.e., a DDS), +R>, X, A, SI ,, we extended the above results for the case of dis- where for every p3S, we have pJ 3SI , given by continuous dynamical systems, +R>, X, A, S,, for uniform 382 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 stability and for uniform asymptotic stability. Continuing (iii) there exists a function h3C[R>, R>] with h(0)"0 h hO" ' this work, we state and prove in the present paper a con- and limF>h( )/ 0 for some constant q 0 such » qN 4 » qN qN qN verse theorem for the case of exponential stability.Itis that (p(t, a, ), t) h( (p( I, a, ), I)) for every ) qN 3 3 qN qN 3 qN3 > possible to establish results of this kind for uniform p( , a, ) S, t ( I, I>), a A and  R . asymptotic stability in the large, exponential stability in the large, uniform boundedness and uniform ultimate Proof. By Problem 3.8.10 in Michel and Wang (1995) 3 boundedness of solutions, instability, and complete insta- there exists a function K defined on [0, r] and a con- bility. However, due to space limitations, we will forego stant a'0 such that this. N qN ( \?(R\O ) We will require the following hypotheses. d(p(t, a, ), M) (d(a, M))e (4.2)

for all p( ) , a,qN)3S and all t3¹ N , whenever + > ,  ? O  Assumption 4.1. Let R , X, A, S be a DDS and assume d(a, M)(r . Let X "+x3A:d(x, M)(r , and let ) 3 J ) 3    that (i) for any p( , a, t) S, there exists a p( , a, t) S 5 " J +a3 ( \ , ' with t t and a p(t, a, t) such that p(t, a, t) A : d(a, M) (r) if (r) r, " 5 A " p(t, a, t) for all t t; and (ii) for any two motions  GX otherwise. ) 3 " ' "  pG( , aG, tG) S, i 1, 2, t t,ifa p(t, a, t), then L ) 3 L 3 ; > there exists p( , a, t) S such that p(t, a, t) For (x, t) X R , define " 3 L " p(t, a, t) for t [t, t) and p(t, a, t) p(t, a, t) »(x, t)"sup +d(p(t, x, t), M)e?(RY\R),. (4.3) 5 for t t. RYYR Now for a3A and qN3R>, we have J )   In part (i) of this assumption, p( , a, t) may be viewed ) » qN " +  qN ?(RY\R), as a partial motion of the motion p( , a, t), and in part (p(t, a, ), t) sup d(p(t , p(t, a, ), t), M)e (ii), pL ( ) , a, t ) may be viewed as a composition of RYYR  " +  qN ?(RY\R), p ( ) , a , t ) and p ( ) , a , t ). With this convention, sup d(p(t , a, ), M)e . (4.4)       RYYR Assumption 4.1 states that (a) any partial motion is a motion in S, and (b) any composition of two motions is For k3N, we have a motion in S. » qN qN qN (p( I>, a, ), I>) N N N Assumption 4.2. Let +R>, X, A, S, be a DDS and assume " +  qN ?(RY\OI) \?(OI>\OI ), sup d(p(t , a, ), M)e e ) qN 3 N that every p( , a, ) S is continuous everywhere on RYYO I> [qN, R), except possibly on an unbounded, closed, N  4 sup +d(p(t, a, qN), M)e?(RY\OI), e\?J discrete set E "+qN, qN, 2, qN(qN(2, (where N  N     RYYO I> ) qN 3 " EN depends on p( , a, ) S), and that l infIZ, NZ1 N +qN !qN,' ¸_ +qN !qN,(R 4 sup +d(p(t, a, qN), M)e?(RY\OI), e\?J 0, and that sup . N  I> I IZ, NZ1 I> I RYYO I> " \?J» qN qN qN We now state and prove the main result of the present e (p( I, a, ), I). section. Letting c_(1/¸)(1!e\?J), we obtain Theorem 4.1. ¸et +R>, X, A, S, be a DDS and let MLA » qN qN qN D (p( I, a, ), I) be a closed invariant set, where A is a neighborhood of M. 1 Suppose that system S satisfies Assumptions 1 and 2 and " (»(p(qN , a, qN), qN ) 3 ; > qN !qN I>  I> that for every (a, t) A R , there exists a unique motion I> I p( ) , a, t )3S. ¸et (S, M) be exponentially stable. ¹hen !» qN qN qN  (p( I, a, ), I)) there exist neighborhoods A and X of M such that A LX LA, and a mapping » : X ;R>PR> which 1    4! ! \?J » p qN a qN qN satisfies the following conditions: qN ! qN(1 e ) ( ( I, , ), I) I> I t t 3 > (i) there exist ,  K (defined on R ) such that 4!c»(p(qN, a, qN), qN). t 4» 4t 3 I  I (d(x, M)) (x, t) (d(x, M)) for all (x, t) ; > 4» 4 X R ; Also, Eqs. (4.2)—(4.4) imply that d(x, M) (x, t) ' 3 ; > (ii) there exists a constant c 0 such that for every (d(x, M)) for all (x, t) X R . By Eq. (4.4) we have for ) qN 3 3 qN qN p( , a, ) S, every t [ I, I>) that N N » qN qN qN 4! » qN qN qN » qN " +  qN ?(RY\O I) \?(R\O I), D (p( I, a, ), I) c (p( I, a, ), I) (4.1) (p(t, a, ), t) sup d(p(t , a, ), M)e e RYYR for k3N, where a3A and where D» is defined in N  4sup +d(p(t, a, qN), M)e?(RY\O I), ¹  heorem 3.2; and RYYR A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 383

N 4sup +d(p(t, a, qN), M)e?(RY\O I), Poincare´ maps have successfully been employed in sev- N  RYYRI eral interesting investigations, including the stabilization "» qN qN qN of mechanical systems (see, e.g., Brogliato et al., 1997 and (p( I, a, ), I). Brogliato, 1996). " " ) The proof is completed by letting h(r) r, q .

We conclude by pointing to the importance of converse References theorems. In general, they are used to establish other qualitative results for dynamical systems which are im- Antsaklis, P. J., Stiver, A., & Lemmon, M. D. (1993). Hybrid system portant in their own right. In the present case, Theorem modeling and autonomous control aystems. In [Grossman et al., 4.1 and the companion results established in Ye (1996), 1993], 366—392. Ye et al. (1998) show that under reasonable additional Bainov, D. D., & Simeonov, P. S. (1989). Systems with impulse effects: hypotheses, the results of Theorem 3.2 are as good as you Stabillity theory and applications. New York: Halsted Press. can expect to obtain, under the given assumptions of that Barabanov, A. T., & Starozhilov, Ye. F. (1988). Investigation of the stability of continuous discrete systems by Lyapunov’s second theorem. methods. Sov. J. Automat Inform. Sci., 21(6), 35—41. Branicky, M. S. (1995). Studies in hybrid systems: Modelling, analysis, and control. Ph.D. dissertation, MIT. 5. Concluding remarks Branicky, M. S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE ¹rans. Automat. Con- trol, 43(4), 475—482. In the present paper we first established several results Branicky, M. S., Brokar, V. S., & Mitter, S. K. (1994). A unified (Theorem 2.1) which are the basis of a general compar- framework for hybrid control. Proc.33rd IEEE Conf. on Decision ison theory for a large class of hybrid dynamical systems and Control (pp. 4228—4234). Lake Beuna Vista, FL. (resp., discontinuous dynamical systems), making use of Brockett, R. W. (1993). Hybrid models for motion control systems. In: H. L. Trentelman, & J. C. Willems (Eds.), Essays on control perspec- stability preserving mappings (between a comparison tives in the theory and its applications (pp. 29—53) Boston: system and the system under investigation). Using Birkhauser. Theorem 2.1, we analyzed several specific classes of dis- Brogliato, B. (1996). Nonlinear impact mechanics: Models, dynamics and continuous dynamical systems (Example 2.1 and The- control. London: Springer. orem 2.2 and Example 2.2 and Corollaries 2.1 and 2.2). Brogliato, B., Niculescu, S. I., & Orhant, P. (1997). On the control of finite-dimensional systems with unilateral constraints. IEEE ¹rans. By utilizing comparison systems, we established some Autom. Control, 42(2), 200—215. of the Principal Lyapunov Theorems (Theorems 3.1, 3.2, Francis, B. A., & Georgiou, T. T. (1988). Stability theory for linear and 3.3). They were proved previously in Ye (1996), Ye et time-invariant plants with periodic digital Controllers. IEEE ¹rans. al. (1995a, b), Ye et al. (1996a), Michel and Hou (1997), Automat. Control, 33(9), 820—832. Hou and Michel (1997), Ye et al. (1998), using different Go¨ llu¨ , A., & Varaiya, P. P. (1989). Hybrid dynamical systems. Proc. 28th IEEE Conf. on Decision and Control (pp. 2708—2712). Tampa, methods. Of particular note are the results in Theorems FL. 3.2 which are quite a bit different from the usual stability Grossman, R., Nerode, A., Ravn, A., & Rischel, H. (Eds.), (1993). Hybrid results. These results have made it possible to reduce the systems. New York: Springer. conservatism of several stability results reported in the Hahn, W. (1967). Stability of motion. Berlin and New York: Springer. literature for special classes of hybrid dynamical systems. Hahn, W. (1967). U® ber stabilita¨ tserhaltende Abbildungen und ljapunovsche Funktionen. J. angewandte Mathematik, 228, Furthermore, Theorem 3.2 enables one to obtain stability 189—192. results for discrete time dynamical systems as well Hou, L. & Michel, A. N. (1997). Stability analysis of a general (Theorem 3.3). class of hybrid dynamical systems. Proc. Amer. Control Conf. Finally, in the spirit of our earlier work, we stated and (pp. 2805—2809) Albuquerque. proved a converse theorem that had not been reported Hou, L., Michel, A.N., & Ye, H. (1997). Some qualitative properties of sampled-data control systems. IEEE ¹rans. Automat. Control, previously (Theorem 4.1). 42(12), 1721—1725. We conclude by pointing out that whereas the class of Lakshmikantham, V., & Leela, S. (1969). Differential and integral in- systems considered in the present paper (and in our equalities (Vol 1 and 2) New York: Academic Press. previous works on this subject) is very large and inclu- Michel, A. N., & Hou, L. (1997). Modeling and qualitative theory for sive, there are of course many important classes of hybrid general hybrid dynamical and control systems. Proc. IFAC/IFIP/ IMACS Conf. on Control of Industrial Systems (Vol. 1, pp. 173—183). and discontinuous dynamical systems that cannot be Belfort, France. analyzed by these results in their present form. As a spe- Michel, A. N., & Wang, K. (1995), Qualitative theory of dynamical cific case, we wish to point to (mechanical) systems with systems. New York: Marcel Dekker, unilateral constraints (refer, e.g., to Brogliato et al., 1997 Passino, K. M., Burgess, K. L., & Michel, A. N. (1995). Lagrange and Brogliato, 1996). For example, when in a robotic stability and boundedness of discrete event systems. Discrete Event Dyn. Systems: ¹heory Appl. 5, 383—403. system a transition (impact) phase occurs, the continuous Passino, K. M., Michel, A. N., & Antsaklis, P. J. (1994). Lyapunov vector field does not necessarily possess any fixed point stability of a class of discrete event systems. IEEE ¹rans. Automat. and the invariant set is usually a limit cycle. In such cases, Control, 39(2), 269—279. 384 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384

Pavlidis, T. (1967). Stability of systems described by differential Anthony N. Michel received the Ph.D. de- equations containing impulses. IEEE ¹rans. Automat. Control, gree in electrical engineering in 1968 from 12(1), 43—45. Marquette University, Milwaukee, WI, Peleties, P., & DeCarlo, R. (1991). Asymptotic stability of m-switched and the D. Sc. degree in applied mathemat- ics from the Technical University of Graz, systems using Lyapunov-like functions. Proc. Amer. Control Conf. Austria, in 1973. (pp. 1679—1684) (Boston, MA). He has seven years of industrial experi- Pettersson, S. (1996). Modelling, control, and stability analysis of hy- ence. From 1968 to 1984 he was on the brid systems. Technical Report, No. 247L. Chalmers University of Electrical Engineering Faculty at Iowa Technology. State University, Ames. In 1984 he became Pettersson, S. & Lennartson, B. (1996). Stability and robustness for Chair of the Department of Electrical En- hybrid systems. Proc.35th IEEE Conf. on Decision and Control gineering, and from 1988 to 1998 he was (pp. 1202—1207) Kobe, Japan. Dean of the College of Engineering at the University of Notre Dame, Rui, C., Kolmanovsky, I., & McClamroch, N. H. (1997). Hybrid control IN. He is currently the Frank M. Freimann Professor of Engineering at the University of Notre Dame. He has coauthored six books and for stabilization of a class of cascade nonlinear systems. Proc. Amer. several other publications. Control Conf. (pp. 2800—2804) Albuquerque. Dr. Michel received (with R.D. Rasmussen) the 1978 Best Transac- Thomas, J. (1964). U® ber die Invarianz der Stabilita¨ t bei einem tions Paper Award of the IEEE Control Systems Society, the 1984 Phasenraum Homo¨ omorphismus. J. angewandte Mathematik, 213, Guillemin-Cauer Prize Paper Award of the IEEE Circuits and Systems 147—150. Society (with R. K. Miller and B. H. Nam), and the 1993 Myril B. Wang, K., Michel, A. N., & Passino, K. M. (1994). Ob Reed Outstanding Paper Award of the IEEE Circuits and Systems Otobrazhenniakh, Sokhraniaiushchikh Ustojchivost’ Dinamiches- Society. He was awarded the IEEE Centennial Medal in 1984, and kikh Sistem, Part I. Avtomatika i ¹elemekhanika, 10, 3—12. in 1992 he was a Fulbright Scholar at the Technical University Wang, K., Michel A. N., & Passino K. M. (1994). Ob of Vienna. He received the 1995 Technical Achievement Award of the IEEE Circuits and Systems Society. He is a past Editor of the Otobrazhenniakh, Sokhraniaiushchikh Ustojchivost’ Dinamiches- IEEE TRANSACTION ON CIRCUITS AND SYSTEMS (1981—1983) ¹ kikh Sistem, Part II. Avtomatika i elemekhanika, 11,49—58. as well as a past President of Technical Affairs (1994, 1995) and a Ye, H. (1996). Stability analysis of two classes of dynamical systems: past Vice President of Conference Activities (1996, 1997) of the General hybrid dynamical systems and neural networks with delays. IEEE Control Systems Society. He is currently an Associate Editor Ph.D. dissertation, University of Notre Dame. at Large for the IEEE TRANSACTIONS ON AUTOMATIC Ye, H., Michel, A. N., & Antsaklis, P. J. (1995). A general model for the CONTROL. He was Program Chair of the 1985 IEEE Conference qualitative analysis of hybrid dynamical systems. Proc.34th IEEE on Decision and Control and General Chair of the 1997 IEEE Conf. on Decision and Control (pp. 1473—1477), New Orleans, Conference on Decision and Control. He was awarded an Alexander Louisiana. von Humboldt Forschungspreis (Research Award) for Senior U.S. Scientists (1998). Ye, H., Michel, A. N., & Hou, L. (1995). Stability of hybrid dynamical systems. Proc.34th IEEE Conf. on Decision and Control (pp. 2679—2684). New Orleans, Louisiana. Ye, H., Michel, A. N., & Hou, L. (1996). Stability analysis of discontin- uous dynamical systems with applications. Proc.13th ¼orld Congr. Bo Hu received the B.S. degree and the Int. Federation of Automat. Control, Vol. E: Nonlinear systems M.S. degree in 1988 and 1992, respectively, (pp. 461—466). San Francisco, CA in applied mathematics from Fudan Uni- Ye, H., Michel, A. N., & Hou, L. (1996). Stability analysis of systems versity, People’s Republic of China and the with impulse effects. Proc.35th IEEE Conf. on Decision and Control M.S. degree in electrical engineering from (pp. 156—161). Kobe, Japan the University of Notre Dame, USA. Cur- rently, he is a Ph.D. candidate in the De- Ye, H., Michel, A. N., & Hou, L. (1998). Stability Theory for Hybrid partment of Electrical Engineering at the ¹ Dynamical Systems. IEEE rans. Automat. Control, 43(4), 461—474. University of Notre Dame. He is the recipi- Yoshizawa, T. (1966). Stability theory by ¸yapunov’s second method. ent of a Center for Applied Mathematics Japan, Tokyo: Math. Soc. Fellowship at Noter Dame. His current Zubov, V. I. (1964). Methods of A. M. ¸yapunov and their applications. research interests are in the qualitative Groningen, The Netherlands: P. Noordhoff Ltd. analysis of dynamical systems.