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