Stability theory for hybrid dynamical systems

Hybrid systems which are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system are of great current interest. In the present paper we first formulate a definition of hybrid dynamical system which covers a very large number of classes of hybrid systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of invariant set (e.g., equilibrium) for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for the uniform stability and the uniform asymptotic stability of an invariant set of a hybrid dynamical system. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). To demonstrate the applicability of the developed theory, we present two specific examples of hybrid dynamical systems and we conduct a stability analysis of one of these examples (a sampled-data feedback control system with a nonlinear (continuous-time) plant and a linear (discrete-time) controller).

[1]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1919, Nature.

[2]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[3]  P. Hartman.  Ordinary Differential Equations , 1965 .

[4]  吉沢 太郎.  Stability theory by Liapunov's second method , 1966 .

[5]  T. Pavlidis.  Stability of systems described by differential equations containing impulses , 1967 .

[6]  A. Fuller,et al.  Stability of Motion , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[7]  L. Tavernini.  Differential automata and their discrete simulators , 1987 .

[8]  B. Francis,et al.  Stability Theory for Linear Time-Invariant Plants with Periodic Digital Controllers , 1988, 1988 American Control Conference.

[9]  Tomomichi Hagiwara,et al.  Design of a stable state feedback controller based on the multirate sampling of the plant output , 1988 .

[10]  B. Francis,et al.  Stability Theory for Linear Time-Invariant Plants with Periodic Digital Controllers , 1988 .

[11]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[12]  A. Michel,et al.  Lyapunov Stability of a Class of Discrete Event Systems , 1991, 1991 American Control Conference.

[13]  R. Decarlo,et al.  Asymptotic Stability of m-Switched Systems using Lyapunov-Like Functions , 1991, 1991 American Control Conference.

[14]  B. Francis,et al.  Input-output stability of sampled-data systems , 1991 .

[15]  John Guckenheimer,et al.  A Dynamical Simulation Facility for Hybrid Systems , 1993, Hybrid Systems.

[16]  Anil Nerode,et al.  Models for Hybrid Systems: Automata, Topologies, Controllability, Observability , 1992, Hybrid Systems.

[17]  Panos J. Antsaklis,et al.  Hybrid System Modeling and Autonomous Control Systems , 1992, Hybrid Systems.

[18]  Roger W. Brockett,et al.  Hybrid Models for Motion Control Systems , 1993 .

[19]  Robert L. Grossman,et al.  Timed Automata , 1999, CAV.

[20]  Dingjun Luo,et al.  Qualitative Theory of Dynamical Systems , 1993 .

[21]  Pravin Varaiya,et al.  Smart cars on smart roads: problems of control , 1991, IEEE Trans. Autom. Control..

[22]  V. Borkar,et al.  A unified framework for hybrid control , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[23]  Pablo A. Iglesias,et al.  On the stability of sampled-data linear time-varying feedback systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[24]  M. Branicky.  Stability of switched and hybrid systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[25]  Panos J. Antsaklis,et al.  A general model for the qualitative analysis of hybrid dynamical systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[26]  Michael S. Branicky,et al.  Studies in hybrid systems: modeling, analysis, and control , 1996 .

[27]  A. Michel,et al.  Some qualitative properties of sampled-data control systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[28]  A. Michel,et al.  Stability Analysis of Discontinuous Dynamical Systems with Applications , 1996 .