Towards a stability theory of general hybrid dynamical systems

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,~). 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.

[1]  Johannes Thomas,et al.  Über die Invarianz der Stabilität bei einem Phasenraum-Homöomorphismus. , 1964 .

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

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

[4]  Stefan Pettersson.  Modelling, control and stability analysis of hybrid systems , 1996 .

[5]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[6]  Kevin M. Passino,et al.  Lagrange stability and boundedness of discrete event systems , 1995, Discret. Event Dyn. Syst..

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

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

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

[10]  W. Walter.  Differential and Integral Inequalities , 1970 .

[11]  B. Brogliato,et al.  On the control of finite-dimensional mechanical systems with unilateral constraints , 1997, IEEE Trans. Autom. Control..

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

[13]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[14]  Wolfgang Hahn.  Über stabilitätserhaltende Abbildungen und Ljapunovsche Funktionen. , 1967 .

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

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

[17]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[18]  Anthony N. Michel,et al.  Modeling and Qualitative Theory for General Hybrid Dynamical and Control Systems , 1997 .

[19]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

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

[21]  V. Lakshmikantham,et al.  Differential and Integral Inequalities , 2019, Springer Optimization and Its Applications.

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

[23]  Ling Hou,et al.  Stability analysis of a general class of hybrid dynamical systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

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

[25]  P. Varaiya,et al.  Hybrid dynamical systems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[26]  Anthony N. Michel,et al.  Stability of Hybrid and Discontinuous Dynamical Systems , 2001 .

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

[28]  Ilya Kolmanovsky,et al.  Hybrid control for stabilization of a class of cascade nonlinear systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

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

[30]  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.

[31]  S. Pettersson,et al.  Stability and robustness for hybrid systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[32]  A. Michel,et al.  Some qualitative properties of sampled-data control systems , 1997, IEEE Trans. Autom. Control..

[33]  Panos J. Antsaklis,et al.  Lyapunov stability of a class of discrete event systems , 1994, IEEE Trans. Autom. Control..

[34]  A. Michel,et al.  Stability analysis of systems with impulse effects , 1996, Proceedings of 35th IEEE Conference on Decision and Control.