On Uniform Asymptotic Stability of Switched Nonlinear Time-Varying Systems

This paper is concerned with the study of, both local and global, uniform asymptotic stability for general nonlinear and time-varying switched systems. Two concepts of Lyapunov functions are introduced and used to establish uniform Lyapunov stability and uniform global stability. With the help of output functions, an almost bounded output energy condition and a persistent excitation (PE) condition related to output functions are then proposed and employed to guarantee uniform (global) asymptotic stability. Based on this result, a generalized version of Krasovskii-LaSalle theorem in time-varying switched systems is proposed. Moreover, it is shown that the proposed PE condition can be verified by checking a zero-state observability condition for switched systems with persistent dwell-time. Particularly, our results can be used in the case where only some parts of the overall switched system are stable. It is shown that several existing results in past literature can be covered as special cases using the proposed criteria.

[1]  Zvi Artstein,et al.  Uniform asymptotic stability via the limiting equations , 1978 .

[2]  Rafal Goebel,et al.  Solutions to hybrid inclusions via set and graphical convergence with stability theory applications , 2006, Autom..

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

[4]  J. P. Lasalle.  Some Extensions of Liapunov's Second Method , 1960 .

[5]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[6]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[7]  K. Narendra,et al.  On the stability and existence of common Lyapunov functions for stable linear switching systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[8]  A. Morse,et al.  Stability of switched systems: a Lie-algebraic condition ( , 1999 .

[9]  João Pedro Hespanha,et al.  Uniform stability of switched linear systems: extensions of LaSalle's Invariance Principle , 2004, IEEE Transactions on Automatic Control.

[10]  Kellen Petersen August.  Real Analysis , 2009 .

[11]  R. Decarlo,et al.  Perspectives and results on the stability and stabilizability of hybrid systems , 2000, Proceedings of the IEEE.

[12]  J. L. Mancilla-Aguilar,et al.  A converse Lyapunov theorem for nonlinear switched systems , 2000 .

[13]  Zhong-Ping Jiang,et al.  On Uniform Global Asymptotic Stability of Nonlinear Discrete-Time Systems With Applications , 2006, IEEE Transactions on Automatic Control.

[14]  Andrea Bacciotti,et al.  An invariance principle for nonlinear switched systems , 2005, Syst. Control. Lett..

[15]  Bor-Sen Chen,et al.  A general stability criterion for time-varying systems using a modified detectability condition , 2002, IEEE Trans. Autom. Control..

[16]  Ti-Chung Lee,et al.  On the equivalence relations of detectability and PE conditions with applications to stability analysis of time-varying systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[17]  David Angeli,et al.  Nonlinear norm-observability notions and stability of switched systems , 2005, IEEE Transactions on Automatic Control.

[18]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[19]  Zhong-Ping Jiang,et al.  Uniform Asymptotic Stability of Nonlinear Switched Systems With an Application to Mobile Robots , 2008, IEEE Transactions on Automatic Control.

[20]  Zhong-Ping Jiang,et al.  A Generalization of Krasovskii-LaSalle Theorem for Nonlinear Time-Varying Systems: Converse Results and Applications , 2005, IEEE Trans. Autom. Control..

[21]  José Luis Mancilla-Aguilar,et al.  An extension of LaSalle's invariance principle for switched systems , 2005, Syst. Control. Lett..