Smooth Lyapunov Functions for Hybrid Systems Part II: (Pre)Asymptotically Stable Compact Sets

It is shown that (pre)asymptotic stability, which generalizes asymptotic stability, of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth Lyapunov function. This result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small, state-dependent perturbations. As a special case, we state a converse Lyapunov theorem for systems with logic variables and use this result to establish input-to-state stabilization using hybrid feedback control. The converse Lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying, weakly jumping parameters, to temporal regularization, to the insertion of jumps according to an ldquoaverage dwell-timerdquo rule, and to the insertion of flow according to a ldquoreverse average dwell-timerdquo rule.

[1]  Andrew R. Teel,et al.  Smooth Lyapunov functions and robustness of stability for difference inclusions , 2004, Syst. Control. Lett..

[2]  D. Liberzon,et al.  On Input-to-State Stability of Impulsive Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[3]  J. Coron,et al.  Adding an integrator for the stabilization problem , 1991 .

[4]  Xuping Xu,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control - s[Book review; W. M. Haddad; V. S. Chellaboina, and S. G. Nersesov] , 2007, IEEE Transactions on Automatic Control.

[5]  Eduardo D. Sontag,et al.  Global Asymptotic Controllability Implies Input-to-State Stabilization , 2003, SIAM J. Control. Optim..

[6]  Karl Henrik Johansson,et al.  Dynamical properties of hybrid automata , 2003, IEEE Trans. Autom. Control..

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

[8]  Peter E. Kloeden,et al.  Pullback Attractors in Dissipative Nonautonomous Differential Equations Under Discretization , 2001 .

[9]  Chaohong Cai,et al.  Smooth Lyapunov Functions for Hybrid Systems—Part I: Existence Is Equivalent to Robustness , 2007, IEEE Transactions on Automatic Control.

[10]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

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

[12]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1998, IEEE Trans. Autom. Control..

[13]  Andrew R. Teel,et al.  Discrete-time homogeneous Lyapunov functions for homogeneous difference inclusions , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[14]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .

[15]  John Tsinias,et al.  Sufficient lyapunov-like conditions for stabilization , 1989, Math. Control. Signals Syst..

[16]  J. Hespanha,et al.  Hybrid systems: Generalized solutions and robust stability , 2004 .

[17]  Andrew R. Teel,et al.  On the Robustness of KL-stability for Difference Inclusions: Smooth Discrete-Time Lyapunov Functions , 2005, SIAM J. Control. Optim..

[18]  S. Sastry,et al.  On the existence of executions of hybrid automata , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[19]  Johannes Schumacher,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[20]  Yu. S. Ledyaev,et al.  Asymptotic Stability and Smooth Lyapunov Functions , 1998 .

[21]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

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

[23]  Ricardo G. Sanfelice,et al.  Hybrid Systems: Limit Sets and Zero Dynamics with a View Toward Output Regulation , 2008 .

[24]  R. Sanfelice,et al.  Results on convergence in hybrid systems via detectability and an invariance principle , 2005, Proceedings of the 2005, American Control Conference, 2005..

[25]  Yu. S. Ledyaev,et al.  A Lyapunov characterization of robust stabilization , 1999 .

[26]  A.R. Teel,et al.  Results on robust stabilization of asymptotically controllable systems by hybrid feedback , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[27]  Christophe Prieur,et al.  Smooth patchy control Lyapunov functions , 2006, CDC.

[28]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[29]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[30]  Pieter Collins,et al.  A Trajectory-Space Approach to Hybrid Systems , 2004 .

[31]  A. Teel,et al.  A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions , 2000 .

[32]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[33]  João Pedro Hespanha,et al.  Stabilization of nonholonomic integrators via logic-based switching , 1999, Autom..

[34]  F. Hoppensteadt.  Singular perturbations on the infinite interval , 1966 .

[35]  Chaohong Cai,et al.  Results on relaxation theorems for hybrid systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[36]  A. Astolfi,et al.  Robust stabilization of chained systems via hybrid control , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[37]  Chaohong Cai,et al.  Converse Lyapunov theorems and robust asymptotic stability for hybrid systems , 2005, Proceedings of the 2005, American Control Conference, 2005..