On the asymptotic stability of switched homogeneous systems

Abstract The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L > 0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.

[1]  K. Narendra,et al.  A common Lyapunov function for stable LTI systems with commuting A-matrices , 1994, IEEE Trans. Autom. Control..

[2]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

[3]  Dennis S. Bernstein,et al.  Geometric homogeneity with applications to finite-time stability , 2005, Math. Control. Signals Syst..

[4]  David W. Lewis,et al.  Matrix theory , 1991 .

[5]  Daniel Liberzon,et al.  Nonlinear Systems , 2004 .

[6]  Jan Lunze,et al.  Handbook of hybrid systems control : theory, tools, applications , 2009 .

[7]  Vincent D. Blondel,et al.  Open Problems in Mathematical Systems and Control Theory , 2011 .

[8]  Michael Margaliot,et al.  Stability Analysis of Second-Order Switched Homogeneous Systems , 2002, SIAM J. Control. Optim..

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

[10]  A. Morse.  Supervisory control of families of linear set-point controllers Part I. Exact matching , 1996, IEEE Trans. Autom. Control..

[11]  Robert Shorten,et al.  Stability Criteria for Switched and Hybrid Systems , 2007, SIAM Rev..

[12]  Daniel Liberzon,et al.  Lie algebras and stability of switched nonlinear systems , 2009 .

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

[14]  Anthony N. Michel,et al.  Stability results involving time-averaged Lyapunov function derivatives , 2009 .

[15]  Victor Solo,et al.  On the stability of slowly time-varying linear systems , 1994, Math. Control. Signals Syst..

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

[17]  Humberto E. Garcia.  Control and supervision of a complex production process using hybrid systems techniques , 1995, Proceedings of First IEEE International Conference on Engineering of Complex Computer Systems. ICECCS'95.

[18]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..

[19]  L. Grujic.  Large Scale Systems Stability under Structural and Singular Perturbations , 1987 .

[20]  Rong Sun,et al.  On Stability of Switched Homogeneous Nonlinear Systems , 2006, 2006 Chinese Control Conference.

[21]  Y. Orlov,et al.  Finite time stability of homogeneous switched systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[22]  A. V. Il'in,et al.  On the Stability of Families of Dynamical Systems , 2002 .

[23]  Y. Pyatnitskiy,et al.  An iterative method of Lyapunov function construction for differential inclusions , 1987 .

[24]  Bo Hu,et al.  Disturbance attenuation properties of time-controlled switched systems , 2001, J. Frankl. Inst..

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

[26]  H. Stanley,et al.  Systematic Application of Generalized Homogeneous Functions to Static Scaling, Dynamic Scaling, and Universality , 1972 .