The set of asymptotically stable switching sequences of linear discrete-time switching systems

In this paper we study the characterization of asymptotic stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals a underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary.

[1]  Symbolic dynamics , 2008, Scholarpedia.

[2]  Xiongping Dai,et al.  Existence of full-Hausdorff-dimension invariant measures of dynamical systems with dimension metrics , 2005 .

[3]  Patrizio Colaneri,et al.  Stability and Stabilization of Continuous-Time Switched Linear Systems , 2006, SIAM J. Control. Optim..

[4]  Pierre-Alexandre Bliman,et al.  Stability Analysis of Discrete-Time Switched Systems Through Lyapunov Functions with Nonminimal State , 2003, ADHS.

[5]  Geir E. Dullerud,et al.  Uniformly Stabilizing Sets of Switching Sequences for Switched Linear Systems , 2007, IEEE Transactions on Automatic Control.

[6]  Yacine Chitour,et al.  Common Polynomial Lyapunov Functions for Linear Switched Systems , 2006, SIAM J. Control. Optim..

[7]  Michael Margaliot,et al.  Stability analysis of switched systems using variational principles: An introduction , 2006, Autom..

[8]  Ji-Woong Lee,et al.  Uniform stabilization of discrete-time switched and Markovian jump linear systems , 2006, Autom..

[9]  Geir E. Dullerud,et al.  Optimal Disturbance Attenuation for Discrete-Time Switched and Markovian Jump Linear Systems , 2006, SIAM J. Control. Optim..

[10]  A. Bicchi,et al.  On the stabilization of linear discrete-time hybrid automata , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[11]  D. Stanford,et al.  Some Convergence Properties of Matrix Sets , 1994 .

[12]  M. Mirzakhani,et al.  Introduction to Ergodic theory , 2010 .

[13]  Panos J. Antsaklis,et al.  Optimal control of switched systems based on parameterization of the switching instants , 2004, IEEE Transactions on Automatic Control.

[14]  Xiongping Dai,et al.  Almost Sure Stability of Discrete-Time Switched Linear Systems: A Topological Point of View , 2008, SIAM J. Control. Optim..

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

[16]  Daizhan Cheng,et al.  Stabilization of planar switched systems , 2004, Syst. Control. Lett..

[17]  A. Besicovitch.  On the sum of digits of real numbers represented in the dyadic system. , 1935 .

[18]  J. Geromel,et al.  Stability and stabilization of discrete time switched systems , 2006 .

[19]  Clyde F. Martin,et al.  A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching , 1999, IEEE Transactions on Automatic Control.

[20]  Ugo V. Boscain,et al.  Stability of Planar Switched Systems: The Linear Single Input Case , 2002, SIAM J. Control. Optim..

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

[22]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[23]  Jamal Daafouz,et al.  Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach , 2002, IEEE Trans. Autom. Control..

[24]  P. Walters.  Introduction to Ergodic Theory , 1977 .

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