Alternative stability conditions for switched discrete time linear systems

Alternative necessary and sufficient conditions for absolute exponential stability are presented, for switched discrete–time linear systems. To reach these results, we exploit concepts from set theory and in specific the forward reachability set mappings. The developed theorem can be utilized to construct iterative numerical procedures for verifying stability. Two examples illustrate the effectiveness of the proposed methodology.

[1]  M. Lazar,et al.  On stability analysis of discrete-time homogeneous dynamics , 2013, 2013 17th International Conference on System Theory, Control and Computing (ICSTCC).

[2]  Yacine Chitour,et al.  On the marginal instability of linear switched systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[3]  Mircea Lazar,et al.  On infinity norms as Lyapunov functions: Alternative necessary and sufficient conditions , 2010, 49th IEEE Conference on Decision and Control (CDC).

[4]  Zhendong Sun,et al.  Stability of piecewise linear systems revisited , 2010, Annu. Rev. Control..

[5]  R. Jungers.  The Joint Spectral Radius: Theory and Applications , 2009 .

[6]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[7]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

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

[9]  Zhendong Sun,et al.  A Note on Marginal Stability of Switched Systems , 2007, IEEE Transactions on Automatic Control.

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

[11]  N. Barabanov.  Lyapunov exponent and joint spectral radius: some known and new results , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[12]  R. Shorten,et al.  A numerical technique for the stability analysis of linear switched systems , 2004 .

[13]  S. Cooper.  Remote Control , 2002 .

[14]  Andrzej Polanski,et al.  Destabilizing strategies for uncertain linear systems , 2000, IEEE Trans. Autom. Control..

[15]  John N. Tsitsiklis,et al.  A survey of computational complexity results in systems and control , 2000, Autom..

[16]  Andrzej Polanski,et al.  On absolute stability analysis by polyhedral Lyapunov functions , 2000, Autom..

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

[18]  G. Gripenberg.  COMPUTING THE JOINT SPECTRAL RADIUS , 1996 .

[19]  L. Gurvits.  Stability of discrete linear inclusion , 1995 .

[20]  G. Ziegler.  Lectures on Polytopes , 1994 .

[21]  P. Bauer,et al.  A necessary and sufficient condition for robust asymptotic stability of time-variant discrete systems , 1993, IEEE Trans. Autom. Control..

[22]  Franco Blanchini,et al.  Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[23]  Robert K. Brayton,et al.  Stability of dynamical systems: A constructive approach , 1979 .

[24]  Marc Jungers,et al.  Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach , 2014, Autom..

[25]  Nikolaos Athanasopoulos,et al.  Scalable stabilization of large scale discrete-time linear systems via the 1-norm , 2013 .

[26]  V. Blondel,et al.  Approximating the joint spectral radius using a genetic algorithm framework , 2011 .

[27]  G. Lamberson.  A constructive approach , 2007 .

[28]  J. Theys.  Joint spectral radius : theory and approximations/ , 2005 .

[29]  A. Rubinov,et al.  The space of star-shaped sets and its applications in nonsmooth optimization , 1986 .