On the Convergence of Linear Switched Systems

This paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems. We provide a collection of results that use weak dwell-time, dwell-time, strong dwell-time, permanent and persistent activation hypothesis. The obtained results are shown to be tight by counterexample. Finally, we apply our result to the three-cell converter.

[1]  Ricardo G. Sanfelice,et al.  Invariance principles for switching systems via hybrid systems techniques , 2008, Syst. Control. Lett..

[2]  I. Daubechies,et al.  Sets of Matrices All Infinite Products of Which Converge , 1992 .

[3]  W. Beyn,et al.  Infinite products and paracontracting matrices , 1997 .

[4]  M.S. Boucherit,et al.  On sliding mode observer for hybrid three cells converter : Experimental results , 2008, 2008 International Workshop on Variable Structure Systems.

[5]  M. Zelikin,et al.  Control theory and optimization I , 1999 .

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

[7]  Ricardo G. Sanfelice,et al.  Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability , 2007, IEEE Transactions on Automatic Control.

[8]  Wolfgang Kliemann,et al.  The dynamics of control , 2000 .

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

[10]  Karl Henrik Johansson.  Hybrid control systems , 2004 .

[11]  M. Neumann,et al.  Generalizations of the projection method with applications to SOR theory for hermitian positive semidefinite linear systems , 1987 .

[12]  L. Máté.  On the infinite product of operators in Hilbert space , 1998 .

[13]  P. E. Kloeden,et al.  Nonautonomous attractors of switching systems , 2006 .

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

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

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

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

[18]  S. Bhat,et al.  An invariance principle for nonlinear hybrid and impulsive dynamical systems , 2003 .

[19]  Thierry Meynard,et al.  Multicell converters: basic concepts and industry applications , 2002, IEEE Trans. Ind. Electron..

[20]  Velimir Jurdjevic,et al.  Control systems on semi-simple Lie groups and their homogeneous spaces , 1981 .

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

[22]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

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

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

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

[26]  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..

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

[28]  Bin Wu,et al.  Multilevel Voltage-Source-Converter Topologies for Industrial Medium-Voltage Drives , 2007, IEEE Transactions on Industrial Electronics.

[29]  日本自動制御協会,et al.  システムと制御 = Systems and control , 1971 .

[30]  Thierry Meynard,et al.  Modeling of multilevel converters , 1997, IEEE Trans. Ind. Electron..

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

[32]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[33]  M. Fadel,et al.  Floating voltages estimation in three-cell converters using a discrete-time Kalman filter , 2001, 2001 IEEE 32nd Annual Power Electronics Specialists Conference (IEEE Cat. No.01CH37230).

[34]  Hans Schneider,et al.  The convergence of general products of matrices and the weak ergodicity of Markov chains , 1999 .

[35]  W. Rudin.  Real and complex analysis , 1968 .

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

[37]  K. Benmansour,et al.  Adaptive Observer for Multi-Cell Chopper , 2006 .

[38]  C. Foias,et al.  Harmonic Analysis of Operators on Hilbert Space , 1970 .