Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications

In this paper, a unified framework is proposed to study the exponential stability of discrete-time switched linear systems and, more generally, the exponential growth rates of their trajectories under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established, and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples.

[1]  V. Kozyakin.  ITERATIVE BUILDING OF BARABANOV NORMS AND COMPUTATION OF THE JOINT SPECTRAL RADIUS FOR MATRIX SETS , 2008, 0810.2154.

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

[3]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

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

[5]  A. Markushevich.  Analytic Function Theory , 1996 .

[6]  Alessandro Abate,et al.  Efficient suboptimal solutions of switched LQR problems , 2009, 2009 American Control Conference.

[7]  Michael Margaliot,et al.  On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws , 2009, IEEE Transactions on Automatic Control.

[8]  Wei Zhang,et al.  Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications , 2011, IEEE Trans. Autom. Control..

[9]  Y. Nesterov,et al.  On the accuracy of the ellipsoid norm approximation of the joint spectral radius , 2005 .

[10]  Vincent D. Blondel,et al.  Overlap-free words and spectra of matrices , 2007, Theor. Comput. Sci..

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

[12]  M. Morari,et al.  Stability and stabilization of piecewise affine and hybrid systems: an LMI approach , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

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

[14]  Yoav Sharon,et al.  Third-Order Nilpotency, Nice Reachability and Asymptotic Stability ? , 2007 .

[15]  F. Wirth.  The generalized spectral radius and extremal norms , 2002 .

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

[17]  Daniel Liberzon,et al.  Lie-algebraic conditions for exponential stability of switched systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[18]  A. Rantzer,et al.  Piecewise linear quadratic optimal control , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[19]  Federico Najson,et al.  State-feedback stabilizability in switched homogeneous systems , 2009, 2009 American Control Conference.

[20]  Jianghai Hu,et al.  Stability of discrete-time conewise linear inclusions and switched linear systems , 2010, Proceedings of the 2010 American Control Conference.

[21]  Harold R. Parks,et al.  A Primer of Real Analytic Functions , 1992 .

[22]  Tamás Terlaky,et al.  A Survey of the S-Lemma , 2007, SIAM Rev..

[23]  Xiangbo Feng.  Lyapunov exponents and stability of linear stochastic systems , 1990 .

[24]  F. Najson,et al.  Conditions for feedback stabilizability in switched linear systems , 2008, 2008 American Control Conference.

[25]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[26]  A. Jadbabaie,et al.  Approximation of the joint spectral radius using sum of squares , 2007, 0712.2887.

[27]  S. Pettersson,et al.  LMI for stability and robustness of hybrid systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[28]  MargaliotMichael.  Stability analysis of switched systems using variational principles , 2006 .

[29]  J. Tsitsiklis,et al.  The boundedness of all products of a pair of matrices is undecidable , 2000 .

[30]  Y. Pyatnitskiy,et al.  Criteria of asymptotic stability of differential and difference inclusions encountered in control theory , 1989 .

[31]  Robert Shorten,et al.  A Numerical Technique for Stability Analysis of Linear Switched Systems , 2004, HSCC.

[32]  Vincent D. Blondel,et al.  Approximations of the Rate of Growth of Switched Linear Systems , 2004, HSCC.

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

[34]  Vincent D. Blondel,et al.  Joint Spectral Characteristics of Matrices: A Conic Programming Approach , 2010, SIAM J. Matrix Anal. Appl..

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

[36]  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).

[37]  John N. Tsitsiklis,et al.  The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate , 1997, Math. Control. Signals Syst..

[38]  Tingshu Hu,et al.  Stabilization of Switched Systems via Composite Quadratic Functions , 2008, IEEE Transactions on Automatic Control.

[39]  Jianghai Hu,et al.  A generating function approach to the stability of discrete-time switched linear systems , 2010, HSCC '10.

[40]  Jianghai Hu,et al.  Stability of switched linear systems on cones: A generating function approach , 2010, 49th IEEE Conference on Decision and Control (CDC).

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

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

[43]  Michael Margaliot,et al.  Nice reachability for planar bilinear control systems with applications to planar linear switched systems , 2009, IEEE Transactions on Automatic Control.

[44]  Alberto Bemporad,et al.  Optimal control of continuous-time switched affine systems , 2006, IEEE Transactions on Automatic Control.

[45]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

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