Approximating the Spectral Abscissa for Switched Linear Systems via Coordinate Transformations

In this paper, an approach of square coordinate transformation is proposed to approximate the spectral abscissa for continuous-time switched linear systems. By applying elementary transformations iteratively, a series of minimums of least μ1 matrix set measures are obtained, which are utilized to approximate the spectral abscissa of the switched system. The approach is developed into tractable numerical algorithms that provide upper bound estimates of the spectral abscissa. Numerical simulations show the effectiveness of the proposed method.

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

[2]  Zhendong Sun,et al.  Accurate estimation of the largest divergence rate for a class of the 3rd-order switched linear systems , 2013 .

[3]  Peng Shi,et al.  Stability and Stabilization of Switched Linear Systems With Mode-Dependent Average Dwell Time , 2012, IEEE Transactions on Automatic Control.

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

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

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

[7]  Mau-Hsiang Shih,et al.  Asymptotic Stability and Generalized Gelfand Spectral Radius Formula , 1997 .

[8]  Zhendong Sun,et al.  Approximation of extreme measure for switched linear systems , 2011, 2011 9th IEEE International Conference on Control and Automation (ICCA).

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

[10]  S. Ge,et al.  Stability Theory of Switched Dynamical Systems , 2011 .

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

[12]  S. Ge,et al.  Switched Linear Systems: Control and Design , 2005 .

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

[14]  Michael Margaliot,et al.  Necessary and sufficient conditions for absolute stability: the case of second-order systems , 2003 .

[15]  Shen Yin,et al.  Switching Stabilization for a Class of Slowly Switched Systems , 2015, IEEE Transactions on Automatic Control.

[16]  M. Fu,et al.  Piecewise Lyapunov functions for robust stability of linear time-varying systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

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

[19]  Shuzhi Sam Ge,et al.  Switched Linear Systems , 2005 .

[20]  Tingshu Hu,et al.  Composite quadratic Lyapunov functions for constrained control systems , 2003, IEEE Trans. Autom. Control..

[21]  Franco Blanchini.  The gain scheduling and the robust state feedback stabilization problems , 2000, IEEE Trans. Autom. Control..

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

[23]  Michael Margaliot,et al.  Absolute stability of third-order systems: A numerical algorithm , 2006, Autom..

[24]  Raphaël M. Jungers,et al.  Is switching systems stability harder for continuous time systems? , 2013, 52nd IEEE Conference on Decision and Control.

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

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

[27]  Derong Liu,et al.  Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems , 2006, IEEE Trans. Circuits Syst. II Express Briefs.