Stability and Stabilization of Two‐Dimensional LTI Switched Systems with Potentially Unstable Focus

This paper investigates stability and stabilization of two-dimensional switched linear time-invariant (LTI) systems with potentially unstable focus. For the case that the origin is a single common focus of all subsystems, we first give continuous positive definite functions related only to the elements of subsystems' state matrices. Then, based on the continuous positive definite functions obtained, this paper proposes several sufficient conditions of stability/asymptotic stability/instability of the kind of switched LTI systems. By means of the stability results proposed, global asymptotic stabilizing controls (GASC), global asymptotic stabilizing switching paths (GASSP) and corresponding algorithms are designed for two-dimensional switched LTI systems with focus. Finally, two illustrative examples and numerical simulations demonstrate the effectiveness of the new stability and stabilization results obtained in this paper.

[1]  Gang Feng,et al.  Stabilisation of second-order LTI switched positive systems , 2011, Int. J. Control.

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

[3]  Jun Zhao,et al.  Passivity and stability of switched systems: A multiple storage function method , 2008, Syst. Control. Lett..

[4]  Jun Zhao,et al.  Stability and L2-gain analysis for switched delay systems: A delay-dependent method , 2006, Autom..

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

[6]  Wei Wang,et al.  Stability Analysis for Linear Switched Systems With Time-Varying Delay , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[7]  Lei Guo,et al.  STABILIZATION OF SWITCHED LINEAR SYSTEMS , 2003, IEEE Trans. Autom. Control..

[8]  State feedback control for uncertain switched systems with interval time‐varying delay , 2011 .

[9]  Xuping Xu,et al.  Stabilization of second-order LTI switched systems , 2000 .

[10]  A. Michel,et al.  Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[11]  Guangming Xie,et al.  Necessary and sufficient conditions for controllability and observability of switched impulsive control systems , 2004, IEEE Transactions on Automatic Control.

[12]  Jun Zhao,et al.  Backstepping Robust H ∞ Control for a Class of Uncertain Switched Nonlinear Systems Under Arbitrary Switchings , 2013 .

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

[14]  Min Han,et al.  Stabilization for switched stochastic neutral systems under asynchronous switching , 2013, Inf. Sci..

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

[16]  Pingyuan Cui,et al.  Stabilization for a class of second-order switched systems , 2005 .

[17]  Jie Lian,et al.  Passivity Analysis and Synthesis for a Class of Discrete‐Time Switched Stochastic Systems with Time‐Varying Delay , 2013 .

[18]  P. Shi,et al.  Delay‐dependent exponential stability for switched delay systems , 2009 .

[19]  R. Guo,et al.  Stability analysis for a class of switched linear systems , 2012 .

[20]  Daizhan Cheng,et al.  Accessibility of Switched Linear Systems , 2006, IEEE Transactions on Automatic Control.

[21]  G. Feng,et al.  Stability analysis of two-dimensional switched systems with unstable subsystems , 2011, Proceedings of the 30th Chinese Control Conference.

[22]  A. Michel,et al.  Robust stabilizing control laws for a class of second-order switched systems , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[23]  Gang Feng,et al.  Stability analysis of piecewise discrete-time linear systems , 2002, IEEE Trans. Autom. Control..

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

[25]  Zhendong Sun,et al.  Stabilizing switching design for switched linear systems: A state-feedback path-wise switching approach , 2009, Autom..

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