A commutation condition for stability analysis of switched linear descriptor systems

Abstract We study the stability analysis problem for switched linear descriptor systems. Assuming that all subsystems are stable and there is no impulse at the switching instants, we establish a new pairwise commutation condition under which the switched system is stable. We also show that when the proposed commutation condition holds, there exists a common quadratic Lyapunov function (CQLF) for the subsystems. These results are natural and significant extensions to the existing results for switched systems in the state space representation.

[1]  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.

[2]  Raymond A. DeCarlo,et al.  Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems , 1998, Eur. J. Control.

[3]  D. Cobb,et al.  Descriptor variable systems and optimal state regulation , 1983 .

[4]  Daniel Liberzon,et al.  On stability of linear switched differential algebraic equations , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[5]  Bo Hu,et al.  Disturbance attenuation properties of time-controlled switched systems , 2001, J. Frankl. Inst..

[6]  Guisheng Zhai,et al.  Stability analysis and design for switched descriptor systems , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

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

[8]  Shuzhi Sam Ge,et al.  Analysis and synthesis of switched linear control systems , 2005, Autom..

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

[10]  F. Lewis.  A survey of linear singular systems , 1986 .

[11]  B. Barmish,et al.  Adaptive stabilization of linear systems via switching control , 1986, 1986 25th IEEE Conference on Decision and Control.

[12]  A. Michel,et al.  Stability and L2 Gain Analysis of Discrete-Time Switched Systems , 2002 .

[13]  Guisheng Zhai,et al.  A unified approach to analysis of switched linear descriptor systems under arbitrary switching , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[14]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[15]  Bo Hu,et al.  Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach , 2001, Int. J. Syst. Sci..

[16]  Guisheng Zhai,et al.  Qualitative analysis of switched discrete-time descriptor systems , 2009 .

[17]  T. Katayama,et al.  A generalized Lyapunov theorem for descriptor system , 1995 .

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

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

[20]  E. Uezato,et al.  Strict LMI conditions for stability, robust stabilization, and H/sub /spl infin// control of descriptor systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[21]  I. Masubuchi,et al.  H∞ control for descriptor systems: A matrix inequalities approach , 1997, Autom..

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

[23]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[24]  Eiho Uezato,et al.  A Strict LMI Condition for H∞ Control of Descriptor Systems , 2000 .

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