Stability analysis and design for switched descriptor systems

In this paper, we consider stability analysis and design for switched systems consisting of linear descriptor systems that have the same descriptor matrix. When all descriptor systems are stable, we show that if the descriptor matrix and all the system matrices are commutative pairwise, then the switched system is stable under arbitrary switching. This is an extension of the existing well known result in [1] for switched linear systems with state space models to switched descriptor systems. Under the same commutation condition, we also show that in the case where all descriptor systems are not stable, if there is a stable convex combination of the unstable descriptor systems, then we can establish a class of switching laws which stabilize the switched system. We finally make some discussion about obtaining the stable convex combination by solving a matrix inequality efficiently, and about relaxing the commutation condition for stabilizability of the switched system.

[1]  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).

[2]  Kumpati S. Narendra,et al.  Improving transient response of adaptive control systems using multiple models and switching , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

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

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

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

[6]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[7]  Hai Lin,et al.  Switched Linear Systems: Control and Design , 2006, IEEE Transactions on Automatic Control.

[8]  G. Papavassilopoulos,et al.  A global optimization approach for the BMI problem , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

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

[10]  G. Zhai,et al.  Decentralized H∞ Controller Design: A Matrix Inequality Approach Using a Homotopy Method , 1998 .

[11]  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).

[12]  Marco H. Terra,et al.  On the Lyapunov theorem for descriptor systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[13]  Guisheng Zhai,et al.  Decentralized Hinfinity controller design: a matrix inequality approach using a homotopy method , 1998, Autom..

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

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

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

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

[18]  Shu-Li Sun,et al.  Distributed optimal component fusion weighted by scalars for fixed-lag Kalman smoother , 2005, Autom..

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

[20]  Johannes Schumacher,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[21]  R. Decarlo,et al.  Construction of piecewise Lyapunov functions for stabilizing switched systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

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

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

[24]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

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

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

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

[28]  A. Morse,et al.  Applications of hysteresis switching in parameter adaptive control , 1992 .

[29]  Hua Xu,et al.  On Lyapunov Theorem for Descriptor Systems , 1996 .

[30]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

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

[32]  Guisheng Zhai,et al.  Stability Analysis for a Class of Switched Systems , 2000 .

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

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