A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching

• Autorzy: Guisheng Zhai , Xuping Xu
• Języki publikacji: EN

Abstrakty

EN

We establish a unified approach to stability analysis for switched linear descriptor systems under arbitrary switching in both continuous-time and discrete-time domains. The approach is based on common quadratic Lyapunov functions incorporated with linear matrix inequalities (LMIs). We show that if there is a common quadratic Lyapunov function for the stability of all subsystems, then the switched system is stable under arbitrary switching. The analysis results are natural extensions of the existing results for switched linear state space systems.

Słowa kluczowe

EN

switched linear descriptor systems; stability; arbitrary switching; linear matrix inequalities (LMIs); common quadratic Lyapunov functions

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2010
• Tom: 20
• Numer: 2
• Strony: 249-259

Daty

wydano

2010

otrzymano

2009-04-24

poprawiono

2009-11-14

Twórcy

autor

Guisheng Zhai

• Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan

autor

Xuping Xu

• Department of Electrical and Computer Engineering, California Baptist University, Riverside, CA 92504, USA

Bibliografia

• Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA.
• Branicky, M.S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43(4): 475-482.
• Cobb, D. (1983). Descriptor variable systems and optimal state regulation, IEEE Transactions on Automatic Control 28(5): 601-611.
• DeCarlo, R., Branicky, M.S., Pettersson, S. and Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88(7): 1069-1082.
• Hespanha, J.P. and Morse, A.S. (2002). Switching between stabilizing controllers, Automatica 38(11): 1905-1917.
• Hu, B., Zhai, G. and Michel, A.N. (2002). Hybrid static output feedback stabilization of second-order linear timeinvariant systems, Linear Algebra and Its Applications 351-352: 475-485.
• Ikeda, M., Lee, T.W. and Uezato, E. (2000). A strict LMI condition for H₂ control of descriptor systems, Proceedings of the 39th IEEE Conference on Decision and Control, CDC 2000, Sydney, Australia, pp. 601-604.
• Ishida, J.Y. and Terra, M.H. (2001). On the Lyapunov theorem for descriptor systems, Proceedings of the 40th IEEE Conference on Decision and Control, CDC 2001, Orlando, FL, USA, pp. 2860-2864.
• Kaczorek, T. (2002). Polynomial approach to pole shifting to infinity in singular systems by feedbacks, Bulletin of the Polish Academy of Sciences: Technical Sciences 50(2): 134-144.
• Kaczorek, T. (2004). Infinite eigenvalue assignment by output feedback for singular systems, International Journal of Applied Mathematics and Computer Science 14(1): 19-23.
• Lewis, F.L. (1986). A survey of linear singular systems, Circuits Systems Signal Process 5(1): 3-36.
• Liberzon, D. (2003). Switching in Systems and Control, Birkhäuser, Boston, MA.
• Liberzon, D., Hespanha, J.P. and Morse, A.S. (1999). Stability of switched systems: A Lie-algebraic condition, Systems & Control Letters 37(3): 117-122.
• Liberzon, D. and Morse, A.S. (1999). Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19(5): 59-70.
• Masubuchi, I., Kamitane, Y., Ohara, A. and Suda, N. (1997). $H_∞$ control for descriptor systems: A matrix inequalities approach, Automatica 33(4): 669-673.
• Narendra, K.S. and Balakrishnan, J. (1994). A common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control 39(12): 2469-2471.
• Sun, Z. and Ge, S.S. (2005a) Analysis and synthesis of switched linear control systems, Automatica 41(2): 181-195.
• Sun, Z. and Ge, S.S. (2005b). Switched Linear Systems: Control and Design, Springer, London.
• Takaba, K., Morihara, N. and Katayama, T. (1995). A generalized Lyapunov theorem for descriptor systems, Systems & Control Letters 24(1): 49-51.
• Uezato, E. and Ikeda, M. (1999). Strict LMI conditions for stability, robust stabilization, and $H_∞$ control of descriptor systems, Proceedings of the 38th IEEE Conference on Decision and Control, CDC 1999, Phoenix, AZ, USA, pp. 4092-4097.
• Xu, S. and Yang, C. (1999). Stabilization of discrete-time singular systems: A matrix inequalities approach, Automatica 35(9): 1613-1617.
• Zhai, G., Hu, B., Yasuda, K. and Michel, A.N. (2001). Disturbance attenuation properties of time-controlled switched systems, Journal of The Franklin Institute 338(7): 765-779.
• Zhai, G., Hu, B., Yasuda, K. and Michel, A.N. (2002). Stability and L₂ gain analysis of discrete-time switched systems, Transactions of the Institute of Systems, Control and Information Engineers 15(3): 117-125.
• Zhai, G., Liu, D., Imae, J. and Kobayashi, T. (2006). Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems, IEEE Transactions on Circuits and Systems II 53(2): 152-156.
• Zhai, G. and Xu, X. (2009). A unified approach to analysis of switched linear descriptor systems under arbitrary switching, Proceedings of the 48th IEEE Conference on Decision and Control, CDC 2009, Shanghai, China, pp. 3897-3902.
• Zhai, G., Kou, R., Imae, J. and Kobayashi, T. (2009a). Stability analysis and design for switched descriptor systems, International Journal of Control, Automation, and Systems 7(3): 349-355.
• Zhai, G., Xu, X., Imae, J. and Kobayashi, T. (2009b). Qualitative analysis of switched discrete-time descriptor systems, International Journal of Control, Automation, and Systems 7(4): 512-519.

Identyfikatory

DOI

10.2478/v10006-010-0018-2