A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching
Guisheng Zhai ; Xuping Xu
International Journal of Applied Mathematics and Computer Science, Tome 20 (2010), p. 249-259 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:207984
@article{bwmeta1.element.bwnjournal-article-amcv20i2p249bwm,
     author = {Guisheng Zhai and Xuping Xu},
     title = {A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching},
     journal = {International Journal of Applied Mathematics and Computer Science},
     volume = {20},
     year = {2010},
     pages = {249-259},
     zbl = {1196.93070},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv20i2p249bwm}
}
Guisheng Zhai; Xuping Xu. A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching. International Journal of Applied Mathematics and Computer Science, Tome 20 (2010) pp. 249-259. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv20i2p249bwm/

[000] Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA. | Zbl 0816.93004

[001] 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. | Zbl 0904.93036

[002] Cobb, D. (1983). Descriptor variable systems and optimal state regulation, IEEE Transactions on Automatic Control 28(5): 601-611. | Zbl 0522.93036

[003] 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.

[004] Hespanha, J.P. and Morse, A.S. (2002). Switching between stabilizing controllers, Automatica 38(11): 1905-1917. | Zbl 1011.93533

[005] 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. | Zbl 1007.93061

[006] 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.

[007] 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.

[008] 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.

[009] Kaczorek, T. (2004). Infinite eigenvalue assignment by output feedback for singular systems, International Journal of Applied Mathematics and Computer Science 14(1): 19-23. | Zbl 1171.93331

[010] Lewis, F.L. (1986). A survey of linear singular systems, Circuits Systems Signal Process 5(1): 3-36. | Zbl 0613.93029

[011] Liberzon, D. (2003). Switching in Systems and Control, Birkhäuser, Boston, MA. | Zbl 1036.93001

[012] 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. | Zbl 0948.93048

[013] Liberzon, D. and Morse, A.S. (1999). Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19(5): 59-70.

[014] Masubuchi, I., Kamitane, Y., Ohara, A. and Suda, N. (1997). H control for descriptor systems: A matrix inequalities approach, Automatica 33(4): 669-673. | Zbl 0881.93024

[015] 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. | Zbl 0825.93668

[016] Sun, Z. and Ge, S.S. (2005a) Analysis and synthesis of switched linear control systems, Automatica 41(2): 181-195. | Zbl 1074.93025

[017] Sun, Z. and Ge, S.S. (2005b). Switched Linear Systems: Control and Design, Springer, London. | Zbl 1075.93001

[018] Takaba, K., Morihara, N. and Katayama, T. (1995). A generalized Lyapunov theorem for descriptor systems, Systems & Control Letters 24(1): 49-51. | Zbl 0883.93035

[019] 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.

[020] Xu, S. and Yang, C. (1999). Stabilization of discrete-time singular systems: A matrix inequalities approach, Automatica 35(9): 1613-1617. | Zbl 0959.93048

[021] 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. | Zbl 1022.93017

[022] 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.

[023] 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.

[024] 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.

[025] 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.

[026] 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.