Stability Analysis of Linear Switched Systems with Mixed Delays

Authors: Xiuyong Ding, Lan Shu

Abstract:

This paper addresses the stability of the switched systems with discrete and distributed time delays. By applying Lyapunov functional and function method, we show that, if the norm of system matrices Bi is small enough, the asymptotic stability is always achieved. Finally, a example is provided to verify technically feasibility and operability of the developed results.

Keywords: Switched system, stability, Lyapunov function, Lyapunov functional, delays.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328166

References:

[1] A. Morse, "Control using logic-based switching," Lecture Notes in Control and Information Science, vol. 222, London: Springer, 1997.
[2] A. Van Der Schaft and H. Schumacher, "An introduction to hybrid dynamical systems," Lecture Notes in Control and Information Science, vol. 251, London: Springer, 2000.
[3] M. S. Branicky, "Multiple Lyapunov functions and other analysis tools for switched and hybrid systems," IEEE Transaction Automatomatic Control, vol. 43, pp. 475-482, 1998.
[4] W. P. Dayawansa, and C. F. Martin, "A converse Lyapunov theorem for a class of dynamical systems which undergo switching," IEEE Transaction Automatomatic Control, vol. 44, pp. 751-760, 1999.
[5] J. Daafouz, P. Riedinger, and C. Lung, "Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach," IEEE Transaction Automatomatic Control, vol. 47, pp. 1883-1887, 2002.
[6] E. Feron, "Quadratic stabilizability of switched systems via state and output feedback," Center for Intelligent Control Systems, Massachusetts Institute of Technology, Cambridge, MA, Technical Report CICS, pp. 468, 1996.
[7] Z. G. Li, C. Y. Wen, and Y. C. Soh, "Stabilization of a class of switched systems via designing switching laws," IEEE Transaction Automatomatic Control, vol. 46, pp. 665-670, 2001.
[8] Z. G. Li, C. Y. Wen, and Y. C. Soh, "Observer-based stabilization of switching linear systems," Automatica, vol. 39, pp. 517-524, 2003.
[9] D. Liberzon, J. P. Hespanha, and A. S. Morse, "Stability of switched systems: a Lie-algebraic condition," Systems & Control Letters, vol. 37, pp. 117-122, 1999.
[10] D. Liberzon and A. S. Morse, "Basic problems in stability and design of switched systems," IEEE Control Systems Magazine, vol. 19, pp. 59-70, 1999.
[11] A. S. Morse, Control Using Logic-Based Switching. London: Springer, 1997.
[12] Y. G. Sun, L. Wang, and G. Xie, "Delay-dependent robust stability and stabilization for discrete-time switched systems with modedependent time-varying delays," Applied Mathematics and Computation, vol. 180, pp. 428-435, 2006.
[13] M. A. Wicks, P. Peleties, and R. A. DeCarlo, "Construction of piecewise Lyapunov functions for stabilizing switched systems," in Proceedings of IEEE Conference on Decision and Control, Lake Buena Vista, December 1994, pp. 3492-3497.
[14] M. A. Wicks, P. Peleties, and R. A. DeCarlo, "Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems," European Journal of Control, vol. 4, pp. 140-147, 1998.
[15] Y. Wang, L. Xie, and C. E. De Souza, "Robust control of a class of uncertain nonlinear systems," Systems & Control Letters, vol. 19, pp. 139-149, 1972.
[16] D. Xie, L. Wang, F. Hao, and G. Xie, "LMI approach to gain analysis and control synthesis of uncertain switched systems," IEE Proceedings- Control Theory and Applications, vol. 151, pp. 21-28, 2004.
[17] G. Xie and L. Wang, "Controllability and stabilizability of switched linear systems," Systems & Control Letters, vol. 48, pp, 135-155, 2003.
[18] G. Xie, D. Zheng, and L. Wang, "Controllability of switched linear systems," IEEE Transaction Automatomatic Control, vol. 47, pp. 1401- 1405, 2002.
[19] G. Zhai, B. Hu, K. Yasuda, and A. N. Michei, "Disturbance attenuation properties of time- controlled switched systems," Journal of Franklin Institute, vol. 338, pp. 765-779, 2001.
[20] P. Peleties, M. Wicks, and R. DeCarlo, "Switched controller synthesis for quadratic stabilization of a pair of unstable linear systems," European Journal of Control, val. 4, pp. 140-147, 1998.
[21] M. S. Branicky, "Stability of hybrid systems: State of the art," in: Proceeding of the 36th Conference on Decision & Control, San Diego, CA, USA, 1997.
[22] M. S. Branicky, "Multiple Lyapunov functions and other analysis tools for switched and hybrid systems," IEEE Transaction Automatic Control, vol. 43, no. 4, pp. 475-482, 1998.
[23] M. A. Wicks and P. Peleties, "Construction of piecewise Lyapunov functions for stabilizing switched systems," in: Proceeding of the 33rd Conference on Decision & Control, Lake Buena, Vista, FL, Dec. 1994, pp. 3492-3497.
[24] A. V. D. Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems. London Limited: Springer-Verlag, 2000.
[25] J. P. Hespanha and A. S. Morse, "Stability of switched systems with average dwell-time," in: Proceeding of the 38th Conference on Decision & Control, Phoenix, Arizona, USA, Dec. 1999, vol. 88, pp. 2660-2655.
[26] D. Liberzon and A. S. Morse, "Basic problems in stability and design of switchedsystems," IEEE Control Systems Magazine, Oct. 1999.
[27] G. Zhai, B. Hu, K. Yasuda, and A. N. Michael, "Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach," in: Proceeding of the American Control Conference, Chicago, Illinois, USA, Jun. 2000, pp. 200-204.
[28] X. Li, C. E. Souza, "Criteria for robust stability and stabilization of uncertain linear system with state delay," Automatica, vol. 33, pp. 1657- 1662, 1997.
[29] H. Kokame, H. Kobayashi, and T. Mori, "Robust performance for linear delay-differential systems with time-varying uncertainties," IEEE Transactions on Automatic Control, vol. 43, pp. 223-226, 1998.
[30] J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993.
[31] S. j. Kim, S. A. Campbell, and X. Z. Liu, "Delay independent stability of linear switching systems with time delay," Journal of Mathematical Analysis and Applications, vol. 339, pp. 785-801, 2008.
[32] J. Hale, S. V. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993.