Stability of A Switched Linear System
DOI:
https://doi.org/10.15282/jmes.3.2012.8.0030Keywords:
Stability; Switched linear system; Lyapunov function.Abstract
Hybrid systems are dynamic systems that arise out of the interaction of continuous state dynamics and discrete state dynamics. Switched systems, which are a type of hybrid system, have been given much attention by control systems research over the past
decade. Problems with the controllability, observability, converseability and stabilizability of switched systems have always been discussed. In this paper, the trend in research regarding the stability of switched systems will be investigated. Then the variety of methods that have been discovered by researchers for stabilizing switched linear systems with arbitrary switching will be discussed in detail.
References
Blanchini, F., Miani, S., & Mesquine, F. (2008). A separation principle for linear switching systems and parametrization of all stabilizing controllers. 47th IEEE Conference on Decision and Control, pp. 953-958.
Brayton, R. K., & Tong, C. H. (1979). Stability of dynamical systems: a constructive approach. IEEE Transactions on Circuits and Systems, 26, 224-234.
Brayton, R. K., & Tong, C. H. (1980). Constructive stability and asymptotic stability of dynamical systems. IEEE Transactions on Circuits and Systems, 27, 1121-1130.
Brockett, R. W. (1993). Hybrid models for motion control systems. In: Trentelman, H. L. & Willems, J. C. (eds.) Essays on control: perspectives in the theory and its applications, Cambridge, MA, Birkhäuser Boston, pp. 29-53.
Davrazos, G., & Koussoulas, N. T. (2002). A general methodology for stability analysis of differential petri nets. Proceedings of the 10th Mediterranean Conference on Control and Automation, pp. 1-7.
Dayawansa, W. P., & Martin, C. F. (1999). A converse Lyapunov theorem for a class of dynamical systems which undergo switching. IEEE Transactions on Automatic Control, 44 (4), 751-760.
Decarlo, R. A., Branicky, M. S., Pettersson, S., & Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88 (7), 1069-1082.
Ezzine, J., & Haddad, A. H. (1988). On the controllability and observability of hybrid systems. American Control Conference, pp. 41-46.
Ge, S. S., Zhendong, S., & Lee, T. H. (2001). Reachability and controllability of switched linear systems. Proceedings of the 2001 American Control Conference, pp. 1898-1903.
Geng, Z. (2010). Switched stability design on canonical forms. IEEE International Conference on Information and Automation (ICIA), pp. 289-293.
Gopal, M. (2003). Control systems principles and design. New Delhi, McGraw Hill.
Guangming, X., Dazhong, Z., & Long, W. (2002). Controllability of switched linear systems. IEEE Transactions on Automatic Control, 47 (8), 1401-1405.
Guangming, X., & Long, W. (2002). Necessary and sufficient conditions for controllability of switched linear systems. Proceedings of the 2002 American Control Conference, pp. 1897-1902.
Guisheng, Z., Derong, L., Imae, J., & Kobayashi, T. (2006). Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems. IEEE Transactions on Circuits and Systems II: Express Briefs, 53 (2), 152-156.
Hespanha, J. P., & Morse, A. S. (2002). Switching between stabilizing controllers. Automatica, 38 (11), 1905-1917.
Hespanha, J. P., Santesso, P., & Stewart, G. (2007). Optimal controller initialization for switching between stabilizing controllers. 46th IEEE Conference on Decision and Control, pp. 5634-5639.
Jianhong, W., Xun, L., Yaping, G., & Guangfeng, J. (2008). An LMI optimization approach to Lyapunov stability analysis for linear time-invariant systems. Chinese Control and Decision Conference, pp. 3044-3048.
King, C., & Shorten, R. (2004). A singularity test for the existence of common quadratic Lyapunov functions for pairs of stable LTI systems. Proceedings of the 2004 American Control Conference, pp. 3881-3884.
Li, Z. G., Wen, C. Y., & Soh, Y. C. (2001). Stabilization of a class of switched systems via designing switching laws. IEEE Transactions on Automatic Control, 46 (4), 665-670.
Liberzon, D., Hespanha, J. P., & Morse, A. S. (1999). Stability of switched systems: a lie-algebraic condition. Systems and Control Letters, 37 (3), 117-122.
Lyapunov, A. M. (1992). General problem of the stability of motion. Bristol, PA, Taylor and Francis.
Mancilla-Aguilar, J. L., & García, R. A. (2000). A converse Lyapunov theorem for nonlinear switched systems. Systems and Control Letters, 41 (1), 67-71.
Martin, C. F., & Dayawansa, W. P. (1996). On the existence of a Lyapunov function for a family of switching systems. Proceedings of the 35th IEEE Decision and Control, pp. 1820-1823.
Mason, O., & Shorten, R. (2003). A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems. American Control Conference, pp. 4469-4470.
Mason, P., Sigalotti, M., & Daafouz, J. (2007). On stability analysis of linear discrete-time switched systems using quadratic Lyapunov functions. 46th IEEE Conference on Decision and Control, pp. 5629-5633.
Montagner, V. F., Leite, V. J. S., Oliveira, R. C. L. F., & Peres, P. L. D. (2006). State feedback control of switched linear systems: an LMI approach. Journal of Computational and Applied Mathematics, 194(2), 192-206.
Nesic, D., & Liberzon, D. (2005). A small-gain approach to stability analysis of hybrid systems. 44th IEEE Conference on Decision and Control and 2005 European Control Conference, pp. 5409-5414.
Qi, F., Guangming, X., & Long, W. (2005). Stability analysis and stabilization synthesis for periodically switched linear systems with uncertaintie. American Control Conference, pp. 30-35.
Stewart, G. E., & Dumont, G. A. (2006). Finite horizon based switching between stabilizing controllers. American Control Conference, pp. 1550-1556.
Sun, Z. (2007). Converse Lyapunov theorem for switched stability of switched linear systems. Chinese Control and Decision Conference, pp. 678-680.
Sun, Z., & Ge, S. S. (2005). Analysis and synthesis of switched linear control systems. Automatica, 41, 181-195.
Vu, L., & Liberzon, D. (2006). On invertibility of switched linear systems. 45th IEEE Conference on Decision and Control, pp. 4081-4086.
Wenxiang, X., Changyun, W., & Zhengguo, L. (2001). Input-to-state stabilization of switched nonlinear systems. IEEE Transactions on Automatic Control, 46 (7), 1111-1116.
Yijing, W., Guangming, X., & Long, W. (2003). Reachability and controllability of switched linear systems with state jumps. IEEE International Conference on Systems, Man and Cybernetics, pp. 672-677.
Zhang, W., Shen, S. Q., & Han, Z. Z. (2008). Sufficient conditions for Hurwitz stability of matrices. Latin American Applied Research, 38, 253-258.
Zhu, Y. H., Cheng, D. Z., & Qin, H. S. (2007). Constructing common quadratic Lyapunov functions for a class of stable matrices. Acta Automatica Sinica, 33(2), 202-204.
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