Quadratic stabilization of a class of switched nonlinear systems via single Lyapunov function. (English) Zbl 1179.93152

Summary: This paper is concerned with the problem of globally quadratic stabilization for a class of switched cascade systems. The system under consideration is composed of two subsystems: a linear switched part and a nonlinear part, which are also switched systems. The feedback control law and the switching law are designed respectively when the first part is stabilized under some switching law and when both parts can be stabilized under some switching laws. We construct the single Lyapunov functions and design the switching laws based on the structure characteristics of the switched system. Also, the designed switching laws are of hysteresis switching form in order to avoid sliding models.


93D21 Adaptive or robust stabilization
93B52 Feedback control
93C15 Control/observation systems governed by ordinary differential equations
93B12 Variable structure systems
Full Text: DOI


[1] P. Peleties, R.A. Decarlo, Asymptotic stability of \(m\)-switched systems using Lyapunov-like functions, in: Proceedings of 1999 American Control Conference, 1999, pp. 1679-1684
[2] Liberzon, D.; Hespanha, H.J.P.; Morse, A.S., Stability of switched systems: A Lie-algebraic condition, Systems control lett., 37, 117-122, (1999) · Zbl 0948.93048
[3] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched hybridsystems, IEEE trans. automat. control, 43, 475-482, (1998) · Zbl 0904.93036
[4] D. Angeli, D. Liberzon, A note on uniform global asymptotically stability of nonlinear switched systems in triangular form, in: Proc. MTNS. 2000
[5] Cheng, D., Stabilization of planar switched systems, Systems control lett., 51, 79-88, (2004) · Zbl 1157.93482
[6] Liberzon, D.; Morse, A.S., Basic problems in stability and design of switched systems, IEEE control syst. mag., 19, 59-70, (1999) · Zbl 1384.93064
[7] Liberzon, D., Switching in systems and control, (2003), Birkhauser Boston · Zbl 1036.93001
[8] Mancilla-Aguilar, J.L., A condition for the stability of switched nonlinear systems, IEEE trans. automat. control, 45, 2077-2079, (2000) · Zbl 0991.93089
[9] Zhao, J.; Dimirovski, G.M., Quadratic stability of a class of switched nonlinear systems, IEEE trans. automat. control, 49, 574-578, (2004) · Zbl 1365.93382
[10] G. Xie, W. Long, New results on stability of switched linear systems, in: Proc. of the 42nd IEEE Conf. on Decision and Control, 2003, pp. 6265-6270
[11] X. Xu, P.J. Antsaklis, Design of stabilizing control laws for second-order switched systems, in: Proc. of the 143th IFAC World Congress, 1999, C, pp. 181-186
[12] Cheng, D.; Guo, L.; Lin, Y.; Wang, Y., Stabilization of switched linear systems, IEEE trans. automat. control, 55, 5, 661-666, (2005) · Zbl 1365.93389
[13] Cheng, D.; Feng, Gang; Xi, Zairong, Stabilization of a class of switched nonlinear systems, Control theory appl., 1, 53-61, (2006) · Zbl 1171.93384
[14] S. Zhao, J. Jun, Global stabilization of a class of cascade switched nonlinear systems, in: IEEE Conf. on Decision and Control, 2004, pp. 2817-2822
[15] N.H. EI-Farra, P.D Christofides, State feedback control of switched nonlinear systems using multiple Lyapunov functions, in: Proc. the IEEE American Control Conference. VA June, 2001, pp. 25-27
[16] EI-Farra, N.H.; Prashant, Mhaskar; Christofides, P.D., Output feedback control of switched nonlinear systems using multiple Lyapunov functions, Systems control lett., 54, 1163-1182, (2005) · Zbl 1129.93497
[17] S. Pettersson, B. Lennartson, Stabilization of hybrid systems using a min-projection strategy, in: Proc. American Control Conf, 2001, pp. 223-228
[18] G. Zhai, Quadratic stabilizability of discrete-time switched systems via state and output feedback, in: IEEE Proc. Conf. on Decision and Control, 2001, pp. 2165-2166
[19] H.J.P. Hespanha, A.S. Morse, Stability of switched systems with average dell-time, in: IEEE Proc. Conf. on Decision and Control, 1999, pp. 2655-2660
[20] Wicks, M.A.; Peleties, P.; Decarlo, R.A., Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems, Eur. J. control, 140-147, (1998) · Zbl 0910.93062
[21] Cheng, D.; Guo, L.; Huang, J., On quadratic Lyapunov functions, IEEE trans. automat. control, 48, 885-890, (2003) · Zbl 1364.93557
[22] Sepulchre, R.; Jankovic, M.; Kokotovic, P.V., Constructive nonlinear control, (1997), Springer Verlag · Zbl 1067.93500
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