Robust \(H_\infty \) stabilization for uncertain switched impulsive control systems with state delay: An LMI approach. (English) Zbl 1163.93386

Summary: This paper deals with the problem of robust \(H_\infty \) state feedback stabilization for uncertain switched linear systems with state delay. The system under consideration involves time delay in the state, parameter uncertainties and nonlinear uncertainties. The parameter uncertainties are norm-bounded time-varying uncertainties which enter all the state matrices. The nonlinear uncertainties meet with the linear growth condition. In addition, the impulsive behavior is introduced into the given switched system, which results a novel class of hybrid and switched systems called switched impulsive control systems. Using the switched Lyapunov function approach, some sufficient conditions are developed to ensure the globally robust asymptotic stability and robust \(H_\infty \) disturbance attenuation performance in terms of certain Linear Matrix Inequalities (LMIs). Not only the robustly stabilizing state feedback \(H_\infty \) controller and impulsive controller, but also the stabilizing switching law can be constructed by using the corresponding feasible solution to the LMIs. Finally, the effectiveness of the algorithms is illustrated with an example.


93D21 Adaptive or robust stabilization
93D15 Stabilization of systems by feedback
93C05 Linear systems in control theory
93B51 Design techniques (robust design, computer-aided design, etc.)
93B36 \(H^\infty\)-control
Full Text: DOI


[1] Liberzon, D.; Morse, A.S., Basic problems in stability and design of switched systems, IEEE control systems magazine, 19, 59-70, (1999) · Zbl 1384.93064
[2] Decarlo, R.; Branicky, M.S.; Lennartson, B., Perspective and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88, 1069-1082, (2000)
[3] Liberzon, D., Switching in systems and control, (2003), Birkhauser Boston, MA · Zbl 1036.93001
[4] Sun, Z.D.; Ge, S.S., Switched linear systems-control and design, (2004), Springer Verlag New York
[5] Li, Z.G.; Soh, Y.C.; Wen, C.Y., Robust stability of quasiperiodic hybrid dynamic uncertain systems, IEEE transactions on automatic control, 46, 107-111, (2001) · Zbl 0992.93068
[6] Li, Z.G.; Soh, Y.C.; Wen, C.Y., Robust stability for a class of hybrid nonlinear systems, IEEE transactions on automatic control, 47, 897-903, (2002) · Zbl 1001.93069
[7] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE transactions on automatic control, 43, 60-66, (1998)
[8] Cheng, D.Z.; Lei, G.; Lin, Y.D.; Wang, Y., Stabilization of switched systems, IEEE transactions on automatic control, 50, 661-666, (2005) · Zbl 1365.93389
[9] Xie, G.M.; Wang, L., Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE transactions on automatic control, 49, 960-966, (2004) · Zbl 1365.93049
[10] Kim, S.; Campbell, S.A.; Liu, X.Z., Stability of a class of linear switching systems with time delay, IEEE transactions on circuits and systems, part I-fundamental theory and applications, 53, 2, 384-393, (2006) · Zbl 1374.94950
[11] Zong, G.D.; Wu, Y.Q., Exponential stability of discrete time perturbed impulsive switched systems, Acta automatica, sinica, 32, 186-192, (2006)
[12] Guan, Z.H.; Qian, T.H.; Yu, X., Controllaibility and observability of linear time-varying impulsive systems, IEEE transactions on circuits and systems, part I-fundamental theory and applications, 49, 1198-1208, (2002) · Zbl 1368.93034
[13] Guan, Z.H.; Hill, D.J.; Shen, X.M., On hybrid impulsive and switching systems and application to nonlinear control, IEEE transactions automatic control, 50, 1058-1062, (2005) · Zbl 1365.93347
[14] Bainov, D.D.; Simeonov, P.S., Systems with impulse effect: stability, theory and applications, (1989), Halsted Press New York · Zbl 0676.34035
[15] Boyd, S.; EL Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelpha · Zbl 0816.93004
[16] Khargoneckar, P.P.; Petersen, I.R.; Zhou, K., Robust stabilization of uncertain linear systems: quadratic stabilizability and \(H_\infty\) control theory, IEEE transactions automatic control, 35, 356-361, (1990) · Zbl 0707.93060
[17] Daafouz, J.; Riedinger, P.; Iung, C., Stability analysis and control synthesis fow switched systems: A switched Lyapunov function approach, IEEE transactions automatic control, 47, 1883-1887, (2002) · Zbl 1364.93559
[18] V.F. Montagner, V.J.S. Leite, S. Tarbouriech, P.L.D. Peres, Stability and stabilizability of discrete-time switched linear systems with state delay, in: American Control Conference, Portland, OR, USA, June 8-10, 2005, pp. 3806-3811
[19] Y. Zhang, G.R. Duan, \(H_\infty\) control for uncertain discrete-time switched systems with time-delay based on LMI, in: Proceedings ACIT (Automation Control And Information Technology) — Automation Control and Applications, Novosibirsk, Russia, June 20-24, 2005
[20] Du, D.S.; Zhou, S.S.; Zhang, B.Y., Generalized \(H_2\) output feedback controller design for uncertain discrete-time switched systems via switched Lyapunov functions, Nonlinear analysis, 65, 2135-2146, (2006) · Zbl 1106.93051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.