## Reliable $$H_{\infty}$$ control for a class of switched nonlinear systems with actuator failures.(English)Zbl 1118.93351

Summary: This paper focuses on the problem of reliable $$H_{\infty }$$ control for a class of switched nonlinear systems with actuator failures among a prespecified subset of actuators. We consider the case in which the never failed actuators cannot stabilize the system. The multiple-Lyapunov-function method is exploited to derive a sufficient condition for the switched nonlinear system to be asymptotically stable with $$H_{\infty }-norm$$ bound. This condition is given in the form of a set of partial differential inequalities. As a special application, a hybrid state feedback strategy is proposed to solve the standard $$H_{\infty }$$ control problem for non-switched nonlinear systems when no single continuous controller is effective.

### MSC:

 93D20 Asymptotic stability in control theory 93B36 $$H^\infty$$-control 93C10 Nonlinear systems in control theory 93D30 Lyapunov and storage functions 93C15 Control/observation systems governed by ordinary differential equations
Full Text:

### References:

 [1] Liberzon, D., Switching in systems and control, (2003), Birkhauser Boston · Zbl 1036.93001 [2] Cheng, D.Z.; Guo, L.; Lin, Y.D.; Wang, Y., Stabilization of switched linear systems, IEEE transactions on automatic control, 50, 5, 661-666, (2005) · Zbl 1365.93389 [3] Zhao, J.; Dimirovski, G.M., Quadratic stability of a class of switched nonlinear systems, IEEE transactions on automatic control, 49, 4, 574-578, (2004) · Zbl 1365.93382 [4] Sun, Z.D.; Ge, S.S., Switched linear systems—control and design, (2004), Springer-Verlag New York [5] Zhao, J.; Spong, M.W., Hybrid control for global stabilization of the cart – pendulum system, Automatica, 37, 12, 1941-1951, (2001) · Zbl 1005.93041 [6] Xie, G.M.; Wang, L., Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE transactions on automatic control, 49, 6, 960-966, (2004) · Zbl 1365.93049 [7] Zhai, G.S.; Hu, B.; Yasuda, K.; Michel, A.N., Disturbance attenuation properties of time-controlled switched systems, Journal of the franklin institute, 338, 7, 765-779, (2001) · Zbl 1022.93017 [8] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE transactions on automatic control, 43, 4, 475-482, (1998) · Zbl 0904.93036 [9] Hespanha, J.P., Uniform stability of switched linear systems: extensions of lasalle’s invariance principle, IEEE transactions on automatic control, 49, 4, 470-482, (2004) · Zbl 1365.93348 [10] H. Ishii, T. Basar, R. Tempo. Synthesis of switching rules for switched linear systems through randomized algorithms, in: Proceedings of the IEEE Conference on Decision and Control, vol. 5, 2003, pp. 4788-4793 [11] El-Farra, N.H.; Mhaskar, P.; Christofides, P.D., Output feedback control of switched nonlinear systems using multiple Lyapunov functions, Systems and control letters, 54, 12, 1163-1182, (2005) · Zbl 1129.93497 [12] J. Zhao, D.J. Hill, Hybrid $$H_\infty$$ control based on multiple Lyapunov functions, in: 6th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2004, vol. 2, 2004, pp. 567-572 [13] Seo, C.J.; Kim, B.K., Robust and reliable $$H_\infty$$ control for linear systems with parameter uncertainty and actuator failure, Automatica, 32, 3, 465-467, (1996) · Zbl 0850.93214 [14] Wang, Z.D.; Qiao, H., $$H_\infty$$ reliable control of uncertain linear state delayed systems, Journal of dynamical and control systems, 10, 1, 55-76, (2004) · Zbl 1073.93021 [15] Yang, G.H.; Lam, J.; Wang, J.L., Reliable $$H_\infty$$ control for affine nonlinear systems, IEEE transactions on automatic control, 43, 8, 1112-1117, (1998) · Zbl 0957.93027 [16] Wang, J.C.; Shao, H.H., Delay-dependent robust and reliable $$H_\infty$$ control for uncertain time-delay systems with actuator failures, Journal of the franklin institute, 337, 6, 781-791, (2000) · Zbl 0996.93030 [17] Yue, D.; Lam, J.; Ho, D.W.C., Reliable $$H_\infty$$ control of uncertain descriptor systems with multiple time delays, IEE proceedings—control theory and applications, 50, 6, 557-564, (2003) [18] M.A. Demetriou, Adaptive reorganization of switched systems with faulty actuators, in: Proc. of the 40rd IEEE conference on Decision and Control, Orlando, FL, 2001, pp. 1879-1884 [19] Wang, R.; Zhao, J., Robust fault-tolerant control for a class of switched nonlinear systems in lower triangular form, Asian journal of control, 9, 1, (2006) [20] J. Zhao, D.J. Hill, Dissipativity theory for switched systems, in: Proc. of the 44rd IEEE conference on Decision and Control, Seville, Spain, 2005, pp. 7003-7008 [21] Hespanha, J.P.; Liberzon, D.; Angeli, A.; Sontag, E.D., Nonlinear norm observability notions and stability of switched systems, IEEE transactions on automatic control, 50, 2, 154-168, (2005) · Zbl 1365.93349 [22] Su, W.Z.; de Souza, C.E.; Xie, L.H., $$H_\infty$$ control for asymptotically stable nonlinear systems, IEEE transactions on automatic control, 44, 5, 989-993, (1999) · Zbl 0956.93018
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.