# zbMATH — the first resource for mathematics

Switched controller design for stabilization of nonlinear hybrid systems with time-varying delays in state and control. (English) Zbl 1298.93290
Summary: This paper deals with the problem of stabilization for a class of hybrid systems with time-varying delays. The system to be considered is with nonlinear perturbation and the delay is time varying in both the state and control. Using an improved Lyapunov-Krasovskii functional combined with Newton-Leibniz formula, a memoryless switched controller design for exponential stabilization of switched systems is proposed. The conditions for the exponential stabilization are presented in terms of the solution of matrix Riccati equations, which allow for an arbitrary prescribed stability degree.

##### MSC:
 93D21 Adaptive or robust stabilization 93C73 Perturbations in control/observation systems 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
Full Text:
##### References:
 [1] Agarwal, R.P.; Grace, S.R., Asymptotic stability of certain neutral differential equations, Math. comput. model., 31, 9-15, (2000) · Zbl 0973.34062 [2] Abou-Kandil, H.; Freiling, G.; Ionescu, V.; Jank, G., Matrix Riccati equations in control and systems theory, (1982), Birkhauser Basel [3] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004 [4] K.C. Goh, M.G. Safanov, G.P. Papavassilopoulos, A global optimization approach for the BMI problem, in: Proceedings of the 33rd IEEE Conference on Decision and Control, Orlando, 1994, pp. 2009-2014. [5] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer New York [6] Hien, L.V.; Phat, V.N., Exponential stabilization for a class of hybrid systems with mixed delays in state and control, Nonlinear anal. hybrid syst., 3, 259-265, (2009) · Zbl 1184.93075 [7] Lee, C.K., Robust stabilization of linear continuous systems subjected to time-varying state delay and perturbations, J. franklin inst., 333, 707-720, (1996) · Zbl 0886.93057 [8] Z.G. Li, W.X. Xie, C.Y. Wen, Y.C. Soh, Globally exponential stabilization of switched nonlinear systems with arbitrary switchings, in: Proceedings of the 39th IEEE Conference on Decision and Control, December 2000, pp. 3610-3615. [9] Liberzon, D., Switching in systems and control, (2003), Birkhauser Boston · Zbl 1036.93001 [10] Margaliot, M.; Liberzon, D., Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions, Syst. control lett., 55, 8-16, (2006) · Zbl 1129.93521 [11] Mondié, S.; Kharitonov, V.L., Exponential estimates for retarded time-delay systems: an LMI approach, IEEE trans. autom. control, 50, 268-273, (2005) · Zbl 1365.93351 [12] Narendra, K.S.; Balakrishnan, V., A common Lyapunov function for stable LTI systems with commuting matrices, IEEE trans. autom. control, 39, 2469-2471, (1994) · Zbl 0825.93668 [13] Phat, V.N.; Nam, P.T., Robust stabilization of linear systems with delayed state and control, J. optim. theory appl., 140, 287-299, (2009) · Zbl 1159.93027 [14] Phat, V.N., Robust stability and stabilizability of uncertain linear hybrid systems with state delays, IEEE trans. circuits syst. II, 52, 94-98, (2005) [15] Savkin, A.V.; Evans, R.J., Hybrid dynamical systems: controller and sensor switching problems, (2002), Birkhauser Boston · Zbl 1015.93002 [16] Sun, Y.J.; Hsieh, J.G., Global exponential stabilization for a class of uncertain nonlinear systems with time-varying delay arguments and input deadzone nonlinearities, J. franklin inst., 332, 619-631, (1995) · Zbl 0862.93046 [17] Sun, Z.; Ge, S.S., Switched linear systems: control and design, (2005), Springer London · Zbl 1074.93025 [18] Udwadia, F.E.; von Bremen, H.; Phohomsiri, P., Time-delayed control design for active control of structures: principles and applications, Struct. control health monit., 14, 27-61, (2007) [19] Uhlig, F., A recurring theorem about pairs of quadratic forms and extensions, Linear algebra appl., 25, 219-237, (1979) · Zbl 0408.15022 [20] P. Wicks, Construction of piecewise Lyapunov function for stabilizing switched systems, in: Proceedings of the 33rd IEEE Conference on Decision and Control, Florida, 1994, pp. 3492-3497. [21] G. Zhai, Y. Sun, X. Chen, A. Michel, Stability and $$L_2$$ gain analysis for switched symmetric systems with time delay, in: Proceedings of American Control Conference, Colorado, USA, June 2003, pp. 2682-2687. [22] Zhai, G.; Ikeda, M.; Fujisaki, Y., Decentralized $$H_\infty$$ controller design: a matrix inequality approach using a homotopy method, Automatica, 37, 565-572, (2001) · Zbl 0982.93035
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.