×

Stability of linear time-varying delay systems and applications to control problems. (English) Zbl 1161.34353

Summary: This paper deals with the stability of a class of linear time-varying systems with multiple delays. Using the Lyapunov function method, we give sufficient delay-dependent conditions for the exponential stability with a given convergence rate, which are described in terms of linear matrix inequalities (LMI) and the solution of Riccati differential equations (RDE). The results are applied to the problem of stabilization of linear time-varying control systems with multiple delays. Numerical examples are given to illustrate the results.

MSC:

34K20 Stability theory of functional-differential equations
93C23 Control/observation systems governed by functional-differential equations
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abou-Kandil, H.; Freiling, G.; Ionescu, V.; Jank, G., Matrix Riccati Equations in Control and Systems Theory (2003), Birkhauser: Birkhauser Basel · Zbl 1027.93001
[2] S. Boyd, El. Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities and control theory, SIAM Studies in Applied Mathematics, vol. 15, SIAM, Philadelphia, PA, 1994.; S. Boyd, El. Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities and control theory, SIAM Studies in Applied Mathematics, vol. 15, SIAM, Philadelphia, PA, 1994. · Zbl 0816.93004
[3] Cao, D. Q.; He, P.; Zhang, K., Exponential stability criteria of uncertain systems with multiple time delays, J. Math. Anal. Appl., 283, 362-374 (2003) · Zbl 1044.34030
[4] Cao, J.; Wanfg, J., Delay-dependent robust stability of uncertain nonlinear systems with time delays, Appl. Math. Comput., 154, 289-297 (2004)
[5] Chen, M. S.; Chen, Y. Z., Static output feedback control for periodically time-varying systems, IEEE Trans. Automat. Control, 44, 218-221 (1999) · Zbl 1056.93543
[6] Fridman, E.; Shaked, U., A descriptor system approach to H-infinity control of linear time-delay systems, IEEE Trans. Automat. Control, 47, 253-270 (2002) · Zbl 1364.93209
[7] Gibson, J. S., Riccati equations and numerical approximations, SIAM J. Contr. Optim., 21, 95-139 (1983) · Zbl 0557.49017
[8] Hartung, F., Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Appl. Math., 174, 201-211 (2005) · Zbl 1077.34074
[9] Jefferey, J. D., Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math., 176, 381-410 (2005) · Zbl 1064.39005
[10] Kalman, R., Contribution to the theory of optimal control, Bol. Soc. Mat. Mexicana, 5, 102-119 (1960)
[11] Kharitonov, V. L., Lyapunov-Krasovskii functionals for scalar time delay equations, Systems Control Lett., 51, 133-149 (2004) · Zbl 1157.34354
[12] Kharitonov, V. L.; Hinrichsen, D., Exponential estimates for time delay systems, Systems Control Lett., 53, 395-405 (2004) · Zbl 1157.34355
[13] Klamka, J., Controllability of Nonlinear Dynamical Systems (1990), Kluwer Academic Publisher: Kluwer Academic Publisher Berlin
[14] Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press: Academic Press London · Zbl 0593.34070
[15] Laub, A. J., Schur techniques for Riccati differential equations, (Feedback Control of Linear and Nonlinear Systems, Lecture Notes in Control and Information Sciences (1982), Springer: Springer Berlin), 165-174
[16] Liu, P. L.; Su, T. J., Robust stability of interval time-delay systems with delay-dependence, Systems Control Lett., 33, 231-239 (1998) · Zbl 0902.93052
[17] Niamsup, P.; Phat, V. N., Asymptotic stability of nonlinear control systems described by differential equations with multiple delays, Electron. J. Diff. Equations, 11, 1-17 (2000) · Zbl 0941.93052
[18] Phat, V. N., New stabilization criteria for linear time-varying systems with state delay and normed bounded uncertainties, IEEE Trans. Atomat. Control, 12, 2095-2098 (2002) · Zbl 1364.93663
[19] Phat, V. N.; Bay, N. S., Stability analysis of nonlinear retarded difference equations in Banach spaces, J. Comput. Math. Appl., 45, 951-960 (2003) · Zbl 1053.39004
[20] Sun, Y. J.; Hsieh, J. G., On \(\alpha \)-stability criteria of nonlinear systems with multiple delays, J. Franklin Inst., 335B, 695-705 (1998) · Zbl 0910.34062
[21] William Thomas, R., Riccati Differential Equations (1972), Academic Press: Academic Press New York
[22] Xu, B., Stability criteria for linear systems with uncertain delays, J. Math. Anal. Appl., 284, 455-470 (2003) · Zbl 1042.34101
[23] Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE Trans. Automat. Control, 50, 384-387 (2005) · Zbl 1365.93376
[24] Xu, S.; Lam, J.; Zou, Y., A simplified descriptor system approach to delay-dependent stability and performance analyses for time-delay systems, IEE Proc.-Control Theory and Applications, 152, 147-151 (2005)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.