Robust exponential stability and stabilizability of linear parameter dependent systems with delays. (English) Zbl 1207.93087

Summary: The robust exponential stability and stabilizability problems are addressed in this paper for a class of linear parameter dependent systems with interval time-varying and constant delays. In this paper, restrictions on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive delay-dependent exponential stability and stabilizability conditions in terms of Linear Matrix Inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to illustrate the effectiveness of our theoretical results.


93D21 Adaptive or robust stabilization
93D20 Asymptotic stability in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory
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[2] Cao, D. Q.; He, P.; Yhang, K. Y., Exponential stability criteria of uncertain systems with multiple time delays, Journal of Computational and Applied Mathematics, 283, 274-362 (2003) · Zbl 1044.34030
[3] Cauët, S.; Rambault, L.; Bachelier, O.; Mehdi, D., Parameter-dependent Lyapunov functions applied to analysis of induction motor stability, Control Engineering Practice, 10, 337-345 (2002)
[4] Chen, Y.; Xue, A.; Lu, R.; Zhou, S., On robustly exponentially stability of uncertain neutral systems with time-varying delays and nonlinear perturbations, Nonlinear Analysis, 68, 2464-2470 (2008) · Zbl 1147.34352
[5] Chi, B. T.; Hua, M. G., Robust passive control for uncertain discrete-time systems with time-varying delays, Chaos, Solitons and Fractals, 29, 331-341 (2006) · Zbl 1147.93034
[6] Gao, H.; Lam, J.; Wang, C., Mixed \(H_2/H_∞\) filtering for continuous-time polytopic systems: a parameter-dependent approach, Circuits Systems Signal Processing, 24, 689-702 (2005) · Zbl 1102.94033
[7] Geromel, J. C.; Colaneri, P., Robust stability of time varying polytopic systems, Systems and Control Letters, 55, 81-85 (2001) · Zbl 1129.93479
[8] Grammel, G.; Maizurna, I., Exponential stability of nonlinear differential equations in the presence of perturbations or delays, Communications in Nonlinear Science and Numerical Simulation, 12, 869-875 (2007) · Zbl 1120.34039
[9] Gu, K.; Kharitonov, V. L.; Chen, J., Stability of Time-Delay System (2003), Birkhauser: Birkhauser Boston · Zbl 1039.34067
[10] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[11] He, Y.; Wang, Q.-G.; Zheng, W.-X., Global robust stability for delayed neural networks with polytopic type uncertainties, Chaos, Solitons and Fractals, 26, 1349-1354 (2005) · Zbl 1083.34535
[12] He, Y.; Wu, M.; She, J.-H.; Liu, G.-P., Parameter-dependent Lyapunov functional for stability of time-delay system with polytopic-type uncertainties, IEEE Transactions on Automatic Control, 49, 828-832 (2004) · Zbl 1365.93368
[13] Jiang, X.; Han, Q.-L., Delay-dependent robust stability for uncertain linear systems with interval time-varying delay, Automatica, 42, 1059-1065 (2006) · Zbl 1135.93024
[14] Jiang, M.; Shen, Y.; Liao, X., On the global exponential stability for functional differential equations, Communications in Nonlinear Science and Numerical Simulation, 10, 705-713 (2005) · Zbl 1070.35113
[15] Liao, X.; Yang, J.; Guo, S., Exponential stability of Cohen-Grossberg neural networks with delays, Communications in Nonlinear Science and Numerical Simulation, 13, 1767-1775 (2008) · Zbl 1221.34144
[16] Nam, P. T.; Phat, V. N., Robust exponentially stability and stabilization of linear uncertain polytopic time-delay systems, Journal Control Theory and Applications, 6, 2, 163-170 (2008)
[17] Peng, C.; Tian, Y.-C., Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay, Journal of Computational and Applied Mathematics, 214, 480-494 (2008) · Zbl 1136.93437
[18] Phat, V. N., Robust stability and stabilizability of uncertain linear hybrid system with state delays, IEEE Transactions Circuits System II, 52, 94-98 (2005)
[19] Phat, V. N.; Niamsup, P., Stability of linear time-varying delay systems and applications to control problems, Journal of Computational and Applied Mathematics, 194, 343-356 (2006) · Zbl 1161.34353
[20] Wu, M.; Liu, F.; Shi, P., Exponential stability analysis for neural networks with time-varying delay, IEEE Transactions on Systems, Man and Cybernetics, 38, 4, 1152-1156 (2008)
[21] Xu, S.; Lam, J.; Ho, D. W.C.; Zou, Y., Delay-dependent exponentially stability for a class of neural networks with time delays, Journal of Computational and Applied Mathematics, 183, 16-18 (2005) · Zbl 1097.34057
[22] Yoneyama, J., New delay-dependent approach to robust stability and stabilization for Takagi-Sugeno fuzzy time-delay systems, Fuzzy and Systems, 158, 2225-2237 (2007) · Zbl 1122.93050
[23] Zhao, W., Global exponential stability analysis of Cohen-Grossberg neural network with delays, Communications in Nonlinear Science and Numerical Simulation, 13, 847-856 (2008) · Zbl 1221.93207
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