Delay-dependent stability criteria for interval time-varying delay systems with nonuniform delay partitioning approach

This paper investigates the conservatism reduction of Lyapunov-Krasovskii based conditions for the stability of a class of interval time-varying delay systems. The main idea is based on the nonuniform decomposition of the integral terms of the Lyapunov-Krasovskii functional. The delay interval is decomposed into a finite number of nonuniform segments with some scaling parameters. Both differentiable delay case and nondifferentiable delay case and unknown delay derivative bound case are taken into consideration. Sufficient delay-dependent stability criteria are derived in terms of matrix inequalities. Two suboptimal delay fractionation schemes, namely, linearization with cone complementary technique and linearization under additional constraints are introduced in order to find a feasible solution set using LMI solvers with a convex optimization algorithm so that a suboptimal maximum allowable delay upper bound is achieved. It is theoretically demonstrated that the proposed technique has reduced complexity in comparison to some existing delay fractionation methods from the literature. A numerical example with case studies is given to demonstrate the effectiveness of the proposed method with respect to some existing ones from the literature.

Anahtar Kelimeler:

Time delay systems, interval time-varying delay, delay partitioning, cone complementary method, linear matrix inequality

Delay-dependent stability criteria for interval time-varying delay systems with nonuniform delay partitioning approach

Keywords:

Time delay systems, interval time-varying delay, delay partitioning, cone complementary method, linear matrix inequality,

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Table 1. The maximum allowable upper bound for h2versus h1. h1 0 Shao (2009) Sun et al. (2009) Theorem 1, N = 2 Theorem 1, N = 5 μ = 0.3 6972 0129 1767 1938 2591 3408 3475 3574 0744 1690 2072 2182 0275 0936 1060 1060 1060 Theorem 1, N = 5
Table 2. The maximum allowable upper bound for h2versus h1with unknown μ . Methods Shao (2009) Sun et al. (2009) Corollary 1, N = 2 Corollary 1, N = 5 h 0 0 0 0 h2 h2 h2 h2 5412 3570 2182 1060 Conclusions
P. Park, J. W. Ko,”Stability and robust stability for systems with a time-varying delay”, Automatica, vol. 43, pp. 1858, 2007.
E. Fridman, M. Gil, ”Stability of linear systems with time-varying delays: A direct frequency domain approach”, Journal of Computational and Applied Mathematics, vol. 200, pp. 61-66, 2007.
E. Shustin, E. Fridman, ”On delay-derivative dependent stability of systems with fast varying delays”, Automatica, vol. 43, pp. 1649-1655, 2007.
E. Fridman, U. Shaked, ”Input-output approach to stability and L2-gain analysis of systems with time-varying delays”, Systems and Control Letters, vol. 55, pp. 1041-1053, 2006.
E. Fridman, S.-I. Niculescu, ”On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast varying delays”, International Journal of Robust and Nonlinear Control, vol. 18, pp. 364-374, 2008.
E. Fridman, U. Shaked, ”Input-output approach to stability and L2-gain analysis of systems with time-varying delays”, Systems and Control Letters, vol. 55, pp. 1041-1053, 2006.
E. Fridman, ”A new Lyapunov technique for robust control of systems with uncertain non-small delays”, IMA Journal of Control and Information, vol. 23, pp. 165-179, 2006.
Y. He, Q. G. Wang, C. Lin, M. Wu, ”Delay-range-dependent stability for systems with time-varying delay”, Automatica, vol. 43, pp. 371-376, 2007a.
Y. He, Q. P. Liu, D. Rees, ”Augmented Lyapunov functional for the calculation of stability interval for time-varying delay”, IET Control Theory and Applications, vol. 1, pp. 381-386, 2007b.
K. Gu, V. L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Boston, MA, Birkhauser, 2003.
H. Shao, ”Improved delay-dependent stability criteria for systems with a delay varying in a range”, Automatica, vol. 44, pp. 3215-3218, 2008.
X. Jiang, Q. L. Han, ”New stability criteria for linear systems with interval time-varying delay”, Automatica, vol. , pp. 2680-2685, 2008.
C. Peng, Y. C. Tian, ”Improved delay-dependent robust stability criteria for uncertain systems with interval time- varying delay”, IET Control Theory and Applications, vol. 2, pp. 752-761, 2008.
L. Zhang, E. K. Boukas, A. Haidar, ”Delay-range-dependent control synthesis for time-delay systems with actuator saturation”, Automatica, vol. 44, pp. 2691-2695, 2008.
J. Sun, G. P. Liu, J. Chen, D. Rees, ”Improved delay-range-dependent stability criteria for linear systems with time-varying delays”, Automatica (2009), doi:10.1016/j.automatica.2009.11.002.
F. Gouaisbaut, D. Peaucelle, ”Delay-dependent stability analysis of linear time-delay systems”, In IFAC Workshop on Time Delay Systems (TDS’06), L’Aquila, Italy, July 2006.
Y. Ariba, F. Gouaisbaut, ”Construction of Lyapunov-Krasovskii functional for time-varying delay systems”, Pro- ceedings of 47th IEEE Conference on Decision and Control, Mexico, pp. 3995-4000, December 2008.
B. Du, J. Lam, Z. Shu, ”A delay-partitioning projection approach to stability analysis of neutral systems”, Proceedings of the 17th IFAC World Congress on Automatic Control, Seoul, Korea, pp. 12348-12353, July 2008.
K. Gu, ”Discretized LMI set in the stability problem of linear uncertain time-delay systems”, International Journal of Control, vol. 68, no.4, pp. 923-934, 1997.
L. El Ghaoui, F. Oustry, M. Ait Rami, ”A cone complementarity linearization algorithm for static output feedback and related problems”, IEEE Transactions on Automatic Control, vol. 42, pp. 1171-1176, 1997.
M. N. A. Parlakci, ”Stability of interval time-varying delay systems: a nonuniform delay partitioning approach”, In IFAC Workshop on Time Delay Systems (TDS’08), Bucharest, Romania, September 2009.
M. N. A. Parlakci, ”Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems”, International Journal of Robust and Nonlinear Control, vol. 16, pp. 599-636, 2006.
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Philadelphia, PA, SIAM Studies in Applied Mathematics, 1994.
D. Peaucelle, D. Henrion, D. Arzeiler, ”Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation”, In 16th IFAC World Congress on Automatic Control, Prague, Czech Republic, July 2005.
B. Du, J. Lam, Z. Shu, Z. Wang, ”A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components”, IET Control Theory and Applications, vol. 3, pp. 383-390, 2009.