×

Stability analysis of systems with time-varying delay via relaxed integral inequalities. (English) Zbl 1338.93290

Summary: This paper investigates the stability of linear systems with a time-varying delay. The key problem concerned is how to effectively estimate single integral term with time-varying delay information appearing in the derivative of Lyapunov-Krasovskii functionals. Two novel integral inequalities are developed in this paper for this estimation task. Compared with the frequently used inequalities based on the combination of Wirtinger-based inequality (or auxiliary function-based inequality) and reciprocally convex lemma, the proposed ones can provide smaller bounding gap without requiring any extra slack matrix. Four stability criteria are established by applying those inequalities. Based on three numerical examples, the advantages of the proposed inequalities are illustrated through the comparison of maximal admissible delay bounds provided by different criteria.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D30 Lyapunov and storage functions
93C05 Linear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Xiong, J.; Lam, J., Stabilization of networked control systems with a logic ZOH, IEEE Trans. Automat. Control, 54, 2, 358-363, (2009) · Zbl 1367.93546
[2] Gu, K.; Kharitonov, V. L.; Chen, J., Stability of time-delay systems, (2003), Birkhauser · Zbl 1039.34067
[3] Sipahi, R.; Olgac, N., A unique methodology for the stability robustness of multiple time delay systems, Systems Control Lett., 55, 10, 819-825, (2006) · Zbl 1100.93033
[4] Fridman, E., Introduction to time-delay systems: analysis and control, (2014), Birkhauser · Zbl 1303.93005
[5] Zhu, J.; Qi, T.; Chen, J., Small-gain stability conditions for linear systems with time-varying delays, Systems Control Lett., 81, 42-48, (2015) · Zbl 1330.93185
[6] Park, M.; Kwon, Q. M.; Park, J. H.; Lee, S. M.; Cha, E. J., Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica, 55, 204-208, (2015) · Zbl 1377.93123
[7] Li, Y.; Gu, K.; Zhou, J.; Xu, S., Estimating stable delay intervals with a discretized Lyapunov-krasovskii functional formulation, Automatica, 50, 6, 1691-1697, (2014) · Zbl 1296.93170
[8] Kim, J. H., Note on stability of linear systems with time-varying delay, Automatica, 47, 9, 2118-2121, (2011) · Zbl 1227.93089
[9] Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866, (2013) · Zbl 1364.93740
[10] Moon, Y. S.; Park, P.; Kwon, W. H.; Lee, Y. S., Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control, 74, 14, 1447-1455, (2001) · Zbl 1023.93055
[11] Fridman, E., New Lyapunov-krasvoskii functionals for stability of linear retarded and neutral type systems, Systems Control Lett., 43, 309-319, (2001) · Zbl 0974.93028
[12] Fridman, E.; Shaked, U., Delay-dependent stability and \(H_\infty\) control: constant and time-varying delays, Internat. J. Control, 76, 48-60, (2003) · Zbl 1023.93032
[13] Briat, C., Linear parameter-varying and time-delay systems: analysis, observation, filtering & control, (2015), Springer-Verlag · Zbl 1395.93003
[14] He, Y.; Wu, M.; She, J. H.; Liu, G. P., Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems Control Lett., 51, 57-65, (2004) · Zbl 1157.93467
[15] Wu, M.; He, Y.; She, J. H.; Liu, G. P., Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40, 8, 1435-1439, (2004) · Zbl 1059.93108
[16] He, Y.; Wang, Q. G.; Lin, C.; Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 2, 371-376, (2007) · Zbl 1111.93073
[17] He, Y.; Wang, Q. G.; Xie, L.; Lin, C., Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Trans. Automat. Control, 52, 2, 293-299, (2007) · Zbl 1366.34097
[18] Zeng, H. B.; He, Y.; Wu, M.; She, J. H., Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Automat. Control, 60, 10, 2768-2772, (2015) · Zbl 1360.34149
[19] Lam, J.; Gao, H.; Wang, C., Stability analysis for continuous systems with two additive time-varying delay components, Systems Control Lett., 56, 1, 16-24, (2007) · Zbl 1120.93362
[20] Wu, L.; Zheng, W. X., Passivity-based sliding mode control of uncertain singular time-delay systems, Automatica, 45, 9, 2120-2127, (2009) · Zbl 1175.93065
[21] Gao, H.; Sun, W.; Shi, P., Robust sampled-data \(H_\infty\) control for vehicle active suspension systems, IEEE Trans. Control Syst. Technol., 18, 1, 238-245, (2010)
[22] Zhang, C. K.; He, Y.; Jiang, L.; Wu, Q. H.; Wu, M., Delay-dependent stability criteria for generalized neural networks with two delay components, IEEE Trans. Neural Netw. Learn. Syst., 25, 7, 1263-1276, (2014)
[23] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of the 39th IEEE Conference on Decision and Control (2010) Sydney, Australia.
[24] Park, P.; Lee, W.; Lee, S. Y., Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Franklin Inst., 352, 1378-1396, (2015) · Zbl 1395.93450
[25] Seuret, A.; Gouaisbaut, F., Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems Control Lett., 81, 1-7, (2015) · Zbl 1330.93211
[26] Jiang, X.; Han, Q. L., New stability criteria for linear systems with interval time-varying delay, Automatica, 44, 10, 2680-2685, (2008) · Zbl 1155.93405
[27] Park, P.; Ko, J., Stability and robust stability for systems with a time-varying delay, Automatica, 43, 10, 1855-1858, (2007) · Zbl 1120.93043
[28] Briat, C., Convergence and equivalence results for the jensen’s inequality-application to time-delay and sampled-data systems, IEEE Trans. Automat. Control, 56, 7, 1660-1665, (2011) · Zbl 1368.26020
[29] Shao, H., New delay-dependent stability criteria for systems with interval delay, Automatica, 45, 3, 744-749, (2009) · Zbl 1168.93387
[30] Sun, J.; Liu, G. P.; Chen, J.; Rees, D., Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica, 46, 2, 466-470, (2010) · Zbl 1205.93139
[31] Fridman, E.; Shaked, U.; Liu, K., New conditions for delay-derivative-dependent stability, Automatica, 45, 11, 2723-2727, (2009) · Zbl 1180.93080
[32] Park, P.; Ko, J.; Jeong, C., Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47, 1, 235-238, (2011) · Zbl 1209.93076
[33] Zeng, H. B.; He, Y.; Wu, M.; She, J. H., New results on stability analysis for systems with discrete distributed delay, Automatica, 60, 189-192, (2015) · Zbl 1331.93166
[34] Seuret, A.; Gouaisbaut, F.; Ariba, Y., Complete quadratic Lyapunov functionals for distributed delay systems, Automatica, 62, 168-176, (2015) · Zbl 1330.93177
[35] Hien, L. V.; Trinh, H., An enhanced stability criterion for time-delay systems via a new bounding technique, J. Franklin Inst., 352, 4407-4422, (2015) · Zbl 1395.93443
[36] Hien, L. V.; Trinh, H., Refined Jensen-based inequality approach to stability analysis of time-delay systems, IET Control Theory Appl., 9, 14, 2188-2194, (2015)
[37] Nam, P. T.; Pathirana, P. N.; Trinh, H., Discrete Wirtinger-based inequality and its application, J. Franklin Inst., 352, 5, 1893-1905, (2015) · Zbl 1395.93448
[38] Seuret, A.; Gouaisbaut, F.; Fridman, E., Stability of discrete-time systems with time-varying delays via a novel summation inequality, IEEE Trans. Automat. Control, 60, 10, 2740-2745, (2015) · Zbl 1360.93612
[39] Zhang, X. M.; Han, Q. L., Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems, Automatica, 57, 199-202, (2015) · Zbl 1330.93213
[40] Zhang, C. K.; He, Y.; Jiang, L.; Wu, M.; Zeng, H. B., Delay-variation-dependent stability of delayed discrete-time systems, IEEE Trans. Automat. Control, (2015)
[41] Ariba, Y.; Gouaisbaut, F., An augmented model for robust stability analysis of time-varying delay systems, Internat. J. Control, 82, 9, 1616-1626, (2009) · Zbl 1190.93076
[42] Y. Ariba, F. Gouaisbaut, Input-output framework for robust stability of time-varying delay systems. In Proceedings of the 48th IEEE Conference on Decision and Control (2009) Shanghai, China. · Zbl 1190.93076
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.