×

zbMATH — the first resource for mathematics

An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay. (English) Zbl 1375.93114
Summary: This paper is concerned with stability of a linear system with a time-varying delay. First, an improved reciprocally convex inequality including some existing ones as its special cases is derived. Compared with an extended reciprocally convex inequality recently reported, the improved reciprocally convex inequality can provide a maximum lower bound with less slack matrix variables for some reciprocally convex combinations. Second, an augmented Lyapunov-Krasovskii functional is tailored for the use of a second-order Bessel-Legendre inequality. Third, a stability criterion is derived by employing the proposed reciprocally convex inequality and the augmented Lyapunov-Krasovskii functional. Finally, two well-studied numerical examples are given to show that the obtained stability criterion can produce a larger upper bound of the time-varying delay than some existing criteria.

MSC:
93D30 Lyapunov and storage functions
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gu, K., Complete quadratic Lyapunov-krasovskii functional: limitations, computational efficiency, and convergence, (Sun, J.-Q.; Ding, Q., Advances in analysis and control of time-delayed dynamical systems, (2013), World Scientific Singapore), 1-19 · Zbl 1308.34094
[2] Gu, K.; Liu, Y., Lyapunov-krasovskii functional for uniform stability of coupled differential-functional equations, Automatica, 45, 3, 798-804, (2009) · Zbl 1168.93384
[3] He, Y.; Wang, Q.-G.; Lin, C.; Wu, M., Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems, International Journal of Robust and Nonlinear Control, 15, 18, 923-933, (2005) · Zbl 1124.34049
[4] 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
[5] Jiang, X.; Han, Q.-L., Delay-dependent robust stability for uncertain linear systems with interval time-varying delay, Automatica, 42, 6, 1059-1065, (2006) · Zbl 1135.93024
[6] Kim, J. H., Further improvement of Jensen inequality and application to stability of time-delayed systems, Automatica, 64, 121-125, (2016) · Zbl 1329.93123
[7] Kwon, O. M.; Park, M. J.; Park, J. H.; Lee, S. M.; Cha, E. J., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute, 351, 5386-5398, (2014) · Zbl 1393.93104
[8] Park, P.; Ko, J.; Jeong, J., Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47, 1, 235-238, (2011) · Zbl 1209.93076
[9] Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866, (2013) · Zbl 1364.93740
[10] Seuret, A.; Gouaisbaut, F., Hierarchy of LMI conditions for the stability of time delay systems, Systems & Control Letters, 81, 1-7, (2015) · Zbl 1330.93211
[11] Seuret, A.; Gouaisbaut, F., Delay-dependent reciprocally convex combination lemma. rapport LAAS n16006, 2016, (2016)
[12] Xu, S.; Lam, J.; Zhang, B.; Zou, Y., New insight into delay-dependent stability of time-delay systems, International Journal of Robust and Nonlinear Control, 25, 7, 961-970, (2015) · Zbl 1312.93077
[13] Zeng, H.-B.; He, Y.; Wu, M.; She, J., Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Transactions on Automatic Control, 60, 10, 2768-2772, (2015) · Zbl 1360.34149
[14] Zeng, H.-B.; He, Y.; Wu, M.; She, J., New results on stability analysis for systems with discrete distributed delay, Automatica, 60, 189-192, (2015) · Zbl 1331.93166
[15] Zhang, X.-M.; Han, Q.-L., Robust \(H_\infty\) filtering for a class of uncertain linear systems with time-varying delay, Automatica, 44, 1, 157-166, (2008) · Zbl 1138.93058
[16] Zhang, X.-M.; Han, Q.-L., Global asymptotic stability for a class of generalized neural networks with interval time-varying delays, IEEE Trans. Neural Netw., 22, 8, 1180-1192, (2011)
[17] Zhang, X.-M.; Han, Q.-L., Novel delay-derivative-dependent stability criteria using new bounding techniques, International Journal of Robust and Nonlinear Control, 23, 13, 1419-1432, (2013) · Zbl 1278.93230
[18] Zhang, X.-M.; Han, Q.-L., Event-based \(H_\infty\) filtering for sampled-data systems, Automatica, 51, 55-69, (2015) · Zbl 1309.93096
[19] 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
[20] Zhang, X.-M.; Han, Q.-L., State estimation for static neural networks with time-varying delays based on an improved reciprocally convex inequality, IEEE Transactions on Neural Networks and Learning Systems, (2017), (in press)
[21] Zhang, C.-K.; He, Y.; Jiang, L.; Wu, M.; Zeng, H.-B., Stability analysis of systems with time-varying delay via relaxed integral inequalities, Systems & Control Letters, 92, 52-61, (2016) · Zbl 1338.93290
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.