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Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems. (English) Zbl 1330.93213
Summary: This paper is concerned with stability of linear discrete time-delay systems. Note that a tighter estimation on a finite-sum term appearing in the forward difference of some Lyapunov functional leads to a less conservative delay-dependent stability criterion. By using Abel lemma, a novel finite-sum inequality is established, which can provide a tighter estimation than the ones in the literature for the finite-sum term. Applying this Abel lemma-based finite-sum inequality, a stability criterion for linear discrete time-delay systems is derived. It is shown through numerical examples that the stability criterion can provide a larger admissible maximum upper bound than stability criteria using a Jensen-type inequality approach and a free-weighting matrix approach.

MSC:
93D99 Stability of control systems
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
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[1] Bromwich, T. J.I’A., An introduction to the theory of infinite series, (1959), Macmillan and Co. London · JFM 39.0306.02
[2] Gao, H.; Chen, T., New results on stability of discrete-time systems with time-varying state delay, IEEE Transactions on Automatic Control, 52, 2, 328-334, (2007) · Zbl 1366.39011
[3] He, Y.; Wu, M.; Liu, G.-P.; She, J.-H., Output feedback stabilization for a discrete-time systems with a time-varying delay, IEEE Transactions on Automatic Control, 53, 10, 2372-2377, (2008) · Zbl 1367.93507
[4] Jiang, X., Han, Q.-L., & Yu, X. (2005). Stability criteria for linear discrete-time systems with interval-like time-varying delay. In Proceedings of the 2005 American control conference (pp. 2817-2822). Portland, OR, USA.
[5] Kwon, O.; Park, M.; Park, J.; Lee, S.; Cha, E., Improved robust stability criteria for uncertain discrete-time systems with interval time-varying delays via new zero equalities, IET Control Theory & Applications, 6, 2567-2575, (2012)
[6] Peng, C., Improved delay-dependent stabilisation criteria for discrete systems with a new finite sum inequality, IET Control Theory & Applications, 6, 448-453, (2012)
[7] Peng, C.; Tian, Y.; Yue, D., Output feedback control of discrete-time systems in networked environments, IEEE Transactions on Systems, Man, Cybernetics A: Systems and Humans, 41, 1, 185-190, (2011)
[8] Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866, (2013) · Zbl 1364.93740
[9] Shao, H.; Han, Q.-L., New stability criteria for linear discrete-time systems with interval-like time-varying delays, IEEE Transactions on Automatic Control, 56, 619-625, (2011) · Zbl 1368.93478
[10] Song, S.-H.; Kim, J.-K., \(H_\infty\) control of discrete-time linear systems with norm-bounded uncertainties and time delay in state, Automatica, 34, 1, 137-139, (1998) · Zbl 0904.93011
[11] Xia, Y.; Liu, G.-P.; Shi, P.; Rees, D.; Thomas, E. J.C., New stability and stabilzation conditions for systems with time-delay, International Journal of Systems Science, 38, 1, 17-24, (2007) · Zbl 1111.93052
[12] Xiong, J.; Lam, J., Stabilization of networked control systems with a logic ZOH, IEEE Transactions on Automatic Control, 54, 2, 358-363, (2009) · Zbl 1367.93546
[13] Xu, S.; Lam, J.; Zhou, Y., Improved conditions for delay-dependent robust stability and stabilization of uncertain discrete time-delay systems, Asian Journal of Control, 7, 3, 344-348, (2005)
[14] Zhang, X.-M.; Han, Q.-L., Event-based \(H_\infty\) filtering for sampled-data systems, Automatica, 51, 55-69, (2015) · Zbl 1309.93096
[15] Zhang, B.; Xu, S.; Zou, Y., Improved stability criterion and its applications in delayed controller design for discrete-time systems, Automatica, 44, 11, 2963-2967, (2008) · Zbl 1152.93453
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