# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
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.