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Comparison of bounding methods for stability analysis of systems with time-varying delays. (English) Zbl 1364.93738
Summary: Integral inequalities for quadratic functions play an important role in the derivation of delay-dependent stability criteria for linear time-delay systems. Based on the Jensen inequality, a reciprocally convex combination approach was introduced by P. Park et al. [Automatica 47, No. 1, 235–238 (2011; Zbl 1209.93076)] for deriving delay-dependent stability criterion, which achieves the same upper bound of the time-varying delay as the one on the use of the Y. S. Moon’s et al. [Int. J. Control 74, No. 14, 1447–1455 (2001; Zbl 1023.93055)] inequality. Recently, a new inequality called Wirtinger-based inequality that encompasses the Jensen inequality was proposed by A. Seuret et al. [“Stability of systems with fast-varying delay using improved Wirtinger’s inequality”, in: Proceedings of the 52nd IEEE conference on decision and control. Los Alamitos, CA: IEEE Computer Society. 946–951 (2013; doi:10.1109/CDC.2013.6760004)] for the stability analysis of time-delay systems. In this paper, it is revealed that the reciprocally convex combination approach is effective only with the use of Jensen inequality. When the Jensen inequality is replaced by Wirtinger-based inequality, the Moon’s et al. inequality [loc. cit.] together with convex analysis can lead to less conservative stability conditions than the reciprocally convex combination inequality. Moreover, we prove that the feasibility of an LMI condition derived by the Moon’s et al. inequality as well as convex analysis implies the feasibility of an LMI condition induced by the reciprocally convex combination inequality. Finally, the efficiency of the methods is demonstrated by some numerical examples, even though the corresponding system with zero-delay as well as the system without the delayed term are not stable.

##### MSC:
 93D99 Stability of control systems 93C05 Linear systems in control theory
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