Abstract
This paper is concerned with the stability analysis problems of linear systems with time-varying delays using integral inequalities. To reduce the conservatism of stability criteria obtained with Lyapunov-Kraosvksii approach, there has been a growing tendency to utilize various integral quadratic functions in the construction of Lyapunov-Krasovskii functionals. Consequently, integral inequalities also have played key roles to derive stability criteria guaranteeing the negativity of the Lyapunov-Krasovskii functional’s derivative. Recently, by utilizing first-degree or second-degree orthogonal polynomials, new free-matrix-based integral inequalities have been proposed for integral quadratic functions containing both a system state variable and its time derivative. This paper tries to generalize these inequalities and their stability criteria with a note on the relation among the existing integral inequalities and the proposed one. In this note, it is shown that increasing a degree of the proposed integral inequality only reduces or maintains the conservatism by deriving the hierarchical stability criteria. Four numerical examples including practical systems demonstrate the effectiveness of the proposed methods in terms of allowable upper delay bounds.
Highlights
Time delays are common phenomena arisen in many practical systems including chemical systems, biological systems, mechanical engineering systems, and networked control systems [1]
Stability analysis for a linear system with time delays is the foundation because analysis techniques for this system can be extended to controller and filter synthesis problems of various systems including switched systems, neural network systems, fuzzy systems with time delays
This paper aims at deriving sufficient linear matrix inequalities (LMIs) conditions guaranteeing that a linear system with a time-varying delay is asymptotically stable with the allowable upper delay bound as large as possible
Summary
Time delays are common phenomena arisen in many practical systems including chemical systems, biological systems, mechanical engineering systems, and networked control systems [1]. In the presence of time-varying delays, the Jensen inequality provides reciprocally convex upper bounds with respect to the length of an integral interval, which leads to non-convex stability conditions. The negative convex upper bound of LKF’s derivative sufficiently guarantees an asymptotic stability of time-delay systems. This paper aims at deriving sufficient LMI conditions guaranteeing that a linear system with a time-varying delay is asymptotically stable with the allowable upper delay bound as large as possible. Four numerical examples of stability analysis for linear systems with time-varying delays are given In these examples including practical systems, the effectiveness of the proposed methods are shown in terms of allowable upper delay bounds for a given lower delay bounds.
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