Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Abstract

SUMMARYIntegral inequalities have been widely used in stability analysis for systems with time-varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger-based inequality, the Bessel-Legendre inequality, the Wirtinger-based double integral inequality, and the auxiliary function-based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time-varying delay are demonstrated with two numerical examples.

Reliable H ∞ static output control of linear time‐varying delay systems against sensor failures

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“…In most situations, it could degenerate the system performance or even cause system instability [1]. Accordingly, in the last decades, it has been studied widely and many important results on stability analysis have been achieved in this field [2][3][4][5][6][7][8][9][10]. Depending on whether the obtained methods involve time-delay bounds, they are categorized into the delayindependent [2] and delay-dependent ones [3][4][5][6][7][8][9][10].…”

Synchronization of decentralized event-triggered uncertain switched neural networks with two additive time-varying delays

“…In recent years, neural networks (NNs) have become the active research field, and their successive application used in various areas, such as image processing and optimization problems [3,6,26]. Time delays, which always cause instability and degrade performance, are ubiquitously present in many NNs due to the signal transmission (see [1,12] and references therein).…”

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

“…Stability criteria for them fall into two categories, ie, delay‐independent and delay‐dependent ones. Since delay‐dependent criteria tend to be less conservative than delay‐independent ones, especially in the case of small delay, a growing body of research works focus on the former, such as in other works …”

“…Since delay-dependent criteria tend to be less conservative than delay-independent ones, especially in the case of small delay, a growing body of research works focus on the former, such as in other works. [2][3][4][5] In view that the frequency domain approach is oriented to constant delay systems, stability analysis for time-varying delay systems has been widely concerned through Lyapunov-Krasovskii functional (LKF) in past few decades. [6][7][8][9][10][11][12] There also exist the "Input-to-State" or "Input-to-Output" approaches, which are related to the robust analysis.…”