A survey of inequality techniques for stability analysis of time‐delay systems

Abstract

During the past decades, much attention has been paid to the stability problem of linear time-delay systems. In order to obtain tractable stability conditions shown in the linear matrix inequality form, a great number of remarkable results have been reported in the literature. This article first gives a survey of inequality techniques recently developed to estimate integral quadratic terms and reciprocally convex combination terms arising in the estimation of the time derivative of a Lyapunov-Krasovskii functional candidate. Emphases are specially placed on the evolution of various integral and matrix inequalities and the relationships among them. Second, several stability conditions are obtained and carefully compared to illustrate the effectiveness of different inequality techniques on reducing the conservatism. Finally, the quadratic negative-definiteness lemmas are insightfully reviewed and meanwhile, a simple method to calculate matrix coefficients of a quadratic matrix-valued function is presented.

Stability analysis of discrete‐time systems with a time‐varying delay via improved methods

“…As a common phenomenon, time delay widely exists in dynamic systems in the real world such as chemical systems and networked control systems [1][2][3]. The existence of a time delay usually induces instability, oscillation and other poor performances [4][5][6][7][8][9][10][11][12]. On the other hand, the introduction of a time delay can make some systems stable that is originally unstable without the delay [2] and can also accelerate the rate of convergence of system states [13].…”

“…where h 1 and h 2 are known integers. The Lyapunov-Krasovskii functional (LKF) method is now widely used to analyse the stability of system (1) [2,4]. By constructing a positive-definite LKF and estimating its forward difference, a delay-dependent stability condition is expected to be obtained in form of linear matrix inequalities (LMIs).…”

General stability criteria for time‐delay systems using a new delay‐dependent reciprocally convex inequality

“…Over the last few decades, researchers have been motivated to explore more efficient methods for stability analysis of TDSs and given the significant impact of time-delays on system behavior, where the comprehensive survey provides an overview of recent contributions in this field [13,14]. Also, the Lyapunov-Krasovskii functional (LKF) method has become widely adopted to address stability problems in TDSs [15][16][17] .…”

Adaptive iterative learning control for continuous systems with non‐repetitive uncertainties and unknown state delays

“…Since time-delay is unavoidable in engineering to degrade system performance or even render instability, therefore, it is meaningful to address this issue in ILC field. 17 In particular, a delay independent convergence condition is built in Reference 18 to overcome input delay. Utilizing the repeatable characteristics of state delay, 19 delivers D 𝛼 -type ILC schemes with fast convergence speed and strong robustness.…”

“…With the help of composite energy functions (CEF), 16 delivers a convergent condition for non‐repetitive disturbances under a locally Lipschitz restriction. Since time‐delay is unavoidable in engineering to degrade system performance or even render instability, therefore, it is meaningful to address this issue in ILC field 17 . In particular, a delay independent convergence condition is built in Reference 18 to overcome input delay.…”