Stability and stabilisation for time-varying delay systems based on flexible augmented Lyapunov–Krasovskii functional

Abstract

This paper studies the problems of stability and stabilisation for time-varying delay systems. A flexible augmented Lyapunov–Krasovskii functional (FALKF) including some triple integral terms and delay-product-type quadratic terms is first constructed, in which the upper and lower limits of integral terms are flexible. Compared with some existing ones, the information of time-varying delay can be fully utilised. A parameter-dependent reciprocally convex inequality (PDRCI) is proposed, which covers some existing ones as its special cases. Based on the FALKF and PDRCI, a less conservative stability condition is obtained to ensure the time-varying delay systems to be asymptotically stable. By using a matrix inequality decoupling technique, the corresponding controller for the closed-loop systems is derived. Compared with some existing works, the constraints on introduced slack matrices are avoided. It directly provides extra free dimensions in the solution space. Two examples are employed to illustrate the effectiveness of the proposed methods.