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.

Keywords:

• Time-varying delay system
• flexible augmented Lyapunov–Krasovskii functional
• delay-product-type functional
• matrix inequality decoupling technique

Acknowledgments

The authors would like to thank the editors and the referees for carefully reading the paper and for their comments which have helped to greatly improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61973070, Liaoning Revitalization Talents Program under Grant XLYC1802010, in part by SAPI Fundamental Research Funds under Grant 2018ZCX22, the Chongqing postdoctoral innovative talents support program under Grant CQBX202205, the Fundamental Research Funds for the Central Universities N2204004, and the National Natural Science Foundation of China under Grant 62003337.