Stability Analysis of Linear Systems With a Time-Varying Delay via Less Conservative Methods

The stability issues of linear systems with time-varying delays are tackled in this article. Several positive augmented Lyapunov-Krasovskii (L-K) functionals are proposed by introducing integral quadratic functions based on the L-K stability theorem. To further reduce the estimation gap caused by the existing integral inequalities, which were applied for dealing with the derived augmented-type integral term from augmented functional, some refined matrix-separation-based inequalities are introduced. With fewer decision variables, the proposed method considers the information among the system state, its derivative, and their related terms. After these, some cubic functions in the time-varying delay appear and show nonconvexity. Then, negative definite conditions are imposed on such cubic terms to obtain the stability criterion in the form of the linear matrix inequality (LMI). Inspired by the Taylor's expansion methodology and the delay-partitioning techniques, we improve the existing negative definite conditions on the cubic function without introducing any decision variable. For linear systems with a time-varying delay, this relaxed condition, the refined matrix-separation-based inequalities, and the constructed L-K functionals, are combined to produce a less conservatism stability criterion. Two numerical examples illustrate the effect of the offered methods and the stability condition.