ISS of Rapidly Time-Varying Systems Via a Novel Presentation and Delay-Free Transformation

We treat input-to-state stability (ISS) of linear continuous-time systems with multiple time-scales. These systems contain rapidly-varying, piecewise continuous and almost periodic coefficients with small parameters (time-scales). Our method relies on a novel delay-free system transformation in conjunction with a new system presentation, where the rapidly-varying coefficients are scalars that have zero average. We employ time-varying Lyapunov functions for ISS analysis. The analysis yields LMI conditions for ISS, leading to explicit bounds on the small parameters, decay rate and ISS gains. The novel system presentation plays a crucial role in the ISS analysis by allowing for essentially less conservative upper bounds on terms containing the small parameters. The derived LMIs are accompanied by suitable feasibility guarantees. Numerical examples demonstrate the efficacy of the proposed approach in comparison to existing methods.