On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

Abstract | The input-to-state stability of time invariant systems described by coupled di®erential and di®erence equations with multiple non commensurate and distributed time delays is investigated in this paper. Such equations include neutral functional di®erential equations in Hale's form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii function-als, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale's form with stable di®erence operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the e®ectiveness of the proposed methodology