On Input-to-State Stability for Nonlinear Systems with Delayed Feedbacks

We analyze a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions are both known. We prove that the closed loop systems are input-to-state stable (ISS) relative to actuator errors when small time delays are introduced in the feedbacks. We explicitly construct ISS Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the known Lyapunov functions for the original undelayed closed-loop dynamics. We also provide a general result on ISS for cascade systems with delays. We demonstrate the efficacy of our results using a generalized pendulum dynamics and other examples.