Robustness of delayed multistable systems

In this chapter sufficient conditions for input-to-state stability (ISS) of delayed systems are presented using Lyapunov-Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in the feedback path. First, the approach is illustrated by establishing the ISS property for the model of a nonlinear pendulum, then delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of a nonlinear pendulum, and corresponding delay-dependent conditions for asymptotic phase-locking are derived for an exemplary microgrid consisting of two droop-controlled inverters.