Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids

Abstract

Motivated by the problem of phase-locking in droop-controlled inverter-based microgrids with delays, the recently developed theory of input-to-state stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using Lyapunov-Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and 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 the nonlinear pendulum. For this case, corresponding delay-dependent conditions for asymptotic phase-locking are given.

Relaxing the conditions of ISS for multistable periodic systems

“…An ISS property of (6) has been shown in Angeli andEfimov (2013, 2015) and an ISS Lyapunov function for (6) has been proposed in Efimov et al (2015). Using that result, for the case |c| < ω 2 , the global convergence to one of the two equilibria [asin(cω −2 ), 0] or [π − asin(cω −2 ), 0] has been proven in Efimov et al (2016) under some restrictions on values of parameters c, κ, ω and using an additional discontinuous Lyapunov function for a local analysis.…”

“…In this work we will show the ISS property of (6) under less restrictive conditions than in Efimov et al (2016) and using the ISS Leonov function framework proposed above. For this purpose, assume that |c| < ω 2 and consider…”

The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: An overview

“…In case of friction saturation, i.e. b(ω) = bω 2 /(1 + ω 2 ), the nonlinear pendulum is Strong iISS and the result is proved by using the same Lyapunov function as in [9,19,18]). …”

“…This paper provides an overview of this current line of research and its related novel contributions. Furthermore, we would like to point out that the developed theory has already shown to be of great potential interest for applications in the domain of power systems [19,18] and robotics [26].…”

“…In case of linear friction, i.e. b(ω) = bω, the nonlinear pendulum has been proved to be ISS with respect to the input d and the invariant set W = {(0, 0), (π, 0)} in [9,19,18]. In case of vanishing friction, i.e.…”