Relaxing the conditions of ISS for multistable periodic systems

The input-to-state stability property of nonlinear dynamical systems with multiple invariant solutions is analyzed under the assumption that the system equations are periodic with respect to certain state variables. It is shown that stability can be concluded via a sign-indefinite function, which explicitly takes the systems’ periodicity into account. The presented approach leverages some of the difficulties encountered in the analysis of periodic systems via positive definite Lyapunov functions proposed in Angeli and Efimov (2013, 2015). The new result is established based on the framework of cell structure introduced in Leonov (1974) and illustrated via the global analysis of a nonlinear pendulum with a constant persistent input

On the robust synchronization of Brockett oscillators

International audienceIn this article, motivated by a recent work of R. Brockett Brockett (2013), we study a robust synchronization problem for multistable Brockett oscillators within an Input-to-State Stability (ISS) framework. Based on a recent generalization of the classical ISS theory to multistable systems and its application to the synchronization of multistable systems, a synchronization protocol is designed with respect to compact invariant sets of the unperturbed Brockett oscillator. The invariant sets are assumed to admit a decomposition without cycles (i.e. with neither homoclinic nor heteroclinic orbits). Contrarily to the local analysis of Brockett (2013), the conditions obtained in our work are global and applicable for family of non-identical oscillators. Numerical simulation examples illustrate our theoretical results

Robust Synchronization for Multistable Systems

International audienceIn this note, we study a robust synchronization problem for multistable systems evolving on manifolds within an Input-to-State Stability (ISS) framework. Based on a recent generalization of the classical ISS theory to multistable systems, a robust synchronization protocol is designed with respect to a compact invariant set of the unperturbed system. The invariant set is assumed to admit a decomposition without cycles, that is, with neither homoclinic nor heteroclinic orbits. Numerical simulation examples illustrate our theoretical results

Robustness of Delayed Multistable Systems with Application to Droop-Controlled Inverter-Based Microgrids

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

On Robust Synchronization of Nonlinear Systems with Application to Grid Integration of Renewable Energy Sources

International audienceMany systems in the natural and physical world often work in unison with similar other systems. This process of simultaneous operation is known as synchronization. In the past few decades, owing to this phenomenon's importance, extensive research efforts have been made. However, many of the existing results consider the systems are identical and/or linear time-invariant, while practical systems are often nonlinear and nonidentical for various reasons. This observation motivated several recent studies on the synchronization of nonidentical (i.e., heterogeneous) nonlinear systems. This paper summarizes some recent results on the synchronization of heterogeneous nonlinear systems, as developed in the thesis [1]. First, the results on the synchronization of a particular class of robustly stable nonlinear systems are presented. Then, these results are applied to an example model known as Brockett oscillator. Finally, using the Brockett oscillator as a common dynamics, output oscillatory synchronization results are given for heterogeneous nonlinear systems of relative degree 2 or higher. An application example of Brockett oscillator for power-grid synchronization is also presented. Some outlooks are provided regarding future research directions

Input-to-state stability for cascade systems with multiple invariant sets

In a recent paper Angeli and Efimov (2015), the notion of Input-to-State Stability (ISS) has been generalized for systems with decomposable invariant sets and evolving on Riemannian manifolds. In this work, we analyze the cascade interconnection of such ISS systems and we characterize the finest possible decomposition of its invariant set for three different scenarios: 1. the driving system exhibits multistability (convergence to fixed points only); 2. the driving system exhibits multi-almost periodicity (convergence to fixed points as well as periodic and almost-periodic orbits) and the driven system is assumed to be incremental ISS; 3. the driving system exhibits multiperiodicity (convergence to fixed points and periodic orbits) whereas the driven system is ISS in the sense of Angeli and Efimov (2015). Furthermore, we provide marginal results on the backward/forward asymptotic behavior of incremental ISS systems and on the response of a contractive system under asymptotically almost-periodic forcing. Three examples illustrate the potentiality of the proposed framework