On Boundedness of Solutions of State Periodic Systems: A Multivariable Cell Structure Approach

Abstract

Many dynamical systems are periodic with respect to several state variables. This periodicity typically leads to the coexistence of multiple invariant solutions (equilibria or limit cycles). As a consequence, while there are many classical techniques for analysis of boundedness and stability of such systems, most of these only permit to establish local properties. Motivated by this, a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented. The proposed method is inspired by the cell structure approach developed by Leonov and Noldus and characterized by two main advances. First, the conventional cell structure framework is extended to the case of dynamics, which are periodic with respect to multiple states. Second, by introducing the notion of a Leonov function the usual definiteness requirements of standard Lyapunov functions are relaxed to sign-indefinite functions. Furthermore, it is shown that under (mild) additional conditions the existence of a Leonov function also ensures input-to-state stability (ISS), i.e., robustness with respect to exogenous perturbations. The performance of the proposed approach is demonstrated via the global analysis of boundedness of trajectories for a nonlinear system. The preliminary formulations of the main results of this paper are given in [17] without proofs.

Global synchronization analysis of droop-controlled microgrids—A multivariable cell structure approach

“…• Derive necessary and sucient conditions for existence of synchronized solutions and their local stability properties. • Provide sucient conditions for global boundedness of trajectories via the multivariable cell structure approach recently proposed in [46,1]. • By using these results, establish almost global asymptotic stability of the desired equilibrium set of the MG, i.e., we show that for all initial conditions, except a set of measure zero, the solutions of the MG converge to an asymptotically stable equilibrium point.…”

“…Yet, these analyses are restricted to the single-machine-innite-bus scenario, because the cell structure approach of Leonov and Noldus is only applicable to systems, whose dynamics are periodic with respect to a scalar state variable. This fundamental drawback has motivated the development of a multivariable cell structure framework and the concept of a Leonov function in [46,1].…”

“…Unlike the original cell structure framework, the approach in [46,1] is applicable to nonlinear systems, whose dynamics are periodic with respect to several state variables and which possess multiple invariant solutions. Clearly, both latter properties are inherent features of MGs.…”

“…In contrast to their more usual denition in S N (i.e., the N -dimensional torus), it is essential for the subsequent analysis to dene the angles in R N . This permits us to construct a continuous Lyapunov-like function (see Section 3.4), which contains linear terms in the angles θ i and is instrumental to establish our main global stability claim 1 . Following the usual approach in frequency synchronization studies [55,30], we assume that the voltage amplitudes at all nodes are positive real constants.…”

“…where k i ∈ R >0 is the frequency droop gain, and thus a design parameter, ω d ∈ R >0 is the nominal electrical frequency, P d i ∈ R is the active power setpoint and the active power ow P i is given by (1). For rotationally interfaced units, M i ∈ R >0 denotes the inertia constant of the machine, while for inverter-interfaced units the virtual inertia constant is given by…”