Strong and Weak Leonov Functions for Global Boundedness of State Periodic Systems

Abstract

We present new results for the analysis of global boundedness of state periodic systems. Thereby, we address both the case of systems, whose dynamics is periodic with respect to a part of the state vector and the case of systems, whose dynamics is periodic with respect to all state variables. To derive the results the notion of strong Leonov functions is introduced. The main results are complemented by a number of relaxations based on the concept of weak Leonov functions.

A Control Leonov Function Guaranteeing Global ISS of Two Coupled Synchronverters

“…Based on [8], [25], we consider the following assumption. Assumption 3: All solutions of (19) belong to the set…”

“…In this section, we present a control law, which solves Problem 1. Since the system dynamics ( 22) are periodic with respect to the variable θ, Problem 1 is hard to solve using ISS methods based on conventional Lyapunov theory, see also the discussion in [5], [6], [19]. Thus, to design the controller, we follow our recent work [14].…”

“…To achieve this objective, we use the concept of Leonov functions [5], [6], [12], [19]. This is motivated by the fact that the dynamics of AC power systems are periodic with respect to a part of the state vector.…”

Design of Controls for Boundedness of Trajectories of Multistable State Periodic Systems

“…To this end, [22] introduced the notion of Leonov functions. This concept has been further developed in [23].…”

“…Recently, a fitting tool for the analysis of boundedness of solutions for multistable state-periodic systems has been proposed in [21], [22] through the concept of Leonov functions. These Leonov functions are useful to establish attractiveness of certain sets of the state space, and boundedness of solutions is concluded by exploiting the periodicity of the system's dynamics [21]- [23]. These results are based on the ideas of [24] (see also [7], [25], [26]), who have proposed the so-called cell structure approach.…”