Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities

TL;DR

This paper introduces a class of evolution variational inequalities (EVIs) combining ODEs and VIs with time-varying set-valued mappings, establishing conditions for solutions and applying the results to output regulation and control in constrained systems.

Contribution

It develops existence and uniqueness conditions for EVIs with time-dependent set-valued mappings and applies these to design output regulation controllers using the internal model principle.

Findings

Established conditions for existence and uniqueness of solutions to EVIs.

Designed static and dynamic output feedback controllers for systems with EVIs.

Applied the framework to regulate outputs in systems with polyhedral constraints and power converters.

Abstract

A class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available, In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters.