Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals

TL;DR

This paper develops new output regulation strategies for linear systems driven by non-smooth, non-periodic signals, including robust methods for uncertain models, using hybrid and immersion techniques.

Contribution

It introduces novel non-resonance conditions, extends internal model concepts to non-smooth signals, and proposes two robust regulation methods based on hybrid and immersion approaches.

Findings

Successfully solves the full-information regulation problem.

Designs a non-robust internal model for error-feedback control.

Proposes two robust internal model methods for uncertain systems.

Abstract

We address the output regulation problem of linear systems with non-smooth and non-periodic exogenous signals. Specifically, we first formulate and solve the full-information problem by designing a state-feedback controller. We study the solvability of the regulator equations, providing a new non-resonance condition. We then focus on the error-feedback problem, for which we design a (non-robust) internal model leveraging the concept of canonical realisation and applying a high-gain method for the stabilisation of the closed-loop system under the minimum-phase assumption. Finally, we study the regulation problem involving model parameter uncertainties. Drawing ideas from both hybrid and time-varying (smooth) output regulation, we propose two methods to establish an internal model that is robust to uncertainties. The first method is an extension of the hybrid internal model, while the second relies on a new concept of immersion. In this non-smooth case, the immersion is established based on integrals rather than derivatives. The effectiveness of the proposed solutions is illustrated by a circuit regulation example.