The Internal Model Principle for Systems with Unbounded Control and Observation

TL;DR

This paper extends the internal model principle to infinite-dimensional systems with unbounded control and observation, providing new theoretical insights and definitions, and illustrating with a heat equation example.

Contribution

It introduces a new definition of the internal model for infinite-dimensional systems and extends existing theories to systems with unbounded operators.

Findings

Established the internal model principle for systems with unbounded control and observation.

Compared different definitions of internal models in the literature.

Demonstrated the theory with a heat equation example.

Abstract

In this paper the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation. As the main result, we present the internal model principle for linear infinite-dimensional systems with unbounded input and output operators. We do this for two different definitions of an internal model found in the literature, namely, the p-copy internal model and the $\mathcal{G}$-conditions. We also introduce a new way of defining an internal model for infinite-dimensional systems. The theoretic results are illustrated with an example where we consider robust output tracking for a one-dimensional heat equation with boundary control and pointwise measurements.