About Robustness of Control Systems Embedding an Internal Model

TL;DR

This paper develops a unifying framework for robustness in control systems, extending the internal model principle to nonlinear systems and highlighting limitations in achieving exact regulation with smooth regulators.

Contribution

It introduces a general framework unifying various notions of uncertainty and control goals, extending classical linear results to nonlinear systems, and demonstrates limitations of smooth regulators.

Findings

Classical linear results are reinterpreted in a nonlinear setting.

A unifying framework for different notions of uncertainty is proposed.

Counter-example shows no smooth finite-dimensional robust regulator always exists.

Abstract

Robustness is a basic property of any control system. In the context of linear output regulation, it was proved that embedding an internal model of the exogenous signals is necessary and sufficient to achieve tracking of the desired reference signals in spite of external disturbances and parametric uncertainties. This result is commonly known as "internal model principle". A complete extension of such linear result to general nonlinear systems is still an open problem, which is exacerbated by the large number of alternative definitions of uncertainty and desired control goals that are possible in a nonlinear setting. In this paper, we develop a general framework in which all these different notions can be formally characterized in a unifying way. Classical results are reinterpreted in the proposed setting, and new results and insights are presented with a focus on robust rejection/tracking of arbitrary harmonic content. Moreover, we show by counter-example that, in the relevant case of continuous unstructured uncertainties, there are problems for which no smooth finite-dimensional robust regulator exists ensuring exact regulation.