About Robustness of Internal Model-Based Control for Linear and Nonlinear Systems

In this paper we propose a general framework in which the robustness properties and requirements of output regulation schemes can be formally described. We introduce a topological definition of robustness relative to arbitrary steady state properties, extending the usual notion of robustness relative to the existence of a steady state in which the regulation error vanishes. We review some of the main control approaches for linear and nonlinear systems, by re-framing their robustness properties within the proposed setting. We show that the celebrated robustness property of the linear regulator, namely the "internal model principle" stated by Francis, Wonham and Davison in the 70's, can be generalized to nonlinear systems in a robustness property relative to the Fourier expansion of the regulation error. We then focus on nonlinear regulation, where we show that only practical regulation can be achieved robustly, while asymptotic regulation is achieved in a quite fragile way. The paper concludes with a conjecture stating that, in a general nonlinear context, asymptotic regulation cannot be achieved in a robust way with a finite dimensional regulator.