An extension of the Georgiou-Smith example: Boundedness and attractivity in the presence of unmodelled dynamics via nonlinear PI control

TL;DR

This paper extends the Georgiou-Smith example to nonlinear systems, demonstrating that nonlinear PI controllers with Nussbaum functions ensure boundedness and attractivity despite unmodelled fast dynamics.

Contribution

It provides new theoretical conditions for global boundedness and attractivity of nonlinear systems with unmodelled dynamics using nonlinear PI control with Nussbaum functions.

Findings

Global boundedness and attractivity are achieved under certain conditions.

Fast unmodelled dynamics are effectively handled by the nonlinear PI controller.

Simulation confirms the theoretical robustness results.

Abstract

In this paper, a nonlinear extension of the Georgiou-Smith system is considered and robustness results are proved for a class of nonlinear PI controllers with respect to fast parasitic first-order dynamics. More specifically, for a perturbed nonlinear system with sector bounded nonlinearity and unknown control direction, sufficient conditions for global boundedness and attractivity have been derived. It is shown that the closed loop system is globally bounded and attractive if (i) the unmodelled dynamics are sufficiently fast and (ii) the PI control gain has the Nussbaum function property. For the case of nominally unstable systems, the Nussbaum property of the control gain appears to be crucial. A simulation study confirms the theoretical results.