Patchy Solution of a Francis-Byrnes-Isidori Partial Differential Equation

TL;DR

This paper introduces a method for approximating solutions to the Francis-Byrnes-Isidori PDEs in nonlinear output regulation, leveraging the periodic properties of two-dimensional analytic center manifolds to ensure uniform convergence.

Contribution

The paper presents a novel approach to compute approximate solutions to FBI equations for systems with hyperbolic zero dynamics and two-dimensional exosystems, ensuring convergence.

Findings

Approximations converge uniformly to true solutions.

Method exploits periodicity of two-dimensional analytic center manifolds.

Applicable to systems with hyperbolic zero dynamics.

Abstract

The solution to the nonlinear output regulation problem requires one to solve a first order PDE, known as the Francis-Byrnes-Isidori (FBI) equations. In this paper we propose a method to compute approximate solutions to the FBI equations when the zero dynamics of the plant are hyperbolic and the exosystem is two-dimensional. With our method we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two-dimensional analytic center manifolds.