Approximate nonlinear output regulation based on the universal approximation theorem

The regulator equations arising from the nonlinear output regulation problem are a set of mixed partial and algebraic equations. Due to the nonlinear nature, it is difficult to obtain the exact solution of the regulator equations. This paper presents an approximation method for solving the regulator equations based on a class of feedforward neural networks. It is shown that a three-layer neural network can solve the regulator equations up to a prescribed arbitrarily small error, and this small error can be translated into a guaranteed steady-state tracking error for the closed-loop system. The method has led to an effective approach to approximately solving the nonlinear output regulation problem.