A New Characterization on the Approximation of Nonlinear Functions via Neural Networks: An Adaptive Control Perspective

Abstract

A novel neural network approximation scheme that is especially appropriate for adaptive control of nonlinear dynamical systems is proposed. In light of the new function approximation characterization, a neuro adaptive control framework for continuous- and discrete-time nonlinear uncertain dynamical systems is also presented. Specifically, the proposed neural network control framework is Lyapunov-based and, unlike standard neural network controllers guaranteeing ultimate boundedness, the framework guarantees partial asymptotic stability of the closed-loop system, that is, asymptotic stability with respect to part of the closed-loop system states associated with the system plant dynamics. The neuro adaptive controllers are constructed without requiring explicit knowledge of the system dynamics other than the assumption that the plant dynamics are continuous and piecewise continuously differentiable, and that the approximation error of uncertain system nonlinearities lie in a small gain-type norm bounded conic sector. This allows us to show that the standard neural network controllers are in fact capable of achieving partial asymptotic stability around the system equilibrium point for continuous-and discrete-time uncertain systems.