Function Approximation and Adaptive Control with Unstructured Uncertainty

Abstract

This chapter presents the fundamental theories of least-squares function approximation and least-squares adaptive control of systems with unstructured uncertainty. The function approximation theory based on polynomials, in particular the Chebyshev orthogonal polynomials, and neural networks is presented. The Chebyshev orthogonal polynomials are generally considered to be optimal for function approximation of real-valued functions. The Chebyshev polynomial function approximation is therefore more accurate than function approximation with regular polynomials. The neural network function approximation theory for a two-layer neural network is presented for two types of activation functions: sigmoidal function and radial basis function. Model-reference adaptive control of systems with unstructured uncertainty is developed in connection with the function approximation theory. Least-squares direct adaptive control methods with polynomial approximation and neural network approximation are presented. Because the least-squares methods can guarantee the parameter convergence, the least-squares adaptive control methods are shown to achieve uniform ultimate boundedness of control signals in the presence of unstructured uncertainty. The standard model-reference adaptive control can be used for systems with unstructured uncertainty using polynomial or neural network approximation. Unlike the least-squares adaptive control methods, boundedness of tracking error is guaranteed but boundedness of adaptive parameters cannot be mathematically guaranteed. This can lead to robustness issues with model-reference adaptive control, such as the well-known parameter drift problem. In general, least-squares adaptive control achieves better performance and robustness than model-reference adaptive control.