Least-Squares Adaptive Control Using Chebyshev Orthogonal Polynomials

where x(t) ∈ D ⊂ Rp and f (x) ∈ R is an unknown function but assumed to be bounded function in x. When the structure of the uncertainty is unknown, function approximation is usually employed to estimate the unknown function. In recent years, neural networks have gained a lot of attention in function approximation theory in connection with adaptive control. Multi-layer neural networks have the capability of approximating an unknown function to within an arbitrary value of the approximation error. The universal approximation theorem for sigmoidal neural networks by Cybenko1 and the Micchelli’s theorem2 for radial basis functions provide a theoretical justification of function approximation using neural networks. The use of multi-layer neural networks can create an additional complexity in the back propagation gradient-based training rules. Polynomial approximation is a well-known regression technique for function approximation. In theory, as the degree of an approximating polynomial increases, the approximation error is expected to decrease. However, increasing the degree of the approximating polynomial beyond a theoretical limit could lead to oscillations in the approximating polynomial due to over-parametrization. Regularization techniques to constrain parameters have been developed to prevent over-parametrization.3 In this paper, we explore the use of a special class of polynomials, known as Chebyshev orthogonal polynomials, as basis functions for function approximation. The Chebyshev polynomials have been shown to provide the “best” approximation of a function over any other types of polynomials.4 The use of the Chebyshev polynomials in the context of adaptive control with unstructured uncertainty is demonstrated in this paper. Simulation results demonstrate a significant improvement in the effectiveness of Chebyshev polynomials in the adaptive control setting over a regular polynomial regression.