Adaptive wavelet networks for nonlinear system identification

Neural networks have been extensively studied in function approximation and system identification problems. Multilayer perceptron (MLP) and radial basis function network (RBF) are the most often used neural net. Wavelet transforms are a means of representing a function in a manner which readily reveals properties of the function in localized regions of the joint time-frequency space. Wavelet basis functions have been used for adaptive control and estimation of nonlinear systems. The basis functions were selected online (structural adaptation) according to the local spatial frequency content of the approximated function, and stable output weight adaptation laws for controller and estimator were derived. These ideas were extended to the adaptive control of robot manipulators. In this paper, a wavelet-based neural network (WNN) is introduced for adaptive nonlinear system identification. Orthogonal scaling functions are used to construct wavelet networks according to the theory of multiresolution analysis. Adaptive weight updating law is derived based on Lyapunov stability theory. It is shown that even in the presence of modeling error between the system and the WNN model, the weight updating law guarantees the boundedness of identification error and the weights.