Nonlinear Adaptive Wavelet Control Using Constructive Wavelet Networks

In this paper, an adaptive wavelet-network-based control approach is proposed for highly nonlinear uncertain dynamical systems. Wavelet network, as a kind of universal approximator, has two novel properties-orthonormality and multiresolution. The orthonormal property ensures that adding a new resolution (new wavelets) does not affect the existing wavelet network that may have been well tuned. In the sequel, the online adjustment of the structure of the nonlinear adaptive wavelet controller (AWC) can be done in a constructive manner by gradually increasing the network resolution. The multiresolution property, on the other hand, assures a guaranteed improvement of the approximation precision when a new resolution is added. In real life problems we are unable to know the adequate size of a network, either a neural network (NN) or a wavelet network, to produce the required approximation precision. By virtue of the novel wavelet network properties, a coarse or very simple structure can be selected first. If the system fails to converge after the elapse of a dwell time, a new wavelet resolution is considered to be necessary and added directly. In this manner, the AWC can be easily constructed and tuned from the coarse to finer levels until the performance requirement is satisfied. The trial and error way of selecting the network structure, which may lead to either an inadequate or a highly redundant structure, can be avoided. In this paper, the proposed adaptive wavelet network is applied first to a class of nonlinear dynamical systems with a partially known model and an affine-in-input structure. Then, the adaptive wavelet network is applied to a class of nonlinear nonaffine dynamical systems