Nonlinear adaptive wavelet control using constructive wavelet networks

In this paper, a nonlinear adaptive wavelet controller (AWC) using constructive wavelet networks is proposed for a class of general nonlinear dynamic systems. Wavelet networks are employed as a universal approximator for the highly nonlinear and non-affine dynamics. By virtue of the novel orthonormal property and multi-resolution property of wavelet networks, the structure of the nonlinear adaptive wavelet controller can be on-line adjusted constructively according to the tracking performance of the dynamic system. The orthonormal property ensures that adding a new resolution (new wavelets) does not affect the existing wavelet network that may have been well tuned. The multi-resolution property assures the improvement of the approximation precision when a new resolution is added. A coarse adaptive wavelet controller can be first constructed with a simple structure. An adaption period is introduced to specify the admissible waiting period for the tracking error to converge under the current wavelet structure. If the system fails to converge after the elapse of the admissible waiting period, a new wavelet resolution is considered to be necessary and added directly. In this manner, the adaptive wavelet controller can be easily constructed and tuned from the coarse to the fine level while retaining the closed-loop stability. It also provides an extra degree of freedom for users to make balance between the network complexity and the convergence rate.