Adaptive fuzzy wavelet network control design for nonlinear systems

This paper presents a new adaptive fuzzy wavelet network controller (A-FWNC) for control of nonlinear affine systems, inspired by the theory of multiresolution analysis (MRA) of wavelet transforms and fuzzy concepts. The proposed adaptive gain controller, which results from the direct adaptive approach, has the ability to tune the adaptation parameter in the THEN-part of each fuzzy rule during real-time operation. Each fuzzy rule corresponds to a sub-wavelet neural network (sub-WNN) and one adaptation parameter. Each sub-WNN consists of wavelets with a specified dilation value. The degree of contribution of each sub-WNN can be controlled flexibly. Orthogonal least square (OLS) method is used to determine the number of fuzzy rules and to purify the wavelets for each sub-WNN. Since the efficient procedure of selecting wavelets used in the OLS method is not very sensitive to the input dimension, the dimension of the approximated function does not cause the bottleneck for constructing FWN. FWN is constructed based on the training data set of the nominal system and the constructed fuzzy rules can be adjusted by learning the translation parameters of the selected wavelets and also determining the shape of membership functions. Then, the constructed adaptive FWN controller is employed, such that the feedback linearization control input can be best approximated and the closed-loop stability is guaranteed. The performance of the proposed A-FWNC is illustrated by applying a second-order nonlinear inverted pendulum system and compared with previously published methods. Simulation results indicate the remarkable capabilities of the proposed control algorithm. It is worth noting that the proposed controller significantly improves the transient response characteristics and the number of fuzzy rules and on-line adjustable parameters are reduced.