Haar wavelet fat-based adaptive controller with self-tuning fuzzy compensation for a piezoelectric-actuated system control

An adaptive sliding controller is proposed in this paper for controlling a piezoelectric-actuated X-Y table system. Due to hysteretical behaviors observed in the piezoelectric actuator (PA), the X-Y table can be viewed as a nonlinear time-varying system. Therefore, reliable and universal hysteresis model is difficult to achieve in order to implement model-based controller design. To cope with this problem, nonlinear hysteresis and the system's uncertainties are firstly lumped into an unknown time-varying function. The variation bound of this function is assumed to be unavailable. Then, the function approximation technique (FAT) levering on Haar wavelets is employed to represent the unknown function. In addition, a fuzzy scheme with online learning ability is augmented to compensate for the finite approximation error and facilitate the controller design. Finally, the Lyapunov direct method is applied to find adaptive laws for updating coefficients in the approximating series and tuning parameter in the fuzzy compensator. The closed-loop stability is also guaranteed. To validate the proposed scheme, control results obtained by using the proposed FAT sliding method augmented with fuzzy compensator are compared with those obtained by using solely the FAT approach.