Optimal Selection of Basis Functions for Robust Tracking Control of Linear Systems using Filtered Basis Functions

There is growing interest in the use of the filtered basis functions (FBF) approach for feedforward tracking control of linear systems. The FBF approach expresses the control input to the plant as a linear combination of basis functions. The basis functions are forward filtered through the plant dynamics and the coefficients of the linear combination are selected such that the tracking error is minimized. It has been demonstrated that the FBF approach is more versatile compared to other methods in the literature. However, the tracking accuracy of the FBF approach deteriorates in the presence of model uncertainty, much like it does with other feedforward control methods. But, unlike other methods, the FBF approach presents flexibility in terms of the choice of the basis functions, which can be used to improve its accuracy in the presence of model uncertainty. This paper analyzes the effect of choice of the basis functions on the tracking accuracy of FBF, in the presence of uncertainty, using the Frobenius norm of the lifted system representation of FBF's error dynamics. Based on the analysis, a methodology for optimal selection of basis functions is presented. The effectiveness of the proposed methodology is demonstrated using examples. Large improvements in robustness are observed using the proposed optimal set of basis functions compared to popular basis functions, viz., B-splines, discrete cosine transform and block pulse functions.