Analysis and synthesis of matched basis function repetitive control

There are many control problems where the disturbances from the environment are periodic. While conventional feedback controllers such as PID controllers are suited to handle broadband disturbances, they have limited capabilities in the presence of deterministic periodic disturbances. Matched Basis Function Repetitive Control (MBFRC) is a control technique which can potentially eliminate a selected number of frequency components of the disturbance completely. Due to the periodic coefficients in the governing equations of MBFRC, typical transform techniques cannot be applied directly. In this dissertation, it is shown that the input-output relationship of MBFRC systems can be represented by an equivalent Linear Time Invariant (LTI) system and methods to investigate the system in the frequency domain are developed. A moving average filter is introduced and its effectiveness to reduce the influence from unaddressed frequencies, called "cross talk," is studied. The stability of the system is analyzed by the equivalent LTI system and computer simulations are conducted to evaluate the performance. Averaging techniques are applied to gain additional insights into the dynamic behavior of the system which cannot be obtained through the study of the equivalent LTI representation. MBFRC requires knowledge of the disturbance frequency of interest which may not be available or too expensive to obtain in practical situations. Two types of MBFRC are proposed in which a Phase-Locked Loop and a gradient algorithm are introduced to estimate the disturbance frequency. The effectiveness of these adaptive controllers for a drifting multiple frequency disturbance is demonstrated by computer simulations, and two variations of the gradient method adaptive MBFRC system are developed employing the normalization and the basis function space concepts. The stability and convergence of the adaptive MBFRC system are studied by averaging techniques and local convergence of the system is proved. Various methods in the literature for rejecting a periodic disturbance with known and unknown frequencies are discussed, and the performance of selected methods is evaluated in comparison with adaptive MBFRC systems.