Matrix factorization for design of Q-filter in iterative learning control

Iterative learning control (ILC) has been extensively used in systems that repeatedly follow the same desired trajectory. The key idea is to incorporate the tracking errors from previous iterations to generate a better feedforward signal for the next iteration. A drawback of ILC is that all disturbances are assumed to be repetitive, while in practice non-repetitive disturbances may also affect the system behaviors. To address this problem, many efforts have been made on designing Q-filters to filter out the non-repetitive effects from the error signal. This paper presents a nonparametric Q-filter design procedure which does not require any explicit specification of the properties of non-repetitive disturbances. Namely, we perform matrix factorization on a set of error signals in the time-frequency domain to construct a non-repetitive error dictionary. The learned dictionary is then used to encode the error signal in each ILC iteration. This in turn results in a low-rank matrix and a sparse matrix that, respectively, describe the undesired non-repetitive effects and the desired repetitive effects. The effectiveness of the proposed method is demonstrated on a laboratory testbed wafer scanner.