BASIS FUNCTION ITERATIVE LEARNING CONTROL: LIMITING ACCUMULATION IN UNADDRESSED ERROR SPACE WITH SINGULAR VECTOR BASIS FUNCTIONS

Iterative Learning Control (ILC) aims to produce high precision tracking of a trajectory learning from repeating trials. The approach can significantly improve the performance of control systems that repeatedly execute a task. Spacecraft applications include repeated scanning with a fine pointing sensor. Basis function ILC restricts the input to a subspace, and aims for zero error in a subspace of the output. The advantages include, reduced computation, avoiding the common difficulty of an unstable inverse for many digital systems, avoiding difficulty from unmodeled high frequency dynamics, and avoiding the need for a zero phase low-pass filter for robustification. This paper identifies a potentially serious issue, that while the ILC is converging to zero error in the chosen or addressed part of the output error space, error can be accumulating in the unaddressed part of the space. A formula is derived to analyze the accumulation, and give the final value of the error. A method to pick the basis functions is presented that can avoid this accumulation. The concept of matched basis functions is presented, and if the model used is correct, there is no accumulation. The basis functions are chosen from the input and the output singular vectors of the singular value decomposition of the input-output matrix of Markov parameters. These basis functions are orthogonal and mapped one to one. They are related to the system frequency response which helps guide the designer in the choice of which singular vectors to include. The design can be made by fmding the Markov parameters using the OKID algorithm directly from data. There is no need to identify a transfer function model.