Multi-Input Square Iterative Learning Control With Bounded Inputs

TL;DR: A very simple modification of the iterative learning control algorithm of S. Arimoto et al. to the case where the inputs are bounded, finding that the usual Jacobian error condition is not sufficient to guarantee the chopping rule will converge to the solution, as proved herein.

Abstract: In this paper, we present a very simple modification of the iterative learning control algorithm of S. Arimoto et al. (1984, "Bettering Operation of Robots by Learning, " J. Robot Syst., 1(2), pp. 123-140) to the case where the inputs are bounded. The Jacobian condition presented in K. Avrachenkov (1998, "Iterative Learning Control Based on Quasi-Newton Methods, " Conference on Decision Control, pp. 170-174) is specified instead of the usual condition specified by Arimoto et al. (1984). (See also K. L. Moore, 1993, Iterative Learning Control for Deterministic Systems, Advances in Industrial Control Series, Springer-Verlag, London, UK.) In particular the former is a condition for monotonicity in the distance to the solution instead of monotonicity in the output error. This observation allows for a simple extension of the methods of Arimoto et al. (1984) to the case of bounded inputs since the process of moving an input back to a bound if it exceeds it does not affect the contraction mapping property; in fact, the distance to the solution, if anything, can only decrease even further The usual Jacobian error condition, on the other hand, is not sufficient to guarantee the chopping rule will converge to the solution, as proved herein. To the best of our knowledge, these facts have not been previously pointed out in the iterative learning control literature.