Bounded-input iterative learning control: Robust stabilization via a minimax approach

TL;DR: In this paper, the authors consider the design problem of making the convergence of the bounded-input, multi-input iterative learning controller presented in their previous work robust to errors in the model-based value of the input-output Jacobian matrix via a minimax min-max or "minimize the worst case" approach.

Abstract: In this paper, we consider the design problem of making the convergence of the bounded-input, multi-input iterative learning controller presented in our previous work robust to errors in the model-based value of the input-output Jacobian matrix via a minimax min-max or 'minimize the worst case' approach. We propose to minimize the worst case largest value of the infinity-norm of the matrix whose norm being less then unity implies convergence of the controller. This matrix is the one associated with monotonicity of a sequence of input error norms. The input-output Jacobian uncertainty is taken to be an additive linear one. Theorem 3.1 and its proof show that the worst-case infinity-norm is actually minimized by choosing either the inverse of the centroid of the set of possible input-output Jacobians or a zero matrix. And an explicit expression is given for both the criteria used to choose between the two matrices and the resulting minimum worst-case infinity norm. We showed previously that the matrix norm condition associated with monotonicity of a sequence of output-error norms is not sufficient to assure convergence of the bounded-input controller. The importance of knowing which norm condition is the relevant one is demonstrated by showing that the set of minimizers of the minimax problem formulated with the wrong norm does not contain in general minimizers of the maximum relevant norm and moreover can lead to a gain matrix that destroys the assured convergence of the bounded-input controller given in previous work. Copyright © 2016 John Wiley & Sons, Ltd.