An observation about monotonic convergence in discrete-time, P-type iterative learning control

In this note we make an observation about the equivalence between the necessary and sufficient condition for convergence and the sufficient condition for monotonic convergence in discrete-time, P-type iterative learning control. Specifically, requirements on the plant are given so that convergence of the learning algorithm ensures monotonic convergence. In particular, for the case where one minus the learning gain times the first Markov parameter is positive, but less than one, it is shown that if the first non-zero Markov parameter of the system has a larger magnitude than the sum of the magnitudes of the next N-1 Markov parameters, then convergence of the learning control algorithm implies monotonic convergence, independent of the learning gain. For the case where one minus the learning gain times the first Markov parameter is negative, but greater than negative one, a condition depending on the learning gain is derived whereby learning convergences also implies monotonic convergence