New connections between positivity and Parameter-Optimal Iterative Learning Control

In this paper new connections between positivity and Parameter-Optimal Iterative Learning Control (ILC) are established. It is in fact shown that if a given plant satisfies a positivity condition, there exists both a simple feedforward and feedback ILC-algorithm that gives convergent learning, and the convergence is monotonic. Furthermore, under the positivity assumption the feedback algorithm results in geometric convergence, which is a very strong property of an ILC algorithm. The convergence properties of the feedback algorithm are derived for the first time in this publication. As another new result it is shown that an important subclass of discrete-time systems can be conditioned with a simple feedback loop so that they become positive. Hence, after the conditioning, also this subclass of systems can be approached with Parameter-Optimal ILC algorithms so that monotonic convergence (geometric convergence with the feedback algorithm) is achieved. The different theoretical findings are illustrated with simulation examples, which support the theory presented in this paper.