PI-type iterative learning control revisited

Iterative learning control (ILC) is a value-added block to enhance the feedback control performance by utilizing the fact that the system is operated repeatedly for the same task. In terms of how to use the tracking error signal of previous iteration to form the control signal of current iteration, ILC updating schemes can be classified as P-type, D-type, PI-type and PID type etc. So far, no one answered this question: what's the use of the error integral in ILC updating law? In this paper, for discrete-time linear time invariant systems, we show that the error integral in ILC updating scheme is helpful in achieving a monotonic convergence in a suitable norm topology other than the exponentially weighted sup-norm. Optimal design of PI-type ILC scheme is presented. Two extreme cases are considered to show that PI-type ILC can be better than P-type ILC in terms of convergence speed. Simulation results are presented to illustrate how to optimal design the PI-type ILC. We also show that when the number of time instants in an iteration is large, I-component in ILC updating law is of little use.