A Q, L factorization of Norm-Optimal Iterative Learning Control

In this paper we consider the norm-optimal iterative learning control (ILC) problem for discrete-time linear multiinput, multioutput systems. The solution to this problem is well known and naturally factors into a form with a filter on the previous control, L_u and a filter on the previous error, L_e. We show that this solution can always be factored into a Q,L form where Q filters the previous control and QL filters the previous error. This latter form is popularized with frequency domain ILC designs, and this common factorization suggests some general relationships between norm-optimal and frequency domain design, which are explored. Although the Q,L factorization is well known for some special cases, the results here are general and include differently dimensioned control and observation windows.