  This work investigates robust monotonic convergent iterative learning control
(ILC) for uncertain linear systems in both time and frequency domains, and the
ILC algorithm optimizing the convergence speed in terms of $l_{2}$ norm of
error signals is derived. Firstly, it is shown that the robust monotonic
convergence of the ILC system can be established equivalently by the positive
definiteness of a matrix polynomial over some set. Then, a necessary and
sufficient condition in the form of sum of squares (SOS) for the positive
definiteness is proposed, which is amendable to the feasibility of linear
matrix inequalities (LMIs). Based on such a condition, the optimal ILC
algorithm that maximizes the convergence speed is obtained by solving a set of
convex optimization problems. Moreover, the order of the learning function can
be chosen arbitrarily so that the designers have the flexibility to decide the
complexity of the learning algorithm.
