We introduce an impulsive fractional order time-delay systems with nonpermutable constant coefficient matrices whose solution is given by delayed perturbation of Mittag-Leffler type matrix function. The existence and uniqueness of the solution of the system is proved by using Banach contraction principle. The Ulam-Hyers stability of the given system below are demonstrated. Construct the iterative learning control (ILC) problem obtaining from the mentioned system. The conditions of convergence of ILC problem of each of P, D, and D alpha-types are presented and proved. A comprehensive example with three different original reference trajectories is given to illustrate some theoretical results. This article provides novel outcomes.