This note studies the effect of variable initial state error in iterative learning control (ILC) systems and proposes a new ILC algorithm based on an average operator. Then, it is shown that, when the proposed algorithm is applied to linear time-invariant (LTI) systems, the effect of the initial state error can be exactly estimated under a specific condition, while the existing algorithms guarantee only the boundness of the error or the convergence from stochastic point of view. To show the validity of the proposed algorithm, a numerical example is given.