In this paper, the previous results that the performance of iterative learning control (ILC) algorithm can be improved by adding a proportional term and/or an integral term of error in D-type ILC algorithm are generalized using an operator. Then, a sufficient condition for convergence and robustness of the generalized ILC algorithm are investigated against initial state error. As a special case of the operator, a non-linear ILC algorithm is also proposed and it is shown that the effect of initial state error can be reached to zero in a given finite time. It is shown that the bound of error reduction can be effectively controlled by tuning gains of the proposed non-linear ILC algorithm. In order to confirm validity of the proposed algorithms, two examples are presented.