Since it was first introduced by Arimoto, iterative learning control (ILC) methods have received lots of attention because of its effectiveness and briefness. In this paper, open-closed-loop PID-type ILC scheme with bounded learning matrixes for the control of nonlinear time-varying systems are studied. Under a few reasonable restrictions that are very common as in other ILC research papers on convergence problem, sufficient condition for guaranteeing the convergence of ILC systems are given and then proved with linear operator theory. Both the system and the ILC scheme described have very general forms. It is shown that the convergence condition is weaker than some known results for similar ILC scheme.