<p>This paper addresses the iterative learning control (ILC) problem for discrete-time linear systems with unknown system matrices and arbitrary initial state shifts in each iteration. A data-driven learning control algorithm is proposed, which leverages the linearity of the system by collecting an output error set after running the controller for a certain number of iterations and then constructing an ideal controller from this dataset to achieve error-free tracking. This approach relaxes the conventional convergence conditions typically required in ILC. It is shown that the proposed algorithm can guarantee complete tracking at all time points except the initial one within a finite number of iterations. The effectiveness of the algorithm is demonstrated through three simulation examples.</p>