On the Convergence of ILC for Hilfer Fractional-Order Systems with Initial State Errors

This paper addresses tracking control for nonlinear Hilfer fractional-order systems (HFOS) with initial state errors. To enhance tracking performance over repeated operations, a closed-loop Dα,β-type and first-order PDα,β-type iterative learning control strategy is proposed. By employing the (1−γ,λ)-norm in conjunction with Hölder’s inequality and Young’s inequality, sufficient conditions are derived to guarantee the convergence of the proposed learning algorithm. It is shown that, under the derived convergence conditions, the tracking error converges to zero as the iteration number increases. Finally, numerical examples are presented to validate the theoretical results and demonstrate the convergence and tracking performance of the proposed control schemes.