On the iterative learning control of random differential equations with Hilfer fractional derivative

This paper studies iterative learning control for a class of Hilfer-type fractional systems subject to random-measure disturbances, motivated by repetitive high-precision tracking problems in flexible robotic systems such as gantry mechanisms. The considered dynamics combine a Hilfer fractional derivative with a finite random measure acting on the state, so that both long-memory effects and jump-type perturbations can be represented in a unified model. To improve tracking performance from trial to trial, we propose a PIα-type iterative learning controller, where the input update contains a proportional correction term together with a fractional integral memory term. In this way, the learning law uses both the current tracking error and past error information in a form consistent with the fractional nature of the system. Sufficient conditions are established for the convergence of the tracking error in a weighted Orlicz–Hilfer space equipped with the Luxemburg norm. Finally, the method is applied to a single-axis flexible gantry robot subject to Poisson-type random disturbances, and the simulations show improved convergence speed, smaller peak tracking errors, and reduced residual vibration compared with a classical P-type ILC scheme.