Data-Driven Controllability and Controller Designs for Nabla Fractional Order Systems

This letter presents a data-driven approach to assess the controllability of nabla discrete fractional-order systems. This letter also proposes a data-driven approximation method for these systems, grounded in behavioral system theory. Instead of identifying a model that replicates the input-output dynamics, a direct modification is applied to the collected input-output data by solving a Hankel-structured low-rank approximation problem. The performance of the approximated nonparametric models in simulating system responses is compared to that of ARX models with the same number of states. The results indicate that the proposed method achieves similar simulation performance with improved accuracy. Furthermore, leveraging this approximation, many data-driven controllers can be designed. As an example, a data-driven predictive control strategy is designed and applied to a discrete fractional-order system. Simulation results demonstrate that the controller successfully drives the system outputs to the desired positions.