# Finite-dimensional control of linear discrete-time fractional-order systems

## TL;DR

This paper develops finite-dimensional feedback controllers for linear discrete-time fractional-order systems with disturbances, ensuring convergence to bounds and effective trajectory tracking using finite approximations.

## Contribution

It introduces new control design conditions and methods for tracking in fractional-order systems, addressing practical approximation issues.

## Key findings

- Controllers guarantee convergence to an ultimate bound.
- Tracking error can be minimized with better approximation accuracy.
- Numerical simulations validate the effectiveness of the proposed methods.

## Abstract

This paper addresses the design of finite-dimensional feedback control laws for linear discrete-time fractional-order systems with additive state disturbance. A set of sufficient conditions are provided to guarantee convergence of the state trajectories to an ultimate bound around the origin with size increasing with the magnitude of the disturbances. Performing a suitable change of coordinates, the latter result can be used to design a controller that is able to track reference trajectories that are solutions of the unperturbed fractional-order system. To overcome the challenges associated with the generation of such solutions, we address the practical case where the references to be tracked are generated as a solution of a specific finite-dimensional approximation of the original fractional-order system. In this case, the tracking error trajectory is driven to an asymptotic bound that is increasing with the magnitude of the disturbances and decreases with the increment in the accuracy of the approximation. The proposed controllers are finite-dimensional, in the sense that the computation of the control input only requires a finite number of previous state and input vectors of the system. Numerical simulations illustrate the proposed design methods in different scenarios.