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.

本文讨论了具有加性状态扰动的线性离散时间分数阶系统的有限维反馈控制律的设计。提供了一组充分的条件，以保证状态轨迹收敛到原点周围的最终边界，并且随着干扰的大小而增大。执行适当的坐标更改，后一个结果可用于设计一个控制器，该控制器能够跟踪作为无扰动分数阶系统解的参考轨迹。为了克服与此类解决方案的生成相关的挑战，我们解决了实际情况，其中生成要跟踪的参考作为原始分数阶系统的特定有限维近似的解决方案。在这种情况下，跟踪误差轨迹被驱动到一个渐近边界，该边界随着干扰量的增加而增加，并且随着近似精度的增加而减小。在控制输入的计算仅需要有限数量的系统的先前状态和输入向量的意义上，所提出的控制器是有限维的。数值模拟说明了在不同情况下提出的设计方法。