  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.
