<p>A notion of fractional discrete linear time-invariant (<i>DLTI</i>) systems is presented. This paper studies the controllability and output controllability of fractional <i>DLTI</i> systems with non-negative input constraints (called positive controllability and positive output controllability, respectively). Using some arguments attached to the geometrics of the system, necessary and sufficient conditions are established for the positive controllability and the positive output controllability of fractional <i>DLTI</i> systems. Additionally, the paper develops and proves sufficient conditions for these concepts to demonstrate their practical evaluative ability. Based on the dimensions of the system matrices, necessary and sufficient conditions are provided for positive controllability and positive output controllability in different cases. Several academic examples are offered to support the theoretical results.</p>