Relative Controllability of Fractional Delayed Quaternion Discrete Systems

This paper develops a relative controllability theory for linear quaternion discrete systems with a fixed state delay governed by a Caputo nabla fractional difference. The history is prescribed on the delay window preceding the first controlled sample, so the dynamics begin where the control begins. Since the control enters by right scalar multiplication, the state space is treated as a right quaternion module. The delayed state and control kernels are defined as the unique causal solutions of their fractional impulse problems, and their transform representations, coefficient shifts, and growth bounds are proved rather than postulated. A coefficientwise convolution theorem for the nabla transform yields a variation of constants formula whose history contribution is written explicitly through the state kernel. A quaternion Hermitian Gramian gives a necessary and sufficient controllability criterion, and the associated input is the unique minimum energy control. For the pure state delay case, an exact kernel expansion identifies the reachable right Krylov submodule and gives the minimal horizon under a right rank condition. Exact reductions to the real delayed Mittag‐Leffler theory, to the nondelayed model, and to the integer‐order quaternion delay system are proved. Three verified numerical studies, including a representative networked unmanned aerial vehicle attitude error regulation model, confirm the theory.