Linear quaternion-valued difference equations: Representation of solutions, controllability, and observability

Abstract

In this paper, we present the fundamental theory of linear quaternion-valued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternion-valued difference equations are controllable and observable in the sense of quaternion-valued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.