On the structure of linear behaviors over quaternions

Some results on the time-domain structure of linear, time-invariant systems over quaternions are presented, both in the continuous and in the discrete-time case. Within the behavioral approach, a system is defined as a set of trajectories (functions or sequences). In this paper, the trajectories are solutions of linear differential or difference equation with constant coefficients which belong to the skew-field of quaternions. As in the real and complex case, the equations may be represented by polynomials whose roots are related to the solutions. However, the properties of the roots and the structure of the corresponding solutions, which are analyzed in the paper, differ in many aspects from the standard commutative case.