Hermitian solution to constraint system of generalized Sylvester quaternion matrix equations

The different systems of Sylvester quaternion matrix equations have prolific functions in system and control. This paper considers a Hermitian solution of a system of Sylvester quaternion matrix equations over a quaternion algebra \(\mathbb {H}\). If some necessary and sufficient conditions are fulfilled, the general solution to these quaternion matrix equations is expressed by explicit representation formulas in terms of generalized inverses. We provide an algorithm and a numerical example based on the original direct method using determinantal representations of the quaternion Moore–Penrose inverse.