Closed-form solution to the non-homogeneous generalised Sylvester matrix equation

A closed-form solution to the non-homogeneous generalised Sylvester matrix equation AV+BW=EVF+R, with the matrix F being in an arbitrary form, is proposed. The solution is expressed in terms of R-controllability matrices and observability matrices, and can provide all the degrees of freedom represented by a free parameter matrix. The proposed approach allows the matrices F and R to be set undetermined, and this feature may give great convenience and advantages in many applications of control systems design. A numerical example is given to illustrate the effect of the proposed approach.