Conjugate direction method to solve system of Sylvester matrix equations related to control systems

Abstract

The coupled Sylvester matrix equations (CSMEs)
$$
\left\{
\begin{array}{ll}
A_1XB_1+C_1YD_1=E_1, & \hbox{} \\
A_2XB_2+C_2YD_2=E_2, & \hbox{}
\end{array}
\right.
$$
appear frequently in
the various fields of mathematics and engineering such as in control theory and signal processing.
In this work, we establish the generalized conjugate direction (GCD) method for solving the CSMEs.
We prove that
the sequence pair generated by the GCD method converges to the solution pair of the CSMEs any restriction on the initial matrix pair
within a finite number of iterations in
the absence of round-off errors.
Also we show that the GCD method with the special initial matrix pair can compute the least Frobenius norm
solution pair.
Finally the GCD method is illustrated through two numerical examples.