There are three ways to do this in MATLAB:
1. Starting with R2014a, the sylvester function for solving this kind of problem is contained within base MATLAB.
2. The Sylvester equation is a special form of the Lyapunov equation. The LYAP function in Control Systems Toolbox solves the Lyapunov equation and therefore can be used to solve the Sylvester equation.
3. Alternatively, It is also possible to rearrange the equation in a linear form using the KRON function as follows:
AX + XB + Q = 0 is equivalent to the linear equation:
(KRON(I, A)+KRON(B',I))x: = -q:
where x: and q: contain the concatenated columns of X and Q.
The following lines of code provides an example in MATLAB:
A=[1 1;2 3];
B=[1 2;3 4];
Q=[1 3;1 1];
K=kron(eye(2),A)+kron(B',eye(2));
Qb=reshape(Q,4,1);
X=K\-Qb;
X=reshape(X,2,2)
X2 = lyap(A,B,Q)
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