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Continuous Lyapunov equation solution
lyap
X = lyap(A,Q)
X = lyap(A,B,C)
X = lyap(A,Q,[],E)
lyap solves the special and general forms of the Lyapunov equation. Lyapunov equations arise in several areas of control, including stability theory and the study of the RMS behavior of systems.
X = lyap(A,Q) solves the Lyapunov equation
$$AX+X{A}^{T}+Q=0$$
where A and Q represent square matrices of identical sizes. If Q is a symmetric matrix, the solution X is also a symmetric matrix.
X = lyap(A,B,C) solves the Sylvester equation
$$AX+XB+C=0$$
The matrices A, B, and C must have compatible dimensions but need not be square.
X = lyap(A,Q,[],E) solves the generalized Lyapunov equation
$$AX{E}^{T}+EX{A}^{T}+Q=0$$
where Q is a symmetric matrix. You must use empty square brackets [] for this function. If you place any values inside the brackets, the function errors out.
The continuous Lyapunov equation has a unique solution if the eigenvalues $${\alpha}_{1},{\alpha}_{2},\mathrm{...},{\alpha}_{n}$$ of A and $${\beta}_{1},{\beta}_{2},\mathrm{...},{\beta}_{n}$$ of B satisfy
$${\alpha}_{i}+{\beta}_{j}\ne 0\text{\hspace{1em}}for\text{\hspace{0.17em}}all\text{\hspace{0.17em}}pairs\text{\hspace{0.17em}}(i,j)$$
If this condition is violated, lyap produces the error message:
Solution does not exist or is not unique.
Solve Lyapunov Equation
Solve the Lyapunov equation
$$AX+X{A}^{T}+Q=0$$
where
$$A=\left[\begin{array}{cc}1& 2\\ -3& -4\end{array}\right]\text{\hspace{1em}}\text{\hspace{1em}}Q=\left[\begin{array}{cc}3& 1\\ 1& 1\end{array}\right]$$
The A matrix is stable, and the Q matrix is positive definite.
A = [1 2; -3 -4]; Q = [3 1; 1 1]; X = lyap(A,Q)
These commands return the following X matrix:
X = 6.1667 -3.8333 -3.8333 3.0000
You can compute the eigenvalues to see that X is positive definite.
eig(X)
The command returns the following result:
ans = 0.4359 8.7308
Solve Sylvester Equation
Solve the Sylvester equation
$$AX+XB+C=0$$
where
$$A=5\text{\hspace{1em}}\text{\hspace{1em}}B=\left[\begin{array}{cc}4& 3\\ 4& 3\end{array}\right]\text{\hspace{1em}}\text{\hspace{1em}}C=\left[\begin{array}{cc}2& 1\end{array}\right]$$
A = 5; B = [4 3; 4 3]; C = [2 1]; X = lyap(A,B,C)
These commands return the following X matrix:
X = -0.2000 -0.0500
[1] Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.
[2] Barraud, A.Y., "A numerical algorithm to solve A XA - X = Q," IEEE^{®} Trans. Auto. Contr., AC-22, pp. 883–885, 1977.
[3] Hammarling, S.J., "Numerical solution of the stable, non-negative definite Lyapunov equation," IMA J. Num. Anal., Vol. 2, pp. 303–325, 1982.
[4] Penzl, T., "Numerical solution of generalized Lyapunov equations," Advances in Comp. Math., Vol. 8, pp. 33–48, 1998.
[5] Golub, G.H., Nash, S. and Van Loan, C.F., "A Hessenberg-Schur method for the problem AX + XB = C," IEEE Trans. Auto. Contr., AC-24, pp. 909–913, 1979.