Continuous Lyapunov equation solution
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
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
have compatible dimensions but need not be square.
X = lyap(A,Q,,E) solves the generalized
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 of A and of B satisfy
If this condition is violated,
the error message:
Solution does not exist or is not unique.
Solve Lyapunov Equation
Solve the Lyapunov equation
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)
X = 6.1667 -3.8333 -3.8333 3.0000
The command returns the following result:
ans = 0.4359 8.7308
Solve Sylvester Equation
Solve the Sylvester equation
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
 Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.
 Barraud, A.Y., "A numerical algorithm to solve A XA - X = Q," IEEE® Trans. Auto. Contr., AC-22, pp. 883–885, 1977.
 Hammarling, S.J., "Numerical solution of the stable, non-negative definite Lyapunov equation," IMA J. Num. Anal., Vol. 2, pp. 303–325, 1982.
 Penzl, T., "Numerical solution of generalized Lyapunov equations," Advances in Comp. Math., Vol. 8, pp. 33–48, 1998.
 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.