Control System Toolbox™

lyap - Solve continuous-time Lyapunov equation

Syntax

lyap X = lyap(A,Q) X = lyap(A,B,C) X = lyap(A,Q,[],E)

Description

lyap solves the special and general forms of the Lyapunov matrix 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

where and are square matrices of identical sizes. The solution X is a symmetric matrix if is.

X = lyap(A,B,C) solves the Sylvester equation

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

where Q is a symmetric matrix. The empty square brackets, [], are mandatory. If you place any values inside them, the function will error out.

Algorithm

lyap transforms the and matrices to complex Schur form, computes the solution of the resulting triangular system, and transforms this solution back[1].

lyap uses SLICOT routines SB03MD and SG03AD for Lyapunov equations and SB04MD (SLICOT) and ZTRSYL (LAPACK) for Sylvester equations.

Limitations

The continuous Lyapunov equation has a (unique) solution if the eigenvalues of and of satisfy

If this condition is violated, lyap produces the error message

Solution does not exist or is not unique.

References

[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] Bryson, A.E. and Y.C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975. pp. 328-338.

[3] Barraud, A.Y., "A numerical algorithm to solve A XA - X = Q," IEEE^® Trans. Auto. Contr., AC-22, pp. 883-885, 1977.

[4] Hammarling, S.J., "Numerical solution of the stable, non-negative definite Lyapunov equation," IMA J. Num. Anal., Vol. 2, pp. 303-325, 1982.

[5] Higham, N.J., "FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation," A.C.M. Trans. Math. Soft., Vol. 14, No. 4, pp. 381-396, 1988.

[6] Penzl, T., "Numerical solution of generalized Lyapunov equations," Advances in Comp. Math., Vol. 8, pp. 33-48, 1998.