Control System Toolbox™

dlyap - Solve discrete-time Lyapunov equations

Syntax

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

Description

X = dlyap(A,Q) solves the discrete-time Lyapunov equation

where and are -by- matrices.

The solution is symmetric when is symmetric, and positive definite when is positive definite and has all its eigenvalues inside the unit disk.

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

where A, B, and C must have compatible dimensions but need not be square.

X = dlyap(A,Q,[],E) solves the generalized discrete-time 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

dlyap uses SLICOT routines SB03MD and SG03AD for Lyapunov equations and SB04QD (SLICOT) for Sylvester equations.

Diagnostics

The discrete-time Lyapunov equation has a (unique) solution if the eigenvalues of satisfy for all .

If this condition is violated, dlyap produces the error message

Solution does not exist or is not unique.

References

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

[2] Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.

[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] 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.

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

[6] 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.