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| R2012a Documentation → Control System Toolbox | |
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X = dlyap(A,Q)
X = dlyap(A,B,C)
X = dlyap(A,Q,[],E)
X = dlyap(A,Q) solves the discrete-time Lyapunov equation
where A and Q are n-by-n matrices.
The solution X is symmetric when Q is symmetric, and positive definite when Q is positive definite and A has all its eigenvalues inside the unit disk.
X = dlyap(A,B,C) solves the Sylvester equation AXBT − X + C = 0
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 AXAT − EXET + Q= 0
where Q is a symmetric matrix. The empty square brackets, [], are mandatory. If you place any values inside them, the function will error out.
dlyap uses SLICOT routines SB03MD and SG03AD for Lyapunov equations and SB04QD (SLICOT) for Sylvester equations.
The discrete-time Lyapunov equation has a (unique) solution if the eigenvalues α1, α2, …, αN of A satisfy αiαj ≠ 1 for all (i, j).
If this condition is violated, dlyap produces the error message
Solution does not exist or is not unique.
[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.
[7] Sima, V. C, "Algorithms for Linear-quadratic Optimization," Marcel Dekker, Inc., New York, 1996.
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