Solve discrete-time Lyapunov equations
X = dlyap(A,Q)
X = dlyap(A,B,C)
X = dlyap(A,Q,,E)
X = dlyap(A,Q) solves the discrete-time
Lyapunov equation AXAT − X + Q =
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 AXB – X + C =
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 =
where Q is a symmetric matrix. The empty
, are mandatory. If you place
any values inside them, the function will error out.
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,
the error message
Solution does not exist or is not unique.
 Barraud, A.Y., "A numerical algorithm to solve A XA - X = Q," IEEE® Trans. Auto. Contr., AC-22, pp. 883-885, 1977.
 Bartels, R.H. and G.W. Stewart, "Solution of the Matrix Equation AX + XB = C," Comm. of the ACM, Vol. 15, No. 9, 1972.
 Hammarling, S.J., "Numerical solution of the stable, non-negative definite Lyapunov equation," IMA J. Num. Anal., Vol. 2, pp. 303-325, 1982.
 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.
 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.
 Sima, V. C, "Algorithms for Linear-quadratic Optimization," Marcel Dekker, Inc., New York, 1996.