control.dlyap

control.dlyap(A, Q, C=None, E=None, method=None)[source]

Solves the discrete-time Lyapunov equation.

X = dlyap(A, Q) solves

$A X A^T - X + Q = 0$

where A and Q are square matrices of the same dimension. Further Q must be symmetric.

dlyap(A, Q, C) solves the Sylvester equation

$A X Q^T - X + C = 0$

where A and Q are square matrices.

dlyap(A, Q, None, E) solves the generalized discrete-time Lyapunov equation

$A X A^T - E X E^T + Q = 0$

where Q is a symmetric matrix and A, Q and E are square matrices of the same dimension.

Parameters:

A, Q2D array_like: Input matrices for the Lyapunov or Sylvestor equation.
C2D array_like, optional: If present, solve the Sylvester equation.
E2D array_like, optional: If present, solve the generalized Lyapunov equation.
methodstr, optional: Set the method used for computing the result. Current methods are ‘slycot’ and ‘scipy’. If set to None (default), try ‘slycot’ first and then ‘scipy’.

Returns:

X2D array (or matrix): Solution to the Lyapunov or Sylvester equation.

Notes

For the generalized Lyapunov equation, method=’slycot’ uses the SLICOT routine SG03AD, based on the generalized Schur method of Penzl [1], which factors the matrix pencil without inverting E. With method=’scipy’, the equation is transformed to a standard Lyapunov equation by inverting E, which requires E to be nonsingular and loses accuracy when E is ill-conditioned (a UserWarning is then issued). The generalized Lyapunov problem is itself ill-conditioned (about cond(E)**2) when E is, so method=’slycot’, though it does not invert E, is not measurably more accurate in that case. Both methods require E nonsingular; a truly singular (descriptor) E is not currently handled by either.

For the Sylvester equation, method=’slycot’ uses the Hessenberg-Schur method of the SLICOT routine SB04QD [2] and method=’scipy’ uses the Bartels-Stewart method [3]; both reduce the coefficient matrices to (Hessenberg-)Schur form and solve the result by back-substitution, with O(n^3 + m^3) cost.

References

[1]

Penzl, T., “Numerical solution of generalized Lyapunov equations”, Advances in Computational Mathematics, 8:33-48, 1998.

[2]

Golub, G.H., Nash, S., and Van Loan, C., “A Hessenberg-Schur method for the problem AX + XB = C”, IEEE Trans. Automatic Control, AC-24, pp. 909-913, 1979.

[3]

Bartels, R.H. and Stewart, G.W., “Solution of the matrix equation AX + XB = C”, Comm. ACM, 15(9), pp. 820-826, 1972.