control.lyap

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

Solves the continuous-time Lyapunov equation.

X = lyap(A, Q) solves

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

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

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

$A X + X Q + C = 0$

where A and Q are square matrices.

X = lyap(A, Q, None, E) solves the generalized continuous-time Lyapunov equation

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

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: 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.

References

[1]

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