scipy.linalg.solve_continuous_are#

scipy.linalg.solve_continuous_are(a, b, q, r, e=None, s=None, balanced=True)[source]#

Solves the continuous-time algebraic Riccati equation (CARE).

The CARE is defined as

\[X A + A^H X - X B R^{-1} B^H X + Q = 0\]

The limitations for a solution to exist are :

All eigenvalues of \(A\) on the right half plane, should be controllable.

The associated hamiltonian pencil (See Notes), should have eigenvalues sufficiently away from the imaginary axis.

Moreover, if e or s is not precisely None, then the generalized version of CARE

\[E^HXA + A^HXE - (E^HXB + S) R^{-1} (B^HXE + S^H) + Q = 0\]

is solved. When omitted, e is assumed to be the identity and s is assumed to be the zero matrix with sizes compatible with a and b, respectively.

Parameters:

a(M, M) array_like: Square matrix
b(M, N) array_like: Input
q(M, M) array_like: Input
r(N, N) array_like: Nonsingular square matrix
e(M, M) array_like, optional: Nonsingular square matrix
s(M, N) array_like, optional: Input
balancedbool, optional: The boolean that indicates whether a balancing step is performed on the data. The default is set to True.

Returns:

x(M, M) ndarray: Solution to the continuous-time algebraic Riccati equation.

Raises:

LinAlgError: For cases where the stable subspace of the pencil could not be isolated. See Notes section and the references for details.

See also

solve_discrete_are: Solves the discrete-time algebraic Riccati equation

Notes

The equation is solved by forming the extended hamiltonian matrix pencil, as described in [1], \(H - \lambda J\) given by the block matrices

[ A    0    B ]             [ E   0    0 ]
[-Q  -A^H  -S ] - \lambda * [ 0  E^H   0 ]
[ S^H B^H   R ]             [ 0   0    0 ]

and using a QZ decomposition method.

In this algorithm, the fail conditions are linked to the symmetry of the product \(U_2 U_1^{-1}\) and condition number of \(U_1\). Here, \(U\) is the 2m-by-m matrix that holds the eigenvectors spanning the stable subspace with 2-m rows and partitioned into two m-row matrices. See [1] and [2] for more details.

In order to improve the QZ decomposition accuracy, the pencil goes through a balancing step where the sum of absolute values of \(H\) and \(J\) entries (after removing the diagonal entries of the sum) is balanced following the recipe given in [3].

Added in version 0.11.0.

References

[1] (1,2)

P. van Dooren , “A Generalized Eigenvalue Approach For Solving Riccati Equations.”, SIAM Journal on Scientific and Statistical Computing, Vol.2(2), DOI:10.1137/0902010

[2]

A.J. Laub, “A Schur Method for Solving Algebraic Riccati Equations.”, Massachusetts Institute of Technology. Laboratory for Information and Decision Systems. LIDS-R ; 859. Available online : http://hdl.handle.net/1721.1/1301

[3]

P. Benner, “Symplectic Balancing of Hamiltonian Matrices”, 2001, SIAM J. Sci. Comput., 2001, Vol.22(5), DOI:10.1137/S1064827500367993

Examples

Given a, b, q, and r solve for x:

>>> import numpy as np
>>> from scipy import linalg
>>> a = np.array([[4, 3], [-4.5, -3.5]])
>>> b = np.array([[1], [-1]])
>>> q = np.array([[9, 6], [6, 4.]])
>>> r = 1
>>> x = linalg.solve_continuous_are(a, b, q, r)
>>> x
array([[ 21.72792206,  14.48528137],
       [ 14.48528137,   9.65685425]])
>>> np.allclose(a.T.dot(x) + x.dot(a)-x.dot(b).dot(b.T).dot(x), -q)
True