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.

The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

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