solve_continuous_are — SciPy v1.15.2 Manual (original) (raw)
scipy.linalg.
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 sis assumed to be the zero matrix with sizes compatible with a andb, 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.
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
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
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