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
ors
is not preciselyNone
, 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 ands
is assumed to be the zero matrix with sizes compatible witha
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.
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