scipy.linalg.solve_continuous_are¶

scipy.linalg.
solve_continuous_are
(a, b, q, r, e=None, s=None, balanced=True)[source]¶ Solves the continuoustime 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
 balanced : bool, 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 continuoustime 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 discretetime 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 2mbym matrix that holds the eigenvectors spanning the stable subspace with 2m rows and partitioned into two mrow 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].
New in version 0.11.0.
References
[1] (1, 2, 3) 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] (1, 2) A.J. Laub, “A Schur Method for Solving Algebraic Riccati Equations.”, Massachusetts Institute of Technology. Laboratory for Information and Decision Systems. LIDSR ; 859. Available online : http://hdl.handle.net/1721.1/1301 [3] (1, 2) 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:
>>> 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