solve_discrete_are#
- scipy.linalg.solve_discrete_are(a, b, q, r, e=None, s=None, balanced=True)[source]#
Solves the discrete-time algebraic Riccati equation (DARE).
The DARE is defined as
\[A^HXA - X - (A^HXB) (R + B^HXB)^{-1} (B^HXA) + Q = 0\]The limitations for a solution to exist are :
All eigenvalues of \(A\) outside the unit disc, should be controllable.
The associated symplectic pencil (See Notes), should have eigenvalues sufficiently away from the unit circle.
Moreover, if
e
ands
are not both preciselyNone
, then the generalized version of DARE\[A^HXA - E^HXE - (A^HXB+S) (R+B^HXB)^{-1} (B^HXA+S^H) + Q = 0\]is solved. When omitted,
e
is assumed to be the identity ands
is assumed to be the zero matrix.- 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
Square matrix
- e(M, M) array_like, optional
Nonsingular square matrix
- s(M, N) array_like, optional
Input
- balancedbool
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 discrete 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_continuous_are
Solves the continuous algebraic Riccati equation
Notes
The equation is solved by forming the extended symplectic matrix pencil, as described in [1], \(H - \lambda J\) given by the block matrices
[ A 0 B ] [ E 0 B ] [ -Q E^H -S ] - \lambda * [ 0 A^H 0 ] [ S^H 0 R ] [ 0 -B^H 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 \(J\) rows/cols (after removing the diagonal entries) is balanced following the recipe given in [3]. If the data has small numerical noise, balancing may amplify their effects and some clean up is required.
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 as la >>> a = np.array([[0, 1], [0, -1]]) >>> b = np.array([[1, 0], [2, 1]]) >>> q = np.array([[-4, -4], [-4, 7]]) >>> r = np.array([[9, 3], [3, 1]]) >>> x = la.solve_discrete_are(a, b, q, r) >>> x array([[-4., -4.], [-4., 7.]]) >>> R = la.solve(r + b.T.dot(x).dot(b), b.T.dot(x).dot(a)) >>> np.allclose(a.T.dot(x).dot(a) - x - a.T.dot(x).dot(b).dot(R), -q) True