scipy.linalg.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 and s are not both precisely None, 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 and s 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
balanced : bool
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 [R149], \(H - \lambda J\) 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 2m rows and partitioned into two m-row matrices. See [R149] and [R150] 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 [R151]. If the data has small numerical noise, balancing may amplify their effects and some clean up is required.
New in version 0.11.0.
References
[R149] (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 [R150] (1, 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 [R151] (1, 2) P. Benner, “Symplectic Balancing of Hamiltonian Matrices”, 2001, SIAM J. Sci. Comput., 2001, Vol.22(5), DOI: 10.1137/S1064827500367993