scipy.linalg.solve_discrete_are¶
- scipy.linalg.solve_discrete_are(a, b, q, r)[source]¶
Solves the discrete algebraic Riccati equation (DARE).
The DARE is defined as
\[X = A'XA - (A'XB) (R+B'XB)^{-1} (B'XA) + Q\]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.
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
Returns: x : 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 [R117], \(H - \lambda J\) given by the block matrices:
[ A 0 B ] [ I 0 B ] [ -Q I 0 ] - \lambda * [ 0 A^T 0 ] [ 0 0 R ] [ 0 -B^T 0 ]
and using a QZ decomposition method.
In this algorithm, the fail conditions are linked to the symmetrycity 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 [R117] and [R118] for more details.
New in version 0.11.0.
References
[R117] (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 [R118] (1, 2) Alan 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