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.

The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

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