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Fixed database typo and removed unnecessary class identifier.
2020-10-14 14:10:37 +00:00
"""Matrix equation solver routines"""
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# February 24, 2012
# Modified: Chad Fulton <ChadFulton@gmail.com>
# June 19, 2014
# Modified: Ilhan Polat <ilhanpolat@gmail.com>
# September 13, 2016
import warnings
import numpy as np
from numpy.linalg import inv, LinAlgError, norm, cond, svd
from .basic import solve, solve_triangular, matrix_balance
from .lapack import get_lapack_funcs
from .decomp_schur import schur
from .decomp_lu import lu
from .decomp_qr import qr
from ._decomp_qz import ordqz
from .decomp import _asarray_validated
from .special_matrices import kron, block_diag
__all__ = ['solve_sylvester',
'solve_continuous_lyapunov', 'solve_discrete_lyapunov',
'solve_lyapunov',
'solve_continuous_are', 'solve_discrete_are']
def solve_sylvester(a, b, q):
"""
Computes a solution (X) to the Sylvester equation :math:`AX + XB = Q`.
Parameters
----------
a : (M, M) array_like
Leading matrix of the Sylvester equation
b : (N, N) array_like
Trailing matrix of the Sylvester equation
q : (M, N) array_like
Right-hand side
Returns
-------
x : (M, N) ndarray
The solution to the Sylvester equation.
Raises
------
LinAlgError
If solution was not found
Notes
-----
Computes a solution to the Sylvester matrix equation via the Bartels-
Stewart algorithm. The A and B matrices first undergo Schur
decompositions. The resulting matrices are used to construct an
alternative Sylvester equation (``RY + YS^T = F``) where the R and S
matrices are in quasi-triangular form (or, when R, S or F are complex,
triangular form). The simplified equation is then solved using
``*TRSYL`` from LAPACK directly.
.. versionadded:: 0.11.0
Examples
--------
Given `a`, `b`, and `q` solve for `x`:
>>> from scipy import linalg
>>> a = np.array([[-3, -2, 0], [-1, -1, 3], [3, -5, -1]])
>>> b = np.array([[1]])
>>> q = np.array([[1],[2],[3]])
>>> x = linalg.solve_sylvester(a, b, q)
>>> x
array([[ 0.0625],
[-0.5625],
[ 0.6875]])
>>> np.allclose(a.dot(x) + x.dot(b), q)
True
"""
# Compute the Schur decomposition form of a
r, u = schur(a, output='real')
# Compute the Schur decomposition of b
s, v = schur(b.conj().transpose(), output='real')
# Construct f = u'*q*v
f = np.dot(np.dot(u.conj().transpose(), q), v)
# Call the Sylvester equation solver
trsyl, = get_lapack_funcs(('trsyl',), (r, s, f))
if trsyl is None:
raise RuntimeError('LAPACK implementation does not contain a proper '
'Sylvester equation solver (TRSYL)')
y, scale, info = trsyl(r, s, f, tranb='C')
y = scale*y
if info < 0:
raise LinAlgError("Illegal value encountered in "
"the %d term" % (-info,))
return np.dot(np.dot(u, y), v.conj().transpose())
def solve_continuous_lyapunov(a, q):
"""
Solves the continuous Lyapunov equation :math:`AX + XA^H = Q`.
Uses the Bartels-Stewart algorithm to find :math:`X`.
Parameters
----------
a : array_like
A square matrix
q : array_like
Right-hand side square matrix
Returns
-------
x : ndarray
Solution to the continuous Lyapunov equation
See Also
--------
solve_discrete_lyapunov : computes the solution to the discrete-time
Lyapunov equation
solve_sylvester : computes the solution to the Sylvester equation
Notes
-----
The continuous Lyapunov equation is a special form of the Sylvester
equation, hence this solver relies on LAPACK routine ?TRSYL.
.. versionadded:: 0.11.0
Examples
--------
Given `a` and `q` solve for `x`:
>>> from scipy import linalg
>>> a = np.array([[-3, -2, 0], [-1, -1, 0], [0, -5, -1]])
>>> b = np.array([2, 4, -1])
>>> q = np.eye(3)
>>> x = linalg.solve_continuous_lyapunov(a, q)
>>> x
array([[ -0.75 , 0.875 , -3.75 ],
[ 0.875 , -1.375 , 5.3125],
[ -3.75 , 5.3125, -27.0625]])
>>> np.allclose(a.dot(x) + x.dot(a.T), q)
True
"""
a = np.atleast_2d(_asarray_validated(a, check_finite=True))
q = np.atleast_2d(_asarray_validated(q, check_finite=True))
r_or_c = float
for ind, _ in enumerate((a, q)):
if np.iscomplexobj(_):
r_or_c = complex
if not np.equal(*_.shape):
raise ValueError("Matrix {} should be square.".format("aq"[ind]))
# Shape consistency check
if a.shape != q.shape:
raise ValueError("Matrix a and q should have the same shape.")
# Compute the Schur decomposition form of a
r, u = schur(a, output='real')
# Construct f = u'*q*u
f = u.conj().T.dot(q.dot(u))
# Call the Sylvester equation solver
trsyl = get_lapack_funcs('trsyl', (r, f))
dtype_string = 'T' if r_or_c == float else 'C'
y, scale, info = trsyl(r, r, f, tranb=dtype_string)
if info < 0:
raise ValueError('?TRSYL exited with the internal error '
'"illegal value in argument number {}.". See '
'LAPACK documentation for the ?TRSYL error codes.'
''.format(-info))
elif info == 1:
warnings.warn('Input "a" has an eigenvalue pair whose sum is '
'very close to or exactly zero. The solution is '
'obtained via perturbing the coefficients.',
RuntimeWarning)
y *= scale
return u.dot(y).dot(u.conj().T)
# For backwards compatibility, keep the old name
solve_lyapunov = solve_continuous_lyapunov
def _solve_discrete_lyapunov_direct(a, q):
"""
Solves the discrete Lyapunov equation directly.
This function is called by the `solve_discrete_lyapunov` function with
`method=direct`. It is not supposed to be called directly.
"""
lhs = kron(a, a.conj())
lhs = np.eye(lhs.shape[0]) - lhs
x = solve(lhs, q.flatten())
return np.reshape(x, q.shape)
def _solve_discrete_lyapunov_bilinear(a, q):
"""
Solves the discrete Lyapunov equation using a bilinear transformation.
This function is called by the `solve_discrete_lyapunov` function with
`method=bilinear`. It is not supposed to be called directly.
"""
eye = np.eye(a.shape[0])
aH = a.conj().transpose()
aHI_inv = inv(aH + eye)
b = np.dot(aH - eye, aHI_inv)
c = 2*np.dot(np.dot(inv(a + eye), q), aHI_inv)
return solve_lyapunov(b.conj().transpose(), -c)
def solve_discrete_lyapunov(a, q, method=None):
"""
Solves the discrete Lyapunov equation :math:`AXA^H - X + Q = 0`.
Parameters
----------
a, q : (M, M) array_like
Square matrices corresponding to A and Q in the equation
above respectively. Must have the same shape.
method : {'direct', 'bilinear'}, optional
Type of solver.
If not given, chosen to be ``direct`` if ``M`` is less than 10 and
``bilinear`` otherwise.
Returns
-------
x : ndarray
Solution to the discrete Lyapunov equation
See Also
--------
solve_continuous_lyapunov : computes the solution to the continuous-time
Lyapunov equation
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter. The default method is *direct* if ``M`` is less than 10
and ``bilinear`` otherwise.
Method *direct* uses a direct analytical solution to the discrete Lyapunov
equation. The algorithm is given in, for example, [1]_. However, it requires
the linear solution of a system with dimension :math:`M^2` so that
performance degrades rapidly for even moderately sized matrices.
Method *bilinear* uses a bilinear transformation to convert the discrete
Lyapunov equation to a continuous Lyapunov equation :math:`(BX+XB'=-C)`
where :math:`B=(A-I)(A+I)^{-1}` and
:math:`C=2(A' + I)^{-1} Q (A + I)^{-1}`. The continuous equation can be
efficiently solved since it is a special case of a Sylvester equation.
The transformation algorithm is from Popov (1964) as described in [2]_.
.. versionadded:: 0.11.0
References
----------
.. [1] Hamilton, James D. Time Series Analysis, Princeton: Princeton
University Press, 1994. 265. Print.
http://doc1.lbfl.li/aca/FLMF037168.pdf
.. [2] Gajic, Z., and M.T.J. Qureshi. 2008.
Lyapunov Matrix Equation in System Stability and Control.
Dover Books on Engineering Series. Dover Publications.
Examples
--------
Given `a` and `q` solve for `x`:
>>> from scipy import linalg
>>> a = np.array([[0.2, 0.5],[0.7, -0.9]])
>>> q = np.eye(2)
>>> x = linalg.solve_discrete_lyapunov(a, q)
>>> x
array([[ 0.70872893, 1.43518822],
[ 1.43518822, -2.4266315 ]])
>>> np.allclose(a.dot(x).dot(a.T)-x, -q)
True
"""
a = np.asarray(a)
q = np.asarray(q)
if method is None:
# Select automatically based on size of matrices
if a.shape[0] >= 10:
method = 'bilinear'
else:
method = 'direct'
meth = method.lower()
if meth == 'direct':
x = _solve_discrete_lyapunov_direct(a, q)
elif meth == 'bilinear':
x = _solve_discrete_lyapunov_bilinear(a, q)
else:
raise ValueError('Unknown solver %s' % method)
return x
def solve_continuous_are(a, b, q, r, e=None, s=None, balanced=True):
r"""
Solves the continuous-time algebraic Riccati equation (CARE).
The CARE is defined as
.. math::
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 :math:`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`` or ``s`` is not precisely ``None``, then the
generalized version of CARE
.. math::
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 and ``s``
is assumed to be the zero matrix with sizes compatible with ``a`` and
``b``, 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 continuous-time 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 discrete-time algebraic Riccati equation
Notes
-----
The equation is solved by forming the extended hamiltonian matrix pencil,
as described in [1]_, :math:`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 :math:`U_2 U_1^{-1}` and condition number of
:math:`U_1`. Here, :math:`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
:math:`H` and :math:`J` entries (after removing the diagonal entries of
the sum) is balanced following the recipe given in [3]_.
.. versionadded:: 0.11.0
References
----------
.. [1] 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`:
>>> 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
"""
# Validate input arguments
a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args(
a, b, q, r, e, s, 'care')
H = np.empty((2*m+n, 2*m+n), dtype=r_or_c)
H[:m, :m] = a
H[:m, m:2*m] = 0.
H[:m, 2*m:] = b
H[m:2*m, :m] = -q
H[m:2*m, m:2*m] = -a.conj().T
H[m:2*m, 2*m:] = 0. if s is None else -s
H[2*m:, :m] = 0. if s is None else s.conj().T
H[2*m:, m:2*m] = b.conj().T
H[2*m:, 2*m:] = r
if gen_are and e is not None:
J = block_diag(e, e.conj().T, np.zeros_like(r, dtype=r_or_c))
else:
J = block_diag(np.eye(2*m), np.zeros_like(r, dtype=r_or_c))
if balanced:
# xGEBAL does not remove the diagonals before scaling. Also
# to avoid destroying the Symplectic structure, we follow Ref.3
M = np.abs(H) + np.abs(J)
M[np.diag_indices_from(M)] = 0.
_, (sca, _) = matrix_balance(M, separate=1, permute=0)
# do we need to bother?
if not np.allclose(sca, np.ones_like(sca)):
# Now impose diag(D,inv(D)) from Benner where D is
# square root of s_i/s_(n+i) for i=0,....
sca = np.log2(sca)
# NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !!
s = np.round((sca[m:2*m] - sca[:m])/2)
sca = 2 ** np.r_[s, -s, sca[2*m:]]
# Elementwise multiplication via broadcasting.
elwisescale = sca[:, None] * np.reciprocal(sca)
H *= elwisescale
J *= elwisescale
# Deflate the pencil to 2m x 2m ala Ref.1, eq.(55)
q, r = qr(H[:, -n:])
H = q[:, n:].conj().T.dot(H[:, :2*m])
J = q[:2*m, n:].conj().T.dot(J[:2*m, :2*m])
# Decide on which output type is needed for QZ
out_str = 'real' if r_or_c == float else 'complex'
_, _, _, _, _, u = ordqz(H, J, sort='lhp', overwrite_a=True,
overwrite_b=True, check_finite=False,
output=out_str)
# Get the relevant parts of the stable subspace basis
if e is not None:
u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m])))
u00 = u[:m, :m]
u10 = u[m:, :m]
# Solve via back-substituion after checking the condition of u00
up, ul, uu = lu(u00)
if 1/cond(uu) < np.spacing(1.):
raise LinAlgError('Failed to find a finite solution.')
# Exploit the triangular structure
x = solve_triangular(ul.conj().T,
solve_triangular(uu.conj().T,
u10.conj().T,
lower=True),
unit_diagonal=True,
).conj().T.dot(up.conj().T)
if balanced:
x *= sca[:m, None] * sca[:m]
# Check the deviation from symmetry for lack of success
# See proof of Thm.5 item 3 in [2]
u_sym = u00.conj().T.dot(u10)
n_u_sym = norm(u_sym, 1)
u_sym = u_sym - u_sym.conj().T
sym_threshold = np.max([np.spacing(1000.), 0.1*n_u_sym])
if norm(u_sym, 1) > sym_threshold:
raise LinAlgError('The associated Hamiltonian pencil has eigenvalues '
'too close to the imaginary axis')
return (x + x.conj().T)/2
def solve_discrete_are(a, b, q, r, e=None, s=None, balanced=True):
r"""
Solves the discrete-time algebraic Riccati equation (DARE).
The DARE is defined as
.. math::
A^HXA - X - (A^HXB) (R + B^HXB)^{-1} (B^HXA) + Q = 0
The limitations for a solution to exist are :
* All eigenvalues of :math:`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
.. math::
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 [1]_, :math:`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 :math:`U_2 U_1^{-1}` and condition number of
:math:`U_1`. Here, :math:`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
:math:`H` and :math:`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.
.. versionadded:: 0.11.0
References
----------
.. [1] 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`:
>>> 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
"""
# Validate input arguments
a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args(
a, b, q, r, e, s, 'dare')
# Form the matrix pencil
H = np.zeros((2*m+n, 2*m+n), dtype=r_or_c)
H[:m, :m] = a
H[:m, 2*m:] = b
H[m:2*m, :m] = -q
H[m:2*m, m:2*m] = np.eye(m) if e is None else e.conj().T
H[m:2*m, 2*m:] = 0. if s is None else -s
H[2*m:, :m] = 0. if s is None else s.conj().T
H[2*m:, 2*m:] = r
J = np.zeros_like(H, dtype=r_or_c)
J[:m, :m] = np.eye(m) if e is None else e
J[m:2*m, m:2*m] = a.conj().T
J[2*m:, m:2*m] = -b.conj().T
if balanced:
# xGEBAL does not remove the diagonals before scaling. Also
# to avoid destroying the Symplectic structure, we follow Ref.3
M = np.abs(H) + np.abs(J)
M[np.diag_indices_from(M)] = 0.
_, (sca, _) = matrix_balance(M, separate=1, permute=0)
# do we need to bother?
if not np.allclose(sca, np.ones_like(sca)):
# Now impose diag(D,inv(D)) from Benner where D is
# square root of s_i/s_(n+i) for i=0,....
sca = np.log2(sca)
# NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !!
s = np.round((sca[m:2*m] - sca[:m])/2)
sca = 2 ** np.r_[s, -s, sca[2*m:]]
# Elementwise multiplication via broadcasting.
elwisescale = sca[:, None] * np.reciprocal(sca)
H *= elwisescale
J *= elwisescale
# Deflate the pencil by the R column ala Ref.1
q_of_qr, _ = qr(H[:, -n:])
H = q_of_qr[:, n:].conj().T.dot(H[:, :2*m])
J = q_of_qr[:, n:].conj().T.dot(J[:, :2*m])
# Decide on which output type is needed for QZ
out_str = 'real' if r_or_c == float else 'complex'
_, _, _, _, _, u = ordqz(H, J, sort='iuc',
overwrite_a=True,
overwrite_b=True,
check_finite=False,
output=out_str)
# Get the relevant parts of the stable subspace basis
if e is not None:
u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m])))
u00 = u[:m, :m]
u10 = u[m:, :m]
# Solve via back-substituion after checking the condition of u00
up, ul, uu = lu(u00)
if 1/cond(uu) < np.spacing(1.):
raise LinAlgError('Failed to find a finite solution.')
# Exploit the triangular structure
x = solve_triangular(ul.conj().T,
solve_triangular(uu.conj().T,
u10.conj().T,
lower=True),
unit_diagonal=True,
).conj().T.dot(up.conj().T)
if balanced:
x *= sca[:m, None] * sca[:m]
# Check the deviation from symmetry for lack of success
# See proof of Thm.5 item 3 in [2]
u_sym = u00.conj().T.dot(u10)
n_u_sym = norm(u_sym, 1)
u_sym = u_sym - u_sym.conj().T
sym_threshold = np.max([np.spacing(1000.), 0.1*n_u_sym])
if norm(u_sym, 1) > sym_threshold:
raise LinAlgError('The associated symplectic pencil has eigenvalues'
'too close to the unit circle')
return (x + x.conj().T)/2
def _are_validate_args(a, b, q, r, e, s, eq_type='care'):
"""
A helper function to validate the arguments supplied to the
Riccati equation solvers. Any discrepancy found in the input
matrices leads to a ``ValueError`` exception.
Essentially, it performs:
- a check whether the input is free of NaN and Infs
- a pass for the data through ``numpy.atleast_2d()``
- squareness check of the relevant arrays
- shape consistency check of the arrays
- singularity check of the relevant arrays
- symmetricity check of the relevant matrices
- a check whether the regular or the generalized version is asked.
This function is used by ``solve_continuous_are`` and
``solve_discrete_are``.
Parameters
----------
a, b, q, r, e, s : array_like
Input data
eq_type : str
Accepted arguments are 'care' and 'dare'.
Returns
-------
a, b, q, r, e, s : ndarray
Regularized input data
m, n : int
shape of the problem
r_or_c : type
Data type of the problem, returns float or complex
gen_or_not : bool
Type of the equation, True for generalized and False for regular ARE.
"""
if not eq_type.lower() in ('dare', 'care'):
raise ValueError("Equation type unknown. "
"Only 'care' and 'dare' is understood")
a = np.atleast_2d(_asarray_validated(a, check_finite=True))
b = np.atleast_2d(_asarray_validated(b, check_finite=True))
q = np.atleast_2d(_asarray_validated(q, check_finite=True))
r = np.atleast_2d(_asarray_validated(r, check_finite=True))
# Get the correct data types otherwise NumPy complains
# about pushing complex numbers into real arrays.
r_or_c = complex if np.iscomplexobj(b) else float
for ind, mat in enumerate((a, q, r)):
if np.iscomplexobj(mat):
r_or_c = complex
if not np.equal(*mat.shape):
raise ValueError("Matrix {} should be square.".format("aqr"[ind]))
# Shape consistency checks
m, n = b.shape
if m != a.shape[0]:
raise ValueError("Matrix a and b should have the same number of rows.")
if m != q.shape[0]:
raise ValueError("Matrix a and q should have the same shape.")
if n != r.shape[0]:
raise ValueError("Matrix b and r should have the same number of cols.")
# Check if the data matrices q, r are (sufficiently) hermitian
for ind, mat in enumerate((q, r)):
if norm(mat - mat.conj().T, 1) > np.spacing(norm(mat, 1))*100:
raise ValueError("Matrix {} should be symmetric/hermitian."
"".format("qr"[ind]))
# Continuous time ARE should have a nonsingular r matrix.
if eq_type == 'care':
min_sv = svd(r, compute_uv=False)[-1]
if min_sv == 0. or min_sv < np.spacing(1.)*norm(r, 1):
raise ValueError('Matrix r is numerically singular.')
# Check if the generalized case is required with omitted arguments
# perform late shape checking etc.
generalized_case = e is not None or s is not None
if generalized_case:
if e is not None:
e = np.atleast_2d(_asarray_validated(e, check_finite=True))
if not np.equal(*e.shape):
raise ValueError("Matrix e should be square.")
if m != e.shape[0]:
raise ValueError("Matrix a and e should have the same shape.")
# numpy.linalg.cond doesn't check for exact zeros and
# emits a runtime warning. Hence the following manual check.
min_sv = svd(e, compute_uv=False)[-1]
if min_sv == 0. or min_sv < np.spacing(1.) * norm(e, 1):
raise ValueError('Matrix e is numerically singular.')
if np.iscomplexobj(e):
r_or_c = complex
if s is not None:
s = np.atleast_2d(_asarray_validated(s, check_finite=True))
if s.shape != b.shape:
raise ValueError("Matrix b and s should have the same shape.")
if np.iscomplexobj(s):
r_or_c = complex
return a, b, q, r, e, s, m, n, r_or_c, generalized_case
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