844 lines
28 KiB
Python
844 lines
28 KiB
Python
"""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
|
|
|
|
from __future__ import division, print_function, absolute_import
|
|
|
|
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 decomp form of a
|
|
r, u = schur(a, output='real')
|
|
|
|
# Compute the Schur decomp 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 decomp 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 2m 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 2m 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
|