/
_solvers.py
372 lines (277 loc) · 9.95 KB
/
_solvers.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
"""Matrix equation solver routines"""
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# February 24, 2012
# Modified: Chad Fulton <ChadFulton@gmail.com>
# June 19, 2014
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.linalg import inv, LinAlgError
from .basic import solve
from .lapack import get_lapack_funcs
from .decomp_schur import schur
from .special_matrices import kron
__all__ = ['solve_sylvester', 'solve_lyapunov', 'solve_discrete_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
"""
# 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_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 : array_like
Solution to the continuous Lyapunov equation
See Also
--------
solve_sylvester : computes the solution to the Sylvester equation
Notes
-----
Because the continuous Lyapunov equation is just a special form of the
Sylvester equation, this solver relies entirely on solve_sylvester for a
solution.
.. versionadded:: 0.11.0
"""
return solve_sylvester(a, a.conj().transpose(), q)
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_lyapunov : computes the solution to the continuous 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://www.scribd.com/doc/20577138/Hamilton-1994-Time-Series-Analysis
.. [2] Gajic, Z., and M.T.J. Qureshi. 2008.
Lyapunov Matrix Equation in System Stability and Control.
Dover Books on Engineering Series. Dover Publications.
"""
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):
"""
Solves the continuous algebraic Riccati equation (CARE).
The CARE is defined as
.. math::
(A'X + XA - XBR^-1B'X+Q=0)
It is solved directly using a Schur decomposition method.
Parameters
----------
a : (M, M) array_like
Input
b : (M, N) array_like
Input
q : (M, M) array_like
Input
r : (N, N) array_like
Non-singular, square matrix
Returns
-------
x : (M, M) ndarray
Solution to the continuous algebraic Riccati equation
See Also
--------
solve_discrete_are : Solves the discrete algebraic Riccati equation
Notes
-----
Method taken from:
Laub, "A Schur Method for Solving Algebraic Riccati Equations."
U.S. Energy Research and Development Agency under contract
ERDA-E(49-18)-2087.
http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
.. versionadded:: 0.11.0
"""
try:
g = inv(r)
except LinAlgError:
raise ValueError('Matrix R in the algebraic Riccati equation solver '
'is ill-conditioned')
g = np.dot(np.dot(b, g), b.conj().transpose())
z11 = a
z12 = -1.0*g
z21 = -1.0*q
z22 = -1.0*a.conj().transpose()
z = np.vstack((np.hstack((z11, z12)), np.hstack((z21, z22))))
# Note: we need to sort the upper left of s to have negative real parts,
# while the lower right is positive real components (Laub, p. 7)
s, u, _ = schur(z, sort='lhp')
(m, n) = u.shape
u11 = u[0:m//2, 0:n//2]
u21 = u[m//2:m, 0:n//2]
u11i = inv(u11)
return np.dot(u21, u11i)
def solve_discrete_are(a, b, q, r):
"""
Solves the discrete algebraic Riccati equation (DARE).
The DARE is defined as
.. math::
X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q
It is solved directly using a Schur decomposition method.
Parameters
----------
a : (M, M) array_like
Non-singular, square matrix
b : (M, N) array_like
Input
q : (M, M) array_like
Input
r : (N, N) array_like
Non-singular, square matrix
Returns
-------
x : ndarray
Solution to the continuous Lyapunov equation
See Also
--------
solve_continuous_are : Solves the continuous algebraic Riccati equation
Notes
-----
Method taken from:
Laub, "A Schur Method for Solving Algebraic Riccati Equations."
U.S. Energy Research and Development Agency under contract
ERDA-E(49-18)-2087.
http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
.. versionadded:: 0.11.0
"""
try:
g = inv(r)
except LinAlgError:
raise ValueError('Matrix R in the algebraic Riccati equation solver '
'is ill-conditioned')
g = np.dot(np.dot(b, g), b.conj().transpose())
try:
ait = inv(a).conj().transpose() # ait is "A inverse transpose"
except LinAlgError:
raise ValueError('Matrix A in the algebraic Riccati equation solver '
'is ill-conditioned')
z11 = a+np.dot(np.dot(g, ait), q)
z12 = -1.0*np.dot(g, ait)
z21 = -1.0*np.dot(ait, q)
z22 = ait
z = np.vstack((np.hstack((z11, z12)), np.hstack((z21, z22))))
# Note: we need to sort the upper left of s to lie within the unit circle,
# while the lower right is outside (Laub, p. 7)
s, u, _ = schur(z, sort='iuc')
(m, n) = u.shape
u11 = u[0:m//2, 0:n//2]
u21 = u[m//2:m, 0:n//2]
u11i = inv(u11)
return np.dot(u21, u11i)