def get_gap_rom(rom):
"""Based on a rom, create model which is used to evaluate H2-Gap norm."""
A = to_matrix(rom.A, format='dense')
B = to_matrix(rom.B, format='dense')
C = to_matrix(rom.C, format='dense')
if isinstance(rom.E, IdentityOperator):
P = spla.solve_continuous_are(A.T,
C.T,
B.dot(B.T),
np.eye(len(C)),
balanced=False)
F = P @ C.T
else:
E = to_matrix(rom.E, format='dense')
P = spla.solve_continuous_are(A.T,
C.T,
B.dot(B.T),
np.eye(len(C)),
e=E.T,
balanced=False)
F = E @ P @ C.T
AF = A - F @ C
mFB = np.concatenate((-F, B), axis=1)
return LTIModel.from_matrices(
AF, mFB, C, E=None if isinstance(rom.E, IdentityOperator) else E)
def _poles_b_c_to_lti(poles, b, c):
r"""Create an |LTIModel| from poles and residue rank-1 factors.
Returns an |LTIModel| with real matrices such that its transfer
function is
.. math::
\sum_{i = 1}^r \frac{c_i b_i^T}{s - \lambda_i}
where :math:`\lambda_i, b_i, c_i` are the poles and residue rank-1
factors.
Parameters
----------
poles
Sequence of poles.
b
|VectorArray| of right residue rank-1 factors.
c
|VectorArray| of left residue rank-1 factors.
Returns
-------
|LTIModel|.
"""
A, B, C = [], [], []
for i, pole in enumerate(poles):
if pole.imag == 0:
A.append(pole.real)
B.append(b[i].to_numpy().real)
C.append(c[i].to_numpy().real.T)
elif pole.imag > 0:
A.append([[pole.real, pole.imag],
[-pole.imag, pole.real]])
bi = b[i].to_numpy()
B.append(np.vstack([2 * bi.real, -2 * bi.imag]))
ci = c[i].to_numpy()
C.append(np.hstack([ci.real.T, ci.imag.T]))
A = spla.block_diag(*A)
B = np.vstack(B)
C = np.hstack(C)
return LTIModel.from_matrices(A, B, C)
def _poles_b_c_to_lti(poles, b, c):
r"""Create an |LTIModel| from poles and residue rank-1 factors.
Returns an |LTIModel| with real matrices such that its transfer
function is
.. math::
\sum_{i = 1}^r \frac{c_i b_i^T}{s - \lambda_i}
where :math:`\lambda_i, b_i, c_i` are the poles and residue rank-1
factors.
Parameters
----------
poles
Sequence of poles.
b
|NumPy array| of shape `(rom.order, rom.dim_input)`.
c
|NumPy array| of shape `(rom.order, rom.dim_output)`.
Returns
-------
|LTIModel|.
"""
A, B, C = [], [], []
for i, pole in enumerate(poles):
if pole.imag == 0:
A.append(pole.real)
B.append(b[i].real)
C.append(c[i].real[:, np.newaxis])
elif pole.imag > 0:
A.append([[pole.real, pole.imag],
[-pole.imag, pole.real]])
B.append(np.vstack([2 * b[i].real, -2 * b[i].imag]))
C.append(np.hstack([c[i].real[:, np.newaxis], c[i].imag[:, np.newaxis]]))
A = spla.block_diag(*A)
B = np.vstack(B)
C = np.hstack(C)
return LTIModel.from_matrices(A, B, C)
def reduce(self, sigma, b, c):
"""Realization-independent tangential Hermite interpolation.
Parameters
----------
sigma
Interpolation points (closed under conjugation), list of length `r`.
b
Right tangential directions, |NumPy array| of shape `(fom.input_dim, r)`.
c
Left tangential directions, |NumPy array| of shape `(fom.output_dim, r)`.
Returns
-------
lti
The reduced-order |LTIModel| interpolating the transfer function of `fom`.
"""
r = len(sigma)
assert isinstance(b, np.ndarray) and b.shape == (self.fom.input_dim, r)
assert isinstance(c,
np.ndarray) and c.shape == (self.fom.output_dim, r)
# rescale tangential directions (to avoid overflow or underflow)
if b.shape[0] > 1:
for i in range(r):
b[:, i] /= spla.norm(b[:, i])
else:
b = np.ones((1, r))
if c.shape[0] > 1:
for i in range(r):
c[:, i] /= spla.norm(c[:, i])
else:
c = np.ones((1, r))
# matrices of the interpolatory LTI system
Er = np.empty((r, r), dtype=complex)
Ar = np.empty((r, r), dtype=complex)
Br = np.empty((r, self.fom.input_dim), dtype=complex)
Cr = np.empty((self.fom.output_dim, r), dtype=complex)
Hs = [self.fom.eval_tf(s, mu=self.mu) for s in sigma]
dHs = [self.fom.eval_dtf(s, mu=self.mu) for s in sigma]
for i in range(r):
for j in range(r):
if i != j:
Er[i, j] = -c[:, i].dot(
(Hs[i] - Hs[j]).dot(b[:, j])) / (sigma[i] - sigma[j])
Ar[i, j] = -c[:, i].dot(
(sigma[i] * Hs[i] - sigma[j] * Hs[j])).dot(b[:, j]) / (
sigma[i] - sigma[j])
else:
Er[i, i] = -c[:, i].dot(dHs[i].dot(b[:, i]))
Ar[i, i] = -c[:, i].dot(
(Hs[i] + sigma[i] * dHs[i]).dot(b[:, i]))
Br[i, :] = Hs[i].T.dot(c[:, i])
Cr[:, i] = Hs[i].dot(b[:, i])
# transform the system to have real matrices
T = np.zeros((r, r), dtype=complex)
for i in range(r):
if sigma[i].imag == 0:
T[i, i] = 1
else:
indices = np.nonzero(
np.isclose(sigma[i + 1:], sigma[i].conjugate()))[0]
if len(indices) > 0:
j = i + 1 + indices[0]
T[i, i] = 1
T[i, j] = 1
T[j, i] = -1j
T[j, j] = 1j
Er = (T.dot(Er).dot(T.conj().T)).real
Ar = (T.dot(Ar).dot(T.conj().T)).real
Br = (T.dot(Br)).real
Cr = (Cr.dot(T.conj().T)).real
return LTIModel.from_matrices(Ar,
Br,
Cr,
D=None,
E=Er,
cont_time=self.fom.cont_time)
def reduce(self, sigma, b, c):
"""Realization-independent tangential Hermite interpolation.
Parameters
----------
sigma
Interpolation points (closed under conjugation), sequence of
length `r`.
b
Right tangential directions, |NumPy array| of shape
`(r, fom.dim_input)`.
c
Left tangential directions, |NumPy array| of shape
`(r, fom.dim_output)`.
Returns
-------
lti
The reduced-order |LTIModel| interpolating the transfer
function of `fom`.
"""
r = len(sigma)
assert b.shape == (r, self.fom.dim_input)
assert c.shape == (r, self.fom.dim_output)
# rescale tangential directions (to avoid overflow or underflow)
b = b * (1 / np.linalg.norm(b)) if b.shape[1] > 1 else np.ones((r, 1))
c = c * (1 / np.linalg.norm(c)) if c.shape[1] > 1 else np.ones((r, 1))
# matrices of the interpolatory LTI system
Er = np.empty((r, r), dtype=np.complex_)
Ar = np.empty((r, r), dtype=np.complex_)
Br = np.empty((r, self.fom.dim_input), dtype=np.complex_)
Cr = np.empty((self.fom.dim_output, r), dtype=np.complex_)
Hs = [self.fom.eval_tf(s, mu=self.mu) for s in sigma]
dHs = [self.fom.eval_dtf(s, mu=self.mu) for s in sigma]
for i in range(r):
for j in range(r):
if i != j:
Er[i, j] = -c[i] @ (Hs[i] - Hs[j]) @ b[j] / (sigma[i] -
sigma[j])
Ar[i, j] = (
-c[i] @ (sigma[i] * Hs[i] - sigma[j] * Hs[j]) @ b[j] /
(sigma[i] - sigma[j]))
else:
Er[i, i] = -c[i] @ dHs[i] @ b[i]
Ar[i, i] = -c[i] @ (Hs[i] + sigma[i] * dHs[i]) @ b[i]
Br[i, :] = Hs[i].T @ c[i]
Cr[:, i] = Hs[i] @ b[i]
# transform the system to have real matrices
T = np.zeros((r, r), dtype=np.complex_)
for i in range(r):
if sigma[i].imag == 0:
T[i, i] = 1
else:
j = np.argmin(np.abs(sigma - sigma[i].conjugate()))
if i < j:
T[i, i] = 1
T[i, j] = 1
T[j, i] = -1j
T[j, j] = 1j
Er = (T @ Er @ T.conj().T).real
Ar = (T @ Ar @ T.conj().T).real
Br = (T @ Br).real
Cr = (Cr @ T.conj().T).real
return LTIModel.from_matrices(Ar,
Br,
Cr,
None,
Er,
cont_time=self.fom.cont_time)
def reduce(self, sigma, b, c):
"""Realization-independent tangential Hermite interpolation.
Parameters
----------
sigma
Interpolation points (closed under conjugation), list of length `r`.
b
Right tangential directions, |NumPy array| of shape `(fom.input_dim, r)`.
c
Left tangential directions, |NumPy array| of shape `(fom.output_dim, r)`.
Returns
-------
lti
The reduced-order |LTIModel| interpolating the transfer function of `fom`.
"""
r = len(sigma)
assert isinstance(b, np.ndarray) and b.shape == (self.fom.input_dim, r)
assert isinstance(c, np.ndarray) and c.shape == (self.fom.output_dim, r)
# rescale tangential directions (to avoid overflow or underflow)
if b.shape[0] > 1:
for i in range(r):
b[:, i] /= spla.norm(b[:, i])
else:
b = np.ones((1, r))
if c.shape[0] > 1:
for i in range(r):
c[:, i] /= spla.norm(c[:, i])
else:
c = np.ones((1, r))
# matrices of the interpolatory LTI system
Er = np.empty((r, r), dtype=complex)
Ar = np.empty((r, r), dtype=complex)
Br = np.empty((r, self.fom.input_dim), dtype=complex)
Cr = np.empty((self.fom.output_dim, r), dtype=complex)
Hs = [self.fom.eval_tf(s) for s in sigma]
dHs = [self.fom.eval_dtf(s) for s in sigma]
for i in range(r):
for j in range(r):
if i != j:
Er[i, j] = -c[:, i].dot((Hs[i] - Hs[j]).dot(b[:, j])) / (sigma[i] - sigma[j])
Ar[i, j] = -c[:, i].dot((sigma[i] * Hs[i] - sigma[j] * Hs[j])).dot(b[:, j]) / (sigma[i] - sigma[j])
else:
Er[i, i] = -c[:, i].dot(dHs[i].dot(b[:, i]))
Ar[i, i] = -c[:, i].dot((Hs[i] + sigma[i] * dHs[i]).dot(b[:, i]))
Br[i, :] = Hs[i].T.dot(c[:, i])
Cr[:, i] = Hs[i].dot(b[:, i])
# transform the system to have real matrices
T = np.zeros((r, r), dtype=complex)
for i in range(r):
if sigma[i].imag == 0:
T[i, i] = 1
else:
indices = np.nonzero(np.isclose(sigma[i + 1:], sigma[i].conjugate()))[0]
if len(indices) > 0:
j = i + 1 + indices[0]
T[i, i] = 1
T[i, j] = 1
T[j, i] = -1j
T[j, j] = 1j
Er = (T.dot(Er).dot(T.conj().T)).real
Ar = (T.dot(Ar).dot(T.conj().T)).real
Br = (T.dot(Br)).real
Cr = (Cr.dot(T.conj().T)).real
return LTIModel.from_matrices(Ar, Br, Cr, D=None, E=Er, cont_time=self.fom.cont_time)