def dTFsensitivity(self):
"""Compute the transfer function sensitivity measure and matrix
Returns
- M: tf sensitivity measure
- MX: tf sensitivity matrix
"""
return _w_norm_prod(dSS(self.AZ, self._M1, self.CZ, self._M2),
dSS(self.AZ, self.BZ, self._N1, self._N2), self.dZ)
def to_dSS(self, form="ctrl"):
"""
Transform the transfer function into a state-space
Parameters:
- form: controllable canonical form ('ctrl') or observable canonical form ('obs')
"""
# TODO: code it without scipy
from fixif.LTI import dSS
if form == 'ctrl':
A = mat(diagflat(ones((1, self.order - 1)), 1))
A[self.order - 1, :] = fliplr(-self.den[0, 1:])
B = mat(r_[zeros((self.order - 1, 1)), atleast_2d(1)])
C = mat(
fliplr(self.num[0, 1:]) -
fliplr(self.den[0, 1:]) * self.num[0, 0])
D = mat(atleast_2d(self.num[0, 0]))
elif form == 'obs':
# TODO: to it manually!!
A, B, C, D = tf2ss(array(self.num)[0, :], array(self.den)[0, :])
else:
raise ValueError(
'dTF.to_dSS: the form "%s" is invalid (must be "ctrl" or "obs")'
% form)
return dSS(A, B, C, D)
def _build_AZtoDZ(self): # compute AZ, BZ, CZ and DZ matrices AZ = self.K * self._invJ * self.M + self.P BZ = self.K * self._invJ * self.N + self.Q CZ = self.L * self._invJ * self.M + self.R DZ = self.L * self._invJ * self.N + self.S # and store them in a dSS object self._dSS = dSS(AZ, BZ, CZ, DZ)
def test_construction(): """ Test the constructor """ # test non-consistency size with pytest.raises(ValueError): dSS([[1, 2], [3, 4], [5, 6]], 1, 2, 3) with pytest.raises(ValueError): dSS([[1, 2], [3, 4]], 1, 2, 3) with pytest.raises(ValueError): dSS([[1, 2], [3, 4]], [[1], [2]], 2, 3) with pytest.raises(ValueError): dSS([[1, 2], [3, 4]], [[1], [2]], [[1, 2], [1, 2]], 3)
def __init__(self,
num=None,
den=None,
A=None,
B=None,
C=None,
D=None,
tf=None,
ss=None,
stable=None,
name='noname'):
"""
Create a Filter from numerator and denominator OR from A,B,C,D matrices
Parameters
----------
num, den: numerator and denominator of the transfer function
A,B,C,D: State-Space matrices
tf: a dTF object
ss: a dSS object
"""
self._dSS = None
self._dTF = None
self._name = name
if A is not None and B is not None and C is not None and D is not None:
self._dSS = dSS(A, B, C, D)
elif num is not None and den is not None:
self._dTF = dTF(num, den)
elif tf is not None:
self._dTF = tf
elif ss is not None:
self._dSS = ss
else:
raise ValueError(
'Filter: the values given to the Filter constructor are not correct'
)
# is the filter stable?
if stable is None:
# TODO: compute the eigenvalues to know if it is stable or not
self._stable = False
else:
self._stable = stable
def Hepsilon(self): """Hepsilon is the state-space system from the roundoff errors to the output""" if self._Hepsilon is None: self._Hepsilon = dSS(self.AZ, self._M1, self.CZ, self._M2) return self._Hepsilon
def Hu(self): """Hu system is the state-space system from the inputs to the intern variables t,x,y""" if self._Hu is None: self._Hu = dSS(self.AZ, self.BZ, self._N1, self._N2) return self._Hu
def WCPG_txy(S):
N1 = np.bmat(np.r_[S.invJ * S.M, S.AZ, S.CZ])
N2 = np.bmat(np.r_[S.invJ * S.N, S.BZ, S.DZ])
SS = dSS(S.AZ, S.BZ, N1, N2)
return SS.WCPG()
def makerhoDFII(filt,
gamma=None,
Delta=None,
transposed=True,
scaling=None,
equiv_dSS=False):
"""
Factory function to make a rho Direct Form II Realisation
- gamma and Delta are free parameters of the rho Direct Form II (Delta can be computed if None, see 'scaling' option, Gamma is set to 1 if None)
Options:
- transposed: (boolean, True as default) transpose (or not) the realization: rhoDirect Form II transposed (True) or rho Direct Form II (False)
- scaling: tell how the Delta parameters should be chosen, WHEN the Deltas are not given (equal to None)
- None: Deltas are all equal to 1 (no scaling)
- 'l2': a l2-scaling is applied to deduce the Deltas
- 'l2-relaxed': a l2-relaxed scaling is applied to deduce the Deltas
- equiv_dSS: if True returns the equivalent State-Space (instead of the Direct Form II itself)
Returns
- a dictionary of necessary infos to build the Realization
"""
# see LI04b and Hila11b for reference
n = filt.order
if gamma is None:
gamma = ones((1, n))
gamma = mat(gamma)
# =====================================
# Step 1: build Valapha_bar, Vbeta_bar
# by considering Delta_k = 1 for all k
# =====================================
# Va and Vb
Va = transpose(filt.dTF.den)
Vb = transpose(filt.dTF.num)
# Build Tbar
Tbar = mat(zeros((n + 1, n + 1)))
Tbar[n, n] = 1
for i in range(n - 1, -1, -1):
Tbar[i, i:n + 1] = poly(gamma[0, i:n].A1)
# Valpha_bar, Vbeta_bar
Valpha_bar = transpose(inv(Tbar)) * Va
Vbeta_bar = transpose(inv(Tbar)) * Vb
# Equivalent state space (Abar, Bbar, Cbar, Dbar)
A0 = diagflat(mat(ones((n - 1, 1))), 1)
A0[:, 0] = -Valpha_bar[1:, 0]
Abar = diagflat(gamma) + A0
Bbar = Vbeta_bar[1:, 0] - Vbeta_bar[0, 0] * Valpha_bar[1:, 0]
Cbar = mat(zeros((1, n)))
Cbar[0, 0] = 1
Dbar = Vbeta_bar[0, 0]
# ============================================
# Step 2: Compute Delta according to scaling
# (when Delta is not given)
# ============================================
if Delta is None:
# We need to compute the Deltas, according to the option 'scaling'
if scaling is None:
Delta = ones((1, n))
else:
Wc = dSS(Abar, Bbar, Cbar, Dbar).Wc
Delta = zeros((1, n))
if scaling == 'l2':
# compute the Deltas that perform a l2-scaling (see Li04b)
Delta[0, 0] = sqrt(Wc[0, 0])
for i in range(1, n):
Delta[0, i] = sqrt(Wc[i, i] / Wc[i - 1, i - 1])
elif scaling == 'l2-relaxed':
# compute the Deltas that perform a l2-scaling (see Hila09b)
Delta[0, 0] = floor2(sqrt(Wc[0, 0]))
for i in range(1, n):
Delta[0, i] = floor2(sqrt(Wc[i, i] / Wc[i - 1, i - 1]))
else:
raise ValueError(
"rhoDFII: the `scaling` parameter should be None, 'l2' or 'l2-relaxed'"
)
# ============================================
# Step 3: Compute the coefficients (Valpha, Vbeta
# ============================================
# compute Valpha and Vbeta
# compute Tbar
Tbar = mat(zeros((n + 1, n + 1)))
Tbar[n, n] = 1
for i in range(n - 1, -1, -1):
Tbar[i, i:n + 1] = poly(gamma[0, i:n].A1) / prod(Delta[0, i:n])
Ka = prod(Delta[0, :])
Valpha = transpose(inv(Ka * Tbar)) * Va
Vbeta = transpose(inv(Ka * Tbar)) * Vb
# ============================
# Step 4 : build SIF
# ============================
if equiv_dSS:
# Equivalent state space (A, B, C, D)
A0 = diagflat(mat(ones((n - 1, 1))), 1)
A0[:, 0] = -Valpha[1:, 0]
Arho = diagflat(gamma) + A0 * diagflat(Delta)
Brho = Vbeta[1:, 0] - Vbeta[0, 0] * Valpha[1:, 0]
Crho = mat(zeros((1, n)))
Crho[0, 0] = Delta[0, 0]
Drho = mat(Vbeta[0, 0])
J = eye(0)
K = zeros((n, 0))
L = zeros((1, 0))
M = zeros((0, n))
N = zeros((0, 1))
P = Arho
Q = Brho
R = Crho
S = Drho
# TODO: build dJtodS matrix, according to gammaExact and DeltaExact options
else:
if array_equal(Delta, ones((1, n))):
# specific SIF when the Delta are all equal to 1 (the temporary variables are not necessary, since t_i = x_i * Delta_i)
J = mat(1)
K = -Valpha[1:n + 1, 0]
L = mat(1)
M = c_[atleast_2d(1), zeros((1, n - 1))]
N = mat(Vbeta[0, 0])
P = mat(diagflat(Delta)) + mat(diagflat(ones((1, n - 1)), 1))
Q = Vbeta[1:n + 1, 0]
R = zeros((1, n))
S = mat(0)
else:
J = eye(n)
K = mat(diagflat(ones((1, n - 1)), 1))
K[:, 0] = -Valpha[1:n + 1, 0]
L = c_[atleast_2d(1), zeros((1, n - 1))]
M = mat(diagflat(Delta))
N = r_[atleast_2d(Vbeta[0, 0]), zeros((n - 1, 1))]
P = mat(diagflat(gamma))
Q = Vbeta[1:n + 1, 0]
R = zeros((1, n))
S = mat(0)
if not transposed:
# matrices J, K, L, M, N, P, Q, R and S were for transposed form
K, M = M.transpose(), K.transpose()
P = P.transpose()
R, Q = Q.transpose(), R.transpose()
L, N = N.transpose(), L.transpose()
J = J.transpose()
S = S.transpose() # no need to really do this, since S is a scalar
# TODO: store gamma !!
return {"JtoS": (J, K, L, M, N, P, Q, R, S)}
def to_dSS(self): from fixif.LTI import dSS return dSS(mpf_to_numpy(self._A), mpf_to_numpy(self._B), mpf_to_numpy(self._C), mpf_to_numpy(self._D))