def distRej():
w = np.logspace(-3,0,1000)
S1 = np.zeros((len(w)))
S2 = np.zeros((len(w)))
Gd1 = np.zeros((len(w)))
distCondNum = np.zeros((len(w)))
condNum = np.zeros((len(w)))
for i in range(len(w)):
U, Sv, V = utils.SVD(S(w[i]*1j))
S1[i] = Sv[0] #S = 1/|L + 1|
S2[i] = Sv[2]
Gd1[i], distCondNum[i] = utils.distRej(G(w[i]*1j), Gd(w[i]*1j))
condNum[i] = utils.sigmas(G(w[i]*1j),)[0]*utils.sigmas(la.inv(G(w[i]*1j)))[0]
plt.figure(5)
plt.clf()
plt.subplot(211)
plt.loglog(w, S1, 'r-', label = 'max $\sigma$S')
plt.loglog(w, S2, 'r-', alpha = 0.4, label = 'min $\sigma$S')
plt.loglog(w, Gd1, 'k-', label = '1/||Gd||$_2$')
plt.axvline(wB, color='green')
plt.ylabel('Magnitude')
plt.xlabel('Frequency [rad/s)]')
plt.text(wB*1.1, 0.015, 'wB = %s rad/s'%(np.round(wB,3)), color='green')
plt.axis([None, None, None, 10])
plt.grid(True)
plt.legend(loc='upper left', fontsize = 12, ncol=5)
fig = plt.gcf()
BG = fig.patch
BG.set_facecolor('white')
plt.subplot(212)
plt.semilogx(w, distCondNum, 'r-', label = 'Dist CondNum')
plt.semilogx(w, condNum, 'k-', label = 'CondNum')
plt.axvline(wB, color='green')
plt.ylabel('Disturbance condtion number')
plt.xlabel('Frequency [rad/s)]')
plt.text(wB*1.1, 0.2, 'wB = %s rad/s'%(np.round(wB,3)), color='green')
plt.axis([None, None, 0, None])
plt.legend(loc='upper left', fontsize = 12, ncol=5)
plt.grid(True)
fig = plt.gcf()
BG = fig.patch
BG.set_facecolor('white')
fig.subplots_adjust(bottom=0.2)
fig.subplots_adjust(top=0.9)
fig.subplots_adjust(left=0.2)
fig.subplots_adjust(right=0.9)
def dis_rejctn_plot(G, Gd, S, axlim=None, w_start=-2, w_end=2, points=100):
'''
A subplot of disturbance conditition number to check for input saturation
and a subplot of to see if the disturbances fall withing the bounds on
the singular values of S.
Parameters
----------
G : numpy array
plant model
Gd : numpy array
plant disturbance model
S : numpy array
Sensitivity function
Returns
-------
fig(Condition number and performance objective) : figure
A figure of the disturbance condition number and the bounds imposed
by the singular values.
Note
----
The disturbance condition number provides a measure of how a disturbance gd
is aligned with the plant G. Alignment can vary between 1 and the condition
number.
For acceptable performance the singular values of S must fall below the
inverse 2-norm of gd.
'''
if axlim is None:
axlim = [None, None, None, None]
w = numpy.logspace(w_start, w_end, points)
s = w * 1j
dim = numpy.shape(G(0))
inv_norm_gd = numpy.zeros((dim[1], points))
condtn_nm_gd = numpy.zeros((dim[1], points))
for i in range(dim[1]):
for k in range(points):
inv_norm_gd[i, k], condtn_nm_gd[i, k] = utils.distRej(
G(s[k]),
Gd(s[k])[:, i])
s_min = [utils.sigmas(S(s[i]))[-1] for i in range(points)]
s_max = [utils.sigmas(S(s[i]))[0] for i in range(points)]
plt.figure('Condition number and performance objective')
plt.clf()
plt.gcf().set_facecolor('white')
plt.subplot(2, 1, 1)
for i in range(dim[1]):
plt.loglog(w,
condtn_nm_gd[i],
label=('$\gamma$$_{d%s}$(G)' % (i + 1)),
color='blue',
alpha=((i + 1.) / dim[1]))
plt.axhline(1., color='red', ls=':')
plt.axis(axlim)
plt.ylabel('$\gamma$$_d$(G)')
plt.axhline(1., color='red', ls=':')
plt.legend()
plt.subplot(2, 1, 2)
for i in range(dim[1]):
plt.loglog(w,
inv_norm_gd[i],
label=('1/||g$_{d%s}$||$_2$' % (i + 1)),
color='blue',
alpha=((i + 1.) / dim[1]))
plt.loglog(w, s_min, label=('$\sigma$$_{MIN}$'), color='green')
plt.loglog(w, s_max, label=('$\sigma$$_{MAX}$'), color='green', alpha=0.5)
plt.xlabel('Frequency (rad/unit time)')
plt.ylabel('1/||g$_d$||$_2$')
plt.axhline(1., color='red', ls=':')
plt.legend()
plt.show()
return
def dis_rejctn_plot(G,
Gd,
S=None,
w_start=-2,
w_end=2,
axlim=None,
points=1000):
"""
A subplot of disturbance condition number to check for input saturation
(equation 6.43, p238). Two more subplots indicate if the disturbances fall
withing the bounds of S, applying equations 6.45 and 6.46 (p239).
Parameters
----------
G : numpy matrix
Plant model.
Gd : numpy matrix
Plant disturbance model.
S : numpy matrix
Sensitivity function (optional, if available).
# TODO test S condition
Returns
-------
Plot : matplotlib figure
Note
----
The disturbance condition number provides a measure of how a disturbance gd
is aligned with the plant G. Alignment can vary between 1 and the condition
number.
For acceptable performance the singular values of S must fall below the
inverse 2-norm of gd.
"""
s, w, axlim = df.frequency_plot_setup(axlim, w_start, w_end, points)
dim = numpy.shape(Gd(0))[1] # column count
inv_norm_gd = numpy.zeros((dim, points))
# row count
yd = numpy.zeros((dim, points, numpy.shape(Gd(0))[0]), dtype=complex)
condtn_nm_gd = numpy.zeros((dim, points))
for i in range(dim):
for k in range(points):
inv_norm_gd[i, k], yd[i, k, :], condtn_nm_gd[i, k] = utils.distRej(
G(s[k]),
Gd(s[k])[:, i])
if S is None:
sub = 2
else:
sub = 3
# Equation 6.43
plt.subplot(sub, 1, 1)
for i in range(dim):
plt.loglog(w, condtn_nm_gd[i], label=('$\gamma_{d%s} (G)$' % (i + 1)))
plt.axhline(1., color='red', ls=':')
plt.axis(axlim)
plt.xlabel('Frequency [rad/unit time]')
plt.ylabel('$\gamma$$_d (G)$')
plt.axhline(1., color='red', ls=':')
plt.legend()
# Equation 6.44
plt.subplot(sub, 1, 2)
for i in range(dim):
plt.loglog(w, inv_norm_gd[i], label=('$1/||g_{d%s}||_2$' % (i + 1)))
if S is not None:
S_yd = numpy.array([
numpy.linalg.norm(S(p) * yd[i, p, :].T, 2)
for p in range(points)
])
plt.loglog(w, S_yd, label='$||Sy_d||_2$')
plt.axis(axlim)
plt.xlabel('Frequency [rad/unit time]')
plt.legend()
if S is not None: # subplot should not be repeated with S is not available
# Equation 6.45
plt.subplot(3, 1, 3)
for i in range(dim):
plt.loglog(w,
inv_norm_gd[i],
label=('$1/||g_{d%s}||_2$' % (i + 1)))
s_min = numpy.array(
[utils.sigmas(S(s[p]), 'min') for p in range(points)])
s_max = numpy.array(
[utils.sigmas(S(s[p]), 'max') for p in range(points)])
plt.loglog(w, s_min, label='$\sigma_{min}$')
plt.loglog(w, s_max, label='$\sigma_{max}$')
plt.axis(axlim)
plt.xlabel('Frequency [rad/unit time]')
plt.legend()
def dis_rejctn_plot(G, Gd, S=None, w_start=-2, w_end=2, axlim=None, points=1000):
"""
A subplot of disturbance condition number to check for input saturation
(equation 6.43, p238). Two more subplots indicate if the disturbances fall
withing the bounds of S, applying equations 6.45 and 6.46 (p239).
Parameters
----------
G : numpy matrix
Plant model.
Gd : numpy matrix
Plant disturbance model.
S : numpy matrix
Sensitivity function (optional, if available).
# TODO test S condition
Returns
-------
Plot : matplotlib figure
Note
----
The disturbance condition number provides a measure of how a disturbance gd
is aligned with the plant G. Alignment can vary between 1 and the condition
number.
For acceptable performance the singular values of S must fall below the
inverse 2-norm of gd.
"""
s, w, axlim = df.frequency_plot_setup(axlim, w_start, w_end, points)
dim = numpy.shape(Gd(0))[1] # column count
inv_norm_gd = numpy.zeros((dim, points))
yd = numpy.zeros((dim, points, numpy.shape(Gd(0))[0]), dtype=complex) # row count
condtn_nm_gd = numpy.zeros((dim, points))
for i in range(dim):
for k in range(points):
inv_norm_gd[i, k], yd[i, k, :], condtn_nm_gd[i, k] = utils.distRej(G(s[k]), Gd(s[k])[:, i])
if S is None:
sub = 2
else:
sub = 3
# Equation 6.43
plt.subplot(sub, 1, 1)
for i in range(dim):
plt.loglog(w, condtn_nm_gd[i], label=('$\gamma_{d%s} (G)$' % (i+1)))
plt.axhline(1., color='red', ls=':')
plt.axis(axlim)
plt.xlabel('Frequency [rad/unit time]')
plt.ylabel('$\gamma$$_d (G)$')
plt.axhline(1., color='red', ls=':')
plt.legend()
# Equation 6.44
plt.subplot(sub, 1, 2)
for i in range(dim):
plt.loglog(w, inv_norm_gd[i], label=('$1/||g_{d%s}||_2$' % (i+1)))
if not S is None:
S_yd = numpy.array([numpy.linalg.norm(S(p) * yd[i, p, :].T, 2) for p in range(points)])
plt.loglog(w, S_yd, label='$||Sy_d||_2$')
plt.axis(axlim)
plt.xlabel('Frequency [rad/unit time]')
plt.legend()
if S is not None: # this subplot should not be repeated with S is not available
# Equation 6.45
plt.subplot(3, 1, 3)
for i in range(dim):
plt.loglog(w, inv_norm_gd[i], label=('$1/||g_{d%s}||_2$' % (i+1)))
s_min = numpy.array([utils.sigmas(S(s[p]), 'min') for p in range(points)])
s_max = numpy.array([utils.sigmas(S(s[p]), 'max') for p in range(points)])
plt.loglog(w, s_min, label='$\sigma_{min}$')
plt.loglog(w, s_max, label='$\sigma_{max}$')
plt.axis(axlim)
plt.xlabel('Frequency [rad/unit time]')
plt.legend()
def dis_rejctn_plot(G, Gd, S, axlim=[None, None, None, None], w_start=-2, w_end=2, points=100):
'''
A subplot of disturbance conditition number to check for input saturation
and a subplot of to see if the disturbances fall withing the bounds on
the singular values of S.
Parameters
----------
G : numpy array
plant model
Gd : numpy array
plant disturbance model
S : numpy array
Sensitivity function
Returns
-------
fig(Condition number and performance objective) : figure
A figure of the disturbance condition number and the bounds imposed
by the singular values.
Note
----
The disturbance condition number provides a measure of how a disturbance gd
is aligned with the plant G. Alignment can vary between 1 and the condition
number.
For acceptable performance the singular values of S must fall below the
inverse 2-norm of gd.
'''
w = numpy.logspace(w_start, w_end, points)
s = w*1j
dim = numpy.shape(G(0))
inv_norm_gd = numpy.zeros((dim[1],points))
condtn_nm_gd = numpy.zeros((dim[1],points))
for i in range(dim[1]):
for k in range(points):
inv_norm_gd[i,k], condtn_nm_gd[i,k] = utils.distRej(G(s[k]), Gd(s[k])[:,i])
s_min = [utils.sigmas(S(s[i]))[-1] for i in range(points)]
s_max = [utils.sigmas(S(s[i]))[0] for i in range(points)]
plt.figure('Condition number and performance objective')
plt.clf()
plt.gcf().set_facecolor('white')
plt.subplot(2,1,1)
for i in range(dim[1]):
plt.loglog(w, condtn_nm_gd[i], label=('$\gamma$$_{d%s}$(G)' % (i+1)), color='blue', alpha=((i+1.)/dim[1]))
plt.axhline(1., color='red', ls=':')
plt.axis(axlim)
plt.ylabel('$\gamma$$_d$(G)')
plt.axhline(1., color='red', ls=':')
plt.legend()
plt.subplot(2,1,2)
for i in range(dim[1]):
plt.loglog(w, inv_norm_gd[i], label=('1/||g$_{d%s}$||$_2$' % (i+1)), color='blue', alpha=((i+1.)/dim[1]))
plt.loglog(w, s_min, label=('$\sigma$$_{MIN}$'), color='green')
plt.loglog(w, s_max, label=('$\sigma$$_{MAX}$'), color='green', alpha = 0.5)
plt.xlabel('Frequency (rad/unit time)')
plt.ylabel('1/||g$_d$||$_2$')
plt.axhline(1., color='red', ls=':')
plt.legend()
plt.show()
return