def sv_dir_plot(G, plot_type, w_start=-2, w_end=2, axlim=None, points=1000):
"""
Plot the input and output singular vectors associated with the minimum and
maximum singular values.
Parameters
----------
G : matrix
Plant model or sensitivity function.
plot_type : string
Type of plot.
========= ============================
plot_type Type of plot
========= ============================
input Plots input vectors
output Plots output vectors
========= ============================
Returns
-------
Plot : matplotlib figure
Note
----
Can be used with the plant matrix G and the sensitivity function S
for controlability analysis
"""
s, w, axlim = df.frequency_plot_setup(axlim, w_start, w_end, points)
freqresp = [G(si) for si in s]
if plot_type == 'input':
vec = numpy.array([V for _, _, V in map(utils.SVD, freqresp)])
d = 'v'
elif plot_type == 'output':
vec = numpy.array([U for U, _, _ in map(utils.SVD, freqresp)])
d = 'u'
else:
raise ValueError('Invalid plot_type parameter.')
dim = numpy.shape(vec)[1]
for i in range(dim):
plt.subplot(dim*2, 1, 2*i + 1)
plt.semilogx(w, abs(vec[:, i, 0]), label='$%s_{max}$' % d, lw=2)
plt.semilogx(w, abs(vec[:, i, -1]), label='$%s_{min}$' % d, lw=2)
plt.axhline(0, color='red', ls=':')
plt.axis(axlim)
plt.ylabel('$|{0}_{1}|$'.format(d, i + 1))
plt.subplot(dim*2, 1, 2*i + 2)
plt.semilogx(w, utils.phase(vec[:, i, 0], deg=True), label='$%s_{max}$' % d, lw=2)
plt.semilogx(w, utils.phase(vec[:, i, -1], deg=True), label='$%s_{min}$' % d, lw=2)
plt.axhline(0, color='red', ls=':')
plt.axis(axlim)
plt.ylabel(r'$\angle {0}_{1}$'.format(d, i + 1))
plt.xlabel('Frequency [rad/unit time]')
def sv_dir_plot(G, plot_type, w_start=-2, w_end=2, axlim=None, points=1000):
"""
Plot the input and output singular vectors associated with the minimum and
maximum singular values.
Parameters
----------
G : matrix
Plant model or sensitivity function.
plot_type : string
Type of plot.
========= ============================
plot_type Type of plot
========= ============================
input Plots input vectors
output Plots output vectors
========= ============================
Returns
-------
Plot : matplotlib figure
Note
----
Can be used with the plant matrix G and the sensitivity function S
for controlability analysis
"""
s, w, axlim = df.frequency_plot_setup(axlim, w_start, w_end, points)
freqresp = [G(si) for si in s]
if plot_type == 'input':
vec = numpy.array([V for _, _, V in map(utils.SVD, freqresp)])
d = 'v'
elif plot_type == 'output':
vec = numpy.array([U for U, _, _ in map(utils.SVD, freqresp)])
d = 'u'
else:
raise ValueError('Invalid plot_type parameter.')
dim = numpy.shape(vec)[1]
for i in range(dim):
plt.subplot(dim*2, 1, 2*i + 1)
plt.semilogx(w, abs(vec[:, i, 0]), label='$%s_{max}$' % d, lw=2)
plt.semilogx(w, abs(vec[:, i, -1]), label='$%s_{min}$' % d, lw=2)
plt.axhline(0, color='red', ls=':')
plt.axis(axlim)
plt.ylabel('$|{0}_{1}|$'.format(d, i + 1))
plt.subplot(dim*2, 1, 2*i + 2)
plt.semilogx(w, utils.phase(vec[:, i, 0], deg=True), label='$%s_{max}$' % d, lw=2)
plt.semilogx(w, utils.phase(vec[:, i, -1], deg=True), label='$%s_{min}$' % d, lw=2)
plt.axhline(0, color='red', ls=':')
plt.axis(axlim)
plt.ylabel(r'$\angle {0}_{1}$'.format(d, i + 1))
plt.xlabel('Frequency [rad/unit time]')
def bodeclosedloop(G,
K,
w_start=-2,
w_end=2,
axlim=None,
points=1000,
margin=False):
"""
Shows the bode plot for a controller model
Parameters
----------
G : tf
Plant transfer function.
K : tf
Controller transfer function.
margin : boolean
Show the cross over frequencies on the plot (optional).
"""
if axlim is None:
axlim = [None, None, None, None]
plt.gcf().set_facecolor('white')
w = numpy.logspace(w_start, w_end, points)
L = G(1j * w) * K(1j * w)
S = utils.feedback(1, L)
T = utils.feedback(L, 1)
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L))
plt.loglog(w, abs(S))
plt.loglog(w, abs(T))
plt.axis(axlim)
plt.grid()
plt.ylabel("Magnitude")
plt.legend(["L", "S", "T"])
if margin:
plt.plot(w, 1 / numpy.sqrt(2) * numpy.ones(len(w)), linestyle='dotted')
plt.subplot(2, 1, 2)
plt.semilogx(w, utils.phase(L, deg=True))
plt.semilogx(w, utils.phase(S, deg=True))
plt.semilogx(w, utils.phase(T, deg=True))
plt.axis(axlim)
plt.grid()
plt.ylabel("Phase")
plt.xlabel("Frequency [rad/s]")
def Bode(G):
"""give the Bode plot along with GM and PM"""
GM, PM, wc, w_180 = margins(G)
# plotting of Bode plot and with corresponding freqeuncies for PM and GM
w = np.logspace(-5, np.log(w_180), 1000)
s = 1j*w
plt.subplot(211)
gains = np.abs(G(s))
plt.loglog(w, gains)
plt.loglog(wc*np.ones(2), [np.max(gains), np.min(gains)])
plt.text(w_180, np.average([np.max(gains), np.min(gains)]), '<G(jw) = -180 Deg')
plt.loglog(w_180*np.ones(2), [np.max(gains), np.min(gains)])
plt.loglog(w, 1*np.ones(len(w)))
# argument of G
plt.subplot(212)
phaseangle = phase(G(s), deg=True)
plt.semilogx(w, phaseangle)
plt.semilogx(wc*np.ones(2), [np.max(phaseangle), np.min(phaseangle)])
plt.semilogx(w_180*np.ones(2), [-180, 0])
plt.show()
return GM, PM
def Bode(G):
"""give the Bode plot along with GM and PM"""
GM, PM, wc, w_180 = margins(G)
# plotting of Bode plot and with corresponding freqeuncies for PM and GM
w = np.logspace(-5, np.log(w_180), 1000)
s = 1j*w
plt.subplot(211)
gains = np.abs(G(s))
plt.loglog(w, gains)
plt.loglog(wc*np.ones(2), [np.max(gains), np.min(gains)])
plt.text(w_180, np.average([np.max(gains), np.min(gains)]), '<G(jw) = -180 Deg')
plt.loglog(w_180*np.ones(2), [np.max(gains), np.min(gains)])
plt.loglog(w, 1*np.ones(len(w)))
# argument of G
plt.subplot(212)
phaseangle = phase(G(s), deg=True)
plt.semilogx(w, phaseangle)
plt.semilogx(wc*np.ones(2), [np.max(phaseangle), np.min(phaseangle)])
plt.semilogx(w_180*np.ones(2), [-180, 0])
plt.show()
return GM, PM
def bode(G, w_start=-2, w_end=2, axlim=None, points=1000, margin=False):
"""
Shows the bode plot for a plant model
Parameters
----------
G : tf
Plant transfer function.
margin : boolean
Show the cross over frequencies on the plot (optional).
Returns
-------
GM : array containing a real number
Gain margin.
PM : array containing a real number
Phase margin.
Plot : matplotlib figure
"""
if axlim is None:
axlim = [None, None, None, None]
plt.clf()
plt.gcf().set_facecolor('white')
GM, PM, wc, w_180 = utils.margins(G)
# plotting of Bode plot and with corresponding frequencies for PM and GM
# if ((w2 < numpy.log(w_180)) and margin):
# w2 = numpy.log(w_180)
w = numpy.logspace(w_start, w_end, points)
s = 1j*w
# Magnitude of G(jw)
plt.subplot(2, 1, 1)
gains = numpy.abs(G(s))
plt.loglog(w, gains)
if margin:
plt.axvline(w_180, color='black')
plt.text(w_180, numpy.average([numpy.max(gains), numpy.min(gains)]), r'$\angle$G(jw) = -180$\degree$')
plt.axhline(1., color='red', linestyle='--')
plt.axis(axlim)
plt.grid()
plt.ylabel('Magnitude')
# Phase of G(jw)
plt.subplot(2, 1, 2)
phaseangle = utils.phase(G(s), deg=True)
plt.semilogx(w, phaseangle)
if margin:
plt.axvline(wc, color='black')
plt.text(wc, numpy.average([numpy.max(phaseangle), numpy.min(phaseangle)]), '|G(jw)| = 1')
plt.axhline(-180., color='red', linestyle='--')
plt.axis(axlim)
plt.grid()
plt.ylabel('Phase')
plt.xlabel('Frequency [rad/unit time]')
return GM, PM
def bode(G, w_start=-2, w_end=2, axlim=None, points=1000, margin=False):
"""
Shows the bode plot for a plant model
Parameters
----------
G : tf
Plant transfer function.
margin : boolean
Show the cross over frequencies on the plot (optional).
Returns
-------
GM : array containing a real number
Gain margin.
PM : array containing a real number
Phase margin.
Plot : matplotlib figure
"""
s, w, axlim = df.frequency_plot_setup(axlim, w_start, w_end, points)
plt.clf()
GM, PM, wc, w_180 = utils.margins(G)
# plotting of Bode plot and with corresponding frequencies for PM and GM
# if ((w2 < numpy.log(w_180)) and margin):
# w2 = numpy.log(w_180)
# Magnitude of G(jw)
plt.subplot(2, 1, 1)
gains = numpy.abs(G(s))
plt.loglog(w, gains)
if margin:
plt.axvline(w_180, color='black')
plt.text(w_180, numpy.average([numpy.max(gains),
numpy.min(gains)]),
r'$\angle$G(jw) = -180$\degree$')
plt.axhline(1., color='red', linestyle='--')
plt.axis(axlim)
plt.grid()
plt.ylabel('Magnitude')
# Phase of G(jw)
plt.subplot(2, 1, 2)
phaseangle = utils.phase(G(s), deg=True)
plt.semilogx(w, phaseangle)
if margin:
plt.axvline(wc, color='black')
plt.text(wc,
numpy.average([numpy.max(phaseangle),
numpy.min(phaseangle)]), '|G(jw)| = 1')
plt.axhline(-180., color='red', linestyle='--')
plt.axis(axlim)
plt.grid()
plt.ylabel('Phase')
plt.xlabel('Frequency [rad/unit time]')
return GM, PM
def bodeclosedloop(G, K, w_start=-2, w_end=2, axlim=None, points=1000, margin=False):
"""
Shows the bode plot for a controller model
Parameters
----------
G : tf
Plant transfer function.
K : tf
Controller transfer function.
margin : boolean
Show the cross over frequencies on the plot (optional).
"""
if axlim is None:
axlim = [None, None, None, None]
plt.gcf().set_facecolor('white')
w = numpy.logspace(w_start, w_end, points)
L = G(1j*w) * K(1j*w)
S = utils.feedback(1, L)
T = utils.feedback(L, 1)
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L))
plt.loglog(w, abs(S))
plt.loglog(w, abs(T))
plt.axis(axlim)
plt.grid()
plt.ylabel("Magnitude")
plt.legend(["L", "S", "T"])
if margin:
plt.plot(w, 1/numpy.sqrt(2) * numpy.ones(len(w)), linestyle='dotted')
plt.subplot(2, 1, 2)
plt.semilogx(w, utils.phase(L, deg=True))
plt.semilogx(w, utils.phase(S, deg=True))
plt.semilogx(w, utils.phase(T, deg=True))
plt.axis(axlim)
plt.grid()
plt.ylabel("Phase")
plt.xlabel("Frequency [rad/s]")
def Rule_7(w_start, w_end):
# Rule 7 determining the phase of GGm at -180 deg.
# This is solved visually from a plot.
w = np.logspace(w_start, w_end, 1000)
Pz = np.polymul(G()[0], Gm()[0])
Pp = np.polymul(G()[1], Gm()[1])
[w, h] = scs.freqs(Pz, Pp, w)
plt.semilogx(w, (180 / np.pi) * (phase(h) + w * Time_Delay()))
plt.show()
def Rule_7(w_start, w_end):
# Rule 7 determining the phase of GGm at -180deg
# this is solved visually from a plot
w = np.logspace(w_start, w_end, 1000)
Pz = np.polymul(G()[0], Gm()[0])
Pp = np.polymul(G()[1], Gm()[1])
[w, h] = scs.freqs(Pz, Pp, w)
plt.semilogx(w, (180 / np.pi) * (phase(h) + w * Time_Delay()))
plt.show()
def Bode_Parametrized(f="8/((s+8)*(s+1))", startBaseFreq=-5, endBaseFreq=37):
"""Give the Bode plot for a given plant f in string format (e.g. 8/((s+8)*(s+1))).
If no argument is provided the example is the default value of f.
The starting frequency and ending frequencies for the plot are provided as minFreq and maxFreq."""
def mod(x):
"""to give the function to calculate |G(jw)| = 1"""
return np.abs(G(f, x)) - 1
# how to calculate the frequency at which |G(jw)| = 1
wc = sc.optimize.fsolve(mod, 0.1)
def arg(w):
"""function to calculate the phase angle at -180 deg"""
return np.angle(G(f, w)) + np.pi
# where the frequency is calculated where arg G(jw) = -180 deg
w_180 = sc.optimize.fsolve(arg, -1)
# Plotting of gain Bode plot
w = np.logspace(startBaseFreq, endBaseFreq, num=1000)
#Set up the plot
fig = plt.figure()
plt.subplot(211)
plt.loglog(w, np.abs(G(f, w)))
#Plot the line for the unit gain
plt.loglog(w, np.ones(1000), 'r--')
plt.ylabel("Magnitude")
plt.title("Figure 2.2: Frequency response (Bode plots) of G(s) = " + f + "\n ", fontsize=12) # \n for a newline so that it doesn't hog the top of the graph.
#Set the axis to the w range - adds in some x-space otherwise, sucks...
plt.xlim(10.0**startBaseFreq, 10.0**endBaseFreq)
# Plotting of phase Bode plot
plt.subplot(212)
phaseangle = phase(G(f, w), deg=True)
plt.semilogx(w, phaseangle)
plt.semilogx(w, -180 * np.ones(1000), 'r--')
#Set the axis to the w range - adds in some x-space otherwise, sucks...
plt.xlim(10.0**startBaseFreq, 10.0**endBaseFreq)
plt.ylabel("Phase")
plt.xlabel("Frequency [rad/s]")
plt.show()
return fig
def Bode():
"""give the Bode plot along with GM and PM"""
def mod(x):
"""to give the function to calculate |G(jw)| = 1"""
return np.abs(G(x)) - 1
# how to calculate the freqeuncy at which |G(jw)| = 1
wc = sc.optimize.fsolve(mod, 0.1)
def arg(w):
"""function to calculate the phase angle at -180 deg"""
return np.angle(G(w)) + np.pi
# where the freqeuncy is calculated where arg G(jw) = -180 deg
w_180 = sc.optimize.fsolve(arg, -1)
PM = np.angle(G(wc), deg=True) + 180
GM = 1/(np.abs(G(w_180)))
# plotting of Bode plot and with corresponding freqeuncies for PM and GM
w = np.logspace(-5, np.log(w_180), 1000)
plt.subplot(211)
plt.loglog(w, np.abs(G(w)))
plt.loglog(wc*np.ones(2), [np.max(np.abs(G(w))), np.min(np.abs(G(w)))])
plt.text(w_180, np.average([np.max(np.abs(G(w))), np.min(np.abs(G(w)))]), '<G(jw) = -180 Deg')
plt.loglog(w_180*np.ones(2), [np.max(np.abs(G(w))), np.min(np.abs(G(w)))])
plt.loglog(w, 1*np.ones(len(w)))
# argument of G
plt.subplot(212)
phaseangle = phase(G(w), deg=True)
plt.semilogx(w, phaseangle)
plt.semilogx(wc*np.ones(2), [np.max(phaseangle), np.min(phaseangle)])
plt.semilogx(w_180*np.ones(2), [-180, 0])
plt.show()
return GM, PM
Kc = 0.05
#plant model
G = 3*(-2*s+1)/((10*s+1)*(5*s+1))
#Controller model
K = Kc*(10*s+1)*(5*s+1)/(s*(2*s+1)*(0.33*s+1))
#closed-loop transfer function
L = G*K
#magnitude and phase
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L(wi)))
plt.loglog(w, np.ones_like(w))
plt.ylabel('Magnitude')
plt.subplot(2, 1, 2)
plt.semilogx(w, phase(L(wi), deg=True))
plt.semilogx(w, -180*np.ones_like(w))
plt.ylabel('Phase')
plt.xlabel('frequency (rad/s)')
plt.figure()
# From the figure we can calculate GM and PM,
# cross-over frequency wc and w180
# results:GM = 1/0.354 = 2.82
# PM = -125.3 + 180 = 54 degC
# w180 = 0.44
# wc = 0.15
#Response to step in reference for loop shaping design
#y = Tr, r(t) = 1 for t > 0
s = w*1j
K = Kc*(1+1/(Tauc*s))
G = 3*(-2*(s)+1)/((10*s+1)*(5*s+1))
L = G*K
S = feedback(1, L)
T = feedback(L, 1)
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L))
plt.loglog(w, abs(S))
plt.loglog(w, abs(T))
plt.ylabel("Magnitude")
plt.legend(["L", "S", "T"],
bbox_to_anchor=(0, 1.01, 1, 0), loc=3, ncol=3)
plt.subplot(2, 1, 2)
plt.semilogx(w, phase(L, deg=True))
plt.semilogx(w, phase(S, deg=True))
plt.semilogx(w, phase(T, deg=True))
plt.ylabel("Phase")
plt.xlabel("Frequency [rad/s]")
# Calculate the frequency at which |L(jw)| = 1
wc = w[np.flatnonzero(np.abs(L) < 1)[0]]
# Calculate the frequency at which Angle[L(jw)] = -180
def Lu_180(w):
s = w*1j
G = 3*(-2*(s)+1)/((10*s+1)*(5*s+1))
w = np.logspace(-2, 1, 1000)
s = w*1j
kc = 0.05
#plant model
G = 3*(-2*s+1)/((10*s+1)*(5*s+1))
#Controller model
K = kc*(10*s+1)*(5*s+1)/(s*(2*s+1)*(0.33*s+1))
#closed-loop transfer function
L = G*K
#magnitude and phase
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L))
plt.loglog(w, 1*np.ones(len(w)))
plt.ylabel('Magnitude')
plt.subplot(2, 1, 2)
plt.semilogx(w, phase(L, deg=True))
plt.semilogx(w, (-180)*np.ones(len(w)))
plt.ylabel('Phase')
plt.xlabel('frequency (rad/s)')
plt.show()
# From the figure we can calculate GM and PM,
# cross-over frequency wc and w180
# results:GM = 1/0.354 = 2.82
# PM = -125.3 + 180 = 54 degC
# w180 = 0.44
# wc = 0.15
Kc = 1.25
tau1 = 1.5
K = Kc * (1 + 1 / (tau1 * s))
L = K * G
S = feedback(1, L)
T = feedback(L, 1)
#Bode magnitude and phase plot - Figure 2.15
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L))
plt.loglog(w, abs(S))
plt.loglog(w, abs(T))
plt.ylabel("Magnitude")
plt.legend(["L", "S", "T"], bbox_to_anchor=(0, 1.01, 1, 0), loc=3, ncol=3)
plt.subplot(2, 1, 2)
plt.semilogx(w, phase(L, deg=True))
plt.semilogx(w, phase(S, deg=True))
plt.semilogx(w, phase(T, deg=True))
plt.semilogx(w, (-180) * np.ones(len(w)))
plt.ylabel("Phase")
plt.xlabel("Frequency [rad/s]")
# Calculate the frequency at which |L(jw)| = 1
wc = w[np.flatnonzero(np.abs(L) < 1)[0]]
# Calculate the frequency at which Angle[L(jw)] = -180
def Lu_180(w):
s = w * 1j
G = 4 / ((s - 1) * (0.02 * s + 1)**2)
Kc = 1.25
Kc = 0.05
#plant model
G = 3 * (-2 * s + 1) / ((10 * s + 1) * (5 * s + 1))
#Controller model
K = Kc * (10 * s + 1) * (5 * s + 1) / (s * (2 * s + 1) * (0.33 * s + 1))
#closed-loop transfer function
L = G * K
#magnitude and phase
plt.subplot(2, 1, 1)
plt.loglog(w, abs(L(wi)))
plt.axhline(1)
plt.ylabel('Magnitude')
plt.subplot(2, 1, 2)
plt.semilogx(w, phase(L(wi), deg=True))
plt.axhline(-180)
plt.ylabel('Phase')
plt.xlabel('frequency (rad/s)')
plt.figure()
# From the figure we can calculate GM and PM,
# cross-over frequency wc and w180
# results:GM = 1/0.354 = 2.82
# PM = -125.3 + 180 = 54 degC
# w180 = 0.44
# wc = 0.15
#Response to step in reference for loop shaping design
#y = Tr, r(t) = 1 for t > 0
def G_GM(w):
Pz = np.polymul(G()[0], Gm()[0])
Pp = np.polymul(G()[1], Gm()[1])
G_w = scs.freqs(Pz, Pp, w)[1]
return np.abs(phase(G_w))-np.pi
def RULES(R, wr):
# rule 1 wc>wd
def Gd_mod_1(w):
return np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1])-1
wd = sc.optimize.fsolve(Gd_mod_1, 10)
wc_min_1 = wd
# rule 2
# rule 3
# for perfect control
def G_Gd_1(w):
f = scs.freqs(G()[0], G()[1], w)[1]
g = scs.freqs(Gd()[0], Gd()[1], w)[1]
return np.abs(f)-np.abs(g)
w_G_Gd = sc.optimize.fsolve(G_Gd_1, 0.001)
plt.figure(1)
if np.abs(scs.freqs(G()[0], G()[1], [w_G_Gd+0.0001])[1])>np.abs(scs.freqs(Gd()[0], Gd()[1], [w_G_Gd+0.0001])[1]):
print "Acceptable control"
print "control only at high frequencies", w_G_Gd, "< w < inf"
w = np.logspace(-3, np.log10(w_G_Gd), 100)
plt.loglog(w, np.abs(scs.freqs(G()[0], G()[1], w)[1]), 'r')
plt.loglog(w, np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1]), 'r.')
max_p = np.max([np.abs(scs.freqs(G()[0], G()[1], w)[1]), np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1])])
w = np.logspace(np.log10(w_G_Gd), 5, 100)
plt.loglog(w, np.abs(scs.freqs(G()[0], G()[1], w)[1]), 'b')
plt.loglog(w, np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1]), 'b.')
min_p = np.min([np.abs(scs.freqs(G()[0], G()[1], w)[1]), np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1])])
if np.abs(scs.freqs(G()[0], G()[1], [w_G_Gd-0.0001])[1])>= np.abs(scs.freqs(Gd()[0], Gd()[1], [w_G_Gd-0.0001])[1]):
print "Acceptable control"
print "control up to frequency 0 < w < ", w_G_Gd
w = np.logspace(-3, np.log10(w_G_Gd), 100)
plt.loglog(w, np.abs(scs.freqs(G()[0], G()[1], w)[1]), 'b')
plt.loglog(w, np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1]), 'b.')
max_p = np.max([np.abs(scs.freqs(G()[0], G()[1], w)[1]), np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1])])
w = np.logspace(np.log10(w_G_Gd), 5, 100)
plt.loglog(w, np.abs(scs.freqs(G()[0], G()[1], w)[1]), 'r')
plt.loglog(w, np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1]), 'r.')
min_p = np.min([np.abs(scs.freqs(G()[0], G()[1], w)[1]), np.abs(scs.freqs(Gd()[0], Gd()[1], w)[1])])
plt.loglog(w_G_Gd*np.ones(2), [max_p, min_p], 'g')
# rule 4
# rule 5
# critical freqeuncy of controller needs to smaller than
wc_5 = Time_Delay()[0]/2.0000
# rule 6
# control over RHP zeros
Pz_G_Gm = np.polymul(G()[0], Gm()[0])
if len(Pz_G_Gm) == 1:
wc_6 = wc_5
else:
Pz_roots = np.roots(Pz_G_Gm)
print Pz_roots
if np.real(np.max(Pz_roots))>0:
if np.imag(np.min(Pz_roots)) == 0:
# it the roots aren't imagenary
# looking for the minimum values of the zeros = > results in the tightest control
wc_6 = (np.min(np.abs(Pz_roots)))/2.000
else:
wc_6 = 0.8600*np.abs(np.min(Pz_roots))
else:
wc_6 = wc_5
# rule 7
def G_GM(w):
Pz = np.polymul(G()[0], Gm()[0])
Pp = np.polymul(G()[1], Gm()[1])
G_w = scs.freqs(Pz, Pp, w)[1]
return np.abs(phase(G_w))-np.pi
w = np.logspace(-3, 3, 100)
plt.figure(2)
Pz = np.polymul(G()[0], Gm()[0])
Pp = np.polymul(G()[1], Gm()[1])
[w, h] = scs.freqs(Pz, Pp, w)
plt.subplot(211)
plt.loglog(w, np.abs(h))
plt.subplot(212)
plt.semilogx(w, phase(h))
wc_7 = np.abs(sc.optimize.fsolve(G_GM, 10))
w_vec = [wc_5, wc_6, wc_7]
wc_min_everything = np.min(w_vec)
print " "
print "maximum value of wc < ", wc_min_everything
# rule 8
# unstable RHP poles
Poles_p = np.roots(G()[1])
vec_p = [wc_min_1]
for p in Poles_p:
if np.real(p) > 0:
vec_p.append(2*np.abs(p))
wc_min_everything = np.max(vec_p)
print "minimum value of wc > ", wc_min_everything
plt.show()
def phase(self):
"""Returns the current phase of the cursor
"""
return utils.phase(self.angle)
