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 rule6(G, Gm, message=False):
"""
This is rule six of chapter five
Calculates if tight control at low frequencies with RHP-zeros is possible
Parameters
----------
G : tf
plant model
Gm : tf
measurement model
message : boolean
show the rule message (optional)
Returns
-------
valid6 : boolean value if rule conditions was met
wc : crossover frequency where | G(jwc) | = 1
wd : crossover frequency where | Gd(jwd) | = 1
"""
GGm = G * Gm
zeros = GGm.zeros()
_, _, wc, _ = margins(GGm)
wz = 0
if len(zeros) > 0:
if np.imag(np.min(zeros)) == 0:
# If the roots aren't imaginary.
# Looking for the minimum values of the zeros =>
# results in the tightest control.
wz = (np.min(np.abs(zeros))) / 2
valid6 = wc < 0.86 * np.abs(wz)
else:
wz = 0.86 * np.abs(np.min(zeros))
valid6 = wc < wz / 2
else:
valid6 = False
if message:
print('Rule 6:')
if wz != 0:
print('These are the roots of the transfer function '
'matrix GGm', zeros)
if valid6:
print(
'The critical frequency of S for the system to be '
'controllable is', wz)
else:
print('No zeros in the system to evaluate')
return valid6, wz
def rule6(G, Gm, message=False):
"""
This is rule six of chapter five
Calculates if tight control at low frequencies with RHP-zeros is possible
Parameters
----------
G : tf
plant model
Gm : tf
measurement model
message : boolean
show the rule message (optional)
Returns
-------
valid6 : boolean value if rule conditions was met
wc : crossover frequency where | G(jwc) | = 1
wd : crossover frequency where | Gd(jwd) | = 1
"""
GGm = G * Gm
zeros = GGm.zeros()
_, _, wc, _ = margins(GGm)
wz = 0
if len(zeros) > 0:
if np.imag(np.min(zeros)) == 0:
# If the roots aren't imaginary.
# Looking for the minimum values of the zeros =>
# results in the tightest control.
wz = (np.min(np.abs(zeros)))/2
valid6 = wc < 0.86*np.abs(wz)
else:
wz = 0.86*np.abs(np.min(zeros))
valid6 = wc < wz/2
else:
valid6 = False
if message:
print('Rule 6:')
if wz != 0:
print('These are the roots of the transfer function '
'matrix GGm', zeros)
if valid6:
print('The critical frequency of S for the system to be '
'controllable is', wz)
else:
print('No zeros in the system to evaluate')
return valid6, wz
def rule5(G, Gm=1, message=False):
"""
This is rule five of chapter five
Calculates constraints for time delay, wc < 1 / theta
Parameters
----------
G : tf
plant model
Gm : tf
measurement model
message : boolean
show the rule message (optional)
Returns
-------
valid5 : boolean
value if rule conditions was met
wtd: real
time delay frequency
"""
GGm = G * Gm
TimeDelay = GGm.deadtime
_, _, wc, _ = margins(GGm)
valid5 = False
if TimeDelay == 0:
wtd = 0
else:
wtd = 1 / TimeDelay
valid5 = wc < wtd
if message:
print('Rule 5:')
if TimeDelay == 0:
print("There isn't any deadtime in the system")
if valid5:
print('wc < 1 / theta :', np.round(wc, 2), '<', np.round(wtd, 2))
else:
print('wc > 1 / theta :', np.round(wc, 2), '>', np.round(wtd, 2))
return valid5, wtd
def rule5(G, Gm=1, message=False):
"""
This is rule five of chapter five
Calculates constraints for time delay, wc < 1 / theta
Parameters
----------
G : tf
plant model
Gm : tf
measurement model
message : boolean
show the rule message (optional)
Returns
-------
valid5 : boolean
value if rule conditions was met
wtd: real
time delay frequency
"""
GGm = G * Gm
TimeDelay = GGm.deadtime
_, _, wc, _ = margins(GGm)
valid5 = False
if TimeDelay == 0:
wtd = 0
else:
wtd = 1 / TimeDelay
valid5 = wc < wtd
if message:
print('Rule 5:')
if TimeDelay == 0:
print("There isn't any deadtime in the system")
if valid5:
print('wc < 1 / theta :', np.round(wc, 2), '<', np.round(wtd, 2))
else:
print('wc > 1 / theta :', np.round(wc, 2), '>', np.round(wtd, 2))
return valid5, wtd
def rule7(G, Gm, message=False):
"""
This is rule seven of chapter five
Calculates the phase lag constraints
Parameters
----------
G : tf
plant model
Gm : tf
measurement model
message : boolean
show the rule message (optional)
Returns
-------
valid7 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
wd : real
crossover frequency where | Gd(jwd) | = 1
"""
# Rule 7 determining the phase of GGm at -180 deg.
# This is solved visually from a plot.
GGm = G*Gm
_, _, wc, wu = margins(GGm)
valid7 = wc < wu
if message:
print('Rule 7:')
if valid7:
print('wc < wu :', wc, '<', wu)
else:
print('wc > wu :', wc, '>', wu)
return valid7, wu
def rule7(G, Gm, message=False):
"""
This is rule seven of chapter five
Calculates the phase lag constraints
Parameters
----------
G : tf
plant model
Gm : tf
measurement model
message : boolean
show the rule message (optional)
Returns
-------
valid1 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
wd : real
crossover frequency where | Gd(jwd) | = 1
"""
# Rule 7 determining the phase of GGm at -180 deg.
# This is solved visually from a plot.
GGm = G * Gm
_, _, wc, wu = margins(GGm)
valid7 = wc < wu
if message:
print('Rule 7:')
if valid7:
print('wc < wu :', wc, '<', wu)
else:
print('wc > wu :', wc, '>', wu)
return valid7, wu
def rule8(G, message=False):
'''
This is rule one of chapter five
This function determines if the plant is open-loop stable at its poles
Parameters
----------
G : tf
plant model
Gd : tf
plant distrubance model
message : boolean
show the rule message (optional)
Returns
-------
valid1 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
'''
#Rule 8 for critical frequency min value due to poles
poles = np.roots(G.denominator)
_,_,wc,_ = margins(G)
wp = 0
if np.max(poles) < 0:
wp = 2 * np.max(np.abs(poles))
valid8 = wc > wp
else: valid8 = False
if message:
print 'Rule 8:'
if valid8:
print 'wc > 2p :', wc , '>' , wp
else:
print 'wc < 2p :', wc , '<' , wp
return valid8, wp
def rule8(G, message=False):
"""
This is rule eight of chapter five
This function determines if the plant is open-loop stable at its poles
Parameters
----------
G : tf
plant model
message : boolean
show the rule message (optional)
Returns
-------
valid1 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
"""
# Rule 8 for critical frequency min value due to poles
poles = G.poles()
_, _, wc, _ = margins(G)
wp = 0
if np.max(poles) < 0:
wp = 2 * np.max(np.abs(poles))
valid8 = wc > wp
else:
valid8 = False
if message:
print('Rule 8:')
if valid8:
print('wc > 2p :', wc, '>', wp)
else:
print('wc < 2p :', wc, '<', wp)
return valid8, wp
def rule8(G, message=False):
"""
This is rule eight of chapter five
This function determines if the plant is open-loop stable at its poles
Parameters
----------
G : tf
plant model
message : boolean
show the rule message (optional)
Returns
-------
valid8 : boolean
value if rule conditions were met
wc : real
crossover frequency where | G(jwc) | = 1
"""
# Rule 8 for critical frequency min value due to poles
poles = G.poles()
_, _, wc, _ = margins(G)
wp = 0
if np.max(poles) < 0:
wp = 2*np.max(np.abs(poles))
valid8 = wc > wp
else:
valid8 = False
if message:
print('Rule 8:')
if valid8:
print('wc > 2p :', wc, '>', wp)
else:
print('wc < 2p :', wc, '<', wp)
return valid8, wp
import matplotlib.pyplot as plt
import utilsplot
import numpy as np
s = tf([1,0], 1)
z = 0.1
tau = 1
L = (-s + z)/(s*(tau*s + tau*z + 2))
T = (-s + z)/((s + z)*(tau*s + 1))
S = 1/(1 + L)
mT = maxpeak(T)
mS = maxpeak(S)
GM, PM, wc, wb, wbt, valid = marginsclosedloop(L)
GM, PM, wc, w_180 = margins(L)
print('GM = ', np.round(GM, 1))
print('PM = ', np.round(PM, 1), 'rad')
print('wB = ', np.round(wb, 3))
print('wC = ', np.round(wc, 3))
print('wBT = ', wbt)
print('w180 = ', np.round(w_180, 3))
print('Ms = ', np.round(mS, 2))
print('Mt = ', np.round(mT, 1))
[t, y] = tf_step(T, 50)
plt.figure('Figure 2.17')
plt.title('Step response for system T')
plt.plot(t, y)
plt.xlabel('time (s)')
plt.ylabel('y(t)')
from utils import tf, margins import Chapter_05 as ch5 s = tf([1, 0], 1) # Example plant based on Example 2.9 and Example 2.16 G = (s + 200) / ((10 * s + 1) * (0.05 * s + 1)**2) G.deadtime = 0.002 Gd = 33 / (10 * s + 1) K = 0.4 * ((s + 2) / s) * (0.075 * s + 1) L = G * K w = np.logspace(-3, 3) GM, PM, wc, wu = margins(G) GM, PM, wd, w_180 = margins(Gd) [valid6, wz] = ch5.rule6(G, 1) [valid5, wtd] = ch5.rule5(G) [valid8, wp] = ch5.rule8(G) print 'wc: ', np.round(wc, 3) print 'wd: ', np.round(wd, 3) print 'wu: ', np.round(wu, 3) wuy = abs(G((1j * wu))) wzy = abs(G((1j * wz))) wpy = abs(G((1j * wp))) wtdy = abs(G((1j * wtd)))
def rule1(G, Gd, K=1, message=False, plot=False, w1=-4, w2=2):
"""
This is rule one of chapter five
Calculates the speed of response to reject distrurbances. Condition require
|S(jw)| <= |1/Gd(jw)|
Parameters
----------
G : tf
plant model
Gd : tf
plant distrubance model
K : tf
control model
message : boolean
show the rule message (optional)
plot : boolean
show the bode plot with constraints (optional)
w1 : integer
start frequency, 10^w1 (optional)
w2 : integer
end frequency, 10^w2 (optional)
Returns
-------
valid1 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
wd : real
crossover frequency where | Gd(jwd) | = 1
"""
_, _, wc, _ = margins(G)
_, _, wd, _ = margins(Gd)
valid1 = wc > wd
if message:
print('Rule 1: Speed of response to reject distrubances')
if valid1:
print('First condition met, wc > wd')
else:
print('First condition no met, wc < wd')
print('Seconds conditions requires |S(jw)| <= |1/Gd(jw)|')
if plot:
plt.figure('Rule 1')
w, mag_s = df.setup_plot(['|S|', '1/R', '$w_r$'], w1, w2, G, K, wr)
inv_gd = 1 / Gd
mag_i = np.abs(inv_gd(s))
plt.loglog(w, mag_s)
plt.loglog(w, mag_i, ls='--')
df.setup_plot(['|S|', '1/|Gd|'])
return valid1, wc, wd
def rule1(G, Gd, K=1, message=False, plot=False, w1=-4, w2=2):
"""
This is rule one of chapter five
Calculates the speed of response to reject disturbances. Condition require
|S(jw)| <= |1/Gd(jw)|
Parameters
----------
G : tf
plant model
Gd : tf
plant distrubance model
K : tf
control model
message : boolean
show the rule message (optional)
plot : boolean
show the bode plot with constraints (optional)
w1 : integer
start frequency, 10^w1 (optional)
w2 : integer
end frequency, 10^w2 (optional)
Returns
-------
valid1 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
wd : real
crossover frequency where | Gd(jwd) | = 1
"""
_, _, wc, _ = margins(G)
_, _, wd, _ = margins(Gd)
valid1 = wc > wd
if message:
print('Rule 1: Speed of response to reject distrubances')
if valid1:
print('First condition met, wc > wd')
else:
print('First condition not met, wc < wd')
print('Second condition requires |S(jw)| <= |1/Gd(jw)|')
if plot:
w = np.logspace(w1, w2, 1000)
s = 1j * w
S = 1 / (1 + G * K)
gain = np.abs(S(s))
inv_gd = 1 / Gd
mag_i = np.abs(inv_gd(s))
plt.figure('Rule 1')
plt.loglog(w, gain, label='|S|')
plt.loglog(w, mag_i, ls='--', label='|1/G_d |')
plt.xlabel('Frequency [rad/s]')
plt.ylabel('Magnitude')
plt.legend(bbox_to_anchor=(0, 1.01, 1, 0), loc=3, ncol=3)
plt.show()
return valid1, wc, wd
import utilsplot
import numpy as np
s = tf([1, 0], 1)
z = 0.1
tau = 1
L = (-s+z)/(s*(tau*s + tau*z + 2))
T = (-s+z)/((s+z)*(tau*s+1))
S = 1/(1+L)
mT = maxpeak(T)
mS = maxpeak(S)
GM, PM, wc, wb, wbt, valid = marginsclosedloop(L)
GM, PM, wc, w_180 = margins(L)
print('GM = ', np.round(GM, 1))
print('PM = ', np.round(PM, 1), 'rad')
print('wB = ', np.round(wb, 3))
print('wC = ', np.round(wc, 3))
print('wBT = ', wbt)
print('w180 = ', np.round(w_180, 3))
print('Ms = ', np.round(mS, 2))
print('Mt = ', np.round(mT, 1))
[t, y] = tf_step(T, 50)
plt.figure('Figure 2.17')
plt.plot(t, y)
plt.xlabel('time (s)')
plt.ylabel('y(t)')
plt.show()
import utils
import numpy as np
import sympy as sp
from scipy.optimize import fsolve
s = utils.tf([1,0],[1])
L = (-s + 2)/(s*(s + 2))
_,_,_,w180 = utils.margins(L)
GM = 1/np.abs(L(1j*w180))
print('w_180',w180)
print('GM = 1/|L(w180)|',GM)
print('From 7.55, kmax = GM = ',GM)
omega = np.logspace(-2,2,1000)
def rk(kmax):
return (kmax - 1)/(kmax + 1)
def kbar(kmax):
return (kmax + 1)/2
def RScondition(kmax):
abs_ineq = [abs(rk(kmax) * kbar(kmax) * L(1j*s)/(1 + kbar(kmax)*L(1j*s))) for s in omega]
max_ineq = max(abs_ineq) - 1
return max_ineq
kcal = fsolve(RScondition,1)
print('From 7.58, kmax = ',np.round(kcal,2))
def rule1(G, Gd, K=1, message=False, plot=False, w1=-4, w2=2):
"""
This is rule one of chapter five
Calculates the speed of response to reject disturbances. Condition require
|S(jw)| <= |1/Gd(jw)|
Parameters
----------
G : tf
plant model
Gd : tf
plant disturbance model
K : tf
control model
message : boolean
show the rule message (optional)
plot : boolean
show the bode plot with constraints (optional)
w1 : integer
start frequency, 10^w1 (optional)
w2 : integer
end frequency, 10^w2 (optional)
Returns
-------
valid1 : boolean
value if rule conditions were met
wc : real
crossover frequency where | G(jwc) | = 1
wd : real
crossover frequency where | Gd(jwd) | = 1
"""
_, _, wc, _ = margins(G)
_, _, wd, _ = margins(Gd)
valid1 = wc > wd
if message:
print('Rule 1: Speed of response to reject disturbances')
if valid1:
print('First condition met, wc > wd')
else:
print('First condition not met, wc < wd')
print('Second condition requires |S(jw)| <= |1/Gd(jw)|')
if plot:
w = np.logspace(w1, w2, 1000)
s = 1j*w
S = 1/(1+G*K)
gain = np.abs(S(s))
inv_gd = 1/Gd
mag_i = np.abs(inv_gd(s))
plt.figure('Rule 1')
plt.loglog(w, gain, label='|S|')
plt.loglog(w, mag_i, ls='--', label='|1/G_d |')
plt.xlabel('Frequency [rad/s]')
plt.ylabel('Magnitude')
plt.legend(bbox_to_anchor=(0, 1.01, 1, 0), loc=3, ncol=3)
plt.show()
return valid1, wc, wd
import Chapter_05 as ch5 s = tf([1, 0], 1) # Example plant based on Example 2.9 and Example 2.16 G = (s + 200) / ((10 * s + 1) * (0.05 * s + 1)**2) G.deadtime = 0.002 Gd = 33 / (10 * s + 1) K = 0.4 * ((s + 2) / s) * (0.075 * s + 1) L = G * K w = np.logspace(-3,3) GM, PM, wc, wu = margins(G) GM, PM, wd, w_180 = margins(Gd) [valid6, wz] = ch5.rule6(G, 1) [valid5, wtd] = ch5.rule5(G) [valid8, wp] = ch5.rule8(G) print 'wc: ' , np.round(wc, 3) print 'wd: ' , np.round(wd, 3) print 'wu: ' , np.round(wu, 3) wuy = abs(G((1j*wu))) wzy = abs(G((1j*wz))) wpy = abs(G((1j*wp))) wtdy = abs(G((1j*wtd)))
def rule1(G, Gd, K=1, message=False, plot=False, w1=-4, w2=2):
"""
This is rule one of chapter five
Calculates the speed of response to reject distrurbances. Condition require
|S(jw)| <= |1/Gd(jw)|
Parameters
----------
G : tf
plant model
Gd : tf
plant distrubance model
K : tf
control model
message : boolean
show the rule message (optional)
plot : boolean
show the bode plot with constraints (optional)
w1 : integer
start frequency, 10^w1 (optional)
w2 : integer
end frequency, 10^w2 (optional)
Returns
-------
valid1 : boolean
value if rule conditions was met
wc : real
crossover frequency where | G(jwc) | = 1
wd : real
crossover frequency where | Gd(jwd) | = 1
"""
_,_,wc,_ = margins(G)
_,_,wd,_ = margins(Gd)
valid1 = wc > wd
if message:
print('Rule 1: Speed of response to reject distrubances')
if valid1:
print('First condition met, wc > wd')
else:
print('First condition no met, wc < wd')
print('Seconds conditions requires |S(jw)| <= |1/Gd(jw)|')
if plot:
plt.figure('Rule 1')
w, mag_s = df.setup_plot(['|S|', '1/R', '$w_r$'], w1, w2, G, K, wr)
inv_gd = 1 / Gd
mag_i = np.abs(inv_gd(s))
plt.loglog(w, mag_s)
plt.loglog(w, mag_i, ls = '--')
df.setup_plot(['|S|', '1/|Gd|'])
return valid1, wc, wd
