W10_FrequencyResponse

!pip install control

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Installing collected packages: control
Successfully installed control-0.10.0

import numpy as np
import control as ctl
import matplotlib.pyplot as plt

Plotting a Nyquist plot (polar plot)

$$\begin{aligned} G_1(s) &= \frac{0.25}{s - 0.5} \\ G_2(s) &= \frac{0.5}{s - 0.5} \\ G_3(s) &= \frac{1.0}{s - 0.5} \end{aligned}$$

s=ctl.tf([1, 0], [1])
G_1 = 0.25/(s-0.5)
G_1.name='G_1'
G_2 = 0.5/(s-0.5)
G_2.name='G_2'
G_3 = 1.0/(s-0.5)
G_3.name='G_3'
fig1 = ctl.nyquist_plot(G_1)
fig2 = ctl.nyquist_plot(G_2)
fig3 = ctl.nyquist_plot(G_3)
plt.show()

<Figure size 640x480 with 1 Axes>

Plotting a Bode plot

$$G(s) = \frac{1}{s^2 + 0.5s + 1}$$

s = ctl.tf([1,0],[1])
G = 1/(s**2+0.5*s+1)
G.name='G(s)'
out = ctl.bode(G)

<Figure size 640x480 with 2 Axes>

Examining frequency response in the time domain

omega = 2   # 0.1 ~ 10
t = np.arange(0,100,0.1)
u = np.sin(omega*t)
t,y = ctl.forced_response(G, t, u)
input = plt.plot(t,u)
output = plt.plot(t,y)
plt.xlabel('Time')
plt.legend(['input','output'])
plt.axis([0, 100, -3, 3])
plt.show()

<Figure size 640x480 with 1 Axes>

current_omega = ctl.bode(G,omega,'ks')
overall = ctl.bode(G,np.arange(0.1,10,0.1))
plt.show()

<Figure size 640x480 with 2 Axes>

Checking bandwidth (cut-off frequency)

ctl.bandwidth(G)

1.4839084518312828

ctl.bandwidth(G, -10)

2.0000000000008864

Plotting a more complex Bode plot

G_dc = ctl.tf([5],[1])
G_dc.name = 'DC'
G_p1 = 1/s
G_p1.name = 'Pole1'
G_p2 = 1/(0.5*s+1)
G_p2.name = 'Pole2'
G_p34 = 1/(s**2/2500+s*0.6/50+1)
G_p34.name = 'Pole3&4'
G_z1 = 0.1*s+1
G_z1.name = 'Zero1'
G_all = G_dc*G_p1*G_p2*G_p34*G_z1
G_all.name = 'All'
G_list = [G_dc, G_p1, G_p2, G_p34, G_z1, G_all]
out = ctl.bode(G_list, plot_phase=False)

<Figure size 640x480 with 1 Axes>

for sys in G_list:
  print(sys.name, ctl.bandwidth(sys))

DC inf
Pole1 nan
Pole2 1.9952566902229083
Pole3&4 72.6529421994053
Zero1 inf
All nan