Control Engineering II, Week 5 - Note 2

Transfer function

Install the control library

!pip install control

Collecting control
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Installing collected packages: control
Successfully installed control-0.10.0

numpy, control, matplotlib import

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

Entering a simple transfer function as variable H

A transfer function is in symbolic form by default

num = np.array([2])
den = np.array([5, 1])
H = ctl.tf(num, den)
H

TransferFunction(array([2]), array([5, 1]))

Connecting two transfer functions in series (multiplication)

num1 = np.array([2])
den1 = np.array([1, 0])
num2 = np.array([3])
den2 = np.array([4, 1])
H1 = ctl.tf(num1, den1)
H2 = ctl.tf(num2, den2)
Hs = ctl.series(H1, H2)
Hs

TransferFunction(array([6]), array([4, 1, 0]))

Hs = H1*H2
Hs

TransferFunction(array([6]), array([4, 1, 0]))

Connecting two transfer functions in parallel (addition)

Hp = ctl.parallel(H1, H2)
Hp

TransferFunction(array([11,  2]), array([4, 1, 0]))

Hp = H1+H2
Hp

TransferFunction(array([11,  2]), array([4, 1, 0]))

Finding poles and zeros of a transfer function

A plot is drawn when plot=True or when the argument is omitted

(p, z) = ctl.pzmap(Hp, plot=False)
p, z

(array([-0.25+0.j,  0.  +0.j]), array([-0.18181818+0.j]))

(p, z) = ctl.pzmap(Hp)

<Figure size 640x480 with 1 Axes>

Feedback connection

(For unity feedback, the feedback gain is 1)

num = np.array([2])
den = np.array([1 , 0])
L = ctl.tf(num, den)
H = ctl.feedback(L, 1)
H

TransferFunction(array([2]), array([1, 2]))

Getting lists of numerator and denominator coefficients from a transfer function

num_, den_ = ctl.tfdata(H)
num_, den_

([[array([2])]], [[array([1, 2])]])

Enter a $2^{nd}$-order transfer function as G

G = ctl.tf([3],[4, 1, 1])
G

TransferFunction(array([3]), array([4, 1, 1]))

Examining the step response

t,y = ctl.step_response(G)
plt.plot(t,y)
plt.xlabel('Time')
plt.title('Response of a Second Order System')

Text(0.5, 1.0, 'Response of a Second Order System')

<Figure size 640x480 with 1 Axes>