Exercises on step responses and dominant poles¶
$$ \newcommand{\eg}{{\it e.g.}} \newcommand{\ie}{{\it i.e.}} \newcommand{\argmin}{\operatornamewithlimits{argmin}} \newcommand{\mc}{\mathcal} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathbf} \newcommand{\minimize}{{\text{minimize}}} \newcommand{\diag}{{\text{diag}}} \newcommand{\cond}{{\text{cond}}} \newcommand{\rank}{{\text{rank }}} \newcommand{\range}{{\mathcal{R}}} \newcommand{\null}{{\mathcal{N}}} \newcommand{\tr}{{\text{trace}}} \newcommand{\dom}{{\text{dom}}} \newcommand{\dist}{{\text{dist}}} \newcommand{\R}{\mathbf{R}} \newcommand{\SM}{\mathbf{S}} \newcommand{\ball}{\mathcal{B}} \newcommand{\bmat}[1]{\begin{bmatrix}#1\end{bmatrix}} $$ ASE3093: Automatic control, Inha University.
Jong-Han Kim (jonghank@inha.ac.kr)
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!pip install control
Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/ Collecting control Using cached control-0.9.3.post2-py3-none-any.whl (432 kB) Requirement already satisfied: numpy in /usr/local/lib/python3.9/dist-packages (from control) (1.22.4) Requirement already satisfied: scipy>=1.3 in /usr/local/lib/python3.9/dist-packages (from control) (1.10.1) Requirement already satisfied: matplotlib in /usr/local/lib/python3.9/dist-packages (from control) (3.7.1) Requirement already satisfied: importlib-resources>=3.2.0 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (5.12.0) Requirement already satisfied: pyparsing>=2.3.1 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (3.0.9) Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (23.1) Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (1.4.4) Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (0.11.0) Requirement already satisfied: contourpy>=1.0.1 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (1.0.7) Requirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (2.8.2) Requirement already satisfied: pillow>=6.2.0 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (8.4.0) Requirement already satisfied: fonttools>=4.22.0 in /usr/local/lib/python3.9/dist-packages (from matplotlib->control) (4.39.3) Requirement already satisfied: zipp>=3.1.0 in /usr/local/lib/python3.9/dist-packages (from importlib-resources>=3.2.0->matplotlib->control) (3.15.0) Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.9/dist-packages (from python-dateutil>=2.7->matplotlib->control) (1.16.0) Installing collected packages: control Successfully installed control-0.9.3.post2
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import numpy as np
import matplotlib.pyplot as plt
from control.matlab import *
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num1 = np.array([1])
den1 = np.array([1, 1])
G1 = tf(num1, den1)
num2 = np.array([100])
den2 = np.array([1, 100])
G2 = tf(num2, den2)
num3 = np.array([10000])
den3 = np.array([1, 2*0.1*100, 10000])
G3 = tf(num3, den3)
t = np.linspace(0, 5, 10000)
y, t = step(G1*G3, t)
y_r, t_r = step(G1, t)
plt.figure(dpi=100)
plt.plot(t, y, label='Full', alpha=0.8)
plt.plot(t_r, y_r, label='Reduced', alpha=0.8)
plt.xlabel(r'$t$ (sec)')
plt.ylabel(r'$y$')
plt.title('Step response')
plt.legend()
plt.grid()
plt.show()
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num1 = np.array([1])
den1 = np.array([1, 1])
G1 = tf(num1, den1)
num2 = np.array([1])
den2 = np.array([1, 0.01])
G2 = tf(num2, den2)
num3 = np.array([1])
den3 = np.array([1, 0])
G3 = tf(num3, den3)
t = np.linspace(0, 5, 10000)
y, t = step(G1*G2, t)
y_r, t_r = step(G1*G3, t)
plt.figure(dpi=100)
plt.plot(t, y, label='Full', alpha=0.8)
plt.plot(t_r, y_r, label='Reduced', alpha=0.8)
plt.xlabel(r'$t$ (sec)')
plt.ylabel(r'$y$')
plt.title('Step response')
plt.legend()
plt.grid()
plt.show()
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ts = np.arange(0, 15, 0.01)
G = tf([1], [1, 1])
K = tf([2, 2.5], [1, 0])
CL = feedback(K*G, 1, -1)
y, t = step(CL, ts)
print (CL)
plt.figure(dpi=100)
plt.plot(t,y)
plt.grid()
2 s + 2.5 --------------- s^2 + 3 s + 2.5
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G1 = tf([2.5], [1, 3, 2.5])
G2 = tf([2, 0], [1, 3, 2.5])
y1, t1 = step(G1, ts)
y2, t2 = step(G2, ts)
plt.figure(dpi=100)
plt.plot(t1, y1)
plt.plot(t2, y2)
plt.plot(t, y)
plt.grid()
plt.show()
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