W6_SystemModeling

!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

Cruise Model

$$\begin{align}m\dot{v} + bv &= u \\ y &= v \end{align}$$

$$\begin{aligned}\dot{\mathbf{x}} = \dot{v} &= [\frac{-b}{m}]v + [\frac{1}{m}]u \\ y&=[1] v \end{aligned}$$

m = 1000
b = 50

A = -b/m
B = 1/m
C = 1
D = 0

G = ctl.ss(A, B, C, D)

input = 500
t,y = ctl.step_response(500*G)
plt.plot(t,y)
plt.xlabel('Time')
plt.title('Response of a Cruise System')

Text(0.5, 1.0, 'Response of a Cruise System')

<Figure size 640x480 with 1 Axes>

Motor Model

$$\begin{align} J \ddot{\theta} + b \dot{\theta} &= K \cdot i \\ L \frac{di}{dt} + R \cdot i &= V - K \dot{\theta} \end{align}$$ $$$$ $$\begin{align} (J \cdot s^2 + b \cdot s) \cdot \Theta(s) &= K \cdot I(s) \\ (L \cdot s + R) \cdot I(s) &= V(s) - K \cdot s \cdot \Theta(s) \end{align}$$ $$$$ $$G(s) = \frac{s \cdot \Theta(s)}{V(s)} = \frac{K}{(J\cdot s+b)(L\cdot s+R)+K^2}$$

J = 0.01
b = 0.1
K = 0.01
R = 1
L = 0.5
s = ctl.tf([1,0],[1])
G = K/((J*s+b)*(L*s+R)+K**2)

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

Text(0.5, 1.0, 'Response of a Motor')

<Figure size 640x480 with 1 Axes>