{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "

Simple, Forced Pendulum

\n", "

MCHE474: Control Systems

\n", "

Dr. Joshua Vaughan
\n", "joshua.vaughan@louisiana.edu
\n", "http://www.ucs.louisiana.edu/~jev9637/

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this notebook, we'll be looking at the simple forced pendulum shown in Figure 1. It has a point mass, $m$, suspended on an inflexible, inextensible, massless rod of length $l$ from a perfect pin joint. There is a pure torque, $T$, acting about that joint.\n", "\n", "

\n", "\t\"Pendulum
\n", " Figure 1: Pendulum with Torque Input\n", "

\n", "\n", "The equation of motion for this system is:\n", "\n", "$ \\quad m l^2 \\ddot{\\theta} + m g l \\sin{\\theta} = T $\n", "\n", "This can be rewritten as:\n", "\n", "$ \\quad \\ddot{\\theta} = - \\frac{g}{l} \\sin{\\theta} + \\frac{T}{m l^2} $\n", "\n", "To simulate the nonlinear system, we need to write this 2nd-order differential equation as a system of first order differential equations.\n", "\n", "To do so, define the state vector $\\bar{w} = \\left[\\theta \\ \\dot{\\theta}\\right]^T$. We can also define an input vector as $\\bar{u} = \\left[T \\right]$. Since we only have one input to this system, the input vector only has one element.\n", "\n", "Then, the system of first-order ODEs we have to solve is:\n", "\n", "$ \\quad \\dot{\\bar{w}} = g(\\bar{w}, \\bar{u}, t) $\n", "\n", "Writing these out, we have:\n", "\n", "$ \\quad \\dot{\\bar{w}} = \\left[\\dot{\\theta} \\right.$\n", "\n", "$\\phantom{\\quad \\dot{\\bar{w}} = \\left[\\right.}\\left. - \\frac{g}{l} \\sin{\\theta} + \\frac{T}{m l^2}\\right] $\n", "\n", "Now, we can use that system of 1st-order differential equations to simulate the system.\n", "\n", "To begin we will import the NumPy library, the matplotlib plotting library, and the ```odeint``` ODE solver from the SciPy library." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np # Grab all of the NumPy functions with namespace np" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "\n", "# Import the plotting functions \n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Import the ODE solver\n", "from scipy.integrate import odeint" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We need to define two functions for the differential equation solver to use. The first is just the system of differential equations to solve. I've defined it below by ```eq_of_motion()```. It is just the system of equations we wrote above. The second is the input force as a function of time. I have called it ```torque()``` below. Here, it is just a pulse input in torque, lasting 0.5 seconds." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def eq_of_motion(w, t, p):\n", " \"\"\"\n", " Defines the differential equations for the forced pendulum system\n", "\n", " Arguments:\n", " w : vector of the state variables:\n", " t : time\n", " p : vector of the parameters:\n", " \"\"\"\n", " theta, theta_dot = w\n", " g, l, StartTime, T_amp = p\n", "\n", " # Create sysODE = (theta', theta_dot')\n", " sysODE = [theta_dot,\n", " -g/l * np.sin(theta) + torque(t, p) / (m * l**2)]\n", " return sysODE\n", "\n", "\n", "def torque(t, p):\n", " \"\"\"\n", " defines the torque input to the system\n", " \"\"\"\n", " g, l, StartTime, T_amp = p\n", " \n", " # Select one of the two inputs below\n", " # Be sure to comment out the one you're not using\n", " \n", " # Input Option 1: \n", " # Just a step in force beginning at t=DistStart\n", " # f = F_amp * (t >= DistStart)\n", " \n", " # Input Option 2:\n", " # A pulse in force beginning at t=StartTime and ending at t=(StartTime + 0.5)\n", " f = T_amp * (t >= StartTime) * (t <= StartTime + 0.5)\n", " \n", " return f" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Define the system parameters\n", "m = 1.0 # mass (kg)\n", "g = 9.81 # acceleration of gravity (m/s^2)\n", "l = 1.0 # length of the pendulum (m)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Set up simulation parameters\n", "\n", "# ODE solver parameters\n", "abserr = 1.0e-9\n", "relerr = 1.0e-9\n", "max_step = 0.01\n", "stoptime = 10.0\n", "numpoints = 10001\n", "\n", "# Create the time samples for the output of the ODE solver\n", "t = np.linspace(0.0, stoptime, numpoints)\n", "\n", "# Initial conditions\n", "theta_init = 0.0 # initial position\n", "theta_dot_init = 0.0 # initial velocity\n", "\n", "# Set up the parameters for the input function\n", "StartTime = 0.5 # Time the f(t) input will begin\n", "T_amp = 2.0 # Amplitude of Disturbance force (N)\n", "\n", "# Pack the parameters and initial conditions into arrays \n", "p = [g, l, StartTime, T_amp]\n", "x0 = [theta_init, theta_dot_init]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we will actually call the ode solver, using the ```odeint()``` function from the SciPy library. For more information on ```odeint```, see [the SciPy documentation](http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html)." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Call the ODE solver.\n", "resp = odeint(eq_of_motion, x0, t, args=(p,), atol=abserr, rtol=relerr, hmax=max_step)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solver returns the time history of each state. To plot an individual response, we just simply pick the corresponding column. Below, we'll plot the position of the mass as a function of time." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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xUX+IP3FWUasCH8N3YKv2ENjJGvOLKO9H3dSSaJTYrD/Uo8Nx5SuBPyHp6wPq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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Make the figure pretty, then plot the results\n", "# \"pretty\" parameters selected based on pdf output, not screen output\n", "# Many of these setting could also be made default by the .matplotlibrc file\n", "\n", "# Set the plot size - 3x2 aspect ratio is best\n", "fig = plt.figure(figsize=(6, 4))\n", "ax = plt.gca()\n", "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n", "\n", "# Change the axis units to serif\n", "plt.setp(ax.get_ymajorticklabels(),family='serif',fontsize=18)\n", "plt.setp(ax.get_xmajorticklabels(),family='serif',fontsize=18)\n", "\n", "ax.spines['right'].set_color('none')\n", "ax.spines['top'].set_color('none')\n", "\n", "ax.xaxis.set_ticks_position('bottom')\n", "ax.yaxis.set_ticks_position('left')\n", "\n", "# Turn on the plot grid and set appropriate linestyle and color\n", "ax.grid(True,linestyle=':',color='0.75')\n", "ax.set_axisbelow(True)\n", "\n", "# Define the X and Y axis labels\n", "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n", "plt.ylabel('Angle (deg)', family='serif', fontsize=22, weight='bold', labelpad=10)\n", "\n", "# Plot the first element of resp for all time. It corresponds to the position.\n", "plt.plot(t, resp[:,0] * 180/np.pi, linewidth=2, linestyle = '-', label=r'Response')\n", "\n", "# uncomment below and set limits if needed\n", "# plt.xlim(0,5)\n", "# plt.ylim(0,10)\n", "\n", "# # Create the legend, then fix the fontsize\n", "# leg = plt.legend(loc='upper right', fancybox=True)\n", "# ltext = leg.get_texts()\n", "# plt.setp(ltext,family='serif',fontsize=18)\n", "\n", "# Adjust the page layout filling the page using the new tight_layout command\n", "plt.tight_layout(pad = 0.5)\n", "\n", "# save the figure as a high-res pdf in the current folder\n", "# It's saved at the original 6x4 size\n", "# plt.savefig('MCHE474_DirectTorquePendulum_nonlinear.pdf')\n", "\n", "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linearized Version\n", "We can also linearize these equations and simluation the linearized system. To do so, we'll assume small angles about an equilibrium angle of $\\theta_{eq} = 0$. The linearized equation of motion is then:\n", "\n", "$ \\quad m l^2 \\ddot{\\theta} + m g l \\theta = T $\n", "\n", "This can be rewritten as:\n", "\n", "$ \\quad \\ddot{\\theta} = - \\frac{g}{l} \\theta + \\frac{T}{m l^2} $\n", "\n", "Like the nonlinear version, to simulate the system, we need to write this 2nd-order differential equation as a system of first order differential equations.\n", "\n", "To do so, we'll use the same state vector as before, $\\bar{w} = \\left[\\theta \\ \\dot{\\theta}\\right]^T$. Also as before, we will an input vector as $\\bar{u} = \\left[T \\right]$. \n", "\n", "Writing these out, we have:\n", "\n", "$ \\quad \\dot{\\bar{w}} = \\left[\\dot{\\theta} \\right.$\n", "\n", "$\\phantom{\\quad \\dot{\\bar{w}} = \\left[\\right.}\\left. - \\frac{g}{l} \\theta + \\frac{T}{m l^2}\\right] $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We again need to define the function representing these equations that the differential equation solver calls. I've defined it below by ```eq_of_motion_linearized()```. The torque function we defined above is reused in this system." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def eq_of_motion_linearized(w, t, p):\n", " \"\"\"\n", " Defines the linearized differential equations for the forced pendulum system\n", "\n", " Arguments:\n", " w : vector of the state variables:\n", " t : time\n", " p : vector of the parameters:\n", " \"\"\"\n", " theta, theta_dot = w\n", " g, l, StartTime, T_amp = p\n", "\n", " # Create sysODE = (theta', theta_dot')\n", " sysODE = [theta_dot,\n", " -g/l * theta + torque(t, p) / (m * l**2)]\n", " return sysODE" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll use all the same simulation parameters from the earlier nonlinear simluation, so we do not need to define them again. So, we're ready to call the ODE solver." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Call the ODE solver.\n", "resp_linearized = odeint(eq_of_motion_linearized, x0, t, args=(p,), atol=abserr, rtol=relerr, hmax=max_step)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, let's plot the nonlinear and linearized responses together, for comparision." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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O8kClj/s/8g48a9ew/ClzW+0kQP+56xMiLMXI2EwMAJutNiKZTsiJSM1Xykuv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ynFSjnHYfUA1sNY9fiWFD/+FillsunHZDQm+V49YPmSCre7tO0OrGCbx8blZRgYmeW6HOs/OBcC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\nm/T1PzEwA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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Make the figure pretty, then plot the results\n", "# \"pretty\" parameters selected based on pdf output, not screen output\n", "# Many of these setting could also be made default by the .matplotlibrc file\n", "\n", "# Set the plot size - 3x2 aspect ratio is best\n", "fig = plt.figure(figsize=(6, 4))\n", "ax = plt.gca()\n", "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n", "\n", "# Change the axis units to serif\n", "plt.setp(ax.get_ymajorticklabels(),family='serif',fontsize=18)\n", "plt.setp(ax.get_xmajorticklabels(),family='serif',fontsize=18)\n", "\n", "ax.spines['right'].set_color('none')\n", "ax.spines['top'].set_color('none')\n", "\n", "ax.xaxis.set_ticks_position('bottom')\n", "ax.yaxis.set_ticks_position('left')\n", "\n", "# Turn on the plot grid and set appropriate linestyle and color\n", "ax.grid(True,linestyle=':',color='0.75')\n", "ax.set_axisbelow(True)\n", "\n", "# Define the X and Y axis labels\n", "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n", "plt.ylabel('Angle (deg)', family='serif', fontsize=22, weight='bold', labelpad=10)\n", "\n", "# Plot the first element of resp for all time. It corresponds to the position.\n", "plt.plot(t, resp[:,0] * 180/np.pi, linewidth=2, linestyle = '-', label=r'Nonlinear')\n", "plt.plot(t, resp_linearized[:,0] * 180/np.pi, linewidth=2, linestyle = '--', label=r'Linearized')\n", "\n", "# uncomment below and set limits if needed\n", "# plt.xlim(0,5)\n", "plt.ylim(-20,25)\n", "\n", "# # Create the legend, then fix the fontsize\n", "leg = plt.legend(loc='upper right', ncol = 2, fancybox=True)\n", "ltext = leg.get_texts()\n", "plt.setp(ltext,family='serif',fontsize=18)\n", "\n", "# Adjust the page layout filling the page using the new tight_layout command\n", "plt.tight_layout(pad = 0.5)\n", "\n", "# save the figure as a high-res pdf in the current folder\n", "# It's saved at the original 6x4 size\n", "# plt.savefig('MCHE474_DirectTorquePendulum_LinearizedComparison.pdf')\n", "\n", "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the small value of torque that we chose, which results in small angles of oscillation, the linearized approximation of the system fairly closely matches the full, nonlinear version. If the system were oscillating over a larger range of angles, then the linearized version would be a poorer approximation.\n", "\n", "To test this, increase the value of ```T_max``` above (in line 6, if you've run all the cells in order) and re-run the comparison. There will be a greater difference between the responses of the full nonlinear and linearized systems.\n", "\n", "We can also plot the error between the response of full, nonlinear model and the response of the linearized version." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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EsRvvux9tyFMDu4XefBNaWvCMGI6jVy9Lzrn99eA95JD4khrLlxNraLCkBpOE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\nTdr6uT/p6MbMIQfJeKy5XFkTdrLRmGXrzu5KW1XdzMxM6hI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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Make the figure pretty, then plot the results\n", "# \"pretty\" parameters selected based on pdf output, not screen output\n", "# Many of these setting could also be made default by the .matplotlibrc file\n", "\n", "# Set the plot size - 3x2 aspect ratio is best\n", "fig = plt.figure(figsize=(6, 4))\n", "ax = plt.gca()\n", "plt.subplots_adjust(bottom=0.17, left=0.17, top=0.96, right=0.96)\n", "\n", "# Change the axis units to serif\n", "plt.setp(ax.get_ymajorticklabels(),family='serif',fontsize=18)\n", "plt.setp(ax.get_xmajorticklabels(),family='serif',fontsize=18)\n", "\n", "ax.spines['right'].set_color('none')\n", "ax.spines['top'].set_color('none')\n", "\n", "ax.xaxis.set_ticks_position('bottom')\n", "ax.yaxis.set_ticks_position('left')\n", "\n", "# Turn on the plot grid and set appropriate linestyle and color\n", "ax.grid(True,linestyle=':',color='0.75')\n", "ax.set_axisbelow(True)\n", "\n", "# Define the X and Y axis labels\n", "plt.xlabel('Time (s)', family='serif', fontsize=22, weight='bold', labelpad=5)\n", "plt.ylabel('Error (deg)', family='serif', fontsize=22, weight='bold', labelpad=10)\n", "\n", "# Plot the first element of resp for all time. It corresponds to the position.\n", "plt.plot(t, (resp[:,0]- resp_linearized[:,0]) * 180/np.pi, linewidth=2, linestyle = '-', label=r'Error')\n", "\n", "# uncomment below and set limits if needed\n", "# plt.xlim(0,5)\n", "plt.ylim(-5,5)\n", "\n", "# # Create the legend, then fix the fontsize\n", "# leg = plt.legend(loc='upper right', ncol = 2, fancybox=True)\n", "# ltext = leg.get_texts()\n", "# plt.setp(ltext,family='serif',fontsize=18)\n", "\n", "# Adjust the page layout filling the page using the new tight_layout command\n", "plt.tight_layout(pad = 0.5)\n", "\n", "# save the figure as a high-res pdf in the current folder\n", "# It's saved at the original 6x4 size\n", "# plt.savefig('MCHE474_DirectTorquePendulum_LinearizedError.pdf')\n", "\n", "fig.set_size_inches(9, 6) # Resize the figure for better display in the notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As you might expect the error grows with time. This can be a problem *or* it might not, depending on the system you're working with and the tolerances that you have. Also, notice that, looking back at the response comparison, the error in repsonse amplitude doesn't tend to grow with time. In many cases, that error will matter more than the pure *temporal* error that we have plotted above." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "#### Licenses\n", "Code is licensed under a 3-clause BSD style license. See the licenses/LICENSE.md file.\n", "\n", "Other content is provided under a [Creative Commons Attribution-NonCommercial 4.0 International License](http://creativecommons.org/licenses/by-nc/4.0/), CC-BY-NC 4.0." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# This cell will just improve the styling of the notebook\n", "from IPython.core.display import HTML\n", "import urllib.request\n", "response = urllib.request.urlopen(\"https://cl.ly/1B1y452Z1d35\")\n", "HTML(response.read().decode(\"utf-8\"))" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.2" } }, "nbformat": 4, "nbformat_minor": 1 }