{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 0,
     "link": "[](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html)",
     "section": ""
    }
   },
   "source": [
    "<!--NOTEBOOK_HEADER-->\n",
    "*This notebook contains material from [CBE30338](https://jckantor.github.io/CBE30338);\n",
    "content is available [on Github](https://github.com/jckantor/CBE30338.git).*\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 0,
     "link": "[](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html)",
     "section": ""
    }
   },
   "source": [
    "<!--NAVIGATION-->\n",
    "< [5.2 Closed-Loop Transfer Functions for Car Cruise Control](https://jckantor.github.io/CBE30338/05.02-Closed--Loop-Transfer-Functions-for-Car-Cruise-Control.html) | [Contents](toc.html) | [Tag Index](tag_index.html) | [5.4 Controller Tuning Rules in Frequency Domain](https://jckantor.github.io/CBE30338/05.04-Controller-Tuning-Rules-in-Frequency-Domain.html) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE30338/blob/master/docs/05.03-Creating-Bode-Plots.ipynb\"> <img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.ipynb\"> <img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 1,
     "link": "[5.3 Creating Bode Plots](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3-Creating-Bode-Plots)",
     "section": "5.3 Creating Bode Plots"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 5.3 Creating Bode Plots\n",
    "\n",
    "This notebook demonstrate uss of the [Python Control Systems Library](https://github.com/python-control/python-control) to create and annotate Bode plots. Documentation of the control systems library is available [here](http://python-control.readthedocs.io/en/latest/)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[5.3.1 Initializations](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.1-Initializations)",
     "section": "5.3.1 Initializations"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 5.3.1 Initializations\n",
    "\n",
    "These are the standard initializations with the [Python Control Systems Library](https://github.com/python-control/python-control). \n",
    "\n",
    "The control library is imported with full prefix `control`.  This is good practice to avoid name conflicts with other libraries.\n",
    "\n",
    "The control library has a bug where it continues to make use of the deprecated `hold` command from matplotlib. This results in warnings being issued. Use the warnings library to turn off these distracting warnings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "nbpages": {
     "level": 2,
     "link": "[5.3.1 Initializations](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.1-Initializations)",
     "section": "5.3.1 Initializations"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import control\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[5.3.2 Creating Bode Plots](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2-Creating-Bode-Plots)",
     "section": "5.3.2 Creating Bode Plots"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## 5.3.2 Creating Bode Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.1 Specify a Transfer Function](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.1-Specify-a-Transfer-Function)",
     "section": "5.3.2.1 Specify a Transfer Function"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 5.3.2.1 Specify a Transfer Function\n",
    "\n",
    "Given a transfer function with time delay\n",
    "\n",
    "$$G(s) = 0.2 \\frac{0.5s + 1}{1.5s^2 + 0.5 s + 1}$$\n",
    "\n",
    "the first task to create an object representing the transfer function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.1 Specify a Transfer Function](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.1-Specify-a-Transfer-Function)",
     "section": "5.3.2.1 Specify a Transfer Function"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "    0.1 s + 0.2\n",
      "-------------------\n",
      "1.5 s^2 + 0.5 s + 1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# requires coefficients of the numerator and denominator polynomials\n",
    "# the coefficients are given starting with the highest power of s\n",
    "\n",
    "G = 0.2*control.tf([0.5,1],[1.5,0.5,1])\n",
    "print(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.2 Adding Time Delay](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.2-Adding-Time-Delay)",
     "section": "5.3.2.2 Adding Time Delay"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 5.3.2.2 Adding Time Delay\n",
    "\n",
    "Time delay is a common feature of process control applications. The current version of the Python Control Systems Library does not provide a specific representation for time delay. However, the library does provide a function `pade` to create high-order Pade approximations to time delay.\n",
    "\n",
    "$$G_p(s) = e^{-0.25s} G(s)$$\n",
    "\n",
    "Here we add a third order pade approximation for a time delay of 0.25 time units."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.2 Adding Time Delay](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.2-Adding-Time-Delay)",
     "section": "5.3.2.2 Adding Time Delay"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        -0.1 s^4 + 4.6 s^3 - 86.4 s^2 + 576 s + 1536\n",
      "-------------------------------------------------------------\n",
      "1.5 s^5 + 72.5 s^4 + 1465 s^3 + 1.205e+04 s^2 + 4800 s + 7680\n",
      "\n"
     ]
    }
   ],
   "source": [
    "(num,den) = control.pade(0.25,3)\n",
    "Gp = control.tf(num,den)*G\n",
    "print(Gp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.3 Bode Plot using Default Options](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.3-Bode-Plot-using-Default-Options)",
     "section": "5.3.2.3 Bode Plot using Default Options"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 5.3.2.3 Bode Plot using Default Options\n",
    "\n",
    "Function `control.bode()` returns values of the magnitude ratio, phase lag, and frequency (in rad/time) for a given transfer function. It also creates a bode plot as a side effect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.3 Bode Plot using Default Options](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.3-Bode-Plot-using-Default-Options)",
     "section": "5.3.2.3 Bode Plot using Default Options"
    }
   },
   "outputs": [
    {
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WcNT3b/0xCfhuiHX0Alb5pV3AJqA3kA6sAAaG6lMoh71s2HS6NWtOtZcNg+VFonnmzJm6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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1061abd68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mag,phase,omega = control.bode(Gp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.4 Specify Frequency Range](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.4-Specify-Frequency-Range)",
     "section": "5.3.2.4 Specify Frequency Range"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 5.3.2.4 Specify Frequency Range\n",
    "\n",
    "The default frequency range created by `bode` is often too wide. Use `numpy.logspace()` to Fortunately, it is possible to specify a desired set of frequencies at which to evaluate the Bode plot. The frequencies are always specified in radians per unit time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.4 Specify Frequency Range](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.4-Specify-Frequency-Range)",
     "section": "5.3.2.4 Specify Frequency Range"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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wC0Bubu7wiRMn+n2M0tJSMjIyuH/OAYrKQvfZHEwozmtfLlKkNtk45eBLUHW3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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112ab3a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = np.logspace(-1.5,1,200)\n",
    "mag,phase,omega = control.bode(Gp,w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.5 Setting Other Plotting Options](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.5-Setting-Other-Plotting-Options)",
     "section": "5.3.2.5 Setting Other Plotting Options"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 5.3.2.5 Setting Other Plotting Options\n",
    "\n",
    "Bode plots can be customized with several key options as demonstrated in this cell. Note that setting `Hz = True` only changes the x-axis of the resulting bode plot, both input and output frequencies are still specified in radians per unit time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.2.5 Setting Other Plotting Options](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.2.5-Setting-Other-Plotting-Options)",
     "section": "5.3.2.5 Setting Other Plotting Options"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HZ3qH5aEnL7e/G2OMqWXheuipstvfVwXyQ4pKSjlYUMreI4WIgAAOEfd62aun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\njDFeWaIwxhjjlSUKY4wxXv0/6X+tvmG39hYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11336d550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mag,phase,omega = control.bode(Gp,w,Hz=True,dB=True,deg=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 2,
     "link": "[5.3.3 Adding Features to the Bode Plot](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.3-Adding-Features-to-the-Bode-Plot)",
     "section": "5.3.3 Adding Features to the Bode Plot"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 5.3.3 Adding Features to the Bode Plot\n",
    "\n",
    "In addition to creating plots, the `bode` function returns numpy arrays containing the magnitude, phase, and frequency.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.3.1 Crossover Frequency and Gain at Crossover](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.3.1-Crossover-Frequency-and-Gain-at-Crossover)",
     "section": "5.3.3.1 Crossover Frequency and Gain at Crossover"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 5.3.3.1 Crossover Frequency and Gain at Crossover\n",
    "\n",
    "This data can be used to annotate or add features to a Bode plot. The following cell interpolates the phase data to find the crossover frequency, then interpolates the magnitude data to find the gain at crossover."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.3.1 Crossover Frequency and Gain at Crossover](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.3.1-Crossover-Frequency-and-Gain-at-Crossover)",
     "section": "5.3.3.1 Crossover Frequency and Gain at Crossover"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Crossover freq =  5.042527421277537  rad/sec\n",
      "Gain at crossover =  0.014611811647525856\n"
     ]
    },
    {
     "data": {
      "image/png": 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t0kSLaGmLNpu5McaYqNRiz6CMMcZEN0tQxhhjopIlKGOMMVHJEpQxxpioZAnK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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1124855c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = np.logspace(-1,1)\n",
    "mag,phase,omega = control.bode(Gp,w);\n",
    "plt.tight_layout()\n",
    "\n",
    "# find the cross-over frequency and gain at cross-over\n",
    "wc = np.interp(-180.0,np.flipud(phase),np.flipud(omega))\n",
    "Kcu = np.interp(wc,omega,mag)\n",
    "\n",
    "print('Crossover freq = ', wc, ' rad/sec')\n",
    "print('Gain at crossover = ', Kcu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.3.1 Crossover Frequency and Gain at Crossover](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.3.1-Crossover-Frequency-and-Gain-at-Crossover)",
     "section": "5.3.3.1 Crossover Frequency and Gain at Crossover"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x112a72d68>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wI0g55eSIjb9gxvcPzj2M+o3132R5lAnaQInI08D+wNvAraq6MGpaGYZhxCkd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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1130e83c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mag,phase,omega = control.bode(Gp,w);\n",
    "plt.tight_layout()\n",
    "\n",
    "ax1,ax2 = plt.gcf().axes     # get subplot axes\n",
    "\n",
    "plt.sca(ax1)                 # magnitude plot\n",
    "plt.plot(plt.xlim(),[Kcu,Kcu],'r--')\n",
    "plt.plot([wc,wc],plt.ylim(),'r--')\n",
    "plt.title(\"Gain at Crossover = {0:.3g}\".format(Kcu))\n",
    "\n",
    "plt.sca(ax2)                 # phase plot\n",
    "plt.plot(plt.xlim(),[-180,-180],'r--')\n",
    "plt.plot([wc,wc],plt.ylim(),'r--')\n",
    "plt.title(\"Crossover Frequency = {0:.3g} rad/sec\".format(wc))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "nbpages": {
     "level": 3,
     "link": "[5.3.3.1 Crossover Frequency and Gain at Crossover](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.3.1-Crossover-Frequency-and-Gain-at-Crossover)",
     "section": "5.3.3.1 Crossover Frequency and Gain at Crossover"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpages": {
     "level": 3,
     "link": "[5.3.3.1 Crossover Frequency and Gain at Crossover](https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.html#5.3.3.1-Crossover-Frequency-and-Gain-at-Crossover)",
     "section": "5.3.3.1 Crossover Frequency and Gain at Crossover"
    }
   },
   "source": [
    "<!--NAVIGATION-->\n",
    "< [5.2 Closed-Loop Transfer Functions for Car Cruise Control](https://jckantor.github.io/CBE30338/05.02-Closed--Loop-Transfer-Functions-for-Car-Cruise-Control.html) | [Contents](toc.html) | [Tag Index](tag_index.html) | [5.4 Controller Tuning Rules in Frequency Domain](https://jckantor.github.io/CBE30338/05.04-Controller-Tuning-Rules-in-Frequency-Domain.html) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE30338/blob/master/docs/05.03-Creating-Bode-Plots.ipynb\"> <img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://jckantor.github.io/CBE30338/05.03-Creating-Bode-Plots.ipynb\"> <img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": "3.7.4"
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 },
 "nbformat": 4,
 "nbformat_minor": 2
}