Inverted Pendulum.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Setup"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy.physics.mechanics as me\n",
      "import sympy\n",
      "import control\n",
      "import matplotlib.animation\n",
      "import matplotlib.pyplot as plt\n",
      "import scipy.integrate\n",
      "import itertools\n",
      "from mpl_toolkits.mplot3d.axes3d import Axes3D\n",
      "\n",
      "# video animation\n",
      "from tempfile import NamedTemporaryFile\n",
      "from IPython.display import HTML\n",
      "\n",
      "%matplotlib inline\n",
      "import numpy as np\n",
      "sympy.init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#plt.rcParams.keys()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.rcParams['lines.linewidth'] = 2\n",
      "plt.rcParams['font.size'] = 12\n",
      "plt.rcParams['animation.writer'] = 'avconv'\n",
      "plt.rcParams['animation.codec'] = 'libx264'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def anim_to_html(anim):\n",
      "    plt.close(anim._fig)\n",
      "    if not hasattr(anim, '_encoded_video'):\n",
      "        with NamedTemporaryFile(suffix='.mp4') as f:\n",
      "            anim.save(f.name, fps=20, extra_args=['-vcodec', 'libx264'])\n",
      "            video = open(f.name, \"rb\").read()\n",
      "        anim._encoded_video = video.encode(\"base64\")\n",
      "    \n",
      "    return \"\"\"<video controls>\n",
      "    <source src=\"data:video/x-m4v;base64,{0}\" type=\"video/mp4\">\n",
      "     Your browser does not support the video tag.\n",
      "    </video>\"\"\".format(anim._encoded_video)\n",
      "matplotlib.animation.Animation._repr_html_ = anim_to_html"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Definition of the System"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we will define the variables and parameters for the system."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta, t, m, g, l = sympy.symbols('theta, t, m, g, l')\n",
      "values = {'g': 9.8, 'l': 0.1, 'm': 1}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we will define the inertial reference frame (i), the body reference frame (b)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "frame_i = me.ReferenceFrame('i') # the inertial frame\n",
      "frame_b = frame_i.orientnew('b', 'Axis', (theta(t), frame_i.z))  # the body frame"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we define the base and end of the pendulum in terms of the given coordinate frames."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "point_o = me.Point('o') # the base of the pendulum\n",
      "point_o.set_vel(frame_i, 0) # the base of the pendulum is fixed in the inertial frame\n",
      "point_p = point_o.locatenew('p', l*frame_b.y) # the end of the pendulum\n",
      "point_p.set_vel(frame_b, 0) # point p is fixed in the body frame\n",
      "point_p.v1pt_theory(point_o, frame_i, frame_b)\n",
      "particle_a = me.Particle('a', point_p, m) # define a point mass particle at the end of the pendulum"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For this simple problem we assume that the only force acting on the pendulum is the weight due to gravity."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weight = -m*g*frame_i.y\n",
      "weight"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  g m\\mathbf{\\hat{i}_y}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD0AAAAcBAMAAAAgvKpJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMlSJdrsime9m\nq0Tz+RmlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFa2PMUvDQBiG35gmJppein8gEdwN\nCi4uWTonu0MzWRehICIISjZBUQPi4JSgP0DroKP+g3ayDgr9CVZ0CEJioiSXtEdc/Jb73ue5u+8O\nYBfHxjk9m89bVkNuVz0Wz5jWEW+ynrUm4xss/m/syK68Soj1Si/H/UqPZrX+y6qT9wvtbZdsXuwM\nSPsE8uLE/A34gSgdQNvHGnA97oURVLvJ6/Ad9Ap+bjmtBcy6sLyGNkSrgeeCz15qPWILaAFXwD3D\n9zwcApfACkjIJfOVh27ta5gdT09+AnfgIszo5+n7fAPdXEPz6iHkCMobphwn9fVICagng3UDxMB0\nH8KTqcYxau+8Rz0gFXb/8NPXgjaR/K1cVkizOMIxTb+d5FIivOzaNKWd3OHNMikn8WOvDMaSvBQk\n5BvIvk4FTXpTjQAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        "- g\u22c5m \u001b[94m\u001b[1mi_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can then calculate the moment acting on the pendulum."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u = sympy.Symbol('u') # input\n",
      "d = sympy.Symbol('d') # disturbance\n",
      "M_o = point_p.pos_from(point_o).cross(weight).dot(frame_b.z) + u(t) + d(t)\n",
      "sympy.Eq(sympy.Symbol('M_o'), M_o)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$M_{o} = g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + d{\\left (t \\right )} + u{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASUAAAAVBAMAAADyROVdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5jajA1AhP5Q0av8Dl7hFGKg7KgoAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": [
        "M\u2092 = g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + d(t) + u(t)"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Deriving Equations of Motion"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Direct Application of Euler's 2nd Law of Motion"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "H__i_o = particle_a.angular_momentum(point_o, frame_i).dot(frame_i.z)\n",
      "sympy.Eq(sympy.Symbol('H__i_o'), H__i_o)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$H^{i}_{o} = l^{2} m \\frac{d}{d t} \\theta{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 10,
       "text": [
        "       2   d       \n",
        "H\u2071\u2092 = l \u22c5m\u22c5\u2500\u2500(\u03b8(t))\n",
        "           dt      "
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can then use Euler's 2nd law of rotational motion."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eom = M_o - H__i_o.diff(t)\n",
      "sympy.Eq(eom, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - l^{2} m \\frac{d^{2}}{d t^{2}}  \\theta{\\left (t \\right )} + d{\\left (t \\right )} + u{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 11,
       "text": [
        "                         2                        \n",
        "                   2    d                         \n",
        "g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) - l \u22c5m\u22c5\u2500\u2500\u2500(\u03b8(t)) + d(t) + u(t) = 0\n",
        "                         2                        \n",
        "                       dt                         "
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This can be simplified by solving for the second derivative of $\\theta$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eom_theta_dd = sympy.solve(eom, theta(t).diff(t,2))[0]\n",
      "sympy.Eq(theta(t).diff(t,2), eom_theta_dd)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{d^{2}}{d t^{2}}  \\theta{\\left (t \\right )} = \\frac{1}{l^{2} m} \\left(g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + d{\\left (t \\right )} + u{\\left (t \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYMAAAAvBAMAAADz47sAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGiLdUtD/9UYFwsHQvBIAAAAASUVORK5CYII=\n",
       "prompt_number": 12,
       "text": [
        "  2                                      \n",
        " d          g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + d(t) + u(t)\n",
        "\u2500\u2500\u2500(\u03b8(t)) = \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "  2                       2              \n",
        "dt                       l \u22c5m            "
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "At equilibrium, if we set the derivatives of theta to zero, we get the equilibrium input."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eom_equil = eom.subs({theta(t).diff(t,2):0})\n",
      "sympy.Eq(eom_equil, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + d{\\left (t \\right )} + u{\\left (t \\right )} = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARUAAAAVBAMAAACat2dlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7FYIviVOid0R7SgR4+HRuwCaGy843EJnAAAAAElFTkSuQmCC\n",
       "prompt_number": 13,
       "text": [
        "g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + d(t) + u(t) = 0"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u_eq = sympy.solve(eom.subs({theta(t).diff(t,2):0}), u(t), theta(t))[0][u(t)]\n",
      "u_eq"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - d{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAVBAMAAAAKgUmDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZcU/2F1Ha4EYH+e9Q9V5R/Ebyk3XDcFS5lIAAAAASUVORK5CYII=\n",
       "prompt_number": 14,
       "text": [
        "-g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) - d(t)"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_vect = sympy.Matrix([theta(t), theta(t).diff(t)])\n",
      "u_vect = sympy.Matrix([u(t), d(t)])\n",
      "y_vect = x_vect"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_vect = sympy.Matrix([theta(t).diff(t), eom_theta_dd])\n",
      "f_vect"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{d}{d t} \\theta{\\left (t \\right )}\\\\\\frac{1}{l^{2} m} \\left(g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + d{\\left (t \\right )} + u{\\left (t \\right )}\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAS8AAAAzBAMAAAA0k5O/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCZzXbdqyK7\nZkQR6/c4AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGlElEQVRYCe1Zb6hTZRj/7c/Z3dx2NlJCkHJI\n6U0iZkgZBE40EiF2CjK1YvsQppY0iLwfrFgZt6uJ3iwQkmiIwhClSeQH/+D8UMo1c0RWUODsm0Lc\n5fWSefWu53nPe7ZzdrbpPTugH3rgPe973uf3PO/vvOc9787vDHDB3kq5kKQ1xez61dauKZ9nphzR\nPcBbr8cxfdkz3VF34H3yDjBTgSjLlsYxYyoRHbBzO/Q77/a7QWzl0Ns6g1crVD/H7WiKjz2YG8SU\nz/3jgoL63RNAuCza+3ogxaFuEAtkKQvbEi2moU8DVpfQJ3qcH9wg5kkGC4LBl4gl8SE1FwPBhOhy\nfHCDWKYUuUDThGgcsQpmUet1QNEn8a4S86TSYsGHskgn1WHisouKfB6cMnNjxtTioo08/mgVO6ve\nLNSDN4tAjzuIG8SMSfn6wIHZiFQAX426fjW6ndVuEtsN/INQEgjkiMxTzggZUb0Ti9V1A67CPwZP\nFQglKH3aGMJZ3TuxxrjKGAJxQSxTpc57gVjgGrNTJhHLi1s5CgW4zH3OzZUZU2roSwEP4hGIxT8P\nC++Nxe/LIaMB68K0uXoLwAYlD6x1Plsc6cqM0d7Fz+DKQWLHG+zxITp7n9M7NxeIKes+Th7Z/ZrB\nYY/eCMeNDme1C8RWp9Ka6QfoF52IN++MkBHlArG1WAJMGAkRKYnmS40OZw0XiE3HNqjNGydfFM86\n49OI6p2YMoFzLwYLLzdSnuBWVJ+3RueUG70Twwz/tUogn5jy0N0DLMT85e7g9t4rWwYrajHV3um4\n10zMv2HYcR7XA83E4Puf2O0nuMuMiff422ZgVIvO/blDkEjYUQn7KxzGsSK+MzG5HyH4VYojTOaf\n2TxhVKvOnZZq+mVLPZW9nRI+LqAcK+I7E2PhKuyirBuV3/R1iFE2nZtrQBsNEpwM7aKE36E3Ovqd\n5VgunYmxcBVGIrGzMcqmc43Q5qaLUc2mhE1eSsIC2VeDQHF8Z2KzyCvMdOOMrmZNKLvOfV76FzVx\n9ArJCc1K2OQlj6ekixiO5WIlFqceaUK4cjt8zegy6vAyoyXevuw6ty+pA0xDbxJQixI2eQnOb+Kh\nBMCxXMzEwutv5MmtG7380dz2bylHa1DnHxsqqv0/Uo9/y1Du+A08e//ybXk6ZdSoTeeKWPIaQ4f3\nfjTGUKsSll5lbh6HNdbHV86P5EVGjjcTo1RNY+GKR3EyO60Mb+QMMjuwgHpeAYYxArybD0/QKaPs\nOjea0/MYxC6VIuMC6quRw1DC0hvwxbETOEce+hYDjuXSkRgLV984fb7x5LE0lMPJBPjxvKQhwcRO\nAbfolFF2nWv8ghhz8gBLYJsSlt4T5PsB+Iyy8WPGsVwkMSlaZVUjAAtXTxnpUjoFLVPFYg0HqTt0\n8xMwsdPAdYmy61w/JyCTQwfGefVwQl5EDcEpvRpd+UyWWPq3GI4VpfGp0z/H8m7HedIVbMUHFEFz\nvB4cDmVePcvE6Bqv0ymh2uhcQWzewMCnAwObGFRGptqihJte2kn89HwRMRKmkhST04kp57WIVaLy\nzF8s4XuMkHilOd4FdYJaxxCdNBEjVBudqw7TCGSL9CqToCvTOKFFCUsvXXKQiNCtVGvQxMPL8cYa\nu0zfJlkbNoyXNU3UGG55gft4+EDuG+Bx4JCJGKPsOrdl8WeSOO0rMdSihA1iI4iUL/Di9+Z8Kdvi\nn0uMLN9nvAUgU5o2gclVCE8iSmI7kQAWaDjL91HeSkbZdW4gC2Fy6L589AuvXQkbxN7EpWwC82l/\nLayiA8VyMWbsPbrFm/V0+lEdprktzhnGYIKnN1iAb28K+PO3n6or6vtX1Pc9/XeBEISy69y+KoTJ\noZXiQ0d3CKhFCRvE1vQvLKZ4HYXP5GlzrerFIEab0hoN0Y3LB4PbU2Dbw4dIlo9dTKDYLxtC5x6W\nAcbQ+qkZwT1Wb6ikoziWiyRGlfLGC4iOVjz7aXLYSLim+GnqblLeCjgjvXk69HOL7C+9ksdWJWz1\nRrM6jGO5SGLBMkL1f/HtVsSqgYSARErecV7m3U3KW5pb/YpZ5yq5tjEmRDu/vllxrIiXxCJ5HVsk\nVc3PNVm47Ds0qDd1X9tjuCy7ZYPTN8haI0wIq0M/018UOVbES2Ie+cIwA9tx0q8jhXBtl8LS10CJ\nRpTy4ncLoHnSRDT7mq1whdscK+KJGP8tGAsKRLiGIv6gjetum/hbkP9IDTwmqKgFbMbRh+82LXqG\n6I/U/wDPNMmXsSpR3QAAAABJRU5ErkJggg==\n",
       "prompt_number": 16,
       "text": [
        "\u23a1          d                  \u23a4\n",
        "\u23a2          \u2500\u2500(\u03b8(t))           \u23a5\n",
        "\u23a2          dt                 \u23a5\n",
        "\u23a2                             \u23a5\n",
        "\u23a2g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + d(t) + u(t)\u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2              2              \u23a5\n",
        "\u23a3             l \u22c5m            \u23a6"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = f_vect.jacobian(x_vect)\n",
      "B = f_vect.jacobian(u_vect)\n",
      "C = y_vect.jacobian(x_vect)\n",
      "D = y_vect.jacobian(u_vect)\n",
      "(A,B,C,D)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\left[\\begin{matrix}0 & 1\\\\\\frac{g}{l} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0 & 0\\\\\\frac{1}{l^{2} m} & \\frac{1}{l^{2} m}\\end{matrix}\\right], & \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right], & \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfMAAAAyBAMAAAC64i/dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nS38UDn25efavAHTt/wMkJUXkmqPVrAAAAABJRU5ErkJggg==\n",
       "prompt_number": 17,
       "text": [
        "\u239b\u23a1     0       1\u23a4, \u23a1 0     0  \u23a4, \u23a11  0\u23a4, \u23a10  0\u23a4\u239e\n",
        "\u239c\u23a2              \u23a5  \u23a2          \u23a5  \u23a2    \u23a5  \u23a2    \u23a5\u239f\n",
        "\u239c\u23a2g\u22c5cos(\u03b8(t))   \u23a5  \u23a2 1     1  \u23a5  \u23a30  1\u23a6  \u23a30  0\u23a6\u239f\n",
        "\u239c\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  0\u23a5  \u23a2\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u23a5                \u239f\n",
        "\u239c\u23a3     l        \u23a6  \u23a2 2     2  \u23a5                \u239f\n",
        "\u239d                  \u23a3l \u22c5m  l \u22c5m\u23a6                \u23a0"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Kane's Method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Kane's method is a systematic way of deriving the equation of motion for a complicated possibly containing complicated constraints. It is overkill to apply it to this problem, but we will use it to introduce the method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Q = sympy.Symbol('Q')\n",
      "force_moment_list = [(point_p, weight), (frame_b, u(t)*frame_b.z)]\n",
      "body_list = [particle_a]\n",
      "q_ind = [theta(t)] # generalized coordinates\n",
      "u_ind = [Q(t)] # generalized speeds\n",
      "kd_eqs = [Q(t) - theta(t).diff(t)] # kinematic differential equations\n",
      "km = me.KanesMethod(frame_i, q_ind, u_ind, kd_eqs=kd_eqs)\n",
      "km.kanes_equations(force_moment_list, body_list)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\left[\\begin{matrix}g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + u{\\left (t \\right )}\\end{matrix}\\right], & \\left[\\begin{matrix}- l^{2} m \\frac{d}{d t} Q{\\left (t \\right )}\\end{matrix}\\right]\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAAbBAMAAACXc8khAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMkS7zRCZdiKJ71Rm\nq90icBAQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGWklEQVRYCdVYbYhUVRh+7sze2TsfOzNFEFi2\n049+KOw6ZH5QSVNuQUhyoTASaeeHJVLpWOBmsjrgj+wDdgJx/6QOVlr0owklUTEnqKwgXLFQKXEN\nC6XYZtfMtTW25z3n3jtzZ3dnhAzshXPuOe955j3Pec4nA9wItnWuXU9jdbLeM6F+MxAcOj/Bfb0c\nnUOFiaH8HYbzbZl6THe9w6mrcDmpGGkgPgXquritgYlh/B3G88ZoPWZvvcOtS7gOqbTY10z8dsKX\nlZkZkuEzJ0l5ggkWsSSz5sQjKWukPsDWeodbl3DhMmsrmDwBNgy77c7X2F11WGkgMOcTOjYpZxtp\nSZrMBEs7xtScOBC9pOBetqRnkVeWQlUpHe5nhpWfeMRjFR+ev7hSdcjcTLMTzB8GTO6KDNskTWKC\nfTqPFjY1Iy5ixXO+GOZZwxnJayu3ldlUVUqHW0vVs/R7xEMZ1qaydWz4DYkSWgvUqAKIQ5KYmVcf\nNxP3YShkM+IiVt3CCGWNoooUfhD4jqUapVS4PiBSot8jHskp/OTZ+1y0RSTKiOQBGeJS4iSJRf2/\nJBbLOZxic8UZKZaFzyKl1gFxmAuZLc36lFLE42W022zziPclBV9j1k1eJTBIYBbtJXTSFy9wX5d0\nEoifuGBxgYlLtZniFGsG7iS2at354BfCq2WQWVvFp5QKF0rjG4Er4vt79uAoMOPkq725GS+luL7X\n92Q2/Y1bdt3amyNIFtWsFDamZGI7HliQUx611NjqEn9qBaz3pCVw/OIJtQaaEe9LGosOzmGIqkWS\n7epU2pGiLzzsU0qFay3KfGri4bT1J44Dhdk583IqWgQeBwaxAFiYs+SYDZaBM6dPfwTczxoXMGIZ\nnVj0iGcPoLWisLIL8HlzxY8iMj5eQWDb5p4Tgf67JVjgxL7n+bH+kEpoxKeUIh64hNnSJooftjGG\n3TBK52BdgeyYmTYKQvxe4KqASsA5gMF+ZU0GHB3UiUWXuFVejmBaYUMZuj9tTpxiiYWD89D9Ig7o\nmnZJtwgO+5RSxI1hPCZtQvwdGKOsm+aInKptaXovvgwhfg9wmYBICrgCYwzYyZosYIOjkyTmLBWT\nY+eiFazsArQ3J+7cFLfFM+gryGL1LK7u00TRp5QizgvrF4GRuHEVrYNcOnLOhQalc5hbxrNCnJPi\nEDfHECKExDm8GuLhrq7F73Z1ZenjJLYnFfHuFKtTEzcXD9HOJSmWMpt4zjr1H9dWQUKJcjjnU0oT\nH/MUj40imA0O2HLOUatZeRubERupIc6lQroJjohLJVCBDTk71PnBnt3NyUFvpBIlbmSYkG2lelLc\nqplMsWMiljLumlXghFUtXpTykF8pV3F3qVDoRClRLiBSAMe+3CphP/BjDXHZnDuxnaG4OcOZaHLS\nzRkcwOyYLdgt+PhaNqeIpYy75gICoxwtoB+1PAh5h3FGapVSxPkoU1uDApijOJNP3JVDX0rW2aNh\n4ICN+bJOnKUSHuDDxpKbZRu3+sCTzLI60VVVPB3YFYZgXzBzwLNNFRexlH0gExrKvCGVboB3ZmDE\n6k9OS/qVUsS5IF8RHImjY912tK5Uk9sLHHqLJ/qRr1PTx7+fPn5sz18kEhgElqy1Ce0km3k53g8p\nnejyiJun1k9fo7Cbeuh+yCFuqpNNgMqkQ8dELDHut1gFLYWCVPbClMfKhiV4+yS/tUop4rEiZgkT\nHWcuS43svNsYz+vS6/xIEnPXuK7BwVpFV/HfpR/PaoiLWGLUhfd89FRSKlvVUQvjeP9PMPxKKeK8\nOSMp4hjH0AeF/Goq+9JtiGV1qZ8fSWKBsvq4mYMN51zigbzbJN8a4rVup8xHrbXqETUC3CfP/lql\nFPGWHEIFHWeHzW3V2IJ5t32+KpgZbg2myczBPsE21RNiPlRD4upRK+8gsQXRDK/nrBS1Sipctw2j\nSAfjHDrynDQ1MivtturnsZBzCLoN3tfBygg18WVekxQaEleP2kRe/2D1eJKFqlI63Lf0/cDUMI4O\nIPmbbtEqS+krJ0l5gilsLE+/Jl72IRp2qB61Mx38HR9KoaqUDneWPpnNhnHkh//GNHF/hIYdqkdt\nr5mv+UlVKUU8VmBbNHujEVeP2hXhGt61RdFBT8BBEv9P/1dhT/XWsEP1qO1YU/8bXe8cYrhnJm/7\nX3j/AVvdxobcPacaAAAAAElFTkSuQmCC\n",
       "prompt_number": 18,
       "text": [
        "\u239b                          \u23a1  2   d       \u23a4\u239e\n",
        "\u239c[g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + u(t)], \u23a2-l \u22c5m\u22c5\u2500\u2500(Q(t))\u23a5\u239f\n",
        "\u239d                          \u23a3      dt      \u23a6\u23a0"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "km_mass = km.mass_matrix_full\n",
      "km_mass_inv= km_mass.inv()\n",
      "\n",
      "km_forcing_full = km.forcing_full\n",
      "km_linear = km.linearize()\n",
      "km_rhs = km.rhs()\n",
      "(km_mass, km_mass_inv, km_forcing_full)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\left[\\begin{matrix}1 & 0\\\\0 & l^{2} m\\end{matrix}\\right], & \\left[\\begin{matrix}1 & 0\\\\0 & \\frac{1}{l^{2} m}\\end{matrix}\\right], & \\left[\\begin{matrix}Q{\\left (t \\right )}\\\\g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + u{\\left (t \\right )}\\end{matrix}\\right]\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAckAAAAyBAMAAADFMz0sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiUSZq1TvELvdZiIy\nds1Wk1T5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAKRUlEQVRoBd1afYhcVxU/M9m9b2bezM5o+oeK\nSSYQ8ANhJxgVoU02IDVqw65Qg0JrRpMSwYROqJZWm+xogpZqyJbaKBTMYEWKrdkRMQY/2Cs2VUpk\npzZKK10zfkTxnzjdJI0mxvV37sf7mHnvzUcFowfmnHPvOee++7v3vvvuO2+IQKkykVhZKUFNpHUr\nr0TZ7155Oao6WDd6ZLCVofTUykqV6FN+zGqo4o4zNb8mWnvgzI4ow11nfh5VHawbPTLYylC6c+Z0\nlSjdtkGpKWhCgvWlZ6I89kZVdteNHhlo6VmlPxWoSVLzVVhPWI8dPIlCgtFfmHXRPWtu9mri+zoR\nsRIGi/Qaj1fc1a95P6xuXbmMSSX6MoUyY+5D5+PsLyTYqStg3XSe7m/auliU7tEIlANF2qaT5FpJ\nY20iHknnMSKog5BC6dylXTMVlkIS3bcpAmWuSqk2GYpD6Rya7kU5UKRtOUkerAPdLqLn4DTe0TLJ\n39gUSjJbyU7VPyFhK0SgzEzRuFcdh5LoXC/KASNNnxLEX9k2TfRJiFVtonu53J80ymxJeT6iuJAQ\nUSiLU5S+rFzAhkI5YKRtO1amltm0TqbnILjTEzOxvkGDRjl+jetyipOQ0KNQzlYofR02RUOhHDDS\nth0rF9ps2lJOVYg2HD3cIlYGII2S1Cxm6ypASIgolAslyntP/aFQDhipLp/E1jXYelRmWO7EL9cG\n608G5XwNrotl5S8kxI2J8niTe/gKZWcgNuM3PscVfcmgnG3B853aW0jIKJSzpZFX7GCR+voJfIVt\n49dJlCEfxi/fAetPBuVEG667tbuQkFEoixUaH3H3GSxSXz+BH2Nbdk6hdHgnGg5lARtP/qpuX0jI\nKJS4b3OjPUkGjNTXT+DTbDsn1YpNd6hGarNNCDAmM5cu9k5GyiQkWBTKQlU9o2AFDbX7DBipW07g\nk+XvvpTHYZl3n1R7XA65+9AnEFrV7QsJGYWSHqV72aZoKJQDRtq2Y2Xhq5XUN5tAOIVDwdQBsEqs\nb9Bg5pK+gDmsa4OQiL71H3wo7qL9S7d5NbEoN939lYbnZZWBIq1zgnz6cy+WSaqF6h5p4VRQTnD2\nTRbluibh0a1ISC2TeSzK5DBYR4/kpjMzOUn0GKugk1r04xblfJkmMTZMQoL1pdH7Onokdyoz9VHw\nx03/1hjZR1iUiy3aOaN9hewTo8yj93X0SL7wqpV/gmeaqhNOW4m+zKLEcsUOrUgYqUtxfPS+jh7J\nfckf48lw66xbsEpPYhZlcYq26PEhIZMCrG30vo4eaa8N+UOlPxmoSVI9lPX/LZRJmHptFqVok33H\nF7LXrbdmyBn5hd/CkJF+4Oiah3KOX8EVCallF199qBasGa6vP3ufHztcpB/3KjSLMlvtgzLVHGsH\nrzNkXz/ixw4Z6QeOrgVQbjFzJWRUc9lm3pxztXXIvt4gKOcIxx9FQmoZ5qLsqqSLrf3voMw3kMP7\nNvrAChFnnfkoFEnPcq222rkUc6HnpbPxeRkMfRGPYj9/xxaF0vlX0IloL4rBBLNv7Z7LqEjlXTgu\n/SilPdPyK05BffOnK8gXqzqVdb7NNwc1t65KyuqhbBMOeYqE5NwY53R9ynXw9hq4mj2NHq/5PtAY\nZTDB7Bu7UVJEpPae94O0tljya5Dxz+8hvORCsVnnCd8c1DgxfX+TlNWiLNZpZ1k7CUn0LaK1uqQ5\n5z75S5FPesWmm34NNKDMBVPTvrEHZW+kcd7sB/VoBQDONugysWKzzqwr+mzInxPTSIApq0WJE17w\ntH4YHrVAkGhRrhIo27nMheoYZWaqa2lrjx6UvZGmqX3hJkMl0VQTcJVYsVlnp2p87gz5cmIaI6as\nFiVO68E3r0tEkzOBoAVJ6+ltgQrz/vSZYJWaSxwV/dS0b93hq3oV9EZqD5cP4yH6sfSKb4H2COUv\nEyvZksk6nzf2EEqVK+EEGFstSryQ4PijSEhyLgJlSxcVn6f8+Q99kdI3/Whpe3rN51Gn+9oI+EDF\nXGKs/NS0Z/3e8e2eHhdJ9MGlD1OuQ+s/8JOzrfUvlBHxyzt/5x6dyv/55iUVj5smvUyFZb57/Kyz\nvZUMyv23kLuH89HpbX9HFFstSjxGMh3dESHJfRnZWQyVR9v4C3WHUpkjNPsr9VlF99Vz0ApQhhLM\nXWZdjIukVN29SGN1Kj3Ycq6Xx6tYbnXK8j0wcYQWmoh+iGjs0ntu7bDiZ51PoMBkUFZ2UKGjUkN8\n56rPlhbltJ/qEREo93ErRKezbYYxDzWur68CJXaCyyRa+Zmv8zBjUimNOWGUSEkVG7joo1inbSq2\nWfGzzvae1yjdxmbKYHRm9J1LbLUokd3CclcksGIxl8EVmzc3S222zLvSNvjFoQylpuFXXAlRQiQy\nbDheLUoHmVY8m8fq8D1+BGsMKAGthSK+rxZLtDjDip91XkSBSaN0aB+GSqVs+c4lthqUnKkMZp0v\nYkFgMCwVqkbD1vxuQjPxKEOpaRP109c1jcYibnyQDy7M0W/UI2IVMssMa//fLiqUdQ/l7Ax/AwBK\nP+usUK69cOHYhQt/REz6CoZKocSchFCqXOwW1IGEJPoakfp0giJTZqqmFWzND1P6mhPf12w9kJrm\noPuuUD2UHI5DiU9umQoddvgRgVmYbNbIbeKswnNpUGKhoueYESh+1pk3XCY9l4QROoeVjUmaJHSU\nrWYuM3PQJ8tgGiVGFIA8KjZKWv8YD+Gq9vfjUWLa+UHmExC2HS8lj/o4lNgrijN0NUWixFg2uzM0\nXqKxRgDlQ+hdOYW+QvGzzifMxQxK+D+Yq2UafLC5I7T7FFvwzFaUu5A8Eg72S4+Kb2I70i3LvClM\nlErxfQ0lmDkGW3r401QcSucaTTdp+QAtlHl/+30KcXso08RE2rm8CTtK62STiBUv6/x2vgzIzmU9\nvSulEtObHHSbrWYu1fI0H0GExKJfcxbco8JGrWJeCjjbbJUJKIMJZo6aaBAdRMc8ikNJG557PdHz\nJV5jZ4lufyNW5Rt++9r89KUHpi+dProbDcDi3PIurfhZ5z+Ypg1KZ+uvn35JJaZPLcHCVoPyT8px\nl+JCKtGHxfa1O67YJOe9yKA+tVRZ+jIbEyIPdQd3lbN2tKxyEg5u1XgZlKZk3jaUVaNM6xdkfXQV\n0vglioS+huPgeM8PSvSNq7S7himKR5nX22Y4OlzyztJWWQN7CstS0VuN1OJxLZRVo8TexjShhJCs\n96OBUb6Ap97KnHtgjpZJLfy4yHU17Bt96EvWrhWnjfJBWxeWGT3vyqpRqv9sYe7Vw1/IsHt0Ka6v\nPd5HdM1YC5fCAyB+Lm//zjt6grsr9MszarWikHjIw85uXZWVVaHU/9lCpTorCanMfdjAKPfq5oQs\n1N0O6wNHRvTAbZhKrTyJUk7PWa+zSkxrq0JpFizvznAWEqwvDdzXTku1tUBjrVy9BH3gyL59GNhB\noXzCc98O7T/8z9GNZdX6BppoultrdAP8c5RWoUv/r/8CrvFo/xsqBRYpVrKxuAAAAABJRU5ErkJg\ngg==\n",
       "prompt_number": 19,
       "text": [
        "\u239b\u23a11   0  \u23a4, \u23a11   0  \u23a4, \u23a1         Q(t)         \u23a4\u239e\n",
        "\u239c\u23a2       \u23a5  \u23a2       \u23a5  \u23a2                      \u23a5\u239f\n",
        "\u239c\u23a2    2  \u23a5  \u23a2    1  \u23a5  \u23a3g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + u(t)\u23a6\u239f\n",
        "\u239c\u23a30  l \u22c5m\u23a6  \u23a20  \u2500\u2500\u2500\u2500\u23a5                          \u239f\n",
        "\u239c           \u23a2    2  \u23a5                          \u239f\n",
        "\u239d           \u23a3   l \u22c5m\u23a6                          \u23a0"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(km_linear, km_rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\begin{pmatrix}\\left[\\begin{matrix}0 & 1\\\\g l m \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right], & \\left[\\begin{matrix}u{\\left (t \\right )}\\end{matrix}\\right]\\end{pmatrix}, & \\left[\\begin{matrix}Q{\\left (t \\right )}\\\\\\frac{1}{l^{2} m} \\left(g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + u{\\left (t \\right )}\\right)\\end{matrix}\\right]\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAo8AAAAyBAMAAADPdSzKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n2p1iof3ju7T1P9MQG6wzDSKHAAAAAElFTkSuQmCC\n",
       "prompt_number": 20,
       "text": [
        "\u239b\u239b\u23a1       0         1\u23a4, \u23a10\u23a4, [u(t)]\u239e, \u23a1         Q(t)         \u23a4\u239e\n",
        "\u239c\u239c\u23a2                  \u23a5  \u23a2 \u23a5        \u239f  \u23a2                      \u23a5\u239f\n",
        "\u239c\u239d\u23a3g\u22c5l\u22c5m\u22c5cos(\u03b8(t))  0\u23a6  \u23a31\u23a6        \u23a0  \u23a2g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + u(t)\u23a5\u239f\n",
        "\u239c                                     \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\u239f\n",
        "\u239c                                     \u23a2          2           \u23a5\u239f\n",
        "\u239d                                     \u23a3         l \u22c5m         \u23a6\u23a0"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "km_A = km_mass_inv*km_linear[0]\n",
      "km_B = km.mass_matrix_full.inv()*km.linearize()[1]\n",
      "km_C = sympy.zeros(2)\n",
      "km_D = sympy.zeros(2,1)\n",
      "km_x = sympy.Matrix([theta(t), Q(t)])\n",
      "km_u = sympy.Matrix([u(t)])\n",
      "km_A, km_B, km_C, km_D, km_x\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\left[\\begin{matrix}0 & 1\\\\\\frac{g}{l} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\\\frac{1}{l^{2} m}\\end{matrix}\\right], & \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right], & \\left[\\begin{matrix}0\\\\0\\end{matrix}\\right], & \\left[\\begin{matrix}\\theta{\\left (t \\right )}\\\\Q{\\left (t \\right )}\\end{matrix}\\right]\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfsAAAAyBAMAAACpNW8pAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEJ8AAAAASUVORK5CYII=\n",
       "prompt_number": 21,
       "text": [
        "\u239b\u23a1     0       1\u23a4, \u23a1 0  \u23a4, \u23a10  0\u23a4, \u23a10\u23a4, \u23a1\u03b8(t)\u23a4\u239e\n",
        "\u239c\u23a2              \u23a5  \u23a2    \u23a5  \u23a2    \u23a5  \u23a2 \u23a5  \u23a2    \u23a5\u239f\n",
        "\u239c\u23a2g\u22c5cos(\u03b8(t))   \u23a5  \u23a2 1  \u23a5  \u23a30  0\u23a6  \u23a30\u23a6  \u23a3Q(t)\u23a6\u239f\n",
        "\u239c\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  0\u23a5  \u23a2\u2500\u2500\u2500\u2500\u23a5                     \u239f\n",
        "\u239c\u23a3     l        \u23a6  \u23a2 2  \u23a5                     \u239f\n",
        "\u239d                  \u23a3l \u22c5m\u23a6                     \u23a0"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "km_forcing_full"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}Q{\\left (t \\right )}\\\\g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + u{\\left (t \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL8AAAAyBAMAAAD/6hczAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhAimatmzbvd\nRHZFqlmGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAErklEQVRYCe1YW2gcVRj+Mjsze53dxRoRpHXe\nmhbdRumDoLRTUh8EMesFFZFmoiKUPnS9oajoKiJiIFnBC/pgR1SEiBgQNCIma4X6Yun2raLQtS+i\naFxsTBNljd8/Z3a7k6zdiWzAB3/4zznzn3++79xmz8fi8tXfsFmmra7msWX/9ZuFD2P/SB4XbxBe\nr8gLL0d8S98AwT0PDgn2IR864URj2ACB9g7wIlHfBYxjgNdvAuN9It5cQtYGYg3gTL8JUlUiJhoY\ncIG4Ry4+RrDoS7SrRjhtEd+xythAajgCPBCZwPxD8OJNbAdOzc/WoZX6S6CtCF56EZ+y2ktPeix6\nW+QZZJYELJfHl6wep8eqLHpbZIIcDw6HXsdWVkfpuh+Q4AUtMkEmLzhfQwiMJpv9JuABBbKLkCWy\nGijCqjLQ2yLPwGqaBWefA9lkzYs5fd9k/HwHdjzPEQ/xsJYPsCj1Hj4zIs8A+kThOHTIh2bO1fmh\n1fpMQLiP5Ec64yrg8Uj4G5gB8WZjHj+wYGkKm0Dw9KpD1I99ZMPzq55F9D0g1I1bBE9dOGlX2r1t\nQwQKzqxI/WpvbD/jXxBERA7S/ifouV7/oSUSvWW8LiO+QYqko1za66xDm3XOIHvCWZO6r34+IMf/\n21v5GZszfvA5luLdTH0qvjbrJMCRtcmj9vkI9ZY+iN/5K1cE7naRYpd4N+vQZiECuWr/yURvZSo4\nqxQXb30JiPt2e1CrSsKxhsoMEWwLpYUfRG+9BKwAl7CDQzHyyv20PaHkTm3WSWDyQgzbfU77WfTW\nV9DPqqtSbv23A5eUMEGnNmsRXL1wLZINWEP3L0xahc/5zk17fjLny/pbDy9MCgT1ltVEtimCy5r4\nkzFGfJfegMDYXsd4MaTNAgJtxjyHxAy09BzGfsFVnP0MMkiXkZrDYZcQvIoTyy8800C6ohYYbzAq\nLhYQxGN5/OrntrVZQLC3yO0bqGMk4+GwLcfJGoQmBOk8coQUMcG+nIfMsBK/uI5RcbGA4GDcw2d+\nblubBQSXQV/CqIPiWA0km+A7J+Zcn4CodT5upayzMTqMgRqpbEZGA2fVIihyjNv83LY2UwT6CrJV\nfM9Mnr8nIUm45cNzPsFMm2BsGA/AJ+AwOgh2Tk9/MT39HkM4XdR5UjiYtjZTBMklpEuYNfzzdxTW\nkgHTxZGaLFFAwCUi7F8cPZfoNJgqZ0VcLFgiDi7bCGszRRBrIDeMFQ24SLjj3r2I2UhUOgi4yQM1\nrQp/k3dit9rgNZuMWQ7olZA2UwTGEqZcNA/AbMphTdk2YoNIuxx+await+rjLmVdGXjMqAOPQjmr\n9gyewMmSHdJmigCnzlwB/GDLV5QtI/asA2vHj1fqU8u3TS2PzF9KCC6G8dQjbDAFhxbY+CBwVm2C\nOwu7J52QNgsIgE8k7wKWcVudx1TDzPOHle5baw/UUytXtJki0NWuq+7uZUtvAa+pBK3O5aL79k1Q\nq6qVW+CjIthV5Hb2MKW3mBQoorvYFO9mKtfw2KcIrnnzoW55oZi6RBgKLhwBaZOGMkParL0Ha1LW\nPyq9JfGDUiRd5dJeZx3aLDrBOpRoARJs+t85m/2H1N/zqCcVyTMKYgAAAABJRU5ErkJggg==\n",
       "prompt_number": 22,
       "text": [
        "\u23a1         Q(t)         \u23a4\n",
        "\u23a2                      \u23a5\n",
        "\u23a3g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + u(t)\u23a6"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can also solve for the equilibrium conditions using the matrices from Kane's method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.solve(km_mass_inv*km_forcing_full, [Q(t), u(t)])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}Q{\\left (t \\right )} : 0, & u{\\left (t \\right )} : - g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAS0AAAAaBAMAAAAQxRd8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 23,
       "text": [
        "{Q(t): 0, u(t): -g\u22c5l\u22c5m\u22c5sin(\u03b8(t))}"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Lagrangian Mechanics"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L = me.Lagrangian(frame_i, particle_a)\n",
      "L"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{l^{2} m}{2} \\left(\\frac{d}{d t} \\theta{\\left (t \\right )}\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIgAAAA4BAMAAADUXq0UAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOJ+65W5yYTdSc/ld1+y+owPxLw8rJ3ihxh4fAAAAAElFTkSuQmCC\n",
       "prompt_number": 24,
       "text": [
        "               2\n",
        " 2   \u239bd       \u239e \n",
        "l \u22c5m\u22c5\u239c\u2500\u2500(\u03b8(t))\u239f \n",
        "     \u239ddt      \u23a0 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "       2        "
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lm = me.LagrangesMethod(L, [theta(t)], forcelist=force_moment_list, frame=frame_i)\n",
      "lm.form_lagranges_equations()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}- g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + l^{2} m \\frac{d^{2}}{d t^{2}}  \\theta{\\left (t \\right )} - u{\\left (t \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUgAAAAmBAMAAABUsx+eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 25,
       "text": [
        "\u23a1                          2             \u23a4\n",
        "\u23a2                    2    d              \u23a5\n",
        "\u23a2-g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + l \u22c5m\u22c5\u2500\u2500\u2500(\u03b8(t)) - u(t)\u23a5\n",
        "\u23a2                          2             \u23a5\n",
        "\u23a3                        dt              \u23a6"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For this simple problem the Lagrangian method is more direct than Kane's method, but both give the same result."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lm.rhs(), km.rhs()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\left[\\begin{matrix}\\frac{d}{d t} \\theta{\\left (t \\right )}\\\\\\frac{1}{l^{2} m} \\left(g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + u{\\left (t \\right )}\\right)\\end{matrix}\\right], & \\left[\\begin{matrix}Q{\\left (t \\right )}\\\\\\frac{1}{l^{2} m} \\left(g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} + u{\\left (t \\right )}\\right)\\end{matrix}\\right]\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAi8AAAAzBAMAAACtcfW+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nNAAAAABJRU5ErkJggg==\n",
       "prompt_number": 26,
       "text": [
        "\u239b\u23a1       d              \u23a4                          \u239e\n",
        "\u239c\u23a2       \u2500\u2500(\u03b8(t))       \u23a5, \u23a1         Q(t)         \u23a4\u239f\n",
        "\u239c\u23a2       dt             \u23a5  \u23a2                      \u23a5\u239f\n",
        "\u239c\u23a2                      \u23a5  \u23a2g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + u(t)\u23a5\u239f\n",
        "\u239c\u23a2g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) + u(t)\u23a5  \u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\u239f\n",
        "\u239c\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5  \u23a2          2           \u23a5\u239f\n",
        "\u239c\u23a2          2           \u23a5  \u23a3         l \u22c5m         \u23a6\u239f\n",
        "\u239d\u23a3         l \u22c5m         \u23a6                          \u23a0"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Linearization"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Given the jacobian, we can linearize the dynamics at the equilibrium conditions. We will linearize about $\\theta = 0$, the pendulum is inverted."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "equil = {theta(t):0}\n",
      "equil.update(values)\n",
      "ss_eq = [ np.array(np.array(mat.subs(equil)), dtype=float) for mat in [A,B,C,D]]\n",
      "sys = control.StateSpace(ss_eq[0], ss_eq[1], ss_eq[2], ss_eq[3])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we see that the system is unstable."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tout, xout, yout = control.forced_response(sys, X0=[0, 0.001], T=np.linspace(0,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(tout, yout.T);\n",
      "plt.legend([r'$\\theta$', r'$\\dot{\\theta}$'], loc='best');\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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bcOHxzRvmfDjF04nkuHSSHGkkO9JJcvjeZ9CFTJIdGSSTSbKRSRKZJBkZJBs9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\n/C8z/TOyFoD3yAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4d9b850>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Control Design"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Classical Frequency Domain Design"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.ss2tf(sys)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "\n",
        "Input 1 to output 1:\n",
        "      100\n",
        "---------------\n",
        "s^2 + -0 s - 98\n",
        "\n",
        "Input 1 to output 2:\n",
        "     100 s\n",
        "---------------\n",
        "s^2 + -0 s - 98\n",
        "\n",
        "Input 2 to output 1:\n",
        "      100\n",
        "---------------\n",
        "s^2 + -0 s - 98\n",
        "\n",
        "Input 2 to output 2:\n",
        "     100 s\n",
        "---------------\n",
        "s^2 + -0 s - 98"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Looking at the root locus plot we see that the system is marginally stable at high gains as both poles approach the real axis. The system must be stabilized with derivative gain to pull the poles to the left hand side of the complex plane."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "    control.root_locus(sys[0,0], kvect=np.linspace(0,1,1000));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x4f03110>"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The bode plot shows that the phase margin is zero and that the system is unstable and we must increase the phase margin to around 70 degress/ gain margin of 10 dB to make the system robust to disturbances."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.bode_plot(sys[0,0], omega=np.logspace(0, 2), );"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Z9X8+FjFAiHKhK7AHMAcYn3JbhPAynJYF61cAZxSrUKmxxcJyFHC/e77QTQtR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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x54a4e50>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We choose a PD type controller because we know that a zero must be added to the left hand plane from observing the root locus plot. We implement the PD controller with proportional gain from the $\\theta$ and $\\dot{\\theta}$ plant states to avoid noise due to numerical differentiation. The gains were iteratively changed until the desired bandwidth and phase/gain margin were achieved."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "kp_th = 2\n",
      "kp_thd = 0.15\n",
      "contr = [kp_th, kp_thd]\n",
      "contr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}2, & 0.15\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAZBAMAAABdgxRAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "prompt_number": 33,
       "text": [
        "[2, 0.15]"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tfm = control.ss2tf(sys)\n",
      "tfm"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "\n",
        "Input 1 to output 1:\n",
        "      100\n",
        "---------------\n",
        "s^2 + -0 s - 98\n",
        "\n",
        "Input 1 to output 2:\n",
        "     100 s\n",
        "---------------\n",
        "s^2 + -0 s - 98\n",
        "\n",
        "Input 2 to output 1:\n",
        "      100\n",
        "---------------\n",
        "s^2 + -0 s - 98\n",
        "\n",
        "Input 2 to output 2:\n",
        "     100 s\n",
        "---------------\n",
        "s^2 + -0 s - 98"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop = sys[:,0].feedback(contr)\n",
      "prescaler = 1.0/np.linalg.norm(closed_loop.horner(0))\n",
      "closed_loop *= prescaler\n",
      "open_loop = prescaler*contr*sys"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.root_locus(closed_loop[0,0], kvect=np.linspace(0,1,100));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x576d510>"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.bode_plot(open_loop[0,0], omega=np.logspace(-1, 2), );"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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8vJa7zQ8C+uwlL8vU+KnCURRFUbJCkGw4IxG7y97Igc0RMeUDkbgM7YFPqG6j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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x595f750>"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gm, pm, sm, gc, pc, sc = control.stability_margins(open_loop[0,0])\n",
      "gm, pm, sm, gc, pc, sc"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "(None,\n",
        " 45.910447258290475,\n",
        " 0.90924251168982306,\n",
        " 13.763953660529673,\n",
        " None,\n",
        " 4.2265522274077174)"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.bode_plot(closed_loop[0,0], omega=np.logspace(-1, 2), );"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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K5lSoANWrQ40a8rptW1t69oSaNaFWLbkC73Nz2zJjBhx6KNSpI6+VKpk9t217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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x54639d0>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop.damp()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "(array([ 10.09950494,  10.09950494]),\n",
        " array([ 0.74261066,  0.74261066]),\n",
        " array([-7.5+6.76387463j, -7.5-6.76387463j]))"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tout, yout, xout = control.forced_response(closed_loop, X0=[1, 1], T=np.linspace(0,1))\n",
      "plt.plot(tout, yout.T);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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teuPz+9j086bGeAnH3bn6TgpLChnfezwDkgc4XY6IhIGQOoUynE+K+nH/jzy2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\nAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4f31d90>"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Linear Quadratic Regulator Design"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "K, S, E = control.lqr(sys.A,sys.B[:,0],np.diag([1, 0]), 1*np.eye(1))\n",
      "contr_lqr = K\n",
      "contr_lqr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "array([[ 2.38014285,  0.21818079]])"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop_lqr = sys[:,0].feedback(contr_lqr)\n",
      "prescaler_lqr = 1.0/np.linalg.norm(closed_loop_lqr.horner(0))\n",
      "closed_loop_lqr *= prescaler_lqr\n",
      "open_loop_lqr = sys[:,0]*contr_lqr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.bode_plot(open_loop_lqr[0,0], omega=np.logspace(-1, 2), );"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6KJBl2mhEwUjTSiNquzTTSA20X26P5XvA51hMqA2x4a61wNGufA02eyHLdtiN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\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4bed650>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gm, pm, sm, gc, pc, sc = control.stability_margins(open_loop_lqr[0,0])\n",
      "gm, pm, sm, gc, pc, sc"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "(None, 0.0, 0.0, 11.832763201615128, None, 11.832763201615132)"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "control.bode_plot(closed_loop_lqr[0,0], omega=np.logspace(-1, 2), );"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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d23J4Yexi1XFSFi4faOH851nIphlOsHHrKuAYxLh1UmjFJUvi79QY2LsX9uyB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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5777310>"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop_lqr.damp()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 47,
       "text": [
        "(array([ 11.8327632,  11.8327632]),\n",
        " array([ 0.92193508,  0.92193508]),\n",
        " array([-10.90903949+4.58335494j, -10.90903949-4.58335494j]))"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "tout, yout, xout = control.forced_response(closed_loop_lqr, X0=[1, 1], T=np.linspace(0,1,1000))\n",
      "plt.plot(tout, yout.T);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7nCpZvG0x/9vzP+Kj4hnVbZTd5YhIHefXIR8eEk5iw0QKPYVs2r/J7nKqpHgK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\nERERERERERERERERERGRoPP/tiiKW8oatn8AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x4dbebd0>"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Nonlinear Simulation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = sympy.lambdify((theta(t), Q(t), u(t)), km_rhs.subs(values))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u_eq.subs(values)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- d{\\left (t \\right )} - 0.98 \\sin{\\left (\\theta{\\left (t \\right )} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMAAAAAVBAMAAADiAfyoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYsD7lMwAAAAASUVORK5CYII=\n",
       "prompt_number": 50,
       "text": [
        "-d(t) - 0.98\u22c5sin(\u03b8(t))"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "feed_forward = sympy.lambdify(theta(t), (u_eq).subs(values).subs(d(t),0))\n",
      "feed_forward(1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-0.824641565112$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAAPBAMAAAABuKDqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niF8jqP8nfwG7/NS5QF7FwQAAAABJRU5ErkJggg==\n",
       "prompt_number": 51,
       "text": [
        "-0.824641565112"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.array(contr).dot(np.array([1,1])) + feed_forward(1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.32535843489$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAPBAMAAAA/sQ3hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nycfkmig9/9X+AKumx2fGKD1KAAAAAElFTkSuQmCC\n",
       "prompt_number": 52,
       "text": [
        "1.32535843489"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def simulate(contr, tf=1.0, dt=1.0/1000.0, dt_cont= 1.0/50.0, x0=[1,1]):\n",
      "    data = {}\n",
      "    sim = scipy.integrate.ode(lambda t,x,u: f(x[0],x[1],u))\n",
      "    sim.set_integrator('dopri5')\n",
      "    sim.set_initial_value(x0)\n",
      "    n_t = int(tf/dt)\n",
      "    data['x'] = np.zeros((n_t, 2))\n",
      "    data['u'] = np.zeros((n_t, 2))\n",
      "    data['t'] = np.zeros(n_t)\n",
      "    i = 0\n",
      "    t_last_control = 0\n",
      "    u_i = contr(x0)\n",
      "    while sim.successful() and i < n_t:\n",
      "        x_i = sim.y\n",
      "        data['x'][i,:] = x_i\n",
      "        data['u'][i,:] = u_i\n",
      "        data['t'][i] = sim.t\n",
      "        if (sim.t - t_last_control) >= dt_cont:\n",
      "            t_last_control = sim.t\n",
      "            u_i = contr(x_i)\n",
      "        sim.set_f_params(u_i)\n",
      "        sim.integrate(sim.t + dt)\n",
      "        i += 1\n",
      "    return data"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = sympy.DeferredVector('x')\n",
      "cont_lqr_f = lambda x: -np.array(contr_lqr).dot(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure(figsize=(15,5))\n",
      "h = []\n",
      "freq_list = [15, 11, 10]\n",
      "for freq in freq_list:\n",
      "    data = simulate(cont_lqr_f,  dt_cont=1.0/freq);\n",
      "    plt.subplot(121)\n",
      "    h.append(plt.plot(data['t'], np.rad2deg(data['x'][:,0])))\n",
      "    plt.subplot(122)\n",
      "    plt.plot(data['t'], data['u'])\n",
      "    \n",
      "plt.subplot(121)\n",
      "plt.legend([h_i[0] for h_i in h ], ['{:g} Hz'.format(freq) for freq in freq_list], loc='best')\n",
      "plt.title('control sampling time effect (output)')\n",
      "plt.xlabel('t, sec')\n",
      "plt.ylabel(r'$\\theta$, deg')\n",
      "\n",
      "plt.subplot(122)\n",
      "plt.xlabel('t, sec')\n",
      "plt.ylabel('u')\n",
      "plt.title('control sampling time effect (control)');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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R9/FTCCGECFk8JIwVwO3Gj7/5QH97w0lsWVlZTocQd2SfhUf2V3hkf4koBDt+igjI5zE8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\nngAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x53bd490>"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Digital Control"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "From the nonlinear simulation, it is clear that we need to account for the control sampling if the control will be sampled below 20 Hz. We do this using digital control design techniques. First, we will discretize the state space system at 10 Hz."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dt = 1.0/10\n",
      "sysd = scipy.signal.cont2discrete((sys.A, sys.B, sys.C, sys.D), dt)\n",
      "sysd = control.ss(sysd[0], sysd[1], sysd[2], sysd[3], dt)\n",
      "sysd"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 56,
       "text": [
        "A = [[  1.531347     0.1171526 ]\n",
        " [ 11.48095456   1.531347  ]]\n",
        "\n",
        "B = [[  0.54219082   0.54219082]\n",
        " [ 11.71525976  11.71525976]]\n",
        "\n",
        "C = [[ 1.  0.]\n",
        " [ 0.  1.]]\n",
        "\n",
        "D = [[ 0.  0.]\n",
        " [ 0.  0.]]\n",
        "\n",
        "dt = 0.1"
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we use the discrete algebraic ricatti equation to solve for the LQR gain for the discrete controller."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Q = np.diag([1, 0])\n",
      "R = np.eye(1)\n",
      "X, L, G1 = control.dare(sysd.A, sysd.B[:,0], Q, R)\n",
      "X = scipy.linalg.solve_discrete_are(sysd.A, sysd.B[:,0], Q, R)\n",
      "G = np.linalg.inv(R + sysd.B[:,0].T.dot(X).dot(sysd.B[:,0])).dot(sysd.B[:,0].T).dot(X).dot(sysd.A)\n",
      "contr_lqrd = G\n",
      "cont_lqrd_f = lambda x: -contr_lqrd.dot(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Why doesn't this give the same answer? It seems that python control might have a bug here, most of the number are the same except for the diagonal."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X2, L2, G2 = control.dare(sysd.A, sysd.B[:,0], Q, R)\n",
      "print X\n",
      "print X2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 3.87377252  0.25714186]\n",
        " [ 0.25714186  0.02409592]]\n",
        "[[-2.87377252  0.25714186]\n",
        " [ 0.25714186 -0.02409592]]\n"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.array(contr_lqr).dot(x[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 59,
       "text": [
        "array([[2.38014284985497*x[0], 0.218180789706838*x[0]]], dtype=object)"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we compare the new discrete and previously designed continuous LQR gains."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure(figsize=(15,5))\n",
      "\n",
      "plt.subplot(121)\n",
      "plt.title('discrete vs. continuous LQR at 10 Hz (output)')\n",
      "plt.xlabel('t, sec')\n",
      "plt.ylabel(r'$\\theta$, deg')\n",
      "data_lqr = simulate(cont_lqr_f, dt_cont=dt, tf=2);\n",
      "hc = plt.plot(data_lqr['t'], np.rad2deg(data_lqr['x'][:,0]));\n",
      "\n",
      "plt.subplot(122)\n",
      "plt.title('discrete vs. continuous LQR at 10 Hz (control)')\n",
      "plt.xlabel('t, sec')\n",
      "plt.ylabel('u')\n",
      "hcc = plt.plot(data_lqr['t'], data_lqr['u']);\n",
      "\n",
      "plt.subplot(121)\n",
      "data_lqrd = simulate(cont_lqrd_f, dt_cont=dt, tf=2);\n",
      "hd = plt.plot(data_lqrd['t'], np.rad2deg(data_lqrd['x'][:,0]));\n",
      "plt.legend([hd[0], hc[0]], ['discrete','continuous']);\n",
      "\n",
      "plt.subplot(122)\n",
      "hdc = plt.plot(data_lqrd['t'], data_lqrd['u'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ijFCeyN+QKoPTkKkHdsTNIkVRFCVpqCjThgSW7s3CNyG2xV58pYyrIcUe/NdVs8c0RVEU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\nIiIiIiIiIiJSrP8PLJer+uUj+iAAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x578ce10>"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Dynamic Inversion Control"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Dynamic inversion control is useful for nonlinear systems that operate around many different equilibirum condiions. Dynamic inversion control allows you to design one controller that will work for all equilbirum conditoins, but it may saturate actuators or use more control effort to follow the desired dynamics."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we will make the pendulum follow the dynamics given by the natural frequency and damping ratio of a second order systems."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s, zeta, wn, Theta = sympy.symbols('s, zeta, \\omega_n, Theta')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "char_poly = Theta*(s**2 + 2*zeta*wn*s + wn**2)\n",
      "char_poly"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Theta \\left(\\omega_{n}^{2} + 2 \\omega_{n} s \\zeta + s^{2}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALgAAAAcBAMAAAAgkGgMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARHZm3VQQq7simTLN\n74m3yPBSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADHklEQVRIDaVUTWgTQRh9yW6SbjZJIyIVBRtS\nRBRKoxdBEXJRpKIGsecuPfgDQkQU9GRO4s+hOaqgXXrwp4isJ0Go1YMWJWBUPOgpKHgotv5UY7VK\nnPlmJrubLiW1H+zMe+97b/g2Oy0QWFMXrUC9TXHReKoce9TmOUG24Pg9adXL5q+gVJsaxTMtZrMk\nhWTa+NrSWwqleFdLImE1hUi9Cf8HsHgqx4OhbFrmR91z9IyLF0XlwC6P72IdbeehgwyyMjzTTpHi\nWa7euuRhLqy50IN4fBN7TlqInic9ZdPGF82FQosex2f1gkKR64keHxWE4iNAmF+Los3FZIGvVOuw\nVqKjYo9ZSFal5N0Oj64XVPoEobieQ7HEeJwu3oQlWoD5ZOCcxAfEnswjNq/a7h59rLD0ERXxeAl9\nOcYjv7m4gzp8STYaXySRoUQ98PAET1N5DxfxcBXDa8bGxs5854Y77Cmx51qaLarckF73d9/3OHC/\ne9PHZapQHfvvNxqNp385ZZ9V4+cWM5zJaobQXfN1E06HHXXvl/JxWUTNebx4yOBHOnyS/fhcTxVE\nl1YVAnb7uyMwLc/9Uj4uizj7Iz/LTzLnOO8HVgB5aGXRpVWFwL6Pr1t8l4bujqF8XBZl/MBgjUFt\nlnM2eQamjRQnVJ8qlb2VynPCG1q6oeE5dDoLfFwWxSbX8wwmqpyzw22EC7jOiSo1UcTGFdndtj1r\nwyhE+8qdZWWD9JEsRPaP1fjG4CC9ymngAiZsPGsmGFCHPwA2yu7mWbzl83SiI9e0Sh/JQmS3BeMZ\naC+J9gLTU29+TtdAswmLmmjfzM286BpDedTR4SAL47UwsVX6SBaixn+ODTN3LaLdFo78cXpfATSb\nsKiJ2IXNy24sE6rCyG4pAEOT9M7MqoYgmaLsBngqqYxiNtGRIY8NSSduO1Iw+yVY6EtkvKF4TTGa\nTZCVSnP3Eeir1OEYl/pCX9FyI+y2VxXzzqY0d+9C+INkUWu1q7egrX5+SlHfbEoM3LO3A2Uu7vF3\njinqmU1JS961mj8Ssf18Wexya3qgVVgGv7GMbDvRf0g51RsEq2GYAAAAAElFTkSuQmCC\n",
       "prompt_number": 62,
       "text": [
        "  \u239b        2                     2\u239e\n",
        "\u0398\u22c5\u239d\\omega_n  + 2\u22c5\\omega_n\u22c5s\u22c5\u03b6 + s \u23a0"
       ]
      }
     ],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "char_poly_theta = char_poly.expand().subs({Theta*s**2: theta(t).diff(t,2), Theta*s: theta(t).diff(t), Theta: theta(t)})\n",
      "char_poly_theta"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\omega_{n}^{2} \\theta{\\left (t \\right )} + 2 \\omega_{n} \\zeta \\frac{d}{d t} \\theta{\\left (t \\right )} + \\frac{d^{2}}{d t^{2}}  \\theta{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARMAAAAvBAMAAAAySDPpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\neYgPo/bK+scxMdcD+wfnKKtLA1ExCQAAAABJRU5ErkJggg==\n",
       "prompt_number": 63,
       "text": [
        "                                           2      \n",
        "        2                     d           d       \n",
        "\\omega_n \u22c5\u03b8(t) + 2\u22c5\\omega_n\u22c5\u03b6\u22c5\u2500\u2500(\u03b8(t)) + \u2500\u2500\u2500(\u03b8(t))\n",
        "                              dt           2      \n",
        "                                         dt       "
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta_dd_des = sympy.solve(char_poly_theta, theta(t).diff(t,2))[0].expand()\n",
      "theta_dd_des"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\omega_{n}^{2} \\theta{\\left (t \\right )} - 2 \\omega_{n} \\zeta \\frac{d}{d t} \\theta{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMYAAAArBAMAAADRb+oyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMrsi70RmdpmJ\nVKuALsSwAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADsElEQVRYCa1WXYhMYRh+zvyc+d2ZsXKBi9my\niaSmFVFoyoVwsUMiuXBcsInYC7FETlHrAjNJ2tqbE0K7fk5KQmmyatOSKe6k3QtKa1u7G/K31vt9\n34zjzMx3Zpnz1nw/z/s8z3fm/HzfC7gawa2mq35VzeKFqrCrYDbhql1Vs76qqLvgInftKt3UkSuT\nlai7SKfmH3fXscJNOYJIUwXqLhAZhzc/DcuB7f//8gUySOq114hqsabaLAkjmccgal+iV/N9kTjU\nhpMFbGrQavICuvq1JklG8Bih1VFZ8m+8gb99s3OErWN4KM3aGqHyvDL8ZP91wXTWew1iBTe+BtQM\n578VKqvde3ezNRGj8qt31AMDTLU2EU/AQ49vpgZPmaEyhBd6GRbrYf/bCic93RqTMVeA9ugbNBgE\nwnmGWBFLINBuTdlI+dBTsCFOeuAMTtFC7YjncIhkd8igzDCQQuyTzRGNun3uqIdv8ZuNgNdEayGY\nIuFK+i22G3jGK9ZYaGdI9X7iqR2BqakxIKvjth41EXz3ebj4hGwu3nFkCGjTBUpMe8j0XURTvgvu\nx9HRE/DngAZaEOftBjTL5kM6dUmDGopkQvR/WpmeX8tzQWsBJuAtQGyjL/9oS4OliLBhlBgsBkRn\nteX6xiUsmoM6vMBFwfsJ3yQCOj2YPAGtDFTmMVpLmo0jGZwGUghpbAasEp3VVtGzZBjYAOQ5T5lE\npJ2vwbdRvgZPFJtbgAGfidKXJ96yqVKMyfT0529CzXMX5SviBr9XWSjASQ5aTYOJ3SbCBexD94Ue\nusflu6hM79d8P+DRhFMH+sCf+WO8qnzm9H+vbkGrifno+sa+om3W+o766LF7gyPHELq/vhd4oJIy\nOgQ8Ugzgod1CXfnsUap/4Piv/rw6JwXaQttydoZMr+xoDh82EMrmAhrm9iZoX0wBu3pIvcDu4KH7\nnpo1kb52FIgZwXZ6Hy69t1Mc9dh5GbQbilgmOpVcZBFIR8w0JdeYVRhS/TA6S/RuMYgaJaCyPwDv\n2TTBPnEd9opaqj+Ih76il1/jg8bitFp3DuGnhM8IZkQ2XmCnQTFkenUMy+lV4lE8o7YUpw5duFcT\nWaqo6TQohkwfHMKNXIm0hw1KH3MJdOzplb9jEf5db2nlo0XiNJAT6s2wilqcBvU6yfW8ouangZxT\nZ0ZU1JGmOm0c5aKinlZR7ejjlBQVNT8NnGh15URFzU+DunycxKKi5qeBE62unKio+WlQl4+TWFTU\n/DQoo/0GYcYI/HcSVjoAAAAASUVORK5CYII=\n",
       "prompt_number": 64,
       "text": [
        "          2                     d       \n",
        "- \\omega_n \u22c5\u03b8(t) - 2\u22c5\\omega_n\u22c5\u03b6\u22c5\u2500\u2500(\u03b8(t))\n",
        "                                dt      "
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eom_des = eom.subs({theta(t).diff(t,2): theta_dd_des.subs({wn:10, zeta:0.8})})\n",
      "eom_des"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - l^{2} m \\left(- 100 \\theta{\\left (t \\right )} - 16.0 \\frac{d}{d t} \\theta{\\left (t \\right )}\\right) + d{\\left (t \\right )} + u{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAhMAAAAyBAMAAAAU4kStAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n0jgmuQAAAABJRU5ErkJggg==\n",
       "prompt_number": 65,
       "text": [
        "                   2   \u239b                 d       \u239e              \n",
        "g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) - l \u22c5m\u22c5\u239c-100\u22c5\u03b8(t) - 16.0\u22c5\u2500\u2500(\u03b8(t))\u239f + d(t) + u(t)\n",
        "                       \u239d                 dt      \u23a0              "
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dyn_inv_f = sympy.solve(eom_des, u(t))[0]\n",
      "dyn_inv_f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- g l m \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - 100.0 l^{2} m \\theta{\\left (t \\right )} - 16.0 l^{2} m \\frac{d}{d t} \\theta{\\left (t \\right )} - d{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAeQAAAArBAMAAABbSeYfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7O1SsvonnJxi51XyfwsKWPAwneFPAAAAAElFTkSuQmCC\n",
       "prompt_number": 66,
       "text": [
        "                          2                2   d              \n",
        "-g\u22c5l\u22c5m\u22c5sin(\u03b8(t)) - 100.0\u22c5l \u22c5m\u22c5\u03b8(t) - 16.0\u22c5l \u22c5m\u22c5\u2500\u2500(\u03b8(t)) - d(t)\n",
        "                                               dt             "
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cont_dyn_inv_f  = sympy.lambdify(x, dyn_inv_f.subs(values).subs({theta(t): x[0], theta(t).diff(t): x[1], d(t):0}))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 67
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure(figsize=(15,5))\n",
      "\n",
      "plt.subplot(121)\n",
      "plt.title('dynamic inversion controller vs. discrete LQR (output)')\n",
      "plt.xlabel('t, sec')\n",
      "plt.ylabel(r'$\\theta$, deg')\n",
      "data_dyn_inv = simulate(cont_dyn_inv_f,  dt_cont=1.0/10, tf=2);\n",
      "h_dyn_inv = plt.plot(data_dyn_inv['t'], np.rad2deg(data_dyn_inv['x'][:,0]));\n",
      "h_lqrd = plt.plot(data_lqrd['t'], np.rad2deg(data_lqrd['x'][:,0]));\n",
      "plt.legend([h_dyn_inv[0], h_lqrd[0]], ['dynamic inversion', 'discrete LQR'])\n",
      "\n",
      "plt.subplot(122)\n",
      "plt.title('(control)')\n",
      "plt.plot(data_dyn_inv['t'], data_dyn_inv['u'])\n",
      "plt.plot(data_lqrd['t'], data_lqrd['u'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 68,
       "text": [
        "[<matplotlib.lines.Line2D at 0x521cd50>,\n",
        " <matplotlib.lines.Line2D at 0x5219e90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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1PX86jRkzhgEDBjBp0iQuu+wy2rdv3+jynaVyBQUFeDwefv/737N27Vr+8Y9/\nMHXqVFf/H5+JEydSU1PDhRdeyPz581m7di3//Oc/ef/995u0nHBKK8855xxyc3O57LLLyM7O5vzz\nz68dN3bsWM4880zGjRvHP/7xD9asWcOiRYt44oknmDZtWsjlPvroo7z88sssXbqUtWvX8swzz5Ca\nmkr//v2DTn/33Xfzxhtv8Mgjj7BixQpee+01pkyZwi9/+UtSU1Nr/x83SmAl1tR7qohIMmu1SeOR\nR0J+PuzdC8NzzwDgozUfxTgqEQF4+OGHueiii7j66qs58cQTKS0t5dZbbw067Q033EBlZWXQqqnB\nSuWcwwYPHswTTzzB//7v/zJo0CAee+wx/vSnP9Wbr7HlNDTM+b5bt27MnTuX9u3bc95553Hsscdy\n7733Bv2fnPM7lxGsN9Vg601JSeGKK67g66+/Zvz48fV6aH377bcZN24cd9xxB0cffTTf//73mT17\nNv369Qv5/+Xk5PDYY48xevRoBg8ezFtvvcUbb7xRe5uNwPjOPfdcnn32WZ5//nmOO+447rjjDm69\n9Vbuu+++kP9TQ/+nJL/a6qleVU8VEYknreGobDV0FfvCC+Htt+H2p2bx+LZLOPXIU5kzYU7QaUUS\njVtt6OLdnXfeyQcffMDixYtjHYq0kFDfbTvZbA3HNrc0eIxsCu9deVjtdrLk2u0M6p3X7OXU1Fik\nTDXXs6vvralLIkVEJCKRHh9bbUkjwMiR5rniu9Px4OHTjZ9yoPJAbIMSkbDs27ePhQsXMm3aNH75\ny1/GOhwRcYEzSaxpBRe9REQShZJG4Jv/duT4/OOpqqnis02fxTYoEQnLhRdeyKmnnsrFF1/s19GL\niCS4GnNqUlOjpFFEJF606qRxxAjz/OWXcNIR5p5wn276NIYRiUi4iouLKS8v55lnnol1KCLiam+n\n9u1rVNIoIhI3WnXS2KkT9O4NBw/CkSmjACWNIiIiMWWZpPFwtTrDERGJF606aQQ47jjz3G63XdK4\n8VNqLB2oREREmsqVjmssVU8VEYk3rT5pPPZY87xteS/ys/IpqShhxe7694MTERGRlqCSRhGReNPq\nk0ZfSeOSbzyM6mlXUd2oKqoiIiIxYalNo4hIvGn1SaOvpHHJEjjpiJMAWLBlQQwjEhERacVUPVVE\nJO60+qRxwABITYXVq+HojsMA+HLblzGOSkREJJG433vq4RpVTxURiRetPmls08YkjpYF6XtN0vj1\n9q+prqna+08QAAAgAElEQVSOcWQiIiKtkEoaRUTiTiIkjcXAQWC//VjmGHcG8B1wAPgY6NWcFfiq\nqG5a0Yme2T0prypXZzgicaaoqIiCgoIG34tI7Hk9LvSeapc0VqukUUQkbiRC0mgBtwLt7cfR9vDO\nwBvAPUBHYCHwanNWcLS9xOXLYVi+qqiKJIJJkybx+eefxzoMbrjhBk4//XRXljV9+nTS0tJCTrN/\n/37uueceBg4cSHp6Orm5uZx33nnMmTPHb7ri4mK8Xm/to0OHDowcOZJXXnnFlVhFosbXEY5KGkVE\n4kYiJI3gu+zobxywBJM4VgJFwBCgf1MX3t+eY8UKGNbNThq3KmkUiWeZmZnk5uZGbfmVlZVRW3Zz\nlZaWMmbMGGbOnMmDDz7IypUr+eSTTygoKOCMM87gueeeqzfPl19+ybZt2/jss8846aSTuOqqq5g3\nb14MohcJj8c+NVHvqSIi8SNRksbfAjuBucBp9rBBwGLHNOXAKuDYpi7cV8PNL2lUSaNIzFRUVHDL\nLbfQoUMHcnNzmThxIocOHfKbJrB66qZNm7jkkkvo0qULGRkZ9O3bl9///ve14w8fPsyUKVPo27cv\n6enp9OjRg5/97Ge1471eL0888QRXXHEFHTp04Oqrrwbgww8/ZMyYMbRr144ePXrw4x//mD179tTG\n8OyzzzJnzpzaEr0ZM2YAUFZWxu23306PHj3IzMzk+OOP580334xou0yePJnVq1fz0Ucfcckll9Cz\nZ0+GDBnC448/zk033cStt97Ktm3b/Obp0qULeXl5DBw4kIceegjLsuKihFaSjYsJnqX7NIqIxJvU\nWAcQhruApZjSxMuBd4ChQCYmkXQqBbKaugLfeeeqVTCka13SaFkWHlfaZ4jEJ8+U6Hy/rfsiO4G8\n++67mTVrFi+88AIDBgzg6aef5sknn6Rr164NzjNx4kQqKir46KOP6NChA2vWrPFLoK6//nref/99\nHnvsMUaPHs3u3buZP3++3zKmTJnClClTePDBB6murubjjz/moosu4ne/+x0zZsygpKSEO++8k3Hj\nxlFcXMykSZNYtWoV69atY9asWQBkZ2djWRYXXHABHo+H1157je7du/Phhx8yfvx4Zs+ezdixY5u8\nTSzL4qWXXuLKK6+kZ8+e9cb/5je/4amnnmLmzJncdtttfvOBKTl96qmn8Hg8jBgxosnrF2k5uk+j\niEi8SYSk8b+O1zMwieN5QBmQHTBtDqazHD9FRUW1rwsLCyksLPSfKQe6doXt28FT2pOO6R3Zc3AP\nW/Zv4YjsI1z5J0QkPAcOHOCpp57iL3/5CxdccAEAjz76KMXFxezbt6/B+TZs2MDFF1/M4MGDAejV\nq65frFWrVvHCCy/w+uuvM27cOACOOuqoesnTxRdfzK233lr7/oYbbuD222/3GzZ9+nR69+7N119/\nzeDBg0lPTyctLY28vLzaaYqLi/nss8/Yvn072dlmN3XjjTfy6aef8sQTTzQrady5cyclJSUMGjQo\n6PgePXqQnZ3NihX+nXgNGDAAj8dDeXk5GRkZvPbaa5x66qlNXn+8Ki4upri4ONZhiM2VjnDs3lOr\nVdIoIhI3EiFpbMhS4FrH+0ygrz3cjzNpbEhBgUkaV670MChvEHM3zOXbnd8qaZSkFmmJYDSsXr2a\nQ4cOMXr0aL/hY8aM4d13321wvp///Of85Cc/Yfbs2RQWFnL++edzyimnAPDFF18A8L3vfS/kuk84\n4QS/9wsWLODzzz/niSee8Bvu8XhYuXJlbYIaaMGCBVRWVnLEEf77j8rKSvr3b3Kz6yZJTfXfrf/r\nX/+ia9euLF26lJtvvpn33nuPSy65JKoxtKTAC4FTpkyJXTAtIxd4BjgL2AXcDQTr3WiCPV25Y9j5\nwL+jHJ8LVNIoIhJv4j1pzAFOAuYAh4HLgFOA24C9wKOYDnHeA+4DvgKada+M/v1h7lxYuRKO6XEM\nczfMZenOpZzV9ywX/g0RibYJEyZwzjnn8P777/PJJ59w7rnncvHFF/PCCy+EvYzMzEy/95Zl8etf\n/7q2faNTqKqyNTU15OTksHDhwnrj2rRpE3Y8Tl26dCE3N5dvvvkm6PiNGzdSWlpaLynt3bs33bt3\np2/fvrRt25Zzzz2XCRMm1CbUknD+ClQAecAw4F1M+/5vg0w7D0i4YmWP5cVCSaOISDyJ945w0oCp\nwA5M+8VbgQsxHd7sAi4BHgT2ACOA8c1dkbMH1UF5pvrXtzuDHYNFJJr69u1LmzZt6vXwOW/evEbb\nGHfr1o0JEybw/PPPM23aNF566SXKyso4/vjjAfjggw+aFMuIESNYsmQJffr0qffwJZht2rShurra\nb76RI0eyd+9eDh48WG++Hj16NCkGH4/Hw5VXXsnLL7/Mhg0b6o1/6KGHyMjI4NJLL21wGWeffTaj\nRo1i6tSpzYpBYi4Tc6H0XkwJ4jzgLaD+VQ0jQRvlqyMcEZF4E+8ljbuAE0KM/4i6+zZGxNmD6vld\njgGUNIrEQmZmJjfffDOTJ0+ma9eu9O/fn2eeeYYVK1b4tRsM9NOf/pTzzz+f/v37U1FRwaxZs+jV\nqxdZWVn069ePK6+8sraznJNOOok9e/bw6aef+vWgGuj+++/ne9/7HnfccQdXX3012dnZrFy5ktdf\nf52//OUvpKen06dPH15//XW+/fZb8vLyyM7OZuzYsZx55pmMGzeORx55hMGDB1NSUsL8+fPJyMjg\nhhtuCLkNFi9eXNuBjc9xxx3H1KlTKS4u5owzzuDhhx9m5MiRlJSU8OyzzzJt2jSef/55OnXqFHLZ\nkyZNYty4cXzxxRe1ybQkjP6YWjerHMMWA4VBprUwJZE7MRdWX8D0RF4dZNrIedwsFTRJY+BvQERE\nYifek8YW07eveV67FgZ1MSWNS3cuVQ+qIjHw8MMPU1FRUVstdPz48dx66628/vrrtdN4PJ56v82f\n//znbNy4kXbt2jFq1Chmz55dO+65557j/vvvZ/LkyWzZsoW8vDx+9KMfhYyjsLCQjz/+mClTpnDa\naadRU1NDr169OOecc0hLSwNMr6yffPIJo0ePprS0lOnTp3PNNdfw9ttvM2XKFO644w42b95Mbm4u\nw4YN484772xwfR6Ph+rqaoYNG1Zv+NatW8nLy2Pu3Lk8/PDD3H333axfv56qqiqysrKYP38+I0eO\nrDdfoAsvvJD+/fvzyCOP8Oqrr4b8/yXuZGF6CXfaD7QPMu2/MbemWo+5FdWrmITz4WgG6AaP3RHO\ng2/OJK996Isg4fjRSaM4a3hB4xOKiEiDWkM2ZIVztXLfPujQATIyoKzMotOjueyt2MuWO7aQ3z6/\nBcIUcZfH49GV+lZg4cKFnH322YwbN46///3vreIiV6jvtv3/J+tGGIa5X7Gz8e2vMO0Wf9DIvJcB\nkzBNOZys++67r/ZNsB7Gw+G9OxcrvYRVN+6mb/fcJs/v1OaXBVRlr2p8wjCl7u9N1e/XurY8EZFE\nENi7uN1RXLOPjypptOXkQMeOUFICO3d6GNRlEPM2zuPbnd8qaRSRuDVixAjmzJnDG2+8weLFixk6\ndGisQ5LoWYE5bvejrorqEGBJmPMHPVkIp4fxlvSb4//EC19GXgp+2KpiQ/b/cbjtDheiEhFJLG73\nLq6k0aF3b5M0rlsHx3Q5hnkb57F051LO6HNGrEMTEWnQsccey7HHHhvrMCT6DgCzgPuBG4DjgQuA\nUUGmPRf4AtgODAQmA6+1TJiRKbryfIquPD/i5ewoOUDXP/+fy+0tRURap3jvPbVFHXWUeV671iSN\nAMt2LothRCIiIn4mAhmYXsVfBG4GlgG9MO0bfd3zjsV0klOGuS3HG8BD0Qsr/hIzrzdZaymLiLQ8\nlTQ69O5tntetg+OON43mV+5ZGbN4REREApQAFwcZvgH/DnEm2Y9Wy1vbvjf+EloRkUSjkkYHZ9JY\n0ElJo4iISFN447EjJlVPFRGJmJJGB2f11KM6HEWKJ4WN+zZScbgitoGJiIhIk9RVT1XSKCISKSWN\nDs6SxrSUNHp36I2FxZqSNbEMS0RERJqottRTJY0iIhFTm0YHX9K4fj3U1JgqqqtLVrNy98rajnFE\nEkXHjh1bxT37pPXp2LFjrEOQBKCSRhER9yhpdMjKgs6dYdcu2L4dCnILeJ/31a5REtKePXtiHYKI\ntBJWHCZmKmkUEXGPqqcG8JU2rl1rkkaAlbuVNIqIiCQSlTSKiLhHSWOAXr3M88aN6kFVRESkKeLp\n3ohx2ZOriEiCUtIYoGdP87xpk6OkUUmjiIhIQqlNYFU9VUQkYkoaA/ToYZ43bYIjOxxJqjeVTaWb\nKK8qj21gIiIi0nRKGkVEIqakMYAzaUz1ptKnYx8AVu9ZHcOoREREpClUPVVExD1KGgM4k0aAfrn9\nAFhdoqRRREQkqDgszXO2r6ypib/4REQSiZLGAIFJ41EdjgJg3d51sQlIREREIlJjKWkUEYmEksYA\n3bub561b4fBh6N2hN6CkUUREpDGeeKsSapl4VNIoIhIZJY0B2rSBrl2huhq2b68raVy7d22MIxMR\nEZEm8SWNKmkUEYmIksYgnFVUVdIoIiKSqFTSKCLiBiWNQQRLGteWrMXSlUoREZEg4vT4aMVZdVkR\nkQSlpDEIZ9KYm5FL+zbt2V+5n5KKktgGJiIiIk2g6qkiIm5Q0hiEM2n0eDyqoioiIhKGeL03oqqn\niohERkljEIG33XBWURUREZEEoY5wRERcoaQxCN2rUUREJBmoIxwRETcoaQzCea9GUA+qIiIiCUkl\njSIirlDSGES3buZ561awLEf1VN2rUUREpD5PvCZlKmkUEXGDksYg2reHdu2gvBzKyuCojqqeKiIi\nknBU0igi4opEShoLgArgBcewM4DvgAPAx0AvN1bk8fiXNjpLGnWvRhERkeC83njrPTXe4hERSUyJ\nlDT+FfgvdXcQ7gy8AdwDdAQWAq+6tbL8fPO8dSt0SO9ATtscyqvK2X1wt1urEBERkahSSaOIiBsS\nJWkcD5QAH1F32XAcsASTOFYCRcAQoL8bK/SVNG7bZp6P7HAkAOv3rndj8SIiItJC1KZRRCQyiZA0\nZgNTgF/gX89kELDY8b4cWAUc68ZKnSWNAEfmmKRR7RpFREQCxWlSpjaNIiKuSISkcSowDdiCOSr5\n9vyZQGnAtKVAlhsrrVfSaCeN6/eppFFERCQxmKRR/RGIiEQmNdYBNGIoprObYfZ7D3WljWWYUkin\nHGC/GysOLGn0dYaj6qkiIiLBeT1x1vGMpVtuiIi4Id6TxtOA3sAG+30WkAIcAzwFXOuYNhPoCywN\nXEhRUVHt68LCQgoLCxtdcUNtGtftWxd28CIiEl3FxcUUFxfHOgxx0Z4dO/jvV1+5sqzhVYNZ712i\n6qkiIhGKs0uC9WQA7e3XHuBXmCTyZvv9KuDHwHvA/cDJwOiAZVjNqZby5Zdw/PFw3HHw9dewYPMC\nTph2AkO6DuGrm905mImIiLs8pqQr3o9t8aRZx8hAnnuyoM0Btt62n265kbUSGTVjBp/1cuUOWgD0\n3rKRf516HgU9Orm2TBGRRBPp8THeSxoP2g+fMvu9774XlwB/AV4EPsP0suoKX/VUX0ljbfVUtWkU\nERGJmg1ZJuk8bd060mtqmr0cC/hXnz5s7tLNpchERFqveE8aA00JeP8RcHQ0VtSlC3i9sHMnVFVB\n53adyUjNYG/FXvZV7CMnPScaqxUREUlA7lX/rEpJAeDVs86i6xFHNHs5Vk0NKcXFVKWlUVVV5VZ4\nIiKtUiL0nhoTKSmQl2de79hhinRr79Wo0kYREZGo8CWNqamRXdf2eL2kV1YCcOjQwUamFhGRUJQ0\nhuDrDCfwXo3qQVVERKQ+N3pPPew1pyZpbdpEvKy2VXbSWF4R8bJERFqzRKue2qLy8+Grr4LcdkMl\njSIiIlFRZZcwupI02iWNq1avINMTeeLYr39/MjIzI16OiEiiUdIYgrN6KqikUUREJNoOu1Q9FaCN\nXdJ4dZcc2Lcv4uUd8/bbLL388oiXIyKSaJQ0huBLGnfuNM+6V6OIiMRQLvAMcBawC7gbeKWBaX8B\n3Am0A14HbgEqoxaZx52OcKyaGqp9SWNaWsTLO+qrd0g/+kzapKb7uptvtiXdu/Ntfj5Vhw6R1rZt\nxLGJiCQSJY0hqKRRRETiyF+BCiAPGAa8CywGvg2Y7mzgLuB0YCvwJqb38btbLNJmOmz3cpp6+DAe\nb+TdLsytKaZm3f+x6IrNHF/QPaJldfznP9mblUXp3r106to14tjcsnrLHp79aC41EdyexOnMwcdy\nxrB+rixLRJKHksYQunQxz76SRrVpFBGRGMkExgGDgHJgHvAWcDX1k8FrgWnAMvv9/cDLQaZzndcb\nWWleld0GMbW62o1w8N3HusaKvCQ0p6KCvVlZDJ07l1QXErSz9u3j7zfcEPFyRj16FTs7zI54OT4P\nf5fDvv7byc5UaaqI1FHSGEJgSWN++3zSvGnsOLCD8qpy2qW1i11wIiLSmvQHDgOrHMMWA4VBpj0G\nU7ro8zXQFegIlEQjuBOrRuHtMJpbX32RlAhKCKsA+vUjzeWk0Q0n7t3L+s6d2dSpkyvLe7pLF756\n/BHaVJRFtJyC/KEcXdUXa8cePKQ0fQGOqsX701ZBSiW/+v0fyc4I5xwn9PY9Kq8LA3t2CT5nE6oL\nV9dYrN9RQk2NZc8b9qz1HK622LZ3H1WHDzc6bbgxNjadx0wU3rLCGBL2tZkw1ukN9hkGnS2MZXnC\nm67hsDxhTBM4hyeM6Rse6TcmnO0VdmCNfyeau6xgc3lc3Nc1REljCL6SRl/S6PV46ZnTkzUla9iw\nbwMDOw+MXXAiItKaZAGlAcP2A+0bmNbZ64tvvvZEKWncNPwmNud15VOXlpdfGvivRsZyoaTxleuu\n45E1a1xZ1ilffsHmrt1YMOTEiJcVDV/GOoBAXqB7Z/eW1yvPvWWJtBJKGkMI7AgHTLvGNSVrWL93\nvZJGERFpKWVAdsCwHEzi2Ni0OfZzsGkpKiqqfV1YWEhhYWGTg6u0O615bMNGurbPavL8gUYPHx7x\nMgCw7OqpNZEnet6UFHoXFES8HAD+egWjM3MZlXsB+R1zGp8+hC8OHWBHevOqkloBJRgHK6s4dDi8\nW5NYjZRsWJ6qEOPCWkWQGb1EUnrs+3/N36aUiEeyzmbPGpP1Nva5hhTBrLFYb+D3v0nzJsB2OvDd\ncsqXLzfzeWB381cLKGkMyVnSaFmmdFjtGkVEJAZWYI7Z/airojoEWBJk2qXAUEyvqb7pttNAKaMz\naYzU2SNP4pijXUqsXOCrsuVGm0Y37Ujdx+a2i/ifwr9w9oj+sQ4nKkoPHOLtz5ZQ7VIHPcP69mRw\nn26uLEukNYq0B2kljSFkZEBWFpSVQWkp5OTU9aC6bu+62AYnIiKtyQFgFqZTmxuA44ELgFFBpp0B\nTAdeArYB9wLPRTO4+ErJnOIzabQ8pj1dRpvIbysSr7Iz23LVGS6VGItIzEXen3WSq3fbDftejSpp\nFBGRFjYRyAB2AC8CN2N6SO2FqXraw57uA+B3wCfAOmA1cF9LBBh2xw4txsTjRjtEN9XYVTfbpuna\nvYgkBu2tGtGlC6xZY5LGggLdq1FERGKmBLg4yPAN1O8Q54/2o0VFWv3JdS62aXSV1y5pbJu8JY0i\nklxU0tiIwM5wfG0aVT1VRETEiKRDiWiK1zaNlkoaRSTBKGlsRGD11B7ZPfB6vGzZv4XK6srYBSYi\nIiKNiM9k1lJJo4gkGCWNjfD1oOoraUxLSaNndk8sLJU2ioiIOMRd9dQ4LWnEa0oa01XSKCIJQnur\nRgSWNAL0ze3L+n3rWb1nNf07JWdX2SIiImGzk0WvN96SRuOC5y8jdXpGxMsZmXs2/+9/JkcekF3S\nmN5Gp2Eikhi0t2pEYJtGgH4d+/Hx2o9ZtWdV8JlEREQk5tofPooS1lDeYZEry/vImotnyr1QHWG1\n0hRT0tguXdVTRSQxKGlshK96amBJI8DqktUxiEhERCQ+eeKsDeGKKW/z2n++cKV66l0f30F5h4Xm\njZ30RSK35ExSU9RKSEQSg5LGRgStntpRSaOIiIiPFWfJok/nnHZM/P7JrizrpxcsoOygex3gtVMn\nOCKSQJQ0NiKwIxyAfrn9AFQ9VUREpBXJymgT6xBERGJC9SIa4Uwaa2rM6z4d+wCwtmQtNVZNjCIT\nERGJL/HXe6qIiLhBSWMj2rSBnByoroZ9+8yw9m3bk5eZx6HqQ2wu3RzbAEVERGLMsnNFr5JGEZGk\npKQxDJ06meddu+qGqYqqiIiIiIi0Bkoaw+BLGnfvrhumznBEREQCqKRRRCQpKWkMQ8ikcY+SRhER\nad0sJYsiIklNSWMYgiWNtdVTS1Q9VUREREREkpeSxjB07mye/Uoac1XSKCIi4uTVzepFRJKS9u5h\nCNYRjrNNo2VZMYhKREQkPlioeqqISDJLhKTxRWArUAqsAe5xjDsD+A44AHwM9IpGAMGqp3Zu15ns\nttmUHiplV/mu4DOKiIiIiIgkuERIGn8LHAVkA+cCtwFnA52BWZgksiOwEHg1GgEESxo9Ho96UBUR\nEXHwqEMcEZGklAhJ41KgwvG+CtgJjAO+Ad4AKoEiYAjQ3+0AgiWNUNcZzsrdK91epYiIiIiISFxI\nhKQR4ElMFdSlwIPAF8AgYLFjmnJgFXCs2ysP1hEOwIBOAwBYvnu526sUERFJGLrlhohIckuUpHEi\nkAWcCTwAnABkYto5OpXa07kqWEc4AAM7DwSUNIqIiAB4lTyKiCSl1FgH0AQWUAzMBC4HyjDtHJ1y\ngP2BMxYVFdW+LiwspLCwsEkrdlZPtSzwHRMHdDYljd/t+q5JyxMREXcUFxdTXFwc6zDEpjaNIiLJ\nKRH37tOAbcB64FrgZHt4Jqat41BghWN6y41bYmRkQEUFlJVBZqYZVnqolJyHc2ib0pYDvzlAijcl\n4vWIiEjz2UlLIh7bYsWVY2TOu/+kNDOLzQUD6X5ENxfCEhERN0V6fIz36qldgPGYhDAF02vqj4C3\ngDcx7RfHAenAfcBX+CeMrgnWGU5222zys/I5VH2IDfs2RGO1IiIiIiIiMRXvSaMF3AxsAnYDU4Gr\ngQXALuASTMc4e4ARmAQzKhrqDEftGkVERAxVTxURSU5NbdM4FZPIgSne9L2uBDYC7wPb3QkNMIlh\nYYjxHwFHu7i+BjXUGc6ATgP4ZN0nfLfrO87pd05LhCIiIhJXLLvGk9erpFFEJBk1NWnsD1wE/BeT\nJPYCRgL/BC7A3Brjh8BsF2OMCw3dq9HXGc7yXSppFBERERGR5NPU6qkeTBXQU4ArMJ3QXApUAydi\nbo3xWzcDjBcNJY2qnioiImKoeqqISHJqatJ4DvB2wLB3gXPt1y8BfSMNKh41WNLYyS5pVNIoIiKt\nlKVkUUQkqTU1aVyNKU10uhlYZb/uDByINKh41FBHOL1yetE2pS1b9m+h9FBpywcmIiIiIiISRU1N\nGq8HfoXpzfRz+3kScIM9vj9wr2vRxZGGOsJJ8aZQ0KkAgBW7o3K3DxERkYSg6qkiIsmpqR3hfAEU\nACcB3YGtwKeY3lMB/m0/kk5D1VPBtGtcsmMJy3ctZ0T3ES0bmIiISIz5qqd6lTSKiCSlpiaNYBLE\npEwMQwmVNPraNX6367sWjEhERERERCT6mlo9FeB7wLOY22wAjADGuhZRnAqVNB7d2dwqctmuZS0Y\nkYiISHxR9VQRkeTU1KTxNuBvwErgVHtYBfCAm0HFo4Y6wgEYlDcIgKU7l7ZgRCIiIvHBUq4oIpLU\nmpo0/gI4E3Mvxmp72DJgoJtBxaOcHEhJgdJSqKz0Hzew80C8Hi8rd6+k4nBFbAIUERERERGJgqYm\njVnAxoBhbYBD7oQTvzweyM01r/fs8R+XnppOv9x+VFvVLN+l+zWKiIjrcoE3gTJgHXB5iGknYC7s\n7nc8Tg0xvWtSUprT6kVEROJdU/fu/wF+HTDsNuATd8KJb74qqoG33QA4Nu9YQFVURUQkKv6KaQ6S\nB1yJaSpyTIjp5wHtHY+odmBnofqpIiLJrDltGi8G1mNKHVcAlwG/dDmuuBQyaexiksYlO5a0YEQi\nItIKZALjMPdBLsckhG8BV4eYR1mciIi4pqm33NiC6S31BKAXpqrqf4Eal+OKS126mOedO+uP83WG\no6RRRERc1h84DKxyDFsMFDYwvQUMA3YCe4AX8O+LIGo8XuWqIiLJKJykcSrmAOQ7Elj2swc4DjjP\nfv8/7oYWf3xJo6qniohIC8oCSgOG7cdUOw3m38AgTK2gY4FXMUnnw9EK0NKtNkREklo4SWNP6hLF\ndOASYAHmYHQkMBJ4IyrRxRlf9dRgJY0FuQWkedNYU7KGA5UHyGyT2bLBiYhIoiqm4Y5q5gI/A7ID\nhudgEsdg1jpeLwHuBybRQNJYVFRU+7qwsJDCwsJGwhURkXhXXFxMcXGxa8sLJ2mc4Hj9f5ge25xJ\n4jjgUtciimOhqqempaQxoPMAluxYwrc7v2XkESNbNjgREUlUhY2Mz8Qcr/tRV0V1CCYhDFeDRYHO\npDFSKd4U15YlIiLNF3gRcMqUKREtr6kd4ZwH/CNg2DvUVVFNaqE6wgFVURURkag4AMzClBi2A04G\nLsC0VQzmXKCr/XogMJn6x25XqXqqiEhya2rSuAr4acCwW/BvnJ+0QpU0Agzqos5wREQkKiYCGcAO\n4EXgZmCZPa4XpqpqD/v9WExHOWXAu5jaQQ+1ZLAiIpJcmtp76vWYq5V3ApuBIzCN68e5HFdcCrek\nUUmjiIi4rARzy6tgNuDfKc4k+9HiPCpxFBFJSk1NGr8ECoCTgO7AVmA+UOVyXHGpsZJGX9L4zY5v\nWigiERGR2LMan0RERBJYU5NGgEpMd96tjrOk0bIg8IJqn459yGqTxZb9W9h5YCddMru0fJAiIiIx\n4pV1OdwAACAASURBVNV9GkVEklJT2zS2ahkZkJkJlZWwP0hH516PlyFdhwDw1bavWjg6ERGR2FL1\nVBGR5KSksYl8VVQbatc4tNtQQEmjiIi0Huo9VUQkuSlpbCJfFdWG2jX6ksYvt33ZQhGJiIiIiIhE\nj5LGJmqsM5xh3YYBKmkUEZHWoaamrhscVU8VEUlObiSNz2FuxZHiwrLiXmO33RiUN4gUTwrLdy+n\nvKq85QITERGJEVVPFRFJbm4kjR7gcuBrF5YV9xoraUxPTefoLkdTY9XwzXbdekNERERERBJbU5LG\njsCNwN3AFYBd5sYE4ExgmKuRxanGShpBVVRFRKR1UvVUEZHk1JSk8XpgL6Ya6r3AGuAVoJc9vtLd\n0OJTYyWNoB5URUREREQkeTQladwDzAQeAP4E5ANvA08BJ7kfGgBtgGeAdUAp8CVwjmP8GcB3wAHg\nY+oS2KhpSkmjelAVEZFkV2NZatMoIpLkmpI0LgeeBoYDFiZRewU4HzjO/dAASAU2AKcC2cBk4DVM\nctgZmAXcg6k6uxB4NUpx1AqnpHFItyEAfL39a6prqqMdkoiIiIiISNSkNmHaecB24BFgJDAUWAYc\nJHpJYzkwxfH+XWAtMAKTNH4DvGGPKwJ2Af2BFVGKJ6ykMTcjl145vdiwbwPLdy/nmC7HRCscERGR\nuOHx6k5eIiLJqKl791XAJZjqqPOBTEyp42SX42pIV0xSuAQYBCx2jCu34zs2mgGEUz0VYET3EQAs\n2LwgmuGIiIjEnNX4JCIiksCae0lwC/Ai8DCmzWGpaxE1LA14CZiOKUnMDLLeUiArmkF06AApKbBv\nH1SG6PrnhO4nALBgi5JGERERERFJXE2pnhpLXuAFoAL4qT2sDNPO0SkH2B84c1FRUe3rwsJCCgsL\nmx+I11RR3bYNduyAHj2CTzfyiJEA/Hfzf5u9LhERCa24uJji4uJYhyE23XJDRCQ5JcLe3QM8i+n8\n5jzgkD38RuBa4GT7fSawE9PW0tmm0bIsdyvODB0KixfDwoUwfHjwafZV7KPjIx1JS0mj9NeltE1t\n62oMIiJSn520JMKxLV5EfIysrKom/T9zsLxeDp98MimpiXI9WkSk9Yj0+JgILdb/BgwEfkBdwgjw\nJqb94jggHbgP+IoodoLj062bed6+veFpctJzGNh5IJXVlXy9/etohyQiIiIiIhIV8Z40HgncBAwB\ntmGqnu4HLsf0lHoJ8CDmHpIjgPEtEVTXruZ527bQ0/mqqKpdo4iItAaqnioikpziPWlcj4mxHdDe\n8XjFHv8RcLQ9fizmno5R50saQ5U0Ql1nOGrXKCIiyczSrTZERJKa9vLNEE71VFBnOCIiIiIikviU\nNDZDuNVTh3QdQpo3je92fUfpoZa4K4mIiEjLqnF0pKPqqSIiyUlJYzOEWz21bWpbhnQbgoXFoi2L\noh+YiIiIiIiIy5Q0NkO4SSPUtWv8fPPnUYxIREREREQkOpQ0NoOvTWNj1VMBRvUcBcC8jfOiGJGI\niEjsedQhjohIUtLevRk6dYKUFCgpgcrK0NOO6TkGgPkb5xPpDZRFRETijVVTE+sQREQkypQ0NoPX\nC126mNc7doSetneH3uRn5bPn4B6W714e/eBERERERERcpKSxmcJt1+jxeBjTy5Q2ztugKqoiIpJc\nampUi0ZEJNkpaWymprRr9FVRVbtGERFJNmp6ISKS/JQ0NlNTelBV0igiIiIiIolKSWMzNSVpHNpt\nKO3S2rFi9wp2HtgZ3cBERERiwKMOcUREkpaSxmZqSvXUtJQ0TjjC3K9x/sb5UYxKRESkZal6qohI\n8lPS2ExNKWkEGN1jNKAqqiIiklxqlDSKiCQ9JY3N1NSk8eReJwPwnw3/iVJEIiIiseNR8igikrSU\nNDaTL2kMp3oqwJheY0jxpLBg8wLKKsuiF5iIiEgLUvVUEZHkp6SxmZrSphEgu202w7sPp9qqZu6G\nudELTERERERExEVKGpupUydIS4O9e+HgwfDmOb336QB8svaTKEYmIiLS8jyxDkBERKJGSWMzeb2Q\nn29eb9kS3jy1SeM6JY0iItIkPwUWAhXAc2FM/wtgK7APeAZoE63ALEu32hARSXZKGiNwxBHmOdyk\ncUyvMaR6U1m0dRH7KvZFLzAREUk2m4GpwLNhTHs2cBcwFv5/e3ceJlV5J3r8W91dTdPNIiASESGK\nILjgkojRGKbRa0CNRmIWFcclkDFGk9GMxDhuqFeTmExu4qO5cV9iNI4LYOJVxxHbZUxMjIhgBAKx\nMSBCQJClaXqpun+81dK0Db3UqTrd1d/P89RzTp3lPT9Ojv3mV+9yGAHsC1ybq8BSKcc0SlKhM2nM\nwtChYblyZfuO71Pah/F7jSeVTjmLqiSpI2YBc4B17Tj2HOBO4G1gA3AdcG7OIstw9lRJKlwmjVno\naEsjbO+iWlVdFX1AkqRC156hgwcA85t9fxMYAgzIRUBpWxolqeCZNGahoy2N4LhGSVJW2pOh9SGM\nZWyyMbPsG304kqSeoCTuALqzppbGjiSNR+19FMmiJPNWzWP91vUM6J2TH34lSd1HFTBhJ/tebrGv\nPS2Nm4F+zb73zyw3tXbwzJkzP1qvrKyksrKyHZf4OLunSlLXUVVVRVVVVWTlmTRmoamlsSPdU8uT\n5Ry999G8sPwF5r4zl9MOOC03wUmSuovKDhzbnszsLeBQ4NHM90OA1cD61g5unjR2RtpkUZK6nJY/\nAl57bXbzodk9NQudaWkEmDRyEgBPL3064ogkSQWqGCgj/NhbDPTKLFtzPzANGEsYx3gV7XtNR6ek\nTBolqeCZNGaheUtjR+rMyftNBuDpZU/7C60kqT2uAmoIr9I4C9gKXJHZN5zQ9XRY5vszwE3A80A1\nsAy4JtcB2j1VkgpXe8ZGdHfpXCZm/frBpk2wbh0MHNi+c1LpFEP/Yyirt6xm4QULOXCPA3MWnyT1\nJIlEAnpG3RaVrOvI1f9Yzyfemk9pXR3bPv/5iMKSJEUp2/rRlsYsdea1G0WJIibtZxdVSVL3Z48Z\nSSp8Jo1Z6sxrNwAmj9zeRVWSJEmSuiqTxix1pqUR4PiRx5MgwYvLX2RL3ZboA5MkKQ9SqdDSaJ9g\nSSpcXT1pvAh4Dajl4zO/HQcsArYAcwkTAeRdZ1sady/fnfF7jaeusY7n3nku+sAkScoDu6dKUuHr\n6knjSuB64O4W23cHHiPMHDeAkFg+nN/Qgs62NAKcPPpkAGYvmh1hRJIkSZIUna6eNM4C5gDrWmz/\nErCQkDjWATMJLy8enc/goPMtjQBTxk4B4InFT9CQaogwKkmS8stXbkhS4erqSWOTlkMlDgTmN/te\nAywFDspbRBlNLY2dSRrH7j6W0YNGs27rOl5a/lK0gUmSlAd2T5WkwtddksaWNVIFsLHFto1An/yE\ns92wzKuUV6zo+LmJRIIpY0Jr46xFsyKMSpIkSZKiURJ3AO3UsqVxM9Cvxbb+wKbWTp45c+ZH65WV\nlVRWVkYW2J57QkkJrF4NtbVQVtax86eMmcKP/udHzF40m59P/nnTizclSe1QVVVFVVVV3GH0aKlM\nS6PdUyWpcHWXDOV6YBhwXub7N4BzgGMy3yuAfwCHAktanJvOddeZffaB6mpYsgRGjerYual0iuH/\nZzgrN63kj9P/yBF7HZGTGCWpJ8j88NZd6rauIOs6cvmK1Xxy6dv0rq2lZvLkiMKSJEUp2/qxq3dP\nLQbKCC2ixUCvzHIWYfzilzL7rwHe4OMJY14Mz7zs4913O35uUaLooy6qD78VywSwkiRJkrRTXT1p\nvIowyc1lwFnAVsJrNtYCpwE3AB8AnwZOjylGRowIy+XLO3f+1HFTAXhwwYM0phojikqSpPxJfGz6\nAUlSoejqSeNMQozNP9dl9j0HjAXKgWOBTrTzRSOblkaAI/c6kpEDRrJq8yqer34+usAkScoxZ0+V\npMLX1ZPGbqEpaexsS2MikeCscWcB8MCbD0QUlSRJuZdOmTRKUqEzaYxAU/fUzrY0Akw9OHRRfezt\nx6ipr4kgKkmS8idh7ihJBcukMQLZdk8FGDVoFEfudSSb6zYze9HsaAKTJCnX0qm4I5Ak5ZhJYwSa\nJ42pLOrOcw45B4A7Xr8jgqgkSZIkKXsmjRGoqIBBg6CuDtas6Xw5U8dNpSJZQVV1FYvWLoouQEmS\ncizhhDiSVLBMGiMSRRfVfr36cebBZwLwy9d+GUFUkiTllrOnSlLhM2mMSLbvamxywacvAOC++fc5\nIY4kqctLmTRKUsEzaYxIFC2NAIfteRjj9xrPhtoNPLjgwewDkyQpD+yeKkmFy6QxIk0tjdXV2Zf1\nnfHfAeAnr/yElLPSSZK6sDQmi5JU6EwaI7LvvmG5bFn2ZX31wK8yvP9wFq9bzG8X/zb7AiVJkiSp\nk0waIzJyZFj+7W/Zl5UsTnLJZy4B4KZXbsq+QEmScixhi6MkFSyTxojss09YvvMONDZmX970w6cz\noGwAr/z9FV5c/mL2BUqSlAON2bygWJLULZg0RqRPHxgyJLyrceXKCMor7cN3jgxjG6+Ye4VTmkuS\nJEmKhUljhJq6qEYxrhHgu0d9l4G9B/Lyuy/zzLJnoilUkqRc8LdNSSpYJo0RinJcI0C/Xv24/JjL\nAfj35/7dmVQlSV2OPWEkqfCZNEYoyhlUm1x4xIUM7TuUee/P44E3H4iuYEmSJElqB5PGCEXdPRWg\nd7I3PzzuhwDMeHYGG2o3RFe4JEkRSdjiKEkFy6QxQrlIGgHOGncWxww/hjVb1nD181dHW7gkSVlo\nbHTohCQVOpPGCEU9prFJIpHg1hNvpThRzK1/upVX/v5KtBeQJEmSpJ0waYzQHntARQWsXx8+URo3\nZByXHn0pqXSKs2edzea6zdFeQJKkLCScPlWSCpZJY4QSidxMhtPk2sprGTdkHMvWL+Pipy+O/gKS\nJHWUuaIkFTyTxoiNGhWWS5ZEX3avkl48MOUBSotLuWveXdz1+l3RX0SSJEmSmjFpjNjYsWH59tu5\nKf/gIQfzixN/AcAFT17g+EZJUqya3tPo7KmSVLhMGiM2ZkxYLlqUu2tMO3wa3x7/bepT9Zzy0Cks\nXLMwdxeTJHUFFwGvAbXAPW0cey7QCGxq9pmQy+AkSYXNpDFi+UgaAX466aecNOok1m1dx3H3H8ei\ntTm+oCQpTiuB64G723n8/wB9m31ezFFcH7U0SpIKl0ljxPbfPyyXLIHGxtxdp6SohEe/+ijH73s8\na7as4XP3fI5XV7yauwtKkuI0C5gDrGvn8YkcxiJJ6mFMGiPWty8MGwZ1dVBdndtrlZWUMfv02Zyw\n3wmsrVnLxPsmMmfRnNxeVJIUp/Ykg2ngMOAfwGLgSqA4l0EBJGxwlKSCZdKYA01dVHM1GU5z5cly\n5pw+h2mHTWNrw1ZOffhUrpx7JQ2phpxcL5WCpUvhxRfhscfgkUfgiSfglVdgzRqwl5Ik5VR7/sq+\nCBwIDAZOA84AZuQsIP/wS1LBK4k7gEI0Zgz893+HcY1f+ELur5csTnLHyXew38D9uGLuFdzw0g28\n/O7L3HfqfYzYbUTW5S9aBLNnw1NPweuvw+bNOz928GCYMAGOPx5OOw123z3ry0tSoati5xPVvNxi\nX3taGt9ptr4QuI6QNP6wtYNnzpz50XplZSWVlZXtuMR2Jo2S1PVUVVVRVVUVWXk9YcxDOt8V2q23\nwkUXwbRpcOedeb00VdVVnPHYGby/+X0qkhXccOwNXDT+IoqLOtYzqb4eHn0UbrkltCI2t9de8MlP\nhgSxpARqa2HVqtAC+eGH248rKYFJk+BrX4NTToH+/bP/90nSriQSCSjsuu16YBhwXgfO+RrwPeBT\nrezLuo58Y8FiDlu3ikEfbmDtF0/NqixJUm5kWz929+6pAwmTA2wGqgldcGLX9K7GXM+g2prKT1by\nxvlv8JUDvsKW+i1c/MzFfPqOT/P00qfb9WtwOg2PPw4HHQRnnhkSxn794JxzQhK5ejWsWAEvvwyz\nZoXuqb/9Lbz2GqxfHyYAuv12OOGEUNaTT8LZZ4cE8wtfgHvuCQmmJKlDioEyQg+hYqAXOx+neAIw\nJLM+hjCmcXauArOlUZIKX3f/NfahzHIaYdD/k8DRwF+aHZP3lsb33gutcQMHwtq1kIjpLj+x+Aku\n/H8XsmLjCgAmjJjAjKNncOKoEylKfPz3gnffhenT4dlnw/eRI2HGDJg6Ffr06fj116wJieajj8IL\nL4TxkE3GjYPPfz58PvtZKC/vzL9QknZUwC2NM4GrW9l2HTAceAsYC6wAfgz8M9AHWA38itBC2dqc\n3lnXkfPeXMThH7xvS6MkdWHZ1o/duWKtAD4gDPZfmtl2H/AecHmz4/KeNKbTMGhQaHlbsSIkkHHZ\nWr+VW/90Kze+dCPra9cDsO+AfZl22DS+csBXGDVoFAAPPADf+hZs2hSS3euug3/5F0gmo4lj9eow\nLnLOnJBA1tRs31dSAoceCkcfDUcdFZZ77x1fsi2p+yrgpDFXoksaN6xn7alTIgpLkhSlnpw0HkaY\nIKCi2bbvApXAKc225T1pBKisDMnRk0/CiSfm/fIf82Hth9z5+p3c8qdbqN5Q/dH2g/cYR9Hyicyf\n80+w6nBOPXZvfvl/ixgyZOdlZWvbttDt9b/+K7Rqzpu3YyskhO6shxwSPoceGpZjxkSXxOZCOh1e\ntVJTA1u2hGVjY9MnTUPmE9ZTNKbSNDSkSKdDglxUFD6JBCSKtn+H9A77WlsnkaYo0ez8Vj4k0s3W\nw3/8uzpmhw/bk/h0OyZvbM8xze9b1CyzZ5YJMGLIbtC967Z8y7qOfO2Nv3DEhjUmjZLUhfXkpPFz\nwH8Cezbb9g3gTGBis22xJI3/+q9w881w441w+eVtH58vjalGnl76NA+/9TBzFs1hY93GHfaXJ8vZ\nf9D+DO8/nGH9hrF3v70Z1m8Yn+jzCQb0HsCAsgEM6D2Afr36tdrFtb0x1DXWUZ+qp66xjg0b63lt\nXj1//HMdf55Xz7wFdWzaUg/FdVBUD8VhvaS0nj32rGPwJ+oZtEcdAwfXU9G3jrI+9ZSV11Pau46i\nknpSiTpSiXrqM9eobwzXqWusp7a+jtr6erbW1bGtoZ66+npqG+qob6xnW0Pm+FQdDal6GtMNpNIp\nUuk06XSaVDpFmrCeJrNOCjJrkMq8qCwNicx601JSfswEunfdlm8mjZLUA2SbNHbnV25sBvq12NYf\n2NTywGynE++MQw4Jy/nzc36pDikuKuak0SfxmUEn8Zcf1fLn1b+n4sAXGDvpZf6+bSGrt6xm3vvz\nmPf+vF2WkyBBRWkFpcWlH32SRUmKi4ppSDXQkGqgvrF++3qz5G2XrVDjMp9WNBD6Hr/XtGFb5rO2\no3ehAzKtcpFJJyBdRIIEUATpRGY9s3tX57Wn7J3t+mgt0Y63vDUrZ6fHZhdPXpg29AzLG+Dd5u+l\nrYstlJ4u4YQ4klSwunPSuIQQ/35sH9N4COGdVDtonjTmS1dNGgE++AAmToQFC8rYZ5+JPPvziYwc\nmdm39QOWrFvCio0rdvi8v/l9NtRuYH3tejbUbmDjto1srtvFCxt3IUEiJJnFyY+SzZbfW9tXlC6l\nbmuSbTVJtm4ppWZTkm1bS9lWk6R2SylbtySpry2lsS5JY10SGkuhMbNMJUmkSilOJOlVkqQsWUpZ\naZLepaX0Lk1S3quUirIk5WVJKsrCekV5CeW9i6goT1BRXkR57wQVFYmwrSJBn4qmfQkqKoroU5Gg\nNJmgKFFEIpFZkiCRSHy0lJRb/nfWcV/L8t1QW1JpGD0qomgkSV1Rd69dHyK0hUwHDgd+BxwFvN3s\nmFi6p9bWhhlH02nYvBl69857CK3avBmOPx7+8IcwRvC552Do0I6X05BqoKa+plnXz/BpSDWQLE5S\nUlRCsiizzHxv3hqZa6lUGF/Y0BDGQSaTTeMDJRUyJ8LpsDTPPx9JQfu9+zf+evbXIylLkhStntw9\nFeBbwN3AGkInxW+yY8IYm7Iy2H9/+MtfYOFCOOKIuCMKCdSXvxwSxuHDwyQ0nUkYAUqKSujXq2Xv\n4K6jqCj8byBJ2rXfbNiQ1fkfbqnlnoW3U11fDZg0SlIh6u5J43qgy466P+SQkDTOn981ksYZM+CZ\nZ8LMpM8+C8OGxR2RJCluXzs1u3crLnvvA85fegYJdosoIklSV2OHvRw67LCw/POf440D4N574Wc/\nC900H38cRo+OOyJJkiRJ3YFJYw4deWRY/uEP8caxYAF885th/dZb4Zhj4o1HklSIHEoqSYXKpDGH\nPvUpKC4OSVtNTTwxbN0KZ54J27bB9OnwjW/EE4ckSZKk7smkMYcqKuDgg6GxMb4uqt//fpiIZ/To\n0D1VkiRJkjrCpDHHPvOZsIyji2pVFdx8M5SUwIMPhiRWkqQopVL5f62VJCm/TBpzrGlc46uv5ve6\n27bB+eeH9SuvDF1lJUmSJKmjTBpzLK7JcH7wA1iyBMaMCV1UJUnKqbQT4UhSoTJpzLH994cBA2Dl\nSnjnnfxc869/DUkjwG23Qa9e+bmuJEmSpMJj0phjRUVQWRnW587NzzUvuwzq6uC882DChPxcU5Ik\nSVJhMmnMg2OPDcvnn8/9tV56CWbNgvJyuOGG3F9PkiRJUmEzacyDiRPDcu5cSOdwkrlUCv7t38L6\n974He+6Zu2tJkgSQymXFJknqEkwa8+CAA2CPPWDVKli8OHfXeewx+NOfQrJ46aW5u44kSZKknsOk\nMQ8SCTjuuLD+1FO5uUYqBddfH9avvtp3MkqS8s3ZUyWpUJk05snJJ4flnDm5KX/OHFiwAIYNCxPg\nSJIkSVIUTBrz5MQTIZkME9WsXRtt2ek0XHddWP/+933FhiRJkqTomDTmSf/+YUKcVAqefDLasn/3\nO3jjjTCWcdq0aMuWJGlXnAhHkgqfSWMenXpqWD7ySLTl/vjHYTljBpSVRVu2JEmSpJ7NpDGPvvzl\n0EX1qafgvfeiKfPVV0OX1912g+nToylTkqSOcyIcSSpUJo15NHgwnHJK6KJ6//3RlPmTn4TlN78J\nfftGU6YkSZIkNTFpzLOvfz0sb7sN6uuzK2vZMnj88dB6+e1vZx+bJEmSJLVk0phnkybB/vtDdTU8\n9FB2Zd10U2i1nDoVhg6NJDxJkiRJ2oFJY54VF4fXYgDccEPnWxuXLoW774aiIrjssujikySpI1Ip\nZ0+VpEJn0hiDqVNh5EhYsiR0U+2Mq66ChgY45xwYMyba+CRJkiSpiUljDJLJ7RPYXHMNrFvXsfOf\new5+8xvo1SucL0lS7NLOnipJhcqkMSZf/CIcdxx88AFccAG0993INTVhplQIrY0jRuQuRkmSJEky\naYxJIgG33w59+sAjj8C997Z9TjoN3/pWGM944IEwY0bOw5QkSZLUw5k0xmjffeHmm8P6+efD3Lm7\nPv5HP4L77oPevcPMq6WluY9RkqRdSbe3q4wkqdsyaYzZeefBJZeEWVRPPjm8d7GlxsbQFfXyy0ML\n5d13w8EH5z9WSZIkST1PSdwBKEyKs3Ej3HUXnHYaTJkCX/86DBsGb70FP/0pvP56eL3GbbfB6afH\nHbEkSS05EY4kFSqTxi6gqAjuuAMOOgiuuAJmzQqf5oYOhfvvD5PnSJIkSVK+dOXuqRcBrwG1wD2t\n7D8OWARsAeYCw/MXWvQSCbj4Yli8GK69FiZMCEnk5Mlwyy3w17+aMEpSD1UK3AVUAxuBecDkNs65\nBFgFfJg511HwkqRO68pJ40rgeuDuVvbtDjwGXAEMICSXD+cvtNwZNgyuvhpeeAEWLICnnoILL4Ty\n8rgj67iqqqq4Q+jRvP/x8d4rYiXAu8AEoB9wJfCfwM5eujQJuAw4NnPMvsC1uQ9THeHfifh47+Pj\nve++unL31KYOmp8GhrXY9yVgISFxBJgJrAVGA0vyEZzaVlVVRWVlZdxh9Fje//h47xWxGnZM+p4E\n3gEOB5a3cvw5wJ3A25nv1wEPApe3VvjIS8/JKri61Fbon1URPZJ/J+LjvY+P97776spJY5PWRtYf\nCMxv9r0GWAochEmjJKmwDSH8SPrWTvYfwPYfXgHezJwzAFjf8uC/9b0/kqBK6wdHUo4kqevpDklj\nay+AqgD+0WLbRqBP7sORJCk2SeDXwL3s/EfSPoSxjE02ZpZ9aSVpnL77vZEE9s9TJkRSjiSp64lr\nfuwqwtiM1rzcYt//BvYCzmu27WeEivPCZtsWAFez46+rEFogR2YRqySp+1gG7Bd3EB1URfvqxCJC\nN9M+wBeBxp2c8wah7nw08313YA0wiI8njdaRktQzZFU/xtXSWNmBY1traXyLMGajSQWh0mutq053\n+z8PkqSepbIdxyQIs6AOBk5k5wkjhLrwULYnjYcAq2mllRHrSElSN1cMlAE/AO4HemW2QfjVdANh\nQpwy4CbglRhilCQpH34J/J7wI2lbJhFetzGWMI6xCrgxZ5FJkhSjmUCqxefqZvuPI8wMV0MBvKdR\nkqSdGEGoA2uATc0+Z2T2D898bz7T+CXA+2x/T2MyX8FKkiRJkiRJUrczkDABzmagmu2/vrbmEkK3\nnaZfX0tzHVyBa++9P5cwBqf5r+ROtdd5FwGvAbXAPW0c6zMfrfbe+3PxmY9aKeEZribMCDoPmLyL\n4332rR/jZP0YH+vI+FhHxsc6sh0eynzKgc8Sxjse0MpxkwjddcYCuwHPE8ZMqvPae+/PBV7MX1gF\nbwph9sRfsOs/yj7z0WvvvT8Xn/molQPXsH04wkmEinFEK8f67AfWj/GxfoyPdWR8rCPjYx3Zhgpg\nGzvOAHcfrf/DHyRMQ95kIiHDVud05N6fC7yUh5h6muvZ9R9ln/ncaeven4vPfD7MJ/yflJZ89q0f\n42T92DVYR8bHOrJriKyOLIowqLiMBhoI75pqMh84sJVjD8jsa/ImMIQwu5w6riP3Pg0cBvwD1ABV\nHQAAA7VJREFUWAxcyfbZcNV5bb1r1Wc+d9q69z7zuTeE8Heotdct+exbP8bJ+rFrsI6Mj3Vk/CKt\nIwshaexDaHptbhPQdyfHftjse9N5rR2rtnXk3r9IqCwHA6cRxnbMyGl0PUNr7zFtzmc+d9q69z7z\nuZUEfg3cCyxpZb/PvvVjnKwfuwbryPhYR8Yr8jqyEJLGzUC/Ftv6E/44t3Vs/8yytWPVto7c+3eA\n5Zn1hcB1wJdzF1qP0dYveT7zudPWvfeZz50i4FeEiRYu2skxPvvWj3GyfuwarCPjYx0Zn5zUkYWQ\nNC4BSthx3MAhhAewpbeAQ1sctxpYn7PoCltH7n1r2vqDora19Uuez3zutHXvW+Mzn70EYZa3pl+n\nG3dynM++9WOcrB+7BuvI+FhHxsM6sg0PEQZ0lgPHEGYoG9vKcZMIgzzHEvrsVgE35ifEgtXee38C\noa80wBhgAXBVPgIsUMVAGWFShfuBXrQ+FsBnPnrtvfc+87nxS+D3hIlGdsVnP7B+jI/1Y3ysI+Nj\nHRkv68g2DGDHdyGdntk+nNDMOqzZsZcQpphteidJMm9RFqb23vsfE+77ZmAZMBMHPGdjJpBq8bka\nn/l8mEn77r3PfPRGEO53DTu+2+sMfPZ3xvoxPtaP8ZmJdWRcZmIdGRfrSEmSJEmSJEmSJEmSJEmS\nJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJBWuauDYuIOQJElSz1UUdwCSdikNJOIO\nQpIkSZLU9fwKaARqgE3Ape04ZzzwGvAh8D7wH832fQZ4BVgPvAH8U7N9A4F7gJXAB8CsLGOXJEmS\nJOXBO3Sse+rvgamZ9XLgyMz6XsBaYHLm+//KfB+U+f4k8BDQHygBPtf5kCVJkiRJ+dLRpPEFYCaw\ne4vtlwH3t9j2NHA2sCehRbN/50KUJElSIXNMo1RYpgGjgbeBPwInZbaPAL5C6Jra9Pks8Algb0KX\n1A/zHawkSZIkKTt/o/Ozp54GbCV0U/0+cPtOjrOlUZIkSTtlS6PUta0GRrbYVk3oVtqas4DBmfUP\nCbOvNgIPACcDnweKgTKgkjDWcRXwFPALYDcgCUyIKH5JkiRJUg6dAiwndCf9LlAKbCR0QW3NrwiJ\n5iZgQeb8JuOBKmAdsAb4LaFrKsAA4F7CjKsfAI9G90+QJEmSJOXLZ4Ffxx2EJEmSJEmSJEmSJEmS\nJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJEmSJElSwfj/a5xaaU1xqckAAAAASUVO\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x5785910>"
       ]
      }
     ],
     "prompt_number": 68
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Dynamical inversion doesn't work perfectly here because we are stil sampling the control at a slow rate."
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Inverted Pendulum Controller Animation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def pend_anim(data_list, data_names, tf=1.0, fps=20, rate=1.0):\n",
      "    fig = plt.figure(figsize=(10,5))\n",
      "    ax = fig.gca()\n",
      "    def update(i):\n",
      "        i_fig = 1\n",
      "        ax.cla()\n",
      "        h = []\n",
      "        colors = itertools.cycle(['b', 'g', 'r', 'c', 'm', 'y', 'k'])\n",
      "        for data in data_list:\n",
      "            theta = data['x'][50*i*rate,0]\n",
      "            x = -1.0*np.sin(theta)\n",
      "            y = 1.0*np.cos(theta)\n",
      "            #ax = plt.subplot(1,3,i_fig)\n",
      "            ax.set_aspect('equal')\n",
      "            ax.axis(1.1*np.array([-1, 1, -1, 1]))\n",
      "            h.append(ax.plot(x, y, 'ro', markersize=10, c=colors.next())[0])\n",
      "            ax.plot([0, x], [0, y], 'k-')\n",
      "            i_fig += 1\n",
      "        ax.legend(h, data_names, loc='lower left')\n",
      "    return matplotlib.animation.FuncAnimation(plt.gcf(), update, frames=int(tf*fps/rate))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pend_anim([data_lqr, data_lqrd, data_dyn_inv], ['LQR', 'discrete LQR', 'dynamic inversoin'], rate=0.2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
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       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 70,
       "text": [
        "<matplotlib.animation.FuncAnimation at 0x5102790>"
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Estimating Disturbance Envelope"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop_lqr.A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 72,
       "text": [
        "matrix([[   0.        ,    1.        ],\n",
        "        [-140.01428499,  -21.81807897]])"
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop_lqr.B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 73,
       "text": [
        "matrix([[  0.],\n",
        "        [100.]])"
       ]
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "closed_loop_lqr.C"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 74,
       "text": [
        "matrix([[ 1.40014285,  0.        ],\n",
        "        [ 0.        ,  1.40014285]])"
       ]
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}