{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## GEKKO Python Example Applications\n",
    "\n",
    "![GEKKO Optimization Suite](http://apmonitor.com/wiki/uploads/Main/gekko_optimization.png)\n",
    "\n",
    "GEKKO is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP). Modes of operation include data reconciliation, real-time optimization, dynamic simulation, and nonlinear predictive control. See the [GEKKO documentation](http://gekko.readthedocs.io/en/latest/overview.html) for additional information.\n",
    "\n",
    "1. Solver Selection\n",
    "2. Solve Linear Equations\n",
    "3. Solve Nonlinear Equations\n",
    "4. Interpolation with Cubic Spline\n",
    "5. Linear and Polynomial Regression\n",
    "6. Nonlinear Regression\n",
    "7. Machine Learning / Artificial Neural Network\n",
    "8. Solve Differential Equation(s)\n",
    "9. Nonlinear Programming Optimization\n",
    "10. Mixed Integer Nonlinear Programming\n",
    "11. Optimal Control with Integral Objective\n",
    "12. Optimal Control with Economic Objective\n",
    "13. Optimal Control: Minimize Final Time\n",
    "14. PID Control Tuning\n",
    "15. Process Simulator\n",
    "16. Moving Horizon Estimation\n",
    "17. Model Predictive Control\n",
    "18. Debugging Resources"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Name: gekko\n",
      "Version: 0.0.4rc3\n",
      "Summary: Optimization software for differential algebraic equations\n",
      "Home-page: https://github.com/BYU-PRISM/GEKKO\n",
      "Author: BYU PRISM Lab\n",
      "Author-email: john_hedengren@byu.edu\n",
      "License: MIT\n",
      "Location: c:\\programdata\\anaconda3\\lib\\site-packages\n",
      "Requires: numpy, flask\n",
      "Required-by: \n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    # import gekko if installed\n",
    "    from gekko import GEKKO\n",
    "except:\n",
    "    # install gekko if error on try\n",
    "    !pip install gekko\n",
    "    from gekko import GEKKO\n",
    "\n",
    "# package information\n",
    "!pip show gekko\n",
    "\n",
    "# upgrade GEKKO to latest version\n",
    "# !pip install --upgrade gekko"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1: Solver selection\n",
    "\n",
    "Solve $y^2=1$ with APOPT solver. See [APMonitor documentation](https://apmonitor.com/wiki/index.php/Main/OptionApmSolver) or [GEKKO documentation](http://gekko.readthedocs.io/en/latest/global.html?highlight=solver#solver) for additional solver options."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y: [1.0]\n"
     ]
    }
   ],
   "source": [
    "m = GEKKO()           # create GEKKO model\n",
    "y = m.Var(value=2)    # define new variable, initial value=2\n",
    "m.Equation(y**2==1)   # define new equation\n",
    "m.options.SOLVER=1    # change solver (1=APOPT,3=IPOPT)\n",
    "m.solve(disp=False)   # solve locally (remote=False)\n",
    "print('y: ' + str(y.value)) # print variable value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2: Solve Linear Equations\n",
    "\n",
    "$3\\,x + 2\\,y=1$\n",
    "\n",
    "$x+2\\,y=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.5] [-0.25]\n"
     ]
    }
   ],
   "source": [
    "m = GEKKO()            # create GEKKO model\n",
    "x = m.Var()            # define new variable, default=0\n",
    "y = m.Var()            # define new variable, default=0\n",
    "m.Equations([3*x+2*y==1, x+2*y==0])  # equations\n",
    "m.solve(disp=False)    # solve\n",
    "print(x.value,y.value) # print solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3: Solve Nonlinear Equations\n",
    "\n",
    "$x+2\\,y=0$\n",
    "\n",
    "$x^2+y^2=1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.8944272, 0.4472136]\n"
     ]
    }
   ],
   "source": [
    "m = GEKKO()             # create GEKKO model\n",
    "x = m.Var(value=0)      # define new variable, initial value=0\n",
    "y = m.Var(value=1)      # define new variable, initial value=1\n",
    "m.Equations([x + 2*y==0, x**2+y**2==1]) # equations\n",
    "m.solve(disp=False)     # solve\n",
    "print([x.value[0],y.value[0]]) # print solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4: Interpolation with Cubic Spline\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15f5b8743c8>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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ljCKPAm7mJl5qMphZ4w6MOJ2IpDsV93hbtgxGjeLd5WPpxHss5lAuYzRf04DN\n1Mb+aw0rEZH4U3GPB3eYOTMMZfzb38CM9+qeSY3iMgC+pMm2Q7V0nYhUB91QjYcvvghT7U6dCr/7\nHSxZwroxE1lZ97AfHaal60SkuqjlXhVffgkPPABLl8Ljj0PTpjBlCvz856GC85/7pVq6TkSioOK+\nO2bPDhN4PfMMlJRA795hObtatXY4d3p+voq5iERD3TKxeuAByMuDF16A3/wGPv00TA1Qq1bUyURE\n/ota7jvzzTdh7dFjjgnTA5x+OvzwAwwYEFY5EhFJYmq5b2/BArjssvAU6XXXhal2IWwPHarCLiIp\nQS33ii64AMaNC10t+flw1VXQrl3UqUREdltmF/fi4rC6UX4+ZGVB+/ZwyCFhLdKf/CTqdCIiVZaZ\nxf2LL2DUKHjoIVi7NiwwffLJodtFRCQNZFaf+5o1cP75YYrG228P49KnTdOydSKSdtK/5b5lS5jv\n5eCDoX79sNj0b34Tpt495JCo04mIJET6FvfvvoPHHgsPHZWUhIWla9eG+fND/7qISBpLv26ZlSth\n2DBo3hyuuCIsMH3XXWHedFBhF5GMEFNxN7OeZrbQzBaZ2bAdvG9mNrL8/Y/N7Gfxj1qJsjADI9On\nw513hn70d94Ja5SedZaKuohklEqLu5llAaOAU4E2QH8za7PdYacCrco/BgIPxDnnjpWVwYsvQo8e\nYQ1SCIV88WKYOBE6d66WGCIiySaWlntHYJG7f+7um4EJQJ/tjukDjPPgPWA/M2uy/YniZuNGGDMG\n2rYN0wIsWhS6XwCys8NoGBGRDBbLDdWmwIoK24XAsTEc0xRYtUfpduass+Cll8JDR+PHh+3s7IRc\nSkQkFVXraBkzG0jotuGgPVmS6IYb4NprwwIZW2+UiojINrEU95VA8wrbzcr37e4xuPsYYAxAXl6e\n71bSirp2rfKXiohkglj63GcBrcyspZnVAvoBk7c7ZjJwfvmomU7AOndPTJeMiIhUqtKWu7uXmtkQ\n4FUgCxjr7nPNbHD5+w8CU4DTgEVAMTAgcZFFRKQyMfW5u/sUQgGvuO/BCq8duDy+0UREpKrS7wlV\nERFRcRcRSUcq7iIiaUjFXUQkDam4i4ikIQsDXSK4sFkRsKyKX94IWBPHOImWSnlTKSukVt5Uygqp\nlTeVssKe5W3h7o0rOyiy4r4nzKzA3fOizhGrVMqbSlkhtfKmUlZIrbyplBWqJ6+6ZURE0pCKu4hI\nGkrV4j4RMrSwAAADHElEQVQm6gC7KZXyplJWSK28qZQVUitvKmWFasibkn3uIiKya6nachcRkV1I\n2eJuZmeZ2VwzKzOzpLxLXtnC4snEzMaa2WozmxN1lsqYWXMzm2pm88p/Bq6KOtOumFkdM3vfzP5d\nnvcPUWeqjJllmdmHZvZi1FkqY2ZLzewTM/vIzAqizrMrZrafmU0yswVmNt/MErbQc8oWd2AOcCYw\nI+ogOxLjwuLJ5DGgZ9QhYlQKXOvubYBOwOVJ/r39ATjB3dsBuUDP8nUPktlVwPyoQ+yGHu6emwLD\nIe8FXnH3w4F2JPB7nLLF3d3nu/vCqHPsQiwLiycNd58BfB11jli4+yp3/6D89QbC/yBNo021c+UL\nx39Xvpld/pG0N7vMrBnQC3gk6izpxMz2BY4HHgVw983u/m2irpeyxT0F7GzRcIkjM8sB2gP/ijbJ\nrpV3c3wErAZed/dkzjsCuA4oizpIjBz4p5nNLl+nOVm1BIqAv5Z3eT1iZvUSdbGkLu5m9k8zm7OD\nj6RtAUv1MbO9geeAoe6+Puo8u+LuW9w9l7C+cEczOzLqTDtiZr2B1e4+O+osu+G48u/tqYQuuuOj\nDrQTNYGfAQ+4e3vgeyBh9+JiWokpKu5+YtQZ9kBMi4ZL1ZhZNqGwj3f356POEyt3/9bMphLubyTj\nzeuuwC/N7DSgDrCPmT3p7r+OONdOufvK8s+rzexvhC7RZLwXVwgUVvirbRIJLO5J3XJPcbEsLC5V\nYGZG6Lec7+53R52nMmbW2Mz2K3+9F3ASsCDaVDvm7je4ezN3zyH8zL6ZzIXdzOqZWf2tr4GTSc5f\nmrj7l8AKM2tdvusXwLxEXS9li7uZnWFmhUBn4CUzezXqTBW5eymwdWHx+cBEd58bbaqdM7OngXeB\n1mZWaGYXR51pF7oC5wEnlA9/+6i8pZmsmgBTzexjwi/919096YcYpogDgJlm9m/gfeAld38l4ky7\ncgUwvvxnIRf4U6IupCdURUTSUMq23EVEZOdU3EVE0pCKu4hIGlJxFxFJQyruIiJpSMVdRCQNqbiL\niKQhFXcRkTT0/wFk/KIDVdTySwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f6568be48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "xm = np.array([0,1,2,3,4,5])\n",
    "ym = np.array([0.1,0.2,0.3,0.5,1.0,0.9])\n",
    "\n",
    "m = GEKKO()             # create GEKKO model\n",
    "m.options.IMODE = 2     # solution mode\n",
    "x = m.Param(value=np.linspace(-1,6)) # prediction points\n",
    "y = m.Var()             # prediction results\n",
    "m.cspline(x, y, xm, ym) # cubic spline\n",
    "m.solve(disp=False)     # solve\n",
    "\n",
    "# create plot\n",
    "plt.plot(xm,ym,'bo')\n",
    "plt.plot(x.value,y.value,'r--',label='cubic spline')\n",
    "plt.legend(loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5: Linear and Polynomial Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x15f65c32748>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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nt3/ChsEbaBbYzI3BFm+aMJRShVdyMvTsCX//bbeNgWnT4IYbXLp854md1H+v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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f66019550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "xm = np.array([0,1,2,3,4,5])\n",
    "ym = np.array([0.1,0.2,0.3,0.5,0.8,2.0])\n",
    "\n",
    "#### Solution\n",
    "m = GEKKO()\n",
    "m.options.IMODE=2\n",
    "# coefficients\n",
    "c = [m.FV(value=0) for i in range(4)]\n",
    "x = m.Param(value=xm)\n",
    "y = m.CV(value=ym)\n",
    "y.FSTATUS = 1\n",
    "# polynomial model\n",
    "m.Equation(y==c[0]+c[1]*x+c[2]*x**2+c[3]*x**3)\n",
    "\n",
    "# linear regression\n",
    "c[0].STATUS=1\n",
    "c[1].STATUS=1\n",
    "m.solve(disp=False)\n",
    "p1 = [c[1].value[0],c[0].value[0]]\n",
    "\n",
    "# quadratic\n",
    "c[2].STATUS=1\n",
    "m.solve(disp=False)\n",
    "p2 = [c[2].value[0],c[1].value[0],c[0].value[0]]\n",
    "\n",
    "# cubic\n",
    "c[3].STATUS=1\n",
    "m.solve(disp=False)\n",
    "p3 = [c[3].value[0],c[2].value[0],c[1].value[0],c[0].value[0]]\n",
    "\n",
    "# plot fit\n",
    "plt.plot(xm,ym,'ko',markersize=10)\n",
    "xp = np.linspace(0,5,100)\n",
    "plt.plot(xp,np.polyval(p1,xp),'b--',linewidth=2)\n",
    "plt.plot(xp,np.polyval(p2,xp),'r--',linewidth=3)\n",
    "plt.plot(xp,np.polyval(p3,xp),'g:',linewidth=2)\n",
    "plt.legend(['Data','Linear','Quadratic','Cubic'],loc='best')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6: Nonlinear Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimized, a = 0.5990964\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x15f65fbfa90>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f6559d940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# measurements\n",
    "xm = np.array([0,1,2,3,4,5])\n",
    "ym = np.array([0.1,0.2,0.3,0.5,0.8,2.0])\n",
    "\n",
    "# GEKKO model\n",
    "m = GEKKO()\n",
    "\n",
    "# parameters\n",
    "x = m.Param(value=xm)\n",
    "a = m.FV()\n",
    "a.STATUS=1\n",
    "\n",
    "# variables\n",
    "y = m.CV(value=ym)\n",
    "y.FSTATUS=1\n",
    "\n",
    "# regression equation\n",
    "m.Equation(y==0.1*m.exp(a*x))\n",
    "\n",
    "# regression mode\n",
    "m.options.IMODE = 2\n",
    "\n",
    "# optimize\n",
    "m.solve(disp=False)\n",
    "\n",
    "# print parameters\n",
    "print('Optimized, a = ' + str(a.value[0]))\n",
    "\n",
    "plt.plot(xm,ym,'bo')\n",
    "plt.plot(xm,y.value,'r-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7: Machine Learning\n",
    "\n",
    "Approximate $y = sin(x)$ with an Artificial Neural Network\n",
    "\n",
    "__Trigonometric Function (trig=True)__\n",
    "\n",
    "* Input: $x$\n",
    "* Layer 1: linear layer, 1 node, $l1 = w1 \\; x$\n",
    "* Layer 2: nonlinear layer, 1 node, cosine function, $l2 = \\cos(w2a+w2b\\;l1)$\n",
    "* Layer 3: linear layer, 1 node, $l3 = w3 \\; l2$\n",
    "* Output: $y = \\sum{l3}$\n",
    "\n",
    "__Artificial Neural Network Description (trig=False)__\n",
    "\n",
    "* Input: $x$\n",
    "* Layer 1: linear layer, 2 nodes, $l1 = w1 \\; x$\n",
    "* Layer 2: nonlinear layer, 2 nodes, hyperbolic tangent activation function, $l2 = \\tanh(w2a+w2b\\;l1)$\n",
    "* Layer 3: linear layer, 2 nodes, $l3 = w3 \\; l2$\n",
    "* Output: $y = \\sum{l3}$\n",
    "\n",
    "See [Online Neural Network Demo](https://playground.tensorflow.org) with TensorFlow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w1[0]: 0.821782978\n",
      "w1[1]: 0.728869969\n",
      "w2a[0]: -1.41731735\n",
      "w2b[0]: 0.548984509\n",
      "w2a[1]: -0.131957576\n",
      "w2b[1]: 0.592683118\n",
      "w2a[2]: -0.330742133\n",
      "w2b[2]: 0.144440713\n",
      "w2a[3]: -2.92831117\n",
      "w2b[3]: 0.66823565\n",
      "w3[0]: -1.5171649\n",
      "w3[1]: -1.5171649\n",
      "w3[2]: 1.62870093\n",
      "w3[3]: 1.62870093\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x15f661ca240>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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RmSUincJQroole/bAokUwcKBrIaSnwwMfDaGeOchVLNRFYsAmgB497BgAFXOcagReCsQb\nYzoDzwHTytpRRMaJSJ6I5O3atcuh8JTr5s2zg7AGDHAthNRUyDjaj8PUZii2N5CvF4n57js7LiI5\n2e1IVISEIwFsA4ov19Q6sO0EY8x+Y8zBwOsMoKaINCntzYwxE4wxicaYxKZuVQUo582ebUfkXnGF\nayHk58Nh4simTyABmBPbfSk72w4C0wQQs8KRAD4D2otIOxGpBYwCZhTfQUSai9hpFkWke6Dc3WEo\nW8UCY2z9f79+rq4zG+zl+CFDOI8NnMuGEtt9JysL6teHyy5zOxIVISEnAGNMIXAHkAl8CUwxxqwW\nkfEiMj6w23XAKhFZDjwLjDJenodaOeuLL2DrVlfr/+HHRWLmYnu8JJPl70VisrKgTx9d/D2GheWT\nDVTrZJy07eVir58Hng9HWSoGZWbaZxfr/+HHMQepf2pPfn4bhsfNpeeE8f5cJGbjRvu45x63I1ER\npCOBlftmz4YLLvBEXUtKCmzaLMTf2p9BteaTMuq42yG5Y948+6z1/zFNE4ByV0EB5Oa6Xv1ziuRk\n2LsXli51OxJ3ZGVBy5bQsaPbkagI0gSg3LVgARw54nr1zyn69bPPc+e6G4cbiorsAjD9+ukSmTFO\nE4ByTXo6TBg1jyPUouMvenprwNXZZ0OXLvZK2G9Wr7ZjAPr2dTsSFWGaAJQrgss/Ju6fz8dcydot\ncd4bdZucbOfCKShwOxJnZWfb5z593I1DRZwmAOWK1FSoXbCbrixjPvZK03Ojbvv3txOhLVzodiTO\nys6Gdu2gbVu3I1ERpglAuSI/H5LIoRrmRAIIbveMnj3tEoh+agcoKrLtMnr17wuaAJQr4uOhL/M5\nSF0W073Eds+Ii4Of/MRf7QArVtiJ+TQB+IImAOWKtDToJ/PJpdeJxd89Oeo2ORmWL4edO92OxBnB\n+v+kJFfDUM7QBKBckZK0jY5mDZ837IsIzi//WFHBnjALFrgbh1Oys+G88+zCOCrmaQJQ7ghcaabO\n6+vO8o8Vdemldi784JVxLDt+3A7K0+of39AEoNwxfz6cdRZ07ep2JKcXXA3LDwlg2TLYt08TgI9o\nAlDOM8bONdOnD1SLgv+CffrAmjXwzTduRxJZWv/vO1Hw7VMxZ+NG298zSkaazjqcBMDoFjmxvU5w\ndrad+6dFC7cjUQ7RBKCcN3++fY6CBJCeDjc+0Y19nEkS2bG7TnBhoR3wplf/vqIJQDkvJweaNYuK\nmSZTU+HA4Rrk0oskcgAPjlgOh88/hwMHNAH4jCYA5SxjbAJISoqKmSaDI5NzSKIDX9GC7SW2x4xg\nN9fevd2NQzlKE4By1oYNsH171FxpBkcmZ2N7xvQhu8T2mJGTAx06QPPmbkeiHKQJQDkrJ8c+R0kC\nCK4TvJwu7KEhSeR4c8RyKI4ft/X/evXvO5oAlLOC9f8dOrgdSYWkpNgRym3aVmcBvelfI9ubI5ZD\nsWwZ7N8fNUlZhY8mAOWcYP1/795RUf8flJJiRyqP+HcSCYUbSOm1xe2Qwkvr/31LE4ByzsaNsG1b\n9F5pBkfIxtq8QAsW2Pl/WrZ0OxLlME0AyjlRVv9/iosvttNXBP+OWBCc/ydaPxMVkrAkABEZKCJr\nRWS9iNxfyu9FRJ4N/H6FiFwSjnJVlMnJsWvtRkH//1JVqwa9esXWHcDKlbB3r1b/+FTICUBEqgMv\nAIOAC4HRInLhSbsNAtoHHuOAl0ItV0WP9HRIaGvY8lYOMw/2Jv3t6Kn/P0Xv3rB+va3KinLp6fCX\nvjkA9Li/d+yNblblCscdQHdgvTFmozHmKDAZGH7SPsOBScZaBDQUEZ1wxAeCi79Xy/+aNmwloyAp\nuqdSCFaVRPldQPBzuXjPAjZwDou2tYnuz0VVSTgSQCugeLeIrYFtld1HxaDUVDt1QnAahRySonsq\nhc6doUGDqG8HSE2FwwVF9CKXBdjqn6j+XFSVeK4RWETGiUieiOTt2rXL7XBUiIJTJiSRw06a8iUX\nlNgedapXj4l2gPx8uIhVNOb7EwkguF35RzgSwDagTbGfWwe2VXYfAIwxE4wxicaYxKZNm4YhPOWm\n4JQJvVkQONFIie1RqXdv+Oor2LHD7UiqLD6+5F1Z8e3KP8KRAD4D2otIOxGpBYwCZpy0zwxgbKA3\n0BXAPmNM9H57VIWlpUHH2ptoS/6JK82on0ohBtoB0tKgb/UFfE0C+bQFYuBzUZUWcgIwxhQCdwCZ\nwJfAFGPMahEZLyLjA7tlABuB9cCrwK9DLVdFh5QUeG1sDgC59Pbu4u+V0bUrnHlmVLcDpIwuYmCd\nBeTVTUKE2PhcVKXVCMebGGMysCf54tteLvbaAL8JR1kq+vQ4ugAaN2bFzk4ebHWqgurV7TrBUZwA\nWL2a2gd3c/0bSVx/k9vBKLfEwtdRed2CBbbhNBrW/62o3r1h7droXSdY5/9RaAJQkZafD19/HXsn\nmmhvB8jJsfU+CQluR6JcpAlARVbwBBlrc81062bbAbKz3Y6k8oqK7OcSa5+JqjRNACqycnLsBGoX\nX+x2JOFVowbbzunJhv9kU62avZCOmlG0X3wB332nCUBpAlARtmCBbTCNpfp/7Mn++dV9OLfwK5qb\n7WzeTPRMpaD1/yogtr6Vylu2brVrAMfglWZqKmQes+sDBAdURc1UCjk5dsSX1v/7niYAFTkxfKWZ\nn//jOsHBheKD2z3NmB/r/6NoVTYVGZoAVOQsWGAnTuvSxe1Iwi4+HoqoTi69SiQAz0+l8MUXsGtX\nTCZlVXmaAFTkZGfb/v/Vq7sdSdilpdmpE7Lpw3lsoDVbomMqhfnz7XO/fu7GoTxBE4CKjK1b7cIp\nwXV0Y0xKip064asWSQBc1zgnOqZSmDcPzjnHjgFQvqcJQEVGsH98DDYAB6WkQMbWztCoEf++Jtv7\nJ//jx20DcN++bkeiPEITgIqM7Gzb/z8G6/9LqFbN1qdHw4Cwzz+Hffu0+kedoAlARUZOjj0xxlj/\n/1IlJcGmTfbhZfPm2ecYrZZTleeDb6dy3ObNdv4fv5xogn+nR+8C0tNtl//M++eztmYn0rOauR2S\n8ghNACr8gidCvySAiy6Cs8+GrCy3IzlFcPH37ZuP0pOFZB7rGz0jllXEaQJQ4ZedDU2aQKdObkfi\nDBFITrYJoKjI7WhKSE21I5SvYBFxHGYe/aJnxLKKOE0AKryMsQnAL/X/Qf37w86dsGqV25GUEByZ\n3Jf5HKfaiWU5PT9iWTnCR99Q5Yivv4YtW/xT/ROUnGyf5851N46TBEcm92U+S7mEfTQssV35myYA\nFV5+q/8Pat0aOnb0XDtAWho0qXOIK1jEPGz3z6gYsawcoQlAhVd2tm0QveACtyNxXnIy5ObCkSNu\nR3JCSgpMuSOXWhwjm766+LsqQROACh9j7BVwv37+nGmyf3/b4vrJJ25HUkKfI7Ohdm0yC3qyaZOe\n/NWPNAGo8Fm9Gr799sf6cL/p3dtOfOexaiAyM+1gtTp13I5EeYwmABU+wROfXxNAgwZw+eXeagje\ntAnWroUBA9yORHmQJgAVPllZ0L69r7uYrGiWzPHFeTSSPd5YJzgz0z4PHOhuHMqTQkoAItJIROaK\nyLrA81ll7LdJRFaKyDIRyQulTOVRx47Z+X/8evWPPdn/NiOZ6hSRRLY31gmePdsm5A4dXAxCeVWo\ndwD3A/OMMe2BeYGfy9LHGNPVGJMYYpnKiz79FA4d8nUCSE2FnCNXsI8zGUwG4PI6wceO2QngBg70\nZ6O8KleoCWA4MDHweiIwIsT3U9EqK8ueZPzW/7+Y/HwopCaZDGAwGQhFJ7a74pNP4MABrf9XZQo1\nATQzxuwIvP4GKGuaQQNkicgSERkXYpnKi7KyIDHRrgHgU8Gmj5kMpSU76MbnJbY7LjPT9krS+f9V\nGcpNACKSJSKrSnkML76fMcZgT/SlucoY0xUYBPxGRHqdprxxIpInInm7du2qzN+i3LJ/Pyxa5Ovq\nH/hxneBZDKIIYQgfujvqdvZs6NHD9k5SqhTlJgBjTLIx5qJSHtOBb0WkBUDgeWcZ77Et8LwTmAp0\nP015E4wxicaYxKZNm1blb1JOy821yw36PAEE1wmu27Ypn3I519aa6d6o2507YelS7f2jTivUKqAZ\nwE2B1zcB00/eQUTqikj94GvgasBbUyaq0GRlQe3acOWVbkfiupQU2/W+x2ND6Hr0M1KSv3UnkAzb\nCK0JQJ1OqAngCaC/iKwDkgM/IyItRSTwP5BmwEcishxYDHxojJkdYrnKSzIzoVcvmwSUNXSofQ6e\niJ02dSq0aQOXXOJO+SoqhJQAjDG7jTH9jDHtA1VF3we2bzfGDA683miM6RJ4dDLG6DyEsWTTJliz\nBgYNcjsSb+nSBVq1gg8/dLTY9HS4IP4Qh2fM4fW9I0h/W7t/qrLpSGAVmlmz7LMmgJJEYMgQe3d0\n9KgjRQaXf7xgSyZ1+IFJB0a6PxBNeZomABWaWbOgXTs4/3y3I/GeoUPh4EHbSO6A4PKPI5nKdzRm\nIT11+Ud1WpoAVNUdOQLz59urfx1peqq+fW27yLRpjhSXnw81OMZQZvIBwzhOjRPblSqNJgBVdQsX\n2ukftPqndHXr2ruAd9+FwsKIFxcfD0nkcBZ7mVZsUL6P5+ZT5dAEoKouIwNq1fL19A/lGjXK9slf\nsCDiRaWlwfU1pnKIOOZwNaDLP6rT0wSgqm7WLLsISt26bkfiXYMHQ716MHlyxItKGV3Ez+pNIzdu\nIEekji7/qMqlCUBVTbD75+DBbkfibXXqwIgR8N57ke8NtHgxcXt3MOiVkRQVocs/qnJpAlBVo90/\nKyyn2Y2wZw9Dz5gb2UVi3nrLNjoHB6EpVQ5NAKpqtPtnhaSnw4gXr2YPDbmRyZFbJOaHH+Dtt2Hk\nSGjYMMxvrmKVJgBVeYcO2XVvhwzR7p/lSE2FfYdr8R4/ZQTTqM3hyPTNnzED9uyBW24J8xurWKYJ\nQFXenDn2inPkSLcj8bxgH/zJjKI+B0+sFBb2vvmvv27n/unbN8xvrGKZJgBVeVOn2oVfepW5rIMK\nCPbBzyGJHTTnVl4rsT0stm2zSXnsWLsAjFIVpAlAVc6xYzBzJgwbBjVquB2N5wUXiTlODV7hlwwh\ng4trrwtv3/xJk6CoCG6+OYxvqvxAE4CqnNxcW9es1T8VElwkpm1bmMAvOUYNpvR+IXzdM42x1T89\ne8J554XpTZVfaAJQlTNtmu3bfvXVbkcSNYKLxGw3Lag55gY6fvK6Xaw9ROnp8NMWH8O6dfz+i1t0\n1k9VaZoAVMUZYxPAgAG2XkNV3p132jWUJ00K6W2CUz///Nt/spcGvLT7ep36WVWaJgBVcUuWwNat\ndmSrqprLL4fLLoPnn7f19lWUmgrtC5Yxguk8zT0cop5O/awqTROAqripU20vEx1pWnUicNdddhqN\nrKwqv01+PvyZx9jHmTzD3SW2K1VRmgBUxRhjE0CvXtC4sdvRRLfrr4dmzeCxx+xxrYKrm6/gp7zP\nM9zNXs46sV2nflaVoQlAVczy5fDll3DDDW5HEv3OOMOe/D/6iHuaplOtGpWeI+iVNo+xn/o8zT0n\ntunUz6qyNAGoinn7bdvv/7rr3I4kJqTXvo28apfxx92/p57ZX7k5glatou3i/5E//C7ObNsIEXTq\nZ1UlYqp4C+qExMREk5eX53YYqqjInmG6doUPPnA7mpiQkABNN3/Gp1zO09zDb3kKsId506bT/MOj\nR2013Jo1sGGDVsepU4jIEmNMYkX21TsAVb6FC23vH728DJv8fMjjMv4ft3MXz9KJVSe2nyw93SaM\natXgtSZ/gE8/hf/8R0/+KmSaAFT50tPtql/DhrkdScwINtb+ib+xjwa8w2iasvOURtxgf//Nm2Gk\neY9bDzzDCzXuJv2HnzoftIo5ISUAEbleRFaLSJGIlHnLISIDRWStiKwXkftDKVM57MgR+N//7NQP\nuvRj2ATnCNpNE27kv5zLBhZKL8b02nriaj8hAe6+GwoKoANreI1bWcTl3Fv4D+3vr8Ii1DuAVcC1\nQG5ZO4hIdeAFYBBwITBaRC4MsVzllNmz7dw/Y8a4HUlMKT5H0HxJ5qZmmcTX2M64t3pSd/NqjDFs\n3gyHdx8Q/tHbAAAL8ElEQVTiUR7ic7pxjJrcwBSOUUv7+6uwCCkBGGO+NMasLWe37sB6Y8xGY8xR\nYDIwPJRylYPS06FpU0hOdjuSmBOcI6ioCN79pic3NJ5PfbOf1VzETs5mNgNYSwce4jGmMYJLWcIW\nbB2R9vdX4eDEfL6tgC3Fft4KXF7WziIyDhgHEK//y931/fe2189tt0HNmm5HE/M+/DaRS1jKUGbS\njc+5hKWs5zxu5L98zE9O7Kf9/VW4lJsARCQLaF7Kr1KNMdPDHZAxZgIwAWw30HC/v6qEiRPtyl+/\n+IXbkfhCfDxs3tyWF/lNie2NG0PberaHUHy8PflrhywVDuUmAGNMqPf+24A2xX5uHdimvMwYePll\n6NEDunRxOxpfSEuzPX4KCn7cFhcHzzyjJ3wVGU50A/0MaC8i7USkFjAKmOFAuSoUOTnw1Vcwfrzb\nkfhG8YZhHd2rnBBqN9CRIrIV6AF8KCKZge0tRSQDwBhTCNwBZAJfAlOMMatDC1tF3EsvQaNGduIy\n5ZjiDcObNunJX0VWSI3AxpipwNRStm8HBhf7OQPICKUs5aBvvrEzf951l139SykVk3QksDrVa69B\nYaGtkFZKxSxNAKqkwkJb8dy3L3To4HY0SqkI0gSgSpoyxU48c+edbkeilIowTQDqR0VF8PjjcOGF\ncM01bkejlIqw2EwA//qXXcBcVc4HH8CqVfDAA3Y2MqVUTIu9b/m+ffDEE5CYCNdeCytXuh1RdDAG\n/vY3aNcORo1yOxqllANiLwE0aADr18Mjj8C8eXYU6+jRdlCTKtv8+bB4Mfzxj3bpR6VUzIu9BAA2\nCTz8MHz9ta3O+OADW699++2lL7mk7DwELVrAzTe7HYlSyiGxmQCCGjWyJ7YNG+COO+DNN6F9e7j3\nXti1y+3ovGPePMjOht/9Ds44w+1olFIOie0EENSsGTz9NKxbBz/7GTz7LJxzDjz6KBw44HZ07ios\ntMtOtWsHv/6129EopRzkjwQQFB9vF9NetQoGDLDtBOecY6dbPHLE7ejcMWECrF4NTz4JtWu7HY1S\nykH+SgBBF1xg17ldvBg6d4Z77rGjXt98E44fdzs653z/PTz0EPTpY9f8VUr5ij8TQNBll9n67zlz\noEkTGDsWLrkEMjJst8hY9+ijdr3fp5+28w8rpXzF3wkgqH9/ezcweTIcOgRDhkBSEixa5HZkkbNs\nGbzwgl3tq3Nnt6NRSrlAE0BQtWpw443w5Zfw4ouwdq1dDWvkSLstlhQUwJgxcPbZurisUj6mCeBk\nNWvCr35lB5M99pitIrroIrj1Vtiypfx/Hw3+8Aeb1CZOtAvOKqV8SRNAWerVgwcfhI0bbSNxerod\nQ3DffdE9hmDmTFv1c++9tupLKeVbmgDK06SJnVxu3Tq7Pt8zz9iuow89BHv3uh1d5Wzfbu9kOne2\n8/4opXxNE0BFBccQrF4NgwbZ6qF27WwdejQMJtu7FwYOhMOH4e23tc+/UkoTQKV17GgXTVm6FHr2\ntNVE7drZGUi9mgh++AGGD4c1a+xav506uR2RUsoDNAFUVbduMGMGfPqpHU/wwAOQkGDvCLxUNXT8\nuK26WrjQDnRLTnY7IqWUR2gCCFX37jBrlk0EV15p7wji4+20yjt2uBvb/v32yv/99+1grxtvdDce\npZSnaAIIl+7d7bTTS5fagWRPPmnvCG65xQ66ctrXX9uENHu2Hddw113Ox6CU8rSQEoCIXC8iq0Wk\nSEQST7PfJhFZKSLLRCQvlDI9r1s3eOcduwDN7bfDu+/abb1725HGP/wQ2fKNgenTbULats0mgF/9\nKrJlKqWiUqh3AKuAa4HcCuzbxxjT1RhTZqKIKeeea/vbb91q7wby8+3KZK1a2emX8/LCP9/QunX2\n7mPECGje3FZLaZ2/UqoMISUAY8yXxpi14QomJjVsCL/9rV2UZu5cO/jq5Zdtw/E559hRubm5cPRo\n1d7fGPjkE3u3cdFF8NFH8NRTtirq/PPD+7copWKKU4u/GiBLRI4DrxhjJjhUrndUq2avxpOT7TTM\n06fb6qF//xv++U+Ii7PdSnv0gIsvto927U5dn9cYW7WzeLF9fPABfPEF1K1rl3N85BG7tKNSSpWj\n3AQgIllA81J+lWqMmV7Bcq4yxmwTkbOBuSKyxhhTarWRiIwDxgHEx8dX8O2jTKNGtnH4lltsl9Gc\nHDvnUHBq6uJVQ/Xrw1ln2USwb599FBba39Wsaev6X33V9vCpX9+VP0cpFZ3EhKEeWkRygN8ZY8pt\n4BWRR4CDxpgny9s3MTHR5OXFdpvxKQ4dslf0K1fa9oM9e+wdw/HjdrH7Bg2gZUt74u/cWUf0KqVK\nEJElFW1rjXgVkIjUBaoZYw4EXl8N/CXS5UatunVt+8Bll7kdiVIqxoXaDXSkiGwFegAfikhmYHtL\nEckI7NYM+EhElgOLgQ+NMbNDKVcppVToQroDMMZMBaaWsn07MDjweiPQJZRylFJKhZ+OBFZKKZ/S\nBKCUUj6lCUAppXxKE4BSSvmUJgCllPIpTQBKKeVTYRkJHCkisgvY7HYcxTQBvnM7iNPQ+ELn9Rg1\nvtB5PcZQ42trjGlakR09nQC8RkTyvDydtcYXOq/HqPGFzusxOhmfVgEppZRPaQJQSimf0gRQOV5f\nx0DjC53XY9T4Quf1GB2LT9sAlFLKp/QOQCmlfEoTQBlE5BER2SYiywKPwWXsN1BE1orIehG53+EY\n/ykia0RkhYhMFZGGZey3SURWBv6OiK+wU94xEevZwO9XiMglkY6pWNltRCRbRL4QkdUicncp+ySJ\nyL5in/1DTsVXLIbTfmYuH8MOxY7NMhHZLyL3nLSP48dQRF4TkZ0isqrYtkYiMldE1gWezyrj30b8\ne1xGfO5+h40x+ijlATyCXeXsdPtUBzYA5wC1gOXAhQ7GeDVQI/D678Dfy9hvE9DEoZjKPSbYqcJn\nAQJcAXzq4DFrAVwSeF0f+KqU+JKAmS7//zvtZ+bmMSzl8/4G2/fc1WMI9AIuAVYV2/YP4P7A6/tL\n+4449T0uIz5Xv8N6BxCa7sB6Y8xGY8xRYDIw3KnCjTFzjDGBBYJZBLR2quzTqMgxGQ5MMtYioKGI\nOLKSvTFmhzFmaeD1AeBLoJUTZYeZa8fwJP2ADcYY1wdsGrvO+PcnbR4OTAy8ngiMKOWfOvI9Li0+\nt7/DmgBO787ArdlrZdw6tgK2FPt5K+6dTG7FXhGWxgBZIrJERMZFOI6KHBNPHDcRSQC6AZ+W8usr\nA5/9LBHp5GhgVnmfmSeOITAKeKeM37l9DAGaGWN2BF5/g12h8GReOZaOf4cjviawl4lIFtC8lF+l\nAi8Bj2EP/GPAv7AfkKNOF6MxZnpgn1SgEEgv422uMsZsE5GzgbkisiZwNeJbIlIPeA+4xxiz/6Rf\nLwXijTEHA20/04D2Dofo+c9MRGoB1wAPlPJrLxzDEowxRkQ82e3Rre+wrxOAMSa5IvuJyKvAzFJ+\ntQ1oU+zn1oFtYVNejCJyMzAU6GcClYWlvMe2wPNOEZmKveWN1MmkIsck4sftdESkJvbkn26Mef/k\n3xdPCMaYDBF5UUSaGGMcmz+mAp+Zq8cwYBCw1Bjz7cm/8MIxDPhWRFoYY3YEqsh2lrKP2/8fb8al\n77BWAZXhpPrUkcCqUnb7DGgvIu0CV0OjgBlOxAe25wLwB+AaY0xBGfvUFZH6wdfYRqfS/pZwqcgx\nmQGMDfRkuQLYV+w2PaJERID/AF8aY54qY5/mgf0Qke7Y78luJ+ILlFmRz8y1Y1jMaMqo/nH7GBYz\nA7gp8PomYHop+7j2PXb9OxzuVuVYeQBvAiuBFdj/DC0C21sCGcX2G4ztSbIBWy3jZIzrsXWXywKP\nl0+OEduzYXngsdqJGEs7JsB4YHzgtQAvBH6/Ekh08Jhdha3WW1HsuA0+Kb47AsdqObZh7kqHP9dS\nPzOvHMNA+XWxJ/QGxba5egyxyWgHcAxbj38b0BiYB6wDsoBGgX0d/x6XEZ+r32EdCayUUj6lVUBK\nKeVTmgCUUsqnNAEopZRPaQJQSimf0gSglFI+pQlAKaV8ShOAUkr5lCYApZTyqf8Pb2Yhswq+x6EA\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f661b25f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from gekko import GEKKO\n",
    "\n",
    "# generate training data\n",
    "x = np.linspace(0.0,2*np.pi,20)\n",
    "y = np.sin(x)\n",
    "\n",
    "# option for fitting function\n",
    "select = False # True / False\n",
    "if select:\n",
    "    # Size with cosine function\n",
    "    nin = 1  # inputs\n",
    "    n1 = 1   # hidden layer 1 (linear)\n",
    "    n2 = 1   # hidden layer 2 (nonlinear)\n",
    "    n3 = 1   # hidden layer 3 (linear)\n",
    "    nout = 1 # outputs\n",
    "else:\n",
    "    # Size with hyperbolic tangent function\n",
    "    nin = 1  # inputs\n",
    "    n1 = 2   # hidden layer 1 (linear)\n",
    "    n2 = 2   # hidden layer 2 (nonlinear)\n",
    "    n3 = 2   # hidden layer 3 (linear)\n",
    "    nout = 1 # outputs\n",
    "\n",
    "# Initialize gekko\n",
    "train = GEKKO() \n",
    "test = GEKKO()\n",
    "\n",
    "model = [train,test]\n",
    "\n",
    "for m in model:\n",
    "    # input(s)\n",
    "    m.inpt = m.Param()\n",
    "\n",
    "    # layer 1\n",
    "    m.w1 = m.Array(m.FV, (nin,n1))\n",
    "    m.l1 = [m.Intermediate(m.w1[0,i]*m.inpt) for i in range(n1)]\n",
    "\n",
    "    # layer 2\n",
    "    m.w2a = m.Array(m.FV, (n1,n2))\n",
    "    m.w2b = m.Array(m.FV, (n1,n2))\n",
    "    if select:\n",
    "        m.l2 = [m.Intermediate(sum([m.cos(m.w2a[j,i]+m.w2b[j,i]*m.l1[j]) \\\n",
    "                                for j in range(n1)])) for i in range(n2)]\n",
    "    else:\n",
    "        m.l2 = [m.Intermediate(sum([m.tanh(m.w2a[j,i]+m.w2b[j,i]*m.l1[j]) \\\n",
    "                                for j in range(n1)])) for i in range(n2)]\n",
    "        \n",
    "    # layer 3\n",
    "    m.w3 = m.Array(m.FV, (n2,n3))\n",
    "    m.l3 = [m.Intermediate(sum([m.w3[j,i]*m.l2[j] \\\n",
    "            for j in range(n2)])) for i in range(n3)]\n",
    "\n",
    "    # output(s)\n",
    "    m.outpt = m.CV()\n",
    "    m.Equation(m.outpt==sum([m.l3[i] for i in range(n3)]))\n",
    "\n",
    "    # flatten matrices\n",
    "    m.w1 = m.w1.flatten()\n",
    "    m.w2a = m.w2a.flatten()\n",
    "    m.w2b = m.w2b.flatten()\n",
    "    m.w3 = m.w3.flatten()\n",
    "    \n",
    "# Fit parameter weights\n",
    "m = train\n",
    "m.inpt.value=x\n",
    "m.outpt.value=y\n",
    "m.outpt.FSTATUS = 1\n",
    "for i in range(len(m.w1)):\n",
    "    m.w1[i].FSTATUS=1\n",
    "    m.w1[i].STATUS=1\n",
    "    m.w1[i].MEAS=1.0\n",
    "for i in range(len(m.w2a)):\n",
    "    m.w2a[i].STATUS=1\n",
    "    m.w2b[i].STATUS=1\n",
    "    m.w2a[i].FSTATUS=1\n",
    "    m.w2b[i].FSTATUS=1\n",
    "    m.w2a[i].MEAS=1.0\n",
    "    m.w2b[i].MEAS=0.5\n",
    "for i in range(len(m.w3)):\n",
    "    m.w3[i].FSTATUS=1\n",
    "    m.w3[i].STATUS=1\n",
    "    m.w3[i].MEAS=1.0\n",
    "m.options.IMODE = 2\n",
    "m.options.SOLVER = 3\n",
    "m.options.EV_TYPE = 2\n",
    "m.solve(disp=False)\n",
    "\n",
    "# Test sample points\n",
    "m = test\n",
    "for i in range(len(m.w1)):\n",
    "    m.w1[i].MEAS=train.w1[i].NEWVAL\n",
    "    m.w1[i].FSTATUS = 1\n",
    "    print('w1['+str(i)+']: '+str(m.w1[i].MEAS))\n",
    "for i in range(len(m.w2a)):\n",
    "    m.w2a[i].MEAS=train.w2a[i].NEWVAL\n",
    "    m.w2b[i].MEAS=train.w2b[i].NEWVAL\n",
    "    m.w2a[i].FSTATUS = 1\n",
    "    m.w2b[i].FSTATUS = 1\n",
    "    print('w2a['+str(i)+']: '+str(m.w2a[i].MEAS))\n",
    "    print('w2b['+str(i)+']: '+str(m.w2b[i].MEAS))\n",
    "for i in range(len(m.w3)):\n",
    "    m.w3[i].MEAS=train.w3[i].NEWVAL\n",
    "    m.w3[i].FSTATUS = 1\n",
    "    print('w3['+str(i)+']: '+str(m.w3[i].MEAS))\n",
    "m.inpt.value=np.linspace(-2*np.pi,4*np.pi,100)\n",
    "m.options.IMODE = 2\n",
    "m.options.SOLVER = 3\n",
    "m.solve(disp=False)\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x,y,'bo')\n",
    "plt.plot(test.inpt.value,test.outpt.value,'r-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 8: Solve Differential Equation(s)\n",
    "\n",
    "Solve the following differential equation with initial condition $y(0) = 5$:\n",
    "\n",
    "$ k \\, \\frac{dy}{dt} = -t \\, y$\n",
    "\n",
    "where $k=10$. The solution of $y(t)$ should be reported from an initial time $0$ to final time $20$. Create of plot of the result for $y(t)$ versus $t$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x15f66195cc0>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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xhN/liEiWUuBngGhhLv/8wdm8VXeErz2rE7JExBsK/AyxbPZ4bllyFt//806eXV/ndzki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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f657da0b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "m = GEKKO() # Initialize gekko\n",
    "a = 0.4\n",
    "m.time=np.linspace(0,20)\n",
    "y = m.Var(value=5.0)\n",
    "m.Equation(y.dt()==-a*y)\n",
    "m.options.IMODE=4\n",
    "m.options.NODES=3\n",
    "m.solve(disp=False)\n",
    "\n",
    "plt.plot(m.time,y.value)\n",
    "plt.xlabel('time')\n",
    "plt.ylabel('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9: Nonlinear Programming Optimization\n",
    "\n",
    "Solve the following nonlinear optimization problem:\n",
    "\n",
    "$\\min x_1 x_4 \\left(x_1 + x_2 + x_3\\right) + x_3$\n",
    "\n",
    "$\\mathrm{s.t.} \\quad x_1 x_2 x_3 x_4 \\ge 25$\n",
    "\n",
    "$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40$\n",
    "\n",
    "$1\\le x_1, x_2, x_3, x_4 \\le 5$\n",
    "\n",
    "with initial conditions:\n",
    "\n",
    "$x_0 = (1,5,5,1)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Results\n",
      "x1: [1.0]\n",
      "x2: [4.743]\n",
      "x3: [3.82115]\n",
      "x4: [1.379408]\n",
      "Objective: 17.0140171\n"
     ]
    }
   ],
   "source": [
    "m = GEKKO() # Initialize gekko\n",
    "# Use IPOPT solver (default)\n",
    "m.options.SOLVER = 3\n",
    "# Change to parallel linear solver\n",
    "m.solver_options = ['linear_solver ma97']\n",
    "# Initialize variables\n",
    "x1 = m.Var(value=1,lb=1,ub=5)\n",
    "x2 = m.Var(value=5,lb=1,ub=5)\n",
    "x3 = m.Var(value=5,lb=1,ub=5)\n",
    "x4 = m.Var(value=1,lb=1,ub=5)\n",
    "# Equations\n",
    "m.Equation(x1*x2*x3*x4>=25)\n",
    "m.Equation(x1**2+x2**2+x3**2+x4**2==40)\n",
    "m.Obj(x1*x4*(x1+x2+x3)+x3) # Objective\n",
    "m.options.IMODE = 3 # Steady state optimization\n",
    "m.solve(disp=False) # Solve\n",
    "print('Results')\n",
    "print('x1: ' + str(x1.value))\n",
    "print('x2: ' + str(x2.value))\n",
    "print('x3: ' + str(x3.value))\n",
    "print('x4: ' + str(x4.value))\n",
    "print('Objective: ' + str(m.options.objfcnval))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10: Mixed Integer Nonlinear Programming"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Results\n",
      "x1: [1.358909]\n",
      "x2: [4.599279]\n",
      "x3: [4.0]\n",
      "x4: [1.0]\n",
      "Objective: 17.5322673\n"
     ]
    }
   ],
   "source": [
    "m = GEKKO() # Initialize gekko\n",
    "m.options.SOLVER=1  # APOPT is an MINLP solver\n",
    "\n",
    "# optional solver settings with APOPT\n",
    "m.solver_options = ['minlp_maximum_iterations 500', \\\n",
    "                    # minlp iterations with integer solution\n",
    "                    'minlp_max_iter_with_int_sol 10', \\\n",
    "                    # treat minlp as nlp\n",
    "                    'minlp_as_nlp 0', \\\n",
    "                    # nlp sub-problem max iterations\n",
    "                    'nlp_maximum_iterations 50', \\\n",
    "                    # 1 = depth first, 2 = breadth first\n",
    "                    'minlp_branch_method 1', \\\n",
    "                    # maximum deviation from whole number\n",
    "                    'minlp_integer_tol 0.05', \\\n",
    "                    # covergence tolerance\n",
    "                    'minlp_gap_tol 0.01']\n",
    "\n",
    "# Initialize variables\n",
    "x1 = m.Var(value=1,lb=1,ub=5)\n",
    "x2 = m.Var(value=5,lb=1,ub=5)\n",
    "# Integer constraints for x3 and x4\n",
    "x3 = m.Var(value=5,lb=1,ub=5,integer=True)\n",
    "x4 = m.Var(value=1,lb=1,ub=5,integer=True)\n",
    "# Equations\n",
    "m.Equation(x1*x2*x3*x4>=25)\n",
    "m.Equation(x1**2+x2**2+x3**2+x4**2==40)\n",
    "m.Obj(x1*x4*(x1+x2+x3)+x3) # Objective\n",
    "m.solve(disp=False) # Solve\n",
    "print('Results')\n",
    "print('x1: ' + str(x1.value))\n",
    "print('x2: ' + str(x2.value))\n",
    "print('x3: ' + str(x3.value))\n",
    "print('x4: ' + str(x4.value))\n",
    "print('Objective: ' + str(m.options.objfcnval))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 11: Optimal Control with Integral Objective\n",
    "\n",
    "__Original Form__\n",
    "\n",
    "$\\min_u \\frac{1}{2} \\int_0^2 x_1^2(t) \\, dt$\n",
    "\n",
    "$\\mathrm{subject \\; to}$\n",
    "\n",
    "$\\frac{dx_1}{dt}=u$\n",
    "\n",
    "$x_1(0) = 1$\n",
    "\n",
    "$-1 \\le u(t) \\le 1$\n",
    "\n",
    "__Equivalent Form for GEKKO with new Variable $x_2$__\n",
    "\n",
    "$\\min_u x_2\\left(t_f\\right)$\n",
    "\n",
    "$\\mathrm{subject \\; to}$\n",
    "\n",
    "$\\frac{dx_1}{dt}=u$\n",
    "\n",
    "$\\frac{dx_2}{dt} = \\frac{1}{2} x_1^2(t)$\n",
    "\n",
    "$x_1(0) = 1$\n",
    "\n",
    "$x_2(0) = 0$\n",
    "\n",
    "$t_f = 2$\n",
    "\n",
    "$-1 \\le u(t) \\le 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x15f663b7710>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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NCUVS2NE65c9z92eUTIBvvoEaNeAvfwk7EinpOneGnj3DjkKkUNT7F41y5YKk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\nUEREJC6UUEREJC6UUEREJC7+P1lGcF7XGuSmAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f6559d668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from gekko import GEKKO\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "m = GEKKO() # initialize gekko\n",
    "nt = 101\n",
    "m.time = np.linspace(0,2,nt)\n",
    "# Variables\n",
    "x1 = m.Var(value=1)\n",
    "x2 = m.Var(value=0)\n",
    "u = m.Var(value=0,lb=-1,ub=1)\n",
    "p = np.zeros(nt) # mark final time point\n",
    "p[-1] = 1.0\n",
    "final = m.Param(value=p)\n",
    "# Equations\n",
    "m.Equation(x1.dt()==u)\n",
    "m.Equation(x2.dt()==0.5*x1**2)\n",
    "m.Obj(x2*final) # Objective function\n",
    "m.options.IMODE = 6 # optimal control mode\n",
    "m.solve(disp=False) # solve\n",
    "plt.figure(1) # plot results\n",
    "plt.plot(m.time,x1.value,'k-',label=r'$x_1$')\n",
    "plt.plot(m.time,x2.value,'b-',label=r'$x_2$')\n",
    "plt.plot(m.time,u.value,'r--',label=r'$u$')\n",
    "plt.legend(loc='best')\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Value')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 12: Optimal Control with Economic Objective\n",
    "\n",
    "__Original Form__\n",
    "\n",
    "$\\max_{u(t)} \\int_0^{10} \\left(E-\\frac{c}{x}\\right) u \\, U_{max} \\, dt$\n",
    "\n",
    "$\\mathrm{subject \\; to}$\n",
    "\n",
    "$\\frac{dx}{dt}=r \\, x(t) \\left(1-\\frac{x(t)}{k}\\right)-u \\, U_{max}$\n",
    "\n",
    "$x(0) = 70$\n",
    "\n",
    "$0 \\le u(t) \\le 1$\n",
    "\n",
    "$E=1, \\, c=17.5, \\, r=0.71$\n",
    "\n",
    "$k=80.5, \\, U_{max}=20$\n",
    "\n",
    "__Equivalent Form for GEKKO__\n",
    "\n",
    "$\\min_{u(t)} -J\\left(t_f\\right)$\n",
    "\n",
    "$\\mathrm{subject \\; to}$\n",
    "\n",
    "$\\frac{dx}{dt}=r \\, x(t) \\left(1-\\frac{x(t)}{k}\\right)-u \\, U_{max}$\n",
    "\n",
    "$\\frac{dJ}{dt} = \\left(E-\\frac{c}{x}\\right) u \\, U_{max}$\n",
    "\n",
    "$x(0) = 70$\n",
    "\n",
    "$J(0) = 0$\n",
    "\n",
    "$0 \\le u(t) \\le 1$\n",
    "\n",
    "$t_f = 10, \\, E=1, \\, c=17.5$\n",
    "\n",
    "$r=0.71, \\, k=80.5, \\, U_{max}=20$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimal Profit: 106.9061\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15f687b56a0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wEP7615Ie+67gxUm/+IVL6Oef79bOebCGPT8/n9TU1FqT5AFEhKZNm7Jnzx6vQ4kriYku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\n7yNeAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f686e9860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from gekko import GEKKO\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# create GEKKO model\n",
    "m = GEKKO()\n",
    "# time points\n",
    "n=501\n",
    "m.time = np.linspace(0,10,n)\n",
    "# constants\n",
    "E,c,r,k,U_max = 1,17.5,0.71,80.5,20\n",
    "# fishing rate\n",
    "u = m.MV(value=1,lb=0,ub=1)\n",
    "u.STATUS = 1\n",
    "u.DCOST = 0\n",
    "x = m.Var(value=70) # fish population\n",
    "# fish population balance\n",
    "m.Equation(x.dt() == r*x*(1-x/k)-u*U_max)\n",
    "J = m.Var(value=0) # objective (profit)\n",
    "Jf = m.FV() # final objective\n",
    "Jf.STATUS = 1\n",
    "m.Connection(Jf,J,pos2='end')\n",
    "m.Equation(J.dt() == (E-c/x)*u*U_max)\n",
    "m.Obj(-Jf) # maximize profit\n",
    "m.options.IMODE = 6  # optimal control\n",
    "m.options.NODES = 3  # collocation nodes\n",
    "m.options.SOLVER = 3 # solver (IPOPT)\n",
    "m.solve(disp=False) # Solve\n",
    "print('Optimal Profit: ' + str(Jf.value[0]))\n",
    "plt.figure(1) # plot results\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(m.time,J.value,'r--',label='profit')\n",
    "plt.plot(m.time,x.value,'b-',label='fish')\n",
    "plt.legend()\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(m.time,u.value,'k--',label='rate')\n",
    "plt.xlabel('Time (yr)')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 13: Optimal Control: Minimize Final Time\n",
    "\n",
    "__Original Form__\n",
    "\n",
    "$\\min_{u(t)} \\; t_f$\n",
    "\n",
    "$\\mathrm{subject \\; to}$\n",
    "\n",
    "$\\frac{dx_1}{dt}=u$\n",
    "\n",
    "$\\frac{dx_2}{dt}=\\cos\\left(x_1(t)\\right)$\n",
    "\n",
    "$\\frac{dx_3}{dt}=\\sin\\left(x_1(t)\\right)$\n",
    "\n",
    "$x(0) = \\left[\\pi/2,4,0\\right]$\n",
    "\n",
    "$x_2\\left(t_f\\right)=0$\n",
    "\n",
    "$x_3\\left(t_f\\right)=0$\n",
    "\n",
    "$-2 \\le u(t) \\le 2$\n",
    "\n",
    "__Equivalent Form for GEKKO__\n",
    "\n",
    "$\\min_{u(t), t_f} \\; t_f$\n",
    "\n",
    "$\\mathrm{subject \\; to}$\n",
    "\n",
    "$\\frac{dx_1}{dt}=t_f \\, u$\n",
    "\n",
    "$\\frac{dx_2}{dt}=t_f \\, \\cos\\left(x_1(t)\\right)$\n",
    "\n",
    "$\\frac{dx_3}{dt}=t_f \\, \\sin\\left(x_1(t)\\right)$\n",
    "\n",
    "$x(0) = \\left[\\pi/2,4,0\\right]$\n",
    "\n",
    "$x_2\\left(t_f\\right)=0$\n",
    "\n",
    "$x_3\\left(t_f\\right)=0$\n",
    "\n",
    "$-2 \\le u(t) \\le 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Final Time: 4.316256\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x15f68817518>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wejXcd5+tpGnZ0k6EUiqnfJO7MWY54FWjfMEidkAyUMyZAyNHOh2F8hKlS8OE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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f65c24fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from gekko import GEKKO\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "m = GEKKO() # initialize GEKKO\n",
    "nt = 501\n",
    "m.time = np.linspace(0,1,nt)\n",
    "# Variables\n",
    "x1 = m.Var(value=np.pi/2.0)\n",
    "x2 = m.Var(value=4.0)\n",
    "x3 = m.Var(value=0.0)\n",
    "p = np.zeros(nt) # final time = 1\n",
    "p[-1] = 1.0\n",
    "final = m.Param(value=p)\n",
    "# optimize final time\n",
    "tf = m.FV(value=1.0,lb=0.1,ub=100.0)\n",
    "tf.STATUS = 1\n",
    "# control changes every time period\n",
    "u = m.MV(value=0,lb=-2,ub=2)\n",
    "u.STATUS = 1\n",
    "m.Equation(x1.dt()==u*tf)\n",
    "m.Equation(x2.dt()==m.cos(x1)*tf)\n",
    "m.Equation(x3.dt()==m.sin(x1)*tf)\n",
    "m.Equation(x2*final<=0)\n",
    "m.Equation(x3*final<=0)\n",
    "m.Obj(tf)\n",
    "m.options.IMODE = 6\n",
    "m.solve(disp=False)\n",
    "print('Final Time: ' + str(tf.value[0]))\n",
    "tm = np.linspace(0,tf.value[0],nt)\n",
    "plt.figure(1)\n",
    "plt.plot(tm,x1.value,'k-',label=r'$x_1$')\n",
    "plt.plot(tm,x2.value,'b-',label=r'$x_2$')\n",
    "plt.plot(tm,x3.value,'g--',label=r'$x_3$')\n",
    "plt.plot(tm,u.value,'r--',label=r'$u$')\n",
    "plt.legend(loc='best')\n",
    "plt.xlabel('Time')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 14: PID Control Tuning\n",
    "\n",
    "A [PID Controller](https://en.wikipedia.org/wiki/PID_controller) has proportional, integral, and derivative terms to determine the controller output ($OP$) based on the set point ($SP$) and process variable ($PV$). A standard PID form has constants $K_c$, $\\tau_I$, and $\\tau_D$.\n",
    "\n",
    "\n",
    "$err = SP-PV$\n",
    "\n",
    "$OP = OP_0 + K_c \\, err + \\frac{K_c}{\\tau_I} \\int err \\, dt - K_c \\, \\tau_D \\frac{d\\,PV}{dt}$ \n",
    "\n",
    "The effect of the tuning constants is shown with the [PID Tuning Notebook](http://nbviewer.jupyter.org/url/apmonitor.com/pdc/uploads/Main/pid_widget.ipynb). This example is an alternative implementation in GEKKO."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15f699b3b70>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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gwgTX97t79/DGZ0x1+fmuqmjYML8jMbEu3FNMzFbVR7wtqkkg0pKS3NrEoSYBcBOQLVvm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\nl1LwoxNTMekAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f69918a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from gekko import GEKKO\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "m = GEKKO()\n",
    "tf = 40\n",
    "m.time = np.linspace(0,tf,2*tf+1)\n",
    "step = np.zeros(2*tf+1)\n",
    "step[3:40] = 2.0\n",
    "step[40:]  = 5.0\n",
    "\n",
    "# Controller model\n",
    "Kc = 15.0                    # controller gain\n",
    "tauI = 2.0                  # controller reset time\n",
    "tauD = 1.0                  # derivative constant\n",
    "OP_0 = m.Const(value=0.0)   # OP bias\n",
    "OP = m.Var(value=0.0)       # controller output\n",
    "PV = m.Var(value=0.0)       # process variable\n",
    "SP = m.Param(value=step)    # set point\n",
    "Intgl = m.Var(value=0.0)    # integral of the error\n",
    "err = m.Intermediate(SP-PV) # set point error\n",
    "m.Equation(Intgl.dt()==err) # integral of the error\n",
    "m.Equation(OP == OP_0 + Kc*err + (Kc/tauI)*Intgl - PV.dt())\n",
    "\n",
    "# Process model\n",
    "Kp = 0.5                    # process gain\n",
    "tauP = 10.0                 # process time constant\n",
    "m.Equation(tauP*PV.dt() + PV == Kp*OP)\n",
    "\n",
    "m.options.IMODE=4\n",
    "m.solve(disp=False)\n",
    "\n",
    "plt.figure()\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(m.time,OP.value,'b:',label='OP')\n",
    "plt.ylabel('Output')\n",
    "plt.legend()\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(m.time,SP.value,'k-',label='SP')\n",
    "plt.plot(m.time,PV.value,'r--',label='PV')\n",
    "plt.xlabel('Time (sec)')\n",
    "plt.ylabel('Process')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 15: Process Simulator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15f69a82080>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Rn7uIzMEm8Nz3Tcj1vQD3ezc0L8mvDz13nzvoZCKlVFAp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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f699fd7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from gekko import GEKKO\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# Generate \"data\" with process simulation\n",
    "nt = 51\n",
    "# input steps\n",
    "u_meas = np.zeros(nt)\n",
    "u_meas[3:10] = 1.0\n",
    "u_meas[10:20] = 2.0\n",
    "u_meas[20:40] = 0.5\n",
    "u_meas[40:] = 3.0\n",
    "# simulation model\n",
    "p = GEKKO()\n",
    "p.time = np.linspace(0,10,nt)\n",
    "n = 1 #process model order\n",
    "# Parameters\n",
    "steps = np.zeros(n)\n",
    "p.u = p.MV(value=u_meas)\n",
    "p.u.FSTATUS=1\n",
    "p.K = p.Param(value=1) #gain\n",
    "p.tau = p.Param(value=5) #time constant\n",
    "# Intermediate\n",
    "p.x = [p.Intermediate(p.u)]\n",
    "# Variables\n",
    "p.x.extend([p.Var() for _ in range(n)])  #state variables\n",
    "p.y = p.SV() #measurement\n",
    "# Equations\n",
    "p.Equations([p.tau/n * p.x[i+1].dt() == -p.x[i+1] + p.x[i] for i in range(n)])\n",
    "p.Equation(p.y == p.K * p.x[n])\n",
    "# Simulate\n",
    "p.options.IMODE = 4\n",
    "p.solve(disp=False)\n",
    "# add measurement noise\n",
    "y_meas = (np.random.rand(nt)-0.5)*0.2\n",
    "for i in range(nt):\n",
    "    y_meas[i] += p.y.value[i]\n",
    "plt.plot(p.time,u_meas,'b:',label='Input (u) meas')\n",
    "plt.plot(p.time,y_meas,'ro',label='Output (y) meas')\n",
    "plt.plot(p.time,p.y.value,'k-',label='Output (y) actual')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 16: Moving Horizon Estimation\n",
    "\n",
    "Run the Process Simulation cell above to generate the data. The MHE application uses a first order model while the process simulation is a second order system. This is done to emulate a realistic case with model mismatch and measurement noise.\n",
    "\n",
    "This demonstrates just one cycle of an MHE application. Typical MHE applications receive an additional measurements, re-optimize parameters and states, and re-inject the parameters into a controller. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15f69b5b358>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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lCpvcCw47mhYxzXLV+noSfPfu3UlLS+Ozz4wCmDk5OYwfP56hQ4fi5eVFQEAA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\nRqG/wbZtpIW1pNqvv/Ph+Yt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      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f688c2c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from gekko import GEKKO\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# Estimator Model\n",
    "m = GEKKO()\n",
    "m.time = p.time\n",
    "# Parameters\n",
    "m.u = m.MV(value=u_meas) #input\n",
    "m.K = m.FV(value=1, lb=1, ub=3)    # gain\n",
    "m.tau = m.FV(value=5, lb=1, ub=10) # time constant\n",
    "# Variables\n",
    "m.x = m.SV() #state variable\n",
    "m.y = m.CV(value=y_meas) #measurement\n",
    "# Equations\n",
    "m.Equations([m.tau * m.x.dt() == -m.x + m.u, \n",
    "             m.y == m.K * m.x])\n",
    "# Options\n",
    "m.options.IMODE = 5 #MHE\n",
    "m.options.EV_TYPE = 1\n",
    "# STATUS = 0, optimizer doesn't adjust value\n",
    "# STATUS = 1, optimizer can adjust\n",
    "m.u.STATUS = 0\n",
    "m.K.STATUS = 1\n",
    "m.tau.STATUS = 1\n",
    "m.y.STATUS = 1\n",
    "# FSTATUS = 0, no measurement\n",
    "# FSTATUS = 1, measurement used to update model\n",
    "m.u.FSTATUS = 1\n",
    "m.K.FSTATUS = 0\n",
    "m.tau.FSTATUS = 0\n",
    "m.y.FSTATUS = 1\n",
    "# DMAX = maximum movement each cycle\n",
    "m.K.DMAX = 2.0\n",
    "m.tau.DMAX = 4.0\n",
    "# MEAS_GAP = dead-band for measurement / model mismatch\n",
    "m.y.MEAS_GAP = 0.25\n",
    "\n",
    "# solve\n",
    "m.solve(disp=False)\n",
    "\n",
    "# Plot results\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(m.time,u_meas,'b:',label='Input (u) meas')\n",
    "plt.legend()\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(m.time,y_meas,'gx',label='Output (y) meas')\n",
    "plt.plot(p.time,p.y.value,'k-',label='Output (y) actual')\n",
    "plt.plot(m.time,m.y.value,'r--',label='Output (y) estimated')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 17: Model Predictive Control"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15f69c21860>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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SSgFDgS7Fr1H9fPyx8c9dFboUFeWj1KWTcmho6da9ePFSskhOvjRYvz9/Hi5cuDQcP268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\nSRBCCCFskgQhhBDCpv8H4C07n2GUhxgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x15f688110f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from random import random\n",
    "from gekko import GEKKO\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "m = GEKKO()\n",
    "m.time = np.linspace(0,20,41)\n",
    "\n",
    "# Parameters\n",
    "mass = 500\n",
    "b = m.Param(value=50)\n",
    "K = m.Param(value=0.8)\n",
    "\n",
    "# Manipulated variable\n",
    "p = m.MV(value=0, lb=0, ub=100)\n",
    "p.STATUS = 1  # allow optimizer to change\n",
    "p.DCOST = 0.1 # smooth out gas pedal movement\n",
    "p.DMAX = 20   # slow down change of gas pedal\n",
    "\n",
    "# Controlled Variable\n",
    "v = m.CV(value=0)\n",
    "v.STATUS = 1  # add the SP to the objective\n",
    "m.options.CV_TYPE = 2 # squared error\n",
    "v.SP = 40     # set point\n",
    "v.TR_INIT = 1 # set point trajectory\n",
    "v.TAU = 5     # time constant of trajectory\n",
    "\n",
    "# Process model\n",
    "m.Equation(mass*v.dt() == -v*b + K*b*p)\n",
    "\n",
    "m.options.IMODE = 6 # control\n",
    "m.solve(disp=False)\n",
    "\n",
    "# get additional solution information\n",
    "import json\n",
    "with open(m.path+'//results.json') as f:\n",
    "    results = json.load(f)\n",
    "\n",
    "plt.figure()\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(m.time,p.value,'b-',label='MV Optimized')\n",
    "plt.legend()\n",
    "plt.ylabel('Input')\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(m.time,results['v1.tr'],'k-',label='Reference Trajectory')\n",
    "plt.plot(m.time,v.value,'r--',label='CV Response')\n",
    "plt.ylabel('Output')\n",
    "plt.xlabel('Time')\n",
    "plt.legend(loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 18: Debugging Resources\n",
    "\n",
    "Applications may need a more detailed inspection to find errors in programming syntax, errors in modeling assumptions, or to generate good initial guess values. The GEKKO or the solver solution reports syntax errors. Setting m.solve(disp=True) displays the solver output with a message on the line of code that is unsuccessful. Naming the variables such as name='state' is helpful to display the equations in a readable form.\n",
    "\n",
    "Other strategies for obtaining a successful solution include:\n",
    "\n",
    "* Increase the number of iterations with MAX_ITER (0-1000+)\n",
    "* Increase the diagnostic level with DIAGLEVEL (0-10)\n",
    "* Change the solver with SOLVER (1-5)\n",
    "* Calculate model SENSITIVITY (1)\n",
    "* Solve a square problem with # Variables = # Equations\n",
    "* Set COLDSTART to initialize problem (0-2)\n",
    "\n",
    "Additional [modeling](http://apmonitor.com/do/index.php/Main/ModelFormulation), [initialization](http://apmonitor.com/do/index.php/Main/ModelInitialization), and [decomposition](https://www.sciencedirect.com/science/article/pii/S0098135415001179) tips may be helpful. There is also an online [discussion group](http://apmonitor.com/wiki/index.php/Main/UsersGroup), [video playlist](https://www.youtube.com/playlist?list=PLLBUgWXdTBDjcqDl2e5F_hcBjEc6vjr1X), [GEKKO documentation](http://gekko.readthedocs.io/en/latest/), and [APMonitor documentation](http://apmonitor.com/wiki/index.php/Main/HomePage) as additional resources."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--------- Follow local path to view files --------------\n",
      "C:\\Users\\johnh\\AppData\\Local\\Temp\\tmp7c4y0_fmgk_model18\n",
      "--------------------------------------------------------\n",
      "apm 10.5.113.123_gk_model18 <br><pre> ----------------------------------------------------------------\n",
      " APMonitor, Version 0.8.1\n",
      " APMonitor Optimization Suite\n",
      " ----------------------------------------------------------------\n",
      " \n",
      " \n",
      " --------- APM Model Size ------------\n",
      " Each time step contains\n",
      "   Objects      :            0\n",
      "   Constants    :            0\n",
      "   Variables    :            2\n",
      "   Intermediates:            0\n",
      "   Connections  :            0\n",
      "   Equations    :            1\n",
      "   Residuals    :            1\n",
      " \n",
      " Number of state variables:              1\n",
      " Number of total equations: -            1\n",
      " Number of slack variables: -            0\n",
      " ---------------------------------------\n",
      " Degrees of freedom       :              0\n",
      " \n",
      " ----------------------------------------------\n",
      " Steady State Optimization with APOPT Solver\n",
      " ----------------------------------------------\n",
      " \n",
      " Iter    Objective  Convergence\n",
      "    0  0.00000E+00  5.00000E+00\n",
      "    1  0.00000E+00  5.00000E+00\n",
      " Successful solution\n",
      " \n",
      " ---------------------------------------------------\n",
      " Solver         :  APOPT (v1.0)\n",
      " Solution time  :   1.300000000628643E-002 sec\n",
      " Objective      :   0.000000000000000E+000\n",
      " Successful solution\n",
      " ---------------------------------------------------\n",
      " \n",
      " \n",
      " Generating Sensitivity Analysis\n",
      " Writing apm_sens_A_matrix.txt\n",
      " Writing sensitivity.txt\n",
      " Writing sensitivity.htm\n",
      "x: [5.0]\n"
     ]
    }
   ],
   "source": [
    "from gekko import GEKKO\n",
    "\n",
    "m = GEKKO()                 # create GEKKO model\n",
    "\n",
    "print('--------- Follow local path to view files --------------')\n",
    "print(m.path)               # show source file path\n",
    "print('--------------------------------------------------------')\n",
    "\n",
    "# test application\n",
    "u = m.FV(value=5,name='u')  # define fixed value\n",
    "x = m.SV(name='state')      # define state variable\n",
    "m.Equation(x==u)            # define equation\n",
    "m.options.COLDSTART = 1     # coldstart option\n",
    "m.options.DIAGLEVEL = 0     # diagnostic level (0-10)\n",
    "m.options.MAX_ITER = 500    # adjust maximum iterations\n",
    "m.options.SENSITIVITY = 1   # sensitivity analysis\n",
    "m.options.SOLVER = 1        # change solver (1=APOPT,3=IPOPT)\n",
    "m.solve(disp=True)          # solve locally (remote=False)\n",
    "print('x: ' + str(x.value)) # print variable value"
   ]
  }
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