{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NOTEBOOK_HEADER-->\n",
    "*This notebook contains course material from [CBE30338](https://jckantor.github.io/CBE30338)\n",
    "by Jeffrey Kantor (jeff at nd.edu); the content is available [on Github](https://github.com/jckantor/CBE30338.git).\n",
    "The text is released under the [CC-BY-NC-ND-4.0 license](https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode),\n",
    "and code is released under the [MIT license](https://opensource.org/licenses/MIT).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [Simulation of an Exothermic CSTR](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/07.04-Simulation-of-an-Exothermic-CSTR.ipynb) | [Contents](toc.ipynb) | [Path Planning for a Simple Car](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/07.06-Path-Planning-for-a-Simple-Car.ipynb) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE30338/blob/master/notebooks/07.05-First-Order-System-in-Pyomo.ipynb\"><img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://raw.githubusercontent.com/jckantor/CBE30338/master/notebooks/07.05-First-Order-System-in-Pyomo.ipynb\"><img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# First Order System in Pyomo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First-Order Differential Equation with Initial Condition\n",
    "\n",
    "The following cell implements a solution to a first-order linear model in the form\n",
    "\n",
    "\\begin{align}\n",
    "\\tau\\frac{dy}{dt} + y & = K u(t) \\\\\n",
    "\\end{align}\n",
    "\n",
    "where $\\tau$ and $K$ are model parameters, and $u(t)$ is an external process input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# ==========================================================\n",
      "# = Solver Results                                         =\n",
      "# ==========================================================\n",
      "# ----------------------------------------------------------\n",
      "#   Problem Information\n",
      "# ----------------------------------------------------------\n",
      "Problem: \n",
      "- Lower bound: -inf\n",
      "  Upper bound: inf\n",
      "  Number of objectives: 1\n",
      "  Number of constraints: 101\n",
      "  Number of variables: 101\n",
      "  Sense: unknown\n",
      "# ----------------------------------------------------------\n",
      "#   Solver Information\n",
      "# ----------------------------------------------------------\n",
      "Solver: \n",
      "- Status: ok\n",
      "  Message: Ipopt 3.12.8\\x3a Optimal Solution Found\n",
      "  Termination condition: optimal\n",
      "  Id: 0\n",
      "  Error rc: 0\n",
      "  Time: 0.024537086486816406\n",
      "# ----------------------------------------------------------\n",
      "#   Solution Information\n",
      "# ----------------------------------------------------------\n",
      "Solution: \n",
      "- number of solutions: 0\n",
      "  number of solutions displayed: 0\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Response of a linear first-order ODE')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1189bc240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% matplotlib inline\n",
    "from pyomo.environ import *\n",
    "from pyomo.dae import *\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "tf = 10\n",
    "tau = 1\n",
    "K = 5\n",
    "\n",
    "# define u(t)\n",
    "u = lambda t: 0 if t < 5 else 1\n",
    "\n",
    "# create a model object\n",
    "model = ConcreteModel()\n",
    "\n",
    "# define the independent variable\n",
    "model.t = ContinuousSet(bounds=(0, tf))\n",
    "\n",
    "# define the dependent variables\n",
    "model.y = Var(model.t)\n",
    "model.dydt = DerivativeVar(model.y)\n",
    "\n",
    "# fix the initial value of y\n",
    "model.y[0].fix(0)\n",
    "\n",
    "# define the differential equation as a constraint\n",
    "model.ode = Constraint(model.t, \n",
    "    rule=lambda model, t: tau*model.dydt[t] + model.y[t] == K*u(t))\n",
    "\n",
    "# transform dae model to discrete optimization problem\n",
    "TransformationFactory('dae.finite_difference').apply_to(model, nfe=50, method='BACKWARD')\n",
    "\n",
    "# solve the model\n",
    "SolverFactory('ipopt').solve(model).write()\n",
    "\n",
    "# access elements of a ContinuousSet object\n",
    "t = [t for t in model.t]\n",
    "\n",
    "# access elements of a Var object\n",
    "y = [model.y[t]() for t in model.y]\n",
    "\n",
    "plt.plot(t,y)\n",
    "plt.xlabel('time / sec')\n",
    "plt.ylabel('response')\n",
    "plt.title('Response of a linear first-order ODE')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!--NAVIGATION-->\n",
    "< [Simulation of an Exothermic CSTR](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/07.04-Simulation-of-an-Exothermic-CSTR.ipynb) | [Contents](toc.ipynb) | [Path Planning for a Simple Car](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/07.06-Path-Planning-for-a-Simple-Car.ipynb) ><p><a href=\"https://colab.research.google.com/github/jckantor/CBE30338/blob/master/notebooks/07.05-First-Order-System-in-Pyomo.ipynb\"><img align=\"left\" src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open in Colab\" title=\"Open in Google Colaboratory\"></a><p><a href=\"https://raw.githubusercontent.com/jckantor/CBE30338/master/notebooks/07.05-First-Order-System-in-Pyomo.ipynb\"><img align=\"left\" src=\"https://img.shields.io/badge/Github-Download-blue.svg\" alt=\"Download\" title=\"Download Notebook\"></a>"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}