notebooks/simulation/Linear_First_Order_System.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear First Order System\n",
    "\n",
    "This notebook demonstrates simulation of a linear first-order system in Pyomo using two distinct approaches. The first uses the `Simulator` class from Pyomo which can employ the "
   ]
  },
  {
   "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": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Response of a linear first-order ODE')"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1114abb38>"
      ]
     },
     "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: 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, 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",
    "tsim, profiles = Simulator(model, package='scipy').simulate(numpoints=100)\n",
    "\n",
    "plt.plot(tsim, profiles)\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": []
  }
 ],
 "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": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}