PID Simulation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import odeint\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PI simulation on first order system\n",
    "==="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = 1\n",
    "m = 1\n",
    "rho = 1\n",
    "A = 1\n",
    "Cd = 1\n",
    "\n",
    "coeff1 = F/m\n",
    "coeff2 = 0.5*rho*A*(Cd/m)\n",
    "\n",
    "\n",
    "def dvcap_dt(vcap, t, params):\n",
    "    Kp, Ki, v_target = params[\"PID\"]\n",
    "    coeff1, coeff2 = params[\"Model\"]\n",
    "\n",
    "    u, v = vcap\n",
    "    dv_dt = coeff1*u - coeff2*(v**2)\n",
    "    du_dt = Kp*(-dv_dt) + Ki*(v_target - v)\n",
    "    \n",
    "    return [du_dt, dv_dt]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "Kp = 10\n",
    "Ki = 10\n",
    "v_target = 20\n",
    "\n",
    "params = {\n",
    "    \"PID\": (Kp, Ki, v_target),\n",
    "    \"Model\": (coeff1, coeff2)\n",
    "}\n",
    "\n",
    "t = np.linspace(0, 30, 1000)\n",
    "res = odeint(dvcap_dt, [0, 0], t, args=(params, ))\n",
    "u, v = res[:, 0], res[:, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f0552eddad0>"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.axhline(y=v_target, color='r', label=\"Target Velocity\")\n",
    "plt.plot(t, v, label=\"Velocity\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PID simulation on second order system\n",
    "==="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plant model equations\n",
    "---\n",
    "\n",
    "We require a function that given a tuple $ x = (y, \\frac{dy}{dt} ) $ and $ t $\n",
    "will return $ [ \\frac{dy}{dt}, \\frac{d^2y}{dt^2} ] $\n",
    "\n",
    "This is a dynamic specific to the model of the given plant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dycap_dt(x, t):\n",
    "    kp = 2.0\n",
    "    tau = 3.0\n",
    "    zeta = 0.05\n",
    "    dx = 10.0\n",
    "\n",
    "    y, dy_dt = x\n",
    "    d2y_dt2 = (-2.0*zeta*tau*dy_dt - y + kp*dx)/tau**2\n",
    "\n",
    "    return [dy_dt, d2y_dt2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Controller equations\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class SecondOrderPIDController:\n",
    "    def __init__(Ki, Kp, Kd, plant_model):\n",
    "        self.params = Ki, Kp, Kd\n",
    "        self.model = plant_model\n",
    "\n",
    "    def get_du_dt(self, ):\n",
    "        \"\"\"\n",
    "            Expects plant_model to be a function like dycap_dt above.\n",
    "        \"\"\"\n",
    "        "
   ]
  },
  {
   "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}