first-order.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 543,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import HTML\n",
    "import jupyprint as jp\n",
    "import numpy as np\n",
    "from pandas import DataFrame\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams.update({\n",
    "    \"text.usetex\": True,\n",
    "    \"font.family\": \"Helvetica\",\n",
    "    \"font.size\": 10,\n",
    "})\n",
    "\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define variables that we are going to use repetitively:\n",
    "$s, t, \\tau$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 544,
   "metadata": {},
   "outputs": [],
   "source": [
    "t   = symbols('t', real=True, nonnegative=True)\n",
    "s   = symbols('s', complex=True)\n",
    "tau = symbols('tau', real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simplify:\n",
    "*  ```laplace_transform(...)``` as ```lapalce(...)```\n",
    "*  ```inverse_laplace_transform(...)``` as ```ilapalce(...)```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 545,
   "metadata": {},
   "outputs": [],
   "source": [
    "def laplace(f):\n",
    "    F = laplace_transform(f, t, s, noconds=True)\n",
    "    return F\n",
    "\n",
    "def ilaplace(F):\n",
    "    f = inverse_laplace_transform(F, s, t)\n",
    "    return f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First-order system equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 546,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Given a first order system:\n"
     ]
    },
    {
     "data": {
      "text/markdown": [
       "$G(s)=\\frac{C(s)}{R(s)}=\\frac{1}{s \\tau + 1}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Given a first order system:\")\n",
    "G = 1 / (tau*s + 1)\n",
    "jp.jupyprint('$G(s)=\\\\frac{C(s)}{R(s)}=' + latex(G)+'$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unit-impulse response"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 547,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit-impulse input:\n"
     ]
    },
    {
     "data": {
      "text/markdown": [
       "$r(t)=\\delta\\left(t\\right)$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$R(s)=1$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$C(s)=\\frac{1}{s \\tau + 1}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$c(t)=\\frac{e^{- \\frac{t}{\\tau}} \\theta\\left(t\\right)}{\\tau}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Unit-impulse input:\")\n",
    "\n",
    "r = DiracDelta(t)\n",
    "jp.jupyprint(\"$r(t)=\"+latex(r)+\"$\")\n",
    "\n",
    "R = laplace(r)\n",
    "jp.jupyprint(\"$R(s)=\"+latex(R)+\"$\")\n",
    "\n",
    "C = G*R\n",
    "jp.jupyprint('$C(s)=' + latex(C) + '$')\n",
    "\n",
    "c = collect(ilaplace(C), Heaviside(t))\n",
    "jp.jupyprint('$c(t)=' + latex(c) + '$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 548,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x250 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "fig = plt.figure(figsize=(5,2.5))\n",
    "ax = fig.add_subplot()\n",
    "ax.axvline(x=0, c=\"k\")\n",
    "ax.axhline(y=0, c=\"k\")\n",
    "\n",
    "T = np.linspace(0, 7, 100)\n",
    "tau_ = 1\n",
    "\n",
    "Y = T * 0.0\n",
    "for k, t_ in enumerate(T):\n",
    "    Y[k] = c.subs([(tau, tau_),(t, t_)])\n",
    "\n",
    "label = '$\\\\tau='+str(tau_)+'$'\n",
    "ax.plot(T,Y, label=label)\n",
    "\n",
    "ax.set_xlabel('$t$');\n",
    "ax.set_ylabel('$c(t)$');\n",
    "ax.legend();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unit-step response"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 549,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit-step input:\n"
     ]
    },
    {
     "data": {
      "text/markdown": [
       "$r(t)=\\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$R(s)=\\frac{1}{s}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$C(s)=\\frac{1}{s \\left(s \\tau + 1\\right)}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$c(t)=\\left(1 - e^{- \\frac{t}{\\tau}}\\right) \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Unit-step input:\")\n",
    "\n",
    "r = Heaviside(t)\n",
    "jp.jupyprint(\"$r(t)=\"+latex(r)+\"$\")\n",
    "\n",
    "R = laplace(r)\n",
    "jp.jupyprint(\"$R(s)=\"+latex(R)+\"$\")\n",
    "\n",
    "C = G*R\n",
    "jp.jupyprint('$C(s)=' + latex(C) + '$')\n",
    "\n",
    "c = collect(ilaplace(C), Heaviside(t))\n",
    "jp.jupyprint('$c(t)=' + latex(c) + '$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 550,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x250 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "fig = plt.figure(figsize=(5,2.5))\n",
    "ax = fig.add_subplot()\n",
    "ax.axvline(x=0, c=\"k\")\n",
    "ax.axhline(y=0, c=\"k\")\n",
    "\n",
    "T = np.linspace(0, 7, 100)\n",
    "tau_ = 1\n",
    "\n",
    "Y = T * 0.0\n",
    "for k, t_ in enumerate(T):\n",
    "    Y[k] = c.subs([(tau, tau_),(t, t_)])\n",
    "\n",
    "label = '$\\\\tau='+str(tau_)+'$'\n",
    "ax.plot(T,Y, label=label)\n",
    "\n",
    "ctau = c.subs([(tau, tau_),(t, 1)])\n",
    "\n",
    "ax.axvline(x=tau_, color='r', linestyle='--', label=f'$t=\\\\tau \\\\rightarrow c(\\\\tau) = {ctau:.4f}$')\n",
    "ax.axhline(y=ctau, color='r', linestyle='--')\n",
    "\n",
    "ax.set_xlabel('$t$');\n",
    "ax.set_ylabel('$c(t)$');\n",
    "ax.legend();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The figure above shows that at $t = \\tau$, $c(\\tau)$ is at $63.21 \\%$ of $c_{ss}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unit-ramp response"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 551,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Unit-ramp input:\n"
     ]
    },
    {
     "data": {
      "text/markdown": [
       "$r(t)=t$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$R(s)=\\frac{1}{s^{2}}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$C(s)=\\frac{1}{s^{2} \\left(s \\tau + 1\\right)}$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "$c(t)=\\left(t - \\tau + \\tau e^{- \\frac{t}{\\tau}}\\right) \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Unit-ramp input:\")\n",
    "\n",
    "r = t\n",
    "jp.jupyprint(\"$r(t)=\"+latex(r)+\"$\")\n",
    "\n",
    "R = laplace(r)\n",
    "jp.jupyprint(\"$R(s)=\"+latex(R)+\"$\")\n",
    "\n",
    "C = G*R\n",
    "jp.jupyprint('$C(s)=' + latex(C) + '$')\n",
    "\n",
    "c = collect(ilaplace(C), Heaviside(t))\n",
    "jp.jupyprint('$c(t)=' + latex(c) + '$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 552,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x250 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "fig = plt.figure(figsize=(5,2.5))\n",
    "ax = fig.add_subplot()\n",
    "\n",
    "ax.axvline(x=0, c=\"k\")\n",
    "ax.axhline(y=0, c=\"k\")\n",
    "\n",
    "T = np.linspace(0, 7, 100)\n",
    "tau_ = 1\n",
    "\n",
    "Y = T * 0.0\n",
    "for k, t_ in enumerate(T):\n",
    "    Y[k] = c.subs([(tau, tau_),(t, t_)])\n",
    "\n",
    "label = '$\\\\tau='+str(tau_)+'$'\n",
    "ax.plot(T,Y, label=label)\n",
    "\n",
    "ax.set_xlabel('$t$');\n",
    "ax.set_ylabel('$c(t)$');\n",
    "ax.legend();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Final Value Theorem\n",
    "\n",
    "The steady state value for each input can be found by using final value theorem: \n",
    "\n",
    "$$\\boxed{c_{ss} = \\lim _{t \\rightarrow \\infty} c(t)=\\lim _{s \\rightarrow 0} s C(s)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unit-impulse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 553,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "$r(t)=\\delta\\left(t\\right)\\rightarrow c_{ss}=0$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r = DiracDelta(t)\n",
    "css = limit(s*G*laplace(r), s, 0)\n",
    "\n",
    "jp.jupyprint('$r(t)=' + latex(r) + '\\\\rightarrow c_{ss}=' + latex(css) + '$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unit-step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 554,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "$r(t)=\\theta\\left(t\\right)\\rightarrow c_{ss}=1$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r = Heaviside(t)\n",
    "css = limit(s*G*laplace(r), s, 0)\n",
    "\n",
    "jp.jupyprint('$r(t)=' + latex(r) + '\\\\rightarrow c_{ss}=' + latex(css) + '$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unit-ramp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 555,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "$r(t)=t\\rightarrow c_{ss}=\\infty$"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r = t\n",
    "css = limit(s*G*laplace(r), s, 0)\n",
    "\n",
    "jp.jupyprint('$r(t)=' + latex(r) + '\\\\rightarrow c_{ss}=' + latex(css) + '$')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "py.3.12.4",
   "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": "3.12.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}