2_Control/1_Conventional_feedback_control/Effect of Proportional Control.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# First-order system with proportional control\n",
    "Consider the simple feedback loop shown below\n",
    "\n",
    "![Simple feedback loop](../../assets/simple_feedback.png)\n",
    "\n",
    "with $G_c=K_c$ and $G_p=\\frac{1}{\\tau s + 1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "G_c = K_C = sympy.Symbol('K_C', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = sympy.Symbol('s')\n",
    "tau = sympy.Symbol('tau', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{s \\tau + 1}$$"
      ],
      "text/plain": [
       "   1   \n",
       "───────\n",
       "s⋅τ + 1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_p = 1/(tau*s + 1)\n",
    "G_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "G_OL = G_p*G_c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from tbcontrol.loops import feedback"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The target is to get $y_{SP} = y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{K_{C}}{K_{C} + s \\tau + 1}$$"
      ],
      "text/plain": [
       "     K_C     \n",
       "─────────────\n",
       "K_C + s⋅τ + 1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_CL = feedback(G_OL, 1).cancel()\n",
    "G_CL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = sympy.Symbol('t', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{K_{C} \\left(e^{\\frac{t \\left(K_{C} + 1\\right)}{\\tau}} - 1\\right) e^{- \\frac{t \\left(K_{C} + 1\\right)}{\\tau}}}{K_{C} + 1}$$"
      ],
      "text/plain": [
       "    ⎛ t⋅(K_C + 1)    ⎞  -t⋅(K_C + 1) \n",
       "    ⎜ ───────────    ⎟  ─────────────\n",
       "    ⎜      τ         ⎟        τ      \n",
       "K_C⋅⎝ℯ            - 1⎠⋅ℯ             \n",
       "─────────────────────────────────────\n",
       "               K_C + 1               "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "general_timeresponse = sympy.inverse_laplace_transform(sympy.simplify(G_CL/s), s, t)\n",
    "general_timeresponse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_func = sympy.lambdify((K_C, tau, t), general_timeresponse, 'numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "smootht = numpy.linspace(0, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def response(K_C=10, tau=10):\n",
    "    y = y_func(K_C, tau, smootht)\n",
    "    e = 1 - y\n",
    "    fig, [ax_y, ax_e] = plt.subplots(2, 1)\n",
    "    ax_y.plot(smootht, y)\n",
    "    ax_y.axhline(1)\n",
    "    ax_y.set_ylabel('Setpoint and y')\n",
    "\n",
    "    ax_e.plot(smootht, e)\n",
    "    ax_e.set_ylabel('Error')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ipywidgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "118002160f1f4352b078a32ae17d936b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=10, description='K_C'), IntSlider(value=10, description='tau', max=20), …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function __main__.response(K_C=10, tau=10)>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "interact(response, K_C=(0, 100), tau=(0, 20))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Offset as function of gain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = 1/s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = r*G_CL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "e = r - y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the final value statement to obtain eventual offset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{K_{C} + 1}$$"
      ],
      "text/plain": [
       "   1   \n",
       "───────\n",
       "K_C + 1"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "steady_offset = sympy.limit(s*e, s, 0)\n",
    "steady_offset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note the steady state offset is not a function of the system dynamics (time constant)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x1141865f8>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.plot(steady_offset, (K_C, 0, 60))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Second order system with proportional control\n",
    "---------------------------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "zeta = sympy.Symbol('zeta')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{s^{2} \\tau^{2} + 2 s \\tau \\zeta + 1}$$"
      ],
      "text/plain": [
       "         1         \n",
       "───────────────────\n",
       " 2  2              \n",
       "s ⋅τ  + 2⋅s⋅τ⋅ζ + 1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = 1/(tau**2*s**2 + 2*tau*zeta*s + 1)\n",
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{K_{C}}{K_{C} + s^{2} \\tau^{2} + 2 s \\tau \\zeta + 1}$$"
      ],
      "text/plain": [
       "           K_C           \n",
       "─────────────────────────\n",
       "       2  2              \n",
       "K_C + s ⋅τ  + 2⋅s⋅τ⋅ζ + 1"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_CL = feedback(G*K_C, 1).cancel()\n",
    "G_CL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def response(new_K_C, new_tau, new_zeta):\n",
    "    real_CL = G_CL.subs({K_C: new_K_C, tau: new_tau, zeta: new_zeta})\n",
    "    timeresponse = sympy.inverse_laplace_transform(sympy.simplify(real_CL/s), s, t)\n",
    "    sympy.plot(timeresponse, 1, (t, 0, 100))\n",
    "    poles = sympy.solve(sympy.denom(sympy.simplify(real_CL)), s)\n",
    "    plt.plot([sympy.re(p) for p in poles], [sympy.im(p) for p in poles], 'x', markersize=10)\n",
    "    plt.axhline(0, color='black')\n",
    "    plt.axvline(0, color='black')\n",
    "    plt.axis([-1, 1, -1, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0c26240192a644158a75cc9ff620d88b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=50.0, description='new_K_C'), FloatSlider(value=5.0, description='new_…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(response, new_K_C=(0., 100), new_tau=(0, 10.), new_zeta=(0, 2.));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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