1_Dynamics/5_Complex_system_dynamics/Block diagram simplification.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simplifying block diagrams\n",
    "============\n",
    "\n",
    "[These notes](http://faculty.mu.edu.sa/public/uploads/1415021770.8406Block_Diagram_Reduction_Rules.pdf) detail some techniques for reducing block diagrams graphically. In this notebook, I will solve the problem using SymPy.\n",
    "\n",
    "Let's reduce this big block diagram to one input-output relationship (Example 4 in the notes linked to above):\n",
    "\n",
    "![](../../assets/bigblockdiagram.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "(R, V1, V2, V3, V4, V5, V6, V7, V8, C,\n",
    " G1, G2, G3, H1, H2, H3) = sympy.symbols('R, V1, V2, V3, V4, V5, V6, V7, V8, C,'\n",
    "                                         'G1, G2, G3, H1, H2, H3')\n",
    "unknowns = V1, V2, V3, V4, V5, V6, V7, V8, C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "eqs = [# Blocks\n",
    "       V2 - G1*V1,\n",
    "       V4 - G2*V3,\n",
    "       C - G3*V5,\n",
    "       V6 - H1*V4,\n",
    "       V7 - H2*V4,\n",
    "       V8 - H3*C,\n",
    "       # Sums\n",
    "       V1 - (R - V6),\n",
    "       V3 - (V2 - V7),\n",
    "       V5 - (V4 + V3 - V8),\n",
    "       ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left \\{ C : \\frac{G_{1} G_{3} R \\left(G_{2} + 1\\right)}{G_{1} G_{2} G_{3} H_{1} H_{3} + G_{1} G_{2} H_{1} + G_{2} G_{3} H_{2} H_{3} + G_{2} H_{2} + G_{3} H_{3} + 1}, \\quad V_{1} : \\frac{R \\left(G_{2} H_{2} + 1\\right)}{G_{1} G_{2} H_{1} + G_{2} H_{2} + 1}, \\quad V_{2} : \\frac{G_{1} R \\left(G_{2} H_{2} + 1\\right)}{G_{1} G_{2} H_{1} + G_{2} H_{2} + 1}, \\quad V_{3} : \\frac{G_{1} R}{G_{1} G_{2} H_{1} + G_{2} H_{2} + 1}, \\quad V_{4} : \\frac{G_{1} G_{2} R}{G_{1} G_{2} H_{1} + G_{2} H_{2} + 1}, \\quad V_{5} : \\frac{G_{1} R \\left(G_{2} + 1\\right)}{G_{1} G_{2} G_{3} H_{1} H_{3} + G_{1} G_{2} H_{1} + G_{2} G_{3} H_{2} H_{3} + G_{2} H_{2} + G_{3} H_{3} + 1}, \\quad V_{6} : \\frac{G_{1} G_{2} H_{1} R}{G_{1} G_{2} H_{1} + G_{2} H_{2} + 1}, \\quad V_{7} : \\frac{G_{1} G_{2} H_{2} R}{G_{1} G_{2} H_{1} + G_{2} H_{2} + 1}, \\quad V_{8} : \\frac{G_{1} G_{3} H_{3} R \\left(G_{2} + 1\\right)}{G_{1} G_{2} G_{3} H_{1} H_{3} + G_{1} G_{2} H_{1} + G_{2} G_{3} H_{2} H_{3} + G_{2} H_{2} + G_{3} H_{3} + 1}\\right \\}$$"
      ],
      "text/plain": [
       "⎧                         G₁⋅G₃⋅R⋅(G₂ + 1)                              R⋅(G₂⋅\n",
       "⎨C: ───────────────────────────────────────────────────────────, V₁: ─────────\n",
       "⎩   G₁⋅G₂⋅G₃⋅H₁⋅H₃ + G₁⋅G₂⋅H₁ + G₂⋅G₃⋅H₂⋅H₃ + G₂⋅H₂ + G₃⋅H₃ + 1      G₁⋅G₂⋅H₁ \n",
       "\n",
       "H₂ + 1)            G₁⋅R⋅(G₂⋅H₂ + 1)                G₁⋅R                    G₁⋅\n",
       "───────────, V₂: ────────────────────, V₃: ────────────────────, V₄: ─────────\n",
       "+ G₂⋅H₂ + 1      G₁⋅G₂⋅H₁ + G₂⋅H₂ + 1      G₁⋅G₂⋅H₁ + G₂⋅H₂ + 1      G₁⋅G₂⋅H₁ \n",
       "\n",
       "G₂⋅R                                    G₁⋅R⋅(G₂ + 1)                         \n",
       "───────────, V₅: ───────────────────────────────────────────────────────────, \n",
       "+ G₂⋅H₂ + 1      G₁⋅G₂⋅G₃⋅H₁⋅H₃ + G₁⋅G₂⋅H₁ + G₂⋅G₃⋅H₂⋅H₃ + G₂⋅H₂ + G₃⋅H₃ + 1  \n",
       "\n",
       "         G₁⋅G₂⋅H₁⋅R                G₁⋅G₂⋅H₂⋅R                               G₁\n",
       "V₆: ────────────────────, V₇: ────────────────────, V₈: ──────────────────────\n",
       "    G₁⋅G₂⋅H₁ + G₂⋅H₂ + 1      G₁⋅G₂⋅H₁ + G₂⋅H₂ + 1      G₁⋅G₂⋅G₃⋅H₁⋅H₃ + G₁⋅G₂\n",
       "\n",
       "⋅G₃⋅H₃⋅R⋅(G₂ + 1)                    ⎫\n",
       "─────────────────────────────────────⎬\n",
       "⋅H₁ + G₂⋅G₃⋅H₂⋅H₃ + G₂⋅H₂ + G₃⋅H₃ + 1⎭"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = sympy.solve(eqs, unknowns)\n",
    "sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution in the notes is factored:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{G_{1} G_{3} R \\left(G_{2} + 1\\right)}{\\left(G_{3} H_{3} + 1\\right) \\left(G_{1} G_{2} H_{1} + G_{2} H_{2} + 1\\right)}$$"
      ],
      "text/plain": [
       "         G₁⋅G₃⋅R⋅(G₂ + 1)         \n",
       "──────────────────────────────────\n",
       "(G₃⋅H₃ + 1)⋅(G₁⋅G₂⋅H₁ + G₂⋅H₂ + 1)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol[C].factor()"
   ]
  }
 ],
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