{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multivariable control\n",
    "You may want to review the [Transfer function matrix](../../1_Dynamics/6_Multivariable_system_representation/Transfer%20function%20matrices.ipynb) notebook before you look at this one.\n",
    "\n",
    "## Closed loop transfer functions for multivariable systems\n",
    "\n",
    "Let's consider a 2 $\\times$ 2 system with feedback control as shown below:\n",
    "\n",
    "![2x2mimo](../../assets/mimo2x2.png)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "G_p11, G_p12, G_p21, G_p22, G_c1, G_c2 = sympy.symbols('G_p11, G_p12, G_p21, G_p22, G_c1, G_c2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrix representation of the system is straigtforwardly handled by `sympy.Matrix`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "G_p = sympy.Matrix([[G_p11, G_p12],\n",
    "                    [G_p21, G_p22]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The controller is a bit harder. Convince yourself you understand how the off-diagonal elements of $G_c$ are zero in the diagram above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "G_c = sympy.Matrix([[G_c1, 0],\n",
    "                    [0, G_c2]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can redraw the block diagram using vectors for the signals and matrices for the blocks.\n",
    "    \n",
    "![Simple feedback](../../assets/simple_feedback.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's derive the closed loop transfer function. There are three equations represented in the above diagram:\n",
    "\n",
    "\\begin{align}\n",
    "E &= Y_{sp} - Y\\\\ \n",
    "U &= G_c E = G_c(Y_{sp} - Y)\\\\\n",
    "Y &= G_p U = G_p G_c(Y_{sp} - Y)\\\\\n",
    "\\end{align}\n",
    "\n",
    "Now, we can solve for $Y$:\n",
    "\n",
    "\\begin{align}\n",
    "Y &= G_p G_c(Y_{sp} - Y)\\\\\n",
    "Y &= G_p G_c Y_{sp} - G_p G_c Y \\\\\n",
    "Y + G_p G_c Y&= G_p G_c Y_{sp} \\\\\n",
    "(I + G_p G_c) Y &= G_p G_c Y_{sp} \\\\\n",
    " Y &= (I + G_p G_c)^{-1} G_p G_c Y_{sp} \\\\\n",
    " Y &= \\Gamma Y_{sp}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can calculate the value of this function easily."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = sympy.Matrix([[1, 0],\n",
    "                  [0, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Gamma = sympy.simplify((I + G_p*G_c).inv()*G_p*G_c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbMAAAAyBAMAAAAkWCh9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny7P3nJbh/Rc6e3V7A4BsCgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{G_{c1} \\left(G_{c2} G_{p12} G_{p21} - G_{p11} \\left(G_{c2} G_{p22} + 1\\right)\\right)}{G_{c1} G_{c2} G_{p12} G_{p21} - \\left(G_{c1} G_{p11} + 1\\right) \\left(G_{c2} G_{p22} + 1\\right)}$$"
      ],
      "text/plain": [
       "     G_c1⋅(G_c2⋅Gₚ₁₂⋅Gₚ₂₁ - Gₚ₁₁⋅(G_c2⋅Gₚ₂₂ + 1))    \n",
       "─────────────────────────────────────────────────────\n",
       "G_c1⋅G_c2⋅Gₚ₁₂⋅Gₚ₂₁ - (G_c1⋅Gₚ₁₁ + 1)⋅(G_c2⋅Gₚ₂₂ + 1)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gamma[0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcQAAAAwBAMAAAB9BReXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+h0SvfFR/gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{G_{c2} G_{p12}}{G_{c1} G_{c2} G_{p12} G_{p21} - \\left(G_{c1} G_{p11} + 1\\right) \\left(G_{c2} G_{p22} + 1\\right)}$$"
      ],
      "text/plain": [
       "                     -G_c2⋅Gₚ₁₂                      \n",
       "─────────────────────────────────────────────────────\n",
       "G_c1⋅G_c2⋅Gₚ₁₂⋅Gₚ₂₁ - (G_c1⋅Gₚ₁₁ + 1)⋅(G_c2⋅Gₚ₂₂ + 1)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gamma[0, 1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We notice that there is a common divisor in all the elements of $\\Gamma$. This is due to the calculation of the inverse of $(I + G_p G_c)$, which involves calculation of the determinant $|I + G_p G_c|$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjMAAAAUBAMAAACKSqFWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6OfbnCUlksFSg3R9pSmbmPwPLA+M/k28SAIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$G_{c1} G_{c2} G_{p11} G_{p22} - G_{c1} G_{c2} G_{p12} G_{p21} + G_{c1} G_{p11} + G_{c2} G_{p22} + 1$$"
      ],
      "text/plain": [
       "G_c1⋅G_c2⋅Gₚ₁₁⋅Gₚ₂₂ - G_c1⋅G_c2⋅Gₚ₁₂⋅Gₚ₂₁ + G_c1⋅Gₚ₁₁ + G_c2⋅Gₚ₂₂ + 1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Delta = (I + G_p*G_c).det()\n",
    "Delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}G_{c1} \\left(G_{c2} G_{p11} G_{p22} - G_{c2} G_{p12} G_{p21} + G_{p11}\\right) & G_{c2} G_{p12}\\\\G_{c1} G_{p21} & G_{c2} \\left(G_{c1} G_{p11} G_{p22} - G_{c1} G_{p12} G_{p21} + G_{p22}\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡G_c1⋅(G_c2⋅Gₚ₁₁⋅Gₚ₂₂ - G_c2⋅Gₚ₁₂⋅Gₚ₂₁ + Gₚ₁₁)                    G_c2⋅Gₚ₁₂   \n",
       "⎢                                                                             \n",
       "⎣                  G_c1⋅Gₚ₂₁                    G_c2⋅(G_c1⋅Gₚ₁₁⋅Gₚ₂₂ - G_c1⋅Gₚ\n",
       "\n",
       "               ⎤\n",
       "               ⎥\n",
       "₁₂⋅Gₚ₂₁ + Gₚ₂₂)⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(Delta*Gamma).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Characteristic equation\n",
    "-----------------------\n",
    "\n",
    "This leads us to conclude that we can calculate the characteristic equation of the closed loop transfer function as $|I + G_p G_c|$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that if we wanted the coupling the other way around, we could have worked with the same controller matrix permuted:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAyBAMAAAC6x3j0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & G_{c1}\\\\G_{c2} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 0    G_c1⎤\n",
       "⎢          ⎥\n",
       "⎣G_c2   0  ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G_c*sympy.Matrix([[0, 1], [1, 0]])"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
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   },
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  }
 },
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