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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Decomposi\u00e7\u00e3o de Kalman\n",
      "Suponha que um par de matrizes $(A,B)$ n\u00e3o seja control\u00e1vel. Ent\u00e3o podemos decompor o sistema numa parte control\u00e1vel e outra sem controle.\n",
      "\n",
      "### Equival\u00eancia de sistemas\n",
      "Diremos que um par de matrizes $(A,B)$ \u00e9 equivalente a um par de matrizes $(A_1,B_1)$ se existirem matrizes invers\u00edveis\n",
      "$P \\in \\mathbb{M}_{n\\times n}$ e $Q\\in \\mathbb{M}_{m\\times m}$ tal que:\n",
      "$$ PAP^{-1} = A_1 \\text{ e } PBK^{-1} = B_1 $$\n",
      "Esta \u00e9 uma *rela\u00e7\u00e3o de equival\u00eancia* no conjunto desses pares de matrizes e, de outra forma, tamb\u00e9m indicam que os sistemas lineares\n",
      "$$ \\begin{gather}\\dot{x} = Ax + Bu \\\\\n",
      "\\dot{z} = A_1 z + B_1 v\\end{gather}$$\n",
      "s\u00e3o obtidos um do outro pelas tranfosma\u00e7\u00f5es lineares $z=Px$ e $ v=Ku.$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Teorema de decomposi\u00e7\u00e3o\n",
      "Suponha que $(A,B)$ n\u00e3o seja control\u00e1vel, e que o posto de $[B|AB|\\cdots|A^{n-1}B]$ seja $k < n$. Ent\u00e3o $(A,B)$ \u00e9 equivalente a um  par $(A_1,B_1)$ que tem as seguintes formas:\n",
      "$$\\begin{gather}\n",
      "A_1 = \\begin{bmatrix} A_{11} & A_{21} \\\\ \\mathbf{0} & A_{22} \\end{bmatrix} \\text{ e }\n",
      "B_1 = \\begin{bmatrix}B_{11} \\\\ \\mathbf{0} \\end{bmatrix}\\end{gather}$$\n",
      "Com $A_{11} \\in \\mathbb{M}_{k\\times k}$, $B_{11}\\in \\mathbb{M}_{k\\times m}$ e $(A_{11},B_{11})$ control\u00e1vel."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Id\u00e9ia da prova\n",
      "Tome $k$ colunas linearmente independentes da matriz $[B|AB|\\cdots|A^{n-1}B]$ e juntamos mais $n-k$ colunas quaisquer formando a matriz $P$, de forma que $P$ seja invers\u00edvel.\n",
      "Verificamos as propriedades para $A_1=P^{-1}AP$ e $B_1= P^{-1}B$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exemplo:\n",
      "$$ A = \\begin{pmatrix} 1 & 1 \\\\ -1 & -1 \\end{pmatrix} \\text{ e } B= \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#rascunho;\n",
      "import numpy as np\n",
      "P = np.array([[1,2],[-1,3]])\n",
      "Pinv = np.linalg.inv(P)\n",
      "Aold = np.array([[0,1],[0,0]])\n",
      "Bold = np.array([[1],[0]])\n",
      "A =np.dot(np.dot(P,Aold),Pinv)\n",
      "B= np.dot(P,Bold)\n",
      "print(A)\n",
      "print(B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 0.2  0.2]\n",
        " [-0.2 -0.2]]\n",
        "[[ 1]\n",
        " [-1]]\n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A=5*A\n",
      "K = np.array([B,np.dot(A,B)])\n",
      "print(K)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[[ 1.]\n",
        "  [-1.]]\n",
        "\n",
        " [[ 0.]\n",
        "  [ 0.]]]\n"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
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  }
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}