Projeto Espaço de Estados.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import * # package for numerical calculations\n",
    "from pylab import * # package for graphics\n",
    "from control import * # package for control theory calculations\n",
    "from control.matlab import * # package for control theory calculations\n",
    "ion() # display graphics automatically"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Representação do sistema em malha aberta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{2}{s^2 + 0.5 s}$$"
      ],
      "text/plain": [
       "\n",
       "     2\n",
       "-----------\n",
       "s^2 + 0.5 s"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys = tf([2], [1, 0.5, 0]) # Define a transfer function\n",
    "sys"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sistema em espaço de Estados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A = [[-0.5 -0. ]\n",
       " [ 1.   0. ]]\n",
       "\n",
       "B = [[1.]\n",
       " [0.]]\n",
       "\n",
       "C = [[0. 2.]]\n",
       "\n",
       "D = [[0.]]"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys_ss = tf2ss(sys)\n",
    "A, B, C, D = sys_ss.A, sys_ss.B, sys_ss.C, sys_ss.D\n",
    "sys_ss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Diagrama de polos e zeros"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.5,  0. ]), array([], dtype=float64))"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pzmap(sys)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resposta ao degrau"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1c19619ba8>]"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "T = np.arange(0,10,0.001)\n",
    "T, y = step_response(sys, T) \n",
    "\n",
    "plt.plot(T, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Representação do sistema em malha fechada"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{2 s^2 + s}{s^4 + s^3 + 2.25 s^2 + s}$$"
      ],
      "text/plain": [
       "\n",
       "        2 s^2 + s\n",
       "------------------------\n",
       "s^4 + s^3 + 2.25 s^2 + s"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sysmf_tf = sys / (1 + sys) \n",
    "sysmf_tf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polos e zeros"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1c1963f080>]"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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E9cDabbdlxWMfy2nXXDPrZzEdcsgh3Hjjjey0005ccMEFAOy3336cdNJJnHXWWZ5VpdnV5y7Mz+lc+HcTcHPzei2deasWb8Gu0THANcBPgbc2bWcAxzavdwQ+BawB/gPYr5/1Ll68uKQtsX79+rroootq/fr1deutt9bU1FTdc88992sfRj133XVXTU1NVedXt4ZWj7Y+wGT1+bc71cd/4Jv7caysqkub90cBRwOfBMarauNB7aEaGxurycnJYZchDUwz5c+wy9BWJMnqqhrrp2/fV45vCA2Aqvoy8PSq+j7wkAdRoyRpjur3Oo5bkrwJuLB5/2Lg181FfOsHUpkkaST1u8fxV3TOepoAPgvs07RtC/zlYEqTJI2ifi8AvInO/cZ7WTNz5UiSRl2/06rvDrwReDxdF9RW1ZEDqkuSNKL6PVR1PvATOvcZP53O6bmrBlSTJGmE9Rscu1XVR4C7quqbVfVK4MkDrEuSNKL6Pavqrub5hiTPozM1yF6DKUmSNMr6DY4zkzwceAPwfmAXYPnAqpIkjax+z6r6fPPyt8AzBleOJGnU9XtW1b50Tsdd1P2Z2oI7AEqS5qZ+D1VNAB8BPodXikvSvNZvcPyhqt430EokSXNCv8ExnuQdwJeBP25orKrLBlKVJGlk9Rsc/xl4KXAk9x2qqua9JGke6Tc4jqdzE6U7B1mMJGn09Xvl+A+BBYMsRJI0N/S7x/FI4CdJVnH/MQ5Px5Wkeabf4HjHQKuQJM0Z/V45/s1BFyJJmhs2GRxJbqNz9tQDFgFVVbsMpCpJ0sjaZHBU1c6zVYgkaW7o96wqSZIAg0OS1JLBIUlqxeCQJLUylOBIsmuSryS5tnn+k2n63ZPk8uZx8WzXKUl6oGHtcbwZ+FpVHQB8rXnfyx1VdXDz8Cp1SRoBwwqO44CPNa8/BiwZUh2SpJaGFRyPrKobAJrnR0zTb8ckk0m+n8RwkaQR0O9cVa0l+SrwqB6L3tpiNftU1VSS/YB/S/J/q+qn02zvZOBkgH322ad1vZKk/gwsOKrqWdMtS/KrJI+uqhuSPBq4cZp1TDXP1yX5BnAI0DM4qups4GyAsbGxXtOkSJJmwLAOVV0MvLx5/XLgsxt3SPInSR7SvF4IHAb8eNYqlCT1NKzg+Cfg2UmuBZ7dvCfJWJIPN30eC0wm+SHwdeCfqsrgkKQhG9ihqk2pqpuBZ/ZonwRe1bz+Lp17nUuSRohXjkuSWjE4JEmtGBySpFYMDklSKwaHJKkVg0OS1IrBIUlqxeCQJLVicEiSWjE4JEmtGBySpFYMDklSKwaHJKkVg0OS1IrBIUlqxeCQJLVicEiSWjE4JEmtGBySpFYMDklSKwaHJKkVg0OS1IrBIUlqxeCQJLVicEiSWjE4JEmtGBySpFaGEhxJXpTkyiTrk4xtot/RSa5OsibJm2ezRklSb8Pa47gCWAp8a7oOSbYFPgg8F3gc8JIkj5ud8iRJ09luGButqqsAkmyq26HAmqq6rul7IXAc8OOBFyhJmtYoj3HsCVzf9X5t09ZTkpOTTCaZXLdu3cCLk6T5amB7HEm+Cjyqx6K3VtVn+1lFj7aarnNVnQ2cDTA2NjZtP0nSlhlYcFTVs7ZwFWuBvbve7wVMbeE6JUlbaJQPVa0CDkiyb5IdgBOAi4dckyTNe8M6Hff4JGuBpwBfSHJp075HkksAqupu4BTgUuAq4JNVdeUw6pUk3WdYZ1WtBFb2aJ8Cjul6fwlwySyWJknajFE+VCVJGkEGhySpFYNDktSKwSFJasXgkCS1YnBIkloxOCRJrRgckqRWDA5JUisGhySpFYNDktSKwSFJasXgkCS1YnBIkloxOCRJrRgckqRWDA5JUisGhySpFYNDGqCqYuXKlVRVX+3SXGBwSAM0MTHB0qVLWb58+b0hUVUsX76cpUuXMjExMeQKpfa2G3YB0tZsyZIlLFu2jPHxcQBWrFjB8uXLGR8fZ9myZSxZsmTIFUrtGRzSACVhxYoVAIyPj98bIMuWLWPFihUkGWZ50oOSrfEY69jYWE1OTg67DOleVcU229x3ZHj9+vVbFBpJHB/RjEqyuqrG+unrGIc0YBvGNLp1j3lIc43BIQ3QhtDYMKaxfv36e8c8DA/NVY5xSAM0MTFxb2hsGNPoHvM4/PDDOf7444dcpdTOUMY4krwIOA14LHBoVfUckEjyc+A24B7g7n6PvznGoVFRVUxMTLBkyZL7jWlM194vxzg009qMcQxrj+MKYCnwv/vo+4yqumnA9UgDkaTnHsV07dJcMJTgqKqrAE9FlKQ5aNQHxwv4cpLVSU7eVMckJyeZTDK5bt26WSpPkuafge1xJPkq8Kgei95aVZ/tczWHVdVUkkcAX0nyk6r6Vq+OVXU2cDZ0xjgeVNGSpM0aWHBU1bNmYB1TzfONSVYChwI9g0OSNDtG9nTcJA8Dtqmq25rXRwFn9PPZ1atX35TkFwMtcOYsBBz8783vZnoLk/jd9ObPzfQ29d08pt+VDOt03OOB9wO7A78BLq+q5yTZA/hwVR2TZD9gZfOR7YB/rap3zXqxA5Zkst9T4OYbv5vp+d1Mz+9mejP13QzrrKqV3BcK3e1TwDHN6+uAJ85yaZKkzRj1s6okSSPG4Bi+s4ddwAjzu5me3830/G6mNyPfzVY5rbokaXDc45AktWJwSJJaMThGQJJ3J/lJkh8lWZlkwbBrGhVJXpTkyiTrk8z7UyyTHJ3k6iRrkrx52PWMkiQfTXJjkiuGXcsoSbJ3kq8nuar5XVq2pes0OEbDV4CDquoJwDXA3w+5nlGyYSbleT9jQJJtgQ8CzwUeB7wkyeOGW9VIOQc4ethFjKC7gTdU1WOBJwOv3dKfG4NjBFTVl6vq7ubt94G9hlnPKKmqq6rq6mHXMSIOBdZU1XVVdSdwIXDckGsaGc08drcMu45RU1U3VNVlzevbgKuAPbdknQbH6Hkl8MVhF6GRtCdwfdf7tWzhHwDNL0kWAYcA/74l6xnZuaq2Nv3MFpzkrXR2K8+fzdqGbYZmUp4Pet3AxvPp1ZckOwGfAV5fVbduyboMjlmyudmCk7wceD7wzJpnF9fMxEzK88RaYO+u93sBU0OqRXNIku3phMb5VXXRlq7PQ1UjIMnRwJuAY6vq98OuRyNrFXBAkn2T7ACcAFw85Jo04tK51epHgKuq6r0zsU6DYzR8ANiZzs2qLk9y1rALGhVJjk+yFngK8IUklw67pmFpTqA4BbiUzgDnJ6vqyuFWNTqSXAB8DzgwydokJw27phFxGPBS4Mjm78vlSY7ZkhU65YgkqRX3OCRJrRgckqRWDA5JUisGhySpFYNDktSKwSHNgCT3NKc5XpHkc1syw3GSnydZOJP1STPJ4JBmxh1VdXBVHURnor3XDrsgaVAMDmnmfY+uyQeT/F2SVc39Vk7vap9Isrq5R8LJQ6lUehAMDmkGNffMeCbNVCBJjgIOoDMl+sHA4iRPb7q/sqoWA2PA65LsNoSSpdYMDmlm/KcklwM3A7vSuTkXwFHN4wfAZcCf0QkS6ITFD+ncg2XvrnZppBkc0sy4o6oOBh4D7MB9YxwB/rEZ/zi4qvavqo8kOQJ4FvCUqnoinWDZcRiFS20ZHNIMqqrfAq8D/raZyvpS4JXNvRBIsmeSRwAPB35dVb9P8md0bukpzQnej0OaYVX1g+YQ1AlV9fEkjwW+15ndmtuBE4EvAa9J8iPgajqHq6Q5wdlxJUmteKhKktSKwSFJasXgkCS1YnBIkloxOCRJrRgckqRWDA5JUiv/H5pr8e8aneKBAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = pzmap(sysmf_tf) \n",
    "plt.plot(a[1][0], a[1][1], 'or') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Projeto do controlador por espaço de estados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Symbol, solve, log, sqrt\n",
    "x = Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.functions.elementary import exponential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "xi = solve(0.1631 - exponential.exp(-x*pi/sqrt(1-x**2)), rational=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.499915723335591]"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 3.5    15.9716]]\n"
     ]
    }
   ],
   "source": [
    "K = place(A,B,[complex(-2,3.46),complex(-2,-3.46)])\n",
    "\n",
    "A = A - dot(B,K)\n",
    "\n",
    "sysmf_ss =  StateSpace(A,B,C,D) #\n",
    "\n",
    "print(K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        2\n",
      "-----------------\n",
      "s^2 + 4 s + 15.97\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = pzmap(sysmf_ss)\n",
    "print(ss2tf(sysmf_ss))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulação da FT de espaço de estados\n",
    "$$ G_f(s) = \\frac{G_P}{1 + K_1 s G_P + K_2 G_p} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "        7.854 s^4 + 7.854 s^3 + 1.964 s^2\n",
      "-------------------------------------------------\n",
      "s^6 + 8.5 s^5 + 39.69 s^4 + 33.82 s^3 + 7.986 s^2\n",
      "\n",
      "3.5 15.971600000000004\n"
     ]
    }
   ],
   "source": [
    "Gp = tf([2], [1, 0.5, 0])\n",
    "s = tf([1, 0], [1]) \n",
    "K1, K2 = K[0][0], K[0][1]\n",
    "\n",
    "Gf = Gp / (1 + K1*s*Gp +K2*Gp) #K0 * \n",
    "print(Gf)\n",
    "print( K1, K2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1c19891ba8>]"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "T = np.arange(0,20,.001)\n",
    "T, y = step_response(Gf, T) \n",
    "\n",
    "plt.plot(T,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "def step_info(t, y): \n",
    "  OS = (y.max()/y[-1]-1)*100\n",
    "  try:\n",
    "      i10 = next(i for i in range(0,len(y)-1) if y[i]>=y[-1]*.10)\n",
    "      Tr = round(t[next(i for i in range(i10,len(y)-1) if y[i]>=y[-1]*.90)]-t[i10],2)\n",
    "  except StopIteration:\n",
    "      Tr = \"unknown\"\n",
    "  try:\n",
    "      Ts = round(\n",
    "          t[next(len(y)-i \n",
    "                 for i in range(2,len(y)-1) \n",
    "                 if (abs(y[-i]/y[-1])>1.02 or abs(y[-i]/y[-1])<0.98))]-t[0],2)\n",
    "  except StopIteration:\n",
    "      Ts = \"unknown\"\n",
    "      \n",
    "  return OS, Tr, Ts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = np.arange(0,20,.001)\n",
    "T, y = step_response(Gf, T) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "OS: 6.166431 %\n",
      "Tr: 0.360000 s\n",
      "Ts: 1.060000 s\n"
     ]
    }
   ],
   "source": [
    "OS, Tr, Ts = step_info(T, y)\n",
    "print(\"OS: %f\" %OS,\"%\")\n",
    "print(\"Tr: %f\" %Tr,\"s\")\n",
    "print(\"Ts: %f\" %Ts,\"s\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}