16_06_Routh_array.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "539b9389",
   "metadata": {},
   "source": [
    "# Routh Array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "28482c77",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sympy version 1.9\n"
     ]
    }
   ],
   "source": [
    "import os.path,  importlib.util\n",
    "\n",
    "if importlib.util.find_spec('tbcontrol') is None:\n",
    "    os.system(f'python -m pip install tbcontrol')\n",
    "\n",
    "from tbcontrol.symbolic import routh\n",
    "\n",
    "import numpy as np\n",
    "import sympy \n",
    "from sympy import fraction, roots, re, N \n",
    "sympy.init_printing()\n",
    "print('sympy version '+sympy.__version__)\n",
    "\n",
    "s = sympy.Symbol('s')\n",
    "a_0, a_1, a_2, a_3, a_4 = sympy.symbols('a_0:5')\n",
    "K_c = sympy.Symbol('K_c')\n",
    "\n",
    "if not os.path.isdir(\"./figs/\"):\n",
    "    os.mkdir(\"./figs\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d14f79c9",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function routh in module tbcontrol.symbolic:\n",
      "\n",
      "routh(p)\n",
      "    Construct the Routh-Hurwitz array given a polynomial in s\n",
      "    \n",
      "    Input: p - a sympy.Poly object\n",
      "    Output: The Routh-Hurwitz array as a sympy.Matrix object\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(routh)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5307a9a7-7dd0-4bd7-9684-caf8a27e1e76",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def display_roots(p):\n",
    "    rr = p.nroots()\n",
    "    for root in rr:\n",
    "        r, i = root.as_real_imag()\n",
    "        if abs(i) > 1e-5:\n",
    "            print('%8.3f %8.3fi' % (r,i))\n",
    "        else:\n",
    "            print('%8.3f' % (r))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8466922-cd81-4c53-9e76-7d9af7c487c4",
   "metadata": {},
   "source": [
    "## Example 8-8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b808f82e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( s^{6} + 7 s^{5} + 13 s^{4} + 8 s^{3} + 5 s^{2} + 10 s + 16, s, domain=\\mathbb{Z} \\right)}$"
      ],
      "text/plain": [
       "Poly(s**6 + 7*s**5 + 13*s**4 + 8*s**3 + 5*s**2 + 10*s + 16, s, domain='ZZ')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 8-8\n",
    "p = s**6 + 7*s**5 + 13*s**4 + 8*s**3 + 5*s**2 + 10*s + 16\n",
    "p = sympy.Poly(p, s)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "cbb6e9f5",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  -4.434\n",
      "  -2.000\n",
      "  -0.907   -0.933i\n",
      "  -0.907    0.933i\n",
      "   0.625   -0.822i\n",
      "   0.625    0.822i\n"
     ]
    }
   ],
   "source": [
    "display_roots(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "db0e4e1d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 13 & 5 & 16\\\\7 & 8 & 10 & 0\\\\\\frac{83}{7} & \\frac{25}{7} & 16 & 0\\\\\\frac{489}{83} & \\frac{46}{83} & 0 & 0\\\\\\frac{1201}{489} & 16 & 0 & 0\\\\- \\frac{45430}{1201} & 0 & 0 & 0\\\\16 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡   1      13   5   16⎤\n",
       "⎢                     ⎥\n",
       "⎢   7      8    10  0 ⎥\n",
       "⎢                     ⎥\n",
       "⎢ 83/7    25/7  16  0 ⎥\n",
       "⎢                     ⎥\n",
       "⎢  489     46         ⎥\n",
       "⎢  ───     ──   0   0 ⎥\n",
       "⎢   83     83         ⎥\n",
       "⎢                     ⎥\n",
       "⎢ 1201                ⎥\n",
       "⎢ ────     16   0   0 ⎥\n",
       "⎢ 489                 ⎥\n",
       "⎢                     ⎥\n",
       "⎢-45430               ⎥\n",
       "⎢───────   0    0   0 ⎥\n",
       "⎢  1201               ⎥\n",
       "⎢                     ⎥\n",
       "⎣  16      0    0   0 ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "routh(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da09ff35",
   "metadata": {},
   "source": [
    "## Example 8-11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "55c6cd1a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( s^{3} + 4 s^{2} + 8 s + 4, s, domain=\\mathbb{Z} \\right)}$"
      ],
      "text/plain": [
       "Poly(s**3 + 4*s**2 + 8*s + 4, s, domain='ZZ')"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = s**3 + 4*s**2 + 8*s + 4\n",
    "p = sympy.Poly(p, s)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "abb159af",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 8\\\\4 & 4\\\\7 & 0\\\\4 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  8⎤\n",
       "⎢    ⎥\n",
       "⎢4  4⎥\n",
       "⎢    ⎥\n",
       "⎢7  0⎥\n",
       "⎢    ⎥\n",
       "⎣4  0⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RA = routh(p)\n",
    "RA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "4513ec3b",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Stable\n"
     ]
    }
   ],
   "source": [
    "msgs = 'Stable' if min(RA[:, 0]) > 0 else 'Unstable'\n",
    "print(msgs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "ef1b7b67",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  -0.704\n",
      "  -1.648   -1.721i\n",
      "  -1.648    1.721i\n"
     ]
    }
   ],
   "source": [
    "display_roots(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45044c7e",
   "metadata": {},
   "source": [
    "## Now check if poles satisfy $Re[s] \\leq h$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "33307679",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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xSCZPxNBdj+QyCTxQyE4vJyYx2yxNkbe2dnGNydzwIu4u78SyEN+85ZFK2SYw7yx2TQm7GS5bVC9t/0LXJ4rT9huFdxVQ9+Dk0Mu7NF/UyaWnt+5jyPTWLb5Jc6T4qw4gNhiJxz0WPfIktzx0Cadp1KfytAVrPGszSaHfEMmIAxieRuVy6FM2zwRW5dPX2FCWjhrOitVBPmV55AjlMrlv6YZs5jwbW/gQFlc0yCz1wTbI6irLbHOzX9lgDmyWevQodvAaD682cVr3RNfAlkyDqxgfnNQBDmSZ27vx3y+U/Y0qAtSc6Hw0dnNRy5ljHK8ZOS8rea2kN8q438os0j+O9zv2SU8G1DtdnGAk0j0AXiu0TrBRHJw4Hn6cGLcdB/0TxTL5Vzopb4WRydjIBk6B6HDWb4JZumeg0wHLQWl8N4Fx94cBfW4YMugGE1Vsd2zl5eFgq8Jq/xKPF0NvPUyMLPTUx84yXbk+irvkiQ9aVbvIJteYbGE+obwXyy1K29/mOFJ2aIucWbp8F+/TiWqeH+NNmV5bJAdbys0U/YQ+Wp4QenmX5sPcc5Dp1RF7XONbbcB8wZehg/UHAQW5xuI+ecpnXmJtxpZASmONY/3l1REIR8LmcMYcTlUi5THv8UiQuSSQ4sjDlnINZ73cKL/0B+C1NPqirZdTdAs2IFsXcfvIYBPva/rAFw6lxMN6VeqrpJMR+gaSbjeKsMbizPFqF+8b0wdqhE03lLn17NkzbnDCCM054+QNDz01mO6rJB7KwYcCEPdMYKGxFCITT9iUIZ1HOdZhdLslpeEMDDqP5dVC8SKTztV8IV08TF4QvADGJEbd1JW8XsXpDHSMgUeue8qAD/b9q+uBrudKT5Oh4tiIfHjghwA4yd8mLfsr+QzE2e/Uqfyh2Fjb5gsGndAGajJYafQLJq3Hig9OdmFS2rliSLvnizJ27vzdJdnX6l8uDFV+bz2xDuqvEZPYPcvwyINXfCdpF9OzFkqnvWMy6t7CfG951UFZL5becYQ8m1OvFGfcMBcybwxIaV6Ze21BsOQxh0HwQ9VxSoaXd2m+Y9R9DJleu2Pd3vFNf4Oaf4xe9brHonhH5SmffvBC1xtdbBxwbOiH9BPSWAvzNc76D2sg703v9B/xM0ex9g++5NY9mxdeBynXVpvT0JXHp8EnUDhVN8ZK2FyrbFpzFW/pY/Ipw6HNznqk9KOT6sWhBBPmBXOm0Zn1EyfX8Ai2KY05gz6w85cVxMuBCO/sfn3r3bt1PCGUMiiLgzIJYPHzj2wxcseRUPpFk2w+yKm7aHCcxnUMnUB1to5AR+CoCMS5qPY6x6x6l5Y3S4le6L0goLbmwASH+pfb76XGSiWqnL+Hlz9ue6T7SQ5dFMtzcffpXkWVc05iZ8XVaT4CHcP52PWSHYGOwHII8C7skvP50vKWs7RLWgwB9Zm7EsahWHhH8M5ikqcLeqQiPGvnmBGFOPKdQziGr3XZY405Ms6yjHD74E4nl26ojuHSiHZ5HYGOwFQE4sLMo81FaGl5iyjVhRwLAfyo8PUvFZzspE51P9fFi5Y8b7+vcM4p3UbleO7PkXX6mEH3nToCHYGOQEegI3AuCDzRGsbL8EvR0vKW0qvLWR4BDrTSodZq3qk7xE4NBo4f+Xca6UXwQ+T1sh2BjkBHoCPQEegIdATWjIB8Hj624K8dpA3BRTh1gC6jOKnjHYLksZLeqSPQEegIdAQ6Ah2BjsAlISBfh69m+Thi8E3BKR+/LoqvDOPxLX/tmU+CO3UEOgIdgY5AR6Aj0BG4VAQ4wMr/nFaw8/9ydvyl284okwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( 1.0 s^{3} + 1.87 s^{2} + 3.8323 s - 0.0215109999999998, s, domain=\\mathbb{R} \\right)}$"
      ],
      "text/plain": [
       "Poly(1.0*s**3 + 1.87*s**2 + 3.8323*s - 0.0215109999999998, s, domain='RR')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = -0.71\n",
    "pa = p.subs(s,s+h)\n",
    "pa = sympy.Poly(pa, s)\n",
    "pa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "55c9324d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0 & 3.8323\\\\1.87 & -0.0215109999999998\\\\3.84380320855615 & 0\\\\-0.0215109999999998 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡        1.0                3.8323       ⎤\n",
       "⎢                                        ⎥\n",
       "⎢       1.87          -0.0215109999999998⎥\n",
       "⎢                                        ⎥\n",
       "⎢ 3.84380320855615             0         ⎥\n",
       "⎢                                        ⎥\n",
       "⎣-0.0215109999999998           0         ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RAa = routh(pa)\n",
    "RAa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "db8b5492",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Poles are not left of s = -0.71\n"
     ]
    }
   ],
   "source": [
    "msgs = 'are' if min(RAa[:, 0]) > 0 else 'are not'\n",
    "print('Poles '+msgs+' left of s = '+str(h))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "364a5c8e",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Example 8-12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "9f682a37",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( 1.0 s^{3} + 0.5 s^{2} + \\left(1.0 K_{c} + 0.5\\right) s + 1.0 - 1.0 K_{c}, s, domain=\\mathbb{R}\\left[K_{c}\\right] \\right)}$"
      ],
      "text/plain": [
       "Poly(1.0*s**3 + 0.5*s**2 + (1.0*K_c + 0.5)*s + 1.0 - 1.0*K_c, s, domain='RR[K_\n",
       "c]')"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gnum = s-1\n",
    "Gden = s**3 + 0.5*s**2 + 0.5*s + 1 \n",
    "p = Gden + K_c*Gnum\n",
    "p = sympy.Poly(p, s)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "b38083f8",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0 & 1.0 K_{c} + 0.5\\\\0.5 & 1.0 - 1.0 K_{c}\\\\3.0 K_{c} - 1.5 & 0\\\\1.0 - 1.0 K_{c} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡     1.0       1.0⋅K_c + 0.5⎤\n",
       "⎢                            ⎥\n",
       "⎢     0.5         1.0 - K_c  ⎥\n",
       "⎢                            ⎥\n",
       "⎢3.0⋅K_c - 1.5        0      ⎥\n",
       "⎢                            ⎥\n",
       "⎣  1.0 - K_c          0      ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RA = routh(sympy.Poly(p, s))\n",
    "RA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "97c5d99e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.5 < K_{c} \\wedge K_{c} < 1.0$"
      ],
      "text/plain": [
       "0.5 < K_c ∧ K_c < 1.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rr = sympy.solve([e > 0 for e in RA[:, 0]], K_c)\n",
    "rr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "5fd3609d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For K_c =   0.0 system is Unstable\n",
      "For K_c =   0.1 system is Unstable\n",
      "For K_c =   0.2 system is Unstable\n",
      "For K_c =   0.3 system is Unstable\n",
      "For K_c =   0.4 system is Unstable\n",
      "For K_c =   0.5 system is Unstable\n",
      "For K_c =   0.6 system is Stable\n",
      "For K_c =   0.7 system is Stable\n",
      "For K_c =   0.8 system is Stable\n",
      "For K_c =   0.9 system is Stable\n",
      "For K_c =   1.0 system is Unstable\n"
     ]
    }
   ],
   "source": [
    "for k_c in np.arange(0,rr.args[1].args[1]*1.1,.1):\n",
    "    r = roots(p.subs(K_c,k_c))\n",
    "    N = len(r)\n",
    "    real_part = [re(e) for e in list(r)[0:N]]\n",
    "    msgs = 'Stable'\n",
    "    if max(real_part) >= 0:\n",
    "        msgs = 'Unstable'        \n",
    "    print(f'For K_c = {k_c:5.1f} system is '+msgs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "6e69ec3c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.pyplot import figure, savefig\n",
    "%matplotlib inline\n",
    "\n",
    "def nicegrid():\n",
    "    plt.grid(True, which='major', color='#666666', linestyle='-')\n",
    "    plt.grid(True, which='minor', color='#999999', linestyle='-', alpha=0.2)\n",
    "    plt.minorticks_on()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5472ad3f",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "RR = np.linspace(0,1,200)\n",
    "R = np.zeros((len(RR),3),dtype=complex)\n",
    "for ii,k_c in enumerate(RR):\n",
    "    r = roots(p.subs(K_c,k_c))\n",
    "    R[ii,:] = [e for e in list(r)[0:len(r)]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "3943c89c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 750x750 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5, 5),dpi=150,constrained_layout = True)\n",
    "plt.plot(np.real(R[:,0]),np.imag(R[:,0]),'b.',linewidth=4,label='$r_1(t)$')\n",
    "plt.plot(np.real(R[:,1]),np.imag(R[:,1]),'r.',linewidth=4,label='$r_2(t)$')\n",
    "plt.plot(np.real(R[:,2]),np.imag(R[:,2]),'g.',linewidth=4,label='$r_3(t)$')\n",
    "\n",
    "nicegrid()\n",
    "plt.legend()\n",
    "plt.xlim(-2,2)\n",
    "plt.ylim(-2,2)\n",
    "plt.xlabel('Real')\n",
    "plt.ylabel('Imag')\n",
    "plt.title('Cubic $D(s)$ Pole Locations for various ${:3.1f}\\leq K \\leq {:3.1f}$'.format(min(RR),max(RR)))\n",
    "savefig(\"./figs/RA_8_12.pdf\", dpi=600)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "1b523c80",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle 0.5 < K_{c} \\wedge K_{c} < 1.0$"
      ],
      "text/plain": [
       "0.5 < K_c ∧ K_c < 1.0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "a1b1ecb9",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 1.5],\n",
       "       [0.5, 0. ],\n",
       "       [1.5, 0. ],\n",
       "       [0. , 0. ]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sympy.lambdify(K_c, RA)\n",
    "f(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "4e027891",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1. , -1.5],\n",
       "       [ 0.5,  3. ],\n",
       "       [-7.5,  0. ],\n",
       "       [ 3. ,  0. ]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c27fd635",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3fbf6412-5ab5-4413-8831-643d03796393",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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