{ "cells": [ { "cell_type": "code", "execution_count": 5, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "## Incluir las siguientes librerias\n", "\n", "import matplotlib.pyplot as plt\n", "import ipywidgets as widgets\n", "from control import * " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Estabilidad de Sistemas en Control\n", "**_Analizar la estabilidad en los sistemas a controlar._**" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "_Un sistema es estable si converge a un valor diferente de infinito_\n", "\n", "La estabilidad puede ser verificada con diferentes métodos:\n", "\n", "- Respuesta Temporal\n", "- Matriz de Routh\n", "- Polos y Ceros\n", "- Mapa de polos y ceros\n", "- Root Locus\n", "\n", "Recomiendo verificar por todos los métodos el sistema para estar seguro de que se tiene un sistema estable. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Verificar Estabilidad en el Sistema masa-resorte-amortiguador \n", "\n", "Para verificar la estabilidad del sistema masa-resorte-amortiguador. Llevemos cada uno de los parametros del sistema a la zona de signos negativos. En el siguiente simulador. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "30fbb59e12504b459a00aa4c07493e76", "version_major": 2, "version_minor": 0 }, "text/plain": [ "HBox(children=(VBox(children=(FloatSlider(value=1.0, description='m', max=3.0, min=-3.0), FloatSlider(value=1.…" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "## Parametros del modelo\n", "\n", "m = widgets.FloatSlider(value=1,min=-3,max=3,step=0.1,description='m')\n", "c = widgets.FloatSlider(value=1,min=-3,max=3,step=0.1,description='c')\n", "k = widgets.FloatSlider(value=1,min=-3,max=3,step=0.1,description='k')\n", "\n", "## Definicion de la simulacion\n", "\n", "def simulacion(m,c,k):\n", " sistema = tf([1],[m,c,k])\n", " tiempo, amplitud = step_response(sistema)\n", " plt.plot(tiempo,amplitud)\n", " plt.title('Simulación masa-resorte-amortiguador')\n", " plt.xlabel('tiempo (t)')\n", " plt.ylabel('amplitud')\n", "\n", "## Presentación de los resultados \n", " \n", "plot_sistema = widgets.interactive_output(simulacion,{'m':m,'c':c,'k':k}) \n", "widgets.HBox([widgets.VBox([m,c,k]),plot_sistema])" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "image/png": 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ff3tiBX9/z6us3lYbdWgiKU+JIE3MXbaFVdtqufG8o8nNTt9/6+hBhfzmuhnc9akTKKvcy8d+/DLfe2ql+iwSOQLpe8TIIE3NMe567j2mDO/LpUnSjUQimRmfPGkkf73xDC4vHcX9C9Zx9n+/yGOvb1J1kchhUCJIA0+/vZUNVXv55rlHp+TloodrYO88/uuTx/HEV2cxckAv/unRZVz6k1dYsEad2IkcCiWCFOfu/OLltYwb3JtzjhkSdTiROH5kf/745VP40aenUbO3kavuW8Tn7l/M8k01UYcmkhKUCFLca2sreWfzLr5w2riMKg20lpVlfPzEEfztpjP4l4uPZdmmGi77yQKunbOEZRuVEEQ6krBEYGb3m1m5mb3TznIzsx+b2RozW25mJyUqlnT2i5fWUtwnj0+cOCLqUJJCQW42Xzh9HK/cfDb/fMEk3thQzey7F3DlvQt5ftV2tSGItCGRF5vPAX4CPNDO8ouAieHjZOBn4V/pove21zJ/9Q5uOu9oCnKzow4nqfTJz+GrZ03gc6eU8NtF6/nVgjKunbOU8YN7c9XJY/jkSSPoX5gXdZgpz92pa4yxu76JvQ1N7KlvZl9jE/saYtQ1NlPfFKO+qZmGphj1TTEam2M0NjtNzTGaYk7MPfgbc5zgMuiYQzD1IcMwAyMo/ZlBlhnZZmRlBX+zsyA7K4ucrGBeTpaRnWXkZtv++TnZwfycrKzwefA3fp3c7Ky41wXT8a9pmZdlwYUL6SBhicDdXzKzkg5WmQ084EGfAQvNrL+ZDXP3rYmKKd384qW1FORmcfXMMVGHkrT65Odw/enj+fysscx7eyv3Lyjj9qdWcsczq7hwylA+cdIITp1QnNaX3HaFu7OnoZnK3fVU7G6gak8D1XsaqNzTQM3eBmr2NlKzL/i7q66JXfsaqa1rZHd9E0dSyMoyyM4yrOWgHh5cDYKjPtCSE2IepAd3aHbH3WkOE0dUDkg6ZmSHiSbLgoTR8jd4zgHzs7KCabMPlxkt89if7IIEGPy95PhhfHp6998nFOXtpyOAjXHTm8J5ByUCM7seuB5g9Oj0u1nqcFTsrudPb23miumjGdBbZ7adyc3OYva0EcyeNoKVW3bx+yUbePzNzcxdtoX+hblcOGUo5085io+OK6ZXXnqVrvY2NLF1Zx3bd9axbVfwKN9Vz/ZddZTX1rMjfOxrZ4S4vJwsBhTm0r9XHv165TKify+OHVpEUUEOfQpy6JOfS5/8bArzcijMy6ZXXja9crMpCB/5OVnkhY/c7CzysrP2n5l3xxm1e5AMmmNBYmh2p7nZaYrFaI4FJY6mcLop5jQ2t5rfHKMx9mEpZf+6zcH2GsPtNDY7zbGgRBNreX0sRnMMmsNtt8w/IJYwWcU8WH7AtDsePvfW82LQ5LEw+QVJsKEpMeN0pEQ/BO5+L3AvQGlpqSp5gT+8vonGZudzp6g0cKgmD+/Ld2dP5TsfO5aX36vgqeVbeHLZFh5espG8nCxmjhvEKeMHMb1kAFNH9CM/J3kTQ11jM1t31rG1Zh9b4v/u3Me2nXVsqdnHrrqmg15XlJ/DkL75DCkq4MTR/RncJ5/BRfkU98lnUJ88BvXOZ2CfPAYU5tIrNzupq0CC0gRJO/hSKogyEWwGRsVNjwznSSfcnYeXbGR6yQAmDCmKOpyUlZ+TzbmTj+LcyUdR39TMknXVzF9dzovv7eCOP68K18li8vC+TB7Wl8nD+zJxSBElgwoZXJSf0INjc8yp3ttA+a56ymuDM/iWs/ntO+uCg//OfVTvPfiO6kG98xjar4CRAwqZMXYgQ/sVMLRvAUP7FTCsXy+O6pufVn1RyZGL8tswF7jBzB4maCTeqfaBrlm4top1FXu44awJUYeSNvJzsjl1YjGnTizm3wiq3paWVbOkrIp3Nu9k7rItPLRow/71C/OyGdG/1/6z6kG98ygqyKVvrxx65+XsrwbJzrL9xfrmmO9vPN3X0Mye+mZ21zdSW9dEdUs9/N5GKvfUU7Wnoc2674G98ziqbwHD+wVn8sPCg/uw/gUM79eLof0KdOGAHLKEJQIz+x1wJlBsZpuAfwdyAdz9HmAecDGwBtgLfD5RsaSbh5dsoKggh4uPGxZ1KGmruE8+F04dyoVThwJBKWxT9T7WVuxhfeUe1lXsYWtNHdtr61i8roqqPQ3t1rF3pFduNn0KcujfK5cBhXmMGVTISWMGMLhP3v6DfkuyGdI3P6mrqSR1JfKqoSs7We7AVxO1/3RVs7eBP7+zjSumj0q7Rs1kZmaMGljIqIGFwOA212lsjrG7rok9DU00NgeNkk3Nvv8KkGwz8nOyyc/NoiAnm9752eRk+NVKkhxUUZhi/vjGZhqaYlyRgEvI5MjkZmcxoHeeruKSlKPTkRQSNBJv4ISR/Zg8vG/U4YhImlAiSCHLN+3kve2703LgGRGJjhJBCnnirS3kZWepkVhEupUSQYpojjlPLt/CmZMG069XbtThiEgaUSJIEYvWVrKjtp7Z09TLqIh0LyWCFPHEW1vonZfNOcdm5uAzIpI4SgQpoL6pmXnvbOWCKUN116iIdDslghTw4uod1NY1cdm09B+YXkR6nhJBCnhi2RYG9c5j1oTiqEMRkTSkRJDkdtc38deV27n4uGEZP3iKiCSGjixJ7vlV5dQ3xbj0BFULiUhiKBEkuWdXbKO4Tx4fGTMg6lBEJE0pESSxusZmXlhVznmTj9LoSyKSMEoESey1DyrZ09DM+VOGRh2KiKQxJYIk9uyKbfTJz+GU8YOiDkVE0pgSQZJqjjnPrdzOmZMGa1QqEUkoJYIk9caGair3NHCBqoVEJMGUCJLUs+9sIy87izMntT0soohId1EiSELuzrMrtzFrwiCKCtTltIgklhJBElq1rZaNVftULSQiPUKJIAk9v6ocgLPV5bSI9AAlgiQ0f1U5x43ox5CigqhDEZEMoESQZGr2NvDGhmrOUiOxiPQQJYIk8/L7FcQczjxG1UIi0jMSmgjM7EIzW21ma8zsljaWjzaz+Wb2ppktN7OLExlPKpi/upwBhbmcMLJ/1KGISIZIWCIws2zgbuAiYDJwpZlNbrXavwKPuPuJwBXATxMVTyqIxZwXV+/gjKMHq5M5EekxiSwRzADWuPtad28AHgZmt1rHgb7h837AlgTGk/Te3ryTyj0NnKVqIRHpQYlMBCOAjXHTm8J58W4DrjazTcA84GttbcjMrjezpWa2dMeOHYmINSnMX12OGZw2UQ3FItJzom4svhKY4+4jgYuB35jZQTG5+73uXurupYMHp+9Bcv7qHUwb1Z+BvfOiDkVEMkgiE8FmYFTc9MhwXrzrgEcA3P01oADIyBHaK3bXs3xTDWdNUrWQiPSsRCaCJcBEMxtrZnkEjcFzW62zATgHwMyOJUgE6Vv304GX39+BO+pkTkR6XMISgbs3ATcAzwLvElwdtMLMbjezy8LVbgK+YGbLgN8B17i7JyqmZPby+xUM7J3H1OH9og5FRDJMTiI37u7zCBqB4+fdGvd8JTArkTGkAnfnlfcrOGX8ILJ02aiI9LCoG4sFeL98N+W19Zw2MSObR0QkYkoESeDl9ysAmDVBiUBEep4SQRJYsKaCscW9GTmgMOpQRCQDKRFErKEpxsK1lZyq0oCIRESJIGJvbqhmb0Mzp6p9QEQiokQQsQVrKsgymDluUNShiEiGUiKI2MtrKjhhVH/69dIg9SISDSWCCO3c18iyjTWcpvYBEYmQEkGEFq6tJOZwqnobFZEIKRFE6NU1FfTKzWbaKI1GJiLRUSKI0GtrKyktGUBejv4NIhIdHYEiUrG7nve27+aj43W1kIhEq8NO58zsxo6Wu/td3RtO5li4thKAj+qyURGJWGe9jxaFfycB0/lwPIFLgcWJCioTLFxbSe+8bI4boW6nRSRaHSYCd/8ugJm9BJzk7rXh9G3A0wmPLo299kEl08cOJCdbtXMiEq2uHoWOAhriphvCeXIYynfV8cGOPaoWEpGk0NWBaR4AFpvZ4+H0x4FfJyak9PdaS/uAGopFJAl0KRG4+3+a2Z+B08JZn3f3NxMXVnpbuLaSooIcpmhYShFJAl1KBGY2GqgAHo+f5+4bEhVYOnvtg0pOHjuQbA1LKSJJoKtVQ08DLYPK9wLGAquBKYkIKp1t3bmPssq9XD1zTNShiIgAXa8aOi5+2sxOAr6SkIjSXMv9A+p2WkSSxWFdu+jubwAnd3MsGWHhB1X0Lchh8rC+UYciIgJ0vY0g/g7jLOAkYEtCIkpzi8uqmDF2IFlqHxCRJNHVEkFR3COfoM1gdqKCSlflu+pYV7GHGWMHRh2KiMh+XW0sXunuj8bPMLPLgUfbWV/asLisCoAZY9U+ICLJo6slgm93cd4BzOxCM1ttZmvM7JZ21vmUma00sxVm9tsuxpOSFq+rojAvm6nD1T4gIsmjs95HLwIuBkaY2Y/jFvUFmjp5bTZwN3AesAlYYmZz3X1l3DoTCRLKLHevNrMhh/c2UsPidVV8ZMwA9S8kIkmlsyPSFmApUAe8HveYC1zQyWtnAGvcfa27NwAPc3C7wheAu929GsDdyw8t/NRRs7eBVdtqOVntAyKSZDrrfXQZsMzMHnL3DksAbRgBbIyb3sTBl5weDWBmC4Bs4DZ3f6b1hszseuB6gNGjRx9iGMlhSVk1oPYBEUk+nVUNPeLunwLeNDNvvdzdj++G/U8EzgRGAi+Z2XHuXtNqP/cC9wKUlpYeFEcqWLyukrycLI4fqf6FRCS5dHbV0DfCv5ccxrY3A6PipkeG8+JtAha5eyOwzszeI0gMSw5jf0lt8boqpo3qT0FudtShiIgcoMM2AnffGv5d39ajk20vASaa2VgzywOu4MMRzlr8iaA0gJkVE1QVrT2M95HUdtc38c6WXWofEJGk1FnVUC0fdjYHYOG0Ae7u7V4H6e5NZnYD8CxB/f/97r7CzG4Hlrr73HDZ+Wa2EmgG/tndK4/oHSWhN9ZX0xxzTlb7gIgkoc4ai4s6Wt4Zd58HzGs179a45w7cGD7S1uJ1VeRkGSeN6R91KCIiB+nqncUtPY6eSlAieEUD03Td4rIqpgzvS2Felz9uEZEe06U7m8zsVoKhKQcBxcAcM/vXRAaWLuqbmnlrYw3TS9Q+ICLJqaunqFcBJ7h7HYCZ3QG8BfxHogJLF+9s3klDU4zpaigWkSTV1b4OtgAFcdP5HHwpqLRh8brgRrLSMQMijkREpG1dLRHsBFaY2XMEbQTnAYtb+h9y968nKL6Ut7SsivGDezOoT37UoYiItKmrieBx4gauB17o/lDSTyzmLF1fzUVTh0YdiohIu7o6ZvGvEx1IOnq/fDc79zWqoVhEklpXrxq6xMzeNLMqM9tlZrVmtivRwaW6loFolAhEJJl1tWroR8AngbfDm8CkC5aWVXFU33xGDewVdSgiIu3q6lVDG4F3lAQOzZJ1VZSWDMRMA9WLSPLqaongW8A8M3sRqG+Z6e53JSSqNLC5Zh9bdtbxRVULiUiS62oi+E9gN8G9BHmJCyd9LFkXtA+Uluj+ARFJbl1NBMPdfWpCI0kzS8qqKMrP4ZihGqheRJJbV9sI5pnZ+QmNJM0sLavmpDEDyM5S+4CIJLeuJoIvA8+Y2T5dPtq5nXsbWb29lumqFhKRFNDVG8qKzGwgwTCSBZ2tn+le39DSPqCGYhFJfl1KBGb2jwTjF48k6HV0JvAqcE7iQktdS8qqyc02ThipgWhEJPl1tWroG8B0YL27nwWcSNARnbRhaVkVU0f0o1eeBqoXkeTX1URQFzcWQb67rwImJS6s1FXX2MyyjTvVrYSIpIyuXj66ycz6A38CnjOzamB94sJKXe9s3klDc0zjD4hIyuhqY/Enwqe3mdl8oB/wTMKiSmFLyoKBaD6iRCAiKeKQR1N39xcTEUi60EA0IpJqutpGIF3QMhCN2gdEJJUoEXSjNTuCgWh0/4CIpBIlgm60ZP9ANGofEJHUkdBEYGYXmtlqM1tjZrd0sN7fmZmbWWki40m0pWXVDC7KZ/TAwqhDERHpsoQlAjPLBu4GLgImA1ea2eQ21isiuGFtUaJi6SmL11UxvWSABqIRkZSSyBLBDGCNu6919w+qaZEAAA6NSURBVAbgYWB2G+t9D/gBUJfAWBJuS80+NtfsU0OxiKScRCaCEQRDXLbYFM7bz8xOAka5+9MdbcjMrjezpWa2dMeOHd0faTdYooHqRSRFRdZYbGZZwF3ATZ2t6+73unupu5cOHjw48cEdhiVlVfTJz+HYYRqIRkRSSyITwWZgVNz0yHBeiyJgKvCCmZUR9Gg6N1UbjDUQjYikqkQmgiXARDMba2Z5wBXA3JaF7r7T3YvdvcTdS4CFwGXuvjSBMSXE/oFo1K2EiKSghCUCd28CbgCeBd4FHnH3FWZ2u5ldlqj9RmHp+ircYfpYtQ+ISOo55L6GDoW7zwPmtZp3azvrnpnIWBJpcVkVudnGtFEaiEZEUo/uLO4GS8uqOW5EPwpyNRCNiKQeJYIjVNfYzPJNNaoWEpGUpURwhN7aWENjszND9w+ISIpSIjhCS8MbyTQQjYikKiWCI7S4rJpJRxXRvzAv6lBERA6LEsERaGqO8XpZFdPHqjQgIqlLieAIrNiyiz0NzZw8dlDUoYiIHDYlgiOwaF0lACePU0OxiKQuJYIjsGhtFeOKezOkqCDqUEREDpsSwWFqjjmLy6pUGhCRlKdEcJje3bqL2romtQ+ISMpTIjhMi9YF9w+oRCAiqU6J4DAtWlvJ6IGFDOvXK+pQRESOiBLBYYi1tA+ofyERSQNKBIfhvfJaavY2cvI4tQ+ISOpTIjgMi9aG7QMqEYhIGlAiOAyL1lUyon8vRg0sjDoUEZEjpkRwiGIxZ9FatQ+ISPpQIjhEq7fXUrmngY+OV/uAiKQHJYJDtGBNBQCzJhRHHImISPdQIjhEr35Qydji3gzvr/sHRCQ9KBEcgsbmGIvWVjJrgqqFRCR9KBEcguWbatjT0Mys8aoWEpH0oURwCBasqcQMNRSLSFpRIjgEr6ypYMrwvhqfWETSSkITgZldaGarzWyNmd3SxvIbzWylmS03s7+Z2ZhExnMk9jY08eaGalULiUjaSVgiMLNs4G7gImAycKWZTW612ptAqbsfDzwG/N9ExXOklpRV09jsnKLLRkUkzSSyRDADWOPua929AXgYmB2/grvPd/e94eRCYGQC4zkir66pIDfbmF4yIOpQRES6VSITwQhgY9z0pnBee64D/tzWAjO73syWmtnSHTt2dGOIXbfggwpOHD2AwrycSPYvIpIoSdFYbGZXA6XAnW0td/d73b3U3UsHDx7cs8EBFbvrWbFlF6eqWkhE0lAiT283A6PipkeG8w5gZucC/wKc4e71CYznsL24egfucPYxQ6IORUSk2yWyRLAEmGhmY80sD7gCmBu/gpmdCPwcuMzdyxMYyxF5fnU5g4vymTysb9ShiIh0u4QlAndvAm4AngXeBR5x9xVmdruZXRaudifQB3jUzN4ys7ntbC4yTc0xXnpvB2dNGkxWlkUdjohIt0toy6e7zwPmtZp3a9zzcxO5/+7w+vpqauuaVC0kImkrKRqLk9nzq8vJzTZ1Oy0iaUuJoBMvrNrB9JKBFBXkRh2KiEhCKBF0YHPNPlZvr+WsSaoWEpH0pUTQgfmrgguZzlL7gIikMSWCDsxfVc6ogb0YP7h31KGIiCSMEkE79jU0s+CDCs6eNAQzXTYqIulLiaAd81eXU9cY44IpQ6MORUQkoZQI2vHksi0U98nn5HEajUxE0psSQRt21zfx/KpyPnbcULJ1N7GIpDklgjb8deV26ptiXHrC8KhDERFJOCWCNjy1fAvD+hVw0mgNQiMi6U+JoJWdext58b0dXHL8MHUyJyIZQYmglWdXbqOx2bnkeFULiUhmUCJo5cllWxg9sJDjR/aLOhQRkR6hRBCnvLaOVz+o5JLjh+kmMhHJGEoEcR5cuIGYO5eXjup8ZRGRNKFEEKprbOahhes555ghjC1W30IikjmUCEJPvLWZyj0NXHvq2KhDERHpUUoEgLtz/ytlHDO0iI+qSwkRyTBKBMCCNZWs3l7LtaeOVSOxiGQcJQLgl6+spbhPHpepSwkRyUAZnwje2byT+at3cNXJYyjIzY46HBGRHpfRiaC+qZmbHlnGkKJ8Pj+rJOpwREQikRN1AFH6n7++z+rttdx/TSn9C/OiDkdEJBIZWyJ4Y0M197z4AZ8qHcnZxxwVdTgiIpFJaCIwswvNbLWZrTGzW9pYnm9mvw+XLzKzkkTG06Jidz3/9OgyhvYt4F8vmdwTuxQRSVoJqxoys2zgbuA8YBOwxMzmuvvKuNWuA6rdfYKZXQH8APh0omLaVdfIfS+t5ZevrKOuKcYD186gb0FuonYnIpISEtlGMANY4+5rAczsYWA2EJ8IZgO3hc8fA35iZubu3t3BPL9qOzc+soyavY1cfNxQbjxvEhOG9Onu3YiIpJxEJoIRwMa46U3Aye2t4+5NZrYTGARUxK9kZtcD1wOMHj36sIIZW9yHaaP6c9N5kzhOXUyLiOyXElcNufu9wL0ApaWlh1VaGFvcmzmfn9GtcYmIpINENhZvBuL7cx4ZzmtzHTPLAfoBlQmMSUREWklkIlgCTDSzsWaWB1wBzG21zlzgc+HzvweeT0T7gIiItC9hVUNhnf8NwLNANnC/u68ws9uBpe4+F/gl8BszWwNUESQLERHpQQltI3D3ecC8VvNujXteB1yeyBhERKRjGXtnsYiIBJQIREQynBKBiEiGUyIQEclwlmpXa5rZDmD9Yb68mFZ3LSe5VIo3lWKF1Io3lWKF1Io3lWKFI4t3jLsPbmtByiWCI2FmS929NOo4uiqV4k2lWCG14k2lWCG14k2lWCFx8apqSEQkwykRiIhkuExLBPdGHcAhSqV4UylWSK14UylWSK14UylWSFC8GdVGICIiB8u0EoGIiLSiRCAikuEyJhGY2YVmttrM1pjZLVHH0x4zG2Vm881spZmtMLNvRB1TV5hZtpm9aWZPRR1LR8ysv5k9ZmarzOxdM/to1DF1xMz+T/g9eMfMfmdmBVHHFM/M7jezcjN7J27eQDN7zszeD/8OiDLGFu3Eemf4XVhuZo+bWf8oY4zXVrxxy24yMzez4u7YV0YkAjPLBu4GLgImA1ea2eRoo2pXE3CTu08GZgJfTeJY430DeDfqILrgf4Bn3P0Y4ASSOGYzGwF8HSh196kE3bknW1ftc4ALW827Bfibu08E/hZOJ4M5HBzrc8BUdz8eeA/4dk8H1YE5HBwvZjYKOB/Y0F07yohEAMwA1rj7WndvAB4GZkccU5vcfau7vxE+ryU4UI2INqqOmdlI4GPAfVHH0hEz6wecTjAOBu7e4O410UbVqRygVziCXyGwJeJ4DuDuLxGMJRJvNvDr8PmvgY/3aFDtaCtWd/+LuzeFkwsJRlJMCu18tgA/BL4FdNuVPpmSCEYAG+OmN5HkB1cAMysBTgQWRRtJp35E8MWMRR1IJ8YCO4BfhdVY95lZ76iDao+7bwb+m+DMbyuw093/Em1UXXKUu28Nn28DjooymENwLfDnqIPoiJnNBja7+7Lu3G6mJIKUY2Z9gD8A33T3XVHH0x4zuwQod/fXo46lC3KAk4CfufuJwB6Sp9riIGHd+myCBDYc6G1mV0cb1aEJh55N+mvUzexfCKplH4o6lvaYWSHwHeDWztY9VJmSCDYDo+KmR4bzkpKZ5RIkgYfc/Y9Rx9OJWcBlZlZGUOV2tpk9GG1I7doEbHL3lhLWYwSJIVmdC6xz9x3u3gj8ETgl4pi6YruZDQMI/5ZHHE+HzOwa4BLgqiQfM308wUnBsvD3NhJ4w8yGHumGMyURLAEmmtlYM8sjaHCbG3FMbTIzI6jDftfd74o6ns64+7fdfaS7lxB8rs+7e1Ketbr7NmCjmU0KZ50DrIwwpM5sAGaaWWH4vTiHJG7cjjMX+Fz4/HPAExHG0iEzu5CgWvMyd98bdTwdcfe33X2Iu5eEv7dNwEnh9/qIZEQiCBuDbgCeJfghPeLuK6KNql2zgM8QnFm/FT4ujjqoNPI14CEzWw5MA74fcTztCksujwFvAG8T/F6TqksEM/sd8Bowycw2mdl1wB3AeWb2PkGp5o4oY2zRTqw/AYqA58Lf2j2RBhmnnXgTs6/kLgmJiEiiZUSJQERE2qdEICKS4ZQIREQynBKBiEiGUyIQEclwSgSSMcKeR78SNz3czB6LMqYWZjaspedWM5sWf8mwmV1iZrdHF52kOyUCyST9gf2JwN23uPvfRxhPvBuBX4TPpwHx9448DVwadjEg0u2UCCST3AGMD28cutPMSlr6eg/HU7jTzJaEfdN/MZx/ppm9aGZPmNlaM7vDzK4ys8Vm9raZjQ/Xm2Nm95jZUjN7L+yDCTMrMLNfheu+aWZntRPb3wHPhHe+3w58Oozz02G3By8QdIMg0u1yog5ApAfdQtD3/DTY37tri+sIevecbmb5wAIza+np8wTgWIIugdcC97n7DAsGDfoa8M1wvRKCLs/HA/PNbALwVYK+144zs2OAv5jZ0e5e17JjMxsLVLt7fTh9K8EYBDfExbcUOA14pHs+CpEPqUQgEjgf+KyZvUXQ7fcgYGK4bEk4TkQ98AHQkiDeJjj4t3jE3WPu/j5BwjgGOBV4EMDdVwHrgaNb7XsYQffYHSkn6IFUpNupRCASMOBr7v7sATPNzgTq42bF4qZjHPgbat1fS1f7b9kHdDYEZUG4nki3U4lAMkktQQdjbXkW+HLYBThmdvRhDFpzuZllhe0G44DVwMvAVS3bBEaH8+O9x4Eli7biPBo4aOxake6gRCAZw90rCer+3zGzO1stvo+gS+o3wgbkn3PoJeYNwGKCUa6+FLYD/BTIMrO3gd8D17S0BcTFtQf4IGxTAJgPTG5pLA7nnUVw9ZBIt1PvoyLdwMzmAE+5+2Hdl2BmnwA+4u7/2sayo4Dfuvs5RxalSNvURiCSBNz9cTMb1M7i0cBNPRmPZBaVCEREMpzaCEREMpwSgYhIhlMiEBHJcEoEIiIZTolARCTD/X+PC5/y2zLE+wAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "simulacion(1,1,1)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**¿Cómo los parámetros influyen en la estabilidad?**\n", "\n", "|masa|resorte|amortiguador|| Estable |\n", "|:---:|:----:|:----------:||:-------:|\n", "|$+$|$+$|$+$|||\n", "|$+$|$+$|$-$|||\n", "|$+$|$-$|$+$|||\n", "|$+$|$-$|$-$|||\n", "|$-$|$+$|$+$|||\n", "|$-$|$+$|$-$|||\n", "|$-$|$-$|$+$|||\n", "|$-$|$-$|$-$|||\n", "\n", "_Si la respuesta temporal de un sistema diverge entonces se considera inestable._" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Matriz de _Routh_\n", "\n", "Para verificar estabilidad con la matriz de _Routh_, primeramente se debe contruir, siguiendo los pasos descritos:\n", "\n", "1. Sacar el polinomio carasteristico (lo que es igual al deominador de la funcion de transferencia).\n", " $$a\\,s^3+b\\,s^2+c\\,s+d = 0$$\n", "2. Ubicar los coefficientes del polinomio en las dos primeras filas de la tabla.\n", " $$\n", " \\begin{array}{ccc}\n", " s^3 & a & c \\\\\n", " s^2 & b & d \\\\\n", " s^1 & \\mathcal{R} & \\\\\n", " s^0 && \\\\\n", " \\end{array}\n", " $$\n", "3. Completar las filas de la tabla hasta la fila $s^0$, siguiendo este cálculo como ejemplo a la tabla anterior:\n", " $$\\mathcal{R} = (b\\cdot c - a\\cdot d)/b$$\n", " " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Matriz de Routh Aplicado al sistema masa-resorte-amortiguador" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$$\n", "\\begin{array}{ccc}\n", "s^2 & m & k \\\\\n", "s^1 & c & \\\\\n", "s^0 & k & \\\\\n", "\\end{array}\n", "$$\n", "\n", "La estabilidad en la matriz de _Routh_ esta representada por los signos de la primer columna en la matriz. Si todos los elementos de dicha columna tienen el mismo signo, entonces el sistema es estable. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Polos y ceros\n", "\n", "Otra forma de verificar la estabilidad del sistema es mirando los polos y los ceros de este. \n", "\n", "- **Cero** : valor de $s$ que hace cero el **numerador** de la función de transferencia.\n", "- **Polo** : valor de $s$ que hace cero el **denominador** de la función de transferencia.\n", "\n", "Todos los polos deben tener parte real negativa para que el sistema sea estable." ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "La funcion de transferencia es :\n", "\n", " 1\n", "-------------\n", "s^2 + 2 s + 4\n", "\n", "Los polos que posee son:\n", "\n", "[-1.+1.73205081j -1.-1.73205081j]\n", "\n", "Los ceros que posee son:\n", "\n", "[]\n" ] } ], "source": [ "m = 1 # Parametros\n", "c = 2\n", "k = 4\n", "s = tf([1,0],1) # Variable de Laplace\n", "G = 1/(m*s**2+c*s+k) # Funcion de transferencia\n", "\n", "print(\"La funcion de transferencia es :\")\n", "print(G)\n", "print(\"Los polos que posee son:\\n\")\n", "print(pole(G))\n", "print(\"\\nLos ceros que posee son:\\n\")\n", "print(zero(G))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "Se puede observar que la parte real de los polos es negativa por lo tanto el sistema es estable. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Mapa de Polos y Ceros\n", "\n", "Este mapa es la represtación gráfica de estos valores en el plano complejo. \n", "\n", "- La parte real en el eje horizontal \n", "- La parte imaginaria en el eje vertical" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "De la función de transferencia anterior:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/plain": [ "(array([-1.+1.73205081j, -1.-1.73205081j]), array([], dtype=float64))" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "pzmap(G)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Ejercicio \n", "\n", "Modelizar y análizar el siguiente sistema:\n", "\n", "\n", "\n", "El sistema tiene tres entradas, la gravedad $g$, la fuerza manipulada $F_m$ y el perfil de la carretera $c$, y tiene cinco parámetros son:\n", "\n", "$$\n", "m=2500\\text{kg} \\quad \n", "m_r=320\\text{kg} \\quad\n", "k_s=80000\\text{N/m} \\quad\n", "k_r=500000\\text{N/m}\\quad\n", "c_S=350\\text{Ns/m}\n", "$$" ] } ], "metadata": { "celltoolbar": "Slideshow", "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": "3.6.5" } }, "nbformat": 4, "nbformat_minor": 4 }