{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "## Libreria de control\n",
    "try:\n",
    "  from control.matlab import *\n",
    "except:\n",
    "  !pip install control\n",
    "  from control.matlab import *\n",
    "    \n",
    "## Libreria para graphicar\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy\n",
    "\n",
    "## Libreria para calculo simbolico\n",
    "import sympy\n",
    "\n",
    "## Libreria de control\n",
    "import control\n",
    "\n",
    "## Libreria para widgets\n",
    "import ipywidgets as widgets\n",
    "\n",
    "## Libreria para animaciones\n",
    "from matplotlib.animation import FuncAnimation\n",
    "plt.rcParams[\"animation.html\"] = \"html5\"#\"jshtml\"\n",
    "\n",
    "## Libreria para importar Iframe de Youtube\n",
    "from IPython.display import IFrame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad de sistemas LTI\n",
    "_Entendamos como un sistema se comporta_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dinámica del sistema\n",
    "\n",
    "![](3-ecuacion.png)\n",
    "\n",
    "- Empecemos ignorando la entrada del sistema, el estado será $x$\n",
    "\n",
    "$$\\array{\\dot{x}=A\\, x &&& x(t_0)=x_0}$$\n",
    "\n",
    "¿Cuál es la respuesta temporal?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Resolviendo la ecuación diferencial ordinaria\n",
    "\n",
    "Si todo fuera escalar:\n",
    "\n",
    "$$\\array{\\dot{x}=a\\, x  &,& x(t_0)=x_0 & \\to & x(t)=e^{a(t-t_0))}x_0 }$$\n",
    "\n",
    "Como sabemos esto? verificamos la posible solución:\n",
    "\n",
    "$$\\array{\n",
    "x(t_0)=e^{a(t_0-t_0))}x_0=e^{0}x_0=x_0 & \\text{condición inicial verificada}\\\\\n",
    "\\frac{d}{dt}x(t_0)=ae^{a(t-t_0))}x_0=ax & \\text{dinámica verificada}\n",
    "} $$\n",
    "\n",
    "Para sistemas de orden superior, tenemos una versión matricial de esto:\n",
    "\n",
    "$$\\array{\\dot{x}=A\\, x &,& x(t_0)=x_0 &\\to& x(t)=e^{A(t-t_0))}x_0}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Exponencial de una matriz\n",
    "\n",
    "La definición es similar a la definición de escalares: \n",
    "\n",
    "$$e^{at}=\\sum_{k=0}^\\infty\\frac{a^kt^k}{k!} \\qquad\\qquad e^{At}=\\sum_{k=0}^\\infty\\frac{A^kt^k}{k!}$$\n",
    "\n",
    "cuya derivada es:\n",
    "\n",
    "$$\\frac{d}{dt} \\sum_{k=0}^\\infty\\frac{A^kt^k}{k!} \n",
    "= 0 + \\sum_{k=0}^\\infty\\frac{kA^kt^{k-1}}{k!} \n",
    "= A\\sum_{k=0}^\\infty\\frac{kA^{k-1}t^{k-1}}{(k-1)!}\n",
    "= A\\sum_{k=0}^\\infty\\frac{A^kt^k}{k!} $$\n",
    "\n",
    "es decir que:\n",
    "\n",
    "$$\\frac{d}{dt}e^{At}=Ae^{At}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Resolviendo la ecuación del controlador\n",
    "\n",
    "La exponencial de una matriz tiene un papel fundamental y por eso tiene su propio nombre. \n",
    "\n",
    "**Matriz de transición de estado**\n",
    "\n",
    "$$e^{A(t-t_0))}=\\Phi(t,t_0)$$\n",
    "\n",
    "para una dinámica de estado $\\dot{x}= A x$ la solución será $x(t)=\\Phi(t,\\tau)x(\\tau)$, tiene las siguientes propiedades: \n",
    "\n",
    "$$\\cases{\\frac{d}{dt}\\Phi(t,t_0)=A\\Phi(t,t_0) \\\\ \\Phi(t,t)=I}$$\n",
    "\n",
    "Cuando tenemos un sistema controlado, tendremos:\n",
    "\n",
    "$$x(t)= \\Phi(t,t_0)x(t_0)+\\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Resolviendo la ecuación del controlador\n",
    "\n",
    "Verifiquemos que la ecuación cumple con la condición inicial:\n",
    "\n",
    "$$x(t_0)= \\Phi(t_0,t_0)x(t_0)+\\int_{t_0}^{t_0}\\Phi(t_0,\\tau)B\\,u(\\tau)d\\tau = I x(t_0) + 0 = x(t_0)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "$$x(t)= \\Phi(t,t_0)x(t_0)+\\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "y cumple para la derivada: \n",
    "\n",
    "$$\\frac{d}{dt}x(t)= A\\Phi(t,t_0)x(t_0)+\\frac{d}{dt}\\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "$$\\frac{d}{dt}\\int_{t_0}^{t}f(t,\\tau)d\\tau = f(t,t) +\\int_{t_0}^{t} \\frac{d}{dt}f(t,\\tau)d\\tau = \\Phi(t,t)B\\,u(t) + \\int_{t_0}^{t} A\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "$$\\frac{d}{dt}x(t)= A\\Phi(t,t_0)x(t_0)+B\\,u(t) + \\int_{t_0}^{t} A\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "$$\\frac{d}{dt}x = Ax + B u$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# En resumen\n",
    "\n",
    "tenemos que\n",
    "\n",
    "$$\\dot{x}=A\\,x+B\\,u \\qquad y \\qquad y=C\\,x$$\n",
    "\n",
    "luego,\n",
    "\n",
    "$$y(t) = C \\Phi(t,t_0)x(t_0) + C \\int_{t_0}^{t}\\Phi(t,\\tau)B\\,u(\\tau)d\\tau$$\n",
    "\n",
    "sabiendo que \n",
    "\n",
    "$$\\Phi(t,\\tau)= e^{A(t-\\tau)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad\n",
    "\n",
    "Lo primero que debemos hacer es entender por que un sistema explota o no.\n",
    "\n",
    "Recordemos que los objetivos en control son: \n",
    "     \n",
    "- Ser estable\n",
    "- Seguir el objetivo\n",
    "- Ser robusto\n",
    "- Entre otros"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en sistemas escalares\n",
    "\n",
    "- Es útil empezar con sistemas escalares para ganar intuición.\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "con $a>0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0,5,num=100)\n",
    "y = numpy.exp(1*t)\n",
    "plt.plot(t,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en sistemas escalares\n",
    "\n",
    "- Es útil empezar con sistemas escalares para ganar intuición.\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "con $a<0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "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 = numpy.linspace(0,5,num=100)\n",
    "y = numpy.exp(-1*t)\n",
    "plt.plot(t,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en sistemas escalares\n",
    "\n",
    "- Es útil empezar con sistemas escalares para ganar intuición.\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "con $a=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "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 = numpy.linspace(0,5,num=100)\n",
    "y = numpy.exp(0*t)\n",
    "plt.plot(t,y);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Tres casos\n",
    "\n",
    "- Asintóticamente estable: $x(t)\\to0, \\forall x(0)$\n",
    "- Inestable: $\\exists x(0) : ||{x(t)}||\\to\\infty$\n",
    "- Críticamente estable: en medio, no explota pero tampoco llega a cero\n",
    "\n",
    "**De**\n",
    "\n",
    "$$\\dot{x}=a\\,x \\quad \\to\\quad x(t)= e^{at}x(0)$$\n",
    "\n",
    "tenemos entonces:\n",
    "\n",
    "$$\\cases{\n",
    "a>0 \\quad: & \\text{inestable} \\\\\n",
    "a<0 \\quad: & \\text{asintóticamente estable}\\\\\n",
    "a=0 \\quad: & \\text{críticamente estable}\n",
    "}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en Matrices\n",
    "\n",
    "$$\\dot{x}=A\\,x \\quad \\to\\quad x(t)= e^{At}x(0)$$\n",
    "\n",
    "No podemos decir que $A>0$, pero podemos usar los valores propios.\n",
    "\n",
    "$$A\\,v = \\lambda\\,v$$\n",
    "\n",
    "donde $\\lambda \\in \\mathscr{C}$ son los valores propios y $v \\in \\mathscr{R}^n$ son los vectores propios. \n",
    "\n",
    "Los valores propios no diran como la matriz $A$ actua en cada dirección. \n",
    "\n",
    "**En MATLAB**\n",
    "\n",
    "``` matlab\n",
    ">> eig(A)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Ejemplo\n",
    "\n",
    "Para la siguiente matriz \n",
    "\n",
    "$$A = \\left[\\array{1&0\\\\0&-1}\\right]$$\n",
    "\n",
    "encontrar los valores y vectores propios ⚠️"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\array{\\lambda_1=1 &,& \\lambda_2=-1 &,& v_1=\\left[\\array{1\\\\0}\\right] &,& v_1=\\left[\\array{0\\\\1}\\right]}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad de Matrices\n",
    "\n",
    "$$\\dot{x}=A\\,x \\quad \\to\\quad x(t)= e^{At}x(0)$$\n",
    "\n",
    "- Asintóticamente estable (si y solo si):\n",
    "  $$Re(\\lambda)<0, \\forall\\lambda \\in \\texttt{eig}(A)$$\n",
    "- Inestable (si):\n",
    "  $$\\exists\\lambda\\in\\texttt{eig}(A) : Re(\\lambda)>0$$\n",
    "- Críticamente estable (solo si):\n",
    "  $$Re(\\lambda)\\le0, \\forall\\lambda \\in \\texttt{eig}(A)$$\n",
    "- Críticamente estable (si):\n",
    "  Un valor propio es cero y el resto tienen parte real negativa **o** Dos valores propios son puramente imaginarios y el resto tienen parte real negativa."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# El cuento de dos pendulos\n",
    "\n",
    "Encontremos las matrices para un péndulo regular y un péndulo invertido. Sin fricción. ⚠️"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$A_{\\text{péndulo regular}} = \\left[\\array{0&1\\\\-1&0}\\right] \\qquad A_{\\text{péndulo invertido}} = \\left[\\array{0&1\\\\1&0}\\right]$$\n",
    "\n",
    "encontremos los valores propios ⚠️"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Para el péndulo regular tenemos:\n",
    "  $$\\lambda_1 = j \\qquad \\lambda_2=-j$$\n",
    "  \n",
    "- Para el péndulo invertido tenemos:\n",
    "  $$\\lambda_1 = -1 \\qquad \\lambda_2=1$$\n",
    "  \n",
    "¿Sistemas estables? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Enjambre de robots\n",
    "\n",
    "Analicemos la estabilidad de un enjambre de robots para resolver el problema de _Rendezvous_\n",
    "\n",
    "- Tenemos una colección de robots que pueden medir su posición relativa a sus vecinos. \n",
    "- Problema: hacer que todos los robots se encuentren en el mismo lugar (no especificada). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Caso de dos robots simples\n",
    "\n",
    "Anteriormente revisamos el caso de dos robots en una linea. \n",
    "\n",
    "![](3-dos-robots.png)\n",
    "\n",
    "Si los robots se dirigen el uno hacia el otro tenemos:\n",
    "\n",
    "$$\\cases{u_1=x_2-x_1\\\\u_2=x_1-x_2}$$\n",
    "\n",
    "La matriz dinámica sera entonces:\n",
    "\n",
    "$$\\dot{x}=\\left[\\array{-1&1\\\\1&-1}\\right]x$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en el caso de dos robots simples\n",
    "\n",
    "$$A = \\left[\\array{-1&1\\\\1&-1}\\right] \\qquad \\lambda_1 = 0 \\qquad \\lambda_2=-2$$\n",
    "\n",
    "En este sistema tenemos un valor propio 0 y todos los demas tienen parte real negativa. Aqui el estado del sistema terminara en algo llamado el ***Espacio nulo (null-space)*** de $A$:\n",
    "\n",
    "$$null(A)=\\{x: Ax = 0\\}$$\n",
    "\n",
    "para el caso particular de esta $A$, el espacio nulo es:\n",
    "\n",
    "$$null(A)=\\{x: x = \\left[\\array{\\alpha\\\\\\alpha}\\right] , \\alpha \\in \\mathscr{R} \\}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Estabilidad en el caso de dos robots simples\n",
    "\n",
    "Si $x_1\\to\\alpha$ y $x_2\\to\\alpha$ entonces tenemos que $(x_1-x_2)\\to0$ por lo que el _Rendezvous_ se logró."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "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": [
    "dosRobots = ss([[-1,1],[1,-1]],[[0],[0]],[[1,0],[0,1]],[[0],[0]])\n",
    "yout, T = step(dosRobots,X0=[-10,10])\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.plot([-12,12],[0,0],color='gray')\n",
    "ax.axis('off')\n",
    "#ax.plot([0,0],[-1,1],color='r')\n",
    "plt.xlim(-12,12);\n",
    "l, = ax.plot([-10],[0],color='b',marker='.',markersize=30)\n",
    "r, = ax.plot([10],[0],color='r',marker='.',markersize=30)\n",
    "\n",
    "def animate(i):\n",
    "    l.set_data([yout[i,0]], [0])\n",
    "    r.set_data([yout[i,1]], [0])\n",
    "\n",
    "ani = FuncAnimation(fig, animate, frames=len(T), interval=1000/40);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
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       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x29dca666ee0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(ani)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Si hay más de dos robots, deberiamos pensar en llevarlos a todos al centroide de sus vecinos (o algo parecido)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# El caso con multiples robots\n",
    "\n",
    "$$\\dot{x}_i = \\sum_{j\\in N_i}(x_j-x_i) \\qquad \\dot{x}=-Lx$$\n",
    "\n",
    "El grafo generado est..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Realimentación del sistema en espacio de estados\n",
    "\n",
    "Sabemos que lo primero que debemos hacer para controlar un sistema es llevarlo a ser asintóticamente estable. Es decir que la parte real de todos sus valores propios sea negativa. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Una particula en una linea recta sin fricción\n",
    "\n",
    "Diseñemos un control proporcional para el sistema propuesto, utilizando el espacio de estados.\n",
    "\n",
    "$$m \\ddot{x} = F$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "La ecuación de estado quedaría asi (con $m=1$). \n",
    "\n",
    "$$\\dot{\\mathbf{x}}=\\left[\\begin{array}{}0&1\\\\0&0\\end{array}\\right]\\mathbf{x}+\\left[\\begin{array}{}0\\\\1\\end{array}\\right]\\mathbf{u}$$\n",
    "\n",
    "La ecuación de salida sería.\n",
    "\n",
    "$$\\mathbf{y}=\\left[\\begin{array}{}1&0\\end{array}\\right]\\mathbf{x}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Control de la particula\n",
    "\n",
    "Si queremos controlar el sistema con un lazo cerrado debemos conectar de alguna forma la salida $\\mathbf{y}$ con la entrada $\\mathbf{u}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "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": [
    "plt.plot([-2,2],[0,0])\n",
    "plt.plot([0,0],[-1,1],color='r')\n",
    "plt.plot([-1],[0],color='k',marker='.',markersize=30)\n",
    "plt.xlim(-2,2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "El objetivo de control es que se mueva al origen. **¿Cómo lo logramos?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\left.\\begin{array}{}u>0 \\text{ si } y<0 \\\\ u<0 \\text{ si } y>0\\end{array}\\right\\}\\qquad \\to \\qquad u=-y$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Modificación de la dinamica del sistema \n",
    "\n",
    "En general tenemos: \n",
    "\n",
    "$$\\mathbf{u}=-K\\mathbf{y} = -KC\\mathbf{x} $$\n",
    "\n",
    "entonces: \n",
    "\n",
    "$$\\dot{\\mathbf{x}}=A\\mathbf{x}+B\\mathbf{u}=A\\mathbf{x}-BKC\\mathbf{x} = \\left(A-BKC\\right)\\mathbf{x}$$\n",
    "\n",
    "tenemos aquí un nuevo sistema, el sistema en lazo cerrado.\n",
    "\n",
    "$$\\dot{\\mathbf{x}} = \\left(A-BKC\\right)\\mathbf{x}=\\hat{A}\\mathbf{x}$$\n",
    "\n",
    "nuestro trabajo ahora es seleccionar $K$ de tal forma que los valores propios de la matriz $\\hat{A}$ den por lo menos un sistema estable. \n",
    "\n",
    "$$Re(\\lambda)<0 \\forall\\lambda \\in \\texttt{eig}(A-BKC)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### La matriz de estado del sistema en lazo cerrado\n",
    "\n",
    "Remplazamos los valores de las matrices y de $K=1$:\n",
    "\n",
    "$$\\hat{A}=\\left(A-BKC\\right)=\\left(\\left[\\begin{array}{}0&1\\\\0&0\\end{array}\\right]-\\left[\\begin{array}{}0\\\\1\\end{array}\\right]1\\left[\\begin{array}{}1&0\\end{array}\\right]\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\-1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ 0, 1],\n",
       "[-1, 0]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mA = sympy.Matrix([[0,1],[0,0]])-sympy.Matrix([0,1])*1*sympy.Matrix([[1,0]])\n",
    "display(mA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Analizando los valores propios del sistema tenemos que: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-I: 1, I: 1}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(mA.eigenvals())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "El sistema es criticamente estable. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Respuesta de la particula al lazo cerrado\n",
    "\n",
    "Con el controlador propuesto, la particula se comporta como se muestra. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0,2*numpy.pi,40)\n",
    "x = numpy.sin(t)\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.plot([-2,2],[0,0])\n",
    "ax.plot([0,0],[-1,1],color='r')\n",
    "plt.xlim(-2,2);\n",
    "l, = ax.plot([0],[0],color='k',marker='.',markersize=30)\n",
    "\n",
    "animate = lambda i: l.set_data([x[i]], [0]);\n",
    "\n",
    "ani = FuncAnimation(fig, animate, frames=len(t), interval=1000/40);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x29dca76f190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(ani)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "**¿Por qué no se queda en el origen si este es el objetivo?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- No tenemos en cuenta la velocidad. \n",
    "- Necesitamos la información del estado (posición,velocidad) del sistema para estabilizarlo."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Estabilizando la particula \n",
    "\n",
    "Para estabilizar la particula necesitamos conocer la información de todos los estados del sistema. Salvo que en nuestro sistema solo tenemos un sensor de posición:\n",
    "\n",
    "$$\\mathbf{y}=\\left[\\begin{array}{}1&0\\end{array}\\right]\\mathbf{x}$$\n",
    "\n",
    "El estado desconocido es la velocidad, el cual puede ser estimado de la posición. Por ahora supongamos que podemos medir ambos estados. \n",
    "\n",
    "$$\\mathbf{y}_{supuesto}=\\left[\\begin{array}{}1&0\\\\0&1\\end{array}\\right]\\mathbf{x}$$\n",
    "\n",
    "con esta nueva matriz $C$, proponemos un controlador $K$:\n",
    "\n",
    "$$K = \\left[\\begin{array}{}k_1&k_2\\end{array}\\right]$$\n",
    "\n",
    "Encontremos la nueva matriz $\\hat{A}$ para el sistema en lazo cerrado."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Nueva matriz de estado del sistema en lazo cerrado\n",
    "\n",
    "Recordemos que aquí estamos suponiendo que podemos medir ambos estados del sistema (posición y velocidad). Remplazamos los valores de las matrices y de $K$:\n",
    "\n",
    "$$\\hat{A}=\\left(A-BKC\\right)=\\left(\\left[\\begin{array}{}0&1\\\\0&0\\end{array}\\right]-\\left[\\begin{array}{}0\\\\1\\end{array}\\right]\\left[\\begin{array}{}k_1&k_2\\end{array}\\right]\\left[\\begin{array}{}1&0\\\\0&1\\end{array}\\right]\\right)$$\n",
    "\n",
    "Luego $\\hat{A}=$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\- k_{1} & - k_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   0,    1],\n",
       "[-k_1, -k_2]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k1,k2 = sympy.symbols('k_1 k_2')\n",
    "mA3 = sympy.Matrix([[0,1],[0,0]])-sympy.Matrix([0,1])*sympy.Matrix([[k1,k2]])*sympy.Matrix([[1,0],[0,1]])\n",
    "display(mA3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Los valores propios o polos del sistema son:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-k_2/2 - sqrt(-4*k_1 + k_2**2)/2: 1, -k_2/2 + sqrt(-4*k_1 + k_2**2)/2: 1}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eigen = mA3.eigenvals()\n",
    "display(eigen)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Escogamos los valores del controlador"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "45a60866e3004a48b8ad32df0c2665ab",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "HBox(children=(FloatSlider(value=1.0, description='k1', max=10.0, orientation='vertical'), FloatSlider(value=2…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Parametros del modelo\n",
    "\n",
    "paramK1 = widgets.FloatSlider(value=1,min=0,max=10,step=0.1,description='k1', orientation=\"vertical\")\n",
    "paramK2 = widgets.FloatSlider(value=2,min=0,max=5,step=0.1,description='k2', orientation=\"vertical\")\n",
    "\n",
    "## Definicion de la simulacion\n",
    "\n",
    "def polos(K1,K2):\n",
    "    P1 = -K2/2+sympy.sqrt(K2**2-4*K1)/2\n",
    "    P2 = -K2/2-sympy.sqrt(K2**2-4*K1)/2\n",
    "    plt.scatter([sympy.re(P1),sympy.re(P2)],[sympy.im(P1),sympy.im(P2)])\n",
    "    plt.grid()\n",
    "    plt.title('Polos de la Particula en lazo cerrado con sensores de posición y velocidad')\n",
    "    plt.xlabel('Real')\n",
    "    plt.ylabel('Imaginario')\n",
    "    plt.ylim(-3,3)\n",
    "    plt.xlim(-6,0)\n",
    "\n",
    "## Presentación de los resultados    \n",
    "    \n",
    "plot_exponencial = widgets.interactive_output(polos,{'K1':paramK1,'K2':paramK2})      \n",
    "widgets.HBox([paramK1,paramK2,plot_exponencial])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "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": [
    "polos(3,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Verifiquemos el comportamiento de la particula\n",
    "\n",
    "- Tomemos los valores para el controlador $k_1=1$ y $k_2=1$\n",
    "- Tomemos los valores para el controlador $k_1=0.1$ y $k_2=1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Verifiquemos el comportamiento de la particula\n",
    "\n",
    "Tomemos los valores para el controlador $k_1=1$ y $k_2=2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.rcParams[\"animation.html\"] = \"html5\"#\"jshtml\"\n",
    "\n",
    "particula = control.ss('0 1;-1 -2','1;0','1 0','0')\n",
    "t,x = control.impulse_response(particula)\n",
    "\n",
    "fig, ax = plt.subplots();\n",
    "\n",
    "ax.plot([-2,2],[0,0])\n",
    "ax.plot([0,0],[-1,1],color='r')\n",
    "plt.xlim(-2,2);\n",
    "l, = ax.plot([0],[0],color='k',marker='.',markersize=30)\n",
    "\n",
    "animate = lambda i: l.set_data([-x[i]], [0]);\n",
    "\n",
    "ani = FuncAnimation(fig, animate, frames=len(t), interval=1000/40);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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    "display(ani)"
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   "source": [
    "# Los valores propios importante\n",
    "\n",
    "- Es claro que algunos valores propios son mejores que otros. \n",
    "  - Algunos causan oscilaciones\n",
    "  - Algunos hacen que el sistema responda lentamente\n",
    "  - etc.\n",
    "\n",
    "- Próximamente veremos como selecionar los valores propios. "
   ]
  }
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