1_Dynamics/4_First_and_second_order_system_dynamics/Standard process inputs.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standard process inputs\n",
    "============"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "import matplotlib.pyplot as plt\n",
    "sympy.init_printing()\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step\n",
    "\n",
    "A step input of magnitude $M$ can be written as\n",
    "$$u_S(t)=\\begin{cases}\n",
    "    0&  t<0,\\\\\n",
    "    M&  t\\geq 0\n",
    "\\end{cases}$$\n",
    "\n",
    "Sympy supplies a unit step function called `Heaviside`, which is typeset as $\\theta(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = sympy.symbols('t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = sympy.Heaviside"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x1178f46a0>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.plot(M*S(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Laplace transform"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sympy can calculate laplace transforms of the step easily:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "M, s = sympy.symbols('M, s')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L(f):\n",
    "    return sympy.laplace_transform(f, t, s, noconds=True)\n",
    "def invL(F):\n",
    "    return sympy.inverse_laplace_transform(F, s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{M}{s}$$"
      ],
      "text/plain": [
       "M\n",
       "─\n",
       "s"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L(M*S(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scaling and translation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can scale and translate the step function in the normal way. Notice that how the time translation is handled."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ipywidgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def translated_step(scale, y_translation, t_translation):\n",
    "    f = scale*S(t - t_translation) + y_translation\n",
    "    print(\"f =\", f, \"  \\u2112(f) =\", L(f))\n",
    "    sympy.plot(f, (t, -10, 10), ylim=(-2, 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "8289be6af90646d18d639f089573ffef",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=1.75, description='scale', max=3.0, min=0.5), FloatSlider(value=0.0, d…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(translated_step, \n",
    "         scale=(0.5, 3.), \n",
    "         y_translation=(-1., 1.), \n",
    "         t_translation=(0., 5.));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rectangular pulse\n",
    "\n",
    "It is now easy to see how we can construct a rectangular pulse, of height $h$ and width $t_w$,\n",
    "\n",
    "$$u_{RP}(t)=\\begin{cases}\n",
    "    0&  t<0,\\\\\n",
    "    h& 0 \\leq t < t_w\\\\\n",
    "    0&  t \\geq t_w\n",
    "\\end{cases}$$\n",
    "\n",
    "by using shifted versions of the step, so that"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "h, t_w = sympy.symbols('h, t_w')\n",
    "u_RP = h*(S(t - 0) - S(t - t_w))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sympy.plot(u_RP.subs({h: 1, t_w: 2}), (t, -1, 3));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{h}{s} - \\frac{h}{s} e^{- s t_{w}}$$"
      ],
      "text/plain": [
       "       -s⋅t_w\n",
       "h   h⋅ℯ      \n",
       "─ - ─────────\n",
       "s       s    "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L(u_RP)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Arbitrary piecewise constant functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can constuct any piecewise constant function by adding together step functions shifted in time. As an example, we can take the function represented below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = [-1, 0, 0, 1, 1, 2, 2, 3]\n",
    "y = [0, 0, 1, 1, 2, 2, 0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(2,0.1,'$D_3=2$')"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y)\n",
    "plt.text(0, 0.5, r'$\\Delta_1=1$')\n",
    "plt.text(1, 1.5, r'$\\Delta_2=1$')\n",
    "plt.text(2, 1, r'$\\Delta_3=-2$')\n",
    "\n",
    "plt.text(0, 0.1, r'$D_1=0$')\n",
    "plt.text(1, 0.1, r'$D_2=1$')\n",
    "plt.text(2, 0.1, r'$D_3=2$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In general piecewise constant functions like the one above can be written as \n",
    "\n",
    "$$f_c(t) = \\sum_{i=1}^{N_d} \\Delta_i S(t - D_i)$$\n",
    "\n",
    "where $S$ is the unit step. We calculate $\\Delta_i$ as the difference between the values at the discontinuities, positive if the function is rising and negative if it is falling. $D_i$ is the time at which the value changes and $N_d$ is the number of discontinuities."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can apply this directly for the example function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPQAAAAVCAYAAABxN8PZAAAABHNCSVQICAgIfAhkiAAABVRJREFUeJztm1uIVVUYx3+a5QwzUGSENVAvWQQWliFKF05p+SBRYUVEOAP1EJkREkSCNUQX7cnwqSCpMMgyuksXoosWZU0JXaZA4ZSlUUOlJZpZ08P3nWa5z1prr73P7L322P7BYc2stb61v/3/r73Ovh2oqak54ukG7ga+Bg4AO4EHgKMd/Z8EfgJ6AsefDYwCN3aWZjSy6gPZNaoqVfKu6HlaVTJ5cBLwBXAQ2ACsBr7UAR6x9D8P+AdY7hhvucZen6h/HtgN9IYkVQDTgJs0j+3AfmAPsAURarIjLqs+4NfIpU8Z5NUgtndQ3jwtg6uBtcBmYK/msT4lJsiDY4BPgD+A8436XuBb4G9geiLmDeA3ZLW0sV4TPCNRP0frV6QkXhQ36/Z3AU8BDwLrkH0ZBTYCkxIxefQBv0YufcogjwYQ37sy52kZbNNt/w4ME3ZAB3mwQjsttbSt0bZrjLrTkVXvUc+Yw5qobbUfRgw4ypdUQVwCXE57XtOB75B9XZxoy6oPpGvk06do8mjQohPvBnTsRo5YKH+eFs3FwAxk8WwQdkBDigfdwK/Iam27BlmpG1pm1K3SuvmW/qu1zfa5Qfvco/8vDEi+TFoTZq1Rl0cfcGsUok9MbBqYdOLdAPkP6BjztEwahB/QbR5MMRqvAo4DHgP+sgR3aXnQqFuAnN58aOk/BDwB9AMfAG8abe9q+b6WlwKvB+xAWbT2/5BRl0cfcGsUok9MbBqYxPIuxjytKm0emAf0Ii37gEFL8AItd2rZA8xCvvb3Wfo/gwjfjwhmO935WMuLUlMvjynAEv37NaM+qz7g1yhEn1i4NDCJ5V2MeVpV2jwwD+gLtLwuZZCvtOxDzt13e/qeq+WnjvY9yOOGU1K2WSargJnAJg7/5smqD6RrlKZPLFwamMTyLsY8rSpOD3qQc/HPHYG9yCmM+e0zT2M2eDa4VeOmevr8gPu0zqSJ+1rH9gm5Bklym8YOA8cb9Xn0gXSNQvQxaRJPAxsh3mXN+XHPWDHnqUmT4nxoZIw5zIPWN3SflrscQQuRGxCvGnX7texq7/7f2GchK+WfnoS6jbF87EBWo1Bc++JiKfAwku984BejLY8+4NcoVB+TmBrYCPFuDXJKazILuAI5xW0m2rZ5xoo5T02K9iELVg9mIqvCS46gV7R9jlF3stZtccScre3rPMlMRh4n7PCmXDy3M7byn2hpz6MP+DUK0adM0jRI0ol3A+S7yx1rnpZJg/BvaKcHJ+ggWy1BczVoU6J+EvIa3c+OjS3B/hjH5Ezt85wv64K5U3P4DNHBRh59wK9RiD5lEaJBkk68GyDfAR1rnpZJg/ADus2D1kP0EeSaaTayYrU4FXgaufi+JTHYKPAeIvJplo1N03KvJ6G5Wr6dnnshrERuAA0hp5gjjn559AG/RiH6lEGoBklieBdrnlaVNg/Mu9z3Ia//vYWsDj3AtYggi2i/1gFZGRYj1y7bE21DWt6PnCrtQ961fdbocxnyfPDFrHsyDvQD9+r2NyM3g5I0GbtJk0cfcGsUok/RZNXAJJZ3MeZp0VypHxh7ZXUeY7qPAHdY4lI96Ed25gDwPfJMrs/VGXmn9kfgI0f7rcA3Ot4oIlqLY5GL+Rc84xfJIOl3J99JxGTVB/wa+fQpg0GyawCdezdA/jfFoNx5WgaD+D1oWmIKO37u0o2ekzFumcZdON4JVZC8GlWViehd7UEgXcgL4i9niOlGbuVvHO9kKkoejarKRPXuf+HBePzK6RDyhs1U5Cdttvdrk8xAnvk9hPyk7Ugnj0ZVZaJ6V3tQU1NTU1NTE4l/Abj2j7NjMMiXAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\theta\\left(t\\right) - 2 \\theta\\left(t - 2\\right) + \\theta\\left(t - 1\\right)$$"
      ],
      "text/plain": [
       "Heaviside(t) - 2⋅Heaviside(t - 2) + Heaviside(t - 1)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = 1*S(t) + 1*S(t-1) - 2*S(t-2)\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or a little more generally using some code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "Delta = [1, 1, -2]\n",
    "D = [0, 1, 2]\n",
    "Nd = len(Delta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sum(Delta[i]*S(t - D[i]) for i in range(Nd))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\theta\\left(t\\right) - 2 \\theta\\left(t - 2\\right) + \\theta\\left(t - 1\\right)$$"
      ],
      "text/plain": [
       "Heaviside(t) - 2⋅Heaviside(t - 2) + Heaviside(t - 1)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's verify it works properly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sympy.plot(f, (t, -1, 3));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{s} + \\frac{e^{- s}}{s} - \\frac{2}{s} e^{- 2 s}$$"
      ],
      "text/plain": [
       "     -s      -2⋅s\n",
       "1   ℯ     2⋅ℯ    \n",
       "─ + ─── - ───────\n",
       "s    s       s   "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.expand(L(f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ramp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A ramp with slope $a$ can be written as\n",
    "\n",
    "$$u_R(t)=\\begin{cases}\n",
    "    0&  t<0,\\\\\n",
    "    a&  t\\geq 0\n",
    "\\end{cases}$$\n",
    "\n",
    "We can construct a unit ($a=1$) ramp by simply multiplying the unit step by $t$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def R(t):\n",
    "    return t*S(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x11a539198>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.plot(R(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAYAAAA/F0BXAAAABHNCSVQICAgIfAhkiAAAARRJREFUSInt1L8rRWEcx/HXFd3CRsrgR7EYxGCQJOU/kDLJaMBiuIz+Af+DzWKzSjLJhIGslLoGRSkJ1/Cc4eT+8NyDpO67vj3nfJ9vn++n833Owy8yhT3cooS5WsVNVfJtOMPKT7nK7KQuGiLlNFfJt2Mw9d6PUdzjOlZ8Whjt59iu2+b/I5espT91EUsPDnEh/NmzWUS6MZI8d+EGrd91do7edKLeYz+GlsRNJjqE7zJRqyiHAq7wjDvsJnt5HGHhq04FXGIGfRjHUiK+g80YuwfYqpCfxDtOUzFcTWQVb9jHMjpjOldiAGtJtwcMZRUiXFaPmI8thnUUcYJXLOJFOO7RInlsCFN5wrEwpWKU7wZlfACXizMu5mZnNAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{s^{2}}$$"
      ],
      "text/plain": [
       "1 \n",
       "──\n",
       " 2\n",
       "s "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L(R(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuous piecewise linear functions\n",
    "\n",
    "We can build any continuous piecewise linear function by using shifted and scaled ramps.\n",
    "\n",
    "$$f(t) = \\sum_{i=1}^{N_s} \\Delta m_i R(t - D_{s,i})$$\n",
    "\n",
    "This time $\\Delta m_i$ represents changes in slopes ($m=\\frac{\\Delta y}{\\Delta x}$). $D_{s,i}$ are the times at which the slopes change and $N_s$ is the number of slope changes.\n",
    "\n",
    "For instance, we can construct a triangular pulse by adding three ramps together."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = t*S(t - 0)\n",
    "r2 = -2*(t - t_w/2)*S(t - t_w/2)\n",
    "r3 = (t - t_w)*S(t - t_w)\n",
    "u_TP = 2/t_w*(r1 + r2 + r3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{2}{t_{w}} \\left(t \\theta\\left(t\\right) + \\left(- 2 t + t_{w}\\right) \\theta\\left(t - \\frac{t_{w}}{2}\\right) + \\left(t - t_{w}\\right) \\theta\\left(t - t_{w}\\right)\\right)$$"
      ],
      "text/plain": [
       "  ⎛                                       ⎛    t_w⎞                           \n",
       "2⋅⎜t⋅Heaviside(t) + (-2⋅t + t_w)⋅Heaviside⎜t - ───⎟ + (t - t_w)⋅Heaviside(t - \n",
       "  ⎝                                       ⎝     2 ⎠                           \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                        t_w                                   \n",
       "\n",
       "    ⎞\n",
       "t_w)⎟\n",
       "    ⎠\n",
       "─────\n",
       "     "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_TP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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1dXX41a9+JVNa/7HaW6GRgOKC8CjokkI9JAk8i6awp6qCPnnyJIxGI7KzsxEZGYlt27Zh7969E4556aWXsGPHDsyfPx8AkJaWJkdUvxFCoNLuxLrsFKQmRMkdxy/SE6OxZnEyrHYnhBByxyEKGFUVtMPhQFZW1vjjzMxMOByOCcdcuHABFy5cwPr167Fu3TpUV1ff8vn27NkDs9kMs9mMzs7OgOWei3pnHy51DYbMznXTtcWkx8WOAZxv55iDwpeqCnqys63P3u7J4/GgsbERx44dQ3l5Ob797W+jp6dn0ufbvn07bDYbbDYbUlNTA5J5rqx2J7QaCcUFGXJH8aviAj00EvfmoPCmqoLOzMxEc3Pz+OOWlhYYDIabjvniF7+IiIgILFmyBDk5OWhsbAx2VL8QQsBqb8Xn7kxBclyk3HH8KjUhCuuyUzjmoLCmqoJes2YNGhsb0dTUBJfLhYqKCpSVlU045ktf+hKOHj0KAOjq6sKFCxeQnZ0tR9w5q3H0orl7CFvCbLwxxmIyoKlrEHWtfXJHIQoIVRW0TqfD7t27UVRUhNzcXGzduhX5+fnYuXMn9u3bBwAoKipCSkoK8vLysHHjRvzyl79ESkqKzMlnx2p3IkIroSg/vMYbY4oLMqDVSFzNQWFLmuOPh/zZ8gaz2QybzSZ3jHFCCHz+fx7FsvR4/P5f1sodJ2Ae+38n0dQ1gLef2njT+wlECjXtP6iqOoNWkzPNPXD0DIXd6o3PshTq0dw9BHtLr9xRiPyOBR2mrGediNRq8EB+utxRAurBvHSsXZyMg3Vtckch8jsWdBjy+QSqapzYsCwVidERcscJqKS4SMRH6/AfZxzw+Thxo/DCgg5Dpz/uxoL4SHx55UK5owSFxaRHa+8wzjRPvl6dKFSxoMOQ9awTjR0D2JCjzItn/G1zXjoitRpuQUphhwUdZrw+garaNmxanob4KJ3ccYIiMToC9+akoqrGyTEHhRUWdJg50XQVnf0jIXvfwdmymPRo7xuB7co1uaMQ+Q0LOsxU2p2IidBi0/LQ3oVvpjbnpiNKxzEHhRcWdBjxeH2orm3D/blpiI1Ux3hjTFyUDpuWp6Gqpg1ejjkoTLCgw8jxS1dxddAFi8rGG2MsJgO6BkZw4tJVuaMQ+QULOoxYzzoRF6nFfTnqGm+M2bQ8DbGRWlhruDcHhQcWdJhwe32ormvDA3npiI7Qyh1HFjGRWtyfm47q2jZ4vD654xDNGQs6TLx7sQu9Q+6w33tjKqWFenQPuvD+RxxzUOhjQYcJ61knEqJ1+MKyBXJHkdV9OamIj9JxNQeFBRZ0GBjxePFmfRsezMtAlE6d440x0RFaPJCXjoN17XB5OOag0MaCDgPvXOhC/7AHlhXqXL3xWWV3GWBMi8d7F5V5I1+i6WJBh4H3PupCclwkPm9U93hjzOfuTMGF9n7s551WKMSxoEPcsNuL1041o6QgAxFa/ucEgCidFkX5GThU144Rj1fuOESzxr/RIe7Y+U4MurwoKgjP+w7OlsWkR/+IB29f6JI7CtGssaBDnNXeiuS4SNyTHZo3tg2U9cYFSIqN4GoOCmks6BB23eXB4XMdKC7IgI7jjQkitBoU52fgrfp2DLs55qDQxL/VIexIQweG3F7V7r0xFYvJgEGXF8fOd8gdhWhWWNAhrNLuRGpCFO5ewvHGZNZlJyMlLpKrOShksaBD1MCIB0caOlBSkAGtRpI7jiLptBoUF2TgyLkOXHd55I5DNGMs6BB1+Fw7Rjw+lKp8742pWEwGDLm9ONLAMQeFHhZ0iLLanchIjIb5jvlyR1G0tUuSkZoQBetZjjko9LCgQ1DfsBv/ON+JkkI9NBxv3JZWI6GkIANHz3dgYIRjDgotLOgQdKiuHS6vj3tvTJNlhQEjHh8On2uXOwrRjLCgQ5DV3oqFSTFYmZUkd5SQsHrRfKw3psB2mXf8ptDCgg4xvdfdeKexC6UmPSSJ443p0Ggk5KQn4t9PNaN3yC13HKJpU11BV1dXIycnB0ajEbt27brlcW+88QYkSYLNZgtiuqkdrGuDxyd4ccoMWVbo4fL68FY9xxwUOlRV0F6vFzt27MCBAwdQX1+P8vJy1NfX33Rcf38/fvOb3+Duu++WIeXt7be3YlFyLAoXzpM7SkhZmZWEhUkx3JuDQoqqCvrkyZMwGo3Izs5GZGQktm3bhr1799503I9//GM8/fTTiI6OliHlrY3da4/jjZmTJAkWkx7vNHah57pL7jhE06KqgnY4HMjKyhp/nJmZCYfDMeGYM2fOoLm5GRaLJdjxpnT4XDvMd8yHpZDjjdkoNenh8QkcrGuTOwrRtKiqoIUQN33s02eiPp8PTz75JJ5//vlpPd+ePXtgNpthNpvR2Rn42yv9/YwDnf0jyDMkBvy1wlHhwnlYlBwLK/fmoBChqoLOzMxEc3Pz+OOWlhYYDJ9cKt3f34/a2lrcd999WLx4MT744AOUlZXd8o3C7du3w2azwWazITU1NaDZO/tH8MEljjfmYmzM8f5HV3F1YETuOERTUlVBr1mzBo2NjWhqaoLL5UJFRQXKysrGPz9v3jx0dXXh8uXLuHz5MtatW4d9+/bBbDbLmHpUda0TPjG6twTNXqlJD69PoJpjDgoBqiponU6H3bt3o6ioCLm5udi6dSvy8/Oxc+dO7Nu3T+54t7Xf7sTStHjkZCTIHSWk5ekTkb0gDpUcc1AIkCaby87AnL44nJjN5oCtmW7vG8a65w7jifuX4geblwXkNdTkhTfPY/fRizjxo81ITYiSOw6pz7RnlKo6gw5VlXYnBMcbflNqMsAngAO1PIsmZWNBh4DKGieWZyTAmBYvd5SwkJORgKVp8VzNQYrHgla41p4hnL5yDVtW8OzZnywmA05d7kZ737DcUYhuiQWtcGNvZpXy4hS/KjXpIQT4ZiEpGgta4az2VhQsTMTiBXFyRwkrxrR4LM9I4N4cpGgsaAVr7r6Osy29fHMwQB5ZuwjN3UNw9AzJHYVoUixoBbNyvBFQ9y5LRefACKo45iCFYkErmNXeihVZSchKjpU7SlhavCAOBQsTOeYgxWJBK1RT1yDqWvuwhRvzB5TFZMDZll40d1+XOwrRTVjQCvVmfRvWLk7GQwUs6EAaGx9xTTQpEQtaof7+oQNeIbBwfozcUcJaVnIsVmQlccxBisSCVqCLHf1oaOvnfQeDZItJj7rWPjR1DcodhWgCFrQCWe1OSBJQwtUbQTH2fa7kWTQpDAtaYYQQsNqdWLs4GemJyronYrgyJMXAfMd8zqFJcVjQCnO+vR8XOwY43giyUpMeDW39uNjRL3cUonEsaIWptDuhkYBirt4IqpJCPSQJ2H+WZ9GkHCxoBRkbb9xzZwo3kg+y9MRorF2cjMoa56Q3FyaSAwtaQcZWEnDvDXlYTHpc7BjA+XaOOUgZWNAKUlnjhFYjoSg/Q+4oqvRQoR6r75iPY+c75Y5CBIAFrRij441WrDcuQHJcpNxxVGlBfBSiIzSoOPkxxxykCCxohahx9KK5e4irN2RmMRlw+ep11LX2yR2FiAWtFFa7ExFaCUV5HG/IqSg/A1qNxDXRpAgsaAUQQuBiRz++vCoT82Ij5I6jaslxkVhvXACrvZVjDpIdC1oBPvy4B0caOnH3kmS5oxBGV3O0XBuCvaVX7iikcixoBai0OxGp0+CBvHS5oxCAorwMRGgl7nBHsmNBy8znE6iqceK+ZalIiOZ4QwnmxUbgC0tTUWl3wufjmIPkw4KWme3KNbT1DaOUqzcUxWLSo7V3GGear8kdhVSMBS0zq70V0REabM7leENJHshLR6ROw9UcJCsWtIy8PoGqmjZsWp6GuCid3HHoUxKiI3DfslRU1XDMQfJhQcvoRNNVdA2MoLSQe28oUalJj/a+EZy63C13FFIpFrSMrHYnYiO12LQ8Te4oNInNuemIjuCYg+SjuoKurq5GTk4OjEYjdu3addPnX3jhBeTl5cFkMuH+++/HlStXApLD4/WhurYN9+emIyZSG5DXoLmJi9Jh0/I0HKh1wssxB8lAVQXt9XqxY8cOHDhwAPX19SgvL0d9ff2EY1auXAmbzQa73Y6HH34YTz/9dECyHL90Fd2DLpTyvoOKtmWFAdmpcTjZdFXuKKRCqirokydPwmg0Ijs7G5GRkdi2bRv27t074ZiNGzciNjYWALBu3Tq0tLQEJIv1rBPxUTrcl5MakOcn/7hvWRpqHX3YxzutkAxUVdAOhwNZWVnjjzMzM+FwOG55/Msvv4yHHnrolp/fs2cPzGYzzGYzOjunv4ewy+NDdV0bHshLR3QExxtKFhOpxf256aiudcLt9ckdh1RGVQU92eY3kiRNeuyrr74Km82Gp5566pbPt337dthsNthsNqSmTv9M+L2LXegdcnO8ESIsJj2uXXfj+Eccc1BwqaqgMzMz0dzcPP64paUFBsPNS9zeeust/Nu//Rv27duHqCj/3xvQanciIVqHLyxb4PfnJv+7d1kq4qN03JuDgk5VBb1mzRo0NjaiqakJLpcLFRUVKCsrm3DMmTNn8J3vfAf79u1DWpr/l7+NuL14p7ETRfkZiNJxvBEKoiO0eCAvHdW1bXB5OOag4FFVQet0OuzevRtFRUXIzc3F1q1bkZ+fj507d2Lfvn0AgKeeegoDAwP4yle+grvuuuumAp+rtxu70DvkQtkKXpwSSiwmPfqGPXjvYpfcUUhFpDluSs7FoTeYzWbYbLYpj/tBxRkcu9CJU/9jMyK0qvr/Y0hzeXxY/cwhPJCXjhe23iV3HAptk7/xNQk2RBANu704VN+O4vwMlnOIidRpUJSfgUN17Rh2e+WOQyrBlgiiY+c7MOjywmLieCMUWUx69I948PaF6S+pJJoLFnQQWe1OpMRFYl02b20VitYbFyApNgKVNbxohYKDBR0k110eHD7XgeKCDOg43ghJEVoNivMz8FY9xxwUHGyKIDnS0IEhN8cboc5iMmDQ5cXRhg65o5AKsKCDxHrWidSEKKzlnbtD2rrsZKTERcLKMQcFAQs6CAZGPDh6vgMlBRnQaqa9woYUSKfVoMSUgY6+YQyOuOWOQ2GOBR0Eh8+1Y8Tjg4UXp4SFkgI9Tl2+hiMNXM1BgcWCDoL9Z53ISIzG6kXz5Y5CfrB2SQpSE6JQyTutUICxoAOsb9iNty90otSkh4bjjbCg1UgoLdTj6PkODIx45I5DYYwFHWCH6trh8vpQauLWouGk1KTHiMeHt+rb5Y5CYYwFHWBWeysWJsVgZVaS3FHIj1Yvmo+MxGjeUJYCigUdQL3X3XinsQsWk/6WNwag0KTRSCg16fH2hU70DnE1BwUGCzqADta1weMTHG+EKYtJD5fXh0Mcc1CAsKADaL+9FYuSY1G4cJ7cUSgA7spKwsKkGFTyTisUICzoAOkedGFg2INH1mZxvBGmJEmCxaTHO41d6LnukjsOhSEWdIBU17bhTHMP7l3m/9tmkXJYTAZ4fAIH69rkjkJhiAUdIFZ7K7IXxCFXnyB3FAqggoWJuCMllqs5KCBY0AHQ2T+CDy5d5eoNFZCk0YtW3v/oKq4OjMgdh8IMCzoAqmud8Alw7w2VsJgM8PoEqjnmID9jQQfAfrsTS9PisSyd4w01yNUnIDs1jntzkN+xoP2svW8Ypy53c2N+FRldzWHAB5euorOfYw7yHxa0n1XanRACsKzgxSlqYjFlYGVWEo408KIV8h8WtJ9Z7a3I1SfiztR4uaNQEC1LT0T/iAd//dAhdxQKIyxoP3L0DOHDj3tg4aXdqmQxGXDqcjfa+4bljkJhggXtR1U33iRiQatTqUkPIcA3C8lvWNB+ZLW3onDhPNyREid3FJLBnanxyNUnwsq9OchPWNB+4vL4cLallzvXqZzFpMeHH/fA0TMkdxQKAyxoPxnbE7i0kAWtZmPjrSqOOcgPWNB+0jPkxl1ZSchKjpU7CsnojpQ4FC6cxzEH+YXqCrq6uho5OTkwGo3YtWvXTZ8fGRnBV7/6VRiNRtx99924fPnylM/Z1DWIYbeXbw4SgNGz6LMtvWjuvi53FApxqipor9eLHTt24MCBA6ivr0d5eTnq6+snHPPyyy9j/vz5uHjxIp588kkLszX6AAAFnklEQVT88Ic/nPJ5jza0QyNJKOF4g4Dx9yF4Q1maK1UV9MmTJ2E0GpGdnY3IyEhs27YNe/funXDM3r178Y1vfAMA8PDDD+Pw4cMQQtz2eV+ztSAmQgtDUkzAslPoyJwfiw1LF+CND5vljkIhTpqqfG7nq//n+Oy/WAadnZ3ovtaNnGU5AID29nb09fdhqXHp+DGnbKdgKjQhKioKAHDixAmsWrUKERERkz6nx+vFR52D6Kt4CpfO2QP/m6CQUH7yCv562gFJAjTccpY+5d+/c8+0/0DMqaCTjSsF3KFz1ZTH44bH60V0VDQAwO1xw+f1IurGYwAYvD6ImJgYaKTRHy4GBwcQGxs36b7ObrcbbrcLAoDobYPJZArK70PJOjs7kZqaKncMRWjv7MIg+FMVMPp35VYnOWpz7eOGOiFEwbQOFkLM+tfq1atFKHn//ffFgw8+OP742WefFc8+++yEYx588EHx/vvvCyGEcLvdIiUlRfh8vimfO9S+F4HC78Mn+L34BL8XnwBgE9PsWFXNoNesWYPGxkY0NTXB5XKhoqICZWVlE44pKyvDH/7wBwDAG2+8gU2bNvGuKEQkC53cAYJJp9Nh9+7dKCoqgtfrxeOPP478/Hzs3LkTZrMZZWVl+Na3voVHH30URqMRycnJqKiokDs2EanUnAp6+/bt/soRNCUlJSgpKZnwsZ///Ofj/x4dHY3XX399xs8bit+LQOD34RP8XnyC34sJ9kz3wDm9SQggpFZxEBEpwLRnpqqaQRMRhZI5FfTrr7+O/Px8aDQa2Gw2f2UKKVNdOq4Wjz/+ONLS0lBQML3VQ+GsubkZGzduRG5uLvLz8/HrX/9a7kiyGR4extq1a7FixQrk5+fjJz/5idyRZCdJklaSpDOSJFmnPHi6yz0m+1VfXy8aGhrEvffeK06dOhWsVSqK4fF4RHZ2tvjoo4/EyMiIMJlMoq6uTu5YsvjHP/4hTp8+LfLz8+WOIrvW1lZx+vRpIYQQfX19YunSpar9c+Hz+UR/f78QQgiXyyXWrl0rjh8/LnMqeQH4LwD+AsAqArnMLjc3Fzk5OXN5ipA2nUvH1WLDhg1ITk6WO4Yi6PV6rFq1CgCQkJCA3NxcOBzqvFehJEmIjx+9P+fohV1uVS9bbWlpAYBSAP93OsfP9U3C0SeRpGMA/qsQQlVzDkmSHgZQLIT49o3HjwK4Wwjxr/Imk4ckSYsxelbAOccNN74nbwMoEEL0yZtGHpIkaQGcBmAE8FshxNQ7kIUpSZLeAPAcgASMdqbldsdPeQYtSdJbkiTVTvLri37KHMomOxXgyhYCAEiSFA/grwB+oNZyBgAhhFcIcReATABrJUlS5f/AJUmyAOgQQpye7tdMuQ5aCLF5TqnCWwuArE89zgTAndoJkiRFYLSc/yyE+JvceZRACNFz46ftYgC1MseRw3oAZZIklQCIBpAoSdKrQoiv3+oLuMxubk4BWCpJ0hJJkiIBbAOwT+ZMJDNpdMj6MoBzQogX5M4jJ0mSUiVJSrrx7zEANgNokDeVPIQQ/10IkSmEWIzRrjhyu3IG5ljQkiT9J0mSWgDcA6BSkqSDc3m+UCOE8AD4VwAHAZwD8JoQok7eVPKQJKkcwHEAOZIktUiS9C25M8loPYBHAWySJOmfN36VTPVFYUoP4KgkSXaMntAcEkJMvbyMAPjpTUIiIvI/jjiIiBSKBU1EpFAsaCIihWJBExEpFAuaiEihWNBEREEiSVKSJEnfn+7xLGgiouBJAsCCJiJSoF0A7rxx8dIvpzqYF6oQEQXJTHd85Bk0EZFCsaCJiBSKBU1EFDz9GN2sf1pY0EREQSKEuArgvRs3PeGbhEREoYpn0ERECsWCJiJSKBY0EZFCsaCJiBSKBU1EpFAsaCIihWJBExEpFAuaiEih/j+tdLyAPCnfYwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x11a6de550>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.plot(u_TP.subs({t_w: 2}), (t, -1, 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{s^{2}} - \\frac{2}{s^{2}} e^{- s} + \\frac{1}{s^{2}} e^{- 2 s}$$"
      ],
      "text/plain": [
       "        -s    -2⋅s\n",
       "1    2⋅ℯ     ℯ    \n",
       "── - ───── + ─────\n",
       " 2      2       2 \n",
       "s      s       s  "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L(u_TP.subs({t_w: 2})).expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that there are three ramps here (one may have expected only two). It becomes more clear when we think about the derivative of this function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x11c5a5710>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.plot(u_TP.diff(t).subs({t_w: 2}), (t, -1, 4), ylim=(-1.1, 1.1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now see that the derivative of a piecewise linear continuous function is a piecewise constant function. We can apply our rule for piecewise constant functions and integrate the steps to ramps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "derivative = 1*S(t-0) - 2*S(t-1) + 1*S(t-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "final = 1*R(t-0) - 2*R(t-1) + 1*R(t-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arbitrary piecewise linear functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can construct any piecewise linear function by adding together ramp functions and steps shifted in time. The general rule now becomes\n",
    "\n",
    "$$f(t) = \\underbrace{\\sum_{i=1}^{N_s} \\Delta m_i R(t - D_{s,i})}_\\text{slope changes} +  \\underbrace{\\sum_{i=1}^{N_d} \\Delta_i S(t - D_i)}_\\text{discontinuities}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = [-1, 0, 2, 2, 3, 4, 5]\n",
    "y = [0, 0, 1, 2, 2, 0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11ac04d30>]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that there are 4 slope changes (at t=0, t=2, t=3 and t=4) and 1 discontinuity. Applying our formula yields:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = 0.5*R(t) - 0.5*R(t-2) - 2*R(t-3) + 2*R(t - 4) + S(t - 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sympy.plot(g, (t, -1, 5));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1.0}{s} e^{- 2 s} + \\frac{0.5}{s^{2}} - \\frac{0.5}{s^{2}} e^{- 2 s} - \\frac{2.0}{s^{2}} e^{- 3 s} + \\frac{2.0}{s^{2}} e^{- 4 s}$$"
      ],
      "text/plain": [
       "     -2⋅s              -2⋅s        -3⋅s        -4⋅s\n",
       "1.0⋅ℯ       0.5   0.5⋅ℯ       2.0⋅ℯ       2.0⋅ℯ    \n",
       "───────── + ─── - ───────── - ───────── + ─────────\n",
       "    s         2        2           2           2   \n",
       "             s        s           s           s    "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.expand(L(g))"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": 2
}