tutorials/tutorial-7-dmdc.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PyDMD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tutorial 7: Dynamic mode decomposition with control"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this tutorial we will show how to extend the dynamic mode decomposition to incorporate the effect of control (this technique has been introduced in the paper [Dynamic mode decomposition with control](https://arxiv.org/abs/1409.6358))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First of all we import the `DMDc` class from the pydmd package, we set matplotlib for the notebook and we import numpy and scipy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy\n",
    "\n",
    "from pydmd import DMDc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we create our dataset: since we want to add the control, the evolution of the complex system can be formally summarized as:\n",
    "$$\n",
    "\\mathbf{x}_{k+1} = \\mathbf{A}\\mathbf{x}_k + \\mathbf{B}\\mathbf{u}_k,\n",
    "$$where the operators $\\mathbf{A}$ and $\\mathbf{B}$ are the ones we will approximate using DMD. So, for a demostrative purpose, we create the original snapshots by using two random operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_system(n, m):\n",
    "    A = scipy.linalg.helmert(n, True)\n",
    "    B = np.random.rand(n, n)-.5\n",
    "    x0 = np.array([0.25]*n)\n",
    "    u = np.random.rand(n, m-1)-.5\n",
    "    snapshots = [x0]\n",
    "    for i in range(m-1):\n",
    "        snapshots.append(A.dot(snapshots[i])+B.dot(u[:, i]))\n",
    "    snapshots = np.array(snapshots).T\n",
    "    return {'snapshots': snapshots, 'u': u, 'B': B, 'A': A}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We got 25 snapshots of the evolving system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(25, 10)\n"
     ]
    }
   ],
   "source": [
    "s = create_system(25, 10)\n",
    "print(s['snapshots'].shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can compute as usually the DMD algorithm on the data: the `fit` method in this version take as arguments the snapshots and the control input (the $\\mathbf{B}$ operator can be also passed). In this case, we do not perform any truncation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<pydmd.dmdc.DMDc at 0x7f970d31e358>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dmdc = DMDc(svd_rank=-1)\n",
    "dmdc.fit(s['snapshots'], s['u'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us visualize the original system and the reconstructed one: also because without truncation, the plots are the same!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1152x432 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(16,6))\n",
    "\n",
    "plt.subplot(121)\n",
    "plt.title('Original system')\n",
    "plt.pcolor(s['snapshots'].real)\n",
    "plt.colorbar()\n",
    "\n",
    "plt.subplot(122)\n",
    "plt.title('Reconstructed system')\n",
    "plt.pcolor(dmdc.reconstructed_data().real)\n",
    "plt.colorbar()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Well, we built the approximation for $\\mathbf{A}$ and for $\\mathbf{B}$; we can now test the system with a different control input: differently by the other versions, we can pass as argument of the `reconstructed_data` method the control input we want to test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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NHQFT6GwxBAA0qGXDpBRDAEBaAzjPMDWKIQAgPYohAKDzWlYMy1klAQBAJnSGAICkLOYMAQBo3TApxRAAkFYLV5NOO2do+1rb221vnvDYCbbvtv1k/8/lzcYEAKA5M1lAc52kDW957LOS7omI0yTd078PAEBPwxf3TW3aYhgR90na+ZaHL5J0ff/r6yV9InEuAECbtawYznbO8OSI2CZJEbHN9sqpnmh7o6SNkjS8hNFUAOiCts0ZNr6AJiJGJI1I0qKVa4t5ew4uyZ2gave7y9o4fHhP7gRHPHvxstwRKpY+nTtB1Y6P78sdYdyar8zPHaHix//7v+aOUHHvH52fO0LFD5o8eDG/7Wdmtifdv2R7lST1/9yeLhIAAIM122J4u6TL+19fLum2NHEAAK1Xd76wxDlD2zdKulDSCttbJV0t6YuSbrJ9haTnJF3SZEgAQLvMuTnDiLh0ir/6SOIsAIC5Yq4VQwAA3q62dYZctQIA0Hl0hgCA9FrWGVIMAQBpZVoRWgfFEACQlPu3NmHOEADQeXSGAID0GCYFAHRd206t6GwxnLc/d4KqsbL2N9ayJ8dyRxi3/d+VNftw/LcP5Y5QcczHyvlmXvh0OZuGS9KioTdzR6hY9vje3BEGh2IIAOi8lhVDFtAAADqPzhAAkFYwZwgAQOuGSSmGAIDk6AwBAGhZMWQBDQCg8+gMAQDJMUwKAOg2rloBAIBaVwyZMwQAdB6dIQAgKYs5w9YYLWxj7BMezZ2g6pWzyhk0WP2tsjbGfvG84dwRKhbfdmLuCOP2nDGaO0LFX/3jT+eOUPGONeVsgC9JerDBYzdcDG1vkPQnkoYkfTkivljneJ0thgCA5jiaq4a2hyT9maT/IGmrpAdt3x4Rj8/2mOV8/AcAzA2R4HZ050p6KiKejog3Jf2NpIvqRKYYAgBKtML2pgm3jRP+brWkH064v7X/2KwxTAoASC7BApodEbF+qsNP8litV6QYAgDSa3YBzVZJayfcXyPphToHZJgUAJCco95tGg9KOs32qbbnS/qkpNvr5KUzBACk12BnGBGHbH9a0l3qnVpxbUQ8VueYFEMAQOtExJ2S7kx1PIohACCtmQ11FoViCABIj2IIAOiyNu5NympSAEDndbYzPHZXWR9bDiyf7BzSfA4eV86GwosfeCZ3hKrLT8mdoGLlzQdzRxj3zC+dkDtCxbIncieo2rt6KHeEwWlwb9ImdLYYAgCa07ZhUoohACCtmW22XRSKIQAgOZcz0zIjLKABAHQenSEAID2GSQEAXccCGgBAt4U4tQIAgLZ1hiygAQB0Hp0hACC9lnWGFEMAQFJt3KibYggASCuCBTRtcezusrZHGHqzrI26j3+mnG/k0XVlbYx94PX5uSNUjD76aO4I4+KXfyJ3hIoTHt+XO0LFCz+5KHcETKGzxRAA0ByGSQEAoBgCALqOzhAA0G0haaxd1ZCT7gEAnUdnCABIr12NIcUQAJAec4YAAHTppHvbz0raI2lU0qGIWJ8iFACg3brYGX4oInYkOA4AAFkwTAoASCvUugU0dU+tCEn/YPsh2xtTBAIAtFvvqhVR6zZodTvDCyLiBdsrJd1t+4mIuG/iE/pFcqMkDS9ZXvPl0nnt1KHcESpc1r7hOlDO/yoteGVp7ggV814s6yPvy/+5oM2xC/s+Hv38ztwRKpZetzB3hMEp7HthOrU6w4h4of/ndkm3Sjp3kueMRMT6iFg/b+HiOi8HAEAjZl0MbS+2vfTw15J+RtLmVMEAAO3VpWHSkyXdavvwcf46Ir6eJBUAoL1auIBm1sUwIp6WdHbCLACAOYEr3QMA0LqT7rlqBQCg8+gMAQDpMUwKAOi0KO/c6elQDAEA6bWsM2TOEADQeXSGAID02tUYUgwBAOnl2EWmjs4Ww2MO5U5Q9doZo7kjVCz9Xjkbmb9xYu4EVfN3OXeEin2rcic4orRFE4vmHcwdoWJ/Ye9PozIWQ9uXSPq8pDMknRsRm6b7N8wZAgDSCvWuWlHnVs9mSb8o6b7pnnhYZztDAMDcFBFbJKm/d/aMUAwBAElZea48UQfFEACQXv1iuML2xLm+kYgYOXzH9j9KOmWSf/c7EXHb230xiiEAIL36xXBHRKyf+vDx0bovMBHFEACQ1uEFNC3CalIAwJxi+2LbWyWdL+nvbd813b+hMwQAJJdzAU1E3Crp1rfzbyiGAID0WE0KAOi2aF0xZM4QANB5dIYAgLRCresMO1sMF20va93vgWXlbIwtSSc+Xs4Gx7vfOZw7QsXud+VOUPWeG17JHWHcE1ctzR2hYseX1+WOUBFl/Zg3q6xfsdPqbDEEADSH7dgAAGhZMWQBDQCg8+gMAQBphaSxdnWGFEMAQGLtO8+QYggASI9iCADovJYVQxbQAAA6j84QAJAWC2gAAAgp2rUFDcUQAJAec4YAALRLZzvDoQNlfWoZPTZ3gqoF3/5e7gjjnrusrJ2x136lrB+b/T9yfO4I44ZeLmsn6uG9ZQ3Vjc1z7giDwZwhAABq3TApxRAAkB7FEADQbe3bjo0FNACAzqMzBACkFZLGylq8NB2KIQAgvZYNk1IMAQDpUQwBAN0WrTvPkAU0AIDOozMEAKQVUrBRNwCg81o2TEoxBACkxwKadtj60dwJqk58OHeCqid+//TcEcbFG2X9UL36vuHcESr2nfd67gjj1p20M3eEih/EmtwRKr532f/OHaFi6O9yJyhHZ4shAKAhEZx0DwAAw6QAgM4LOkMAQLdx1QoAAFqHzhAAkFaI8wwBABA70AAAuiwkRcs6w1pzhrY32P4320/Z/myqUACAFovodYZ1bgM262Joe0jSn0n6OUlnSrrU9pmpggEAMCh1hknPlfRURDwtSbb/RtJFkh5PEQwA0F5tGyatUwxXS/rhhPtbJf14vTgAgDmhZQtoHLM8MdL2JZJ+NiL+U//+ZZLOjYjfeMvzNkra2L97lqTNs487p62QtCN3iILx/kyN9+boeH+mdnpELE19UNtfV+99r2NHRGxIkWcm6nSGWyWtnXB/jaQX3vqkiBiRNCJJtjdFxPoarzln8d4cHe/P1Hhvjo73Z2q2NzVx3EEWsVTqrCZ9UNJptk+1PV/SJyXdniYWAACDM+vOMCIO2f60pLskDUm6NiIeS5YMAIABqXXSfUTcKenOt/FPRuq83hzHe3N0vD9T4705Ot6fqfHe9M16AQ0AAHMFV60AAHTeQIoh27ZNzfZa29+0vcX2Y7avzJ2pNLaHbH/H9h25s5TG9jLbX7X9RP976PzcmUph+7/0f6Y2277R9oLcmXKyfa3t7bY3T3jsBNt3236y/+fynBlzarwYsm3btA5JuioizpB0nqRP8f78f66UtCV3iEL9iaSvR8T7JJ0t3idJku3Vkn5T0vqIOEu9RX6fzJsqu+skvfWUh89KuiciTpN0T/9+Jw2iMxzfti0i3pR0eNs2SIqIbRHxcP/rPer9MludN1U5bK+R9POSvpw7S2lsHyfppyRdI0kR8WZE7MqbqijzJC20PU/SIk1yHnSXRMR9kna+5eGLJF3f//p6SZ8YaKiCDKIYTrZtG7/sJ2F7naRzJD2QN0lR/ljSb0lq195Og/EuSS9L+sv+MPKXbS/OHaoEEfG8pD+U9JykbZJei4h/yJuqSCdHxDap98Fc0srMebIZRDH0JI+xhPUtbC+RdLOkz0TE7tx5SmD745K2R8RDubMUap6kD0j684g4R9Lr6vAw10T9ua+LJJ0q6R2SFtv+lbypULJBFMMZbdvWZbaH1SuEN0TELbnzFOQCSb9g+1n1htc/bPsreSMVZaukrRFxeCThq+oVR0gflfRMRLwcEQcl3SLpJzJnKtFLtldJUv/P7ZnzZDOIYsi2bUdh2+rN+WyJiC/lzlOSiPjtiFgTEevU+775RkTw6b4vIl6U9EPbp/cf+oi4hNphz0k6z/ai/s/YR8TiosncLuny/teXS7otY5asau1AMxNs2zatCyRdJulR24/0H/tcf3cfYDq/IemG/gfNpyX9auY8RYiIB2x/VdLD6q3Y/o46vtuK7RslXShphe2tkq6W9EVJN9m+Qr0PEJfkS5gXO9AAADqPHWgAAJ1HMQQAdB7FEADQeRRDAEDnUQwBAJ1HMQQAdB7FEADQeRRDAEDn/T8jnlewrqN91wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 576x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "new_u = np.exp(s['u'])\n",
    "\n",
    "plt.figure(figsize=(8,6))\n",
    "plt.pcolor(dmdc.reconstructed_data(new_u).real)\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can also use a different timestep for the reconstruction, obviously passing the proper control input (the number of inputs we pass has to be the number of reconstructed snapshots we try to compute, except for the first snapshots). We continue halving the timestep and create a new random input."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dmdc.dmd_time['dt'] = .5\n",
    "new_u = np.random.rand(s['u'].shape[0], dmdc.dynamics.shape[1]-1)\n",
    "\n",
    "plt.figure(figsize=(8,6))\n",
    "plt.pcolor(dmdc.reconstructed_data(new_u).real)\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}