{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Duffing-stuffing parameter estimation\n", "\n", "Goals:\n", "\n", " * Estimate parameters for a Duffing-like model such that it describes the behavior of the system with low error in different experimental schemes (varying resonnance frequencies, degradation, etc.).\n", " * Simulate the system and perform various analyses (sensitivity, stability, etc.)\n", " \n", "\n", "## Table of contents\n", "\n", " 1. [Empirical data](#empiric-data)\n", " 2. [Frequency response](#frequency-response)\n", " 3. [Model](#model)\n", " 4. [Loss function](#loss-function)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "\n", "import os\n", "import numpy as np\n", "import random\n", "import itertools\n", "from scipy.stats import skew\n", "from scipy.interpolate import interp1d\n", "from tqdm import tqdm_notebook as tqdm\n", "from toolz import curry\n", "from scipy import signal\n", "from scipy.optimize import minimize\n", "from scipy.io import loadmat\n", "\n", "# PyDSTool requires scipy 0.X\n", "# However, solve_ivp was introduced in scipy 1.X.\n", "from scipy.integrate import odeint, solve_ivp\n", "#from pydstool_integrator import simulate as ds_simulate\n", "\n", "import matplotlib.pyplot as plt\n", "#from matplotlib import animation\n", "#plt.rcParams[\"animation.html\"] = \"jshtml\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Empirical data\n", "\n", "Data is measured through frequency scans at lab.\n", "\n", "Read data from `.mat`-files as a dict of numpy arrays. We focus primarily on the XY-trace data, containing a stable-state period of 100 observations per frequency, for 5 experiments total with variying resonnance frequencies." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def read_xy(matfile, experiment_no):\n", " \"\"\"Experiment number in [0, 5]\"\"\"\n", " xy = loadmat(matfile)['XYPost'][0, experiment_no]\n", " print(\"Variables (rows x observations): \", xy.dtype.names)\n", " xy_data = dict([(k, xy[i]) for i, k in enumerate(xy.dtype.names)])\n", " t_min, t_max = xy_data['t'][0,0], xy_data['t'][-1,0]\n", " f_min, f_max = xy_data['f'][0,0], xy_data['f'][-1,0]\n", " print(\"Resonnance frequencies: (%d, %d)\" % (xy_data['XResfFreq'][0,0], xy_data['YResfFreq'][0,0]))\n", " print(\"Resonnance amplitudes: (%.2f, %.2f)\" % (xy_data['XResAmp'][0,0], xy_data['YResAmp'][0,0]))\n", " print(\"T = %.2f\" % (t_max - t_min,))\n", " print(\"t in [%.2f, %.2f]\" % (t_min, t_max))\n", " print(\"f in [%.1f, %.1f]\" % (f_min, f_max))\n", " print(\"x shape: %d x %d\" % xy_data['x'].shape)\n", " print(\"y shape: %d x %d\" % xy_data['y'].shape)\n", " return xy_data\n", "\n", "\n", "def read_amp(matfile):\n", " ds_name = os.path.splitext(os.path.basename(matfile))[0]\n", " print(\"Reading ds '%s'\" % ds_name)\n", " amp = loadmat(matfile)[ds_name]\n", " _, n_vars = amp.shape\n", " amp_data = dict([(amp[0,i][1][0][0][0], amp[0,i][0][:,0]) for i in range(n_vars)])\n", " print(\"Variables: %s\" % ','.join(amp_data.keys()))\n", " return amp_data" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Reading first experiment\n", "Variables (rows x observations): ('x', 'y', 's', 'f', 'v', 't', 'XResAmp', 'XResfFreq', 'YResAmp', 'YResfFreq')\n", "Resonnance frequencies: (7600, 8100)\n", "Resonnance amplitudes: (1.03, 1.10)\n", "T = 55.78\n", "t in [0.00, 55.78]\n", "f in [0.0, 10000.0]\n", "x shape: 101 x 100\n", "y shape: 101 x 100\n" ] } ], "source": [ "# Read first experiment\n", "print(\"Reading first experiment\")\n", "xy_data = read_xy('data/XYPost.mat', 0)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "scrolled": false }, "outputs": [], "source": [ "#amp_pre = read_amp('data/APre.mat')\n", "#amp_post = read_amp('data/APost.mat')\n", "#amp_postb = read_amp('data/APostB.mat')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Plotting\n", "\n", "Three types of plots:\n", "\n", " * Frequency scan with amplitude mean/std.\n", " * Trajectory plot in (x, y)-plane.\n", " * Trajectory over time for x and y." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "def plot_std_freqscan(xy_data):\n", " linewidth = 0.6\n", " fig, (ax11, ax12) = plt.subplots(\n", " nrows=1,\n", " ncols=2,\n", " figsize=(8, 4)\n", " )\n", " rows = list(filter(\n", " lambda r: xy_data['f'][r,0] != 0,\n", " range(xy_data['f'].shape[0])\n", " ))\n", " ax11.errorbar(\n", " xy_data['f'][rows,0],\n", " np.mean(xy_data['x'][rows, :], axis=1),\n", " np.std(xy_data['x'][rows, :], axis=1),\n", " linewidth=linewidth\n", " )\n", " ax11.axvline(x=xy_data['XResfFreq'][0,0], linestyle='--', color='red', linewidth=0.8)\n", " ax11.axvline(x=xy_data['YResfFreq'][0,0], linestyle='--', color='red', linewidth=0.8)\n", " ax11.set_xlabel(r'$f$')\n", " ax11.set_ylabel(r'$x$')\n", " ax12.errorbar(\n", " xy_data['f'][rows,0],\n", " np.mean(xy_data['y'][rows, :], axis=1),\n", " np.std(xy_data['y'][rows, :], axis=1),\n", " linewidth=linewidth\n", " )\n", " ax12.axvline(x=xy_data['XResfFreq'][0,0], linestyle='--', color='red', linewidth=0.8)\n", " ax12.axvline(x=xy_data['YResfFreq'][0,0], linestyle='--', color='red', linewidth=0.8)\n", " ax12.set_xlabel(r'$f$')\n", " ax12.set_ylabel(r'$y$')\n", " plt.suptitle(\"Amplitude mean and standard deviation per frequency\\nRed lines are resonnance frequencies\")\n", " #plt.tight_layout()\n", " \n", " \n", "def plot_xy(rows, xy_data):\n", " linewidth = 0.6\n", " fig, ((ax11, ax12), (ax21, ax22)) = plt.subplots(\n", " nrows=2,\n", " ncols=2,\n", " figsize=(8, 6)\n", " )\n", " for row in rows:\n", " ax11.plot(xy_data['x'][row, :], linewidth=linewidth)\n", " ax12.plot(xy_data['y'][row, :], linewidth=linewidth)\n", " ax21.plot(xy_data['x'][row, :], xy_data['y'][row, :], linewidth=linewidth)\n", " ax22.plot(xy_data['f'][row, :])\n", " \n", " ax11.set_ylabel(r'$x$')\n", " ax12.set_ylabel(r'$y$')\n", " ax21.set_xlabel(r'$x$')\n", " ax21.set_ylabel(r'$y$')\n", " ax22.set_ylabel(r'$f$')\n", " plt.suptitle(\"XY-data plots for given frequencies\")\n", " #plt.tight_layout()\n", " \n", " \n", "def plot_xyt(rows, xy_data, normalizer=lambda x: x, sim_xy_data=None):\n", " N = len(rows)\n", " linewidth = 0.6\n", " fig, axes = plt.subplots(\n", " nrows=N,\n", " ncols=2,\n", " #sharey=True,\n", " #sharex=True,\n", " figsize=(8, 3*N)\n", " )\n", " for i in range(N):\n", " axes[i,0].plot(normalizer(xy_data['x'][rows[i], :]), linewidth=linewidth)\n", " axes[i,1].plot(normalizer(xy_data['y'][rows[i], :]), linewidth=linewidth)\n", " if sim_xy_data is not None:\n", " axes[i,0].plot(normalizer(sim_xy_data['x'][rows[i], :]), linewidth=linewidth)\n", " axes[i,1].plot(normalizer(sim_xy_data['y'][rows[i], :]), linewidth=linewidth)\n", " axes[i,0].set_ylabel('$x$ ($f$=%d)' % xy_data['f'][rows[i], 0])\n", " axes[i,1].set_ylabel('$y$ ($f$=%d)' % xy_data['f'][rows[i], 0])\n", " plt.suptitle(\"Stable-state XY-data plots for given frequencies\")\n", " #plt.tight_layout()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Variables (rows x observations): ('x', 'y', 's', 'f', 'v', 't', 'XResAmp', 'XResfFreq', 'YResAmp', 'YResfFreq')\n", "Resonnance frequencies: (7600, 8100)\n", "Resonnance amplitudes: (1.03, 1.10)\n", "T = 55.78\n", "t in [0.00, 55.78]\n", "f in [0.0, 10000.0]\n", "x shape: 101 x 100\n", "y shape: 101 x 100\n" ] }, { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def right(gamma, alpha, beta, delta, z, omega):\n", " denom = (omega**2 - alpha - (3/4)*beta*z**2)**2 + (delta*omega)**2\n", " return gamma/np.sqrt(denom)\n", "\n", "gamma = 1\n", "alpha = 1\n", "beta_mul = 0.01\n", "delta = 0.1\n", "\n", "z, omega = np.meshgrid(\n", " np.linspace(0, 15, 500),\n", " np.linspace(0, 2, 500)\n", ")\n", "\n", "plt.figure()\n", "for beta in (-0.3, 0.0, 1, 4):\n", " G = right(gamma, alpha, beta*beta_mul, delta, z, omega)\n", " plt.contour(omega, z, (z-G), [0])\n", " plt.xlabel(\"$\\omega$\")\n", " plt.ylabel(\"$z$\")" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "def right(gamma, alpha, beta, delta, z, f):\n", " omega = 2*np.pi*f\n", " denom = (omega**2 - alpha - (3/4)*beta*z**2)**2 + (delta*omega)**2\n", " return gamma/np.sqrt(denom)\n", "\n", "def get_pointmap(gamma, alpha, beta, delta, z, f):\n", " fig = plt.figure()\n", " G = right(gamma, alpha, beta, delta, z, f)\n", " cs = plt.contour(f, z, (z-G), [0])\n", " p = cs.collections[0].get_paths()[0]\n", " v = p.vertices\n", " plt.close(fig)\n", " return v[:,0][::-1], v[:,1][::-1]\n", "\n", "def clip_pointmap(xs, ys):\n", " # If softening spring, mirror the distribution across 0 and back\n", " if skew(xs) > 0:\n", " # Is order/direction different when softening?\n", " raise NotImplementedError()\n", " _xs, _ys = clip_pointmap(-1*xs, ys)\n", " return -1*_xs, _ys\n", " \n", " # Find peak and continue s.t. x is monotonically increasing\n", " indexes = []\n", " N = len(xs)\n", " max_x = xs[0] - 1e-9\n", " for i in range(0, N):\n", " if xs[i] > max_x:\n", " indexes.append(i)\n", " max_x = xs[i]\n", " return xs[indexes], ys[indexes]\n", "\n", "def resample_pointmap(xs, ys, n):\n", " lin = interp1d(xs, ys)\n", " new_xs = np.linspace(xs[0], xs[-1], n)\n", " return new_xs, lin(new_xs)\n", "\n", "def sample_frequency_response(n, gamma, alpha, beta, delta, z, f):\n", " plt.ioff()\n", " xs, ys = get_pointmap(gamma, alpha, beta, delta, zs, fs)\n", " plt.ion()\n", " xs, ys = clip_pointmap(xs, ys)\n", " return resample_pointmap(xs, ys, n)" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "[]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f_resonance = 8e3\n", "omega_resonance = 2*np.pi*f_resonance\n", "\n", "gamma = 1e7\n", "alpha = omega_resonance**2\n", "beta = 2e8\n", "delta = 200\n", "\n", "zs, fs = np.meshgrid(\n", " np.linspace(0, 1.2, 500),\n", " np.linspace(7e3, 9e3, 500)\n", ")\n", "\n", "plt.figure()\n", "G = right(gamma, alpha, beta, delta, zs, fs)\n", "plt.contour(fs, zs, (zs-G), [0])\n", "plt.xlabel(\"$\\omega$\")\n", "plt.ylabel(\"$z$\")\n", "\n", "n = 100\n", "xs, ys = sample_frequency_response(n, gamma, alpha, beta, delta, zs, fs)\n", "plt.figure()\n", "plt.plot(xs, ys)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "@curry\n", "def frequency_objective(omega, p):\n", " obj = 0.0\n", " # Simulate and return MSE(xy_data, sim_xy_data)\n", " return obj\n", "\n", "p0 = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1)\n", "bounds = [\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2)\n", "]\n", "#omegas = xy_data['f'][:,0].tolist()\n", "#solution = minimize(objective(omega), p0, method='SLSQP', bounds=bounds)\n", "#p = solution.x\n", "\n", "# Simulate with updated values\n", "#t, X, dt, pstep = model(T, t_trans, dt_per_period, x0, v0, omega, p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Model\n", "\n", "Model derived from Duffing equations:\n", "\n", "\\begin{align}\n", "m_1 \\ddot{y}_1 &= F_1 - \\dot{y}_1(c_1 + c_3) + \\dot{y}_2c_3 - y_1(k_1 + k_3) + y_2k_3 - \\alpha_1y_1^3 + \\alpha_3(y_2 - y_1)^3 \\\\\n", "m_2 \\ddot{y}_2 &= F_2 - \\dot{y}_2(c_2 + c_3) + \\dot{y}_1c_3 - x_2(k_2 + k_3) + y_1k_3 - \\alpha_2y_2^3 + \\alpha_3(y_2 - y_1)^3 \\\\\n", "\\end{align}\n", "\n", "where $F = Ce^{i\\omega t}$. Transform to first-order form by variable substitutions $x_3 = \\dot{y}_1, x_1 = y_1$ and $x_4 = \\dot{y}_2, x_2 = y_2$:\n", "\n", "\\begin{align}\n", "\\dot{x}_1 &= x_3 \\\\\n", "\\dot{x}_2 &= x_4 \\\\\n", "m_1\\dot{x}_3 &= F_1 - x_3(c_1 + c_3) + x_4c_3 - x_1(k_1 + k_3) + x_2k_3 - \\alpha_1x_1^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", "m_2\\dot{x}_4 &= F_2 - x_4(c_2 + c_3) + x_3c_3 - x_2(k_2 + k_3) + x_1k_3 - \\alpha_2x_2^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", "\\end{align}\n", "\n", "\n", "For some reason, this model doesn't work out when working backwards from resonnance frequencies. I may be missing something obvious, otherwise the fact that mass goes into the estimations may mess things up.\n", "\n", "\n", "### Transformed model\n", "\n", "Eliminate mass, + easier to reason about physical constants:\n", "\n", "\\begin{align}\n", "\\dot{x}_1 &= x_3 \\\\\n", "\\dot{x}_2 &= x_4 \\\\\n", "\\dot{x}_3 &= \\frac{1}{m_1}F_1 - x_3(c_1 + c_3) + x_4c_3 - x_1(k_1 + k_3) + x_2k_3 - \\alpha_1x_1^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", "\\dot{x}_4 &= \\frac{1}{m_2}F_2 - x_4(c_2 + c_3) + x_3c_3 - x_2(k_2 + k_3) + x_1k_3 - \\alpha_2x_2^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", "\\end{align}\n", "\n", "With the Jacobian $\\mathbf{J} = \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{X}}$\n", "\n", "\\begin{bmatrix}\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & 1 \\\\\n", "-k_1 - k_3 - 3\\alpha_1x_1^2 - 3\\alpha_3(x_2 - x_1)^2 & k_3 + 3\\alpha_3(x_2 - x_1)^2 & -c_1 - c_3 & c_3 \\\\\n", "k_3 - 3\\alpha_3(x_2 - x_1)^2 & -k_2 - k_3 - 3\\alpha_2x_2^2 + 3\\alpha_3(x_2 - x_1)^2 & c_3 & -c_2 - c_3\n", "\\end{bmatrix}\n", "\n", "Use the harmonic oscillator identities\n", "\n", " * Undamped angular frequency:\n", "\n", "\\begin{align}\n", "\\omega_0 &= \\sqrt{\\frac{k}{m}}\n", "\\end{align}\n", "\n", " * Damping ratio:\n", "\n", "\\begin{align}\n", "\\zeta &= \\frac{c}{2\\sqrt{mk}}\n", "\\end{align}\n", "\n", " * Resonant freqency:\n", "\n", "\\begin{align}\n", "\\omega_r &= \\omega_0\\sqrt{1-2\\zeta^2}, \\zeta < \\frac{1}{\\sqrt{2}}\n", "\\end{align}\n", "\n", "and express the constants subject to estimation as\n", "\n", "\\begin{align}\n", "c_1 &= 2 \\zeta_1 \\omega_{0,1} \\\\\n", "c_2 &= 2 \\zeta_2 \\omega_{0,2} \\\\\n", "c_3 &= g_c(c_1, c_2) \\\\\n", "k_1 &= \\omega_{0,1}^2 \\\\\n", "k_2 &= \\omega_{0,2}^2 \\\\\n", "k_3 &= g_k(k_1, k_2) \\\\\n", "\\alpha_1 &= f_1(\\mathbf{X} ; \\theta) \\\\\n", "\\alpha_2 &= f_2(\\mathbf{X} ; \\theta) \\\\\n", "\\alpha_3 &= g_{\\alpha}(\\alpha_1, \\alpha_2)\n", "\\end{align}\n", "\n", "\n", "With the model expressed this way, things make sense and we get resonnance where it should be.\n", "\n", "\n", "### Notes on solvers\n", "\n", " * We use SciPy's `solve_ivp` to simulate the system. Different methods (RK45, LSODA, Radeau) has been tested with no noticable differences.\n", " * The standard `odeint` from SciPy is super shit. It easily diverges and is unstable. They claim to use the standard LSODA solver (same as `solve_ivp` with method='LSODA') but the results are entirely different.\n", " * Once we hit the right parameters, the simulation is considerable slower because these are adaptive solvers.\n", " * There is an ODE implementation from the [PyDSTool package](https://github.com/robclewley/pydstool) which compiles to C and is much much faster.\n", " \n", "#### PyDSTool solver\n", "\n", "The model is implemented with this solver. It's slightly faster, but notoriously more complicated to use. Besides, it requires SciPy version `<1.0` which is not compatible with the rest of this code. Let's stick to scipy's modern solvers." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "@curry\n", "def model1(omega, p, t, X):\n", " x1, x2, xd1, xd2 = X\n", " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", " F1 = C1*np.cos(omega*t)\n", " F2 = C2*np.cos(omega*t)\n", " xdd1 = F1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*x1**3 + a3*(x2 - x1)**3\n", " xdd2 = F2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*x2**3 + a3*(x2 - x1)**3\n", " return xd1, xd2, m1*xdd1, m2*xdd2\n", "\n", "@curry\n", "def model2(omega, p, t, X):\n", " x1, x2, xd1, xd2 = X\n", " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", " xdd1 = C1*np.cos(omega*t)/m1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*x1**3 + a3*(x2 - x1)**3\n", " xdd2 = C2*np.cos(omega*t)/m2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*x2**3 + a3*(x2 - x1)**3\n", " return xd1, xd2, xdd1, xdd2\n", "\n", "@curry\n", "def jacobian2(omega, p, t, X):\n", " x1, x2, xd1, xd2 = X\n", " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", " return np.array([\n", " [ 0 , 0 , 1 , 0 ],\n", " [ 0 , 0 , 0 , 1 ],\n", " [-k1-k3-3*a1*x1**2-3*a3*(x2-x1)**2, k3+3*a3*(x2-x1)**2 , -c1-c3 , c3 ],\n", " [ k3-3*a3*(x2-x1)**2 , -k2-k3-3*a2*x2**2+3*a3*(x2-x1)**2 , c3 , -c2-c3]\n", " ])\n", "\n", "def odeint_integrate(model, jac, dt, T, X0, rtol, atol):\n", " t = np.linspace(0, T, int(T/dt))\n", " X = odeint(model, X0, t, Dfun=jac, tfirst=True, rtol=rtol, atol=atol)\n", " return X.T\n", "\n", "def solve_ivp_integrate(model, jac, dt, T, X0, rtol, atol):\n", " t = np.linspace(0, T, int(T/dt))\n", " sol = solve_ivp(model, [0, T], X0, t_eval=t, jac=jac, method='Radau', first_step=dt, rtol=rtol, atol=atol)\n", " return sol.y\n", "\n", "# Rely on the same functional signatures by mimicing the data structure of lab measurement.\n", "def simulate_xy_data(integrator, rows, xy_data, T, t_trans, t_scale, steps, x0, v0, p, progress=True):\n", " sim_xy_data = {\n", " 'x': np.zeros((101, 100)),\n", " 'y': np.zeros((101, 100)),\n", " 'f': np.zeros((101, 100)),\n", " 'XResfFreq': xy_data['XResfFreq'],\n", " 'YResfFreq': xy_data['YResfFreq']\n", " }\n", " _rows = tqdm(rows) if progress else rows\n", " for i in _rows:\n", " f = xy_data['f'][i,0]\n", " omega = 2*np.pi*f/t_scale\n", " X = integrator(\n", " model2(omega, p),\n", " jacobian2(omega, p),\n", " (T-t_trans)/steps,\n", " T,\n", " x0 + v0,\n", " 1e-3,\n", " [1e-4, 1e-4, 1e-2, 1e-2]\n", " )\n", " x1, x2, xd1, xd2 = X\n", " sim_xy_data['x'][i,:] = x1[-steps:]\n", " sim_xy_data['y'][i,:] = x2[-steps:]\n", " sim_xy_data['f'][i,:] = f\n", " return sim_xy_data\n", "\n", "def simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True):\n", " # Amplitude at resonnance should be 1\n", " # Aim for better numerical stability by setting C and m to approx. the same numeric precision\n", " t_scale = 1\n", " C1 = 1e7/t_scale\n", " C2 = 1e7/t_scale\n", " m1 = 1\n", " m2 = 1\n", "\n", " # Fetch resonnance frequencies from data\n", " f_r1 = xy_data['XResfFreq']/t_scale\n", " f_r2 = xy_data['YResfFreq']/t_scale\n", " omega_r1 = 2*np.pi*f_r1\n", " omega_r2 = 2*np.pi*f_r2\n", " #omega_01 = omega_r1/(np.sqrt(1-2*zeta1**2))\n", " #omega_02 = omega_r2/(np.sqrt(1-2*zeta2**2))\n", "\n", " # Compute parameters from identities\n", " #c1 = 2*zeta1*omega_01\n", " #c2 = 2*zeta2*omega_02\n", " c1 = 200\n", " c2 = 200\n", " c3 = gc(c1, c2)\n", " k1 = omega_r1**2\n", " k2 = omega_r2**2\n", " k3 = gk(k1, k2)\n", " #a1 = f1(k1, k2)\n", " #a2 = f2(k1, k2)\n", " #a3 = ga(k1, k2)\n", " a1 = 1e8\n", " a2 = 1e8\n", " a3 = 0\n", "\n", " p = (\n", " C1, C2,\n", " m1, m2,\n", " c1, c2, c3,\n", " k1, k2, k3,\n", " a1, a2, a3\n", " )\n", "\n", " if verbose:\n", " print(\"Parameters:\")\n", " print(\"Omega_r 1 = %.3f\" % omega_r1)\n", " print(\"Omega_r 2 = %.3f\" % omega_r2)\n", " #print(\"Omega0 1 = %.3f\" % omega_01)\n", " #print(\"Omega0 2 = %.3f\" % omega_02)\n", " print(\"c1 = %.3f\" % c1)\n", " print(\"c2 = %.3f\" % c2)\n", " print(\"c3 = %.3f\" % c3)\n", " print(\"k1 = %.3f\" % k1)\n", " print(\"k2 = %.3f\" % k2)\n", " print(\"k3 = %.3f\" % k3)\n", " print(\"a1 = %.3f\" % a1)\n", " print(\"a2 = %.3f\" % a2)\n", " print(\"a3 = %.3f\" % a3)\n", "\n", " # Start from any state, the system stabilize quickly\n", " x0, v0 = (-0.002, 0.01), (-0.004, 0.03)\n", " #x0, v0 = (0.0, 0.0), (0.0, 0.0)\n", " \n", " # Set transient period to 0.1 seconds and simulate 100 steps over 0.5 seconds\n", " t_trans = 0.1*t_scale\n", " T = t_trans + 0.5*t_scale\n", " steps = 100\n", "\n", " # Simulate mostly around resonnance\n", " return simulate_xy_data(solve_ivp_integrate, rows, xy_data, T, t_trans, t_scale, steps, x0, v0, p, progress=verbose)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Harmonic oscillator\n", "\n", "Set $c_3 = k_3 = \\alpha_1 = \\alpha_2 = \\alpha_3 = 0$ for simulating a standard driven harmonic oscillator with no coupling between x- and y-components. Assume damping $\\zeta_1 = \\zeta_2 = 0.1$ and use resonnance frequencies from lab data to estimate parameters." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "exp_no = 0\n", "print(\"Read data, experiment %d\" % exp_no)\n", "xy_data = read_xy('data/XYPost.mat', exp_no)\n", "\n", "def gc(c1, c2):\n", " return 0.0\n", "\n", "def gk(k1, k2):\n", " return 0.0\n", "\n", "def f1(k1, k2):\n", " return 0.0\n", "\n", "def f2(k1, k2):\n", " return 0.0\n", "\n", "def ga(k1, k2):\n", " return 0.0\n", "\n", "rows = [50, 65, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 90, 100]\n", "zeta1 = 0.1\n", "zeta2 = 0.1\n", "xyhat_data = simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True)\n", "\n", "plot_std_freqscan(xyhat_data)\n", "plot_xy(rows, xyhat_data)\n", "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### With duffing term\n", "\n", "The duffing term has to be pretty large to see any stiffening effect. Set $\\alpha_1 = 1500k_1$ and $\\alpha_2 = 1500k_2$ (still without coupling)." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "exp_no = 0\n", "print(\"Read data, experiment %d\" % exp_no)\n", "xy_data = read_xy('data/XYPost.mat', exp_no)\n", "\n", "def gc(c1, c2):\n", " return 0.0\n", "\n", "def gk(k1, k2):\n", " return 0.0\n", "\n", "def f1(k1, k2):\n", " return 1.5e3*k1\n", "\n", "def f2(k1, k2):\n", " return 1.5e3*k2\n", "\n", "def ga(k1, k2):\n", " return 0.0\n", "\n", "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", "zeta1 = 0.1\n", "zeta2 = 0.1\n", "xyhat_data = simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True)\n", "\n", "plot_std_freqscan(xyhat_data)\n", "plot_xy(rows, xyhat_data)\n", "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### With coupling" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "exp_no = 0\n", "print(\"Read data, experiment %d\" % exp_no)\n", "xy_data = read_xy('data/XYPost.mat', exp_no)\n", "\n", "def gc(c1, c2):\n", " return 0.05*(c1 + c2)\n", "\n", "def gk(k1, k2):\n", " return 0.05*(k1 + k2)\n", "\n", "def f1(k1, k2):\n", " return 1.5e3*k1\n", "\n", "def f2(k1, k2):\n", " return 1.5e3*k2\n", "\n", "def ga(k1, k2):\n", " return 0.05*(a1 + a2)\n", "\n", "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", "zeta1 = 0.1\n", "zeta2 = 0.1\n", "xyhat_data = simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True)\n", "\n", "plot_std_freqscan(xyhat_data)\n", "plot_xy(rows, xyhat_data)\n", "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Loss function\n", "\n", "Given the two multivariate signals, one empirical and one simulated, we need a distance metric $d(\\mathbf{S}(\\omega), \\hat{\\mathbf{S}}(\\omega))$ that quantifies the error of our simulation.\n", "\n", "Consider first a single frequency $\\mathbf{S}(\\omega=x) \\in \\mathbb{R}^2$. Treat each component individually, perform autocorrelation do find the shift, then simply use mean squared error as the distance metric between the two common periods of $\\mathbf{S}$ and $\\hat{\\mathbf{S}}$. We extend this to the multivariate case by simply averaging the loss for each frequency.\n", "\n", "TODO: Assert that both components of $\\mathbf{S}(\\omega=x)$ have the same shift." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "def normalize_signal(s):\n", " return (s - np.mean(s)) / np.std(s)\n", "\n", "def autocorrelate_1d(s1, s2):\n", " corr = np.correlate(s1, s2, mode='same') / (np.linalg.norm(s1)*np.linalg.norm(s2))\n", " corr_half = corr[int(len(s1)/2):]\n", " idx = np.argmax(corr_half)\n", " return idx, corr_half[idx]\n", "\n", "def loss_1d(s1, s2, normalize=True):\n", " assert len(s1) == len(s2)\n", " N = len(s1)\n", " if normalize:\n", " _s1 = normalize_signal(s1)\n", " _s2 = normalize_signal(s2)\n", " else:\n", " _s1 = s1\n", " _s2 = s2\n", " idx, coeff = autocorrelate_1d(_s1, _s2)\n", " return idx, coeff, np.mean((_s1[idx:]-_s2[:N-idx])**2)\n", "\n", "def xy_loss(rows, xy_data, xyhat_data, normalize=True, verbose=False):\n", " # Calculate correlation coefficients and MSE for both x and y for the specified set of rows\n", " x_idxs, x_coeffs, x_mses = zip(*[loss_1d(xy_data['x'][i,:], xyhat_data['x'][i,:], normalize=normalize) for i in rows])\n", " y_idxs, y_coeffs, y_mses = zip(*[loss_1d(xy_data['y'][i,:], xyhat_data['y'][i,:], normalize=normalize) for i in rows])\n", " # Print some statistics\n", " if verbose:\n", " print('\\n'.join(map(\n", " lambda var: \"%s: %.4f mean, %.4f std\" % (var[0], np.mean(var[1]), np.std(var[1])),\n", " [\n", " ('X idx', x_idxs),\n", " ('Y idx', y_idxs),\n", " ('X coeffs', x_coeffs),\n", " ('Y coeffs', y_coeffs),\n", " ('X MSEs', x_mses),\n", " ('Y MSEs', y_mses)\n", " ]\n", " )))\n", " # Return the sum of means of both components\n", " return np.mean(x_mses) + np.mean(y_mses)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Loss function test\n", "\n", "Random signals and sines." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "# Test loss function\n", "t = np.linspace(0, 100, 100)\n", "s1 = {\n", " 'f': np.array([\n", " np.ones((100,)),\n", " 2*np.ones((100,))\n", " ]),\n", " 'x': np.array([\n", " np.random.randn(100),\n", " np.array([1*np.sin(i/np.pi) for i in t]),\n", " ]),\n", " 'y': np.array([\n", " np.random.randn(100),\n", " np.array([1*np.cos(i/np.pi) for i in t]),\n", " ])\n", "}\n", "s2 = {\n", " 'f': np.array([\n", " np.ones((100,)),\n", " 2*np.ones((100,))\n", " ]),\n", " 'x': np.array([\n", " np.random.randn(100),\n", " np.array([1*np.cos(i/np.pi) for i in t]),\n", " ]),\n", " 'y': np.array([\n", " np.random.randn(100),\n", " np.array([1*np.sin(i/np.pi) for i in t]),\n", " ])\n", "}\n", "\n", "plot_xyt([0,1], s1, sim_xy_data=s2)\n", "\n", "print(\"Loss: %.4f\\n\" % xy_loss([0], s1, s2, verbose=True))\n", "print(\"Loss: %.4f\\n\" % xy_loss([1], s1, s2, verbose=True))\n", "print(\"Loss: %.4f\" % xy_loss([0, 1], s1, s2, verbose=True))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Loss for model\n", "\n", "Plotting normalized signals." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "print(\"Loss: %.4f\\n\" % xy_loss(rows, xy_data, xyhat_data, verbose=True))\n", "plot_xyt(rows, xy_data, normalizer=normalize_signal, sim_xy_data=xyhat_data)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": false }, "outputs": [], "source": [ "#C_grid = [1e1] # Amplitude of driving force (gamma)\n", "#c_grid = [1e2] # Damping (delta)\n", "#a_grid = [1e-1] # Non-linear restoring force (beta)\n", "#k_grid = [1e6] # Linear stiffness (alpha)\n", "\n", "#xy_data = read_xy('data/XYPost.mat', 0)\n", "#rows = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]\n", "#rows = [10, 65, 70, 75, 80, 85, 90]\n", "\n", "#losses = []\n", "#best_loss = 1e9\n", "#best_p = None\n", "#for c, a, k, C in tqdm.tqdm(itertools.product(*[c_grid, a_grid, k_grid, C_grid])):\n", "# print(\"c1=c2=c3=%.4f, a1=a2=a3=%.4f, k1=k2=k3=%.4f, C1=C2=%.4f\" % (c, a, k, C))\n", "# _damping = 0.001\n", "# _omegar = 8e3\n", "# _omega0 = _omegar/(np.sqrt(1-2*_damping**2))\n", "# C = 1e5\n", "# c2 = 2*_damping*_omega0\n", "# k2 = _omega0**2\n", "# a = 0.5\n", "# p = (C, C, 0.03*c2, c2, 0.01*c2, 0.01*a, 0.9*a, 0.02*a, 0.5*k2, k2, 0.1*k2)\n", "# #p = (C, C, c, c, c, a, a, a, k, k, k)\n", "# xyhat_data = simulate_xy_data(rows, xy_data, p)\n", "# loss = xy_loss(rows, xy_data, xyhat_data, verbose=True)\n", "# losses.append(loss)\n", "# if loss < best_loss:\n", "# best_loss = loss\n", "# best_p = p\n", "# print(\"Loss: %.4f\\n\" % loss)\n", "# plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)\n", "# break" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "@curry\n", "def objective(omega, p):\n", " obj = 0.0\n", " # Simulate and return MSE(xy_data, sim_xy_data)\n", " return obj\n", "\n", "p0 = (0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1)\n", "bounds = [\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2),\n", " (0.1, 0.2)\n", "]\n", "#omegas = xy_data['f'][:,0].tolist()\n", "#solution = minimize(objective(omega), p0, method='SLSQP', bounds=bounds)\n", "#p = solution.x\n", "\n", "# Simulate with updated values\n", "#t, X, dt, pstep = model(T, t_trans, dt_per_period, x0, v0, omega, p)" ] } ], "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": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 1 }