|
| 1 | +{ |
| 2 | + "cells": [ |
| 3 | + { |
| 4 | + "attachments": {}, |
| 5 | + "cell_type": "markdown", |
| 6 | + "metadata": {}, |
| 7 | + "source": [ |
| 8 | + "# Welcome to ProgPy's Linear Model Example" |
| 9 | + ] |
| 10 | + }, |
| 11 | + { |
| 12 | + "attachments": {}, |
| 13 | + "cell_type": "markdown", |
| 14 | + "metadata": {}, |
| 15 | + "source": [ |
| 16 | + "The goal of this notebook is to instruct users on how to use ProgPy Model LinearModel.\n", |
| 17 | + "\n", |
| 18 | + "This example shows the use of the LinearModel class, a subclass of PrognosticsModel for models that can be described as a linear time series, which can be defined by the following equations:\n", |
| 19 | + "\n", |
| 20 | + "\n", |
| 21 | + "\n", |
| 22 | + "#### _<b>The State Equation<b>_:\n", |
| 23 | + "$$\n", |
| 24 | + "\\frac{dx}{dt} = Ax + Bu + E\n", |
| 25 | + "$$\n", |
| 26 | + "\n", |
| 27 | + "#### _<b>The Output Equation<b>_:\n", |
| 28 | + "$$\n", |
| 29 | + "z = Cx + D\n", |
| 30 | + "$$\n", |
| 31 | + "\n", |
| 32 | + "#### _<b>The Event State Equation<b>_:\n", |
| 33 | + "$$\n", |
| 34 | + "es = Fx + G\n", |
| 35 | + "$$\n", |
| 36 | + "\n", |
| 37 | + "$x$ is `state`, $u$ is `input`, $z$ is `output`, and $es$ is `event state`" |
| 38 | + ] |
| 39 | + }, |
| 40 | + { |
| 41 | + "attachments": {}, |
| 42 | + "cell_type": "markdown", |
| 43 | + "metadata": {}, |
| 44 | + "source": [ |
| 45 | + "Linear Models are defined by creating a new model class that inherits from progpy's LinearModel class and defines the following properties:\n", |
| 46 | + "* $A$: 2-D np.array[float], dimensions: n_states x n_states. <font color = 'teal'>The state transition matrix. It dictates how the current state affects the change in state dx/dt.</font>\n", |
| 47 | + "* $B$: 2-D np.array[float], optional (zeros by default), dimensions: n_states x n_inputs. <font color = 'teal'>The input matrix. It dictates how the input affects the change in state dx/dt.</font>\n", |
| 48 | + "* $C$: 2-D np.array[float], dimensions: n_outputs x n_states. The output matrix. <font color = 'teal'>It determines how the state variables contribute to the output.</font>\n", |
| 49 | + "* $D$: 1-D np.array[float], optional (zeros by default), dimensions: n_outputs x 1. <font color = 'teal'>A constant term that can represent any biases or offsets in the output.</font>\n", |
| 50 | + "* $E$: 1-D np.array[float], optional (zeros by default), dimensions: n_states x 1. <font color = 'teal'>A constant term, representing any external effects that are not captured by the state and input.</font>\n", |
| 51 | + "* $F$: 2-D np.array[float], dimensions: n_es x n_states. <font color = 'teal'>The event state matrix, dictating how state variables contribute to the event state.</font>\n", |
| 52 | + "* $G$: 1-D np.array[float], optional (zeros by default), dimensions: n_es x 1. <font color = 'teal'>A constant term that can represent any biases or offsets in the event state.</font>\n", |
| 53 | + "* __inputs__: list[str] - `input` keys\n", |
| 54 | + "* __states__: list[str] - `state` keys\n", |
| 55 | + "* __outputs__: list[str] - `output` keys\n", |
| 56 | + "* __events__: list[str] - `event` keys" |
| 57 | + ] |
| 58 | + }, |
| 59 | + { |
| 60 | + "attachments": {}, |
| 61 | + "cell_type": "markdown", |
| 62 | + "metadata": {}, |
| 63 | + "source": [ |
| 64 | + "We will now utilize our LinearModel to model the classical physics problem throwing an object into the air! We can create a subclass of LinearModel which will be used to simulate an object thrown, which we will call the ThrownObject Class.\n", |
| 65 | + "\n", |
| 66 | + "\n", |
| 67 | + "First, some definitions for our Model!\n", |
| 68 | + "\n", |
| 69 | + "#### __Events__: (2)\n", |
| 70 | + "* `falling: The object is falling`\n", |
| 71 | + "* `impact: The object has hit the ground`\n", |
| 72 | + "\n", |
| 73 | + "#### __Inputs/Loading__: (0)\n", |
| 74 | + "* `None`\n", |
| 75 | + "\n", |
| 76 | + "#### __States__: (2)\n", |
| 77 | + "* `x: Position in space (m)`\n", |
| 78 | + "* `v: Velocity in space (m/s)`\n", |
| 79 | + "\n", |
| 80 | + "#### __Outputs/Measurements__: (1)\n", |
| 81 | + "* `x: Position in space (m)`" |
| 82 | + ] |
| 83 | + }, |
| 84 | + { |
| 85 | + "attachments": {}, |
| 86 | + "cell_type": "markdown", |
| 87 | + "metadata": {}, |
| 88 | + "source": [ |
| 89 | + "Now, for our keyword arguments:\n", |
| 90 | + "\n", |
| 91 | + "* <font color = green>__thrower_height : Optional, float__</font>\n", |
| 92 | + " * Height of the thrower (m). Default is 1.83 m\n", |
| 93 | + "* <font color = green>__throwing_speed : Optional, float__</font>\n", |
| 94 | + " * Speed at which the ball is thrown (m/s). Default is 40 m/s" |
| 95 | + ] |
| 96 | + }, |
| 97 | + { |
| 98 | + "attachments": {}, |
| 99 | + "cell_type": "markdown", |
| 100 | + "metadata": {}, |
| 101 | + "source": [ |
| 102 | + "With our definitions, we can now create the ThrownObject Model.\n", |
| 103 | + "\n", |
| 104 | + "First, we need to import the necessary packages." |
| 105 | + ] |
| 106 | + }, |
| 107 | + { |
| 108 | + "cell_type": "code", |
| 109 | + "execution_count": null, |
| 110 | + "metadata": {}, |
| 111 | + "outputs": [], |
| 112 | + "source": [ |
| 113 | + "import numpy as np\n", |
| 114 | + "from progpy import LinearModel" |
| 115 | + ] |
| 116 | + }, |
| 117 | + { |
| 118 | + "attachments": {}, |
| 119 | + "cell_type": "markdown", |
| 120 | + "metadata": {}, |
| 121 | + "source": [ |
| 122 | + "Now we'll define some features of a ThrownObject LinearModel. Recall that all LinearModels follow a set of core equations and require some specific properties (see above). In the next step, we'll define our inputs, states, outputs, and events, along with the $A$, $C$, $E$, and $F$ values." |
| 123 | + ] |
| 124 | + }, |
| 125 | + { |
| 126 | + "attachments": {}, |
| 127 | + "cell_type": "markdown", |
| 128 | + "metadata": {}, |
| 129 | + "source": [ |
| 130 | + "First, let's consider state transition. For an object thrown into the air without air resistance, velocity would decrease literally by __-9.81__ \n", |
| 131 | + "$\\dfrac{m}{s^2}$ due to the effect of gravity, as described below:\n", |
| 132 | + "\n", |
| 133 | + " $$\\frac{dv}{dt} = -9.81$$\n", |
| 134 | + "\n", |
| 135 | + " Position change is defined by velocity (v), as described below:\n", |
| 136 | + " \n", |
| 137 | + " $$\\frac{dx}{dt} = v$$" |
| 138 | + ] |
| 139 | + }, |
| 140 | + { |
| 141 | + "attachments": {}, |
| 142 | + "cell_type": "markdown", |
| 143 | + "metadata": {}, |
| 144 | + "source": [ |
| 145 | + "Note: For the above equation x is position not state. Combining these equations to the model $\\frac{dx}{dt}$ equation defined earlier yields the A and E matrix defined below. Note that there is no B defined because this model does not have an inputs." |
| 146 | + ] |
| 147 | + }, |
| 148 | + { |
| 149 | + "cell_type": "code", |
| 150 | + "execution_count": null, |
| 151 | + "metadata": {}, |
| 152 | + "outputs": [], |
| 153 | + "source": [ |
| 154 | + "class ThrownObject(LinearModel):\n", |
| 155 | + " events = ['impact']\n", |
| 156 | + " inputs = [] \n", |
| 157 | + " states = ['x', 'v']\n", |
| 158 | + " outputs = ['x']\n", |
| 159 | + " \n", |
| 160 | + " A = np.array([[0, 1], [0, 0]])\n", |
| 161 | + " C = np.array([[1, 0]])\n", |
| 162 | + " E = np.array([[0], [-9.81]])\n", |
| 163 | + " F = None" |
| 164 | + ] |
| 165 | + }, |
| 166 | + { |
| 167 | + "attachments": {}, |
| 168 | + "cell_type": "markdown", |
| 169 | + "metadata": {}, |
| 170 | + "source": [ |
| 171 | + "Note that we defined our `A`, `C`, `E`, and `F` values to fit the dimensions that were stated at the beginning of the notebook! Since the parameter `F` is not optional, we have to explicitly set the value as __None__.\n", |
| 172 | + "\n", |
| 173 | + "Next, we'll define some default parameters for our ThrownObject model." |
| 174 | + ] |
| 175 | + }, |
| 176 | + { |
| 177 | + "cell_type": "code", |
| 178 | + "execution_count": null, |
| 179 | + "metadata": {}, |
| 180 | + "outputs": [], |
| 181 | + "source": [ |
| 182 | + "class ThrownObject(ThrownObject): # Continue the ThrownObject class\n", |
| 183 | + " default_parameters = {\n", |
| 184 | + " 'thrower_height': 1.83,\n", |
| 185 | + " 'throwing_speed': 40,\n", |
| 186 | + " }" |
| 187 | + ] |
| 188 | + }, |
| 189 | + { |
| 190 | + "attachments": {}, |
| 191 | + "cell_type": "markdown", |
| 192 | + "metadata": {}, |
| 193 | + "source": [ |
| 194 | + "In the following cells, we'll define some class functions necessary to perform prognostics on the model." |
| 195 | + ] |
| 196 | + }, |
| 197 | + { |
| 198 | + "attachments": {}, |
| 199 | + "cell_type": "markdown", |
| 200 | + "metadata": {}, |
| 201 | + "source": [ |
| 202 | + "The `initialize()` function sets the initial system state. Since we have defined the `x`and `v` values for our ThrownObject model to represent position and velocity in space, our initial values would be the thrower_height, and throwing_speed parameters, respectively." |
| 203 | + ] |
| 204 | + }, |
| 205 | + { |
| 206 | + "cell_type": "code", |
| 207 | + "execution_count": null, |
| 208 | + "metadata": {}, |
| 209 | + "outputs": [], |
| 210 | + "source": [ |
| 211 | + "class ThrownObject(ThrownObject):\n", |
| 212 | + " def initialize(self, u=None, z=None):\n", |
| 213 | + " return self.StateContainer({\n", |
| 214 | + " 'x': self.parameters['thrower_height'],\n", |
| 215 | + " 'v': self.parameters['throwing_speed']\n", |
| 216 | + " })" |
| 217 | + ] |
| 218 | + }, |
| 219 | + { |
| 220 | + "attachments": {}, |
| 221 | + "cell_type": "markdown", |
| 222 | + "metadata": {}, |
| 223 | + "source": [ |
| 224 | + "For our `threshold_met()`, we define the function to return True for event 'falling' when our thrown object model has a velocity value of less than 0 (object is 'falling') and for event 'impact' when our thrown object has a distance from of the ground of less than or equal to 0 (object is on the ground, or has made 'impact').\n", |
| 225 | + "\n", |
| 226 | + "`threshold_met()` returns a _dict_ of values, if each entry of the _dict_ is __True__, then our threshold has been met!" |
| 227 | + ] |
| 228 | + }, |
| 229 | + { |
| 230 | + "cell_type": "code", |
| 231 | + "execution_count": null, |
| 232 | + "metadata": {}, |
| 233 | + "outputs": [], |
| 234 | + "source": [ |
| 235 | + "class ThrownObject(ThrownObject):\n", |
| 236 | + " def threshold_met(self, x):\n", |
| 237 | + " return {\n", |
| 238 | + " 'falling': x['v'] < 0,\n", |
| 239 | + " 'impact': x['x'] <= 0\n", |
| 240 | + " }" |
| 241 | + ] |
| 242 | + }, |
| 243 | + { |
| 244 | + "attachments": {}, |
| 245 | + "cell_type": "markdown", |
| 246 | + "metadata": {}, |
| 247 | + "source": [ |
| 248 | + "Finally, for our `event_state()`, we will calculate the measurement of progress towards the events. We normalize our values such that they are in the range of 0 to 1, where 0 means the event has occurred." |
| 249 | + ] |
| 250 | + }, |
| 251 | + { |
| 252 | + "cell_type": "code", |
| 253 | + "execution_count": null, |
| 254 | + "metadata": {}, |
| 255 | + "outputs": [], |
| 256 | + "source": [ |
| 257 | + "class ThrownObject(ThrownObject):\n", |
| 258 | + " def event_state(self, x): \n", |
| 259 | + " x_max = x['x'] + np.square(x['v'])/(9.81*2)\n", |
| 260 | + " return {\n", |
| 261 | + " 'falling': np.maximum(x['v']/self.parameters['throwing_speed'],0),\n", |
| 262 | + " 'impact': np.maximum(x['x']/x_max,0) if x['v'] < 0 else 1\n", |
| 263 | + " }" |
| 264 | + ] |
| 265 | + }, |
| 266 | + { |
| 267 | + "attachments": {}, |
| 268 | + "cell_type": "markdown", |
| 269 | + "metadata": {}, |
| 270 | + "source": [ |
| 271 | + "With these functions created, we can now run our ThrownObject Model!\n", |
| 272 | + "\n", |
| 273 | + "In this example, we will initialize our ThrownObject as `m`, and we'll use the `simulate_to_threshold()` function to simulate the movement of the thrown object in air. For more information, see the [Simulation](https://nasa.github.io/progpy/prog_models_guide.html#simulation) documentation." |
| 274 | + ] |
| 275 | + }, |
| 276 | + { |
| 277 | + "cell_type": "code", |
| 278 | + "execution_count": null, |
| 279 | + "metadata": {}, |
| 280 | + "outputs": [], |
| 281 | + "source": [ |
| 282 | + "m = ThrownObject()\n", |
| 283 | + "save = m.simulate_to_threshold(print = True, save_freq=1, threshold_keys='impact', dt=0.1)" |
| 284 | + ] |
| 285 | + }, |
| 286 | + { |
| 287 | + "attachments": {}, |
| 288 | + "cell_type": "markdown", |
| 289 | + "metadata": {}, |
| 290 | + "source": [ |
| 291 | + "__Note__: Because our model takes in no inputs, we have no need to actually define a future loading function! As a result, we are simply passing in an empty Input Container. However, for most models, there would be inputs, thus a need for a future loading function. For more information on future loading functions and when to use them, please refer to the ProgPy [Future Loading](https://nasa.github.io/progpy/prog_models_guide.html#future-loading) Documentation." |
| 292 | + ] |
| 293 | + }, |
| 294 | + { |
| 295 | + "attachments": {}, |
| 296 | + "cell_type": "markdown", |
| 297 | + "metadata": {}, |
| 298 | + "source": [ |
| 299 | + "We'll also demonstrate how this looks plotted on a graph." |
| 300 | + ] |
| 301 | + }, |
| 302 | + { |
| 303 | + "cell_type": "code", |
| 304 | + "execution_count": null, |
| 305 | + "metadata": {}, |
| 306 | + "outputs": [], |
| 307 | + "source": [ |
| 308 | + "import matplotlib.pyplot as plt\n", |
| 309 | + "save.outputs.plot(title='generated model')\n", |
| 310 | + "plt.show()" |
| 311 | + ] |
| 312 | + }, |
| 313 | + { |
| 314 | + "attachments": {}, |
| 315 | + "cell_type": "markdown", |
| 316 | + "metadata": {}, |
| 317 | + "source": [ |
| 318 | + "Notice that that plot resembles a parabola, which represents the position of the ball through space as time progresses!" |
| 319 | + ] |
| 320 | + }, |
| 321 | + { |
| 322 | + "attachments": {}, |
| 323 | + "cell_type": "markdown", |
| 324 | + "metadata": {}, |
| 325 | + "source": [ |
| 326 | + "#### Conclusion\n", |
| 327 | + "\n", |
| 328 | + "In this example, we will initialize our ThrownObject as `m` and use the `simulate_to_threshold()` function to simulate the movement of the thrown object in air. For more information, see the [Linear Model](https://nasa.github.io/progpy/api_ref/prog_models/LinearModel.html) Documentation." |
| 329 | + ] |
| 330 | + }, |
| 331 | + { |
| 332 | + "cell_type": "markdown", |
| 333 | + "metadata": {}, |
| 334 | + "source": [] |
| 335 | + } |
| 336 | + ], |
| 337 | + "metadata": { |
| 338 | + "kernelspec": { |
| 339 | + "display_name": "Python 3.11.0 64-bit", |
| 340 | + "language": "python", |
| 341 | + "name": "python3" |
| 342 | + }, |
| 343 | + "language_info": { |
| 344 | + "codemirror_mode": { |
| 345 | + "name": "ipython", |
| 346 | + "version": 3 |
| 347 | + }, |
| 348 | + "file_extension": ".py", |
| 349 | + "mimetype": "text/x-python", |
| 350 | + "name": "python", |
| 351 | + "nbconvert_exporter": "python", |
| 352 | + "pygments_lexer": "ipython3", |
| 353 | + "version": "3.11.0" |
| 354 | + }, |
| 355 | + "orig_nbformat": 4, |
| 356 | + "vscode": { |
| 357 | + "interpreter": { |
| 358 | + "hash": "aee8b7b246df8f9039afb4144a1f6fd8d2ca17a180786b69acc140d282b71a49" |
| 359 | + } |
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| 362 | + "nbformat": 4, |
| 363 | + "nbformat_minor": 2 |
| 364 | +} |
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