Skip to content

Commit

Permalink
04 - Linear model
Browse files Browse the repository at this point in the history
  • Loading branch information
@teubert
teubert committed Aug 28, 2023
1 parent a904490 commit 51d0541
Showing 1 changed file with 83 additions and 78 deletions.
161 changes: 83 additions & 78 deletions examples/linear_model.ipynb → examples/04_New Models.ipynb
View file
Original file line number Diff line number Diff line change
@@ -1,47 +1,51 @@
{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"# Welcome to ProgPy's Linear Model Example"
"# 4. Defining new Physics-Based Prognostic Models"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"The goal of this notebook is to instruct users on how to use ProgPy Model LinearModel.\n",
"\n",
"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",
"All of the past sections describe how to use an existing model. In this section we will describe how to create a new model. This section specifically describes creating a new physics-based model, for training and creating data-driven models see 5. Data-driven Models."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Linear Models"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The easiest model to build is a linear model. Linear models are defined as a linear time series, which can be defined by the following equations:\n",
"\n",
"\n",
"\n",
"#### _<b>The State Equation<b>_:\n",
"**The State Equation**:\n",
"$$\n",
"\\frac{dx}{dt} = Ax + Bu + E\n",
"$$\n",
"\n",
"#### _<b>The Output Equation<b>_:\n",
"**The Output Equation**:\n",
"$$\n",
"z = Cx + D\n",
"$$\n",
"\n",
"#### _<b>The Event State Equation<b>_:\n",
"**The Event State Equation**:\n",
"$$\n",
"es = Fx + G\n",
"$$\n",
"\n",
"$x$ is `state`, $u$ is `input`, $z$ is `output`, and $es$ is `event state`"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"$x$ is `state`, $u$ is `input`, $z$ is `output`, and $es$ is `event state`\n",
"\n",
"Linear Models are defined by creating a new model class that inherits from progpy's LinearModel class and defines the following properties:\n",
"* $A$: 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",
"* $B$: 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",
Expand All @@ -57,32 +61,29 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"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",
"\n",
"First, some definitions for our Model:\n",
"\n",
"First, some definitions for our Model!\n",
"\n",
"#### __Events__: (2)\n",
"**Events**: (2)\n",
"* `falling: The object is falling`\n",
"* `impact: The object has hit the ground`\n",
"\n",
"#### __Inputs/Loading__: (0)\n",
"**Inputs/Loading**: (0)\n",
"* `None`\n",
"\n",
"#### __States__: (2)\n",
"**States**: (2)\n",
"* `x: Position in space (m)`\n",
"* `v: Velocity in space (m/s)`\n",
"\n",
"#### __Outputs/Measurements__: (1)\n",
"**Outputs/Measurements**: (1)\n",
"* `x: Position in space (m)`"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand All @@ -95,7 +96,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand All @@ -115,34 +115,21 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"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."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"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.\n",
"\n",
"First, let's consider state transition. For an object thrown into the air without air resistance, velocity would decrease literally by __-9.81__ \n",
"$\\dfrac{m}{s^2}$ due to the effect of gravity, as described below:\n",
"\n",
" $$\\frac{dv}{dt} = -9.81$$\n",
"\n",
" Position change is defined by velocity (v), as described below:\n",
" \n",
" $$\\frac{dx}{dt} = v$$"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"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."
" $$\\frac{dx}{dt} = v$$\n",
"\n",
" 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."
]
},
{
Expand All @@ -164,7 +151,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand All @@ -187,18 +173,11 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following cells, we'll define some class functions necessary to perform prognostics on the model."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following cells, we'll define some class functions necessary to perform prognostics on the model.\n",
"\n",
"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."
]
},
Expand All @@ -217,7 +196,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand All @@ -241,7 +219,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand All @@ -264,13 +241,10 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"With these functions created, we can now run our ThrownObject Model!\n",
"\n",
"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."
"With these functions created, we can now use the `simulate_to_threshold()` function to simulate the movement of the thrown object in air. For more information, see 1. Simulation."
]
},
{
Expand All @@ -280,58 +254,89 @@
"outputs": [],
"source": [
"m = ThrownObject()\n",
"save = m.simulate_to_threshold(print = True, save_freq=1, threshold_keys='impact', dt=0.1)"
"save = m.simulate_to_threshold(print=True, save_freq=1, threshold_keys='impact', dt=0.1)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"__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."
"__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 future loading section in 1. Simulation.\n",
"\n",
"Let's late a look at the outputs of this model"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"fig = save.outputs.plot(title='generated model')"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"We'll also demonstrate how this looks plotted on a graph."
"Notice that that plot resembles a parabola, which represents the position of the ball through space as time progresses!"
]
},
{
"cell_type": "code",
"execution_count": null,
"cell_type": "markdown",
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"save.outputs.plot(title='generated model')\n",
"plt.show()"
"For more information on Linear Models, see the [Linear Model](https://nasa.github.io/progpy/api_ref/prog_models/LinearModel.html) Documentation."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice that that plot resembles a parabola, which represents the position of the ball through space as time progresses!"
"## New State Tranisition Models"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Conclusion\n",
"\n",
"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."
"## Direct Models"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Derived Parameters"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": []
"source": [
"## Matrix Models"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## State Limits"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Custom Events"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Conclusions"
]
}
],
"metadata": {
Expand Down

0 comments on commit 51d0541

Please sign in to comment.