Kalman Filter: How to Define Inputs and Outputs of a Model

Question

I'm a software engineer with a CS degree working in machine learning. I'm trying to learn about Kalman Filters.

In this short YouTube video from Mathworks, there's a discussion on a Kalman Filter with regard to a rocket:

• We want to measure a rocket engine's internal temperature $T_{in}$.
• We can't put a temperature sensor directly inside the exhaust port because it's too hot.
• Instead, we put an external temperature sensor outside the engine exhaust to measure $T_{ext}$.
• We know how much fuel we're using: $W_{fuel}$.

The video says that we want to reduce the error between the measured external temperature $T_{ext}$ and an estimate of the external temperature $\hat{T}_{ext}$. In turn, that will reduce the error between the unobserved temperature $T_{in}$ and its estimate $\hat{T}_{in}$.

I work in machine learning, so I'm confused by what are the inputs and outputs of this system. My specific questions are:

1. $T_{ext}$ is an input to the system using an external sensor. Why is it shown as an output of this system?

2. We are trying to predict $\hat{T}_{in}$. Presumably it is a function of $W_{fuel}$ and $T_{ext}$; that is, $\hat{T}_{in} = f(W_{fuel}, T_{ext})$. Why is this system even trying to predict the external temperature $\hat{T}_{ext}$ if the true external temperature $T_{ext}$ is measured?

3. If I were to solve this problem using machine learning, I'd implement a regression model to predict exactly $\hat{T}_{in} = f(W_{fuel}, T_{ext})$ with $W_{fuel}$ and $T_{ext}$ as input features to linear regression or a neural network model. Why do we need to set up this Kalman Filter system at all?

Thanks for any help.

Answer 1

The Kalman Filter is basically a framework to fuse 2 things:

1. Measurement.
2. Dynamic Model (Dynamic in the sense we can predict next value from a current value).

In your case the model is composed of two things:

1. Model which connects $ {T}_{ext} $ and $ {W}_{fuel} $ to $ {T}_{in} $.
2. Measurement of $ {T}_{ext} $.

Now, the reason you estimate $ {T}_{ext} $ is because no sensor is perfect. The model of a sensor is $ {T}_{ext} \left[ k \right] = {T}_{ext - gt} \left[ k \right] + v \left[ k \right] $. Namely we get a measurement with added noise.

What's missing in the model of the video is the actual model. I have no experience in the field of heat transfer but let's assume the model is something like:

$$ {T}_{in} \left[ k \right] = {T}_{in} \left[ k - 1 \right] + \alpha {w} \left[ k - 1 \right] $$

And:

$$ {T}_{ext} \left[ k \right] = \beta {T}_{in} \left[ k \right] $$

Then you can build the model for the Kalman Filter and it will fuse the knowledge about $ {T}_{in} $ from the model which relates to $ {T}_{out} $ and the model which given the $ {T}_{in} $ of the previous iteration how it should be in the next.

Regarding your experience with Machine Learning.
Kalman Filter assumes a Bayesian Model. Hence it is not a parameterized.
You could estimate $ {T}_{in} $ from measurements of $ {T}_{ext} $ and $ W $ but then you'll miss the prior you're given: The dynamic model.