With Kalman filter, you not only get your output filtered, but along with the state of the system at each step as well. No amount of low pass filtering can achieve that.
Can't a simple IIR "low pass" do that for certain systems once they reach a steady state? I don't remember exactly when the KF becomes an IIR but intuitively for certain systems I kinda think the constants would stabilize after a while and then if you just use those you get to optimality if the system stabilizes.
Also, while it’s true the covariance matrix is initialized with diagonals (we often don’t have good priors), the values are being dynamically re-estimated from measurements at every iteration. The initial parameters are just that — initial.
If certain assumptions are true, ie Gaussian noise and LTI model is approximately correct, the covariance matrix converges to a stable covariance matrix that reflects the current reality. Some of those assumptions are relaxed with EKFs and UKFs but they’re essentially built on a Kalman framework (the competing one being MHE).
The Kalman filter is a state estimator, not just a filter. And it is used in industrial applications with advanced control systems (MPC).
The P covariance matrix does not depend on online input though, right? So the initial value of covariance matrix Q and R together with model H and F decides what P will end at. I mean, there is no online parameter estimation (dependent on input). Or am I getting it wrong?
I sure agree to that Kalman filter could be useful. But I don't like how they are presented.
Like the author of the blog writes: "At times its ability to extract accurate information seems almost magical"
Like most things, in practical implementations there’s a bunch of extra things that go beyond the basic theory you have to do like reestimating the initial P every few iterations as well as reestimating Q and R with online data.
It’s not magic and yes it is a form of recursive linear regression but it does work in real life.
If instead of making such posts you would like to learn more then: The P matrix is changed both through the model and the Kalman gain, where in both of these steps it depends on online input (model can depend on the state) and the Kalman gain depends on sensor measurements.
You would intuitively expect that if your two measurements are very far, that you should be less confident in the estimate. A Kalman filter does not do that.
What you can calculate is the likelihood of your data given the estimate, and that is going to be small if your two measurements are far.
The way you make it sound is that a KF would trust a measurement more or less depending on its value. That is simply not true. It does not, e.g., reject outliers.
My point was and still is that thinking about Kalman filters just as a measurement filtering system misses the point. An important value of measurements(especially in the more interesting formulations of KF such as the UKF) is updating the covariance matrices over time which affects the priori distributions of the state over time and thus affects your state estimation.
This is exactly why we use Kalman filters.