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Assume that we have a nonlinear dynamical model, e.g called transition function

$$\hat x = f(x, u)$$

And $y = x$ as our observability function.

According to Mathworks

Kalman filters are used to optimally estimate the variables of interests when they can't be measured directly, but an indirect measurement is available. They are also used to find the best estimate of states by combining measurements from various sensors in the presence of noise.

In this case, practical case, the observability function is often an identity matrix because states are assumed as they were close as measurements.

So my question is:

If Mathworks says that Kalman filters are used when the user need to measure the states, when the output is only known. Then what if the states is the outputs. I'm most cases, the states are the output. Especially in system identification.

So what's is the use case for kalman filter then?

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As stated by the second sentence of the quoted text, in this case a Kalman filter can still be beneficial to minimize the impact of noise. Noises can both act on the state dynamics as the measurements. If there is no noise on the measurements then directly using the measurements will be best. If there is no noise on the dynamics then a Kalman filter will eventually use the measurements less and less and more and more rely on the prediction steps. However, in practice both will act to some degree on actual systems. Though, it might be that the not all assumptions need for a Kalman filter are satisfied, such as that noises are Gaussian zero mean white noise and that dynamics can be described by a linear model. But in many cases those assumptions give a close enough approximation of the actual system and thus Kalman filters at least yield some improvement over just using the output directly in this case.

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  • $\begingroup$ I would also like to note that it might be that the measurement noise might be small compared to the sensor resolution, in which case the linearity assumption is not a good approximation. So in that case one can probably improve on a standard Kalman filter. $\endgroup$ Commented Dec 20, 2021 at 22:11
  • $\begingroup$ Thank you. Good answer. But to find the dynamical system, system identification must be done first. Right? No kalman filter without model. Can it be so that kalman filters does not create a phase delay between the output and trajectory because it's optimal estimated, and not "filtered"? $\endgroup$
    – euraad
    Commented Dec 20, 2021 at 22:45
  • $\begingroup$ @MrYui a Kalman filter would be an online state estimation optimization problem solver. While system identification is normally done offline and would in your case only try to find a mapping from $y_i$ to $y_{i+1}$ for many $i$ at the same time. The effect of the measurement noise should for certain model types average to zero and thus can be omitted. If this isn't the case then instead one could also try to find a mapping between $y_i-e_i$ to $y_{i+1}-e_{i+1}$ and also try to minimize the correlation between $e_i$ and $e_{i+1}$. $\endgroup$ Commented Dec 20, 2021 at 23:51
  • $\begingroup$ Yes I know that. But if I want to have a kalman filter on an object, then I need to find a model of that object first. In this case, by measurement data (system identification). That leds to the question: system identification only identify a transition model dx = f(x,u), not observability model y = h(x). If I assume that the observability model is equal as identity matrix, then the measurement y and the estimated state xhat will be the same, right? Then if you is noisy, the xhat will contain noise too? $\endgroup$
    – euraad
    Commented Dec 21, 2021 at 22:48
  • $\begingroup$ @MrYui system identification should also identify $y=h(x)$. At least I assume that you apply system identification to a set input ($u$) and output ($y$) data? Note that the model is not unique, since one can always find an equivalent model by defining the state differently (also see similarity transformations for linear systems). $\endgroup$ Commented Dec 21, 2021 at 23:34

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