Why do we need to estimate the true position in Kalman filters?

I am following a probably well-known tutorial about Kalman filter.

From these lines of code:

figure;
plot(t,pos, t,posmeas, t,poshat);
grid;
xlabel('Time (sec)');
ylabel('Position (feet)');
title('Figure 1 - Vehicle Position (True, Measured, and Estimated)')

I under开发者_如何学Cstand that x is the true position, y is measured position, xhat is estimated position. Then, if we can compute x (this code: x = a * x + b * u + ProcessNoise;), why do we need to estimated x anymore?

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OK, after a second look at the referenced article, I think I see the confusion. Apparently, the program in the article is a simulation of a linear system (and thus, it repeatedly generates new x's as new states in the simulated system). Then it also simulates a "noisy" measurement of x, and from that (simulated) noisy measurement, then demonstrates using a Kalman Filter on the noisy data to try to estimate the actual (simulated) x's.

So, the exact x calculations that you ask about are just part of the Simulation, and not part of the Kalman Filter itself or the data that is available to the Kalman Filter algorithm.

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Kalman filter (and stochastic filters in general) do not provide you directly with estimators of the hidden process: they provide you with the conditional law of the hidden process, given observations (the so called filter law) (*).

If you want an estimator of the hidden process, you then have to do it yourself (posterior maximum, posterior expected value). For the Kalman filter, the filter law you compute is gaussian, and you just update its mean value and covariance matrix. You can take the mean value as an estimator of the signal value, and the covariance matrix as an error estimate.

Make sure to make the difference between the filter law (the output of a filter method) and an estimator of the hidden signal.

(*) actually for the Kalman filter, it is the linear conditional law, but if you do the hypothesis of everything being linear, and the noises being gaussian white noise, it is the actual conditional law. Particle filters, to the contrary, approximate the true filter law by discrete measures.