control//state estimation//Kalman filter//estimate covariance

\(P\) is not measured, it is propagated. The **estimate covariance** \(P\) is how uncertain the filter is about its own estimate, the variance of the estimation error \(\hat x-x\). That error can never be computed in operation, because computing it needs the truth \(x\), which is exactly what nobody has. So the filter does not recalculate \(P\) by watching its errors: it carries it forward mathematically, tick after tick, and only needs someone to hand it the first value (initial covariance).

PPP is not measured, it is propagated. The estimate covariance PPP is how uncertain the filter is about its own estimate, the variance of the estimation error x^−x\hat x-xx^−x. That error can never be computed in operation, because computing it needs the truth xxx, which is exactly what nobody has. So the filter does not recalculate PPP by watching its errors: it carries it forward mathematically, tick after tick, and only needs someone to hand it the first value (initial covariance).

Two moves change it. Predicting inflates it, P−=FPFT+QP^-=FPF^{\mathsf T}+QP−=FPFT+Q: every step taken without measuring adds the model's doubt (process noise). Correcting deflates it, P=(1−K)P−P=(1-K)P^-P=(1−K)P−, and the sensor's noise is hidden inside KKK. Substitute KKK and watch where RRR appears:

P=(1−K) P−=(1−P−P−+R)P−=RP−+R P−=P−RP−+R.P=(1-K)\,P^-=\Big(1-\frac{P^-}{P^-+R}\Big)P^-=\frac{R}{P^-+R}\,P^-=\frac{P^-R}{P^-+R}.P=(1−K)P−=(1−P−+RP−​)P−=P−+RR​P−=P−+RP−R​.

RRR enters PPP through KKK. Every time the sensor speaks, PPP shrinks, and RRR decides by how much: a precise sensor shrinks it a lot, a noisy one barely (inverse-variance weighting).

RRR and PPP are different things. RRR is a property of the sensor, measured once in a lab (measurement noise); PPP is a property of your belief, and it changes at every step even though RRR does not. The sensor knows nothing about your PPP.

For a vector state PPP is a matrix, and its off-diagonal terms say how the errors of different variables move together (covariance propagation).

?Then the covariances between state variables can be known without a reference?

Yes, as the filter's own bookkeeping: they come out of the propagation, not out of observing errors. Whether that bookkeeping is realistic is a separate question, answered offline, in an experiment with ground truth where the real error can be computed and compared with what PPP claimed, and online by checking that the innovation is as large as the filter expects.