When you have a unit (identity) state transition matrix, all the states evolve independent of each other. In principle they act as individual subsystems without any interactions between them. Hence the only way to estimate the states is by measuring all of them (or having n linearly independent measurements with the combinations of the states). Kalman observability criterion will also lead to the same conclusion. However there exist exceptions if the unobserved states are detectable for a decoupled systems with only diagonal elements (in this case convergence speed cannot be changed by any estimation schemes- for an identity matrix this is not satisfied by the way).

Dear Pranav,

The uncontrollable systems are sometimes desirable for Kalman filtering, but a Kalman filter built around a system with unobservable states will simply not work. And your identity STM matrix makes the system unobservable, i.e. all the states are independent of each other. By definition, an unobservable state is one about which no information may be obtained through the observation equations; in the absence of information, the filter estimate for that state will not converge on a meaningful solution. For more understanding, kindly follow this paper:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.2399&rep=rep1&type=pdf

Also refer the following page for realtime implementation of KF on Python:

http://nbviewer.jupyter.org/github/balzer82/Kalman/blob/master/Kalman-Filter-CA-Ball.ipynb?create=1

Hope, this information will help you !!!

Other comments are welcome in this regard.

In addition, if the system is not observable, the state covariance matrix would be unbounded or greater than a predefined threshold value. It is justified by this paper:

https://pdfs.semanticscholar.org/ea5e/84449d805454d9ff952e2d3ffc622cb6062e.pdf