Kalman Filter Decomposition in the Time Domain Using Observability Index

Considering the TKalman–Bucy Filter (KBF) from an engineering point of view it is always important to know in advance, before KBF implementation, which variables are practically "good" and which are "bad" observable and how long it will take to estimate all of them in the presence of measurement noise to some appropriate (not necessarily theoretically optimal) level. This article presents an approach to measuring the observability by a special index that has the physical meaning of signal to nose ratio. This approach leads to the decomposition of the KBF in the time domain into two filters applied consecutively in time: the filter estimating the transitional process caused by the uncertainty in initial conditions and the filter estimating the system steady state. In turn; this results in mitigation of the computational requirements and in a simplification of the filter implementation by the engineers.