The feedback theorem for sub-optimal filters

This paper concerns the filtering problem of stochastic nonlinear systems that depend either on some external input (open-loop system) or on the system output (closed-loop system), through a controller. Such systems are denoted feedback systems. The following result is proven: for feedback systems, the minimum variance estimator belonging to a specific class of functions of the output in the open loop case has better or equal performance than the minimum variance estimator of the same class for the corresponding closed-loop system. Some consequences of this general result are illustrated together with a simple example.