Discrete-time piecewise linear filtering with small observation noise

A problem of discrete-time piecewise linear filtering with small observation noise is considered. The case in which the observation function is not one-to-one and has two intervals of linearity is studied, in particular when this function is symmetric. Under some detectability assumption, a procedure is presented to detect the intervals of linearity of the observation function. During such time intervals, one can then approximate the optimal filter by the corresponding Kalman-Bucy filter.