On the existence of an optimal observer in singular measurement systems

An essential stage in each of the existing methods for singular optimal estimation is to obtain a nonsingular measurement vector which is a linear combination of the original measurement vector and a finite number of its derivatives. A necessary and sufficient condition for the existence of such a nonsingular measurement is presented. If this condition is satisfied, then the existence of an optimal observer is guaranteed. The new condition is shown to be considerably simpler and of much wider applicability than a recently published condition which is just a sufficient one. The new condition requires just a rank check of a single composite matrix consisting of the system and noise matrices. Unlike the other condition, the existence of an optimal estimator is determined here without carrying out the complex procedures for singular optimal estimation.