On the Time-Varying Riccati Difference Equation of Optimal Filtering

This paper studies the time-varying Riccati difference equation (RDE) for the filtering problem. In particular, existence, stabilizability, and attractiveness properties of the real symmetric solutions that remain bounded on ( - ∞ , + ∞ ) (infinite-time solutions) are investigated. Under the assumption of uniform detectability, conditions for the existence of the maximal and stabilizing solutions are given. Analogous results are worked out for the minimal and antistabilizing solutions by making reference to the uniform antidetectability notion. Moreover, it is shown that, under uniform observability, the set of all symmetric infinite-time solutions constitute an infinite number of lattices with common minimal and maximal elements.