Optimal structurally partitioned filter for undisturbable stochastic systems Part II. Stability and its asymptotic behaviour

This paper investigates the stability of an optimal structurally partitioned filter and its asymptotic behaviour in undisturbable stochastic systems. In fact, these filters cannot assure the exponential asymptotic stability and therefore cannot proceed directly to the steady state problem because they include a fixed point smoother as one estimation mechanism in the parallel processors. It is shown, howover, that these types of estimator are asymptotically stable in the Lyapunov sense by introducing a generalized disturbability (or controllability) matrix (Anderson 1971). It is also shown that after analysing the asymptotic behaviour of such filters, a steady state partitioned filter, which takes the gains as the practical constant, is able to generate the unbiased estimate.