Observability and nonlinear filtering

Abstract

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces

An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

“…See also [44,18,73] and the early review paper [45] for related literature. A (marginally) stable solution of the algebraic Riccati equation exists under detectability conditions; see [53,54,73,62] and [71]. The discussion in [9, Chapter 2 and 3] is also of general interest here; as is [48].…”

“…In [25,19,59] convergence to a (marginally) stable solution was studied under relaxed conditions (with necessity also explored). In [71], given only detectability, the stability of a time-varying "closed loop", e.g. (A − φ t (Q)S), is proven even when the limiting Riccati solution, or (A − P ∞ S), is only marginally stable.…”

When Do Nonlinear Filters Achieve Maximal Accuracy?

“…On the other hand, the results in [20] are easily adapted to show that if the model is reconstructible, then the time-reversed noiseless filter is stable in the sense that…”

“…Our definition 2.4 in the stochastic setting is close to a similar notion that plays an important role in the realization theory of stationary Gaussian processes [18,13]. Reconstructibility is essentially the time reversed counterpart of the notion of observability [20], though as discussed above we must restrict to probability measures µ, ν ≪ π.…”

Discrete time nonlinear filters with informative observations are stable

“…Somewhat surprisingly, assuming only ergodicity of the signal is not sufficient to guarantee stability (see [5, section 5]); both the mixing condition and the condition of Chigansky and Liptser are strictly stronger than ergodicity. Under a mild nondegeneracy assumption on the observations, however, ergodicity of the signal is already sufficient to ensure stability of the filter [15].…”