On Stability of a Class of Filters for Nonlinear Stochastic Systems

Abstract

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Taylor Moment Expansion for Continuous-Discrete Gaussian Filtering

“…It is important and useful that the filter is in some sense stable. Some classical stability results for linear Kalman filters can be found in [8], [21] while more recent results on the stability of different approximate Gaussian (Kalman) filters for non-linear systems have been analysed in [22]- [24]. In this section, we follow [24] and prove that the TME Gaussian filter is stable in the mean-square sense if a number of assumptions on the system and the sigma-point approximation, verifiable before the filter is run, are satisfied.…”

“…An example to satisfy the assumptions is that the drift function f is smooth enough and all of its partial derivatives up to certain orders are uniformly bounded. More practical examples that satisfy Assumption 10 can be found in [24]. It is typically necessary that the measurement model matrix H is a scaled identity matrix and the discretised dynamics a in (27) defines an exponentially stable system.…”

“…It is typically necessary that the measurement model matrix H is a scaled identity matrix and the discretised dynamics a in (27) defines an exponentially stable system. Although these assumptions are quite restrictive, stability results with more general assumptions are currently not yet available in literature [24].…”

“…Because we have assumed that 4λ 2 f < 1, the claim then follows from the discrete Grönwall's inequality (e.g., [24,Theorem IV.2]).…”

On the convergence of the unscented Kalman filter

“…In the stochastic framework, a novel stochastic analysis based on exponential concentration inequalities and uniform χ-square type estimates for stochastic quadratic type systems, has been adapted to complement the above-mentioned techniques. For more details, see [10,26].…”