Unscented Filtering Algorithm for Discrete-Time Systems with Uncertain Observations and State-Dependent Noise

Abstract

Most conventional filtering algorithms address situations in which the signal to be estimated is always present in the observations. However, in many real situations, usually the measurement device or the transmission of such measurements can be subject to random failures, generating observations which may consist of noise only. More specifically, there is a positive probability (false alarm probability) that the signal to be estimated is not present in the corresponding observation; that is, the observations may be only noise (uncertain observations). Since it is not generally knownwhether the observation used for estimation contains the signal or it is only noise, and only the probabilities of occurrence of such cases are available to the estimation, the observation equation is designed by including a random multiplicative noise described by a sequence of Bernoulli random variables, whose values one or zero indicate the presence or absence of the signal in the observations, respectively. The least-squares optimal estimation problem in systems with uncertain observations is not easily treatable in general, due to the fact that the multiplicative noise perturbing the observations causes that the joint distribution of the signal and the observations is not gaussian (even if the signal and additive noises are gaussian processes). For this reason, the research on the estimation problem in these systems has been focused on the search of suboptimal estimators for the signal that can be easily derived. Nahi (1969) was the first who described this observation model and analyzed the linear least-squares estimation problem in linear systems with independent uncertainty. After that, numerous studies have been developed in this context, assuming different hypotheses on the Bernoulli random variables modelling the uncertainty when the state-space model is known and, also, when only covariance information is available (see Nakamori et al. (2005) and references therein). On the other hand, there are many practical applications in communication theory (phase modulation of analog communication systems, object tracking in video sequences, robot navigation, location tracking, navigation sensors, etc.) where the observations are not linear function of the signal to be estimated. Although the estimation problem in discrete-time systems from uncertain observations has been extensively studied in linear systems, the literature on nonlinear filtering with uncertainty, which is the focus of this chapter, is fairly Unscented Filtering Algorithm for Discrete-Time Systems with Uncertain Observations and State-Dependent Noise