Gaussian Filtering for Simultaneously Occurring Delayed and Missing Measurements

Abstract

Approximate filtering algorithms in nonlinear systems assume Gaussian prior and predictive density and remain popular due to ease of implementation as well as acceptable performance. However, these algorithms are restricted by two major assumptions: they assume no missing or delayed measurements. However, practical measurements are frequently delayed and intermittently missing. In this paper, we introduce a new extension of the Gaussian filtering to handle the simultaneous occurrence of the delay in measurements and intermittently missing measurements. Our proposed algorithm uses a novel modified measurement model to incorporate the possibility of the delayed and intermittently missing measurements. Subsequently, it redesigns the traditional Gaussian filtering for the modified measurement model. Our algorithm is a generalized extension of the Gaussian filtering, which applies to any of the traditional Gaussian filters, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), and cubature Kalman filter (CKF). A further contribution of this paper is that we study the stochastic stability of the proposed method for its EKF-based formulation. We compared the performance of the proposed filtering method with the traditional Gaussian filtering (particularly the CKF) and three extensions of the traditional Gaussian filtering that are designed to handle the delayed and missing measurements individually or simultaneously.INDEX TERMS Delayed measurements, Gaussian filtering, missing measurements, nonlinear Bayesian filtering.

Improved Gaussian filtering for handling concurrent delayed and missing measurements

“…This section presents a comprehensive analysis of proposed and existing delays and missing filters (handling the concerning irregularities individually or simultaneously). In this regard, we compare the simulation results of the GFJDM with state-of-the-art delay and missing filters abbreviated as CQKF_RD [19], CQKF_GRD [16], MDCQKF [26], MLCQKF [14], and EKF_M [12]. Please note that we simulated the results by implementing the CQKF-based extensions (using third-order spherical and three-degree radial rules) of the proposed method and Gaussian filtering techniques reported in [14,16,19,26].…”

“…with 0 d further representing an array of d zeroes. It should be stressed that the existing delay works (e.g., [14,19,26]) rely on a large number of Bernoulli random variables to model the measurement delay, prompting the imperative knowledge on a range of delay probabilities. We characterize the delay extent by a single geometric random variable to relax these constraints.…”

“…On the other hand, the model used in [14,19] describes larger delays with the help of a sequence of Bernoulli random variables. In [26], a similar approach is adopted for jointly handling the occurrence of delay and missing measurement. However, the resulting filter suffers from poor accuracy due to the (i) need of prior information of large number of delay probabilities, which are difficult to obtain, and (ii) need of ambiguously fixing an upper bound of delay.…”

“…Our method for addressing this problem involves mathematically characterizing irregularities, particularly the delayed and missing measurements. This is done by constructing a stochastic model with fewer prerequisite delay probabilities compared to the recently presented [26].…”

“…Please refer to [26] for formulating the error dynamics in detail. Recalling Remark 5, the error dynamics (20) should satisfy the conditions presented in Equation (15) for the EKFJDM to be exponentially bounded in the mean square.…”