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.

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Traditionally, Kalman filter (KF) is designed with the assumptions of non-delayed measurements and additive white Gaussian noises. However, practical problems often fail to satisfy these assumptions and the conventional Kalman filter suffers from poor estimation accuracy. This paper proposes a modified Kalman filter to address both the problems of delayed measurements and non-Gaussian noises. The proposed filter is updated using correntropy maximization criterion, which is suitable for non-Gaussian noise environments. It falls short of a closed-form solution due to analytically complex equations that appear during the filtering. We use fixed-point iterative method to find an approximate solution. The delayed measurement problem is addressed by implementing a likelihood-based approach to identify the delay. Based on the identified delay information, the measurement is used to update the desired state in the subsequent past instant. To perform real-time filtering, the estimated state is further updated up to the current time instant using the process dynamics. The performance analysis validates the improved accuracy of the proposed method compared to the ordinary Kalman filter and its existing extensions.

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Particle filtering is probably the most widely accepted methodology for general nonlinear filtering applications. The performance of a particle filter critically depends on the choice of proposal distribution. In this paper, we propose using a wrapped normal distribution as a proposal distribution for angular data, i.e. data within finite range (−π, π]. We then use the same method to derive the proposal density for a particle filter, in place of a standard assumed Gaussian density filter such as the unscented Kalman filter. The numerical integrals with respect to wrapped normal distribution are evaluated using Rogers-Szegő quadrature. Compared to using the unscented filter and similar approximate Gaussian filters to produce proposal densities, we show through examples that wrapped normal distribution gives a far better filtering performance when working with angular data. In addition, we demonstrate the trade-off involved in particle filters with local sampling and global sampling (i.e. by running a bank of approximate Gaussian filters vs running a single approximate Gaussian filter) with the former yielding a better filtering performance than the latter at the cost of increased computational load. INDEX TERMSNonlinear dynamical systems, Angular data, Particle filtering, Wrapped normal distribution, and Rogers-Szegő quadrature rule.