This paper reports a new extended Kalman filter where the underlying nonlinear functions are linearized using a Gaussian orthogonal basis of a weighted $\mathcal {L}_{2}$ space. As we are interested in computing the states’ mean and covariance with respect to Gaussian measure, it would be better to use a linearization, that is optimal with respect to the same measure. The resulting first-order polynomial coefficients are approximately calculated by evaluating the integrals using (i) third-order Taylor series expansion (ii) cubature rule of integration. Compared to direct integration-based filters, the proposed filter is far less susceptible to the accumulation of round-off errors leading to loss of positive definiteness. The proposed algorithms are applied to four nonlinear state estimation problems. We show that our proposed filter consistently outperforms the traditional extended Kalman filter and achieves a competitive accuracy to an integration-based square root filter, at a significantly reduced computing cost.

In this manuscript, an underwater target tracking problem with passive sensors is considered. The measurements used to track the target trajectories are (i) only bearing angles, and (ii) Doppler-shifted frequencies and bearing angles. Measurement noise is assumed to follow a zero mean Gaussian probability density function with unknown noise covariance. A method is developed which can estimate the position and velocity of the target along with the unknown measurement noise covariance at each time step. The proposed estimator linearises the nonlinear measurement using an orthogonal polynomial of first order, and the coefficients of the polynomial are evaluated using numerical integration. The unknown sensor noise covariance is estimated online from residual measurements. Compared to available adaptive sigma point filters, it is free from the Cholesky decomposition error. The developed method is applied to two underwater tracking scenarios which consider a nearly constant velocity target. The filter’s efficacy is evaluated using (i) root mean square error (RMSE), (ii) percentage of track loss, (iii) normalised (state) estimation error squared (NEES), (iv) bias norm, and (v) floating point operations (flops) count. From the simulation results, it is observed that the proposed method tracks the target in both scenarios, even for the unknown and time-varying measurement noise covariance case. Furthermore, the tracking accuracy increases with the incorporation of Doppler frequency measurements. The performance of the proposed method is comparable to the adaptive deterministic support point filters, with the advantage of a considerably reduced flops requirement.

In this paper, an approximate Gaussian state estimator is developed based on generalised polynomial chaos expansion for target tracking applications. Motivated by the fact that calculating conditional moments in an approximate Gaussian filter involves computing integrals with respect to Gaussian density, the authors approximate the non‐linear dynamics using polynomial chaos expansion. Second‐order as well as third‐order polynomial chaos expansions were used for approximate filtering, to derive the necessary recursive algorithm and also provide certain algebraic simplifications which reduce the computational burden without significantly affecting the filtering performance. Two comprehensive numerical experiments for multivariate systems, including one for a multi‐model system, demonstrate the potential of the new algorithms.

This paper deals with a remote state estimation problem for a nonlinear system. In a typical networked control system (NCS) scenario, the estimator and controller are remotely located, and they are connected with the plant through a common communication network. Traditional Bayesian filters assume that the measurements are always available. However, this may not be the case in reality. As the sensor measurements are transmitted to the remotely located estimator through an unreliable communication channel, delay may arise during data transfer. Similarly, the control signal is also applied remotely, and it reaches to the plant through a similar unreliable communication channel, and due to which here also delay may occur. In this paper, the authors develop a generalized framework of nonlinear filtering where the states can be estimated in the presence of arbitrary random delay in (i) transmission of measurement from sensor to the estimator and (ii) transmission of input from the remotely located controller to the system. The filtering algorithm in such a scenario is realized with deterministic sample points. The performance of the proposed method is tested experimentally on one simulation problem. With the help of the simulation result, it is shown that the developed method performs better than traditional non-delayed nonlinear filters in the presence of arbitrary delay in measurement and input.