The direction of arrival (DOA) tracking problem based on an angle sensor is an important topic in many fields. In this paper, a nonlinear filter named the feedback M-estimation based robust cubature Kalman filter (FMR-CKF) is proposed to deal with measurement outliers from the angle sensor. The filter designs a new equivalent weight function with the Mahalanobis distance to combine the cubature Kalman filter (CKF) with the M-estimation method. Moreover, by embedding a feedback strategy which consists of a splitting and merging procedure, the proper sub-filter (the standard CKF or the robust CKF) can be chosen in each time index. Hence, the probability of the outliers’ misjudgment can be reduced. Numerical experiments show that the FMR-CKF performs better than the CKF and conventional robust filters in terms of accuracy and robustness with good computational efficiency. Additionally, the filter can be extended to the nonlinear applications using other types of sensors.

In multi-target tracking, the outliers-corrupted process and measurement noises can reduce the performance of the probability hypothesis density (PHD) filter severely. To solve the problem, this paper proposed a novel PHD filter, called Student’s t mixture PHD (STM-PHD) filter. The proposed filter models the heavy-tailed process noise and measurement noise as a Student’s t distribution as well as approximates the multi-target intensity as a mixture of Student’s t components to be propagated in time. Then, a closed PHD recursion is obtained based on Student’s t approximation. Our approach can make full use of the heavy-tailed characteristic of a Student’s t distribution to handle the situations with heavy-tailed process and the measurement noises. The simulation results verify that the proposed filter can overcome the negative effect generated by outliers and maintain a good tracking accuracy in the simultaneous presence of process and measurement outliers.

For states estimation problem of continuous–discrete systems, the numerical approximation methods with high order of accuracy are commonly used to build the continuous–discrete filtering algorithms. However, there is a common contradiction between the computational efficiency and the accuracy. In order to improve the efficiency of state estimation in the continuous–discrete filtering method, continuous–discrete extended cubature Kalman filtering based on Adams–Bashforth–Moulton (ABM) numerical approximation is proposed. ABM is the linear multi-step numerical method, which can achieve the fourth-order accuracy for solving the differential state equation, and its ‘predictor–corrector’ mathematic structure is relatively simple. The performances of ABM method are theoretically analysed; the mixed-type filtering method for continuous–discrete nonlinear states estimation is proposed to integrate the best features of extended Kalman filtering and cubature Kalman filtering. More precisely, the time updates are deduced by extended Kalman filtering whereas the measurement updates are conducted by the third-degree spherical-radial cubature rule. The superior performances of proposed method are illustrated in the simulations under the conditions of different step-sizes and sampling periods.

Bearings-only tracking only adopts measurements from angle sensors to realize target tracking, thus, the accuracy of the state prediction has a significant influence on the final results of filtering. There exist unpredictable approximation errors in the process of filtering due to state propagation, discretization, linearization or other adverse effects. The idea of online covariance adaption is proposed in this work, where the post covariance information is proved to be effective for the covariance adaption. With theoretical deduction, the relationship between the posterior covariance and the priori covariance is investigated; the priori covariance is modified online based on the feedback rule of covariance updating. The general framework integrates the continuous-discrete cubature Kalman filtering and the feedback rule of covariance updating. Numerical results illustrated that the proposed method has advantages over decreasing unpredictable errors and improving the computational accuracy and efficiency.

The model of bearings-only tracking is generally described by discrete–discrete filtering systems. Discrete robust methods are also frequently used to address measurement uncertainty problems in bearings-only tracking. The recently popular continuous–discrete filtering system considers the state model of the target to be continuous in time, and is more suitable for bearings-only tracking because of its higher mathematical solution accuracy. However, the sufficient evaluation of robust methods in continuous–discrete systems is not available. In addition, in the different continuous–discrete measurement environments, the choice of a robust algorithm also needs to be discussed. To fill this gap, this paper firstly establishes the continuous–discrete target tracking model, and then evaluates the performance of proposed robust square-root continuous–discrete cubature Kalman filter algorithms in the measurement of uncertainty problems. From the simulation results, the robust square-root continuous–discrete maximum correntropy cubature Kalman filter algorithm and the variational Bayesian square-root continuous–discrete cubature Kalman filter algorithm have better environmental adaptability, which provides a promising means for solving continuous–discrete robust problems.