# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
###### Abstract  
For the bearing-only target motion analysis (TMA), the pseudolinear Kalman filter (PLKF) solves the complex nonlinear estimation of the motion model parameters but suffers serious bias problems. The pseudolinear Kalman filter under the minimum mean square error framework (PL-MMSE) has a more accurate tracking ability and higher stability compared to the PLKF. Since the bearing signals are corrupted by non-Gaussian noise in practice, we reconstruct the PL-MMSE under Gaussian mixture noise. If some prior information, such as state constraints, is available, the performance of the PL-MMSE can be further improved by incorporating state constraints in the filtering process. In this paper, the mean square and estimation projection methods are used to incorporate PL-MMSE with linear constraints, respectively. Then, the linear approximation and second-order approximation methods are applied to merge PL-MMSE with nonlinear constraints, respectively. Simulation results demonstrate that the constrained PL-MMSE algorithms result in lower mean square errors and bias norms, which demonstrates the superiority of the constrained algorithms. b 2022 2022 2022 2022 2  
filter (EKF). [PERSON] et al. [12; 13] proposes the method of the unscented Kalman filter. The particle filter [14; 15; 16] (PF) is also used for the bearing-only target motion analysis. [PERSON] et al. [17] proposes an initial value optimization method for inverse smoothing filtering. This method effectively solves the problem that Kalman filtering methods are sensitive to initial value selection and reduces the estimation error. The third category is to linearize the nonlinear angle measurement equation by using the pseudolinear estimator (PLE) method [18]. The pseudolinear Kalman filter (PLKF) [19; 20] is produced by combining the Kalman filter with the pseudolinear estimator method. Compared with other filtering methods, the main advantages of PLKF are high stability, good tracking performance, and small initial error under lower computational complexity [21]. However, the PLKF has a large bias due to the correlation between the measurement matrix and the pseudo-linear noise variable. Hence, several methods have been proposed to improve the performance of PLKF by compensating or reducing the pseudolinear estimation bias, including the modified pseudolinear estimator (MPLE) [22], bias-compensated PLKF (BC-PLKF) [23], IV Kalman filter (IVKF) [24] and IVKF based on the selective-angle-measurement (SAM-IVKF) [25] strategy. These variants of PLKF based on bias compensation are not always perfect when the measurement noise is large and the geometry is unfavorable. Based on the PLKF, [PERSON] et al. [26] proposes a new pseudolinear filter under the minimum mean square error (PL-MMSE) framework without offset compensation, which shows better tracking performance under the large measurement noise than the above algorithms. If the prior information, such as linear or nonlinear constraints on motion state, is available, we can take these conditions into consideration to improve the state estimation [27]. For example, tracking the vehicle driven on a straight or curved road is a constrained state estimation problem with the available road information [28]. Similar models also appear in other engineering applications, including the compartmental models method [29], turbofan engine health estimation [30] and so on. To estimate the states in such systems, some methods have been proposed, e.g., the model parameters reduction method [31], perfect measurements approach [32], estimation projection [33], linear approximation [33], second-order approximation [34; 35]. For linear constraints, the model parameters' reduction method [31] transforms the constrained state estimation to the unconstrained state estimation. However, the reduction of the state constrained equations makes the interpretation such as the physical meaning of the states more difficult. The perfect measurements approach [32] adds state equality constraints into the measurement equation.