# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
###### Abstract
The method increases the dimension of the state estimation problem and hence increases the computation effort. Estimation projection [33] incorporates the equality constraints into the state estimation frame. It projects the unconstrained state estimation to the constrained surface. For nonlinear constraints, linear approximation [33] uses the Taylor series expansion to the nonlinear state constraints. This method linearizes the nonlinear state constraints by keeping only the first-order terms. Distinct from the linear approximation, second-order approximation [34; 35] keeps both the first-order and second-order terms to maintain the nonlinearity of constraints. In practice, the bearing noise of the sensor is not always Gaussian. For example, the measurement disturbance is described by distribution with impulsive (heavy-tailed) properties in [36]. The performance of standard Kalman filters based on the MMSE framework does not behave well under such noise [37]. To study such heavy-tailed signals, [PERSON]. [38] proposes a suitable method to approximate the heavy-tailed gamma distribution of random telegraph noise by Gaussian mixture distribution. Inspired by [38], the PL-MMSE is extended to estimate the bearings-only target motion model parameters in the presence of Gaussian mixture noise as the first contribution of this paper. This contribution can be deemed as the application of PL-MMSE under heavy-tailed noise with adaptive adjustment of the noise weights. Secondly, we focus on merging the PL-MMSE with constraints by four approaches to address TMA. The mean square method is applied with the PL-MMSE for linear constraints by minimizing the conditional mean square error subject to the state constraints. The estimation projection method is incorporated into the PL-MMSE by pro jecting the unconstrained estimate onto the constrained surface. For nonlinear constraints, the linear approximation method is to linearly approximate the nonlinear constraints by using Taylor series expansion. Then, the estimation projection method follows to equip with the PL-MMSE filter. The second-order approximation method views the nonlinear constraint function as a second-order approximation to the nonlinearity. It constructs an extra optimization step after the PL-MMSE by projecting an unconstrained state estimation onto a nonlinear constrained surface and solves this optimization to realize the estimation. Finally, the PL-MMSE filter with state constraints is tested for TMA on the straight line and the arc section. Experimental results confirm that the behavior of our constrained method is better than other competitors. The rest of this paper is organized as follows. In Section 2, the PL-MMSE under Gaussian mixture noise is designed after the notations are introduced. Sections 3 and 4 combine constrained estimation technologies with the PL-MMSE to derive the PL-MMSE filter with linear and nonlinear state constraints, respectively. Section 5 simulates the two bearings-only TMA examples to show the sound performance of the constrained PL-MMSE algorithms. Section 6 concludes the whole paper and points out future research directions.