Sequential linear filtering with non‐linear position and Doppler measurements for target tracking

IET Radar, Sonar & NavigationEarly View ORIGINAL RESEARCHOpen Access Sequential linear filtering with non-linear position and Doppler measurements for target tracking Ting Cheng, Corresponding Author Ting Cheng citrus@uestc.edu.cn orcid.org/0000-0003-1922-8083 School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, China Correspondence Ting Cheng, School of Information and Communication Engineering, University of Electronic Science and Technology of China, Qingshuihe Campus, No. 2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, Sichuan 611731, China. Email: citrus@uestc.edu.cnSearch for more papers by this authorLifu Li, Lifu Li orcid.org/0000-0001-7370-2423 School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, ChinaSearch for more papers by this author Ting Cheng, Corresponding Author Ting Cheng citrus@uestc.edu.cn orcid.org/0000-0003-1922-8083 School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, China Correspondence Ting Cheng, School of Information and Communication Engineering, University of Electronic Science and Technology of China, Qingshuihe Campus, No. 2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, Sichuan 611731, China. Email: citrus@uestc.edu.cnSearch for more papers by this authorLifu Li, Lifu Li orcid.org/0000-0001-7370-2423 School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, ChinaSearch for more papers by this author First published: 10 December 2021 https://doi.org/10.1049/rsn2.12209AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract For radar target tracking with non-linear measurements, a sequential linear filtering method is proposed in this study, which includes a linear filter based on the position measurements and a linear sequential filter based on the range rate measurement. The linear radar measurement equation in the sequential filter is estimated based on the position filtering result. Furthermore, the feedback information during tracking is adaptively selected according to the quality of the angle measurements. When tracking the manoeuvring target, the proposed sequential linear filtering with adaptive information feedback is combined with the interacting multiple model (IMM) framework, and the IMM filter based on sequential linear filtering with adaptive model probability selection is proposed. Simulation results demonstrate the effectiveness of the proposed algorithms in non-manoeuvring and manoeuvring target tracking. Compared with traditional algorithms, the tracking performance is improved by the proposed algorithms in both typical non-manoeuvring and manoeuvring scenes. 1 INTRODUCTION In radar target tracking, the target's state equation is usually modelled in the Cartesian coordinates, while the measurements are always obtained in the polar or spherical coordinates, including the measurements of the range, azimuth, elevation and Doppler, where the Doppler measurement is also called as the range rate measurement. The measurements are non-linear with the target's state; therefore, target tracking with these measurements is a non-linear filtering problem. The extended Kalman filter (EKF) [1] is a classical method designed to solve such a non-linear problem, where Taylor expansion is adopted to approximate the non-linear measurement function linearly. But it may cause poor tracking performance when the degree of non-linearity is high. The Unscented Kalman filter [2] and Cubature Kalman filter [3] are both approximate Bayesian filters, which approximate the system state distribution by constructing sigma points or cubature points, respectively, but the algorithms' performance depends on the selection of parameters. The Particle filter [4] can solve the non-linear filtering problem effectively and performs well in the strong non-linear case. However, it has disadvantage of large computational complexity and is not suitable for practical applications. The Converted measurement Kalman filter (CMKF) converts the measurements in the polar or spherical coordinates to the Cartesian coordinates to perform the linear Kalman filter, but the converted measurement is biased. In order to remove the deviation after measurements conversion, a series of modified measurements conversion methods have been proposed successively. The Debiased CMKF [5] uses subtraction operation to eliminate the conversion bias. The Unbiased CMKF (UCMKF) [6] is proposed because the bias is found to be multiplicative essentially and therefore multiplication operation is adopted to remove the bias. In the above two algorithms, the statistical characteristics of the converted measurement errors are calculated based on the measurement information, which leads to the correlation between the converted error covariance and the measurement noise. The Decorrelated UCMKF [7] is proposed to eliminate the correlation, where the statistical characteristics are calculated based on the predicted information. When the range rate measurement is also available, the tracking accuracy can be further improved, but the degree of non-linearity between the measurements and target's state increases. The sequential filtering method [8, 9] processes the position measurements and the range rate measurement sequentially through a linear position filter and a non-linear filter. In our previous work [10], an unbiased measurements conversion-based sequential filtering algorithm with measurements in the direction cosine coordinates is proposed. In the above work, the input of the sequential filter is the pseudo measurement, which is constructed as the multiplication of the range and range rate measurements. Although the non-linearity between the pseudo measurement and the target state is reduced compared with the original range rate measurement, the non-linearity still exits. Therefore, all the filters involved in the sequential filtering are non-linear. To construct a linear measurement equation related to the range rate measurement, the pseudo state is introduced in the statically fused CMKF (SF-CMKF) [11, 12], where two linear filters process the converted position measurements and pseudo measurement in parallel. The outputs of them are statically fused to obtain the final result. Although the two filters in SF-CMKF are linear, the non-linearity still exists in the fusion process. What's more, the state equation of the pseudo state is highly related to the target motion mode. Therefore, it is difficult to apply it in manoeuvring target tracking. On the other hand, a non-informative cross range rate measurement is introduced to develop a linear measurement equation in the CMKF with range rates [13]. However, prior information of this non-informative measurement is hard to be obtained, which limits the application of this method. Actually, the linear mapping between the target state and range rate can be built based on the direction cosine information. In Ref. [14], the direction cosines are estimated based on the measured or predicted angle information to construct the linear measurement matrix and two UCMKF algorithms with range rates (UCMKF-R) are proposed. Because the linear measurement matrix depends on the measured or predicted angle information, the corresponding algorithm is easy to diverge without precise measured or predicted information. Furthermore, conventional best linear unbiased estimator [15] with only position measurements is developed in Ref. [16], and the best linear unbiased estimation algorithm with Doppler measurement that is named as BLUEPM is obtained. During the deduction of BLUEPM, many approximations are involved to get the analytical form of the estimator. Existing studies lay us a solid foundation to deal with the target tracking problem with non-linear positions and Doppler measurements. However, each algorithm has its own drawback as mentioned before. Although with the sequential filtering technique both positions and Doppler measurements are utilized, non-linearity still exists in the algorithms in Refs. [8, 10]. To deal with the non-linear problem with linear filtering, the linear measurement equation is constructed in Refs. [11, 14] using different ways. The linear measurement equations in Refs. [11, 13] are based on the introduction of the virtual state or measurement. On the contrary, the linear measurement equation in Ref. [14] essentially exists. However, the linear measurement matrix is related to the unknown angle information. When the measured or predicted angle information is used, the corresponding algorithm will be unstable in the case of large measurement errors. Inspired by the framework of sequential filtering, the target state can be estimated with the converted position measurements firstly. Therefore, a relatively more accurate angle information can be obtained based on the estimated state at this time, and it can be used to replace the measured or predicted angle information in the linear measurement matrix. Based on this idea, the sequential linear filtering technique for target tracking with non-linear positions and Doppler measurements is proposed in this study. The main contributions of this study are as follows: (1) The technique of sequential linear filtering is proposed, which sequentially processes the position measurements and range rate measurement linearly. The linear measurement matrix in the sequential filter is constructed based on the filtering result of the linear position filter, which makes the linear measurement matrix more accurate than the one constructed based on measured or predicted angle information (2) Based on the sequential linear filtering proposed above, adaptive information feedback is proposed to further solve the problem of filtering divergence caused by inaccurate measurements, and a sequential linear filtering with adaptive information feedback (ASLF) is proposed (3) For manoeuvring target tracking, the proposed ASLF is imbedded in the IMM framework [17], and an IMM based on sequential linear filtering with adaptive model probability selection (ASLF-IMM) is proposed, where the updated model probabilities are also selected adaptively. The rest of this paper is organized as follows. The non-linear measurement model is given in Section 2. The ASLF is proposed in Section 3. The ASLF-IMM is proposed in Section 4. Simulations results are given in Section 5 to demonstrate the effectiveness of the proposed algorithms. The conclusions are drawn in Section 6. 2 NON-LINEAR MEASUREMENT MODEL In radar target tracking, the measurements are generally obtained in the polar/spherical coordinates. Assume that a three-dimensional radar is located at the origin, which can obtain the target's range, azimuth, elevation and range rate measurements. The measurement equation can be written as follows. Z ( k ) = h [ X ( k ) ] + V ( k ) (1)where Z ( k ) = [ r m ( k ) θ m ( k ) φ m ( k ) r ˙ m ( k ) ] T h [ X ( k ) ] = [ r ( k ) θ ( k ) φ ( k ) r ˙ ( k ) ] T V ( k ) = [ r ˜ m ( k ) θ ˜ m ( k ) φ ˜ m ( k ) r ˙ m ( k ) ] T where r m ( k ) , θ m ( k ) , φ m ( k ) and r ˙ m ( k ) are the radar measurements of the true target range r ( k ) , azimuth θ ( k ) , elevation φ ( k ) and range rate r ˙ ( k ) , respectively. r ˜ m ( k ) , θ ˜ m ( k ) , φ ˜ m ( k ) and r ˜ m ( k ) are the corresponding measurements' noise, which are assumed to be zero-mean white Gaussian noise with variances σ r 2 , σ θ 2 , σ φ 2 and σ r ˙ 2 . r ˜ m ( k ) , θ ˜ m ( k ) and φ ˜ m ( k ) are statistically independent of each other while r ˙ m ( k ) and r ˜ m ( k ) are correlated with the correlation coefficient ρ . 3 SEQUENTIAL LINEAR FILTERING WITH ADAPTIVE INFORMATION FEEDBACK Since the state equation is usually modelled in the Cartesian coordinates, for non-linear measurements the intuitive idea is to convert them into the Cartesian coordinates. For position measurements (range, azimuth and elevation measurements), the decorrelation unbiased converted measurement method in Ref. [7] is utilized for measurement conversion and the results are as follows: x c ( k ) = e σ θ 2 / 2 e σ φ 2 / 2 r m ( k ) cos ( θ m ( k ) ) cos ( φ m ( k ) ) (2) y c ( k ) = e σ θ 2 / 2 e σ φ 2 / 2 r m ( k ) sin ( θ m ( k ) ) cos ( φ m ( k ) ) (3) z c ( k ) = e σ φ 2 / 2 r m ( k ) sin ( φ m ( k ) ) (4) For range rate measurement, it is ideal to directly use the raw measurement for filtering. The relationship between the range rate measurement and the true value is as follows: r ˙ m ( k ) = cos ( θ ( k ) ) cos ( φ ( k ) ) x ˙ ( k ) + sin ( θ ( k ) ) cos ( φ ( k ) ) y ˙ ( k ) + sin ( φ ( k ) ) z ˙ ( k ) + r ˙ m ( k ) (5) Based on Equations (2)–(5), a linear measurement equation can be constructed as follows. Z c ( k ) = H ( k ) X ( k ) + V c ( k ) (6)where X ( k ) = [ x ( k ) x ˙ ( k ) x ¨ ( k ) y ( k ) y ˙ ( k ) y ¨ ( k ) z ( k ) z ˙ ( k ) z ¨ ( k ) ] T (7) Z c ( k ) = [ x c ( k ) y c ( k ) z c ( k ) r ˙ m ( k ) ] T (8) V c ( k ) = [ x ˜ c ( k ) y ˜ c ( k ) z ˜ c ( k ) r ˙ ˜ m ( k ) ] T (9) H ( k ) is the linear measurement matrix, which is shown as below. H ( k ) = [ 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 cos ( θ ) cos ( φ ) 0 0 sin ( θ ) cos ( φ ) 0 0 sin ( φ ) 0 ] (10) The true values of θ and φ are not available in practice; therefore, the measured or predicted values are usually adopted [14]. It is found that the resulting filter is sensitive to the accuracy of the measured or predicted values [18, 19]. The inaccurate angle measurements or predicted angle values will lead to performance reduction or even filter divergence. How to construct an accurate linear measurement matrix is a key problem. Consider based on the position measurements, the estimated azimuth and elevation can be obtained. They are more accurate than the measured or predicted ones; therefore, they can be utilized to estimate H ( k ) . An ASLF is proposed in this study, which includes a linear position filter and a linear sequential filter. The sequential linear measurement matrix is estimated based on the output of the position filter. Adaptive information feedback can be performed at the same time in the proposed algorithm in order to adapt to different noise scenarios. The structure of ASLF is shown in Figure 1. FIGURE 1Open in figure viewerPowerPoint Structure of sequential linear filtering with adaptive information feedback Assume the feedback information after update at time k − 1 is X ^ ( k − 1 ) and the corresponding error covariance matrix is P ( k − 1 ) . The steps of the ASLF are shown as follows. 3.1 Measurement conversion According to Equations (2)–(5)-(2)–(5), the converted measurement Z c ( k ) can be obtained. The mean μ c ( k ) and covariance R c ( k ) of V c ( k ) can be deduced based on the predicted information of the target, which results in μ c ( k ) = [ 0 0 0 0 ] T (11) R c ( k ) = [ R x x ( k ) R x y ( k ) R x z ( k ) R x r ˙ ( k ) R x y ( k ) R y y ( k ) R y z ( k ) R y r ˙ ( k ) R x z ( k ) R y z ( k ) R z z ( k ) R z r ˙ ( k ) R r ˙ x ( k ) R r ˙ y ( k ) R r ˙ z ( k ) R r ˙ r ˙ ( k ) ] (12)where the elements R x x ( k ) , R x y ( k ) , R x z ( k ) , R y y ( k ) , R y z ( k ) , and R z z ( k ) are the same as in Ref. [7]; obviously R r ˙ r ˙ ( k ) is σ r ˙ 2 , and the remaining elements are derived as follows: R x r ˙ ( k ) = cov [ x ˜ c , r ˙ ˜ m | r p , θ p , φ p , r ˙ p ] = E [ { e σ θ 2 / 2 e σ φ 2 / 2 ( r p − r ˜ p + r ˜ m ) × cos ( θ p − θ ˜ p + θ ˜ m ) cos ( φ p − φ ˜ p + φ ˜ m ) − ( r p − r ˜ p ) cos ( θ p − θ ˜ p ) cos ( φ p − φ ˜ p ) } × r ˙ ˜ m ] = e σ θ 2 / 2 e σ φ 2 / 2 E [ r ˜ m r ˙ ˜ m ] × E [ cos ( θ p ) cos ( θ ˜ p ) cos ( θ ˜ m ) × cos ( φ p ) cos ( φ ˜ p ) cos ( φ ˜ m ) ] = ρ σ r σ r ˙ cos ( θ p ) cos ( φ p ) e − σ θ p 2 / 2 e − σ φ p 2 / 2 (13) R y r ˙ ( k ) = cov [ y ˜ c , r ˙ ˜ m | r p , θ p , φ p , r ˙ p ] = E [ { e σ θ 2 / 2 e σ φ 2 / 2 ( r p − r ˜ p + r ˜ m ) × sin ( θ p − θ ˜ p + θ ˜ m ) cos ( φ p − φ ˜ p + φ ˜ m ) − ( r p − r ˜ p ) sin ( θ p − θ ˜ p ) cos ( φ p − φ ˜ p ) } × r ˙ ˜ m ] = e σ θ 2 / 2 e σ φ 2 / 2 E [ r ˜ m r ˙ ˜ m ] × E [ sin ( θ p ) cos ( θ ˜ p ) cos ( θ ˜ m ) × cos ( φ p ) cos ( φ ˜ p ) cos ( φ ˜ m ) ] = ρ σ r σ r ˙ sin ( θ p ) cos ( φ p ) e − σ θ p 2 / 2 e − σ φ p 2 / 2 (14) R z r ˙ ( k ) = cov [ z ˜ c , r ˙ ˜ m | r p , θ p , φ p , r ˙ p ] = E [ { e σ φ 2 / 2 ( r p − r ˜ p + r ˜ m ) sin ( φ p − φ ˜ p + φ ˜ m ) − ( r p − r ˜ p ) sin ( φ p − φ ˜ p ) } × r ˙ ˜ m ] = e σ φ 2 / 2 E [ r ˜ m r ˙ ˜ m ] × E [ sin ( φ p ) cos ( φ ˜ p ) cos ( φ ˜ m ) ] = ρ σ r σ r ˙ sin ( φ p ) e − σ φ p 2 / 2 (15)where r p , θ p , φ p , r ˙ p are the predicted range, azimuth, elevation and range rate. r ˜ p , θ ˜ p , φ ˜ p , r ˙ ˜ p and σ θ p 2 , σ φ p 2 are the corresponding predicted errors and variances, and they can be calculated according to the predicted target state and error covariance [7]. 3.2 Decorrelation From Equation (12), it can be seen that R c ( k ) is not block diagonal, which indicates that the converted measurement errors of the position and range rate measurement error are correlated. It is necessary to decorrelate before sequential linear filtering. In order to realize decorrelation, R c ( k ) is divided into four blocks as follows: Then construct B ( k ) as follows: B ( k ) = [ I 3 0 3 × 1 L ( k ) 1 ] (17)where L ( k ) = − ( R c p r ˙ ( k ) ) T ( R c p o s ( k ) ) − 1 (18) Multiplying B ( k ) on both sides of Equation (6) we have B ( k ) Z c ( k ) = B ( k ) H ( k ) X ( k ) + B ( k ) V c ( k ) (19)where the covariance of B ( k ) V c ( k ) can be calculated as B ( k ) V c ( k ) [ B ( k ) V c ( k ) ] T = B ( k ) R c ( k ) B T ( k ) = I 3 0 3 × 1 L ( k ) 1 × 3 1 [ R c p o s ( k ) 3 × 3 R c p r ˙ ( k ) 3 × 1 R c r ˙ p ( k ) 1 × 3 R c r ˙ r ˙ ( k ) ] I 3 0 3 × 1 L ( k ) 1 × 3 1 T = R c p o s ( k ) 3 × 3 R c p r ˙ ( k ) 3 × 1 [ L ( k ) R c p o s ( k ) ] 1 × 3 + R c r ˙ p ( k ) 1 × 3 L ( k ) R c p r ˙ ( k ) + R c r ˙ r ˙ ( k ) · I 3 L T ( k ) 3 × 1 0 1 × 3 1 Based on the matrices multiplication in the above equation, put the expression of L ( k ) in Equation (18) and finally we have B ( k ) V c ( k ) [ B ( k ) V c ( k ) ] T = [ R c p o s ( k ) 3 × 3 0 3 × 1 0 1 × 3 R c r ˙ r ˙ ( k ) − ( R c p r ˙ ( k ) ) T ( R c p o s ( k ) ) − 1 R c p r ˙ ( k ) ] (20) It can be seen that the covariance of B ( k ) V c ( k ) is a block diagonal matrix; the decorrelation is realized and Equation (19) can be written as two separate equations accordingly as follows. The detailed form of Equation (19) is [ I 3 0 3 × 1 L ( k ) 1 ] [ x c ( k ) y c ( k ) z c ( k ) r ˙ m ( k ) ] = [ I 3 0 3 × 1 L ( k ) 1 ] H ( k ) X ( k ) + [ I 3 0 3 × 1 L ( k ) 1 ] [ x ˜ c ( k ) y ˜ c ( k ) z ˜ c ( k ) r ˙ ˜ m ( k ) ] (21) Let L ( k ) = [ L 1 ( k ) , L 2 ( k ) , L 3 ( k ) ] , and H ( k ) and X ( k ) can refer to Equations (10) and (7); after the multiplication of matrices, Equation (21) can be further expressed as follows: [ x c ( k ) y c ( k ) z c ( k ) ε c ( k ) ] = [ x ( k ) y ( k ) z ( k ) ε ( k ) ] + [ x ˜ c ( k ) y ˜ c ( k ) z ˜ c ( k ) ε ˜ c ( k ) ] (22)where ε c ( k ) = L 1 ( k ) x c ( k ) + L 2 ( k ) y c ( k ) + L 3 ( k ) z c ( k ) + r ˙ m ( k ) ε ( k ) = L 1 ( k ) x ( k ) + cos ( θ ) cos ( φ ) x ˙ ( k ) + L 2 ( k ) y ( k ) + sin ( θ ) cos ( φ ) y ˙ ( k ) + L 3 ( k ) z ( k ) + sin ( φ ) z ˙ ( k ) ε ˜ c ( k ) = L 1 ( k ) x ˜ c ( k ) + L 2 ( k ) y ˜ c ( k ) + L 3 ( k ) z ˜ c ( k ) + r ˙ ˜ m ( k ) According to the diagonal block matrix form in Equation (20), it is known that E [ ε ˜ c ( k ) x ˜ c ( k ) ] = 0 , E [ ε ˜ c ( k ) y ˜ c ( k ) ] = 0 and E [ ε ˜ c ( k ) z ˜ c ( k ) ] = 0 , which means that ε ˜ c ( k ) is uncorrelated to x ˜ c ( k ) , y ˜ c ( k ) and z ˜ c ( k ) . However, in Equation (6), r ˙ ˜ m ( k ) is related to x ˜ c ( k ) , y ˜ c ( k ) and z ˜ c ( k ) because E [ x ˜ c ( k ) r ˙ ˜ m ( k ) ] = R x r ˙ ( k ) ≠ 0 , E [ y ˜ c ( k ) r ˙ ˜ m ( k ) ] = R y r ˙ ( k ) ≠ 0 and E [ z ˜ c ( k ) r ˙ ˜ m ( k ) ] = R z r ˙ ( k ) ≠ 0 as shown in Equations (13)–(15)-(13)–(15). Each vector in Equation (22) is blocked and further expressed as x c ( k ) y c ( k ) z c ( k ) ε c ( k ) ‾ = [ x ( k ) y ( k ) z ( k ) ε ( k ) ‾ ] + [ x ˜ c ( k ) y ˜ c ( k ) z ˜ c ( k ) ε ˜ c ( k ) ‾ ] ⇒ [ Z c p o s ( k ) ε c ( k ) ] = [ H p o s ( k ) H ε ( k ) ] X ( k ) + [ V c p o s ( k ) ε ˜ c ( k ) ] (23)where Z c p o s ( k ) = [ x c ( k ) y c ( k ) z c ( k ) ] T H p o s ( k ) = [ 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ] V c p o s ( k ) = [ x ˜ c ( k ) y ˜ c ( k ) z ˜ c ( k ) ] T H ε ( k ) = [ L 1 ( k ) cos ( θ ( k ) ) cos ( φ ( k ) ) 0 L 2 ( k ) sin ( θ ( k ) ) cos ( φ ( k ) ) 0 L 3 ( k ) sin ( φ ( k ) ) 0 ] Therefore, after decorrelation we have obtained two uncorrelated linear measurement equations: Z c p o s ( k ) = H p o s ( k ) X ( k ) + V c p o s ( k ) (24) ε c ( k ) = H ε ( k ) X ( k ) + ε ˜ c ( k ) (25) 3.3 Linear position filter After the decorrelation, the linear position filter is performed based on the position converted measurements as follows: X ^ ( k | k − 1 ) = F ( k − 1 ) X ^ f ( k − 1 ) (26) P ( k | k − 1 ) = F ( k − 1 ) P f ( k − 1 ) F T ( k − 1 ) + G ( k − 1 ) Q ( k − 1 ) G T ( k − 1 ) (27) e p o s ( k ) = Z c p o s ( k ) − H p o s ( k ) X ^ ( k | k − 1 ) (28) S p o s ( k ) = H p o s ( k ) P ( k | k − 1 ) ( H p o s ( k ) ) T + R c p o s ( k ) (29) K p o s ( k ) = P ( k | k − 1 ) ( H p o s ( k ) ) T ( S p o s ( k ) ) − 1 (30) X ^ p o s ( k ) = X ^ ( k | k − 1 ) + K p o s ( k ) e p o s ( k ) (31) P p o s ( k ) = [ I − K p o s ( k ) H p o s ( k ) ] P ( k | k − 1 ) (32)where X ^ f ( k − 1 ) and P f ( k − 1 ) are the feedback information at k − 1 , which will be explained in Section 3.5. 3.4 Linear sequential filter Unlike the traditional sequential filter, which needs to construct the pseudo measurement and use EKF for state estimation, the linear sequential filter here adopts the original range rate measurement and uses the constructed linear measurement matrix to realize the completely linear sequential filter. As shown in Section 3.2, after the decorrelation, a linear measurement equation related to the range rate measurement is obtained, which is Equation (25). However, in the linear measurement matrix H ε ( k ) , the azimuth θ ( k ) and elevation φ ( k ) are unknown. Here, the filtering result from above linear position filter is utilized to estimate the azimuth and elevation. Based on X ^ p o s ( k ) obtained in Equation (31), the azimuth and elevation can be estimated as θ ^ p o s ( k ) = tan − 1 ( y ^ p o s ( k ) x ^ p o s ( k ) ) (33) φ ^ p o s ( k ) = tan − 1 ( z ^ p o s ( k ) ( x ^ p o s ( k ) ) 2 + ( y ^ p o s ( k ) ) 2 ) (34)where x ^ p o s ( k ) , y ^ p o s ( k ) and z ^ p o s ( k ) are the position terms in X ^ p o s ( k ) . Then the linear sequential filter is performed as follows: e ε ( k ) = ε c ( k ) − H ^ ε ( k ) X ^ p o s ( k ) (35) S ε ( k ) = H ^ ε ( k ) P p o s ( k ) ( H ^ ε ( k ) ) T + R c ε ε ( k ) (36) K ε ( k ) = P p o s ( k ) ( H ^ ε ( k ) ) T ( S ε ( k ) ) − 1 (37) X ^ ε ( k ) = X ^ p o s ( k ) + K ε ( k ) e ε ( k ) (38) P ε ( k ) = [ I − K ε ( k ) H ^ ε ( k ) ] P p o s ( k ) (39)where H ^ ε ( k ) = [ L 1 ( k ) cos ( θ ^ p o s ( k ) ) cos ( φ ^ p o s ( k ) ) 0 L 2 ( k ) sin ( θ ^ p o s ( k ) ) cos ( φ ^ p o s ( k ) ) 0 L 3 ( k ) sin ( φ ^ p o s ( k ) ) 0 ] (40) X ^ ε ( k ) and P ε ( k ) are the outputs of the linear sequential filter, which are also the final outputs at k . Therefore, X ^ ( k ) = X ^ ε ( k ) and P ( k ) = P ε ( k ) . In general, Equations (26)–(32)-(26)–(32) are the filtering process corresponding to the position measurement equation in Equation (24), and Equations (35)–(40)-(35)–(40) are the filtering processes corresponding to the range rate measurement equation in Equation (25). However, in the linear measurement matrix H ε ( k ) , the azimuth and elevation of the target is unknown. The advantage of splitting the filtering into two separate filters is that we can utilize the filtering result obtained in the first filter to estimate the unknown azimuth and elevation and estimate the linear measurement matrix for the second filter. This is the basic idea of this manuscript. 3.5 Adaptive information feedback Compared with the method of directly utilizing measured or predicted angle information to approximate the linear measurement matrix H ( k ) in Equation (10), the algorithm proposed in this study has more advantages in tracking accuracy. However, when the angle measurement