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Square-root cubature Kalman filter (SCKF) is more effective for nonlinear state estimation than an unscented Kalman filter.
In this paper, we study the design of nonlinear filters based on SCKF for the system with one step noise correlation and abrupt
state change. First, we give the SCKF that deals with the one step correlation between process and...
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... simulation results are given in Fig. 3, Fig. 4, and Table 2. The results show that SCKF-STF can achieve a better tracking effect besides the target state with abrupt change, and the tracking performance of SCKF is reduced because the abrupt state change will harm its estimation accuracy. ...
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Citations
... The adaptive current statistical (CS) model and the improved interacting multiple model (IMM) were combined with the SRCKF respectively in [11], [12], which effectively improved the maneuvering target tracking accuracy. The second is to develop adaptive filtering algorithms to obtain better filter precision, among which the strong tracking filter (STF) based on the fading memory filtering theory [13]- [16] can effectively resist the bad influence of model errors. On the other hand, it is necessary to use robust techniques for controlling the measurement outliers such as the Huber-based Kalman filter (HKF) [17] and the robust UKF with gain correction [18], [19]. ...
To solve the problem of decreased filtering accuracy and even filter divergence for the case that the model errors and measurement outliers exist simultaneously, an adaptively robust square-root cubature Kalman filter (SRCKF) based on amending is proposed in this paper. Based on the error analysis, the judgment criterion and the amending criterion related to the innovation are set. Then the filter could overcome the influence of the measurement outliers with the robust amendment of the measurement noise covariance matrix based on the principle of the innovation covariance matching. To further deal with model errors, the new method of the adaptive amendment of the predicted state based on the innovation is developed and combined. Finally, the proposed algorithm can balance the effect of the prior predicted value and the posterior feedback measurement value on the filtering process and reduce the state estimation error. The simulation results show that the proposed algorithm can effectively suppress the negative impact of the model errors and the measurement outliers and can obtain better estimation performance and obviously decreasing running time compared with the adaptively robust unscented Kalman filter (ARUKF) and the robust multiple fading factors cubature Kalman filter (RMCKF).
... This type of STF-based method has drawn attention from many researchers. [23][24][25][26][27][28][29][30][31][32] Based on the theoretical relationship between state prediction covariance matrix and crosscovariance matrix, Wang et al. 23 obtained the equivalent calculation method of Jacobian matrix in UKF, and then established the equivalent calculation method of fading factor in nonlinear filtering algorithms. Equivalent calculation avoids the dependence of strong tracking UKF on the Jacobian matrix, and expands its applicable scale. ...
... For this reason, the fading factor is introduced into the state prediction covariance, which can force Eq. (5) to hold. The recursive equations of TSTCKF can be represented as follows: 28 (1) Suppose that the state estimationx k and the state covariance matrix P k at the moment k of the system are known. (2) Time updating Calculate the state predictionx kþ1jk and the state prediction covariance P s kþ1jk at the moment k þ 1 usingx k and P k : ...
... These two introduction methods are both used in the current researches, 23,24,31 and there are not clear differences between the two methods. 28 Therefore, Q kþ1 is ignored from the analysis. Then, the value of c kþ1 and k 2 1;kþ1 will decide whether the two fading factor calculation methods are equivalent or not. ...
The fading factor exerts a significant role in the strong tracking idea. However, traditional fading factor introduction method hinders the accuracy and robustness advantages of current strong-tracking-based nonlinear filtering algorithms such as Cubature Kalman Filter (CKF) since traditional fading factor introduction method only considers the first-order Taylor expansion. To this end, a new fading factor idea is suggested and introduced into the strong tracking CKF method. The new fading factor introduction method expanded the number of fading factors from one to two with reselected introduction positions. The relationship between the two fading factors as well as the general calculation method can be derived based on Taylor expansion. Obvious superiority of the newly suggested fading factor introduction method is demonstrated according to different nonlinearity of the measurement function. Equivalent calculation method can also be established while applied to CKF. Theoretical analysis shows that the strong tracking CKF can extract the third-order term information from the residual and thus realize second-order accuracy. After optimizing the strong tracking algorithm process, a Fast Strong Tracking CKF (FSTCKF) is finally established. Two simulation examples show that the novel FSTCKF improves the robustness of traditional CKF while minimizing the algorithm time complexity under various conditions. Keywords: Algorithm time complexity, Cubature Kalman filter, Nonlinear filtering, Robustness, Strong tracking filter
... However, the performance of the nonlinear filters will be greatly influenced when faced with the inaccurate model or bad measurement caused by the aircraft maneuver. In order to improve the tracking stability on the abrupt change, references [19]- [25] introduce the fading factor in the state error covariance matrix and structure the strong tracking (ST) nonlinear filter. However, the fading factor in [19]- [25] has two problems. ...
... In order to improve the tracking stability on the abrupt change, references [19]- [25] introduce the fading factor in the state error covariance matrix and structure the strong tracking (ST) nonlinear filter. However, the fading factor in [19]- [25] has two problems. The first is the arbitrariness of the introduced position, which is due to a lack of mathematical deduction to the simultaneous satisfaction of the orthogonality principle of residual error sequences and of the least mean-square error of the output estimation. ...
... Last but not the least, two simulation scenarios conducted show that the proposed algorithm is able to accurately track highly maneuvering motion and weak maneuvering motion at the condition of lacking prior knowledge. The performance of the proposed algorithm is better than the multiple-fading-factor SCKF [24] based on the CS model and the SCKF-STF [25] based on the modified CS model [6]. Moreover, the proposed algorithm decreases the runtime by 40% while maintaining the commensurate performance compared with the interacting-multiple-model SCKF (IMM-SCKF). ...
A novel strong tracking square-root cubature Kalman filter (ST-SCKF) based on the adaptive current statistical (CS) model is proposed aiming at the maneuvering aircraft tracking problem. The Jerk input estimation is introduced on the basis of the modified input estimation (MIE) algorithm in order to make the connection with the state process noise and the state error covariance matrix. Thus, the online-adaptive adjustment of the CS model is achieved. Additionally, the introduced position of the fading factor is re-deduced and a novel calculation method is designed in order to overcome the invalidity problem of the traditional fading factor. Two simulation scenarios are conducted to verify the effectiveness of the proposed algorithm. The simulation results show that the proposed algorithm possesses better adaptability and tracking precision than two state-of-the-art single model filters. Moreover, the proposed algorithm decreases the runtime by 40% while maintaining the comparable performance compared with the interacting-multiple-model SCKF (IMM-SCKF). OAPA
... For the first case, literatures [21,22] have, respectively, proposed the networked STF and STF with randomly delayed measurements (STF/RDM). For the second case, a novel nonlinear filter derived from square-root CKF and the idea of STF was proposed in [23]. However, when the above two cases exist simultaneously, these existing STFs [17][18][19][20][21][22][23] are not suitable for dealing with the filtering problem in the above two coupled cases, and little attention has been paid to the study of deriving the corresponding STF. ...
... For the second case, a novel nonlinear filter derived from square-root CKF and the idea of STF was proposed in [23]. However, when the above two cases exist simultaneously, these existing STFs [17][18][19][20][21][22][23] are not suitable for dealing with the filtering problem in the above two coupled cases, and little attention has been paid to the study of deriving the corresponding STF. Consequently, there is a great demand to further improve the STF for the nonlinear discrete-time stochastic dynamic systems with randomly delayed measurements and correlated noises, which motivate this study. ...
This paper proposes a novel strong tracking filter (STF), which is suitable for dealing with the filtering problem of nonlinear systems when the following cases occur: that is, the constructed model does not match the actual system, the measurements have the one-step random delay, and the process and measurement noises are correlated at the same epoch. Firstly, a framework of decoupling filter (DF) based on equivalent model transformation is derived. Further, according to the framework of DF, a new extended Kalman filtering (EKF) algorithm via using first-order linearization approximation is developed. Secondly, the computational process of the suboptimal fading factor is derived on the basis of the extended orthogonality principle (EOP). Thirdly, the ultimate form of the proposed STF is obtained by introducing the suboptimal fading factor into the above EKF algorithm. The proposed STF can automatically tune the suboptimal fading factor on the basis of the residuals between available and predicted measurements and further the gain matrices of the proposed STF tune online to improve the filtering performance. Finally, the effectiveness of the proposed STF has been proved through numerical simulation experiments.
... The strong tracking filtering has attracted considerable amount of attention from researchers and engineers in various fields, such as fault detection and diagnosis, target tracking, integrated navigation, observer design, and networked control, and many associated methods have been presented based on the STF [10][11][12][13][14]. It has been extensively used to design various state estimators and fusion estimators by combing other nonlinear filters [15,16]. Unfortunately, the effectiveness of the strong tracking filtering is usually validated by only simulations and experiments and theoretical analysis of the STF is very lacking. ...
... The idea of the STF is to compute a time-variant fading factor in a heuristic way to adjust one step state prediction error covariance matrix in real time. For the linear dynamic system given by (3), the corrected state prediction error covariance matrix is [6,7,15,16] ...
... Clearly, the modification of the state model depends closely on the fading factor and the real-time filtering process for this view. The presented view helps to explain the phenomenon that the STF is particularly effective for inaccurate state models and sudden change of the state [7,13,15]. In fact, the state model is improved in the STF by correcting the computation of the prediction error covariance. ...
Strong tracking filtering (STF) is a popular adaptive estimation method to effectively deal with state estimation for linear and nonlinear dynamic systems with inaccurate models or sudden change of state. The key of the STF is to use a time-variant fading factor, which can be evaluated based on the current measurement innovation in real time, to forcefully correct one step state prediction error covariance. The strong tracking filtering technology has been extensively applied in many practical systems, but the theoretical analysis is highly lacking. In an effort to better understand STF, a novel analysis framework is developed for the strong tracking filtering and some new problems are discussed for the first time. For this, we propose a new perspective that correcting the state prediction error covariance by using the fading factor can be thought of directly modifying the state model by correcting the covariance of the process noise. Based on this proposed point of view, the conditions for the STF function to be effective are deeply analyzed in a certain linear dynamic system. Meanwhile, issues of false alarm and alarm failure are also briefly discussed for the strong tracking filtering function. Some numerical simulation examples are demonstrated to validate the results.
... (1) and (2) and the modified DUKF described by Eqs. (4)-(8), (10)- (13), and (25). Let the following assumptions hold for every k ≥ 0. ...
... The bearings-only tracking system [25] is also considered to verify the effectiveness of the proposed technique. The bearings-only tracking system is composed of two sensors to track a moving target, where each sensor obtains only an angle measurement related to the system state. ...
This is the second of two consecutive papers focusing on the filtering algorithm for a nonlinear stochastic discrete-time system with linear system state equation. The first paper established a derivative unscented Kalman filter (DUKF) to eliminate the redundant computational load of the unscented Kalman filter (UKF) due to the use of unscented transformation (UT) in the prediction process. The present paper studies the error behavior of the DUKF using the boundedness property of stochastic processes. It is proved that the estimation error of the DUKF remains bounded if the system satisfies certain conditions. Furthermore, it is shown that the design of the measurement noise covariance matrix plays an important role in improvement of the algorithm stability. The DUKF can be significantly stabilized by adding small quantities to the measurement noise covariance matrix in the presence of large initial error. Simulation results demonstrate the effectiveness of the proposed technique.
... However, the cumbersome calculation of Jacobian matrices and linearization errors would degrade its performance. Some versions of nonlinear STF utilizing sigma points have been derived, but some linearization steps still exist in the given procedures (Ge et al. 2011; Yang et al. 2011 ). In this subsection , the formation of fading factor will be embedded in STF into the framework of SUKF without linearization. ...
An autonomous navigation method for low-thrust interplanetary vehicle is proposed. In the proposed navigation system, one inertial navigation system (INS) is employed to continuously estimate the position and attitude of vehicle, and three X-ray sensors observing X-ray pulsars are utilized to reduce the long-term effects of the errors in the INS. In addition, a modified square-root unscented Kalman filter (MSUKF) is proposed. The MSUKF adopts a fading factor to compensate the impact of modeling error, and contains no linearization steps. The results have shown that the proposed navigation system outperforms the traditional celestial-inertial method and the MSUKF could guarantee a faster convergence compared with former proposed nonlinear filters. (C) 2015 American Society of Civil Engineers.
... As is known to all, Bearings-only tracking is a representatively passive tracking problem and it is very popular in current research on target tracking all the time (Li and Ge, 2010;Musicki et al.,2012;Ge and Li, 2011). Actually, the nature of Bearings-only tracking is nonlinear filtering and fusion. ...
In this paper, we study the nonlinear data fusion problem for a kind of Bearings-only tracking network system with correlations among measurement noises. As a result, a decentralized cubature Kalman fusion algorithm is proposed by using an information filtering form (CIF) of cubature Kalman filter (CKF) and noise de-correlation technology. Based on the CIF and the augmented measurement, a centralized cubature Kalman fusion algorithm is firstly established and the problem on computational performance is pointed out. In order to improve the computational performance of the centralized fusion method, the noise de-correlation technology is used to obtain a functionally equivalent Bearings-only tracking fusion system without the noise correlations, namely the variance of augmented measurement noise is diagonalized. Accordingly, the decentralized fusion method based on the CIF can be used to achieve a more effective tracking fusion estimate.
... For decades, many research findings have been obtained in the field of maneuvering target localization and tracking (Blom and Bar-Shalom, 1988;Li and Bar-Shalom, 1993;Munir and Atherton, 1995;Li et al., 2006;Peng, 2007;Zhang and Chen, 2010;Messaoudi et al., 2010;Foo and Ng, 2011;Ge et al., 2011;Zhang et al., 2012). Current research is focused mainly on the multiple model method, which is generally thought to be the best method for maneuvering target tracking (Magill, 1965;Blom and Bar-Shalom, 1988;Li and Bar-Shalom, 1993;Munir and Atherton, 1995;Xu et al., 2003;Zhu, 2008;Liu et al., 2009;Foo and Ng, 2011;Yuan and Zheng, 2011). ...
The trajectory of a shipborne radar target has a certain complexity, randomness, and diversity. Tracking a strong maneuvering target timely, accurately, and effectively is a key technology for a shipborne radar tracking system. Combining a variable structure interacting multiple model with an adaptive grid algorithm, we present a variable structure adaptive grid interacting multiple model maneuvering target tracking method. Tracking experiments are performed using the proposed method for five maneuvering targets, including a uniform motion — uniform acceleration motion target, a uniform acceleration motion — uniform motion target, a serpentine locomotion target, and two variable acceleration motion targets. Experimental results show that the target position, velocity, and acceleration tracking errors for the five typical target trajectories are small. The method has high tracking precision, good stability, and flexible adaptability.
In this paper, an adaptive maximum correntropy cubature Kalman filter based on multiple fading factors (MAMCKF) is proposed to address the problem of inaccurate process noise covariance and unknown measurement noise covariance together with outliers in target tracking. Although there are many adaptive filters and robust filters have been proposed to handle unknown measurement noise covariance or measurement outliers, most filters cannot deal with both unknown noise covariance and outliers simultaneously. In this article, we propose an adaptive and robust cubature Kalman filter. The modified measurement noise covariance matrix (MNCM) and innovation covariance matrix are used to construct multiple fading factors for correcting the prediction error covariance matrix (PECM), which can achieve adaptability. Then, the maximum correntropy criterion (MCC) is introduced to suppress outliers, which further enhances the robustness. Compared with the existing approaches, the proposed approach improves the performance by at least 5% in unknown time-varying noise, unknown time-varying heavy-tailed noise, and non-Gaussian heavy-tailed noise scenarios. The simulation results show that the proposed approach can effectively suppress inaccurate process noise covariance and unknown time-varying measurement noise together with outliers. Compared with the existing filtering approaches, the proposed approach exhibits both adaptability and robustness.
