Kalman filter with recursive covariance estimation for protection against system uncertainty

IET Control Theory & ApplicationsVolume 14, Issue 15 p. 2097-2105 Research ArticleFree Access Kalman filter with recursive covariance estimation for protection against system uncertainty Xuan Xiao, Xuan Xiao School of Automation, Beijing Institute of Technology, South Street of Zhongguancun, Beijing, People's Republic of ChinaSearch for more papers by this authorKai Shen, Corresponding Author Kai Shen kshen@bit.edu.cn orcid.org/0000-0002-1798-3977 School of Automation, Beijing Institute of Technology, South Street of Zhongguancun, Beijing, People's Republic of ChinaSearch for more papers by this authorYuan Liang, Yuan Liang School of Automation Science and Electrical Engineering, Beihang University, Xueyuan Road, Beijing, People's Republic of ChinaSearch for more papers by this authorTingxin Liu, Tingxin Liu School of Automation, Beijing Institute of Technology, South Street of Zhongguancun, Beijing, People's Republic of ChinaSearch for more papers by this author Xuan Xiao, Xuan Xiao School of Automation, Beijing Institute of Technology, South Street of Zhongguancun, Beijing, People's Republic of ChinaSearch for more papers by this authorKai Shen, Corresponding Author Kai Shen kshen@bit.edu.cn orcid.org/0000-0002-1798-3977 School of Automation, Beijing Institute of Technology, South Street of Zhongguancun, Beijing, People's Republic of ChinaSearch for more papers by this authorYuan Liang, Yuan Liang School of Automation Science and Electrical Engineering, Beihang University, Xueyuan Road, Beijing, People's Republic of ChinaSearch for more papers by this authorTingxin Liu, Tingxin Liu School of Automation, Beijing Institute of Technology, South Street of Zhongguancun, Beijing, People's Republic of ChinaSearch for more papers by this author First published: 02 September 2020 https://doi.org/10.1049/iet-cta.2019.1476Citations: 5AboutSectionsPDF 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 onFacebookTwitterLinkedInRedditWechat Abstract This study is intended to design a novel adaptive Kalman filter (KF) that can solve the filtering problem with unknown noise statistics. The proposed method named as measurement sequence adaptive KF (MSAKF) can adaptively estimate the unknown parameters of noise statistics via the information from measurement sequences. In order to enhance the computational efficiency, algorithm optimisation via recursive covariance estimation is introduced. In addition, stability analysis of the MSAKF is also made under some given conditions. The estimation process is proved to be stable and its filtering results converge to the ones of the ideal KF with exact system parameters. To demonstrate the effectiveness of the MSAKF algorithm, the simulation based on a navigation signal-tracking model is presented, and the results show that the MSAKF algorithm possesses low calculation complexity, fast convergence, high-precision and adaptability in complex application environments. 1 Introduction With the rapid development of advanced navigation technology and vast deployment of low-cost sensors, Kalman filter (KF) has become the most important estimation technique applied in multisensor fusion and integration [1], such as robots positioning, integrated navigation and network commutation. It has been proved that classical or standard KF is optimal when accurate statistical information of process and measurement noises can be obtained [2, 3]. However, since the noises are dependent upon the uncertainties of environment and system dynamics, the restricting conditions of KF are too strong to be always satisfied in real application scenarios [4]. The uncertainty in process noise does have a crucial impact on KF and may result in filter divergence. To overcome this problem, various extensions on KF have been developed to mitigate the impact of inaccurate statistical information. Typically, two approaches to KF modification, namely robust KF (RKF) and adaptive KF (AKF) [5-9], can be used to solve the filtering problem with unknown noise statistics. RKF has been developed to suppress the influence of statistics errors. filter [10-12] as the most well-known RKF, can guarantee bounded state estimation error through criterion. Its method for adaptation is to use a scale or a matrix factor as a multiplier to the process noise covariance matrix so that the adaptation against the system uncertainty is possible. However, its filtering results are sub-optimal and worse than those generated by the KF in a least-squares sense. Strong-tracking filter (STF) [13] is widely used for dealing with the uncertainty in process noise, because of its real-time-tracking ability and strong robustness. STF is actually an extension of KF by introducing a time-varying suboptimal fading factor into the prediction covariance. The drawback of STF is the selection of softening factor. Risk-sensitive KF (RSKF) [14, 15] is originated to minimise the cost function. As for RSKF, the risk-sensitive criterion is used for bounding the estimation errors and maintaining robustness characteristics. Generally speaking, RKF is designed to avoid large estimation errors and filter divergence. Whereas, its estimation results are always suboptimal, which lead to occasional failures when facing practical demands. AKF guarantees the accuracy of state estimation via adaptively modifying noise-related parameters. For developing AKFs in practice, two strategies are usually used for estimating the noise statistics in real-time. The first strategy is to apply an adaptive law, while the second is to use the law of large numbers [16, 17]. As for the first strategy, the noise covariance matrices are determined by an adaptive law, such as the fuzzy method-based AKF [18-20] and the neural network-based AKF (NNAKF) [21, 22]. In these algorithms, both the adaptive law and the empirical coefficients play important roles in sequentially estimating noise covariance. For example, in a NNAKF [23], it appears that proper gains of nodal points guarantee good estimation results. However, selecting proper coefficients requires an enormous amount of experimental data for training and it is sometimes troublesome to collect sufficient data. Additionally, the adaptive law and empirical coefficients are always sensitive to system characteristics and application environment, which indicates that these algorithms may hold poor robustness. As for the second strategy, the noise covariance matrices are adaptively estimated based on either innovation sequence AKF (ISAKF) or residual sequence AKF (RSAKF) [24-26]. Both of them are widely used as adaptive estimation techniques [27-29]. According to the law of large numbers, when given enough samples, the estimations on the noise statistics will converge to real values. In ISAKF and RSAKF, the innovation sequence or the residual sequence serves as sample information, and there are various methods for estimation, such as Bayesian, maximum likelihood, window average method and so on. In addition, if the estimations on noise covariance matrices are precise enough in ISAKF and RSAKF, the state estimations should also be accurate. However, in our investigation, such methods can hardly reach the expected results. Let us take this type of AKFs as an example for further discussion (as shown in Fig. 1). Fig. 1Open in figure viewerPowerPoint Flow chart of ISAKF Generally, the KF can be considered as a model-based algorithm that is used to recursively estimate both the state mean vector and the covariance matrix [24, 30]. The flow chart of KF is shown in Fig. 2. As we can see from the figure, the process of acquiring Kalman gain is completely unaffected by any information from the measurement update procedure, which indicates that the filter gain update is independent of the measurement update procedure. As for ISAKF illustrated in Fig. 1, it is clear that is influenced by the innovation sequence and the residual sequence, which states clearly that the independence between two procedures has been removed. In this scheme, when there are abnormal measurement conditions, a harmful loop, marked in blue in Fig. 1, may degrade the performance of the filter. The estimation error on Q can corrupt the estimation on state, and then the estimation error on the state may degrade the estimation of Q in turn, which does undermine the stability and optimality of the filter. Fig. 2Open in figure viewerPowerPoint Flow chart of KF To overcome this problem, we propose a novel AKF in which the process noise covariance matrix is adaptively estimated based on the measurement sequence AKF (MSAKF), while the measurement noise covariance matrix may be adaptively estimated based on the innovation sequence. The brief introduction to MSAKF is shown in Fig. 3. As we can see from Fig. 3, in order to achieve satisfactory optimal estimation results, the algorithm acquires covariance estimation Q and R via the information from measurement sequence and innovation sequence, respectively. Therefore, the measurement update does help to correct the filter gain update. The scheme of MSAFK can avoid the possible divergence on the state estimations. Fig. 3Open in figure viewerPowerPoint Flow chart of MSAKF Our main contributions are described as follows. First, we propose a novel AKF algorithm, i.e. MASKF, which can deal with unknown noise statistics through estimating them via the measurement and/or innovation sequences. Second, a recursive covariance estimation algorithm is designed to estimate the unknown covariance matrix of input process noise from a sample of information collected in the measurement sequence, while its recursive form can help us to increase the calculation speed and to optimise the calculation procedure. Third, we have proved some useful properties of the MSAKF. In a MSAKF, the estimation on Q converges to the real value; the filtering results of MSAKF are stable and the filtering results of the MSAKF converge to the ones of the ideal KF [2, 4] with accurate system parameters. We believe that MSAKF investigated in this study favours the development of estimation-related tracking and fusion in integrated navigation for unmanned vehicles as well as advanced vehicle networking. The rest of the paper is organised as follows. In Section 2, the problem of state estimation for linear stochastic systems is formulated. In Section 3, the adaptation strategies are provided to estimate the noise covariance matrices. In Section 4, the MSAKF algorithm is introduced. In Section 5, the stability and optimality of MSAKF algorithm are analysed. In Section 6, a numerical simulation on navigation and commutation is carried out to illustrate the effectiveness of the MSAKF. The conclusion is drawn in Section 7. 2 Problem formulation The discrete system that we are interested in is represented by the following equations (1) where , and are the state transfer matrices, is the measurement matrix and is the input matrix. The process noise and measurement noise with appropriate dimension are white Gaussian noises [1]. To clearly present sequel text, some assumptions are provided below. Assumption 1.The input and measurement noises and are uncorrelated zero mean white Gaussian noises with (2) (3) (4) where Q is the covariance matrix of input process noise , R is the covariance matrix of measurement noise , is the Kronecker function. Assumption 2.The initial state is independent of input process and measurement noises and , and (5) (6) For the system represented by (1), the implementation of classical linear KF (LKF) [4, 31] with accurate system parameters is (7) where is the optimal estimation of , is the Kalman gain and and denote the covariance matrices of estimation errors of and , respectively. However, the LKF usually fails in practice. Compared with other uncertainty factors, accurate statistic characteristics of noises are harder to be acquired. What is more, inaccurate statistic characteristics can worsen the filtering results. In practice, acquiring accurate statistic characteristics Q in advance is quite difficult, while acquiring R previously is relatively easy to realise. Moreover, scholars have proved that the uncertainty of Q may even result in filter divergence theoretically [32-34]. Therefore, an efficient way for avoiding the unknown Q and R degrading the performance of KF is in great demand. 3 Recursive covariance estimation In order to prevent the divergence of the algorithm, AKFs are highly required to be formulated. When reliable statistical information about noises and is limited, the estimation of covariance matrices Q and R are introduced in this section. This idea is the foundation of the MSAKF. 3.1 Estimation based on sample sequence Lemma 1.It is supposed that an over determined system has the following form (8) The unique least-squares solution [35-37] to (4) is (9)For a better description, a simple but not trivial problem is given, how can we obtain the covariance of a random vector from its sample sequence in real-time? For a random vector , it is supposed that (10) where , , the covariance matrix , and are uncorrelated. When we have an independent identical distributed (i.i.d.) sample of vector , how can we obtain the accurate covariance matrix of from sample in real-time? Theorem 1.The covariance of can be estimated as (11) where T represents the transpose of a matrix. Proof.According to basic statistical knowledge, we can get (12) Since , and . According to the law of large numbers, we can get (13)Since and are uncorrelated, we know that (14)Thus, can be obtained by (15)Therefore, for recursively estimating the covariance matrix of , we can use the following statistical result (16) □ 3.2 Estimation of matrix Q via noise sequence This basic idea of estimation based on sample sequence is applied to the estimation of Q for system represented by (1). To accomplish the estimation, what we are concerned about is the way to construct a sample sequence of noise. In this paper, the sequence is constructed by the measurement sequence. From (1), it is not difficult to formulate a new measurement [30] based on the measurement sequence from initial moment to time k (17) where represents the measurement sequence represents the process noise sequence represents the formulated measurement noise sequence represents the observability matrix of the system and represents extended input matrix. Consequently, the measurement sequence can be decomposed into two parts. First, the initial state of the system. Considering the difficulty of obtaining the accurate value of in advance, may be considered as an unknown constant disturbance to a certain extent. Second, the noise sequences and contain necessary noise statistics information and can be considered as sample sequences. Therefore, we can construct the noise sequences and by extracting useful information from the measurement sequence . Assuming that there exists a proper constant matrix that satisfies the following equation (18) In this case, we can get (19) The newly constructed measurement only contains noise sequences. Since and are zero-mean and uncorrelated noised, it is clear that and . In addition, the noise processes and are also uncorrelated. Furthermore, we can obtain (20) (21) Noticing that and , we know that (22) (23) where represents the diagonal matrix. Denoting as the i.i.d. sample of , when R may be considered as a prior knowledge, based on (15) and (19)–(23), it is obtained that (24) For a better description of (24), we introduce the following notations (25) (26) Thus, it is not different to obtain that (27) Let us introduce a new notation as (28) When the rank of equals l, there exists a matrix that satisfies the following equation (29) where represents an identity matrix, may be considered as a pseudoinverse matrix of the matrix [38]. Therefore, we can get (30) Based on (30), Q can be estimated in real time with the system measurements. Considering as the estimate of Q and as the real-time calculation of , we can finally obtain (31) (32) 3.3 Numerical optimisation of estimation process The calculation method introduced above is efficient in theory. Yet, it may not perform very well in practical applications due to the following reasons. First, in terms of the calculation speed of the algorithm, it is hard to obtain all the vector samples at a certain time. As illustrated in (31), the accumulation of samples requires all past measurements. Thus, the calculation time goes up with time increasing, which cannot meet real-time calculation needs in practice. Second, if Q is a matrix with zero off-diagonal elements, the off-diagonal elements are useless. Thus, we can increase the calculation speed significantly by setting zero values for off-diagonal elements. Finally, in terms of the accuracy of the algorithm, some errors may corrupt the measurements, and thus degrade the performance of the estimation algorithm. Therefore, it is necessary to find a numerical optimisation for estimation. In this subsection, a recursive form of estimation and the least-square calculation method are introduced for optimising the algorithm and improve its calculation performance. 3.3.1 Recursive form of estimation Supposing that is the estimated covariance of by utilising the samples up to and including the time k, is the estimated value of at time k. Then, we can get the following relationship as (33) (34) With the arrival of a new sample, the sample sequence should be updated in order to obtain new estimation (35) Thus, we can obtain a recursive form of estimation on . The covariance matrix of can be calculated recursively in real-time with increasing sample sequence. Similarly, can also be estimated in a recursive way as (36) Therefore, can be estimated based on (37) 3.3.2 Least-squares solution to estimation When the matrix Q is with zero off-diagonal elements, (27) can be simplified further. Introducing a notation (38) where are vectors of size. Then, we can get (39) where are diagonal elements of matrix Q. Since are scalars, we can, thus, get (40) It is equivalent to the following equation (41) where the matrix can be considered as a coefficient matrix for achieving the least-squares solution and the matrix is a submatrix of ; , ); the matrix can be considered as a matrix that has an indirect relationship with the system measurement and is a submatrix of ; is a column vector formed by diagonal elements of matrix Q and represents the element in i th row and j th column. Equation (41) does provide us with an indirect relationship between the covariance matrix of input process noise Q and the measurement of system, which favours the the estimation of matrix Q on the basis of the measurement sequence. If , which can be easily satisfied in practice, can be acquired directly from (41). However, since the uncertain model errors and other errors may worsen the accuracy of the information, (41) may become overdetermined. Hence, it is necessary to mitigate the impact of error-corrupted information. It is well known that least square performs well in acquiring the minimum mean square error solution of overdetermined equations. According to Lemma 1, the least-square solution [37] to (41) is (42) Hence, it can be obtained that (43) where and denote the estimates of and , respectively. Therefore, we can finally obtain the estimate (44) Summarising the above discussions, can be acquired more efficiently and more accurately than the calculation method illustrated in (32). Numerical optimisation enhances the practical performance of MSAKF. 3.4 Estimation of matrix R via innovation sequence Let us consider the case of unknown accurate statistical information of measurement noise covariance R. By definition, the covariance matrix R has a relationship with the innovation sequence as (45) For a linear time-varying system, we may easily get the following estimate of (46) or the recursive covariance estimation form as (47) In this case, substituting the estimate into the (45), we can finally obtain the estimate of the covariance matrix R (48) 4 Kalman filter with recursive covariance estimation In general, the successful use of the LKF is highly restricted by the strict requirements on the statistics information of the process and measurement noises. Considering practical needs in wide applications, recursive covariance estimation is essential to cope with system uncertainties. On the basis of this idea, a new KF algorithm is presented to resolve this challenging problem. In this algorithm, when the measurement matrix H is of column full rank, we can find a proper that satisfies the (18) and the rank of equals l. In order to ensure the stability of KF algorithm, the system should be controllable and observable [23, 34]. What is more, the system measurement should be bounded, which is also satisfied by most of physical systems. Based on the estimation method above, covariance matrices Q and R can be estimated in real-time and it can serve for the application of the KF in (7) for estimating system states. In this section, we intend to introduce the MSAKF via pseudocode based on some assumptions about the system. The MSAKF algorithm is illustrated in Fig. 4. Fig. 4Open in figure viewerPowerPoint Algorithm 1: MSAKF algorithm 5 Stability analysis In this section, the stability analysis [4, 34] of the MSAKF is presented to show that this algorithm is stable and the state estimate obtained by MSAKF is convergent to the ideal KF with accurate system parameters. Lemma 2.Considering the discrete algebraic Riccati equation (DARE) [39-41], we can get (49) where is a posteriori estimation error covariance matrix, is a priori estimation error covariance matrix, and they are positive definite Hermitian matrices. When the system is observable and controllable, the DARE has a unique semi-positive definite solution . Theorem 2.For the system as in (1), using the estimation method in MSAKF, is convergent to Q as . Proof.Introducing the notation (50) Based on (30) and (32), we can get (51)Thus, we know that (52)Therefore, almost surely, which indicates that is convergent to Q as . □ Theorem 3.For the system as in (1), using the estimation method in MSAKF, as , where , is the covariance matrix of estimation errors in MSAKF, is the covariance matrice of estimation errors in the ideal KF with exact system parameters [2-4]. Proof.Considering the DARE in (49), we can get (53) Introducing the notation (54) then, we can get (55)According to Theorem 2, we can obtain (56)As for the ideal KF with exact system parameters, it is supposed that (57)Similarly, introducing the notation (58) then, we can get (59)According to Lemma 2, the solutions of the Riccati equation as in (55) and (59) are unique. Hence, we can get (60)Therefore, we can finally obtain (61) □ Theorem 4.For the system as in (1), using the estimation method in MSAKF, as , where , is the Kalman gain in MSAKF, is the Kalman gain in the ideal KF with exact system parameters. Proof.According to the ideal KF with exact system parameters in (3), we can get (62) Introducing the notation (63) thus, we can get (64)Similarly, according to the estimation method in MSAKF, we can get (65)Introducing the notation (66) then, we can obtain (67)According to Theorem 3 (68) hence, it is obtained that (69)Therefore, we can finally obtain (70) □ Theorem 5.For the system as in (1), using the estimation method in MSAKF, as , where , is the estimation on states in MSAKF, is the estimation on states in the ideal KF with exact system parameters. Lemma 3.When (A,H) is observable and (A,G) is controllable, the ideal KF described in [23, 34] is stable. The eigenvalues of have a magnitude 1. Lemma 4.Considering a dynamic error system: (71) Assume that as , where F is a stable matrix. If as , then as . Proof.For the system as in (1), we can get the estimate [4, 14] (72) According to the estimation method in MSAKF, the system state estimate is (73)Thus, we can get (74)Introducing the notation (75) (76) thus, it is obtained that (77)According to Lemma 3, since the system is observable and controllable, is bounded and the innovation sequence will be bounded as well.In Theorem 4: (78) hence, we can get (79)According Lemma 3, we can finally obtain (80)In summary, According to Theorems 2–5, it is clear that when , , , and as . □ 6 Results and analysis In this section, we numerically evaluate the performance of the MSAKF and compare it with the LKF and the well-known particle filter (PF), for fully analysing the effectiveness and computational load of our method. In order to illustrate the effects of the MSAKF algorithm, a navigation signal-tracking model is selected to be used for simulation. 6.1 Navigation signal tracking model High-precision positioning and navigation are of critical importance for autonomous unmanned systems and advanced vehicle networking. In order to track Global Navigation Satellite Systems (GNSS) signals in signal degraded environments, a phase-locked loop (PLL) [42] can be used in practice. For a PLL system, we may apply a KF-based algorithm to estimate the information parameters of a frequency-modulated signal. The signals of a PLL system [33] may be described in a form of differential equation as (81) In a discrete-time form, the signals are (82) where is the sampling time period. Hence, the PLL system can be represented by a discrete-time system as (83) where is the transmission coefficient of a phase detector. 6.2 Simulation results For demonstrating the effectiveness of the MSAKF algorithm, we use the new proposed MSAKF algorithm to estimate the information parameters of a PLL frequency-modulated signal with unknown noise statistics. Simulation parameters have the following numerical values: , , , and the initial covariance of estimation . The input sine signal is equal to , where . We implement MSAKF, LKF, PF and ISAKF algorithms using Matlab and compare the performance of the MSAKF algorithm with LKF, PF and ISAKF algorithms during tracking signals in PLL. The simulation results are presented in Figs. 5-8. Fig. 5Open in figure viewerPowerPoint Estimation of with MSAKF, LKF, PF and ISAKF algorithms Fig. 6Open in figure viewerPowerPoint Estimation of with MSAKF, LKF, PF and ISAKF algorithms Fig. 7Open in figure viewerPowerPoint The RMSE of X estimation with MSAKF, LKF, PF and ISAKF algorithms Fig. 8Open in figure viewerPowerPoint Estimation of with MSAKF, LKF, PF and ISAKF algorithms Figs. 5 and 6 illustrate that ISAKF, PF and MSAKF do have better performance than that of LKF during the navigation signal-tracking process. Here, the LKF is the filtering algorithm proposed by Rudolf E. Kalman in 1960s. In LKF, the initial values of the covariance matrices Q and R should be determined according to rich engineering experiences. Although we try our best to select appropriate initial values for LKF, the performance of LKF algorithm is poor because of unknown noise statistics in practice. As for ISAKF, it does not work very well at first, but it can adaptively estimate and correct the unknown or uncertain system model parameters and noise statistics, so the final result is much better than that of LKF. In terms of PF, a Monte-Carlo-based method is used. The core idea is to approximate the probability density function of the system variable by a series of particles from the state space, and replace the integral operation with the mean of the particles, so as to obtain the minimum variance estimation of the state. Thus, in our simulation, the state estimation obtained by the PF can track our signal very well in theory. In the MSAKF algorithm, although we cannot accurately know noise statistical characteristics, we may acquire them via the information from measurement sequence. Hence, MSAKF algorithm can also track the navigation signals fast and accurately. Fig. 7 illustrates the root mean square error RMSE between the estimated value and the true value obtained by the four filtering algorithms, and the RMSE of PF and MSAKF are very close, indicating that the performance of the two filtering algorithms is similar. For ISAKF, its filtering results are slightly worse than that of the two algorithms above. However, compared with the LKF, ISAKF's performance is much better. Fig. 8 illustrates the locking-in process of navigation signal. When the deviation between the estimated and carrier phases is no more than 5% from the standard value, we may consider that the locking-in process is done. Hence, it is quite obvious that the locking-in process lasts for 2.5 s for MSAKF, 3.5 s for PF, 4 s for ISAKF and 5 s for LKF, respectively. Fig. 7 shows that the tracking effect of PF is slightly better than that of MSAKF. However, as the PF uses a large number of particles to estimate the state, the computation load of the algorithm is much larger. In our simulations, the running time of the algorithms were calculated in Matlab. Among them, the running time of MSAKF, ISAKF and LKF are , and , respectively. It can be seen that when we use the online estimation of Q and R, compared with LKF, the running time of MSAKF and ISAKF increases, whereas the running time of MSAKF is relatively shorter. However, when the particle number was set to be N =200, the running time of PF was =19.87 s, which was definitely much longer than the running time of the other three filtering algorithms. What is more, with the increase of time, particle degradation may occur, which reduces the efficiency of the PF algorithm. Therefore, the novel proposed MSAKF algorithm possesses superior performance of fast convergence, high-precision and adaptability under complex application environments. 7 Conclusion In order to solve the problem of completely unknown covariance matrix of process and measurement noises during filtering process, a novel algorithm named as the MSAKF is proposed in this paper. This algorithm adaptively estimates the unknown noise covariance matrix based on the measurement sequence. To enhance computational efficiency, two methods for optimising the algorithm have been proposed. Additionally, the stability analysis of the MSAKF has been introduced under some given conditions, and the state estimates acquired by MSAKF do converge to the optimal state estimates of the ideal KF. For demonstrating the effectiveness of the MSAKF algorithm, a navigation signal-tracking model is selected to be used for simulation. 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