A novel strong tracking cubature Kalman filter and its application in maneuvering target tracking

An ZHANG, Shuid BAO, Fei GAO, Wenho BI

a School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

b School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710129, China

KEYWORDS

Abstract 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 thirdorder 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.

1. Introduction

Extended Kalman Filter (EKF) has nowadays been widely used in navigation, guidance and control,1,2aircraft attitude estimation,3target tracking,4,5battery state-of-charge estimation6-8and many other fields.9-12It is developed based on Kalman Filter(KF)and uses the first-order expansion to linearize the nonlinear function.When the nonlinearity of the system is weak, and the model parameters basically match the process parameters, the EKF can still achieve an approximate unbiased estimation as long as the initial state is selected properly.However,in some practical applications,due to the high complexity and strong nonlinearity of the system,a first-order Taylor expansion is insufficient to well approximate the physical system.There exist large differences between the model parameters and the process parameters,which could lead to a significant reduction of the EKF’s filtering accuracy, or even divergence.In order to break this limitation,Zhou et al.13combined the optimal fading Kalman filter used in linear system with EKF, and proposed Strong Tracking Filter (STF). Based on the orthogonal principle,STF introduces the fading factor into the prediction covariance matrix to make full use of the effective information in the residual sequences. This improvement enhances the robustness of EKF regarding model uncertainty and state saltation. Currently, the STF has been widely used in various situations.14-16Yin et al.14proposed a symmetric STF for the drive induction of a motor’s wireless sensor,where multiple fading factors were introduced into the prediction covariance matrix. The Cholesky triangular decomposition ensured the symmetry of the prediction covariance matrix and gained better stability during the iteration. For the problems of parameter perturbation and unknown input in sensor networks, the networked orthogonal principle was proposed by He et al.15to establish the networked STF, which was applied to a practical Internet-based three-tank system and showed better performance than networked EKF. Ge et al.16found that the superiority of STF relative to EKF is usually based on simulation and previous experiments while lacking theoretical analysis. Accordingly, they compared STF and EKF from three aspects: (A) theoretical performance, (B)inconsistency in the estimation performance and (C) mismatching of performance measure. The remarkable conclusion, that the structure of traditional EKF is vulnerable and the practicability is severely restricted, revealed the great significance to study STF in engineering applications.

Among the nonlinear filtering algorithms, Unscented Kalman Filter (UKF)17and Cubature Kalman Filter (CKF)18are the most representative ones. Compared with EKF,UKF and CKF do not need to solve the Jacobian matrix.Higher filtering accuracy enables them to gradually replace EKF and have been used.19-21Therein, CKF adopts thirdorder spherical-radial cubature rule.Its simple frame with little parameters and rigorous mathematical derivation effectively overcomes UKF’s instability dealing with high-order nonlinearity problems.22Inheriting from EKF,however,the filtering capabilities of UKF and CKF decrease when the model and the physical system do not match. So similarly, one can also introduce the concept of strong tracking into UKF or CKF.This type of STF-based method has drawn attention from many researchers.23-32Based on the theoretical relationship between state prediction covariance matrix and crosscovariance matrix,Wang et al.23obtained 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. Hu et al.24adjusted the introduction position of the fading factor in UKF to avoid solving the Jacobian matrix. The hypothesis test method was adopted to improve the recognition validity of faults such as inaccurate models. These studies have proved that strong tracking UKF/CKF have better robustness and better filtering performance than UKF/CKF when there is model uncertainty.Nevertheless, these studies are still within the architecture of STF. In spite of the preceding improvements on introduction position and calculation method of the fading factor,the existing strong tracking nonlinear filtering algorithms are still imperfect mainly in the following aspects. (A) STF is based on EKF, which only considers the first-order Taylor expansion.Under this framework,strong tracking nonlinear filtering algorithms cannot fully utilize the original advantage that some nonlinear filtering algorithms could approximate any nonlinear function with second-order accuracy; (B) the Jacobian matrix needs to be solved additionally when calculating the fading factor, even if there is an approximate calculation method. This will lead to greater algorithm time consumption and robustness loss; (C) introducing fading factor into UKF and CKF will affect the one-step prediction measurement,and results in a large number of repeated calculations.

In order to solve the aforementioned problems,the authors have proposed a low-cost strong tracking CKF formerly.33The calculation of Jacobian matrix is sidestepped and fading factor is directly applied to the relevant variables,which avoids additional sampling operation and significantly reduces the algorithm time complexity. However, the low-cost strong tracking CKF does not utilize the advantage that CKF could approximate any nonlinear function with second-order accuracy. Therefore, in this paper, we further analyze the mechanism of strong tracking idea within nonlinear filtering framework, and extend the idea of low-cost strong tracking CKF.The main work of this paper includes(A)a novel fading factor introduction and calculation method that is generally suitable for nonlinear filtering algorithms, (B) the equivalent fading factor calculation method based on cubature transform,(C) the strong tracking CKF based on traditional process and(D) a fast strong tracking CKF by making full use of the two fading factors. Suggestions are also given on what situations should the Jacobian matrix be used in the strong tracking algorithm to improve the filtering performance.

The forthcoming content is organized as follows.After giving a description of the problem,Section 2 analyzes the limitations and shortcomings of the traditional fading factor introduction method in Traditional Strong Tracking CKF(TSTCKF) in detail. The novel fading factor introduction and calculation method are then established in Section 3. Section 4 deduces the equivalent fading factor calculation method based on cubature transform and puts forward the process of two strong tracking CKF algorithms. Two simulation examples are presented in Section 5 with a conclusion of the paper coming after in the last section.

2. Problem description

Consider the following nonlinear discrete system:

where xk∈Rnis the system state vector,yk+1∈Rmis the measurement vector, and fk（·） and hk+1（·） are the state function and measurement function of the system, respectively.wk∈Rnis the system noise,and vk+1∈Rmis the measurement noise. They are uncorrelated Gaussian white noises. The covariance matrices are Qkand Rk+1.

According to recursive Bayesian estimation, the posterior probability density function of xk+1is with Yk+1=｛y1,y2,···,yk+1｝.

Then xk+1with the largest P（xk+1|Yk+1）is the optimal state estimation.32However,the corresponding integral only has an accurate analytical solution when the system model is linear,i.e., Kalman filter. When the system model is nonlinear, the nonlinear filtering algorithms mainly use certain approximation rules to fit the integral in recursive Bayesian estimation to obtain a suboptimal solution, such as Central Difference Kalman Filter (CDKF), Gauss-Hermite Filter (GHF), UKF and CKF. These nonlinear filtering algorithms can theoretically approximate the posterior mean and covariance of any nonlinear Gaussian system with second-order accuracy.

The main idea of STF can be expressed as13

(1) Suppose that the state estimation^xkand the state covariance matrix Pkat the moment k of the system are known.

(2) Time updating

where l=2n.

(3) Fading factor calculation

First calculate the variables needed in the fading factor calculation:

Then,calculate fading factor γk+1.If the measurement function is complex and it is not easy to obtain the exact solution of Hk+1, then Hk+1can be equivalently calculated as24

where tr（·）is the operator of the matrix trace,and the weakening factor β is equal to or greater than 1 to avoid overregulation. β can be set by experience. Vk+1is the covariance matrix of the actual output residual sequences,and can be estimated by13

where ρ is the forgetting factor,which satisfies 0 ＜ρ ≤1.Usually, we have ρ=0.95.

The updated prediction covariance Pk+1|kand updated square root Sk+1|kcan be obtained as

(4) Measurement updating

Compared with EKF, CKF has two major advantages:

(1) CKF can achieve at least second-order approximation accuracy for nonlinear models, while the accuracy of EKF can only reach first order.

(2) CKF can avoid solving the Jacobian matrix of the model with the lower requirements, which can be applied to black box systems.

As shown above, the combination of strong tracking concept with CKF still follows the concept of STF.The advantage of CKF which is able to reach second-order accuracy is not considered. The Jacobian matrix is only used in calculating the fading factor. Thus, the filtering performance of CKF has not been truly improved relative to EKF to some extent.

3. A new fading factor introduction method

3.1. Feasibility analysis

Based on the orthogonal principle, fading factor γk+1is introduced into the state prediction covariance Pk+1|k.The information of the residual can be fully extracted to achieve the purpose of adjusting the filtering gain Kk+1in real time. Kk+1can be calculated as

Therefore, the key to analyze how the fading factor γk+1adjusts the filtering gain Kk+1is to determine the effect of γk+1on the cross-covariance Pxy,k+1|kand the measurement covariance Pyy,k+1|k.

After the fading factor γk+1of STF is introduced into Pk+1|k,there is a linear transfer relation in Pxy,k+1|kand Pyy,k+1|k,13i.e.,

As shown in Eqs. (18) and (19), the linear transfer relation between γk+1and Pyy,k+1|kis used in the calculation of γk+1.

In nonlinear filter, the one-step state prediction error and the one-step measurement prediction error are defined as

3.2. Fading factor calculation method

The fading factor introduction method based on λ1,k+1and λ2,k+1should satisfy the requirements of Eqs. (4) and (5), that is, the orthogonal principle must be satisfied.

Theorem 1 has been proven in Ref.32

Substituting Eq. (28) into the above equation, there is

Substituting Eq. (36) into Eq. (40), then the calculation of λ2,k+1is

where β is the weakening factor to avoid over-regulation.

To establish the relation between λ2,k+1and λ1,k+1,Theorem 2 is given as follows.

Theorem 2. Let

Proof. From Eq. (33), there is

Substituting Eqs. (35) and (36) into Eq. (47), we have

Then, summing both sides and substituting a, b and c into Eq. (46), Eq. (49) is obtained.

The proof of Theorem 2 is now complete.

Obviously, when λ2,k+1is determined, λ1,k+1is the root of the unary quadratic equation, and 1 ≤λ1,k+1. The calculation of a, b and c is related to the approximation rule adopted by the nonlinear filter. The detailed derivation in CKF is presented in Section 4.

It should be noted that the values of a,b and c are related to the measurement function hk+1（·）.When the order of hk+1（·）is low, that is, abs（a）＜ε ?b (where abs（·） is absolute value function and ε is the threshold),assuming a=0 and substituting it into Eq. (46), we have λ1,k+1=λ2,k+1. When the order of hk+1（·）is high,that is,abs（a）≥ε and x～k+1|kmeets the requirements of Taylor expansion, we get solutions for Eq. (46), and 1 ≤λ1,k+1. In practical applications, due to the influence factors such as the truncation error, it is recommended to determine the value of a and whether there is a root for Eq. (46)before extracting roots of Eq. (46). When the requirements are not met, let λ1,k+1=λ2,k+1. In general, the threshold ε should be a small value and is related to the calculation accuracy. It is recommended to determine the value of ε according to the experiment or experience.

3.3. Comparison of two fading factor introduction methods

As shown in Theorem 2, a in Eq. (46) is main determinant of determining whether λ2,k+1and λ1,k+1are the same as γk+1.It is determined only by the second-order and high-order term of the Taylor expansion of the measurement function hk+1（·）,and is independent of the state function fk（·）.abs（a）＜ε represents that the measurement function hk+1（·）is linear or weakly nonlinear, where abs（a）≥ε represents a strong nonlinear one.For different values of a, the advantages and disadvantages of the two fading factor introduction methods are as follows.

(1) abs（a）＜ε

(2) abs（a）≥ε

It is clear that they are not equal.

The equivalent calculation method of γk+1is shown in Eqs. (17)-(19). The high-order terms are not effectively used,resulting in the inappropriate value of γk+1, which causes the lower accuracy of the traditional fading factor introduction method.

In summary, when the measurement function hk+1（·） is weakly nonlinear, that is, the second-order and higher-order terms of the Taylor expansion are relatively small with respect to the first-order terms of the Taylor expansion,the new fading factor introduction method does not require the calculation of λ1,k+1and thus avoids using the Jacobian matrix. It has the same accuracy as the traditional strong tracking fading factor introduction method.When the high-order terms of the Taylor expansion are not negligible, although the new fading factor introduction method also needs to use the Jacobian matrix,it is expected to have better filtering performance.In any case,the new strong tracking fading factor introduction method is superior to the traditional strong tracking fading factor introduction method.

4. Strong tracking method based on cubature transform

4.1. Fading factor introduction method based on cubature transform

In order to obtain the equivalent calculation method for λ1,k+1and establish the equivalent solution for a,b and c based on cubature transform, Theorem 3 is given as follows.

Substituting a, b and c into Eq. (59), there is

Combining the equations in Eq. (61), Eqs. (56)-(58) are obtained.

The Taylor second-order term of a is

where

Substituting Eq. (63) into Eq. (62), the sum of the secondorder terms of a can be obtained as 0.

The proof of Theorem 3 is now complete.

Theorem 3 shows that when the equivalent measurement equation hk+1（·） is only second-order differentiable and a=0,there is λ1,k+1=λ2,k+1,which means that the calculation of λ1,k+1and the use of the Jacobian matrix are not required.Only when the measurement equation h（·）is third-order differentiable or higher,it is necessary to calculate λ1,k+1,and extract the third-order or high-order information of the residual.Thus the filtering accuracy is improved.

In order to illustrate the rationality and correctness of selecting the first n cubature points to calculate a, b and c in Theorem 3, Theorem 4 is given as follows.

Theorem 4. Let the coefficients obtained using the first n cubature points be A1, B1and C1, and the coefficients obtained using the last n cubature points be A2,B2and C2.Then,there is

Proof. Substituting Eq. (55) into Eq. (56), there is

Obvious∑ly, |B1|=|B2|, B1∑ =-B2.

C1is calculated as

Similarly, C2is calculated as

Obviously, |C1|=|C2|, C1=-C2.

Based on Eq. (60), there is

where |A1|=|A2|, A1=-A2.

The proof of Theorem 4 is now complete.

As shown in the proof of Theorem 4, the third-order and high-order terms of the Taylor expansion in 2n cubature points are fully extracted in a,b and c,and the obtained fading factor λ1,k+1would be closer to the optimal solution than λk+1. Eq.(46) then becomes a simple cubic equation:

It can be known from the Cardano formula that real root for Eq. (74) exists regardless of the values of a, b and c.

However, Cardano formula is quite complicated and will increase the algorithm time complexity. Another better way to solve cubic equation is using iteration method such as Newton iteration method. Donate the cubic equation by

Calculating derivative of φ（λ1,k+1） with respect to λ1,k+1,there is

After few iterations, the termination condition is

4.2. Strong tracking CKF process

Based on the equivalent fading factor calculation method established in Section 4.1 and the traditional strong tracking method process, the Normal Strong Tracking CKF(NSTCKF) is proposed. The method process of NSTCKF is as follows:

Step 2 Time updating

Step 3 Fading factor calculation

First calculate the variables needed in the fading factor calculation with Eqs. (10)-(16). Then, calculating fading factor λ1,k+1and λ2,k+1. The updated prediction covariance Pk+1|kand updated square root Sk+1|kcan be obtained as

(4) Measurement updating

4.3. Fast strong tracking CKF process

There are two obvious shortcomings of NSTCKF during its process:

(1) NSTCKF is based on the traditional strong tracking CKF idea, that is, after the fading factor introduction,the cubature point set is recalculated,and the fading factor is transferred through the measurement function to calculate relevant variables and complete the measurement updating. This results in repeated calculations of multiple variables in steps 3 and 4,and additional transfer of the set of cubature points, which may greatly increase the time complexity of the algorithm.

(2) NSTCKF actually only introduces one fading factor λ1,k+1.λ2,k+1becomes an intermediate variable in the calculation of λ1,k+1,which cannot reflect the advantages of two different fading factors with two different introduction positions.

Therefore, this paper proposes a new fast strong tracking CKF (FSTCKF) algorithm, and its process is as follows:

(2) Time updating

(3) Measurement updating

Firstly, calculate the variables needed in the fading factor calculation as shown in Eqs. (10)-(16), and calculate fading factor λ1,k+1and λ2,k+1.

Then,update Yk+1|k,χk+1|k,and^yk+1|kusing λ1,k+1,and λ2,k+1as follows:

FSTCKF introduces two fading factors while keeping the original structure of CKF as much as possible.It can not only maintain the high precision and stability of CKF, but also effectively enhance the robustness of CKF to problems such as model inaccuracy.

4.4. Algorithm time complexity analysis

Compared with CKF, the algorithm time complexity increase of TSTCKF, NSTCKF and FSTCKF is mainly composed of three parts:

(1) Calculating the variables needed in the fading factor calculation

TSTCKF,NSTCKF and FSTCKF need the same variables in the fading factor calculation, and have the same algorithm time complexity increase.

(2) Calculating fading factor

(A)abs（a）＜ε

NSTCKF and FSTCKF avoid the dependence on the Jacobian matrix, so their algorithm time complexity increase is lower than that of TSTCKF.

(B)abs（a）＞ε

(3) Introducing fading factor

In TSTCKF, additional matrix triangular factorization operation is O（n3） and some variables are recalculated. In NSTCKF, the recalculated variables are the same with TSTCKF. However in FSTCKF, the fading factors are directly introduced to relevant variables.

In conclusion, the algorithm time complexity increase of TSTCKF and NSTCKF is dependent on the measurement function and the total number of Newton iteration.In general,the algorithm time complexity increase of NSTCKF is lower than that of TSTCKF, while FSTCKF has the lowest algorithm time complexity increase.

5. Experimental analysis

In order to illustrate the validity of FSTCKF and NSTCKF, CKF, TSTCKF and TSTCKF234are selected as comparisons. TSTCKF2 uses hypothesis testing to identify the model uncertainty. They are applied to onedimensional strong nonlinear models with time-varying noise and multi-dimensional maneuvering target tracking respectively. In the experiment, ρ=0.95, and β is set by our experience to 3.

5.1. High-order example

Consider the following strong nonlinear model with timevarying noise:35

where wkand vkare uncorrelated Gaussian white noise, with variance

The Root Mean Square Error (RMSE) is used as the comparison criterion in the simulation. Suppose that the simulation time is N=200, and the Monte Carlo simulation time is M=50. The RMSE in one Monte Carlo simulation is defined as

The RMSE of the Monte Carlo simulation when p=2 is shown in Fig.1. In this situation, the measurement function is only second-order differentiable, λ2,k+1=λ1,k+1in NSTCKF and TSTCKF, and the filtering performance of FSTCKF,NSTCKF, TSTCKF and TSTCKF2 are the same.

Fig.1 RMSE of five algorithms when p=2.

Using MATLAB 2016b, the running time of CKF,TSTCKF,TSTCKF2,NSTCKF and FSTCKF,using a laptop with an intel i5-7300HQ processor and 8 GB RAM, is shown in Table 1. Due to the need to calculate the Jacobian matrix,a more complex fading factor calculation method and a large number of repetitive operations,TSTCKF has the largest running time. TSTCKF2 avoids calculating the fading factor when there is no model uncertainty,and has less running time.NSTCKF does not need to calculate the Jacobian matrix and has less running time than TSTCKF2. FSTCKF avoids the repetitive operation in TSTCKF, and the running time is less than NSTCKF. As the complexity of the measurement function hk+1（·） increases, the time complexity difference between FSTCKF and NSTCKF will become more significant.

The RMSE of the Monte Carlo simulation when p=4 is shown in Fig.3. For nonlinear system with p ＜4, TSTCKF can perform relatively well, and its performance gradually approaches CKF. But the filtering performance of TSTCKF decreases sharply with the increase of p. This indicates that p=4 may be a critical point for the traditional strong tracking methods. This is because the proportion of the high-order terms in the residual increases gradually with an increasing p, while TSTCKF can only extract information of the firstorder terms. TSTCKF2 has worse filtering performance compared with TSTCKF. This means that the hypothesis testing may not accurately identify model uncertainty when p=4.As p increases, the NSTCKF’s filtering performance also decreases, but it is still obviously better than TSTCKF.

Fig.2 RMSE of five algorithms when p=3.

Table 1 Running time of five algorithms.

Fig.3 RMSE of five algorithms when p=4.

Fig.4 RMSE of five algorithms when p=5.

The RMSE of the Monte Carlo simulation when p=5 is shown in Fig.4. The filtering performance of TSTCKF and TSTCKF2 further confirms the analysis for Fig.3.FSTCKF is a compromise between performance and time complexity of NSTCKF. However, its performance has no significant difference with NSTCKF until p=5.

The value of λ2,k+1-λ1,k+1in NSTCKF at moment k=90-110 with different values of p is shown in Fig.5. When p=2,λ2,k+1-λ1,k+1=0 which is in accordance with theoretical analysis.As the proportion of the high-order terms in the measurement function hk+1（·） becomes larger with an increasing p, the difference between λ1,k+1and λ2,k+1is the reason why FSTCKF and NSTCKF are better than TSTCKF. That also illustrates the irrationality of the traditional fading factor calculation method when the nonlinearity of the measurement function hk+1（·） is strong.

Fig.5 Value of λ2,k+1-λ1,k+1 in NSTCKF when p has different values.

Table 2 Value of α- and θ when p has different values.

5.2. Maneuvering target tracking

The following two-dimensional plane tracking model is used as the simulation example:18

where the sampling time T=1, φ1and φ2are adjustment parameters, and φ1=1 m2·s-3,φ2=1.75×10-3rad2·s-3.

The radar is placed at the origin(0,0),and then the distance σ and azimuth θ of the target can be obtained as

where atan2 is the inverse tangent function,and vkis an independent Gaussian white noise with a mean of 0 and a covariance of R=diag［1 000 m2100 m rad2］.The initial condition is

RMSE is used as the comparison criterion in this simulation. Suppose that the simulation time is N=100, and the Monte Carlo simulation time is M=50. Then the RMSE at each moment in the simulation is defined as

The measurement function hk+1（·） in this model is theoretically third-order differentiable. However, a ?b is found in the simulation. Thus, let a=0, and then there is λ1,k+1=λ2,k+1in NSTCKF and FSTCKF, with no need to solve the Jacobian matrix. Therefore, smaller time complexity is achieved than TSTCKF. In theory, TSTCKF, NSTCKF and FSTCKF should have the same performance. However,TSTCKF and TSTCKF2 need to approximately calculate the Jacobian matrix, while FSTCKF is an approximate form of NSTCKF. This will result in slightly different performance among TSTCKF, TSTCKF2, NSTCKF and FSTCKF.

5.2.1. Incorrect initial state

Obviously,when the initial value is wrong,CKF has a large position estimation error at the beginning,and its convergence speed is slower than other filters, indicating the ability of the fading factor to quickly track the abnormal state. However,over time,CKF gradually converges to the normal error interval, with no significant difference from other filters. At the same time, in the first few periods, TSTCKF, TSTCKF2,NSTCKF and FSTCKF have significant fluctuations in the estimation of the x axis velocity and the turn rate. This is because the single fading factor can only achieve the average optimization of multiple states and optimize the position estimation, which results in the over-adjustment of velocity and turn rate.

The running time and increased percentage of other filters relative to CKF are shown in Table 3. Due to the need to calculate the Jacobian matrix, a more complex fading factor calculation method and a large number of repetitive operations, TSTCKF has the largest running time with an increase of 41.24% in algorithm time complexity. TSTCKF2 avoids unnecessary fading factor calculation, and the algorithm time complexity growth is 28.48%. NSTCKF does not need to calculate the Jacobian matrix, which simplifies the fading factor calculation process, and the algorithm time complexity growth is 18.09%, which shows that operations such as equivalent calculation of the Jacobian matrix greatly increase the time complexity of the algorithm. FSTCKF avoids the repetitive operation in TSTCKF, and the 8.76% running time increment is the least. As the complexity of the measurement function hk+1（·） increases, the time complexity difference between FSTCKF and NSTCKF will become more significant.

Fig.6 RMSE under incorrect initial values.

Table 3 Running time and increased percentage of five algorithms.

5.2.2. State saltation

When k=15,the simulated aircraft position is abruptly changed as follows:

After the change occurs, all other filters converge to the normal error interval at the next moment k=16,and have faster response speed than CKF.CKF not only converges slowly,but also generates certain fluctuations in subsequent state estimation until k=40 to return to the normal error interval.

Fig.7 RMSE under state saltation.

5.2.3. Inaccurate model

Suppose that due to the complexity of external conditions,there is the following extremely small inaccuracy in the model:

with other conditions remaining the same, and the RMSE of each algorithm is shown in Fig.8.

It should be noted that before the inaccuracy occurs, CKF has reached a convergence state, and the gain matrix Kk+1tends to be a minimum value.After the model becomes inaccurate, the residual ek+1increases, and the gain matrix Kk+1of CKF remains at the minimum value, causing CKF to lose the capability to track inaccurate model. As shown in Fig.8,the position estimation error of CKF becomes significantly larger after k=35,and the error remains larger than other filters after k=60, with no tendency to converge. When 35 ＜k ＜50, the velocity and turn rate estimation of CKF is close to other filters. However, after k ＞50, since the gain matrix Kk+1remains at the minimum value, the estimations of CKF cannot converge to the normal error interval. On the contrary, due to the introduction of the fading factor,TSTCKF, TSTCKF2, NSTCKF and FSTCKF can adjust the gain matrix Kk+1in real time according to the change of the residual ek+1after the model becomes inaccurate. When 35 ＜k ＜50, TSTCKF, TSTCKF2, NSTCKF, and FSTCKF have no significant change on the x axis position estimation,and the estimation error for the velocity and turn rate become slightly larger.When k=50,the model inaccuracy disappears,the estimation error decreases rapidly, and it is completely stable to the previous error interval at k=55. Their ability to track inaccurate models is better than CKF, which shows that strong tracking filter can always track the state of the system accurately.

Fig.8 RMSE under inaccurate model.

Table 4 RMSE value of x position estimation of five algorithms at k=15.

The similar filtering performance of TSTCKF, TSTCKF2,NSTCKF and FSTCKF proves that the fading factor of nonlinear filtering established in this paper is equivalent to the traditional fading factor introduction method when high-order terms of measurement function hk+1（·） are relatively small,i.e. a ?b.

6. Conclusions

A new fading factor introduction method suitable for nonlinear filtering algorithm is proposed in this paper. The number of fading factors expand from one to two,and the introduction position is reselected. In theory, the new fading factor introduction method could extract the information of the secondorder term of the Taylor expansion. It is proved that the new fading factor introduction method is superior than traditional fading factor introduction method under different conditions. Then, the equivalent fading factor calculation method in CKF is established. Based on the characteristics of the cubature transform, the fading factor can extract information of the third-order term of Taylor expansion of the residual sequences, and the novel strong tracking CKF is proposed. Focusing on the shortcomings of the traditional strong tracking method process,this paper proposes FSTCKF,which fully reflects the advantages of two fading factors and reduces time complexity of the algorithm.The results of the two simulation examples indicate that NSTCKF and FSTCKF are superior than TSTCKF.Furthermore, the fading factor introduction method proposed in this paper is applicable to other nonlinear filtering algorithms such as UKF, CDKF and GHF. One of the future research topics would be to develop modified multiple fading factors strong tracking method that is more suitable for CKF.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 61573283).