Three State Estimation Fusion Methods Based on the Characteristic Function Filtering

There are three state estimation fusion methods for a class of strong nonlinear measurement systems, based on the characteristic function filter, namely the centralized filter, parallel filter, and sequential filter. Under ideal communication conditions, the centralized filter can obtain the best state estimation accuracy, and the parallel filter can simplify centralized calculation complexity and improve feasibility; in addition, the performance of the sequential filter is very close to that of the centralized filter and far better than that of the parallel filter. However, the sequential filter can tolerate non-ideal conditions, such as delay and packet loss, and the first two filters cannot operate normally online for delay and will be invalid for packet loss. The performance of the three designed fusion filters is illustrated by three typical cases, which are all better than that of the most popular Extended Kalman Filter (EKF) performance.


Introduction
In practical, filtering methods play an important role in state estimation, such as fault diagnosis, target tracking, signal processing, computer vision, communication, navigation, and other fields [1]. The traditional Kalman Filter (KF) has several good advantages, such as real-time, recursive, and optimal. It is only suitable for linear systems and Gaussian white noises [2]. However, these conditions are difficult to meet in the actual situation. In 1961, Bucy established a filtering method for nonlinear system based on Taylor expansion, called as Extended Kalman Filter (EKF); it was established by the first-order linear approximation to convert it to the standard KF form [3,4]. However, as the nonlinearity increases, the performance gradually decreases. In 1995, Julier established the Unscented Kalman Filter (UKF) based on the unscented transformation [5,6]. In 2009, Arasaratnam and Haykin established the Cubature Kalman Filter (CKF) based on the approximation of points [7,8]. Both the UKF and CKF use sigma interpolation to design a form similar to Kalman filter, so as to improve the influence caused by truncation error, and can achieve the second-order approximation [9][10][11][12]. However, no matter EKF, UKF, or CKF, it still cannot show better performance in strongly nonlinear systems.
When the modeling error of the system is described by the density function, Gordon et al. developed the particle filter (PF), employing the density function of the error as the objective function [13,14]. The PF can solve general non-Gaussian problems [15]. However, since PF is based on conditional probability density, the implementation of PF relies on a large number of particle samplings, which makes the calculation complexity high [16]. The degradation of particles during the resampling process will reduce the speed and accuracy of filters. The later-developed Ensemble Kalman Filter (EnKF) improved the computational complexity of high-dimensional [17], Maximum Correlation Entropy Kalman Filter (MCKF) canceled the requirement for white noise [18], etc., but they still

Problem Description
Consider a type of dynamic system whose state is linear and measured by several sensors [22]. For example, for a moving target in larger space area, there are multiple radars on the ground, distributed in different places, to simultaneously observe the target. The distribution between each radar and the target is far or near, and the transmission of information may be fast or slow. Moreover, after the radar collects the information, it needs to be transmitted to the fusion center for data processing, as shown in Figure 2. function [23]. Moreover, 1, 2, , i N =  represents the number of sensors. The measurement model of Equation (2) can be rewritten as a concise form, as follows:

Problem Description
Consider a type of dynamic system whose state is linear and measured by several sensors [22]. For example, for a moving target in larger space area, there are multiple radars on the ground, distributed in different places, to simultaneously observe the target. The distribution between each radar and the target is far or near, and the transmission of information may be fast or slow. Moreover, after the radar collects the information, it needs to be transmitted to the fusion center for data processing, as shown in Figure 2.

Problem Description
Consider a type of dynamic system whose state is linear and measured by several sensors [22]. For example, for a moving target in larger space area, there are multiple radars on the ground, distributed in different places, to simultaneously observe the target. The distribution between each radar and the target is far or near, and the transmission of information may be fast or slow. Moreover, after the radar collects the information, it needs to be transmitted to the fusion center for data processing, as shown in Figure 2. Therefore, a multi-dimensional state space model is established as follows:  (2) can be rewritten as a concise form, as follows: Therefore, a multi-dimensional state space model is established as follows: where X(r) ∈ R n×1 and Y i (r) ∈ R m i ×1 are state vector and measurement vector respectively; w(r) and v i (r) are system noises; Γ(r, r − 1) is the state transition matrix; λ(r, r − 1) is the known process drive noise; and h i (·) is a continuous smooth nonlinear function [23]. Moreover, i = 1, 2, · · · , N represents the number of sensors. The measurement model of Equation (2) can be rewritten as a concise form, as follows: Correspondingly, the multiple nonlinear measurement models in Equation (2) become the following: where Y(r) ∈ R m , m = m 1 + m 2 + · · · + m N .

Centralized Characteristic Function Filter Design
On the basis of the Equations (1) and (6), design filters in the form of Equations (7) and (8) are as follows: whereX(r|r) ∈ R n×1 andŶ(r|r − 1) ∈ R m×1 are state estimate vector and measurement prediction vector respectively. K(r) ∈ R n×m is the filter gain matrix to be designed.
In probability theory, the characteristic function of any random variable has been clearly given to completely define its probability distribution. Therefore, on the basis of KF, replace the probability density with characteristic function, and a new form of CFF is used to study the fusion algorithm. For any random variable X, we denote its characteristic function as ψ X (t), which is defined as follows: ψ X (t) = E(e itX ) (15) where, t represents any real number, and E represents the expected value. In Equation (15), the right side of the equation is given by the Riemann-Steelches integral: where F X (x) is the distribution function of random variable X.
If the probability density function of the random variable X exists, Equation (16) can be further written as follows: where f X (x) is the probability density function of random variable X.
In Reference [19], two lemmas of characteristic function are given: Assuming multidimensional vector x ∈ R n , z ∈ R n and ψ z (x) is the characteristic function of the strict system output. Define The characteristic function expression of random variable X is ψ X (t), t ∈ R m . Then the characteristic function of X can be expressed as follows.
Lemma 2. For two independent random variables, X 1 and X 2 , let X = X 1 + X 2 , X 1 , X 2 ∈ R n , and then we get the following: Usually, it is more complicated to directly use the probability density function to solve the analytical solution of the K(r) [24]. Here, we use the characteristic function to replace the probability density function to solve Equation (14). First, combine the definition of characteristic function and Equation (18), take the probability density function on both sides of Equation (14) at the same time, and then take the characteristic function at the same time, and we can get the following: where p means probability density function, and ψ means characteristic function. Combining Lemmas 1 and 2, Equation (20) becomes the following: By combining Lemmas 1 and 2 and Equation (21), the characteristic function propagation equation of the error recurrence equation is obtained:

Establishment of Filter Performance Index
It is pointed out in References [20,25] that K-L divergence can be used to quantify the difference between two different probability distributions. On the premise that the conditional estimation error characteristic function is obtained, and an objective function is given, the filter design can be carried out by describing the difference between the conditional estimation error characteristic function and the objective function. Therefore, the performance index of the characteristic function filter is designed as follows: where M(r) represents the filter performance index, M 0 (r) is the parameter, R(r) is a positive definite weight matrix, and K(r) is the gain matrix to be estimated. In Equation (23), let the following be [21]: where Λ(t) represents the weight function, ψ g (t) is the characteristic function of the objective, and ψ γ(r) (t) is the characteristic function of the error. (24), Λ(t) is introduced to ensure that M 0 (r) is bounded. Because M 0 (r) is a parameter, in order to ensure that it is non-negative, a transpose is multiplied to the right-hand side in Equation (24). For example, when the target is measured, the distance is non-negative, and the solution to the radial distance is in the form of a square, so the non-negativity of the parameters is guaranteed by multiplying by a transpose. The term log in Equation (24) can be understood as information entropy. The closer the values of ψ g (t) and ψ γ(r) (t) are, the better the performance of the filter. Moreover, in Equation (8), the measurement model of the system is multi-dimensional, so the objective function is in the form of a matrix. Then in Equation (23), M(r) is a multi-dimensional form, so the gain matrix K(r) is also multi-dimensional.

Establishment of Equation for Solving Filter Gain Matrix K(r)
Bring Equation (22) into Equation (24), and then expand Equation (24). In the expanded Equation (24), let the following be: Combining Equations (23)- (26), Equation (23) can be rewritten as follows: By solving the first-order partial derivative of Equation (27) and taking it to be zero, we have the following: Then, taking the second-order partial derivative of Equation (27), we have the following: In Equation (29), when the second-order partial derivative is greater than zero, the solution of K(r) in Equation (28) is the minimum value. Bring the concentrated gain matrix K(r) into Equation (27), and the obtained estimated value is the optimal estimated value of the centralized fusion method.

Parallel Filter Design
When the distribution position of each sensor and the distance from the fusion center is different, the centralized use of these sensors will increase the communication cost [26]. Moreover, as the number of sensors, N, increases, the filtering iteration process will become more complicated. Therefore, a class of distributed parallel filters is designed. For each sensor, there are the following: where i represents the ith sensor. Based on a network composed of multiple distributed sensors, design a parallel characteristic function filter as shown in Equation (32).
The parallel fusion filtering process is shown in Figure 3.

Solve Each Gain Matrix K i (r) in the Parallel Filter Group
Follow and repeat the process of Sections 3.2-3.4. This subsection establishes an iterative numerical solution algorithm for the gain matrix, K i (r), based on the fixed point principle [27,28] with Equation (28), and construct a fixed point equation K i t (r) = d(K i t (r)), to iteratively solve the K i (r): where t = 0, 1, 2, · · · , T i (r) represents the iteration steps. Therefore, based on the ith group of sensors, the gain matrix, K i (r), of its characteristic function filter is obtained, which is substituted into the parallel filter Equation (32), and the filter in the form of Equation (34) is obtained.X The estimated value obtained at this time is the optimal solution of the parallel fusion method.

Sequential Characteristic Function Fusion Filtering under Multi-Dimensional Observation
Considering that the distance between the sensor and the fusion center is different, the problem of transmission data delay and even packet loss due to network bandwidth constraints will occur. Then, on the basic of Equation (2), considering the reasons for network delay and packet loss, the measured value of the sensor data transmitted to the fusion center through the wireless network is marked as follows and the sensor fusion process is shown in Figure 4 [29]: where, N means that we share N sets of sensors for measurement; L is the number of sensors that transmit data to the fusion center, after taking into account packet loss; and j i is the order in which the sensors arrive at the fusion center.
substituted into the parallel filter Equation (32), and the filter in the for is obtained.
The estimated value obtained at this time is the optimal solution of method.

Sequential Characteristic Function Fusion Filtering under Multi-D servation
Considering that the distance between the sensor and the fusion the problem of transmission data delay and even packet loss due to n constraints will occur. Then, on the basic of Equation (2), considering t work delay and packet loss, the measured value of the sensor data tra sion center through the wireless network is marked as follows and the cess is shown in Figure 4 [29]: where, N means that we share N sets of sensors for measurement of sensors that transmit data to the fusion center, after taking into accou i j is the order in which the sensors arrive at the fusion center.
Remark: In this paper, we divide packet loss into two types. One is mitted to the fusion center in time, and the other is indeed lost data.

Sequential Filter Design
Considering the phenomenon of packet loss, it is assumed that the urement from each sensor arrives at the fusion center is 1, 2, , ( L L  quential characteristic function fusion filter as follows:

Remark 4.
In this paper, we divide packet loss into two types. One is the data not transmitted to the fusion center in time, and the other is indeed lost data.

Sequential Filter Design
Considering the phenomenon of packet loss, it is assumed that the order of the measurement from each sensor arrives at the fusion center is 1, 2, · · · , L (L ≤ N). Design a sequential characteristic function fusion filter as follows: where Y j i (r) ∈ R m j i ×1 , K j i (r) ∈ R n×m j i , and j i indicate that the order of group i sensors reaching the fusion center is j.

Remark 5.
The sequential consideration of the order of information arrival, and first come first fusion. So use j i to indicate that the data collected by the i-th sensor is the j-th transmission to the fusion central. Moreover, m j i represents the dimensionality of the observation value of the i-th sensor. Because the observation is multi-dimensional and the number of sensors is also multiple, thus the observation value of each sensor is also multi-dimensional. In order to add distinguish and clearly describe, we use m j i to represent it.

The Establishment of Error Recurrence Equation
Follow and repeat the process of Section 3.2, to get the following: Then the error recurrence equation is as follows: Equation (38) simplifies to become the following: Take the probability density function and characteristic function on both sides of Equation (39), we can get the following: Combine the two lemmas given in Section 3 and Equation (39), and there is the following:

Establishment of Filter Performance Index
Follow and repeat the process of Sections 3.3 and 3.4, and the filter performance index can be described as follows: Then M j i can be rewritten as follows: Solve the first two-order partial derivative of Equation (43). The second-order partial derivative is greater than zero, and the filter gain matrix obtained by first-order partial is the optimal solution under the minimized performance index, M j i .

Remark 6.
When the performance index has multiple poles, traditional solving methods may lead to local extremes and cause large errors, so we still introduce the fixed point equation as in Section 4.2.

Establishment of Equation for Solving Gain Matrix K j i (r)
According to Section 4.2, construct a fixed point equation like K j i (r) = d(K j i (r)), to iteratively solve the filter gain matrix: where t = 0, 1, 2, · · · , T j i (r) represents the iteration steps of the fixed point method. (45), the iterative process is ended.
Then Equation (35) can be rewritten as follows: Until the L-th sensor transmits the information to the fusion center and completes filtering, the estimated valueX L i (r|r) obtained at this time is the optimal estimated value of the sequential fusion method.

Implementation Process of Sequential Characteristic Function Fusion Filtering Algorithm under Multi-Dimensional Observation
(1) Initialization:X (2) Set arrival order: At the r moment, assume that the measured value of the sensor transmitted to the fusion center via the wireless network is Y 1 (r), Y 2 (r), · · · , Y i (r), · · · , Y L (r), and then we get the following: (3) Filter design: Step 1: Design filters based on Equations (35) and (36).
Step 3: Solve the error characteristic function propagation equation according to two lemmas.
Step 4: Obtain the performance index function according to Equation (42).
Step 5: Establish the filter gain solving equation according to Equation (43).
Step 6: Solve the filter gain matrix K i t+1 (r) according to Equation (44).
Step 7: Substitute the solved K j i (r) into Equation (49). (4) Repeat the above process.
Step 1: Obtain the state prediction value of the first sensor arriving at the fusion center Step 2: Solve the filter gain of the first sensor that reaches the fusion center Step 3: Calculate the state estimate value of the first sensor that reaches the fusion center.X Step 4: Take the first arrival state estimation valueX 1 i (r|r) obtained in Step 3 as the second arrival state prediction value.
After another round of cyclic Equations (56)-(58), the state estimation value of the second sensor that reaches the center can be obtained, denoted asX 2 i (r|r).
Step 5: Repeat the above steps, until all measurements in the L(L ≤ N) group are transmitted to the fusion center, the iteration ends. The corresponding estimated value is the optimal estimated value of the sequential characteristic function filtering. That iŝ X(r|r) =X L i (r|r).

Simulation
This paper uses three typical nonlinear systems to illustrate the effectiveness of the proposed three fusion methods. The first category is to imitate actual target tracking, the second category is from real industrial devices, and the third category is a general nonlinear model. Table 1 is used to present the application of three typical cases and the reasons why these cases were selected. Generally, the measurement of space targets is carried out in the polar coordinate system, and it is usually necessary to perform a unified transformation in the rectangular coordinate system before further data processing. The method based on characteristic function in this paper, avoids the error caused by conversion and also avoids the rounding error caused by the linearization process.

Case 2 Industrial device measurement system
For high-precision industrial devices, the measurement equations must be complicated, and may be super-nonlinear equations. In order to obtain very accurate parameter results, the coordinated measurement of multiple sensors can be used.

Case 3 General nonlinear system
In order to make the fusion method universal, a general case is used to demonstrate the effectiveness of the proposed methods.
Case 1: Given a class of target tracking system, which is composed by where X 1 , X 2 , X 3 , X 4 , X 5 , X 6 represent the position and velocity on the x, y, z axes, and Y 1 , Y 2 , Y 3 respectively represent the radial distance between the target and sensors, and the two direction angles formed by the coordinate axis. In this case, three sensors are used to carry out the experiment. We simulate different measurement environments by setting different measurement noise covariance.
We perform characteristic function filtering on the three sensors, respectively. In order to make the result analysis clearer, we only give the result graph of x1 in Case 1, and focus on it. The analysis of state x2-x6 is the same as x1, and the numerical results are all given in the table. The estimation error of x1 is shown in Figure 5. nsors 2020, 20, x Due to the random noise generated in the simulation experiment, w Monte Carlo average value of 100 times for the filtering result, and the mean is recorded in Table 2.  Due to the random noise generated in the simulation experiment, we obtain the Monte Carlo average value of 100 times for the filtering result, and the mean square error is recorded in Table 2. In Table 2, it can be seen that the accuracy of each sensor is different. Then select the highest precision sensor, and perform CFF and EKF, to further study the filtering effect of CFF in nonlinear systems, as shown in Figure 6.
Due to the random noise generated in the simulation experiment, we ob Monte Carlo average value of 100 times for the filtering result, and the mean squa is recorded in Table 2. In Table 2, it can be seen that the accuracy of each sensor is different. Then s highest precision sensor, and perform CFF and EKF, to further study the filtering CFF in nonlinear systems, as shown in Figure 6. From Figure 6, we can clearly see that the CFF filtering effect is better than better analyze the results, we recorded estimation error in Table 3, and also recor accuracy improvement ratio of using the most accurate sensor for CFF, as comp EKF.  From Figure 6, we can clearly see that the CFF filtering effect is better than EKF. To better analyze the results, we recorded estimation error in Table 3, and also recorded the accuracy improvement ratio of using the most accurate sensor for CFF, as compared to EKF. From the experimental results in Table 3, it can be analyzed that the effect of using CFF is better than that of EKF in nonlinear systems.
To further improve the estimation accuracy of CFF in nonlinear systems, we study the fusion method based on CFF. When using sequential fusion filtering method, the accuracy of the sensor is carried out, from low to high, to simulate the order in which sensors transmit information to the fusion center, as shown in Figure 7.
From the experimental results in Table 3, it can be analyzed that the effect o CFF is better than that of EKF in nonlinear systems.
To further improve the estimation accuracy of CFF in nonlinear systems, we the fusion method based on CFF. When using sequential fusion filtering method, curacy of the sensor is carried out, from low to high, to simulate the order in which s transmit information to the fusion center, as shown in Figure 7. It can be directly observed from Figure 7 that the estimation accuracy of the c ized fusion method is significantly higher than that of the parallel fusion metho higher than or even close to the sequential fusion method. To make the results mo vincing, we also recorded the numerical results, as shown in Table 4. From the data in Table 4, it can be concluded that the filtering effect of the three methods is better than that of using only one sensor, and the centralized fusion m has the highest accuracy.
For space-moving targets, especially high-speed moving targets, when the t velocity increases, the target's motion state will change, and the nonlinear charact of the system will also increase. In order to further explore the filtering effect of the method in nonlinear system, we further study by changing the initial velocity a given characteristic function, as shown in Table 5. It can be directly observed from Figure 7 that the estimation accuracy of the centralized fusion method is significantly higher than that of the parallel fusion method, and higher than or even close to the sequential fusion method. To make the results more convincing, we also recorded the numerical results, as shown in Table 4. From the data in Table 4, it can be concluded that the filtering effect of the three fusion methods is better than that of using only one sensor, and the centralized fusion method has the highest accuracy.
For space-moving targets, especially high-speed moving targets, when the target's velocity increases, the target's motion state will change, and the nonlinear characteristics of the system will also increase. In order to further explore the filtering effect of the fusion method in nonlinear system, we further study by changing the initial velocity and the given characteristic function, as shown in Table 5. It can be analyzed from the data in Table 5 that, with the enhancement of system nonlinearity, the filtering effect of CFF is significantly better than that of EKF. At the same time, no matter which method of multiple sensor fusion, with the enhancement of nonlinearity, it is better than the filtering effect of using only a single sensor.

Remark 8.
When conducting multiple sets of experiments, we only discussed the changes of the initial velocity and the target characteristic function in this case. This is because, when the velocity is very large, the measurement equation is almost a super-nonlinear equation, so the change of velocity can cause a very obvious change in the degree of system nonlinearity. We also tried to change µ and Q in the weight function Λ(t), but we found that the change has very little effect on the result. This is because the appearance of Λ(t) is to ensure that M 0 is bounded, so as long as M 0 is bounded, ans changes in µ and Q will not significantly affect the results.
Case 2: Given a parameter identification system for a type of industrial device.
In the actual measurement of industrial devices, as shown in Figure 8, the state equations are generally not complicated, but in order to obtain very accurate parts size and other parameters, the measurement equations are often very complicated and exhibit nonlinear characteristics. Therefore, multiple sensor fusion methods are usually considered to further improve the estimation accuracy of parameters, especially in nonlinear systems that require very high accuracy.
sin(αX 1 (r)) + cos(αX 2 (r)) sin(βX 2 (r)) + cos(βX 1 (r)) It can be analyzed from the data in Table 5 that, with the nonlinearity, the filtering effect of CFF is significantly better than time, no matter which method of multiple sensor fusion, with the earity, it is better than the filtering effect of using only a single se Remark: When conducting multiple sets of experiments, changes of the initial velocity and the target characteristic func because, when the velocity is very large, the measurement equ nonlinear equation, so the change of velocity can cause a very ob gree of system nonlinearity. We also tried to change μ and Q ( ) t Λ , but we found that the change has very little effect on the re appearance of ( ) t Λ is to ensure that 0 M is bounded, so as long changes in μ and Q will not significantly affect the results.

Case 2: Given a parameter identification system for a type of ind
In the actual measurement of industrial devices, as shown in tions are generally not complicated, but in order to obtain very other parameters, the measurement equations are often very comp linear characteristics. Therefore, multiple sensor fusion methods to further improve the estimation accuracy of parameters, especia that require very high accuracy.  The detailed analysis steps are the same as in Case 1. Thus, in Cases 2 and 3, we did not give a very detailed description as in Case 1, but simplified the expression. When α = β = 0.1, the three sensors perform CFF separately and, at the same time, perform EKF on the sensor with the highest accuracy. The results are shown in Figures 9 and 10. over, given a weight function  Similarly, in order to make the results more clearly presented, all num are recorded in Table 6. From the data in Table 6, we can conclude that the filtering effect of C cantly better than that of EKF. We also compared the three fusion methods b unit matrix.
The detailed analysis steps are the same as in Case 1. Thus, in Cases 2 not give a very detailed description as in Case 1, but simplified the expr = =0.1 α β , the three sensors perform CFF separately and, at the same time on the sensor with the highest accuracy. The results are shown in Figures 9 Figure 9. x1 estimation error. Similarly, in order to make the results more clearly presented, all num are recorded in Table 6. From the data in Table 6, we can conclude that the filtering effect of cantly better than that of EKF. We also compared the three fusion methods Similarly, in order to make the results more clearly presented, all numerical results are recorded in Table 6. From the data in Table 6, we can conclude that the filtering effect of CFF is significantly better than that of EKF. We also compared the three fusion methods based on CFF, as shown in Figures 11 and 12. ensors 2020, 20, x  The numerical results are also compared with the estimated results of t sor with the highest accuracy. All numerical results are recorded in Table 7    The numerical results are also compared with the estimated results of t sor with the highest accuracy. All numerical results are recorded in Table 7   The numerical results are also compared with the estimated results of the single sensor with the highest accuracy. All numerical results are recorded in Table 7.  Table 7, we can get that, no matter which fusion method, the filtering effect is better than that of using only a single sensor. In order to further explore the filtering effect of the fusion method in the nonlinear system, we further study by changing α, β, and the given characteristic function ψ g (t), as shown in Table 8. From the data in the Table 8, it can be seen that, when the measurement model of the system is a more complex nonlinear model, the multi-sensor fusion algorithm based on CFF can obtain higher accuracy. At the same time, no matter which fusion method is used, the estimation accuracy is higher than that of only one sensor.

Case 3: Given a general nonlinear system
In case 3, the characteristic function fusion filtering algorithm is further extended to systems with weakly nonlinear state models. In order to better describe weak nonlinearity, we introduce trigonometric functions in the state equation. The general nonlinear model is shown as follows.
In Case 3, three sensors are also used for simulation, and the parameters of the three sensors are set as α 1 = β 1 = 0.1, α 2 = β 2 = 0.2, α 3 = β 3 = 0.3. The three sensors perform CFF separately and, at the same time, perform EKF on the sensor with the highest accuracy. The results are shown in Figures 13 and 14. sors 2020, 20, x  Similarly, all numerical results are recorded in Table 9. It can be obtained from the data in Table 9 that, for general nonlinear eq can still obtain better filtering effects than EKF. Three fusion methods based also compared, as shown in Figures 15 and 16. Moreover, the numerical resu  Similarly, all numerical results are recorded in Table 9. It can be obtained from the data in Table 9 that, for general nonlinear eq can still obtain better filtering effects than EKF. Three fusion methods based also compared, as shown in Figures 15 and 16. Moreover, the numerical res pared with the estimated results of the single sensor with the highest accu Similarly, all numerical results are recorded in Table 9. It can be obtained from the data in Table 9 that, for general nonlinear equations, CFF can still obtain better filtering effects than EKF. Three fusion methods based on CFF are also compared, as shown in Figures 15 and 16. Moreover, the numerical results are compared with the estimated results of the single sensor with the highest accuracy. All numerical results are recorded in Table 10.   From the data in the Table 10, it can be seen that, for the general nonlin ment equation, the multi-sensor fusion algorithm based on CFF can obtain racy. It also shows that three fusion methods based on CFF not only achieve g effects in systems where the state is linear and measured as nonlinear, but als where the state model is weakly nonlinear.
Generally speaking, in the actual system, according to different require   From the data in the Table 10, it can be seen that, for the general nonlin ment equation, the multi-sensor fusion algorithm based on CFF can obtain racy. It also shows that three fusion methods based on CFF not only achieve g effects in systems where the state is linear and measured as nonlinear, but al where the state model is weakly nonlinear.
Generally speaking, in the actual system, according to different requir error error Figure 16. Case 3 x2 fusion estimation error. From the data in the Table 10, it can be seen that, for the general nonlinear measurement equation, the multi-sensor fusion algorithm based on CFF can obtain higher accuracy. It also shows that three fusion methods based on CFF not only achieve good filtering effects in systems where the state is linear and measured as nonlinear, but also in systems where the state model is weakly nonlinear.
Generally speaking, in the actual system, according to different requirements, such as high precision, being easy to implement, or as close to the actual situation as possible, we can choose different fusion methods for state estimation.

Conclusions
In this study, three fusion filters were designed for a class of strong nonlinear measurement systems based on CFF, namely as centralized, parallel, and sequential. They were designed to meet the different needs of the systems, such as accuracy, being easy to implement, or matching with the actual environment. The performances of the three fusion filters are illustrated by three typical cases. Since EKF is the most popular method in the nonlinear system, we compared CFF with EKF. The results show that, under the same conditions, all filters show good performance, but the performance of the three fusion filters we designed is better than the most popular EKF performance, respectively. The fundamental reason is that the performance of CFF is better than that of EKF: (1) On the basic of introducing new performance indicators, the proposed CFF avoids the large truncation error caused by Taylor expansion like EKF. (2) CFF relaxes the requirements for the statistical characteristics of the error, and EKF's requirement for the ideal white noise of the error is replaced by the characteristic function of the error in the CFF.
Motivated by the results of this paper, we need to further think about the following issues: (1) These results were all established under the condition that the characteristic function of the target was given. In complex situations, how to make an accurate characteristic function or whether the characteristic function of the actual system can be obtained through a certain solution method still needs further study. (2) We imitated the EKF to solve the weakly nonlinear state, but in the face of a large number of strongly nonlinear state models, how to design a CFF model suitable for strong nonlinearity is still worth studying. (3) The fusion methods in this paper were all established under the condition that the error characteristic functions were independent. How to design the corresponding CFF when the error characteristic functions are related is worthy of further study.