A method and apparatus for smooth estimation of target tracking state based on maximizing correlation entropy

Abstract

This invention discloses a target tracking state smoothing estimation method and apparatus based on maximizing correlation entropy. The method involves acquiring a first target state vector before the current time and a second target state vector after the current time. Forward recursive filtering is performed based on the first target state vector, and backward recursive filtering is performed based on the second target state vector to obtain the estimated second target state vector at the current time. The first and second target state estimation vectors are then fused to obtain the fused target state estimation vector at the current time. This invention incorporates the maximization of correlation entropy method into the generation process of the fused target state estimation vector, considering all even-numbered terms of the estimation error statistical moments. This makes it highly robust to suppressing outliers with heavy tailing distribution characteristics and achieves satisfactory estimation accuracy in non-Gaussian noise environments.

Description

Technical Field

[0001] This invention belongs to the field of information fusion and target tracking technology, and particularly relates to a target tracking state smoothing estimation method and apparatus based on maximizing relevant entropy. Background Technology

[0002] Target tracking, especially point target tracking, refers to the process of obtaining measurement information of the observed target using sensing methods and estimating and optimizing the target's state based on target motion modeling (i.e., a dynamic model describing the evolution of variables of interest such as the target's position and velocity over time). Target state estimation, as the cornerstone of target tracking, refers to the process of reconstructing the state of the detected target from a noisy measurement sequence by designing reasonable optimization criteria. Target tracking state estimation methods are widely used not only in military fields such as long-range early warning, missile defense, and airborne optoelectronic detection, but also in civilian fields such as autonomous driving, integrated navigation, fault diagnosis, and air traffic control.

[0003] Traditional methods for smoothing the state of target tracking in non-Gaussian noise environments generally fall into two categories: Kalman-like smoothing and particle smoothing. Particle smoothing, which approximates the posterior probability distribution using a large number of weighted particles, can effectively handle model nonlinearity and non-Gaussian noise. However, the design of the required importance sampling function is often complex, especially when the system noise distribution is unknown. More importantly, particle smoothers are computationally intensive, making them unsuitable for practical applications with limited computing power, such as target tracking. For Kalman-like smoothing, Gaussian mixture smoothing algorithms are often used to effectively handle non-Gaussian noise. While Gaussian mixture smoothing algorithms achieve state smoothing recursion under the Gaussian mixture approximation of the posterior probability distribution by jointly executing multiple extended Kalman smoothers, unscented Kalman smoothers, or volumetric Kalman smoothers, resulting in lower computational complexity, they are highly sensitive to outliers caused by non-Gaussian noise because they are based on the MMSE criterion, leading to significant estimation errors.

[0004] In recent years, Kalman-like smoothing methods based on the maximization of correlation entropy criterion have emerged and been applied in practical engineering applications such as dynamic system state estimation. Unlike the MMSE criterion, which only considers the second-order statistical moment of the estimation error, the maximization of correlation entropy criterion considers all even-numbered terms of the statistical moment of the estimation error, thus exhibiting strong robustness in suppressing outliers caused by non-Gaussian noise. Existing Kalman-like smoothing methods based on the maximization of correlation entropy criterion are all based on the RTS structure. Although they can suppress outliers caused by non-Gaussian noise, their correlation entropy calculation process always uses the process noise covariance to replace the covariance of the difference between the smoothed estimate and the state prediction at the same time. This approximation is clearly ineffective in characterizing the covariance of the estimation difference when the target motion model is relatively accurate (i.e., the standard deviation of each dimension of the process noise is small), resulting in inaccurate calculation of the correlation entropy maximization cost function and a degradation in estimation accuracy. Summary of the Invention

[0005] The purpose of this invention is to provide a target tracking state smoothing estimation method and apparatus based on maximizing relevant entropy, so as to improve target tracking accuracy.

[0006] This invention adopts the following technical solution: a target tracking state smoothing estimation method based on maximizing relevant entropy, comprising the following steps:

[0007] The measurement sequence, the first target state vector before the current time, and the second target state vector after the current time are obtained respectively.

[0008] Based on the first target state vector, a forward recursive filter is performed to obtain the first target state estimation vector at the current time; based on the second target state vector, a backward recursive filter is performed to obtain the second target state estimation vector at the current time.

[0009] The first target state estimation vector and the second target state estimation vector are fused to obtain the fused target state estimation vector at the current time. During the fusion process, different weights are assigned to the first target state estimation vector and the second target state, and the weights are calculated based on the method of maximizing the relevant entropy.

[0010] Furthermore, both forward recursive filtering and backward recursive filtering are implemented based on the method of maximizing correlation entropy.

[0011] Furthermore, the first target state estimation vector is:

[0012]

[0013] in, Let be the first target state estimation vector at time k. Let be the predicted value of the first target state estimation vector at time k. i* This is the last iteration of the fixed-point iteration method. P is the first target state estimation vector after the last iteration. k|k-1 Let be the forward prediction error covariance at time k. The Gaussian kernel for the correlation entropy, z k Let k be the measurement value at time k in the measurement sequence. R is the estimated measurement value at time k. k Let k be the measurement covariance.

[0014] Furthermore, the first target state estimation vector is obtained through... It is derived that;

[0015] in, x k h is the true state vector of the target. k (x k ) represents the measurement value corresponding to the true state vector.

[0016] Furthermore, the second target state estimation vector is:

[0017]

[0018] in, Let be the second target state estimation vector at time k. Let be the predicted value of the second target state estimation vector at time k. M is the estimated vector of the second target state after the last iteration. k|k-1 Let be the backward prediction error covariance at time k.

[0019] Furthermore, the second target state estimation vector is obtained through... It is derived that;

[0020] in,

[0021] Furthermore, the fused target state estimation vector is:

[0022]

[0023] in, Let be the fused target state estimation vector at time k and in the (i+1)th iteration. Let P be the fused target state estimation vector for the i-th iteration at time k. k|k Let be the forward error covariance at time k.

[0024] Furthermore, the fused target state estimation vector is obtained through... It is derived that;

[0025] in,

[0026] Another technical solution of the present invention: a target tracking state smoothing estimation device based on maximizing relevant entropy, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-mentioned target tracking state smoothing estimation method based on maximizing relevant entropy.

[0027] The beneficial effects of this invention are as follows: By introducing the maximum correlation entropy method into the generation process of the fused target state estimation vector, this invention considers all even-numbered terms of the estimation error statistical moments, making it highly robust to the suppression of outliers with heavy tail distribution characteristics. It can achieve satisfactory estimation accuracy in non-Gaussian noise environments, and it has a recursive structure similar to the Kalman smoothing method, inheriting the advantage of low computational complexity, thus balancing high accuracy and low complexity. Attached Figure Description

[0028] Figure 1 This is a block diagram illustrating the principle of a target tracking state smoothing estimation method based on maximizing relevant entropy, according to an embodiment of the present invention.

[0029] Figure 2 This is a comparison chart of RMSE curves for position estimation and velocity estimation using different filtering / smoothing algorithms under the maximum correlation entropy criterion in the verification embodiment of this invention.

[0030] Figure 3 A comparison chart of RMSE curves for position estimation and velocity estimation using different smoothing algorithms (and EKF algorithm) in the verification embodiments of this invention;

[0031] Figure 4 This is a schematic diagram comparing the confidence regions of different algorithm state estimations in the verification embodiments of the present invention;

[0032] Figure 5 This is a comparison chart of the ARMSE curves of position estimation and speed estimation of various comparison algorithms under different core bandwidths in the verification embodiment of the present invention;

[0033] Figure 6 This is a schematic diagram comparing the ARMSE of position and velocity estimation of various comparison algorithms under different iteration numbers in the verification embodiment of the present invention. Detailed Implementation

[0034] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0035] For a target motion modeled as a stochastic linear system perturbed by Gaussian white noise, a Kalman estimator, including filtering and smoothing, is presented, providing state estimation results under the Minimum Mean Square Error (MMSE) criterion. This method has been widely used due to its simple recursive structure and low implementation complexity. However, in engineering applications, real-world systems are often nonlinear, making optimal nonlinear estimation based on the MMSE criterion difficult to achieve. Therefore, many scholars and engineers have designed corresponding nonlinear Kalman filter and smoothing methods using various approximation techniques. Generally, nonlinear Kalman filter methods can be divided into two categories. The first category is function approximation, which uses piecewise linear functions or low-order polynomials to replace the nonlinear state evolution model and measurement model; the extended Kalman filter is a representative algorithm of this category. The other category is posterior probability distribution approximation, which uses Gaussian or Gaussian mixture distributions to approximate the posterior probability density distribution of the state, resulting in a nonlinear Kalman filter with a recursive analytical form. This includes common unscented Kalman filters, volumetric Kalman filters, Gaussian-Hermitian filters, as well as Gaussian mixture filters and adaptive Gaussian mixture filters.

[0036] Compared to filtered estimation, smoothing estimation, by utilizing "future" measurements, can further improve estimation accuracy. It is primarily applied in scenarios where accuracy is far more critical than real-time performance, such as backtracking-based broken track adhesion in target tracking. Existing recursive smoothing algorithms based on the Kalman filter framework mainly have two structures: Rauch-Tung-Striebel Smoother (RTS) and Two-Filter Smoother (TFS). The former first performs forward filtering, then uses the difference between the smoothed estimate and the predicted estimate at the next time step to update the current filtering result to obtain the current smoothed estimate. The latter independently runs forward and backward filtering, then weights and fuses the irrelevant forward-filtered state estimates and backward-filtered state predictions to obtain the smoothed estimate at the current time step. For linear systems under Gaussian white noise, the state smoothing estimates obtained by RTS and TFS are consistent. For nonlinear systems, these two types of smoothing structures can obtain corresponding nonlinear RTS or TFS by using nonlinear Kalman filtering to process the nonlinearity of the model, including extended Kalman smoothing, odorless Kalman smoothing, fused Kalman smoothing, and Gaussian mixture smoothing.

[0037] For nonlinear systems perturbed by Gaussian white noise, the aforementioned Kalman-like filtering / smoothing methods can achieve satisfactory results. However, when non-Gaussian noise, especially impulse noise or outliers, is present, their estimation accuracy significantly decreases. In fact, such impulse noise or outliers often exhibit a heavily tailed distribution. In recent years, the Maximum Correntropy criterion has gained widespread attention in adaptive filtering because it considers all even-numbered terms of the statistical moments of the estimation error, unlike the MMSE criterion which only considers the second moment of the estimation error. This makes it highly robust to suppressing outliers with heavily tailed distributions. Some researchers have constructed Kalman filters under the Maximum Correntropy criterion, which have a recursive structure similar to Kalman filters and can achieve satisfactory filtering accuracy in non-Gaussian noise environments. Furthermore, by utilizing the nonlinear form of nonlinear Kalman-like filtering to handle the nonlinearity of the model, a series of nonlinear filters under the Maximum Correntropy criterion have been proposed.

[0038] Furthermore, based on the fact that the estimation accuracy of smoothing is always better than that of filtering, some scholars have proposed an RTS based on the maximization of correlation entropy criterion for nonlinear systems. This method, based on the extended Kalman filter-like approach of maximizing correlation entropy criterion in forward filtering, uses the difference between the smoothed estimate and the state prediction at the next time step to construct the corresponding correlation entropy maximization criterion and obtains the final state smoothness at the current time step through optimization. Meanwhile, some scholars have designed an RTS based on the volume rule under the maximization of mixed correlation entropy criterion, where a single correlation entropy is replaced by the sum of two identical correlation entropies with different weights to improve flexibility.

[0039] The aforementioned state smoothing methods under the correlation entropy maximization criterion are all based on the RTS structure. Considering that forward filtering and backward filtering in the TFS structure are performed independently, the final state smoothing result depends on the weighted fusion of the forward filtering estimate and the backward prediction estimate, and the two are independent of each other. This structure is highly advantageous for the accurate calculation of correlation entropy because all involved variables have prior known or online calculated covariance matrices. Considering that reliable and accurate correlation entropy calculation is a prerequisite for achieving high-precision state estimation, this invention designs a target tracking state smoothing estimation method based on the correlation entropy maximization criterion under the TFS structure, to achieve high-precision, low-complexity target state smoothing recursive estimation while retaining the advantages of the aforementioned methods.

[0040] In this invention, the point target dynamics model and sensor observation model are characterized as the following nonlinear system:

[0041] x k =f k (x k-1 )+w k-1 (1)

[0042] z k =h k (x k )+v k (2)

[0043] in, and Let f represent the n-dimensional state vector (i.e., the state of the point target, including position, velocity, etc.) and the m-dimensional measurement vector (i.e., sensor measurements, including radial moment, azimuth angle, etc.), respectively, and the nonlinear mapping f k (·) and h k (·) Assume that w is known k It has a mean of 0 and a standard covariance of Q. k White noise, v k It has a mean of 0 and a standard covariance R. k White noise. Meanwhile, for any k and l, we have... Furthermore, w k and v k All of these are non-Gaussian noise, and their probability distributions can be illustrated as follows:

[0044]

[0045] Here, ε k ∈[0,1] represents an unknown random variable. This indicates that the mean is 0 and the covariance is R. k Gaussian distribution, The representative value is 0 and the covariance is D. k The arbitrary unknown distribution. Obviously, Equation (3) describes a variety of different forms of probability distribution to model possible outliers, including Gaussian mixture noise, glint noise or heavy tail noise, etc.

[0046] In this embodiment of the invention, the initial state x0 has a known mean. The sum and covariance P0, and with {w k} and {v k} are irrelevant. Let Z be the measurement sequence at time t and before time t. 1:t Z 1:t ={z1,z2,…,z t The purpose of this invention is to utilize the measurement sequence Z. 1:t The state x is recursively estimated by maximizing the correlation entropy criterion. k Since k ≤ t, we first give the definition of correlation entropy. The correlation entropy between two random vectors x and y is defined as follows:

[0047] C(x,y)=E(κ(x,y))=∫∫κ(x,y)f(x,y)dxdy (4)

[0048] In the formula, f(x,y) is the joint probability distribution of x and y, and κ(x,y) is the Mercer kernel, usually represented by the Gaussian kernel function, i.e.:

[0049]

[0050] Where σ is the kernel bandwidth, e = xy, and Σ is the covariance of the error e.

[0051] Based on the above definition of entropy, state x k The smoothing estimation can be decomposed into three processes: forward filtering, backward filtering, and estimation fusion. Its recursive structure diagram is shown below. Figure 1 As shown in Table 1, the implementation flow of the method is as follows. A detailed implementation description is given below.

[0052] Table 1

[0053]

[0054]

[0055] Backward recursive filtering based on the maximization of correlation entropy criterion (recursively applying from time k+1 to time k)

[0056] d. Backward state prediction: According to equations (15) and (16), respectively... and M k|k+1 ;

[0057] e. Backward state update:

[0058] e.1 Order calculate and And calculate from this

[0059] e.2 Let i=1,…,i * Calculate in sequence and And calculate from this

[0060] f. Back-filtered output estimation: Let M is obtained by calculating from equation (23) k|k .

[0061]

[0062] Specifically, this invention discloses a target tracking state smoothing estimation method based on maximizing relevant entropy, comprising the following steps: acquiring a measurement sequence, a first target state vector before the current time, and a second target state vector after the current time; performing forward recursive filtering based on the first target state vector to obtain the first target state estimation vector at the current time; performing backward recursive filtering based on the second target state vector to obtain the second target state estimation vector at the current time; fusing the first target state estimation vector and the second target state estimation vector to obtain the fused target state estimation vector at the current time; wherein, during the fusion process, different weights are assigned to the first target state estimation vector and the second target state, and the weights are calculated based on the method of maximizing relevant entropy.

[0063] This invention introduces the maximum correlation entropy method into the generation process of the fused target state estimation vector, considering all even-numbered terms of the estimation error statistical moments. This makes it highly robust to the suppression of outliers with heavy tail distribution characteristics, and can achieve satisfactory estimation accuracy in non-Gaussian noise environments. Furthermore, it has a recursive structure similar to the Kalman smoothing method, inheriting the advantage of low computational complexity, and can balance high accuracy and low complexity.

[0064] 1. Forward recursive filtering based on the criterion of maximizing correlation entropy.

[0065] This forward recursive filtering process is implemented based on the method of maximizing correlation entropy. Let the measurement sequence Z be the data at time k-1. 1:k-1 ={z1,…,z k-1} state x k The estimated value is Its corresponding estimation error covariance is here Based on equation (1), the state prediction and prediction error covariance are calculated as follows:

[0066]

[0067]

[0068] in, f k (x k-1 At point The first-order Taylor expansion at that point is: Here, := indicates that the content on the left side of the equals sign is calculated using the content on the right side. In actual operation, let... for

[0069] Consider measuring z at time k k The data may contain outliers. We will now update the real state by maximizing the following related entropy:

[0070]

[0071] This embodiment includes:

[0072]

[0073] And there are Where, x k Let x be the true state vector of the target, x represent any error vector, and P be the corresponding covariance matrix.

[0074] make The optimal solution to equation (8) can be obtained by solving the following equation:

[0075]

[0076] here, Obviously, due to complex nonlinear effects, equation (10) is difficult to solve analytically. By using h... k (x k At point Perform a first-order Taylor expansion at that point, i.e. and Formula (10) can be reconstructed as: in, And H k Depend on Substitution. Due to α f,k and β f,k It still contains unknown x k Therefore, a fixed-point iteration strategy is adopted to obtain the asymptotic solution of equation (10), that is...

[0077]

[0078] Among them, the forward filter weighting weights and

[0079] Furthermore, and The superscript i indicates the i-th iteration. * This is the last iteration of the fixed-point iteration method. When i = 0,

[0080] After a series of equivalent transformations, time k is based on the measurement sequence Z 1:k State estimation (i.e., the first target state vector) can be calculated as follows:

[0081]

[0082] in, i* This indicates the last iteration. Simultaneously, the corresponding estimation error covariance is:

[0083]

[0084] in, P is the predicted value of the first target state estimation vector at time k. k|k-1 Let be the forward prediction error covariance at time k. For the Mercer kernel of the correlation entropy, a Gaussian kernel is chosen here, z k For the observation value at time k in the measurement sequence, R is the estimated measurement value at time k. k Let be the measurement covariance at time k.

[0085] 2. Backward recursive filtering based on the criterion of maximizing correlation entropy.

[0086] This backward recursive filtering process is implemented based on the method of maximizing correlation entropy. Let the measurement sequence Z be the data at time k+1. k+1:t ={z k+1 ,…,z t} state x k+1 The backward estimation is Its corresponding estimation error covariance matrix is and At this point, the backward filtering to obtain the state estimate at time k from time k+1 can be performed as follows. From equation (1), we can obtain:

[0087]

[0088] in, It is f k+1 The inverse function of (·). Through the point... and w k Performing a first-order Taylor expansion of equation (13) at the point = 0, we have:

[0089]

[0090] in, and here,

[0091] Based on equations (13) and (14), the backward state prediction is as follows:

[0092]

[0093]

[0094] Here, the linearized matrix and By order Obtained through calculation.

[0095] Considering the presence of non-Gaussian measurement noise in equation (2), the backward state update is achieved by maximizing the following correlation entropy:

[0096]

[0097] in

[0098]

[0099] make The optimal solution to equation (17) can be obtained by solving the following equation:

[0100]

[0101] in, and

[0102] Furthermore, h k (x k At point ( For x k The first-order Taylor expansion at the backward state estimate) Substituting into equation (19), we get:

[0103]

[0104] in, In equation (19), using Replace H k To obtain a concrete value rather than a symbolic expression.

[0105] At this time, α b,k and β b,k Still contains unknown state x k Therefore, the following fixed-point iteration is used to obtain the corresponding asymptotic solution:

[0106]

[0107] here, Let be the second target state estimation vector at time k. M is the predicted value of the second target state estimation vector at time k. k|k-1 The backward prediction error covariance at time k, and the backward filtering weights. and

[0108] Furthermore, The superscript i indicates the i-th iteration. When i = 0,

[0109] Through a finite number of iterations and equivalent transformations, time k is based on the measurement sequence Z. k:t backward state estimation The following can be calculated:

[0110]

[0111] here, i * This indicates the last iteration. Simultaneously, the corresponding backward estimation error covariance is:

[0112]

[0113] 3. Fusion smoothing based on the criterion of maximizing relevant entropy.

[0114] Based on Z 1:k Forward state estimation P k|k and based on Z k+1:t backward state prediction M k|k+1 Considering the presence of non-Gaussian noise, the estimation fusion of forward filtering and backward filtering is achieved using the following maximization of correlation entropy criterion to obtain the final smooth estimate, namely:

[0115]

[0116] here,

[0117]

[0118] make The optimal solution of equation (25) is obtained by solving the following equation:

[0119]

[0120] in Obviously, due to α k and β k Contains unknown x k This makes it difficult to obtain the analytical solution of equation (26).

[0121] Therefore, the following fixed-point iteration method is used to obtain The asymptotic solution is:

[0122]

[0123] Among them, the fusion smooth weighting weight and When i = 0 After a finite number of iterations, the final smoothed estimate is:

[0124]

[0125] Here, i * This indicates the last iteration. Also, since it's based on Z... 1:k Forward filtering Compared with Z-based k+1:t backward prediction Irrelevant, that is, Therefore, the output state is smooth. The estimation error covariance is:

[0126]

[0127] This completes the method of this invention embodiment. To illustrate the effectiveness of the method, the following verification embodiments were also performed.

[0128] The designed algorithm was verified through target backtracking simulation in the o-ξη plane under non-Gaussian noise. The tracked target moves in approximately uniform linear motion, and its state vector... The corresponding kinetic evolution model is x k =F k-1 x k-1 +w k-1 ,in T = 1s. w k It is zero-mean Gaussian white noise, and its covariance matrix is... Here, q ξ =q η =5m / s 2 The system runs for 150 seconds, and the initial state x0 = (50000m, -200m / s, 60000m, -250m / s). T .

[0129] The tracked target is detected and tracked by a radar located at the origin of the o-ξη coordinate system, and the corresponding measurement vector z k =(r k ,θ k ) T ,in Here, the measurement noise v k =(v r,k ,v θ,k ) T The probability distribution of follows equation (3), where Representing a uniform distribution, the random variable ε k It follows a Bernoulli distribution with a parameter of 0.1. Furthermore, R... k =diag{200 2 m 2 0.01 2 rad2}, In other words, for measurement noise v k For each element in a dimension, if the corresponding Gaussian distribution is... The corresponding uniform distribution is u(-25σ) k ,25σ k ).

[0130] A. Comparison of estimation accuracy.

[0131] The first-order Taylor expansion is used to handle the nonlinearity of the measurement model. The proposed method, MRTFS, is compared with EKF, RTS, TFS, MCEKF, and MCRTS. The full names and abbreviations of each algorithm are shown in Table 2. All algorithms have the same initial estimate. P 0|0 =10 5 ×diag{1,0.01,1,0.01}, where Δx0=(300,50,350,35) T The values ​​were selected randomly. Simultaneously, based on the measurement value at the last sampling time t=150, the backward initial state estimates of TFS and MRTFS were set to... The corresponding estimation error covariance is in, And ΔM t :=diag{0,10 10 ,0,10 10}. Here, the superscript "+" indicates the Moore-Penrose inverse operation, ΔM t It is any matrix whose second and fourth diagonal elements are large enough to represent it. This does not contain any useful velocity information. The coordinate transformation described above is a common method in target tracking to convert measurements in polar coordinates to states in the o-ξη coordinate system, without requiring any additional information. Furthermore, in MRTFS, MCEKF, and MCRTS, σ = 10 and i* = 5. In MCEKF and MCRTS, the solution to maximize the relevant entropy is also approximated using an iterative process; the final state estimate output is the state estimate and corresponding estimation error covariance of the last iteration.

[0132] Table 2

[0133]

[0134] For clarity, this is based on 1000 Monte Carlo simulations. Figure 2 The root mean square error (RMSE) curves for position and velocity estimation using different filtering and smoothing algorithms (MCEKF, MCRTS, and MRTFS) under the maximization of correlation entropy criterion are presented. Figure 3The root mean square error (RMSE) curves for position and velocity estimation using different smoothing algorithms (RTS, TFS, MCRTS, and MRTFS) and the EKF algorithm are presented. Here, the RMSE curve for the MRTFS method (MCRTS method) is shown in... Figure 2 and Figure 3 The results are consistent. Meanwhile, the average RMSE (ARMSE) of the position and velocity estimates for each comparison algorithm is shown in Table 3. For state estimation... For example, the calculation of RMSE and ARMSE of the q-th dimension element is defined as follows:

[0135]

[0136]

[0137] Wherein, the superscript i represents the i-th Monte Carlo simulation implementation and the subscript q represents the q-th dimension element; This represents the state estimation in the i-th Monte Carlo simulation. The q-th dimension element Represents the true state in the i-th Monte Carlo simulation. The q-th dimension element; N represents the total number of Monte Carlo simulations; k represents the current sampling time; and t represents the total number of system running times.

[0138] Table 3 shows the ARMSE of each comparative algorithm for position and velocity estimation.

[0139]

[0140] like Figure 2-3 As shown in Table 3, EKF clearly has the highest RMSE because it neither utilizes "future" measurements to improve filtering accuracy nor considers the existence of outliers. RTS and TFS have the second highest RMSEs because they utilize "future" measurement information, resulting in lower RMSEs than EKF, and the RMSE curves of these two algorithms are roughly the same. Meanwhile, the RMSEs of MCEKF, MCRTS, and MRTFS are all lower than those of EKF, RTS, and TFS. This is because the former are designed using the correlation entropy maximization criterion, which is robust to outlier suppression, while the latter are designed based on the minimum mean square error criterion and do not consider non-Gaussian noise. Furthermore, due to the utilization of "future" measurement information, MCRTS has a slightly lower RMSE than MCEKF.

[0141] Furthermore, the MRTFS designed in this invention implements the recursion of backward filtering by maximizing the correlation entropy criterion, thereby avoiding one weight being too large and another too small during the correlation entropy maximization optimization process. Simultaneously, the designed MRTFS achieves a weighted fusion of forward filtering estimation and backward prediction estimation under the correlation entropy maximization criterion, thus obtaining the final smooth estimate. This not only ensures that the smoothed state result of the fused output does not favor one side, but also improves the robustness to outlier suppression. Therefore, the RMSE of MRTFS is the smallest, far lower than that of MCRTS.

[0142] B. Confidence comparison.

[0143] Figure 4 A comparison of the confidence regions for positional state estimations by MCEKF, MCRTS, and MRTFS at times k = 20, 35, 50, 65, 80, 95, 110, 125, and 140 (from left to right and top to bottom in the figure) is presented. The confidence regions are calculated as follows: Where, x p,k Represents the target location variable. Indicates target location estimation, P p,k This represents the corresponding estimation error covariance, with parameter l = 3.

[0144] like Figure 4 As shown, the confidence region of MRTFS is smaller than that of MCEKF and MCRTS at all time points. The confidence region of MCRTS is either covered by the confidence region corresponding to MCKEF or overlaps with the confidence region of MCEKF. This means that the confidence of MRTFS smooth estimation is better than that of MCRTS, while the result of MCRTS smooth estimation is better than that of MCEKF filtering estimation.

[0145] C. Comparison of estimation accuracy under different kernel bandwidths.

[0146] Figure 5 The ARMSE curves for position and velocity estimation of MCEKF, MCRTS, and MRTFS are presented under 1000 Monte Carlo simulations with different kernel width σ values. Here, the σ values ​​are as follows: σ τ =1.0+0.15τ(τ=0,1,2,…,60),σ τ =σ τ-1+0.5(τ-60) (τ = 61, 62, ..., 120), where τ is a non-negative integer. Since ARMSE changes rapidly when σ is small and slowly when σ is large, the increment for the first 60 data points of σ is 0.15, and the increment for the next 60 data points is 0.5(τ-60) to characterize the wide range of kernel bandwidth values. Furthermore, to more clearly show the variation of ARMSE for different algorithms when σ is small, the ARMSE for σ varying between 1.45 and 32.5 (i.e., τ = 3, 4, ..., 69) is plotted on... Figure 5 In the subgraph.

[0147] like Figure 5 As shown, the ARMSE of MRTFS is always less than that of MCRTS until they coincide. Meanwhile, regardless of how σ changes, the ARMSE of MCRTS is always less than that of MCEKF. The changes in ARMSE of the three comparison algorithms as σ increases are as follows. When σ is small, the weight values ​​(α) calculated during the relevance entropy maximization optimization process... f,k and β f,k α b,k and β b,k and α k and β k The small values ​​of σ weaken the robustness of each algorithm to outlier suppression, resulting in very large ARMSE values ​​for all compared algorithms, indicating low estimation accuracy. As σ increases, the weight values ​​calculated during the relevance entropy maximization optimization process fall into a reasonable range. At this point, the ARMSE of each algorithm decreases rapidly until it reaches its minimum value near σ = 4. With further increases in σ, the ARMSE of each algorithm gradually increases until it reaches its second peak around σ = 50. Subsequently, the ARMSE of MCEKF and MRTFS remains almost unchanged, while the ARMSE of MCRTS decreases again until it converges with the ARMSE of MRTFS around σ = 600, and eventually remains unchanged. In summary, the MRTFS designed in this invention exhibits high state estimation accuracy across a wide range of kernel bandwidth σ, demonstrating its robustness to the kernel bandwidth parameter.

[0148] D. Comparison of estimation accuracy under different iteration numbers.

[0149] at last, Figure 6 The ARMSE curves for position and velocity estimation of MCEKF, MCRTS, and MRTFS are presented for different iteration numbers under 1000 Monte Carlo simulations, where the iteration number varies from i to n. * =1 to i * =10, with an increment of 1. Here, Figure 6The left vertical axis corresponds to the MRTFS algorithm, and the right vertical axis corresponds to MCEKF and MCRTS. Furthermore, σ = 3.5.

[0150] like Figure 6 As shown, with the number of iterations i * As the value increases, the ARMSE of each algorithm first decreases, and then decreases at i. * =2 or i * The algorithm rapidly reaches its minimum value at point 3, then slightly increases, and finally stabilizes. This means that both the designed algorithm and existing algorithms require only a small number of iterations to achieve stable, high-precision estimation, avoiding redundant computational costs. More importantly, regardless of the number of iterations, the ARMSE of MRTFS is lower than that of MCRTS and MCEKF, and the ARMSE of MCRTS is lower than that of MCEKE. This indicates that the accuracy of smoothing estimation is always better than that of filtering estimation. Furthermore, at different numbers of iterations, the estimation accuracy of the designed algorithm is superior to that of existing smoothing algorithms.

[0151] In summary, this invention designs a recursive backward filtering based on the maximization of correlation entropy criterion. For the nonlinear dynamic evolution model, a first-order Taylor expansion is used at the backward state estimation point to construct a reverse "linear" evolution model of the system state at the next and current time steps. Based on backward state prediction and current time measurement, a correlation entropy maximization criterion for backward state updating is designed, and an asymptotic iterative optimization solution for backward state estimation is achieved through a fixed-point iteration strategy.

[0152] Simultaneously, a fusion rule based on the maximization of correlation entropy criterion was constructed to obtain the final smooth estimate at the current time. Considering that the forward filtering estimation and backward state prediction based on the maximization of correlation entropy criterion are independent of each other because they do not share any measurement information, a weighted fusion framework based on the maximization of correlation entropy criterion was designed. Through a fixed-point iteration strategy, the fusion weights were learned online, and finally the recursive state smoothness under non-Gaussian noise environment was obtained.

[0153] The present invention provides a target tracking state smoothing estimation method based on the correlation entropy maximization criterion under non-Gaussian noise. Within a bidirectional filtering smoothing framework, it constructs a recursive backward filter based on the correlation entropy maximization criterion and designs a fusion rule based on the correlation entropy maximization criterion to obtain the final smoothed estimation result. On one hand, the proposed method is based on the correlation entropy maximization criterion, considering all even-numbered terms of the estimation error statistical moments. This makes it highly robust to suppressing outliers with heavy tailing distribution characteristics, achieving satisfactory estimation accuracy in non-Gaussian noise environments. Furthermore, it possesses a recursive structure similar to the Kalman smoothing method, inheriting the advantage of low computational complexity. On the other hand, the method proposed in this invention is based on a bidirectional filtering smoothing structure, in which forward filtering and backward filtering are performed independently. The final state smoothing result depends on the weighted fusion of the forward filtering estimate and the backward prediction estimate. Moreover, the two are independent of each other because they do not share any measurement. This structure is very beneficial for the accurate calculation of correlation entropy because each variable involved has a priori known or online calculated covariance matrix. This further ensures the robustness of the maximum correlation entropy criterion for outlier suppression and effectively improves the accuracy of state smoothing estimation under non-Gaussian noise environment.

[0154] The present invention also discloses a target tracking state smoothing estimation device based on maximizing relevant entropy, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the above-mentioned target tracking state smoothing estimation method based on maximizing relevant entropy.

[0155] It should be noted that the information interaction and execution process between the above-mentioned devices are based on the same concept as the method embodiments of the present invention. For details on their specific functions and technical effects, please refer to the method embodiments section, which will not be repeated here.

[0156] The device can be a computing device such as a desktop computer, laptop, handheld computer, radar, or cloud server. The device may include, but is not limited to, a processor and memory. Those skilled in the art will understand that it may include more or fewer components, or a combination of certain components, or different components; for example, it may also include input / output devices, network access devices, etc.

[0157] The processor referred to can be a Central Processing Unit (CPU), but it can also be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. A general-purpose processor can be a microprocessor or any conventional processor.

[0158] In some embodiments, the memory may be an internal storage unit of the extraction device, such as the hard drive or memory of the extraction device. In other embodiments, the memory may be an external storage device of the extraction device, such as a plug-in hard drive, Smart Media Card (SMC), Secure Digital (SD) card, or Flash Card equipped on the extraction device. Furthermore, the memory may include both internal storage units and external storage devices of the extraction device. The memory is used to store operating systems, applications, bootloaders, data, and other programs, such as the program code of the computer program. The memory can also be used to temporarily store data that has been output or will be output.

Claims

1. A target tracking state smoothing estimation method based on correlation entropy maximization, characterized in that, Includes the following steps: The measurement sequence, the first target state vector before the current time, and the second target state vector after the current time are obtained respectively. Based on the first target state vector, a forward recursive filtering is performed to obtain the first target state estimation vector at the current time. Based on the second target state vector, a backward recursive filtering is performed to obtain the second target state estimation vector at the current time. The first target state estimation vector and the second target state estimation vector are fused to obtain the fused target state estimation vector at the current time. During the fusion process, different weights are assigned to the first target state estimation vector and the second target state, and the weights are calculated based on the method of maximizing the relevant entropy. Both the forward recursive filtering and the backward recursive filtering are implemented based on the method of maximizing the correlation entropy; The fusion target state estimation vector is: ， wherein, is the fusion target state estimate vector at the kth time instant for the i+1th iteration, , is the fusion target state estimate vector at the kth time instant for the ith iteration, is the forward error covariance at the kth time instant, ; is the first target state estimate vector at the kth time instant, is the Gaussian kernel of the relevant entropy, is the prediction of the second target state estimate vector at the kth time instant, is the backward prediction error covariance at the kth time instant; The fusion target state estimation vector is obtained through It is derived that; in, 。 2. The target tracking state smoothing estimation method based on maximizing relevant entropy as described in claim 1, characterized in that, The first target state estimation vector is: ， in, Let be the predicted value of the first target state estimation vector at time k. , , This is the last iteration of the fixed-point iteration method. This is the estimated vector for the first target state after the last iteration. Let be the forward prediction error covariance at time k. , This represents the measurement value at time k in the measurement sequence. The estimated measurement value at time k. Let k be the measurement covariance. , .

3. The target tracking state smoothing estimation method based on maximizing relevant entropy as described in claim 2, characterized in that, The first target state estimation vector is obtained through It is derived that; in, , Let be the true state vector of the target. These are the measurement values ​​corresponding to the true state vector.

4. The target tracking state smoothing estimation method based on maximizing relevant entropy as described in claim 2, characterized in that, The second target state estimation vector is: ， in, Let be the second target state estimation vector at time k. , , This is the estimated vector of the second target state after the last iteration. , , .

5. The target tracking state smoothing estimation method based on maximizing relevant entropy as described in claim 4, characterized in that, The second target state estimation vector is obtained through It is derived that; in, 。 6. A target tracking state smoothing estimation device based on maximizing relevant entropy, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes a computer program, it implements the target tracking state smoothing estimation method based on maximizing relevant entropy as described in any one of claims 1-5.

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