Measurement conversion Kalman filtering algorithm based on predicted value second-order Taylor expansion for transmitting and receiving separated radar

Abstract

The invention belongs to the technical field of radar target tracking, and provides a prediction value second-order Taylor expansion-based measurement conversion Kalman filtering algorithm for a transmitting and receiving separated radar. In order to solve the problem of nonlinearity between two-way distance and angle measurement and a target state in transmitting and receiving split radar target tracking, distance measurement information based on a receiving node is obtained through a geometrical relationship, and linear conversion measurement in a Cartesian coordinate system is constructed in combination with angle measurement. A second-order Taylor series expansion method is adopted to analyze a mean value of measurement conversion results, unbiased measurement conversion is obtained through depolarization processing, and second-order statistical characteristics of measurement conversion errors are analyzed at the same time. Wherein the prediction value is introduced to further improve the calculation precision of the statistical magnitude. Simulation results show that the method shows higher convergence speed and better steady-state error, has better convergence performance, and can obtain better tracking performance compared with a traditional method.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar target tracking, and in particular relates to a target tracking system and method based on measurement conversion Kalman filtering of a transmitting and receiving split radar. Background Art

[0002] The modern battlefield environment is becoming increasingly complex, and various anti-radar measures are emerging one after another, highlighting the importance of combat equipment survivability. Compared with co-located radars, multi-base radar systems with separate transmitters and receivers can be passive when the transmitting radar is active, which improves the safety and concealment of the radar, can better deal with enemy electronic reconnaissance on the military battlefield, and have better survivability.

[0003] When tracking a target, the measurement information of the split-transmitter radar includes the dual-base distance measurement and the receiving node angle measurement. Therefore, split-transmitter radar target tracking is a nonlinear problem. At this time, the use of standard linear filtering algorithms such as Kalman filtering is prone to cause large errors and divergence, which cannot meet engineering needs. Therefore, a large number of scholars have proposed various suboptimal nonlinear filtering algorithms in two main directions, namely, the approximation of nonlinear functions and the measurement conversion of nonlinear measurements, to solve this problem.

[0004] The approximation of nonlinear functions is mainly divided into two paths: linearization of nonlinear functions and approximation of the probability density function of nonlinear functions. The extended Kalman filter (EKF) algorithm (V. Sastry. Decomposition of the extended Kalman filter [J]. IEEE Trans. On Automation Control, 1971, 16 (3): 260-261) approximates nonlinear functions based on Taylor expansion to achieve the purpose of linearization, and then processes them according to the standard Kalman filter algorithm. However, when the nonlinearity of the nonlinear system is strong, the linearization error will increase. In severe cases, the filter estimation result will diverge. In addition, when the nonlinear function is more complex, it is difficult to solve the Jacobian matrix, which increases the difficulty of implementing the algorithm. The unscented Kalman filter (UKF) algorithm (SJJulier, JKUhlmann. Unscented filtering and nonlinear estimation [J]. Proc. Of the IEEE, 2004, 92 (3): 401-422) approximates the probability density function of a nonlinear function by sampling through an unscented transformation. It is suitable for systems of any degree of nonlinearity, but the amount of calculation is greater than that of the EKF algorithm. For non-Gaussian noise scenarios, the particle filter (PF) algorithm (MSArulampalam, S. Maskell, N. Gordon, et al. A tutorial on particle filters for online nonlinear / non-Gaussian Bayesian tracking [J]. IEEE Trans. on Signal Processing, 2002, 50 (2): 174-188) has excellent tracking performance. A set of random samples is generated in the state space, and the posterior probability density function is approximated in the form of a weighted sum. When the number of particles tends to infinity, the accuracy of the algorithm approaches the optimal estimate. However, this algorithm has problems with importance function selection and resampling sample exhaustion, and the amount of calculation is extremely large.

[0005] For the measurement conversion of nonlinear measurement, H. Marom et al. proposed a method for converting unbiased and statistically consistent position measurements of bistatic radar to Cartesian coordinates based on Taylor expansion (H. Marom, Y. Bar-Shalom and B. Milgrom. Converting bistatic radar measurements to Cartesian position for tracking [J]. Defense + Commercial Sensing, 2022), and extended it to three-dimensional space (H. Marom, Y. Bar-Shalom and B. Milgrom. Unbiased Conversion of 3-D Bistatic Radar Measurements to Cartesian Position [J]. IEEE Transactions on Aerospace and Electronic Systems, vol. 59, no. 2, pp. 1613-1623, April 2023, doi: 10.1109 / TAES.2022.3203958). However, when deriving the statistical characteristics of the measurement error, only the angle measurement error is considered, and the influence of the distance measurement error is ignored. At the same time, after the target tracking is stable, the target prediction information is often more accurate than the measurement information. Therefore, when calculating the statistical characteristics of the measurement error, the prediction information is used to derive. However, in the existing methods, on the one hand, the prediction error caused by the prediction value is not considered, and on the other hand, the measurement error is still used in some places, which does not match the prediction information.

[0006] Based on this, the present invention takes into account both the angle measurement error and the distance measurement error in two-dimensional space, and adopts the prediction error to express the influence of introducing the prediction value, and proposes a measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the prediction value of the transmitting and receiving radar to realize the unbiased conversion of the measurement value from polar coordinates to Cartesian coordinates, so as to linearize the nonlinear measurement equation, so that the standard Kalman filter can be used for target tracking. Summary of the invention

[0007] Assume that a bistatic radar is tracking a target. At time k, the measurement information obtained by the radar is the bistatic distance measurement r k and the receiving node angle measurement θ k , and its corresponding measurement error ω r,k and ω θ,k is additive Gaussian white noise with zero mean, and the measurement error variances are and

[0008] A measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the predicted value for a transmitting and receiving split radar. The filtering steps from time k-1 to time k are as follows:

[0009] Step 1: Perform state prediction and measurement prediction for the state at time k based on the state estimation at time k-1.

[0010]

[0011] Among them, the set Z k-1 ={z1,z2,...,z k-1}, z k-1 =[r k-1 θ k-1 ] T is the actual measurement value received by the radar at time k-1. is the predicted value of the target state at time k obtained based on the measurement information at time k-1, is the state estimate obtained by filtering at time k-1, is the predicted value of the target measurement at time k. k,k-1 is the state transfer matrix from time k-1 to time k, and h(·) is the nonlinear measurement function:

[0012]

[0013] Where T is the discrete time interval, arctan(·) is the inverse trigonometric function of the tangent function, and the distances between the tracking target and the transmitting and receiving nodes are and and for The position component in the Cartesian coordinate system. The coordinates of the sending and receiving nodes in the Cartesian coordinate system are [x RX y RX ] T and [x TX y TX ] T .

[0014] Step 2: Obtain the prediction error covariance at time k based on the state estimation error covariance matrix at time k-1

[0015]

[0016] Among them, P k|k-1 is the prediction error covariance at time k, P k-1 is the state estimation error covariance matrix at time k-1, Q 1,k-1 is the process noise covariance matrix at time k-1, Γ k,k-1 is the error transfer matrix from time k-1 to time k.

[0017] Step 3: Calculate the innovation error covariance and Kalman gain at the moment.

[0018]

[0019] Among them, S k is the innovation error covariance at time k, K k is the Kalman gain at time k, C k is the measurement conversion matrix between the measurement and the target state after the conversion from polar coordinates to Cartesian coordinates at time k, is the conversion error covariance matrix of the CM2ap measurement conversion algorithm used in the present invention. Specifically,

[0020]

[0021] Where f(·) is the conversion function from the measured value to Cartesian coordinates, Q 2,k is the measurement error covariance matrix at time k, J k is the Jacobian matrix of the prediction error obtained by converting the Cartesian coordinate system to the polar coordinate system at time k. It is specifically expressed as

[0022]

[0023]

[0024]

[0025] Step 4: Calculate the state estimate and state estimate error covariance matrix.

[0026]

[0027] in, is the target state estimate at time k, P k is the state estimation error covariance matrix at time k, The CM2ap measurement conversion of this invention converts the measurement value into an unbiased estimate of the Cartesian coordinate system.

[0028]

[0029] Where diag(·) is a column vector function that takes the diagonal elements of the matrix.

[0030] Principle of the Invention

[0031] Based on the geometric relationship, the cosine theorem of the transmitting node-receiving node-tracking target triangle is solved to obtain the distance r from the receiving node to the tracking target at time k. RX,k , specifically manifested as

[0032]

[0033] Among them, the distance between the sending and receiving nodes The projection of the sending and receiving node path on the receiving node-tracking target a=(x TX -x RX )cosθ m,k +(y TX -y RX )sinθ m,k

[0034] Establish the measurement value z at time k m,k Conversion equation to Cartesian coordinates:

[0035]

[0036] Among them, x m,k and m,k The position component of the target state vector in the Cartesian coordinate system corresponding to time k, r m,k and θ m,k is the measured value z m,k , f(·) is the conversion function from the measured value to Cartesian coordinates, [x RX y RX ] T and [x TX y TX ] T are the coordinates of the sending and receiving nodes in the Cartesian coordinate system, r RX is the distance between the receiving node and the tracking target.

[0037] x m,k and m,k The actual measurement value z at time k t,k Perform a second-order Taylor expansion on the and Respectively expressed as:

[0038]

[0039]

[0040] Among them, r t,k and θ t,k is the actual measurement value z at time k t,k The components of the dual-base distance measurement error ω r,k =r m,k -r t,k , receiving node angle measurement error ω θ,k =θ m,k -θ t,k. Their expected values ​​are expressed as:

[0041]

[0042] in, and Respectively represent the error variance of the dual-base measurement and the receiving node measurement. From equations (20) and (21), it can be seen that the expected value in this case is different from the true value, so an unbiased correction is required. After correction, the true measurement value z at time k is obtained. t,k An unbiased transformation of:

[0043]

[0044] Arranged

[0045]

[0046] in,

[0047] At this time, the conversion error can be obtained as

[0048]

[0049] Therefore, the transformed covariance matrix is

[0050]

[0051] Since the true value z t,k It cannot be obtained in the actual solution process, so the measured value z is used m,k To solve, the corresponding standard unbiased transformation results, transformation errors and transformation error covariance are specifically expressed as

[0052]

[0053] When calculating the first and second order derivatives, the measured value z is used m,k It will introduce errors related to the measurement error, thereby reducing the estimation accuracy, so the predicted measurement value z is used p,k This is avoided by evaluating in the predicted state. The predicted measurement value z p,k It is calculated from the state prediction value obtained in step 1. The unbiased conversion result, conversion error and conversion error covariance based on the prediction value are specifically expressed as follows:

[0054]

[0055] At the same time, if we first use the true value z in equation (25) t,k The calculated first derivative is used to predict the measured value z p,k Then, we use the predicted measured value z p,kCalculation can get higher accuracy. The approximate formula is specifically expressed as

[0056]

[0057] Among them, the prediction distance error ω p,r,k =r p,k -r t,k , predicted angle error ω p,θ,k =θ p,k -θ t,k The new conversion error and transformation error covariance matrix can be obtained:

[0058]

[0059] Among them, the measurement error ω k =[ω r,k ω θ,k ] T , prediction error ω p,k =[ω p,r,k ω p,θ,k ] T , J k The prediction error ω is obtained by converting the Cartesian coordinate system to the polar coordinate system at time k p,k The Jacobian matrix of

[0060]

[0061] Among them, x p,k and p,k They are the Cartesian coordinate positions corresponding to the state prediction values.

[0062] Because the new transformed error covariance R CM2ap The measurement conversion result To further refine the conversion error, for overall consideration, Rename to The specific performance is

[0063]

[0064] Equations (36) and (38) are used to transform the target tracking measurement equation in the polar coordinate system into the Cartesian coordinate system, thereby transforming the tracking problem into a linear problem, which is specifically expressed as

[0065]

[0066] Among them, x k is the target state at time k, C k is the measurement conversion matrix between the measurement and the target state after the conversion from polar coordinates to Cartesian coordinates at time k, as shown in equation (7). In step 3, RCM2ap Solve for the innovation error covariance to get the Kalman gain. In step 4, use The new information in the Cartesian coordinate system is calculated, and the state estimation of the target is obtained through a linear process. Therefore, the standard Kalman filter algorithm can be directly used to achieve target tracking, thus obtaining a Kalman filter method based on measurement conversion.

[0067] The present invention proposes a measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the predicted value, which realizes the unbiased conversion of the measured value from polar coordinates to Cartesian coordinates. The specific conversion method is shown in equations (45) and (47). The measurement equation is converted to the result shown in equation (48), thereby linearizing the nonlinear problem and solving it using the standard Kalman filter method. The measurement conversion based on the predicted value can obtain a more accurate error covariance matrix than the measured value, and the first-order derivative approximation can obtain higher accuracy, thereby improving the target tracking accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a flow chart of the measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the predicted value of the present invention;

[0069] Figure 2 This is the RMSE performance comparison of the algorithm location in scenario 1;

[0070] Figure 3 This is the RMSE performance comparison of the algorithm location in scenario 2;

[0071] Figure 4 This is the RMSE performance comparison of the algorithm location in scenario 3;

[0072] Figure 5 This is the RMSE performance comparison of the algorithm location in scenario 4; DETAILED DESCRIPTION

[0073] Considering that there is a target moving in a uniform straight line in the scene, its initial position is (8km, 8km), initial speed is (100m / s, 100m / s), and the target movement duration is 50s. The present invention sets up a transmitting radar and a receiving radar, which are located at (0km, 4km) and (4km, 0km) respectively. The sampling period of the radar is 1s. The measurements obtained by the radar include the radial distance and azimuth of the dual-base radar and the receiving radar. Assume that each measurement noise is Gaussian zero-mean white noise, and its noise standard deviation is defined as shown in Table 1. The process noise is assumed to be Gaussian white noise, and its standard deviation is q = 1m / s 2 The number of Monte Carlo cycles for the entire simulation is 500.

[0074] Table 1 Simulation scenario parameters

[0075] Simulation scenario Scenario 1 Scenario 2 Scene 3 Scene 4 <![CDATA[σ r (m)]]> 30 30 200 200 <![CDATA[σ θ (°)]]> 2 4 2 4

[0076] The new measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of predicted values ​​(CM2ap new, hereinafter referred to as CM2apn) proposed in the present invention is used to achieve target tracking. At the same time, in order to illustrate the advantages of the algorithm of the present invention, it is compared with the standard second-order Taylor expansion measurement conversion Kalman filter algorithm (CM2) and its improved algorithm (CM2a), the measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of predicted values ​​(CM2p) and its improved algorithm that does not consider the prediction error (CM2apold, hereinafter referred to as CM2apo) (H. Marom, Y. Bar-Shalom and B. Milgrom. Converting bistatic radar measurements to Cartesian position for tracking [J]. Defense + Commercial Sensing, 2022) for performance comparison. The following compares the tracking performance of the algorithm from the perspective of the root mean square error (RMSE) of the position estimation.

[0077] Figure 1 The Kalman filter algorithm flow chart of the present invention is used to show the measurement conversion Kalman filter algorithm flow based on the second-order Taylor expansion of the predicted value.

[0078] In the four simulation scenarios, the position RMSE performance comparison of the CM2 algorithm, CM2a algorithm, CM2p algorithm, CM2apo algorithm, and CM2apn algorithm (the algorithm in this paper) is as follows: Figures 2 to 5 As shown:

[0079] The location RMSE performance comparison of each algorithm in scenario 1 is as follows Figure 2 As shown in the figure, the CM2p algorithm and CM2apo algorithm have good steady-state error, but slow convergence speed; while the CM2 algorithm and CM2a algorithm have fast convergence speed, but poor steady-state error. The CM2apn algorithm proposed in this paper has both good steady-state error and fast convergence speed, and has the best convergence performance.

[0080] Scenario 2 is similar to Scenario 1. Figure 3 As shown, it is more obvious that the convergence speed disadvantages of the CM2p algorithm and CM2apo algorithm in the high angle error scenario, while the CM2apn algorithm can solve this problem well.

[0081] The location RMSE performance comparison of each algorithm in scenario 3 is as follows Figure 4 As shown, the five algorithms have similar convergence performance.

[0082] The location RMSE performance comparison of each algorithm in scenario 4 is as follows Figure 5 As shown in the figure, the convergence speeds of the five algorithms are almost the same, while the CM2p algorithm, CM2apo algorithm, and CM2apn algorithm have better steady-state errors than the CM2 algorithm and CM2a algorithm.

[0083] Combining the experimental results with the above, it can be concluded that in various noise scenarios, the algorithm of the present invention can perform real-time adaptive target tracking. Compared with other algorithms, it has better steady-state error and better convergence speed, and has an estimation result that is closer to the actual state of the target. The algorithm of the present invention has better tracking performance.

[0084] The present invention can obtain higher state estimation accuracy than similar algorithms in different noise scenarios. In summary, the measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the predicted value proposed by the present invention is an effective measurement conversion algorithm for the nonlinearity of the measurement equation in dual-base radar target tracking.

Claims

1. A measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the predicted value for a transmitting and receiving split radar, the specific technical solution is as follows: Assume that a bistatic radar is tracking a target. At time k, the measurement information obtained by the radar is the bistatic distance measurement r k and the receiving node angle measurement θ k , and its corresponding measurement error ω r,k and ω θ,k is additive Gaussian white noise with zero mean, and the measurement error variances are and A measurement conversion Kalman filter algorithm based on the second-order Taylor expansion of the predicted value for a transmitting and receiving split radar. The filtering steps from time k-1 to time k are as follows: Step 1: Perform state prediction and measurement prediction for the state at time k based on the state estimation at time k-1. in, Collection Z k-1 ={z1,z2,...,z k-1 }, z k-1 =[r k-1 θ k-1 ] T is the actual measurement value received by the radar at time k-1. is the predicted value of the target state at time k obtained based on the measurement information at time k-1, is the state estimate obtained by filtering at time k-1, is the predicted value of the target measurement at time k. k,k-1 is the state transfer matrix from time k-1 to time k, and h(·) is the nonlinear measurement function: Where T is the discrete time interval, arctan(·) is the inverse trigonometric function of the tangent function, and the distances between the tracking target and the transmitting and receiving nodes are and and for The position component in the Cartesian coordinate system. The coordinates of the sending and receiving nodes in the Cartesian coordinate system are [x RX y RX ] T and [x TX y TX ] T . Step 2: Obtain the prediction error covariance at time k based on the state estimation error covariance matrix at time k-1 Among them, P k|k-1 is the prediction error covariance at time k, P k-1 is the state estimation error covariance matrix at time k-1, Q 1,k-1 is the process noise covariance matrix at time k-1, Γ k,k-1 is the error transfer matrix from time k-1 to time k. Step 3: Calculate the innovation error covariance and Kalman gain at the moment. Among them, S k is the innovation error covariance at time k, K k is the Kalman gain at time k, C k is the measurement conversion matrix between the measurement and the target state after the conversion from polar coordinates to Cartesian coordinates at time k, is the conversion error covariance matrix of the CM2ap measurement conversion algorithm used in the present invention. Specifically, Where f(·) is the conversion function from the measured value to Cartesian coordinates, Q 2,k is the measurement error covariance matrix at time k, J k is the Jacobian matrix of the prediction error obtained by converting the Cartesian coordinate system to the polar coordinate system at time k. It is specifically expressed as Step 4: Calculate the state estimate and state estimate error covariance matrix. P k =[I-K k C k ]P k|k-1 (23) in, is the target state estimate at time k, P k is the state estimation error covariance matrix at time k, The CM2ap measurement conversion of this invention converts the measurement value into an unbiased estimate of the Cartesian coordinate system. Where diag(·) is a column vector function that takes the diagonal elements of the matrix.

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