A Co-located MIMO Radar Array Optimization and Allocation Method for Multi-Target Tracking

Abstract

The present invention discloses a co-located MIMO radar array optimization allocation method for multi-target tracking. An array allocation optimization model is constructed based on the constructed co-located MIMO radar system model, and antenna selection is performed using convex relaxation and local search algorithms. Finally, an optimized allocation scheme for the next moment is obtained, and the target tracking for the next moment is guided based on this optimized allocation scheme. It can be seen from the simulation results that this algorithm is superior to the heuristic algorithm to a certain extent in terms of both tracking performance indicators and calculation time. The exhaustive search method has a slightly higher accuracy than the proposed algorithm, but it takes longer time and has lower efficiency, and cannot meet the actual combat requirements.

Description

Technical Field

[0001] The present invention relates to the technical field of array optimization, and particularly to a co-located MIMO radar array optimization allocation method for multi-target tracking. Background Art

[0002] Similar to the element allocation in distributed MIMO radars, for co-located MIMO radars, by selecting transmitting and receiving antennas in real time, not only can the element loss be reduced to meet the military's low intercept requirement, but also the system performance can be improved. At present, there have been studies on the array configuration of co-located MIMO radars, but the relevant research is very few, limited to the literature [1] "(A.A. Gorji, T. Kirubarajan, R. Tharmarasa, Optimal antenna allocation in MIMO radars with collocated antennas[J]. IEEE Trans. Aerosp. Electron. Syst., 2014, 50(1): 542–557.)", and it still has the following problems:

[0003] First, the scenario targeted by the literature [1] is static target localization. In actual applications, the prior information of the target is often unknown. Therefore, the target tracking scenario is more in line with the actual needs. In target localization, the literature [1] obtained the CRLB for time delay and DOA estimation. However, in target tracking, speed is also a factor that needs to be concerned about;

[0004] Second, the literature [1] studied the antenna placement problem. When the design of the co-located MIMO radar is completed, the antenna position is often fixed. Therefore, the dynamic selection model of the antenna is more in line with the actual situation;

[0005] Third, the existing literature already has the CRLB for DOA estimation. However, in the derivation process, it is assumed that the received signal is a single pulse case, and the CRLB under multi-pulse conditions has not been discussed. Under multi-pulse conditions, the radar can improve the SNR of the signal through coherent processing to obtain better detection and tracking performance. In addition, in DOA estimation, multi-pulse accumulation is often required to obtain high estimation accuracy. Summary of the Invention

[0006] In view of the above problems, the present invention proposes a co-located MIMO radar array optimization allocation method for multi-target tracking. To achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A co-located MIMO radar array optimization allocation method for multi-target tracking, characterized by comprising the following steps:

[0008] Step 1: Establish a co-located MIMO radar system model for measuring multiple moving targets to obtain measurement data;

[0009] Step 2: Process the obtained measurement data using a square root cubature Kalman filter to obtain the target state estimate at the current moment;

[0010] Step 3: Based on the target state estimate obtained in Step 2, derive the PCRLB for Doppler-DOA and construct an array optimization allocation model;

[0011] Step 4: Solve the array optimization allocation model based on an algorithm combining convex relaxation and local search;

[0012] Step 5: Obtain the optimization allocation scheme for the next moment according to the solution result of Step 4, and use this optimization allocation scheme to guide the target tracking at the next moment.

[0013] Furthermore, the establishment of the co-located MIMO radar system model in Step 1 includes a signal model, a target motion model, and a measurement model.

[0014] Furthermore, the specific operation steps of Step 3 include:

[0015] Step 31: Introduce a binary vector to represent the selection or not of the transmitting and receiving antennas at time k, and the binary vector is:

[0016]

[0017] where, represents whether the m-th transmitting antenna is selected at time k, represents whether the n-th receiving antenna is selected at time k, and

[0018] Step 32: For the tracking accuracy of multiple targets, take the sum of the PCRLBs of multiple targets as the cost function, and the cost function is:

[0019]

[0020] where tr(.) is the matrix trace operation;

[0021] Step 33: Consider the following constraints:

[0022] 1)

[0023] 2)

[0024] where G is the total number of selected antennas;

[0025] Step 34: Construct the optimization allocation model as:

[0026]

[0027] Furthermore, the specific operation steps of Step 4 include:

[0028] Step 41: Use linear programming to relax the binary vector and rewrite Equation (26) as:

[0029]

[0030] Step 42: Since Equation (27) is convex, use the convex optimization algorithm to solve it and obtain the optimal solution;

[0031] Step 43: If the optimal solution is an integer, directly output the optimal antenna selection result; if the optimal solution is a decimal, use the local search algorithm to obtain an approximate optimal solution and output the selection result.

[0032] Furthermore, the specific steps of using the local search algorithm to obtain an approximate optimal solution in Step 43 include:

[0033] Step 431: Assume that the currently selected antenna set is S cur ={S cur (1), S cur (2), …, S cur (g)}, where S cur (g) represents the position of the selected antenna, then all neighborhood sets can be expressed as: S i nei ={S i nei (1), S i nei (2), …, S i nei (g)}, where S i nei represents all feasible solutions after replacing S cur (g) with the remaining antennas in the current antenna set;

[0034] Step 432: Use the antennas with higher rankings in S i nei to replace the antennas with lower rankings in S cur in turn until the objective function value decreases and a single loop terminates;

[0035] Step 433: When all antennas in the initial solution set have been examined, the local search algorithm terminates.

[0036] The beneficial effects of the present invention are as follows:

[0037] The array optimization allocation algorithm in multi-target tracking of co-located MIMO radar proposed by the present invention can further exert the combat potential of co-located MIMO radar, improve the radar working efficiency, and improve the target tracking accuracy. And the proposed algorithm has a low computational complexity and a high computational accuracy, which well meets the engineering requirements. Description of the Drawings

[0038] Figure 1 is the geometric relationship between the MIMO radar and the target;

[0039] Figure 2 (a)- Figure 2 (c) are the antenna selection results at the 5th frame, the 15th frame, and the 25th frame when the target RCS conforms to the type I model;

[0040] Figure 3 (a) and Figure 3 (b) are respectively the comparison result diagrams of PCRLB and RMSE of different algorithms when the target RCS conforms to the type I model;

[0041] Figure 4 is the comparison result diagram of the CPU running time of different solution algorithms;

[0042] Figure 5 (a) and Figure 5 (b) are the schematic diagrams of the changes of PCRLB and RMSE of different algorithms with the adjustment of SNR;

[0043] Figure 6 (a)- Figure 6 (c) are the antenna selection results at the 5th frame, the 15th frame, and the 25th frame when the target RCS conforms to the type II model;

[0044] Figure 7 (a) and Figure 7 (b) are the comparison result diagrams of PCRLB and RMSE of different algorithms when the target RCS conforms to the type II model;

[0045] Figure 8 (a) and Figure 8 (b) are respectively the schematic diagrams of the changes of PCRLB and RMSE of different algorithms with SNR when the target RCS conforms to the type II model. Detailed Embodiments

[0046] In order to enable those of ordinary skill in the art to better understand the technical solution of the present invention, the technical solution of the present invention will be further described below in conjunction with the drawings and embodiments.

[0047] First, a system model of the co-located MIMO radar needs to be established, including establishing the signal model, target motion model, and measurement model of the MIMO radar. By using these established models, multiple moving targets are measured respectively, specifically including:

[0048] 1. Signal model

[0049] To simplify the problem, the following assumptions are made:

[0050] (1) The co-located MIMO radar is located in the x-y two-dimensional plane, and the antenna spacing is much smaller than the distance between the radar and the target;

[0051] (2) The echo signals of multiple batches of targets can be separated, and the propagation time from each batch of targets to the radar is always aligned with the zero time delay of the radar;

[0052] (3) The Doppler frequency shift does not affect the orthogonality of the transmitted signals.

[0053] Assume a single co-located MIMO radar, whose structural center is located at the point (0, 0). The positions of the m-th transmitting antenna and the n-th receiving antenna are (x tm , y tm ) and (x rn , y rn ), respectively. Among them, m = 1, 2,..., M; n = 1, 2,..., N. M and N are the total numbers of transmitting and receiving antennas respectively. Each transmitting antenna emits mutually orthogonal coherent pulse train signals, the pulse width is t c , and the pulse repetition period is T p . The transmitted signal of the m-th antenna is s m (t), the carrier wavelength is λ, and the number of coherent pulse trains emitted in a single transmission is L. In the k-th sampling interval, the baseband signal of the l-th pulse emitted by the m-th transmitting antenna can be expressed as:

[0054] s k,m,l (t) = s k,m (t′ + (l - 1)T p ), 0 ≤ t′ ≤ t c (1)

[0055] where t and t′ represent fast time and slow time respectively.

[0056] Assume that there are Q targets in the radar warning airspace. The arrival angle of the q-th target in the k-th sampling interval is The RCS of the target can be regarded as a constant value within a pulse repetition period.

[0057] Then, through time delay compensation, the received echo signal of the l-th pulse can be expressed as:

[0058]

[0059] In the formula, is the RCS of the target; is the Doppler shift of the target; and are the receiving and transmitting steering vectors respectively; (.) H represents the Hermitian conjugate transpose operation; s k,l (t) is the transmitted signal vector; and there is:

[0060]

[0061] In the formula, [.] i represents the i-th element in the vector. and s k,l (t) have dimensions of N×1, M×1, and M×1 respectively. n k,l (t) is the noise term.

[0062] Substituting Equation (1) into Equation (2), we can get:

[0063]

[0064] Process the received signal through matched filtering:

[0065]

[0066] Based on Assumption (3), the Doppler shift does not affect the orthogonality of the signal, so there is

[0067]

[0068] Then, stretch the matrix by columns into an NM×1 dimensional vector:

[0069]

[0070] In the formula, vec(.) represents the operation of stretching a matrix into a column vector; I MN is an MN×MN dimensional identity matrix. g k The expression of is:

[0071]

[0072]

[0073] In the formula, is the Kronecker product; represents a 1×N dimensional all-ones vector; represents a 1×M dimensional all-ones vector. btm is the position vector of the transmitting antenna m; b rn is the position vector of the receiving antenna n.

[0074] 2. Target motion model

[0075] For simplicity, it is assumed that the target motion model conforms to a constant acceleration motion model. Let Q uniformly moving targets be located in the radar warning area. At the k-th sampling interval, the position and velocity of the q-th target at the k-th sampling moment can be expressed as and Then the motion equation of the q-th target can be written as:

[0076]

[0077] where and the letter x represents the x-direction parameter, y represents the y-direction parameter, represents the position, represents the velocity, represents the acceleration; F and respectively represent the state transition matrix and the process noise;

[0078] and Q q represents the covariance matrix of the noise;

[0079] and the specific forms of F q and Q q are as follows:

[0080]

[0081]

[0082] where T s is the sampling interval; n f is the process noise intensity, is the Kronecker product, and I2 is the second-order identity matrix.

[0083] 3. Measurement model

[0084] The measurement function obtained from the received signal shown in Equation (7) to obtain target information is:

[0085]

[0086] where h(.) is a non-linear transformation with respect to the target velocity and angle, satisfying and:

[0087]

[0088] Here, the square-root cubature Kalman filter (SCKF) with a simple structure and relatively high estimation accuracy is selected to handle the tracking problem under non-linear measurements. If the RCS of the target is regarded as a deterministic known quantity, then can be regarded as obeying a Gaussian distribution with zero mean and covariance matrix . Under high signal-to-noise ratio and multi-pulse conditions, can be approximated by the CRLB matrix, and its PCRLB is derived as follows:

[0089] The PCRLB determines a performance bound for the target tracking algorithm. For the above-established signal model and moving target model, if it is assumed that is an unbiased estimate of , then the following equation always holds:

[0090]

[0091] where denotes the mean operation on the target state and the observation value ;

[0092] is the Bayesian Fisher information matrix (BFIM) about the target state. According to the previous derivation, the BFIM can be written as:

[0093]

[0094] In the formula, is the Jacobian matrix of the measurement function h(x k ) with respect to the target state, and there is:

[0095]

[0096] where denotes the partial derivative of with respect to ; and:

[0097]

[0098]

[0099] The second term in Equation (16) is a process of taking the expectation, which means that to solve the PCRLB, the Monte Carlo method needs to be used to obtain the mean value. To meet the real-time requirements in resource allocation, the following approximation is made here:

[0100]

[0101] In the formula, and respectively represent the approximation of the Jacobian matrix of the measurement function with respect to the target state and the measurement covariance matrix at the target state prediction point .

[0102] According to the system model established above, the derivation process of

[0103] is as follows: Under the condition of correlation between pulses within a single irradiation period, the RCS of the target follows the following distribution:

[0104]

[0105] In the formula, Ω follows a uniform distribution; the PDF of Σ is:

[0106]

[0107] In the formula, is the average reflection coefficient. α remains unchanged during a pulse train.

[0108] Assume that the target signal is a deterministic signal, and there is correlation between pulses within a single irradiation period, but no correlation between different irradiation periods. Then there is:

[0109] α k,l = α k

[0110] (C-3)

[0111] Combining Equation (7), Equation (13), and (C-3), the measurement vector at time k can be written as:

[0112] r k = α k G k c k + v k

[0113] (C-4)

[0114] In the formula,

[0115] G k = diag{g k , g k ,..., g k} L

[0116] (C-5)

[0117]

[0118] where diag{.} L denotes that the matrix contains L diagonal elements.

[0119] It can be obtained from Equation (C-4) that

[0120]

[0121] where For simplified representation, an intermediate variable μ is introduced: Ik = α k G k c Ik then the [p q]-th element of

[0122]

[0123] When p = 2 and q = 2,

[0124]

[0125] The calculation of is equivalent to the calculation of and the analytical formula of the latter is easily obtained:

[0126]

[0127] where

[0128] Δb mn = b tm - b rn (C-11)

[0129] Thus, Equation (C-9) is rewritten as

[0130]

[0131] In addition, when p = 1 and q = 1, it is easily obtained that:

[0132]

[0133] However, the CRLB of velocity and angle may be at different magnitude levels. It can be seen from the analytical formula of that the dynamic selection of antennas directly affects the CRLB of DOA estimation. Under this condition, the small influence of antenna selection is likely to be masked by a large velocity error. To avoid this problem, a diagonal matrix is introduced here to balance the CRLB of velocity and angle, then the BFIM becomes:

[0134]

[0135] Among them, Λ = blkdiag(Λ1, Λ2), which is a 2×2 dimensional diagonal matrix.

[0136] Secondly, an array optimization and allocation model is constructed according to the co-located MIMO radar system model established above.

[0137] 2. Array Optimization and Allocation Model

[0138] To meet the low interception requirements in military applications, reduce the losses of transmitting and receiving array elements, and maximize the system performance of the co-located MIMO radar, the real-time selection of antennas needs to be considered during the tracking process. By establishing a closed-loop feedback from the transmitter to the receiver, the changes in the external environment can be sensed in real time and the operating parameters can be adjusted adaptively, which is widely applied to the resource allocation of the co-located MIMO radar. Therefore, the present invention establishes a closed-loop feedback system from the transmitter to the receiver. Since the PCRLB is predictable, the target information obtained at the previous moment can be used to predict the PCRLB at the current moment, and the antenna configuration can be adjusted accordingly. It includes the following steps:

[0139] First, a binary vector is introduced to represent whether the transmitting and receiving antennas at time k are selected. The binary vector is:

[0140]

[0141] In the formula,

[0142] Secondly, for the tracking accuracy of multiple targets, the present invention selects the sum of the PCRLBs of multiple targets as the cost function, that is:

[0143]

[0144] In the formula, tr(.) is the matrix trace operation, and J is the BFIM;

[0145] Since antenna selection only affects the PCRLB of the target position, the PCRLB of the target position is extracted as the cost function;

[0146] Thirdly, consider the constraint conditions for the selection of the antenna array: In actual operation, the selection of the antenna array needs to meet multiple constraints:

[0147] For example, when the usage period is long, the antenna loss becomes the main consideration factor. Therefore, the total number of selected antennas is often fixed, that is, it is necessary to meet the constraint:

[0148]

[0149] In the formula, G is the total number of selected antennas;

[0150] In addition, to ensure the effectiveness of the system, at least one transmitting antenna and one receiving antenna should be selected, that is, the constraint should be satisfied:

[0151]

[0152] Finally, the optimization problem is described as:

[0153]

[0154] Considering the duality of the selection vector, the optimization model shown in Equation (26) is an NP-hard problem. Although the optimal solution can be obtained through exhaustive search, its computational complexity is quite considerable.

[0155] 3. Solving the Optimization Model

[0156] The present invention uses an algorithm combining convex relaxation and local search to solve this NP-hard problem. The specific solution algorithm includes the following steps:

[0157] Step 1: Use linear programming to relax the binary variables and rewrite Equation (26) as:

[0158]

[0159] The optimized model after relaxation is convex. Therefore, Equation (27) can be solved by a standard convex optimization algorithm, such as the Zoutendijk feasible direction method. It should be noted that the optimal selection variables obtained from Equation (27) can be decimals. Therefore, the relaxed problem is no longer equivalent to the original problem. However, since the feasible solution set of the original problem is included in the feasible solutions of the relaxed problem, the minimum value of the latter will not be greater than the minimum value of the original problem. Based on this, Equation (27) is processed as follows.

[0160] Step 2: When the optimal solution of Equation (27) is an integer, the optimal antenna selection result can be directly obtained. However, in most cases, the optimal solution of Equation (27) is a decimal. At this time, local search is used to obtain an approximate optimal solution.

[0161] The flow of the above entire algorithm is shown in Table 1. It can be seen that: First, the convex relaxation algorithm is used to obtain a high-quality initial solution, and its essence is to select the most promising antenna subset under the condition of satisfying the constraints; then, the algorithm uses the local search strategy to improve the solution accuracy. Considering that the neighborhood structure of the solution set affects the efficiency of local search, in each iteration process, the algorithm only changes a single antenna in the neighborhood of the current solution. Assume that the currently selected antenna set is S cur ={S cur (1),S cur (2),…,S cur (g)}, where Scur Let (g) represent the position of the selected antenna, then all neighborhood sets can be expressed as: S i nei ={S i nei (1), S i nei (2), …, S i nei (g)}. Among them, S i nei represents all feasible solutions after replacing S cur (g) from the current antenna set and replacing it with the remaining antennas. For example, if S cur (1) is replaced by any one of the remaining M + N - G antennas, then there are M + N - G kinds of S i nei . Therefore, if any one antenna in the neighborhood is used to replace any one antenna in S cur , there are a total of (M + N - G) × G feasible solutions.

[0162] Since the solution set of Equation (27) characterizes the suitability of each antenna being selected, the local search algorithm will use the antennas with higher rankings in S i nei to replace the antennas with lower rankings in S cur in turn until the objective function value decreases and a single loop terminates. When all antennas in the initial solution set have been examined, the local search algorithm terminates.

[0163] Table 1 Antenna Selection Algorithm Based on Convex Relaxation and Local Search

[0164]

[0165]

[0166] When applying the search strategy in Table 1, there are the following remarks.

[0167] Remark 1: The proposed algorithm is convergent. First, since Equation (4.27) is a convex optimization problem, the optimal solution of the problem can be obtained using standard convex optimization algorithms. Second, during the local search process, when no better solution set can be found in the neighborhood, the algorithm will stop iterating. Therefore, the algorithm is convergent.

[0168] Remark 2: To obtain a solution set with higher precision, more antennas can be replaced simultaneously from the current solution set S cur , but increasing the number of neighborhoods will also bring a greater computational burden. If two antennas are replaced simultaneously, the number of neighborhoods is In the formula, is the number of combinations of choosing 2 from G candidate solutions without order.

[0169] Annotation 3: In a single loop, the stopping condition of the search can be replaced by other criteria. For example, the objective function value in Equation (4.26) no longer decreases. That is, for the position of a certain antenna in the current solution, all remaining antennas must be traversed, but this also means a larger computational amount.

[0170] As can be seen from Table 1, solving the relaxed problem accounts for a relatively large amount of computation in the entire algorithm, while the local search algorithm is only used as a supplement. When using the standard convex optimization algorithm, at most iterations are required to meet the required accuracy. The amount of computation in each iteration is while the exhaustive search algorithm and the heuristic search algorithm require and iterations respectively. Considering the amount of computation in each iteration, there is a comparison of the computational amounts in Table 2.

[0171] Table 2 Comparison of Computational Amounts of Different Algorithms

[0172]

[0173] Since the measurement data of the radar system is obtained in the polar coordinate system, and the Kalman filter is only applicable to linear systems, therefore, it is necessary to process the non-linear transformation. Recently, a series of filtering schemes based on Gaussian sampling points have been proposed, such as the unscented Kalman filter (UKF), the Gauss-Hermite quadrature filter (GHQF), and the cubature Kalman filter (CKF), etc. Compared with other filters, CKF has the advantages of small computational amount and strong robustness. The SCKF algorithm introduces orthogonal triangular decomposition on the basis of CKF, and through the iterative operation of the square root of the error covariance matrix, the symmetry and positive semi-definiteness of the covariance matrix are guaranteed.

[0174] To describe the filtering process of SCKF, first, the following definition is made for the Tria(.) operation: Let R be the upper triangular matrix obtained by performing QR decomposition on matrix A T Then the QR decomposition algorithm of matrix A can be expressed as S = Tria(a) = R T , where S is a lower triangular matrix. Then the SCKF algorithm can be summarized as:

[0175] Step 1 Time update

[0176] Step 1.1 Decompose the state estimation error covariance matrix:

[0177] P k|k = S k|k (S k|k ) T (28)

[0178] Step 1.2 Calculate the volume points and perform non - linear transformation:

[0179]

[0180]

[0181] Wherein, is the volume point; m is the total number of volume points; n is the dimension of the state vector x; [1] is the point set generated by full permutation or inversion of the unit vectors in the n - dimensional space, satisfying:

[0182]

[0183] Step 1.3 Calculate the square root of the state prediction mean and the prediction error covariance matrix.

[0184]

[0185]

[0186] Wherein, Chol(.) represents the Cholesky decomposition of the matrix. The weighted central matrix is defined as:

[0187]

[0188] Step 2 Measurement update

[0189] Step 2.1 Calculate the volume points and perform non - linear transformation according to the measurement model:

[0190]

[0191]

[0192] Step 2.2 Calculate the measurement prediction mean:

[0193]

[0194] Step 2.3 Calculate the innovation (residual) covariance matrix square root coefficient:

[0195]

[0196] Wherein, the weighted central matrix z k+1|k is:

[0197]

[0198] Step 2.4 Calculate the innovation covariance matrix and the measurement autocovariance matrix:

[0199]

[0200] Step 2.5 Calculate the cross-covariance matrix between the state and the measurement:

[0201]

[0202] Weighted central matrix X k+1|k Satisfy:

[0203]

[0204] Step 2.6 Calculate the filtering gain:

[0205]

[0206] The posterior state estimate value and the square root coefficient of the estimation error covariance matrix are respectively:

[0207]

[0208] S k+1|k+1 = Tria([x k+1|k - K k+1 z k+1|k ,K k+1 Chol(R k+1 )]) (45).

[0209] The target state can be estimated through formulas (28) - (45).

[0210] Embodiment

[0211] To further verify the effectiveness of the algorithm proposed by the present invention, a single co-located MIMO radar (M = N = 25) is selected for analysis. Its antennas are uniformly arranged within a range of 4×4 m, with a spacing of 1 m. Each antenna can transmit or receive orthogonal signals. The carrier length is λ = 0.3 m, and the number of antennas to be selected is G = 20. The number of targets Q = 3, and the sampling interval T s = 5 s, and 30 frames of data are used for each simulation. The geometric relationship between the targets and the radar is as Figure 1 shown. The initial states of the targets are shown in Table 3, and their motions conform to the CA model. The number of pulses for a single illumination is L = 300, and the pulse repetition interval T p = 0.01 s. The reference SNR is set to 0 dB at a distance of 500 km from the radar. The following algorithms are used for simulation comparison:

[0212] (1) Random configuration, which randomly selects 20 transmitting and receiving antennas;

[0213] (2) Exhaustive search algorithm, which uses exhaustive search to solve Equation (26);

[0214] (3) Heuristic search algorithm, which uses a multi-start greedy search algorithm to solve Equation (26);

[0215] The algorithm evaluation metrics are PCRLB and the corresponding RMSE, and the latter is defined as:

[0216]

[0217] where N sim is the number of simulation times; is the estimated value of the position of the q-th target in the j-th simulation.

[0218] Table 3 Initial parameters of the target

[0219]

[0220] The following is a comparative analysis of type I targets and type II targets respectively:

[0221] 1. Type I targets

[0222] When the RCS of the target conforms to the type I model, we have The balancing matrix is set to Λ = blkdiag(1, 1). Figure 2 For the typical results of antenna selection, the algorithm performances are compared as Figure 3 shown.

[0223] It can be seen from the figure that under the condition of the same initial error, compared with the random configuration and the heuristic search algorithm, the proposed algorithm achieves lower PCRLB and RMSE. Moreover, the performance is similar to that of the exhaustive search algorithm. This is because the dynamic antenna configuration can provide the minimum measurement error according to the predicted PCRLB to obtain higher tracking accuracy. This point can also be seen from Figure 3 (a) The subgraph in which the true PCRLB is very close to the predicted PCRLB, indirectly proving the effectiveness of the proposed algorithm. The random configuration fails to effectively utilize the target information to adjust the working parameters. Therefore, the obtained PCRLB and RMSE are the largest. In addition, since the proposed algorithm combines convex relaxation and local search techniques, the former can provide a high-quality initial solution set, and the latter can further improve the solution accuracy. Therefore, the performance is significantly improved compared with the heuristic search algorithm.

[0224] Then, we compared the CPU running times of the three solution algorithms, as Figure 4As shown, it can be seen that compared with the heuristic search and exhaustive search algorithms, the proposed algorithm has a shorter running time. Specifically, the proposed algorithm takes about 5 s to select the best transceiver antenna subset, while the heuristic search and exhaustive search take 10 s and 10 3 s to obtain the results. Therefore, the proposed algorithm is more efficient.

[0225] To further analyze the influence of SNR on the algorithm performance, Figure 5 the variations of PCRLB and RMSE of different algorithms with the adjustment of SNR are given. It can be seen that when SNR increases, the PCRLB and RMSE of all algorithms decrease significantly. However, the performance of the proposed algorithm is still better than that of the random configuration and heuristic search algorithms, and is close to that of the exhaustive search algorithm, thus proving the robustness of the proposed algorithm.

[0226] 2. Type II target

[0227] When the target RCS conforms to the Class II model, for each pulse within a single irradiation time, they are independent of each other. Assume Since under this condition, there is a large difference in the order of magnitude between the CRLB of the target speed and angle, set Λ = blkdiag(1, 10 -2 ). Figure 6 is a typical antenna selection result, Figure 7 is the performance comparison of four algorithms.

[0228] It can be seen from the figure that compared with the random configuration and heuristic search algorithms, the proposed algorithm obtains lower PCRLB and RMSE. This is because the proposed algorithm can dynamically adjust the antenna configuration according to the feedback information in the tracking, making the PCRLB of each frame reach the minimum as much as possible. In addition, the performance of the proposed algorithm is close to that of the exhaustive search algorithm. Figure 7 The subfigure in (a) compares the predicted PCRLB and the true PCRLB. Obviously, they basically coincide, verifying the theoretical correctness of the proposed algorithm.

[0229] Figure 8 shows the variation of the algorithm performance with the increase of SNR. Obviously, the tracking performance of all algorithms has been significantly improved with the increase of SNR. Although the difference in performance between algorithms gradually decreases with the increase of SNR, the performance of the proposed algorithm has always been better than that of the random configuration and heuristic search algorithms. In addition, the performance of the proposed algorithm is close to that of the exhaustive search algorithm, proving the effectiveness of the proposed algorithm under different SNR conditions.

[0230] Through the above simulations, the following conclusions can be drawn:

[0231] (1) Whether the RCS of the target follows the Swerling I or Swerling II distribution, the performance gap between the dynamic antenna configuration and the random antenna configuration will decrease with the increase of SNR;

[0232] (2) Compared with the random configuration and the heuristic search algorithm, the proposed antenna selection algorithm has better performance;

[0233] (3) Although the exhaustive search algorithm can provide the optimal result, due to the limitation of computational complexity, it is difficult to be applied in practice. The proposed algorithm has lower computational complexity and higher computational accuracy, and is more suitable for engineering requirements.

[0234] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments, and what is described in the above embodiments and the specification only illustrates the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of the present invention claimed is defined by the appended claims and their equivalents.

Claims

1. A co-located MIMO radar array optimization allocation method for multi-target tracking, characterized in that, It includes the following steps: Step 1: Establish a co-located MIMO radar system model for measuring multiple moving targets to obtain measurement data; Step 2: Use a square root cubature Kalman filter to process the obtained measurement data to obtain the target state estimate value at the current moment; Step 3: Based on the target state estimate value obtained in Step 2, derive the PCRLB regarding Doppler-DOA and construct an array optimization allocation model; Step 4: Solve the array optimization allocation model based on an algorithm combining convex relaxation and local search; Step 5: Obtain the optimized allocation scheme for the next moment according to the solution result of Step 4, and guide the target tracking at the next moment based on this optimized allocation scheme; Among them, the specific operation steps of Step 3 include: Step 31: Introduce a binary vector to represent whether the transmitting and receiving antennas at time k are selected. The binary vector is: （22） wherein, represents whether the m-th transmitting antenna is selected at the k-th moment, represents whether the n-th receiving antenna is selected at the k-th moment, and ; Step 32: For the tracking accuracy of multiple targets, take the sum of the PCRLBs of multiple targets as the cost function. The cost function is: （23） Among them, is the matrix trace operation, and J is the BFIM; Step 33: Consider the following constraint conditions: 1） （24） 2） （25） Among them, G is the total number of selected antennas; Step 34: Construct the optimization allocation model as: （26）。 2. The co-located MIMO radar array optimization allocation method for multi-target tracking according to claim 1, characterized in that, The establishment of the co-located MIMO radar system model in Step 1 includes a signal model, a target motion model, and a measurement model.

3. The co-located MIMO radar array optimization allocation method for multi-target tracking according to claim 1, characterized in that, The specific operation steps of Step 4 include: Step 41: Use linear programming to relax the binary vector and rewrite Equation (26) as: （27）； Step 42: Since Equation (27) is convex, use a convex optimization algorithm to solve it to obtain the optimal solution; Step 43: If the optimal solution is an integer, directly output the optimal antenna selection result; if the optimal solution is a decimal, use a local search algorithm to obtain an approximate optimal solution and output the selection result.

4. The co-located MIMO radar array optimization allocation method for multi-target tracking according to claim 3, characterized in that, The specific steps of using the local search algorithm to obtain an approximate optimal solution in Step 43 include: Step 431: Assume that the currently selected antenna set is S cur ={ S cur (1), S cur (2),…, S cur (g)}, where S cur (g) represents the positions of the selected antennas. Then all neighborhood sets can be expressed as: S i nei = { S i nei (1), S i nei (2),…, S i nei (g)}, where, S i nei represents all feasible solutions after replacing S cur (g) with the remaining antennas after removing it from the current antenna set. Step 432: Use S i nei the antenna pair with a higher rank in S cur successively replace the antenna with a lower rank in it until the objective function value decreases, and a single loop terminates; Step 433: When all antennas in the initial solution set have been examined, the local search algorithm terminates.

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