CN107728139B - Phased array radar networking system resource management method based on multi-target tracking - Google Patents

Abstract

The invention discloses a resource management method for a phased array radar networking system based on multi-target tracking, belongs to the technical field of phased array radar networking resource management, and relates to multi-target tracking. Firstly, the topology between the radar network and the target is studied, and the influence of the angle and spatial diversity gain on the echo signal-to-noise ratio of different targets in the multi-beam working mode of the polyphased array radar network is analyzed. Then, on the premise that the tracking accuracy of each target meets the predetermined requirements, the beam pointing and beam dwell time of each radar are optimized to minimize the total dwell time of the radar's networking beam for tracking. Due to the difference in target position, angle, RCS and radar networking space diversity gain, the demand for system resources for each target to maintain the predetermined tracking accuracy changes, resulting in the problem that several targets cannot be effectively tracked, which saves resources at the same time. , to complete the orderly tracking of multiple targets by the system.

Description

A Resource Management Method of Phased Array Radar Networking System Based on Multi-target Tracking

technical field

The invention belongs to the technical field of phased array radar networking resource management, and relates to multi-target tracking and research on the joint management technology of beam and dwell time resources of a multi-radar system in a multi-target tracking mode.

Background technique

Phased array radar is an important radar that is widely researched and developed. Because its beam can be pointed arbitrarily and can be agile in microseconds to hundreds of microseconds, it has multi-function, multi-target and highly adaptive capabilities. It has the characteristics of great flexibility and so on. The combination of phased array radar and computer control technology can adaptively change the relevant working parameters and working methods of the radar to adapt to the changing working environment of the outside world, such as changing the antenna beam shape, beam dwell time and power distribution, etc. Therefore, the management of phased array radar resources according to the external environment has extensive research value.

For the target tracking and observation radar network composed of multiple phased array radars, the resource management problem is not only the time resource management problem, but also the space resource (radar allocation) management problem. The radar allocation problem is the so-called sensor allocation problem. Therefore, for a networked tracking system composed of phased array radars, the resource management issues include not only the management of beam pointing and dwell time, but also the correspondence between sensors and targets (which radars track which targets). In the paper "Multi-target tracking distributed MIMO radar transceiver station joint selection optimization algorithm, Acta Radar, 2017, 6(1): 73～80", the author will construct the station selection as a Boolean programming problem and relax it as After the semi-definite programming problem (SDP), the block coordinate descent iteration method is used to obtain the approximate optimal solution of the joint selection. This method effectively solves the problem of selecting the number of radar transceiver stations, but does not consider the resource allocation of each station. Therefore, this method The problem to be solved is relatively simple, and the influence of different radar sites and target positions and different target scattering cross sections (RCS) on site assignment is not analyzed. The document "VariableDwell Time Task Scheduling for Multifunction Radar, IEEE TASE, 2014, 11(2): 463-472." After quantifying the residence time based on the task, an effective heuristic scheduling method is proposed to make the radar system in a More tasks can be completed within the scope of the time axis, but this method is aimed at the macro task management of the system. For the problem of consuming as little resources as possible to complete multi-target tracking, the effect of this method is not obvious.

SUMMARY OF THE INVENTION

The purpose of the present invention is to study and design a resource management method for phased array radar networking based on multi-target tracking in view of the defects in the background technology, so as to solve the problem when the phased array radar network tracks multiple targets in the multi-beam working mode. , due to the difference in target position, angle, RCS and radar space diversity gain, each target has different requirements for phased array radar networking beam and dwell time in order to maintain the predetermined tracking accuracy, resulting in waste of system resources and some targets cannot be used. Effectively tracked issues.

The solution of the present invention is as follows: firstly, the topology structure between the radar network and the target is studied, and under the multi-beam working mode of the polyphased array radar network, the angle and space diversity gain of different targets are different to the echo signal. effect of noise ratio. Then, on the premise that the tracking accuracy of each target meets the predetermined requirements, the beam pointing and beam dwell time of each radar are optimized, so that the total dwell time of the radar network beam for tracking is minimized. Aiming at this problem, a transformation algorithm is proposed. The algorithm first assigns a fixed dwell time value to each target, and selects several radars with rich data information to track the target. The convex problem is transformed into a convex problem, and then the selected dwell time of each radar beam is allocated according to the predetermined tracking accuracy of each target, and finally the resource allocation of the phased array radar networking system is realized. Finally, according to the target observation model, the extended Kalman filter algorithm is used to realize the tracking of multiple targets by the radar network. The method effectively solves the problem of changing radar resource requirements for different targets to maintain a predetermined tracking threshold, thereby realizing a reasonable match between system resources and targets, and completing orderly tracking of multiple targets while saving resources.

The present invention provides a resource management method for a phased array radar networking system based on multi-target tracking, the method comprising the steps of:

Step 1: Determine the radar and target topology and managed resource variables;

和

则在第k个跟踪时刻，目标q的位置和速度分别为

和

在k时刻，每部雷达可发射B_m个波束，且有

个波束被选出用于目标跟踪，每个跟踪时刻每个波束只能跟踪一个目标，由于并不能确定雷达m的波束是否会用来跟踪目标q，引入二元变量

A radar network consisting of M phased array radars, the position of the mth radar is (x _m , y _m ), m=1,2,...,M, and Q targets are widely distributed in the monitoring area. The radar system Track these targets, assuming that each target moves at a uniform speed, the initial position and speed of the target q are respectively

and

Then at the kth tracking moment, the position and velocity of the target q are respectively

and

At time k, each radar can transmit B _m beams, and there are

A number of beams are selected for target tracking, and each beam can only track one target at each tracking time. Since it is not certain whether the beam of radar m will be used to track target q, binary variables are introduced.

个脉冲照射到目标q上，则雷达波束在该目标的驻留时间为

表示脉冲个数，T_pri表示脉冲重复周期，因此，本发明对雷达系统的波束指向和驻留时间进行管控；因此确定出被管理资源变量：1.每个时刻每部雷达用于跟踪的波束数量

2.每个目标如何选择来自哪些雷达的波束照射，3.来自不同雷达的波束照射驻留时间

分；In order to maintain the tracking of the target, at each tracking moment, the radar beam needs to transmit a certain amount of pulses to the target to obtain target information. If the beam of radar m at time k transmits a series of pulse signals with a repetition period of T _pri ,

If pulses are irradiated on the target q, the dwell time of the radar beam on the target is

Represents the number of pulses, and T _pri represents the pulse repetition period. Therefore, the present invention controls the beam pointing and dwell time of the radar system; therefore, the managed resource variables are determined: 1. The beam used by each radar for tracking at each moment quantity

2. How each target chooses which radar beams to illuminate from, 3. Beam illumination dwell time from different radars

Minute;

Step 2: Establish a resource optimization model;

则其动态方程和来自雷达m的目标量测方程分别为：The target q moves at a uniform speed, and its state at time k is:

Then its dynamic equation and the target measurement equation from radar m are:

为均值为零、方差为Q_q,k-1的高斯白噪声，量测

为从回波信号中提取的目标与雷达的距离和角度信息，量测噪声

为零均值、方差为

的高斯白噪声，

表示量测，且该方差与回波信噪比有关；where F _k represents the state transition matrix, process noise

is Gaussian white noise with zero mean and variance Q _q,k-1 , measured

Measure the noise for the distance and angle information of the target and the radar extracted from the echo signal

zero mean, variance is

of white Gaussian noise,

represents the measurement, and the variance is related to the echo signal-to-noise ratio;

表示所有雷达对目标q的照射情况，

表示所有雷达对在目标q上的驻留时间，二者关系为：

For the convenience of the following description, two sets of variables are defined, the beam selection variable at time k Φ _k =[Φ _1,k ,...,Φ _q,k ,...,Φ _Q,k ] ^T and the dwell time variable ΔT _k =[T _1,k ,…,T _q,k ,…,T _Q,k ] ^T , where,

represents the exposure of all radars to the target q,

Represents the dwell time of all radar pairs on the target q, and the relationship between the two is:

Since the Bayesian Cramero bound provides a lower bound for the target state estimation minimum mean square error MSE, and has a certain predictability; therefore, the Bayesian Cramero bound is used as the criterion for tracking performance, and its expression is:

表示贝叶斯克拉美罗界，

表示目标状态

的贝叶斯信息矩阵，为：

represents the Bayesian Cramero bound,

Indicates the target state

The Bayesian information matrix of , is:

表示目标先验信息的费歇尔信息矩阵，

为目标q在k时刻来自于雷达m的数据费歇尔信息矩阵，

表示目标量测对于目标状态的雅克比行列式；

表示量测方差的倒数，

表示求数学期望操作，因为目标贝叶斯克拉美罗界的对角线元素可反映目标状态向量各个分量估计方差的下界，将下式作为各个目标跟踪精度的指标：in,

is the Fisher information matrix representing the target prior information,

is the Fisher information matrix of the target q from the radar m at time k,

represents the Jacobian determinant of the target measurement for the target state;

represents the inverse of the measurement variance,

Represents the mathematical expectation operation, because the diagonal elements of the target Bayesian Cramero bound can reflect the lower bound of the estimated variance of each component of the target state vector, and the following formula is used as an indicator of the tracking accuracy of each target:

where C _CRLB (1,1) and C _CRLB (3,3) represent the first and third components on the diagonal of the Bayesian Cramero boundary, respectively;

结合波束

和驻留时间

约束，建立优化问题模型为：The optimization purpose is determined as follows: in the radar network composed of phased array radars, the radar beam pointing and beam dwell time are reasonably allocated to ensure that the tracking accuracy of all targets meets the predetermined requirement η _q , so that all beams are used for tracking. has the least dwell time; therefore the objective function is

combined beam

and dwell time

Constraints, the optimization problem model is established as:

需要少于雷达形成的波束总数B_m；第四约束表示若是某个目标的预测跟踪性能比较好，则要使其满足预定跟踪精度可能并不需要来自所有雷达的波束照射，一个雷达数量子集即可，因此，k时刻目标q上的波数数量L_q,k不大于雷达总数量M；第五约束表示若目标不被波束照射则驻留时间不存在；第六约束表示驻留时间存在，但其不是任意的，还需要满足一个上下界限，上界为

下界为

第七约束表示对于每个雷达而言用于跟踪的时间上限为

Among them: the first constraint indicates that each target needs to meet its predetermined tracking accuracy η _q ; the second constraint indicates that the beam variable is a binary variable composed of 0 and 1; the third constraint indicates that the radar beam should not only perform tracking but also The search is carried out in the monitoring area, so the total number of beams used by radar m for tracking at time k

Requires less than the total number of beams _Bm formed by the radar; the fourth constraint indicates that if the predicted tracking performance of a target is good, beam illumination from all radars may not be required to make it meet the predetermined tracking accuracy, a subset of the number of radars That is, therefore, the number of wave numbers L _q,k on the target q at time k is not greater than the total number of radars M; the fifth constraint indicates that the dwell time does not exist if the target is not illuminated by the beam; the sixth constraint indicates that the dwell time exists, But it is not arbitrary, it also needs to satisfy an upper and lower bound, the upper bound is

The lower bound is

The seventh constraint expresses that the time limit for tracking for each radar is

Step 3: Propose a beam and dwell time allocation strategy for radar networking, first allocate beam pointing based on radar data information, and then allocate a dwell time algorithm based on optimization theory to achieve resource allocation, and obtain the allocation result;

的大小，对每个雷达的波束给定一个约束范围内的固定时间，即假设

计算出对于目标q，来自于每个雷达的数据信息

然后求出矩阵

的迹

Step 3.1: At time k, in order to reflect the information of each radar data

The size of , given a fixed time within a constraint range for each radar beam, i.e. assuming

Calculate the data information from each radar for the target q

Then find the matrix

trace

并对

的各个元素进行从大到小排序，分类结果如下：Among them: Tr[ ] represents the operation of finding the matrix trace, let

and to

The elements are sorted from large to small, and the classification results are as follows:

表示迹排序结果和每个结果所在的位置，I_q,k表示每个结果所在的位置；

表示排序操作；in:

Indicates the trace sorting result and the position of each result, I _q,k represents the position of each result;

Represents a sorting operation;

Step 3.2: Let the number of wave numbers L _q,k = 0 on the target q at time k, for i = 1, 2, ... M,

Step 3.2.1,

表示驻留时间为T_fix时目标q上来自雷达I_q,k(i)的数据费歇尔信息矩阵，

表示目标q上来自i个雷达的贝叶斯信息矩阵之和，

表示驻留时间为T_fix时目标q上的贝叶斯克拉美罗界，

表示驻留时间为T_fix时目标q的跟踪性能指标；Among them, I _q,k (i) represents the i-th variable of the matrix I _q,k ,

represents the Fisher information matrix of the data from the radar I _q,k (i) on the target q when the dwell time is T _fix ,

represents the sum of Bayesian information matrices from i radars on target q,

represents the Bayesian Cramero bound on the target q when the dwell time is T _fix ,

Indicates the tracking performance index of the target q when the dwell time is T _fix ;

和跟踪门限η_q作对比，如果

则

返回步骤3.2.1；直到

或者i达到M，循环停止；Step 3.2.2, put

Compared with the tracking threshold η _q , if

but

Go back to step 3.2.1; until

or i reaches M, the loop stops;

和跟踪门限η_q作对比，如果

则

返回步骤3.2.1；直到

或者i达到M，循环停止，记录此时i的大小，令L_q,k＝i；Step 3.2.3, put

Compared with the tracking threshold η _q , if

but

Go back to step 3.2.1; until

Or when i reaches M, the loop stops, and the size of i is recorded at this time, so that L _q,k =i;

若

则此时

Step 3.3: For each radar m=1,2,...,M, calculate the total amount of beams used for tracking by each radar at this time

like

then at this time

其中I_q,k(1:L_q,k)表示矩阵I_q,k的前L_q,k个变量；Count the total number of beams L _q,k for each target, and obtain the beam selection result from the radar I _q,k (i) on the target q:

where I _q,k (1:L _q,k ) represents the first L _q,k variables of the matrix I _q,k ;

组成的向量,且有L_q,k个波束被选择来跟踪目标q，对Φ_q,k进行排序，得排序后的波束变量Υ_q,k：Through steps 3.1 to 3.3, the beam selection results from all radars on the target q at time k are obtained Φ _q,k , Φ _q,k represents the beam selection results from all radars on the target q, which is a multi-scalar

, and there are L _q,k beams selected to track the target q, sort Φ _q,k to get the sorted beam variable Υ _q,k :

[Υ _q,k ]=sort(Φ _q,k ,'descend') (10)

Then the beam on the final target q can be written as:

And only L _q,k beams need to illuminate the target q, so the Bayesian information matrix can be written as

表示目标q上来自雷达I_q,k(i)的波束驻留时间；in:

represents the beam dwell time from radar I _q,k (i) on target q;

When the beam allocation is completed, the optimization problem (6) is transformed into the following form:

只能是脉冲重复周期的整数倍，故通过四舍五入，将驻留时间近似为脉冲重复周期的整数倍，记为

The formula (12) is solved by the gradient projection method, and the residence time distribution ΔT _k is obtained; although the residence time value obtained by this method is optimal, it is an arbitrary value between the upper and lower limits, and the residence time time

It can only be an integer multiple of the pulse repetition period, so by rounding, the dwell time is approximated to an integer multiple of the pulse repetition period, which is recorded as

Finally, the beam and dwell time assignment results of the multi-radar system based on multi-target tracking at each tracking moment are obtained.

The invention provides a method for managing the network resources of a phased array radar based on multi-target tracking, and analyzes that in the multi-beam working mode of the multi-phased array radar network, for different targets, the angle and space diversity gain are different for echoes. The effect of the signal-to-noise ratio. Then, construct an optimization problem that ensures the tracking accuracy of each target while reducing the resource consumption of the phased array radar networking system. For this target, a fixed dwell time value is first given to each target, and the data is selected. Several radars with rich information are used to track the target. After establishing the beam pointing, the original non-convex problem is transformed into a convex problem, and then the selected radar beam dwell time is allocated according to the predetermined tracking accuracy of each target. Finally, according to the target observation model, the extended Kalman filter algorithm is used to realize the tracking of multiple targets by the radar network. The advantage of the invention is that when multiple radars perform multiple tracking tasks, due to the difference in target position, angle, RCS and radar networking space diversity gain, the requirements of each target on system resources to maintain predetermined tracking accuracy vary. , resulting in the problem that several targets cannot be effectively tracked, realizing the orderly tracking of multiple targets by the system while saving resources.

Description of drawings

Figure 1 is a flow chart of the joint management of radar networking beam and dwell time based on multi-target tracking.

FIG. 2 is a schematic diagram of a multi-beam working mode of a multi-radar system.

Fig. 3 is the distribution map of target track and radar position.

Figure 4 shows the number of pulses on target 1 for each radar.

Figure 5 shows the number of pulses on target 2 for each radar.

Figure 6 shows the number of pulses on target 3 for each radar.

FIG. 7 is the number of pulses on target 4 for each radar.

Figure 8 shows the time each radar used for tracking.

Figure 9 is a comparison of the total tracking consumption time based on the method in this paper and the traditional greedy algorithm.

Detailed ways

The specific implementation of the present invention is given below according to a MATLAB simulation example.

Since the number of pulses is proportional to the beam dwell time, the present invention will use the number of pulses to reflect the dwell time.

Step 1: Study the radar and target topology and establish managed resource variables

Initialize the system parameters, and the given radar position and target initial state are shown in Table 1 and Table 2, respectively. Considering the operability, the beam pointing Φ _k and the dwell time ΔT _k are selected as the variables of this resource management.

Table 1 Radar locations

Table 2 Target initial state

Step 2: Establishment of resource optimization model

The Bayesian Cramero bound is introduced, and the tracking accuracy criterion Equation (5) is derived from this. Combined with the beam and dwell time constraints, the optimization problem is established as shown in Equation (6).

Step 3: Propose the beam and dwell time allocation strategy of the radar networking system, and get the allocation results

Given resource optimization model parameters: pulse repetition period T _pri = 1ms, transmit power P _av = 2e ⁴ w, the total time for tracking at each tracking moment is T _track = 0.4s, and the beam dwell time constraint is 0.005T _track ≤ΔT _q,k ≤0.9T _track , the tracking threshold η _1:Q =[0.027,0.027,0.027,0.027] ^T , the target process noise is consistent during the tracking process. According to the proposed beam selection and dwell time allocation algorithm, the resource allocation results are obtained

Figure 4, Figure 5, Figure 6 and Figure 7 show the beam and dwell time assignment results for targets 1, 2, 3 and 4, respectively. Figure 8 shows the total time consumed by each radar in this tracking. To reflect the effectiveness of the present invention, we compare a traditional greedy algorithm with the method proposed by the present invention. The time consumption diagram of the two methods for tracking is shown in Figure 9. It can be seen that the method proposed by the present invention is more economical resources, about 25% less resources than the greedy algorithm.

Step 4: Use the extended Kalman filter algorithm to achieve multi-target tracking

代入目标动态模型和量测模型(2)，得出量测噪声协方差和回波信噪比，再根据目标跟踪的预测和更新过程，得到目标的状态估计。目标的实际航迹和估计航迹如图3所示。Allocating resources

Substitute the target dynamic model and measurement model (2) to obtain the measurement noise covariance and echo signal-to-noise ratio, and then obtain the target state estimation according to the target tracking prediction and update process. The actual track and estimated track of the target are shown in Figure 3.

It can be seen from the specific embodiments of the present invention that, compared with using the greedy algorithm to allocate resources, the present invention can reduce the resource consumption of the phased array radar system for tracking tasks on the premise of ensuring the tracking accuracy of all targets. , saving about 25% of resources.

Claims (1)

1. A multi-target tracking based resource management method for a phased array radar networking system comprises the following steps:

step 1: determining the topological structures and managed resource variables of the radar and the target;

a radar network consisting of M phased array radars, the M-th radar being located at (x)_m,y_m) M is 1,2, …, M, Q targets are widely distributed in a monitoring area, the radar system tracks the targets, and assuming that each target moves at a constant speed, the initial position and the speed of the target Q are respectively

And

q is 1, …, Q, and at the kth tracking time, the position and velocity of the target Q are respectively

And

at time k, each radar may transmit B_mA beam of waves having

Selecting a beam for target tracking, wherein each beam can only track one target at each tracking moment, and introducing a binary variable because whether the beam of the radar m is used for tracking the target q cannot be determined

In order to maintain the tracking of the target, at each tracking moment, the radar beam needs to transmit a certain amount of pulses to the target to acquire target information, and if the beam of the radar m at the moment k transmits a series of repeated cycles with a period of T_priPulse of lightA signal, and is provided with

The pulse irradiates on the target q, and the residence time of the radar beam on the target is

Indicating the number of pulses, T_priThe pulse repetition period is represented, so that the beam pointing direction and the residence time of the radar system are controlled; thus, the managed resource variables are determined: 1. number of beams used for tracking per radar per time instant

2. How each target chooses which radar beam to illuminate from, 3. dwell time of beam illumination from different radars

Dividing;

step 2: establishing a resource optimization model;

the target q moves at a constant speed, and the state at the moment k is as follows:

then the dynamic equation and the target measurement equation from radar m are respectively:

wherein, F_kRepresenting state transition matrix, process noise

Is a mean of zero and a variance of Q_q,k-1White Gaussian noise of (1), measurement

Measuring noise for range and angle information of target and radar extracted from echo signal

Is zero mean and variance of

The white gaussian noise of (a) is,

representing a measurement and the variance is related to the echo signal-to-noise ratio;

for convenience of the following description, two sets of variables are defined, the beam selection variable Φ at time k_k＝[Φ_1,k,…,Φ_q,k,…,Φ_Q,k]^TAnd a dwell time variable Δ T_k＝[T_1,k,…,T_q,k,…,T_Q,k]^TWherein

representing the illumination of the target q by all the radars,

representing the residence time of all radar pairs on the target q, the relationship is as follows:

the Bayes Cramer-Rao bound provides a lower bound for the minimum Mean Square Error (MSE) of the target state estimation, and has certain predictability; therefore, the bayesian clar-merome boundary is adopted as a criterion of tracking performance, and the expression is as follows:

representing the bayesian clarmeo bound,

representing target states

The bayesian information matrix of (a) is:

wherein,

a fisher information matrix representing the target prior information,

the fisher information matrix of data from radar m at time k for target q,

a Jacobian determinant representing target measurements versus target states;

the inverse of the measured variance is represented,

the mathematical expectation operation is expressed, because the diagonal elements of the target bayesian clar-merome boundary can reflect the lower bound of the estimation variance of each component of the target state vector, and the following formula is taken as the index of each target tracking precision:

wherein, C_CRLB(1,1) and C_CRLB(3,3) respectively representing a first component and a third component on a diagonal of the Bayesian Cramer-Lo boundary;

the optimization purpose is determined as follows: in a radar networking formed by phased array radars, radar beam pointing and beam residence time are reasonably distributed, and all target tracking accuracy is ensured to meet a preset requirement eta_qMinimizing the dwell time of all beams for tracking; the objective function is thus

Combining beams

And residence time

Constraining, and establishing an optimization problem model as follows:

wherein: the first constraint represents that each target needs to meet its predetermined tracking accuracy η_q(ii) a The second constraint indicates that the beam variable is a binary variable consisting of 0 and 1; the third constraint represents the total number of beams used by the radar m for tracking at time k, considering that the radar beams are to perform not only tracking but also searching in the monitored area

Requiring less than the total number of beams B formed by the radar_m(ii) a The fourth constraint indicates that if the predicted tracking performance of a target is better, it may not require beam shots from all radars for it to meet the predetermined tracking accuracy, since one subset of the number of radars is neededHere, the number of wave numbers L on the target q at the time k_q,kNot greater than the total number M of radars; a fifth constraint indicates that the dwell time does not exist if the target is not illuminated by the beam; the sixth constraint indicates that the residence time exists, but it is not arbitrary and also requires that an upper and lower bound be satisfied, the upper bound being

Lower boundary is

The seventh constraint represents an upper time limit for tracking for each radar of

And step 3: a beam and residence time distribution strategy of the radar networking is provided, beam pointing is distributed based on radar data information, and then resource distribution is realized according to an algorithm for distributing residence time based on an optimization theory to obtain a distribution result;

step 3.1: time k, in order to represent the respective radar data information

Given a fixed time within a constraint range for each radar beam, i.e. assuming

Data information from each radar for target q is calculated

Then, the matrix is obtained

Trace of

Wherein: tr [. to]Indicating an operation of determining a trace of a matrix

And to

The elements of (2) are sorted from large to small, and the classification result is as follows:

wherein:

indicating the trace-ordering results and where each result is located, I_q,kIndicating the location of each result;

representing a sort operation;

step 3.2: let the number L of wave numbers on the target q at time k_q,k0, for i-1, 2, … M,

step 3.2.1,

Wherein, I_q,k(i) Representation matrix I_q,kThe (c) th variable of (a),

denotes a dwell time of T_fixFrom radar I on time target q_q,k(i) The fischer information matrix of the data of (a),

representing the sum of bayesian information matrices from i radars on target q,

denotes a dwell time of T_fixThe bayesian cramer-pero boundary on the time target q,

denotes a dwell time of T_fixTracking performance index of the time target q;

step 3.2.2, mixing

And a tracking threshold η_qIn contrast, if

Then

Returning to the step 3.2.1; up to

Or i reaches M, and the cycle stops;

step 3.2.3, mixing

And a tracking threshold η_qIn contrast, if

Then

Returning to the step 3.2.1; up to

Or i reaches M, the cycle stops, recording i at that timeSize, order L_q,k＝i；

Step 3.3: for each radar M being 1,2, …, M, the total beam amount used for tracking by each radar at the moment is calculated

If it is

Then this time

Counting the total beam quantity L of each target_q,kObtaining the data from radar I on target q_q,k(i) Beam selection result of (2):

wherein I_q,k(1:L_q,k) Representation matrix I_q,kFront L of_q,kA variable;

obtaining a wave beam selection result phi from all radars on the target q at the moment k through the steps 3.1-3.3_q,k，Φ_q,kRepresenting the beam selection results from all radars on target q, is a plurality of scalars

A vector of components and having L_q,kEach beam is selected to track target q, for Φ_q,kSequencing to obtain sequenced beam variable gamma_q,k：

[Υ_q,k]＝sort(Φ_q,k,'descend′) (10)

The beam on the final target q can be written as:

and only L_q,kThe individual beams need to illuminate the target q, so the Bayesian information matrix can be written as

Wherein:

representing the origin from radar I on target q_q,k(i) Beam dwell time of (a);

when the beam allocation is complete, the optimization problem (6) is transformed into the following form:

solving the formula (12) by a gradient projection method to obtain residence time distribution delta T_k(ii) a Although the residence time value obtained by the method is optimal, the value is any value between the upper limit and the lower limit, and the residence time is

Can only be an integer multiple of the pulse repetition period, so by rounding off, the dwell time is approximated as an integer multiple of the pulse repetition period, denoted

Finally, the multi-radar system wave beam and residence time distribution result based on multi-target tracking at each tracking moment is obtained

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