A maneuvering resource allocation method for target tracking in airborne radar networks

Abstract

The present invention relates to the field of radar resource allocation, and more specifically, to a method for allocating maneuverable resources for target tracking in an airborne radar network. This method employs a three-step solution involving the elimination of non-positive constraints and the linearization of positive constraints. Using the proximal alternating direction multiplier method, the method solves a nonconvex objective function with multiple linear inequality / equality constraints while ensuring convergence. This method also avoids the high computational cost of applying gradient projection methods onto polyhedrons. The method fully utilizes maneuverable resources and achieves excellent target tracking accuracy in a shorter computation time.

Description

Technical Field

[0001] The present invention relates to the field of radar resource allocation, and in particular to a method for allocating maneuverable resources for target tracking in an airborne radar network. Background Art

[0002] With the advancement of integrated circuit technology, more and more radars are being installed on airborne platforms, creating a new type of radar resource: mobile resources. However, when radars are operating, there is a conflict between radar resource budget and tracking performance. Therefore, resource allocation is crucial for target tracking in radar network systems. Resources from multiple nodes must be coordinated to ensure that the radar network can complete tracking tasks within a limited resource budget.

[0003] To achieve this, many radar resource allocation methods for tracking tasks have been proposed. These methods formulate resource allocation as an optimization problem, optimizing the maneuverability of each node and designing its trajectory to achieve better radar spatial observation diversity and target tracking performance, thereby maximizing resource utilization efficiency within the radar network. However, most existing methods do not consider radar maneuverability, and existing trajectory planning work also fails to account for radar detection context. Summary of the Invention

[0004] In view of the problems existing in the prior art, the purpose of the present invention is to provide a maneuver resource allocation method for target tracking in an airborne radar network, so as to fully utilize the maneuver resources and obtain better radar target tracking performance.

[0005] In order to achieve the above objectives, the present invention adopts the following technical solutions to achieve them.

[0006] A method for allocating maneuverable resources for target tracking in an airborne radar network comprises the following steps:

[0007] Step 1: Establish an airborne radar network with i=1, 2, ..., N nodes, where N is an integer greater than 0; define an allocation interval T0; let k represent the kth time, the initial value of k is 1, k∈{1, 2, ..., K}, K is a pre-set maximum tracking time, and K is a positive integer greater than 0;

[0008] Step 2: N radar nodes fuse the received measurements. The fusion center builds a target measurement model based on the echo data and determines the target state and node state at the kth moment.

[0009] Step 3: Determine the motion model based on the target state and node state at the kth moment;

[0010] Step 4: Determine the objective function based on the Bayesian Cramer-Rao bound; determine the constraint conditions of the objective based on the actual situation of the objective environment;

[0011] Step 5, eliminate non-positive constraints;

[0012] Step 6, linearize the positive constraints;

[0013] Step 7: Use the proximal alternating direction multiplier method to solve the optimization problem and obtain the speed of each node radar in the airborne radar network in the x direction and y direction; then obtain the level flight speed of each node radar through the speed of the radar in the x direction and y direction. and angular velocity

[0014] Step 8: When k does not reach the preset maximum tracking time K, add 1 to k and return to step 2; when k reaches the preset maximum tracking time K, the iterative execution process stops and the level flight speed of each node radar at the first moment is obtained. and angular velocity The level flight speed of each node radar at the Kth moment and angular velocity It is recorded as the result of maneuver resource allocation for target tracking in an airborne radar network.

[0015] Compared with the existing technology, the beneficial effects of the present invention are: solving a non-convex objective function problem with multiple linear inequality / equality constraints while ensuring convergence, while avoiding the high computational cost when applying the gradient projection method to project on a polyhedron; it can fully utilize maneuvering resources and obtain good target tracking accuracy in a shorter computing time. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0017] Figure 1 Schematic diagram of the flow chart of the mobile resource allocation method for target tracking in the airborne radar network of the present invention

[0018] Figure 2 Spatial relationship diagram of each radar, target location and threat area in the airborne radar network set for the simulation experiment of the present invention

[0019] Figure 3 This is a diagram illustrating that the third constraint in the mathematical expression of each allocated time unit k of the present invention is in a non-active state;

[0020] Figure 4 FIG. 4 is an illustration of a non-active state of the fourth constraint in the mathematical expression for each allocated time unit k of the present invention;

[0021] Figure 5 This is an illustration of the extension point;

[0022] Figure 6 It is an illustration of the extension points under the constraints;

[0023] Figure 7 A diagram illustrating the approximate range of limitations for level flight speed and angular velocity;

[0024] Figure 8 1 is a comparison chart of the results of mobile resource allocation under the method of the present invention, the method of control solution 1, and the method of control solution 2;

[0025] Figure 9 A diagram comparing the distances between two close radars under the method of the present invention, the method of comparative solution 1, and the method of comparative solution 2;

[0026] Figure 10 A comparison diagram of the Bayesian Clamer-Rao lower bound for target tracking under the method of the present invention, the method of control solution 1, and the method of control solution 2;

[0027] Figure 11 It is a comparison chart of the calculation time under the method of the present invention and the method of control scheme 2. DETAILED DESCRIPTION

[0028] The embodiments of the present invention will be described in detail below with reference to examples. However, those skilled in the art will understand that the following examples are only used to illustrate the present invention and should not be construed as limiting the scope of the present invention.

[0029] refer to Figure 1 , which is a flow chart of a method for allocating maneuverable resources for target tracking in an airborne radar network according to the present invention; the method for allocating maneuverable resources for target tracking in an airborne radar network comprises the following steps:

[0030] Step 1: Establish an airborne radar network with i=1, 2, ..., N nodes, where N is an integer greater than 0; define an allocation interval T0; let k represent the kth time, the initial value of k is 1, k∈{1, 2, ..., K}, K is a pre-set maximum tracking time, and K is a positive integer greater than 0;

[0031] Step 2: N radar nodes fuse the received measurements. The fusion center builds a target measurement model based on the echo data and determines the target state and node state at the kth moment.

[0032] In the kth allocation interval, radar i can generate a series of measurements M for the target i，k ≥1. Definition For radar i in k i，m The mth measurement value of (representing a time unit).

[0033] The target measurement model is expressed as formula (1):

[0034]

[0035] In formula (1), represents the state vector corresponding to the mth measurement of the i-th radar at time k, is a composite state vector, which includes the state vector of the target and the state vector of the airborne radar; the state vector of the target at time k is expressed as The subscript T represents the target, x T，k is the horizontal position of the target, is the horizontal velocity of the target, y T，k is the vertical position of the target, is the vertical velocity of the target; the state vector of the airborne radar is expressed as in represents the spatial azimuth of radar i, R represents the radar, is the horizontal position of radar i, is the vertical position of radar i; link the N radar state vectors into a single vector, and we get

[0036] In formula (1), Different measurements are linked, where represents the distance of the target in the mth measurement obtained by the i-th radar at the k-th moment, represents the azimuth of the target in the mth measurement obtained by the i-th radar at the k-th time, and represents the position of radar i in this measurement;

[0037] In formula (1), is the measurement noise of radar i, let is a zero-mean Gaussian noise, and its covariance is formula (2):

[0038]

[0039] In formula (2), is the radar's level flight speed, is the radar angular velocity; and is the corresponding error variance of the GPS measured information, which is independent of the target signal-to-noise ratio; and is the error variance of the distance and azimuth measurement, and is inversely linear with the target signal-to-noise ratio, as shown in formula (3):

[0040]

[0041] In formula (3), the signal-to-noise ratio is the signal-to-noise ratio of the i-th radar node when it obtains the m-th measurement in the k-th time unit, which is directly related to the trajectory control vector d of each node i，k Related;d i，k is the mobile radar resource that needs to be allocated, which is the trajectory control vector of the airborne radar network; β i 2 and is the transmission signal bandwidth and 3dB receiving beamwidth of radar i.

[0042] Step 3: Determine the motion model based on the target state and node state at the kth moment;

[0043] Specifically, link the target state vector to In this case, we can get an extended vector And the dimension of this state vector is 3N+4; for ξ k , its state transfer equation is as follows:

[0044] ξ k =f(ξ k-1 )+u k (4)

[0045] In formula (4), where f T (·) is the known state transition function; u k is a zero-mean Gaussian process noise with covariance Q k ,; is the trajectory control vector of the airborne radar network; f R (·) represents the airborne radar motion function, which is expressed as formula (5):

[0046]

[0047] In formula (5),

[0048] Step 4: Determine the objective function based on the Bayesian Cramer-Rao bound; determine the constraint conditions of the objective based on the actual situation of the objective environment;

[0049] Intuitively, moving towards the target will definitely increase the signal-to-noise ratio of the target and reduce the error variance of its measurement. However, due to some physical constraints, multiple nodes may not move directly towards the target. Mathematically, the maneuver resource allocation scheme can be formulated as a problem of maximizing target tracking accuracy, subject to some maneuver resource constraints. The parameters considered are the maneuver resource variables of multiple airborne radars.

[0050] Specifically, by using composite measurement, the Cramer-Rao lower bound matrix can be approximated as Equation (6):

[0051]

[0052] In formula (6), F is the Jacobian matrix of the state transfer function; It is k+1 The Jacobian matrix of the mth measurement estimate is as follows; J is the Fisher information matrix;

[0053] In order to maximize the target accuracy, the mathematical representation of the maneuver resource allocation scheme at each allocation time unit k can be expressed as formula (7):

[0054]

[0055] Its objective function is shown in formula (8), which is used to describe the tracking accuracy of the target.

[0056]

[0057] In formula (8), Λ is a normalized matrix, as shown in formula (9):

[0058]

[0059] In formula (9), represents the Kronecker operator, and I2 represents the identity matrix of order 2.

[0060] The constraint of formula (7) means that the level flight speed and angular velocity are subject to minimum and maximum limits. Each airborne radar cannot pass through multiple threat zones because it needs to avoid being detected by the enemy radar and any two airborne radars need to be separated by a minimum distance D in each allocated time unit. min .

[0061] In the model, each enemy radar in the threat zone is considered to be stationary. and To represent the detection radius and center of the enemy's lth radar.

[0062] Step 5: Eliminate non-positive constraints.

[0063] The mathematical expression of the mobile resource allocation scheme at each allocated time unit k is typically a difficult nonlinear optimization problem due to a nonconvex objective function and numerous nonconvex constraints. Despite the nonconvexity of the objective function, an efficient solution still faces two difficulties. On the one hand, the nonlinearity and nonconvex constraints make it difficult to generate feasible iterations through the optimization algorithm. On the other hand, the large number of constraints significantly increases the computational burden. To address these two difficulties, a three-step solution technique consisting of non-active constraint elimination and active constraint linearization is designed to solve the original nonconvex optimization problem.

[0064] In practice, because the distance between the radar and the threat area and between radars may be relatively far, many constraints may be non-positive. Eliminating non-positive constraints will greatly reduce the amount of calculation, so it is necessary to eliminate non-positive constraints.

[0065] In the following proposition, the Sufficient conditions for determining whether the last two constraints are positive.

[0066] Proposition 1: Let r be the maximum distance the radar can move within the allocated interval, Assign the known position of radar i at the kth time unit. Then the last two constraints are not positive under the following conditions:

[0067] a) For if Then the third constraint is negative.

[0068] b) For if Then the fourth constraint is negative.

[0069] The proof of Proposition 1 is obvious, because we only need to prove that the eliminated non-positive constraints are always valid for the given first two constraints. The most intuitive explanation of Proposition 1 is as follows Figure 3 shown.

[0070] refer to Figure 3 , the black cross represents the position of radar i, the red circle area represents the threat area, and the blue sector area represents the possible position that the radar can reach within the next allocated time. If the radar is far away from the threat area, as long as the first two constraints remain unchanged, the third constraint is always a non-active constraint. Figure 4 , a similar conclusion can be drawn. The fourth constraint is the collision avoidance constraint. If the radars are moving away from each other, they will maintain a safe distance to avoid collision.

[0071] Therefore, the mathematical expression of the mobile resource allocation scheme in each allocation time unit k can be rewritten as formula (10).

[0072]

[0073] In formula (10), φ i，k and μ i，k There are two sets defined as follows:

[0074] For the threat area constraint of node i, if Then we call the constraint positive, and the set of all threat regions l for the i-th radar in the k-th time period is φ i，k ; For the collision constraint of node i, if Then we call the constraint positive, and the set of all nodes j that may collide with the i-th radar in the k-th time period is μ i，k .

[0075] Step 6: Linearize the positive constraints.

[0076] Specifically, a set of auxiliary variables are introduced:

[0077]

[0078]

[0079] in, and They represent the velocity components of the radar in the x and y directions respectively.

[0080] Based on the auxiliary vectors, the constraints can be approximated as linear constraints.

[0081] The mathematical expression of the mobile resource allocation scheme at each allocation time unit k has four constraints.

[0082] Substep 6.1, linearize the third constraint.

[0083] According to the inequality x 2 +y 2 ≥2xy, the third nonlinear constraint can be relaxed to a linear constraint. Specifically, for any have Therefore, the left side of the third nonlinear constraint allows a linear lower bound, as shown in Equation (11):

[0084]

[0085] In formula (11), and It is an extension point.

[0086] According to Equation (11), the third constraint can be relaxed as a linear constraint, as shown in Equation (12):

[0087]

[0088] The boundary line of the third constraint can be specified as formula (13):

[0089]

[0090] Mathematically, the linearization process has the following properties:

[0091] Proposition 2: For any Must satisfy the equation The boundary line of the above formula is tangent to the boundary of the first threat area. is a tangent point.

[0092] prove: The distance from the boundary line can be expressed as:

[0093]

[0094] Therefore, the boundary line is tangent to the threat area. Substitute the boundary line and verify that it is a point on the boundary line. In addition, Located on the border of the threat zone, so see is a tangent point. This completes the proof.

[0095] refer to Figure 5 , it can be seen that the best expansion point is chosen in the middle of the arc, which is located at the boundary where the threat area intersects the reachable area ( Figure 5 After linearization, the area outside the red circle is approximated to the area outside the black line.

[0096] The auxiliary variable g i，k And the state transfer function expression of airborne radar Substituting the relaxed third constraint into equation (13):

[0097] χ i，l，k ·g i，k ≥ζ i，l，k (13)

[0098] In formula (13), χ i，l，k and ζ i，l，k Related to predefined parameters, such as extension points Radar location The center and detection radius of the enemy radar, etc.

[0099] Substep 6.2, linearize the fourth constraint.

[0100] To handle the fourth nonlinear constraint, we also turn to the inequality x 2 +y 2 ≥2xy. On this basis, the fourth constraint can be relaxed to formula (14):

[0101]

[0102] From formula (14), we can see that the fourth constraint is is a linear relationship and is located in four-dimensional Euclidean space; when given or When , the left side of Equation (14) specifies two parallel lines in two-dimensional Euclidean space.

[0103] Therefore, the fourth constraint can be viewed as two parallel linear constraints in a two-dimensional Euclidean space, where the feasible region corresponds to the exterior of the two parallel lines, as Figure 6 shown.

[0104] These two parallel lines can be written as formula (15):

[0105]

[0106] By choosing the right and The line specified by equation (15) has the following properties.

[0107] Proposition 3: For any and The distance between the two parallel lines in the above formula is equal to D if the following two conditions are met: min , both lines are perpendicular to the point and Determined line.

[0108]

[0109]

[0110] Proof: By and The slope of the determined line is At the same time, due to and If the first condition is met, the slope of the two parallel lines can be derived as

[0111] Therefore, we have:

[0112]

[0113] This means that the two parallel lines are perpendicular to the point and In addition, similar to the proof of Proposition 2, it can be calculated when and When the second condition is met, the distance between the two parallel lines is D min . This completes the proof.

[0114] Proposition 3 shows that if the extension point and If the conditions are met, the fourth constraint (linear constraint) can be relaxed to ensure that and The distance between them is greater than or equal to D min .

[0115] The auxiliary variable g i，k And the state transfer function expression of airborne radar Substituting the relaxed fourth constraint, we get formula (16):

[0116]

[0117] In formula (16), γ i，j，k and ε i，j，k Related to multiple predefined parameters, such as extension points and Two radars and location, and allocation time interval T0, etc.

[0118] Substep 6.3, linearize the first and second constraints.

[0119] Use g i，k The first and second constraints are reformulated and linearized. The above linearization process can effectively convert nonlinear constraints into linear constraints.

[0120] Use g i，k replace and Then we have:

[0121]

[0122]

[0123] The linear level flight speed and angular velocity constraints are expressed as nonlinear constraints, which are difficult to handle.

[0124] Then, the non-convex sector region represented by the first two constraints is approximated as a convex trapezoidal region, as Figure 7 To get the four sides of the trapezoidal area, we need to get the coordinates of vertices A, B, and O.

[0125]

[0126]

[0127]

[0128] at the same time, Figure 7 The lines l1 and l2 in can be written as:

[0129]

[0130]

[0131] According to OA, OB, l1 and l2, we can get Figure 7 The area of the trapezoid in .

[0132] The area of this trapezoid is expressed by four linear constraints, which can be abbreviated as Equation (19):

[0133]

[0134] In formula (19), A i，k and c i，k and and Related.

[0135] In sub-step 6.4, combined with the above linearization process, the mathematical expression of the mobile resource allocation scheme in each allocation time unit k can be written as Equation (20).

[0136]

[0137] Step 7: Use the proximal alternating direction multiplier method to solve the optimization problem and obtain the speed of each node radar in the airborne radar network in the x direction and y direction; then obtain the level flight speed of each node radar through the speed of the radar in the x direction and y direction. and angular velocity

[0138] Equation (20) is a nonconvex objective function with multiple linear inequality / equality constraints. To ensure convergence and solve this type of problem, gradient projection methods are often used. However, projection onto a polyhedron (corresponding to the linear inequality / equality constraints in Equation (20)) results in a very large computational load.

[0139] Specifically, in formula (20), the two variables g i，k 、g j，k It is linearly coupled.

[0140] Introducing decoupled variables To decouple, the problem to be optimized can be written as formula (21):

[0141]

[0142] Use the proximal alternating direction multiplier method to convert the auxiliary variable e k , s k Introduce inequality constraints and rewrite Equation (21) as Equation (22):

[0143]

[0144] A notable feature of the proximal alternating direction multiplier method is the introduction of a smooth iterative (i.e., exponentially weighted) sequence, and in each iteration an augmented Lagrangian function containing a quadratic proximal term centered around the current smoothed primitive iteration.

[0145] make:

[0146]

[0147] Formula (23) is the augmented Lagrangian function, where χ k =[...,χ i，l，k , ...] and ζ k =[...,ζ i，l，k , ...] represents a set of χ i，l，k and ζ i，l，k , B k g k =z k Represents a set of constraints y=[y e ,y s ,y w ] represents the dual variable; ρ e , ρ s , ρ w >0 is the original penalty parameter; p is a predefined positive parameter that ensures L(g k , z k , e k , s k , b k ;y) in g k is strongly convex; b k It is a smooth iterative (i.e., exponentially weighted) sequence introduced. By inserting this sequence into the augmented Lagrangian function, it can be guaranteed that the next iteration will not deviate too much, ensuring the stability of the inexact augmented Lagrangian multiplier method on non-convex problems.

[0148] The proximal multiplier alternating direction algorithm can be summarized as follows:

[0149] The first step is to set ρ e ,ρ s ,ρw >0,λ e ,λ s ,λ w >0, 0<β≤1, c1, c2>0;

[0150] The second step is to initialize y e，0 ,y s，0 ,y w，0 ,in and All are greater than or equal to 0;

[0151] The third step is to determine φ i，k and μ i，k ;

[0152] Step 4: Calculate χ i，l，k and γ i，l，k and

[0153] Step 5: For t = 1, 2, 3, ..., start looping and loop the following formula:

[0154]

[0155]

[0156]

[0157]

[0158] and Update via gradient projection method;

[0159] in, and Calculated by the following three formulas:

[0160]

[0161]

[0162]

[0163] in,[·] + It is an operator that projects a vector into the non-negative quadrant, ensuring that all elements of the vector are positive.

[0164] Step 6: End the loop;

[0165] Get the speed of each node radar in the airborne radar network in the x and y directions;

[0166] The speed of the radar in the x and y directions is calculated according to Calculate the level flight speed of each node radar and angular velocity

[0167] Step 8: When k does not reach the preset maximum tracking time K, add 1 to k and return to step 2; when k reaches the preset maximum tracking time K, the iterative execution process stops and the level flight speed of each node radar at the first moment is obtained. and angular velocity The level flight speed of each node radar at the Kth moment and corners It is recorded as the result of maneuver resource allocation for target tracking in an airborne radar network.

[0168] The effects of the present invention are further verified and explained through simulation experiments below.

[0169] (1) Simulation conditions:

[0170] The simulation conditions were performed on a computer with a 2.9 GHz Intel(R) Core(TM) i7-4790 CPU and MATLAB 9.8.0 (2020a) installed.

[0171] (2) Simulation content and result analysis:

[0172] refer to Figure 2 , which is a spatial relationship diagram of each radar, target position and threat area in the airborne radar network set in the simulation experiment of the present invention.

[0173] The number of airborne radars is N=5, and the maximum angular velocity is w max =0.03rad / s, minimum level flight speed v min and maximum level flight speed v max Set to (25, 100) m / s.

[0174] Assume that the radar cross section of the target is non-fluctuating, so the signal-to-noise ratio of the target is only related to the radar-to-target distance. When the radar-to-target distance is 120km, we set the signal-to-noise ratio of a reference target to 12dB. Assume that the target moves at a constant speed of 50m / s. Set the interval between consecutive allocation time units to T0 = 20s, and use 20 allocation intervals to support our simulation. For collision avoidance, D min Should be greater than 2v max T0, so we set D min =5km.

[0175] The initial state of the radar, the location of the target and threat areas are as follows: Figure 2 shown.

[0176] To initialize the parameters of the proximal alternating direction multiplier method, the original step sizes c1 = 0.05 and c2 = 10 are set. 3 . Set the penalty parameter to ρ e =2,ρ s =1×10 -4 ,ρ w =3×10 -3 to balance the different terms of the gradient during the iteration. The maximum number of iterations is set to 120. In addition, the values of the dual and proximal step sizes λ are initialized e =4,λ s =5×10 -5 ,λ w =1.5×10 -3 and β=0.9 (0<β≤1).

[0177] To illustrate the superiority of our invention, we compare the resource allocation results, the corresponding Cramer-Rao lower bound and the computation time with two control schemes.

[0178] In contrast to Scheme 1, the actual situation of each node resource is scheduled in advance. In Scheme 1, it is assumed that the airborne radar moves at a constant speed of 80m / s, and its initial position and direction angle are as follows: Figure 2 shown.

[0179] In contrast to Solution 2, a genetic algorithm is used to solve the problem. The initial conditions of the genetic algorithm are as follows: population size is 200, crossover rate is 0.8, mutation rate is 0.2, and maximum number of generations is 100.

[0180] Figure 8 The results of maneuver resource allocation for the three methods are shown below. As can be seen, Radar 3 can avoid the threat area and continue moving toward the target in both our solution and Control Solution 2, demonstrating the correctness of our invention. In Control Solution 1, Radar 3 moves on a constant trajectory, so it may not reach the target. Even worse, if its direction angle is set incorrectly, it will pass through the threat area.

[0181] The distances between radar 1 and radar 2 corresponding to these three methods are as follows: Figure 9 As shown. Both our solution and control solution 2 can ensure that the distance is greater than D min In contrast, in the control scheme 1, since the radars have already predetermined trajectories to move, the distance between the two radars cannot be guaranteed.

[0182] from Figure 10 It can be seen from the figure that, compared with the control scheme 2, the control scheme 1 has the worst tracking accuracy because it uses pre-determined maneuvering parameters.

[0183] In order to analyze the computational load, the computational time of the present invention was compared with that of the control scheme 2. The average running time over 10 Monte Carlo experiments, the corresponding results are shown in Figure 2. Figure 11 Reference Figure 11 It can be seen that compared with the control scheme 2, the present invention can achieve equivalent target tracking accuracy while consuming less computing time.

[0184] In summary, the simulation experiments have verified the correctness, effectiveness and reliability of the present invention.

[0185] Although this specification has provided a detailed description of the present invention using general descriptions and specific embodiments, it will be apparent to those skilled in the art that modifications and improvements may be made based on the present invention. Therefore, such modifications and improvements, which do not depart from the spirit of the present invention, are intended to be within the scope of protection claimed herein.

Claims

1. A method for allocating maneuverable resources for target tracking in an airborne radar network, characterized in that: The following steps are involved: Step 1: Establish a set of airborne radar networks with i=1, 2, ..., N nodes, where N is an integer greater than 0; define an allocation interval T0; let k represent the kth time, the initial value of k is 1, k∈{1, 2, L, K}, K is the preset maximum tracking time, and K is a positive integer greater than 0; Step 2: N radar nodes fuse the received measurement data. The fusion center builds a target measurement model based on the measurement data and determines the target state and node state at the kth moment. Step 3: Determine the motion model based on the target state and node state at the kth moment; The target state is the position of the target at the kth moment, and the node state is the position of all radar nodes at the kth moment. The motion model is a functional relationship that derives the target position and node position at the k+1th moment from the target position and node position at the kth moment. Therefore, the motion model is determined as: k =f(ξ k-1 )+u k , Step 4: Determine the objective function based on the Bayesian Cramer-Rao bound; determine the constraint conditions of the objective based on the actual situation of the objective environment; The objective function is: The target constraints are the limits of level flight speed and handover speed, the position of the enemy radar, the scanning radius, and the minimum distance between two radar nodes, which are expressed as: Step 5, eliminate non-positive constraints; Step 6, linearize the positive constraints; Step 7: Use the proximal alternating direction multiplier method to solve the optimization problem and obtain the speed of each node radar in the airborne radar network in the x direction and y direction; then obtain the level flight speed of each node radar through the speed of the radar in the x direction and y direction. and angular velocity Step 8: When k does not reach the preset maximum tracking time K, add 1 to k and return to step 2; when k reaches the preset maximum tracking time K, the iterative execution process stops and the level flight speed of each node radar at the first moment is obtained. and angular velocity The level flight speed of each node radar at the Kth moment and angular velocity It is recorded as the result of maneuver resource allocation for target tracking in an airborne radar network.

2. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 1, characterized in that: In step 2, the target measurement model is expressed as formula (1): In formula (1), For radar i in k i,m The mth measurement value of ; In formula (1), represents the state vector corresponding to the mth measurement of the i-th radar at time k, is a composite state vector, which includes the state vector of the target and the state vector of the airborne radar; the state vector of the target at time k is expressed as The subscript T represents the target, x T,k is the horizontal position of the target, is the horizontal velocity of the target, y T,k is the vertical position of the target, is the vertical velocity of the target; the state vector of the airborne radar is expressed as in represents the spatial azimuth of radar i, R represents the radar, is the horizontal position of radar i, is the vertical position of radar i; link the N radar state vectors into a single vector, and we get In formula (1), Different measurements are linked, where represents the distance of the target in the mth measurement obtained by the i-th radar at the k-th moment, represents the azimuth of the target in the mth measurement obtained by the i-th radar at the k-th time, and represents the position of radar i in this measurement; In formula (1), is the measurement noise of radar i, let is a zero-mean Gaussian noise, and its covariance is formula (2): In formula (2), is the radar's level flight speed, is the radar angular velocity; and is the corresponding error variance of the GPS measured information, which is independent of the target signal-to-noise ratio; and is the error variance of the distance and azimuth measurement, and is inversely linear with the target signal-to-noise ratio, as shown in formula (3): In formula (3), the signal-to-noise ratio is the signal-to-noise ratio of the i-th radar node when it obtains the m-th measurement in the k-th time unit, which is directly related to the trajectory control vector d of each node i,k Related;d i,k is the mobile radar resource that needs to be allocated, which is the trajectory control vector of the airborne radar network; and is the transmission signal bandwidth and 3dB receiving beamwidth of radar i.

3. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 2, wherein: Step 3, specifically, link the target state vector to In the example above, we get an extended vector For ξ k , its state transfer equation is as follows: x k =f(ξ k-1 )+u k (4) In formula (4), where f T (·) is the known state transition function; u k is a zero-mean Gaussian process noise with covariance Q k ; is the trajectory control vector of the airborne radar network; f R (·) represents the airborne radar motion function, which is expressed as formula (5): In formula (5), 4. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 1, wherein: In step 4, specifically, by using composite measurement, the Cramer-Rao lower bound matrix is approximated as formula (6): In formula (6), F is the Jacobian matrix of the state transfer function; It is k+1 The Jacobian matrix of the mth measurement estimate is as follows; J is the Fisher information matrix; Then the mathematical expression of the mobile resource allocation scheme in each allocation time unit k can be expressed as formula (7): Its objective function is shown in formula (8), which is used to describe the tracking accuracy of the target; In formula (8), Λ is a normalized matrix, as shown in formula (9): In formula (9), represents the Kronecker operator, and I2 represents the identity matrix of order 2.

5. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 1, wherein: Eliminate the non-positive constraints in step 5. Specifically, the mathematical expression of the mobile resource allocation scheme at each allocation time unit k is Equation (10) after eliminating the non-positive constraints. In formula (10), φ i,k and μ i,k There are two sets defined as follows: For the threat area constraint of node i, if Then we call the constraint positive, and the set of all threat regions l for the i-th radar in the k-th time period is φ i,k ; For the collision constraint of node i, if Then we call the constraint positive, and the set of all nodes j that may collide with the i-th radar in the k-th time period is μ i,k .

6. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 1, wherein: Step 6, specifically, introduce a set of auxiliary variables: in, and Represent the velocity components of the radar in the x-direction and y-direction respectively; Sub-step 6.1, linearize the third constraint; Sub-step 6.2, linearize the fourth constraint; Sub-step 6.3, linearize the first and second constraints; In sub-step 6.4, combined with the above linearization process, the mathematical expression of the mobile resource allocation scheme at each allocation time unit k can be written as Equation (20):

7. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 6, characterized in that: Sub-step 6.1, specifically, according to the inequality x 2 +y 2 ≥2xy, relax the third nonlinear constraint to a linear constraint; the left side of the third nonlinear constraint allows a linear lower bound, as shown in Equation (11): In formula (11), and It is an extension point; According to Equation (11), the third constraint can be relaxed as a linear constraint, as shown in Equation (12): The auxiliary variable g i,k And the state transfer function expression of airborne radar Substituting the relaxed third constraint into equation (13): x i,l,k ·g i,k ≥ζ i,l,k (13) In formula (13), χ i,l,k and ζ i,l,k Related to predefined parameters.

8. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 7, characterized in that: Sub-step 6.2, specifically, according to the inequality x 2 +y 2 ≥2xy, the fourth constraint is relaxed to formula (14): The auxiliary variable g i,k And the state transfer function expression of airborne radar Substituting the relaxed fourth constraint, we get formula (16): In formula (16), γ i,j,k and ε i,j,k Associated with several predefined parameters.

9. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 8, wherein: Sub-step 6.3, specifically, use g i,k Re-express the first and second constraints and linearize the re-expressed formula; use g i,k replace and Then we have: Then, the non-convex sector region represented by the first two constraints is approximated as a convex trapezoidal region, which can be expressed as: In formula (19), A i,k and c i,k and and Related.

10. The method for allocating maneuverable resources for target tracking in an airborne radar network according to claim 1, wherein: Step 7: Specifically, in formula (20), the two variables g i,k 、g j,k It is linearly coupled; Introducing decoupled variables To decouple, the auxiliary variable e k , s k Introducing inequality constraints, the optimization problem (20) is rewritten as (22): make: Formula (23) is the augmented Lagrangian function, where χ k =[K,χ i,l,k ,K] and ζ k =[K,ζ i,l,k ,K] represents a set of χ i,l,k and B k g k =z k Represents a set of constraints y=[y e ,y s ,y w ] represents the dual variable; ρ e ,ρ s ,ρ w >0 is the original penalty parameter; p is a predefined positive parameter to ensure that L(g k ,z k ,e k ,s k ,b k ;y) in g k is strongly convex; b k is the introduced smooth iterative sequence; The proximal multiplier alternating direction algorithm can be summarized as follows: The first step is to set ρ e ,ρ s ,ρ w >0,λ e ,λ s ,λ w >0, 0<β≤1, c1, c2>0; The second step is to initialize y e,0 ,y s,0 ,y w,0 ,in and All are greater than or equal to 0; The third step is to determine φ i,k and μ i,k ; Step 4: Calculate χ i,l,k and γ i,l,k and Step 5: For t=1, 2, 3, L, start looping and loop the following formula: and Update via gradient projection method; in, and Calculated by the following three formulas: in,[·] + It is an operator that projects a vector into the non-negative quadrant, ensuring that all elements of the vector are positive; Step 6: End the loop; Get the speed of each node radar in the airborne radar network in the x direction and y direction; The speed of the radar in the x and y directions is calculated according to Calculate the level flight speed of each node radar and angular velocity

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