A collaborative interception multi-target allocation method based on reachable domain coverage

Abstract

A collaborative interception multi-target allocation method based on reachable domain coverage, the present invention relates to a collaborative interception multi-target allocation method based on reachable domain coverage. The purpose of the present invention is to solve the problem of long calculation time of the existing multi-target allocation method in the "many-to-many" collaborative interception process. The process is: 1. Predict the reachable domain of the target aircraft at the moment of interception; 2. Construct an interceptor missile reachability judgment database; 3. Obtain a trained deep neural network for judging interception reachability; Use the grid method to sample the reachable domain of a target, input the sampled data into the trained deep neural network, and the trained deep neural network outputs 0 or 1; The set of reachable sampling points of the interceptor missile corresponding to 1 represents the maneuverable reachable domain of the interceptor missile; 4. Establish a many-to-many interception target allocation optimization model, and use a two-stage multi-objective integer differential evolution algorithm to solve the allocation scheme. The present invention is used in the field of collaborative interception multi-target allocation.

Description

Technical Field

[0001] The invention relates to a collaborative interception multi-target allocation method based on reachable domain coverage. Background Art

[0002] With the development of scramjet technology, thermal insulation material technology and advanced control methods, more and more countries have invested a lot of energy in the research and development of near-space hypersonic aircraft. Russia's "Zircon", "Vanguard" and "Dagger" hypersonic missiles are already in service; the "hypersonic air-breathing weapon concept" (HAWC) developed by the United States flew more than 550km at a speed of more than 5Mach in a flight test in March 2022. The characteristics of near-space hypersonic aircraft with high speed and strong maneuverability have brought considerable pressure to the existing air defense system. With the development of clustered attack mode, the simultaneous attack of multiple near-space hypersonic aircraft will pose a serious threat to the safety of personnel and property of the defender. In the traditional air defense mode, a single missile is often used to conduct a "one-to-one" interception against a single target. Even if multiple interceptors are launched, there is no interaction and coordination between the interceptors. When multiple enemy aircraft attack at the same time, continuing to use the traditional "one-to-one" interception mode will pose a huge challenge to the launch frequency and decision-making speed of the air defense system.

[0003] Under this circumstance, researchers in related fields have proposed a "many-to-many" interception mode, that is, using multiple missiles to coordinately intercept multiple target aircraft. The primary technical difficulty faced by many-to-many coordinated interception is the weapon-target allocation strategy. Formulating the optimal target allocation plan can maximize the interception capability of the interceptor group; at the same time, in order to meet the timeliness of combat, the calculation time of target allocation must be shortened as much as possible. In existing research on weapon-target allocation, it is often necessary to assume that the interception probability of each interceptor missile for each target is known before solving the allocation strategy, which is quite different from the actual combat scenario, and the research mainly focuses on the algorithm level. A rapid allocation strategy that comprehensively considers the interception coverage capability of the interceptor missile for the target and the interception probability needs to be studied urgently.

[0004] The present invention designs a collaborative interception multi-target allocation method based on reachable domain coverage, which can quickly solve the target allocation scheme in the scenario where 12 interceptor missiles cooperate to intercept 4 targets. The method consists of four parts: target reachable domain prediction, interceptor missile reachability database establishment, interceptor missile maneuverable reachable domain rapid estimation, and allocation scheme optimization solution. It maximizes the interception coverage capability of the interceptor missile group while ensuring a short calculation time. It is suitable for online target allocation in multi-interceptor missile multi-target engagement scenarios. Summary of the invention

[0005] The purpose of the present invention is to solve the problem of long calculation time of the existing multi-target allocation method in the "many-to-many" collaborative interception process, and proposes a collaborative interception multi-target allocation method based on reachable domain coverage.

[0006] A collaborative interception multi-target allocation method based on reachable domain coverage has the following specific process:

[0007] Step 1: Obtain measurement information of the target aircraft based on radar;

[0008] The measurement information of the target aircraft is the position and speed of the target aircraft;

[0009] The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained;

[0010] The state of the target aircraft is the position and speed of the target aircraft;

[0011] The state estimation result and covariance matrix of the target aircraft are introduced into the maneuver model of the target, and the error propagation is performed with the help of the covariance analysis describing function method to predict the reachable domain of the target aircraft at the interception time.

[0012] Step 2: For a given interceptor missile and target point, considering the angle of attack constraint and overload constraint, the interception trajectory between the interceptor missile and the target point is solved by using the virtual domain dynamic inverse trajectory optimization method to satisfy the existence of the interception trajectory with the angle of attack constraint and overload constraint, and to characterize the interception accessibility of the interceptor missile to the corresponding target point;

[0013] Repeat step 2 to build an interceptor missile reachability judgment database;

[0014] Step 3: construct a deep neural network for judging interception reachability, and train the constructed deep neural network based on the database obtained in step 2 to obtain a trained deep neural network for judging interception reachability;

[0015] Use the grid method to sample the target reachable domain in step 1, input the sampled data into the trained deep neural network for judging interception reachability, and the trained deep neural network for judging interception reachability outputs 0 or 1;

[0016] 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain;

[0017] The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile;

[0018] Step 4. Based on the target reachable domain of step 1 and the maneuverable reachable domain of the interceptor missile in step 3, the coverage capability of the interceptor missile for each target reachable domain is quantitatively evaluated with the help of position coverage theory. The coverage capability value of the interceptor missile for each target reachable domain is brought into the established many-to-many interception target allocation optimization model, and the two-stage multi-objective integer differential evolution algorithm is used to solve the allocation plan.

[0019] The beneficial effects of the present invention are:

[0020] The present invention proposes a collaborative interception multi-target allocation method based on reachable domain coverage. In the present invention, the state of the target is first tracked, and then the reachable domain of the target is predicted with the help of error propagation theory; secondly, the reachability of the interceptor missile to the target point is solved with the help of a ballistic optimization algorithm, and a reachability judgment database is established through a large number of simulations; then, a reachability judgment neural network is designed and trained based on the reachability judgment database, and the neural network is used to judge the reachability of the interceptor missile to the sampling points in the reachable domain of the target, so as to obtain the maneuverable reachable domain of the interceptor missile; finally, based on the coverage of the maneuverable reachable domain of the interceptor missile to the reachable domain of the target, a many-to-many interception allocation optimization model is established, and the target allocation scheme is solved with the help of a swarm intelligence optimization algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is a flow chart of the present invention;

[0022] Figure 2 This is a schematic diagram of many-to-many target allocation based on reachable domain coverage;

[0023] Figure 3 This is a schematic diagram of the calculation process of step 1;

[0024] Figure 4 This is a schematic diagram of the process of quickly solving the maneuverable reachable domain of the interceptor missile;

[0025] Figure 5 It is a schematic diagram of the neural network structure;

[0026] Figure 6 This is the x-axis overload diagram for target 1;

[0027] Figure 7 This is a schematic diagram of the y-axis overload of target 1;

[0028] Figure 8 This is a schematic diagram of the z-axis overload of target 1;

[0029] Fig. 9 Schematic diagram of target trajectory tracking and prediction effect

[0030] Fig.10 It is the ellipsoid diagram of the target reachable domain at the terminal moment;

[0031] Fig.11This is a schematic diagram of the dynamic inverse trajectory optimization results in the virtual domain;

[0032] Fig.12 This is a graph showing the decrease of the cross entropy loss function with the number of iterations;

[0033] Fig.13 This is a graph showing the accuracy of the neural network in determining reachability versus the number of iterations;

[0034] Fig.14 This is a diagram showing the coverage of the maneuverable reachable areas of interceptor missiles 1, 2, and 3 over the reachable area of ​​target 1;

[0035] Fig.15 This is the trajectory diagram of the target and interceptor missile engagement. DETAILED DESCRIPTION

[0036] Specific implementation method 1: This implementation method is a collaborative interception multi-target allocation method based on reachable domain coverage. The specific process is as follows:

[0037] Step 1: Obtain measurement information of the target aircraft based on radar;

[0038] The measurement information of the target aircraft is the position and speed of the target aircraft;

[0039] The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained;

[0040] The state of the target aircraft is the position and speed of the target aircraft;

[0041] The state estimation result and covariance matrix of the target aircraft are introduced into the maneuver model of the target, and the error propagation is performed with the help of the covariance analysis describing function method to predict the reachable domain of the target aircraft at the interception time.

[0042] Step 2: For a given interceptor missile and target point, considering the angle of attack constraint and overload constraint, the interception trajectory between the interceptor missile and the target point is solved by using the virtual domain dynamic inverse trajectory optimization method to satisfy the existence of the interception trajectory with the angle of attack constraint and overload constraint, and to characterize the interception accessibility of the interceptor missile to the corresponding target point;

[0043] Repeat step 2 to build an interceptor missile reachability judgment database;

[0044] Step 3: construct a deep neural network for judging interception reachability, and train the constructed deep neural network based on the database obtained in step 2 to obtain a trained deep neural network that can quickly judge interception reachability;

[0045] Use the grid method to sample the target reachable domain in step 1, input the sampled data into a trained deep neural network that can quickly determine interception reachability, and the trained deep neural network that can quickly determine interception reachability outputs 0 or 1;

[0046] 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain;

[0047] The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile;

[0048] Step 4. Based on the target reachable domain of step 1 and the maneuverable reachable domain of the interceptor missile in step 3, the coverage capability of the interceptor missile for each target reachable domain is quantitatively evaluated with the help of position coverage theory. The coverage capability value of the interceptor missile for each target reachable domain is brought into the established many-to-many interception target allocation optimization model, and the two-stage multi-objective integer differential evolution algorithm is used to solve the allocation scheme (which interceptor missile is assigned to which target, an assignment problem).

[0049] Specific implementation method 2: This implementation method is different from specific implementation method 1 in that: in step 1, the measurement information of the target aircraft is obtained based on radar;

[0050] The measurement information of the target aircraft is the position and speed of the target aircraft;

[0051] The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained;

[0052] The state of the target aircraft is the position and speed of the target;

[0053] The state estimation result and covariance matrix of the target aircraft are introduced into the maneuver model of the target aircraft, and the error propagation is performed with the help of the covariance analysis describing function method to predict the reachable domain of the target aircraft at the interception time.

[0054] The specific process is:

[0055] In step 1, the state of the target is first tracked using filtering technology, and then the target reachable domain at a specified time in the future is predicted using the error propagation algorithm (see Figure 3 ).

[0056] Step 1

[0057] In step 1, the characteristic that near-space hypersonic aircraft mainly rely on aerodynamic force for maneuvering is taken into account, and the 9th-order differential equation shown in formula (1) is used as the target maneuvering model (see Zhang Bolun, Zhou Di, Wu Shikai. Maneuvering model and trajectory prediction of near-space hypersonic aircraft [J]. Systems Engineering and Electronics, 2019, 41(9).);

[0058] The maneuver model of the target aircraft is:

[0059]

[0060] Among them, x T ,y T 、z T are the x, y, and z axis coordinates of the target aircraft in the observation inertial system; V Tx 、V Ty 、V Tz are the velocity components of the target aircraft in the x, y, and z axes of the observation inertial system respectively; x , Z y , Z z are the aerodynamic coefficients of the target aircraft under the x, y, and z axes of the observed inertial system, respectively; are the first-order derivatives of each variable with respect to time; ω x ,ω y ,ω z is Gaussian white noise, representing Z x , Z y , Z z The slow time-varying random walk process; ρ is the atmospheric density; R e is the radius of the Earth; μ is the Earth's gravitational constant; θ v and ψ v are the velocity inclination and velocity deviation of the target aircraft in the observed inertial system; V T It indicates the three-axis combined velocity of the target aircraft in the observation inertial system;

[0061] The subscript T indicates that the variable is defined in the observation inertial system; the subscripts x, y, and z indicate that the variable is the component on the x, y, and z axes of the observation inertial system, respectively; the subscript v indicates that the variable is determined by the direction of the velocity;

[0062] Step 1 and 2

[0063] Acquire measurement information of target aircraft based on radar;

[0064] The measurement information of the target aircraft is the position and speed of the target aircraft;

[0065] The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained;

[0066] The state of the target aircraft is the position and speed of the target;

[0067] Based on radar measurement information, the unscented Kalman filter method is used to filter and track the target. The unscented Kalman filter algorithm introduces the unscented transform into the Kalman filter theory framework, and uses the unscented transform to approximate the nonlinear transformation of the mean and covariance during time update and measurement update.

[0068] Step 1-3

[0069] Based on the covariance description function method (CADET), the maneuver model of the target aircraft is linearized (see Zheng Xu. Research on initial error propagation and influence characteristics of long-range rocket launch [D]. Harbin Institute of Technology, 2018.), and the maneuver model of the target aircraft after linearization is obtained;

[0070] Then, the estimated value of the target state at the end of the state estimation of the target aircraft is used as the initial value, and the covariance of the estimation process is used as the initial covariance. The position ellipsoid at the specified time is obtained by using the error propagation algorithm, that is, the target reachable domain;

[0071] The basic idea of ​​CADET is to use the describing function theory to first perform statistical linearization on the system, and then use covariance analysis to obtain the mean and covariance propagation differential equations of the random state variables for the linearized system mathematical model. Using these two differential equations, the statistical characteristics of the system state variables can be determined with just one solution.

[0072] Formula 1-7 is the linearization process of the maneuver model of the target aircraft;

[0073] The general continuous-time nonlinear system can be expressed as

[0074]

[0075] Where t is time, f(x,t) is the nonlinear vector function of the n-dimensional state vector x(t), w(t) is the p-dimensional random input vector, and G(t) is the deterministic function matrix.

[0076] The random state variable x(t) is given by its mean The idea of ​​statistical linearization is to find a quasi-linear expression for the nonlinear vector function f(x, t) of the random state variable x(t): The mean square error E{e T e} reaches a minimum, and then the description function is obtained

[0077]

[0078] in, is called the expectation vector, N δx(t) is the dynamic matrix of the quasi-linear system, p(x) is the probability density function of x(t), and P(t) is the covariance matrix. Specifically,

[0079]

[0080] Using the principle of linear system covariance analysis, under the interference of white noise, the random state mean can be obtained And the propagation equation of the covariance matrix P(t):

[0081]

[0082] Where Q(t) is the process noise matrix.

[0083] Combined with the target maneuver model of equation (1), the expected vector is obtained by using the CADET method for:

[0084]

[0085] in The expected vectors are 1st to 6th components of ; are the means of the corresponding variables.

[0086] The other steps and parameters are the same as those in the first embodiment.

[0087] Specific implementation method three: This implementation method is different from specific implementation methods one or two in that: the aerodynamic coefficient Z x , Z y , Z z After the filter estimate converges, it is obtained through first-order power series fitting and substituted into the subsequent recursive prediction in the form of a time-varying variable. The specific form is shown in formula (8).

[0088] Z i =ae bt +c (8)

[0089] Among them, Z i are the means of the three-axis aerodynamic coefficients, i = x, y, z, a, b, c are the power series coefficients obtained by fitting, and t is the time.

[0090] The other steps and parameters are the same as those in the first or second embodiment.

[0091] Specific implementation method 4: This implementation method is different from any one of specific implementation methods 1 to 3 in that: in step 2, for a given interceptor missile and a target point, the attack angle constraint and the overload constraint are considered, and the interception trajectory between the interceptor missile and the target point is solved by means of a virtual domain dynamic inverse trajectory optimization method, so as to satisfy the existence of the interception trajectory with the attack angle constraint and the overload constraint, and characterize the interception accessibility of the interceptor missile to the corresponding target point;

[0092] Repeat step 2 to build an interceptor missile reachability judgment database;

[0093] The specific process is:

[0094] The trajectory optimization algorithm is used to determine the reachability of the interceptor missile to the target point under constraints, and then a large number of simulations are used to establish an intercept reachability judgment database.

[0095] Step 21. Set up the optimization model (Formula 9-16);

[0096] Step 22: Therefore, the virtual arc length τ can be selected f , the ballistic inclination angle θ of the interceptor missile at the terminal moment f , trajectory deviation angle ψ f , normal overload n y and lateral overload n z is the decision variable;

[0097] In order to constrain the state quantity of the interceptor missile during flight, the penalty function is designed as follows:

[0098] C=k1C1+k2C2 (17)

[0099] Among them, C1 and C2 represent the overload constraint and the angle of attack constraint penalty terms respectively, k1 represents the weight coefficient of C1, and k2 represents the weight coefficient of C2;

[0100] Step 2 and 3

[0101] Based on the decision variables and penalty function, the genetic algorithm is used to solve the optimization model (Formula 9-16) to obtain the penalty function value of the trajectory;

[0102] If the penalty function value of the trajectory is higher than the threshold C set by the user max , then the trajectory is considered to not satisfy the process constraints, that is, the interceptor missile is unreachable to the target point, otherwise it is reachable;

[0103] Step 24:

[0104] Repeat steps 21 to 23 N times, based on the interceptor missile's initial three-axis position e0, k0, m0 in the northeast celestial coordinate system, the interceptor missile's initial flight speed v0, the interceptor missile's initial ballistic inclination θ0, the interceptor missile's initial ballistic deviation ψ0, and the interceptor missile's three-axis terminal position e0 in the northeast celestial coordinate system.f , k f 、m f And the corresponding reachability labels (0 / 1) are obtained to build a reachability judgment sample database;

[0105] N is a positive integer.

[0106] The other steps and parameters are the same as those in Specific Embodiments 1 to 3.

[0107] Specific implementation method 5: This implementation method is different from specific implementation methods 1 to 4 in that: in step 21, an optimization model (Formula 9-16) is set;

[0108] The specific process is:

[0109] Step 2: Describe the dynamic model of the mid-range guided interceptor missile in the northeast celestial coordinate system as follows:

[0110]

[0111] Where: e, k, m are the positions of the interceptor missile in the northeast celestial coordinate system; v is the flight speed of the interceptor missile; θ is the ballistic inclination angle; ψ is the ballistic deviation angle; They are e, The first-order derivatives of v, θ, ψ with respect to time; n x 、n y 、n z are the components of interceptor overload in the axial, normal and lateral directions of the ballistic coordinate system, and g is the gravitational acceleration;

[0112] Step 212

[0113] The trajectory planning problem of guided interception in interceptor missile is described as follows:

[0114] The state equation is expressed as

[0115]

[0116] Among them, X is the 6-dimensional interceptor missile state vector including e, k, m, v, θ, and ψ. is the first-order derivative of X with respect to time; u is the derivative including n y 、n z 2D control quantity; t is time, t0 is the initial time, t f is the terminal moment; f(X(t),u(t),t) represents the function of the first-order derivative of the state quantity calculated by the state quantity, control quantity and time;

[0117] The boundary conditions at the initial and terminal moments are expressed as

[0118]

[0119] Among them, X(t0) represents the state vector of the interceptor at the initial moment; X(t f ) represents the state vector of the interceptor missile at the terminal moment; u(t0) represents the control quantity at the initial moment; u(t f ) represents the terminal moment control quantity; the superscript T represents the transposition; e0, k0, m0, v0, θ0, ψ0, n y0 、n z0 They are e, k, m, v, θ, ψ, and n respectively. y 、n z The value at the initial moment; e f , k f 、m f 、v f ,θ f , f 、n yf 、n zf They are e, k, m, v, θ, ψ, and n respectively. y 、n z The value at the terminal moment;

[0120] In the guidance phase of interception, the process constraints are mainly considered as the angle of attack constraint and the overload constraint, which can be expressed as:

[0121]

[0122] Where α represents the angle of attack, α max represents the maximum value of the angle of attack, represents the combined normal and lateral overload of the interceptor missile, n max Indicates the maximum value of overload;

[0123] Therefore, the fast trajectory planning problem of medium-range interception can be described as follows: At a given terminal interception time t f After determining the position of the terminal interception point, a feasible trajectory that meets various process constraints can be quickly planned.

[0124] The interception trajectory optimization solution under constraint conditions adopts the virtual domain dynamic inverse trajectory optimization algorithm (see Yakimenko O A. Direct method for rapid prototyping of near-optimal aircraft trajectories [J]. Journal of Guidance, Control, and Dynamics, 2000, 23 (5): 865-875.);

[0125] Step 2-3

[0126] Design a virtual variable τ, and fit the position of the interceptor missile to τ five times. The expression is as follows

[0127]

[0128] In the formula, Indicates the position of the interceptor missile in the northeast celestial coordinate system,

[0129] represents the fitting coefficient,

[0130] The fitting coefficients From the initial condition X(t0) and the terminal condition X(t f ) is determined by the algebraic equation (14)

[0131]

[0132] Among them, τ f is the virtual arc length, and are the initial coordinates and terminal coordinates of the interceptor missile respectively. The initial coordinates of the interceptor missile The first-order derivative in the virtual domain, The initial coordinates of the interceptor missile The second-order derivative in the virtual domain, Terminal coordinates of the interceptor missile The first-order derivative in the virtual domain, Terminal coordinates of the interceptor missile The second-order derivative in the virtual domain;

[0133] The superscripts ′ and ″ respectively represent the first-order and second-order derivatives of the variables in the dummy domain;

[0134] Step 2-4

[0135] The mapping relationship between the virtual domain and the time domain is established through the virtual speed λ, and the mapping relationship is defined as

[0136]

[0137] So there is

[0138]

[0139] Where v′(τ) is the first-order derivative of the interceptor missile velocity with respect to the dummy variable τ, is the first-order derivative of the interceptor missile's flight speed with respect to time t;

[0140] Since equations (13) and (14) have already satisfied the boundary conditions in the virtual domain, when changing the virtual arc length τ fWhen , it will not affect the boundary conditions, but only the state quantity X(t) and control quantity u(t) in the process. At the same time, when solving the polynomial coefficients in equation (14), the second-order differential information of each state quantity of the interceptor missile is required. However, for this interception problem, only the position at the terminal moment is required, so there is also room for optimization.

[0141] The other steps and parameters are the same as those in Specific Embodiments 1 to 4.

[0142] Specific implementation method 6: This implementation method is different from any one of specific implementation methods 1 to 5 in that: in step 3, a deep neural network for judging interception reachability is constructed, and the constructed deep neural network is trained based on the database obtained in step 2 to obtain a trained deep neural network that can quickly judge interception reachability;

[0143] Use the grid method to sample the target reachable domain in step 1, input the sampled data into a trained deep neural network that can quickly determine interception reachability, and the trained deep neural network that can quickly determine interception reachability outputs 0 or 1;

[0144] 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain;

[0145] The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile;

[0146] The specific process is:

[0147] Construct a deep neural network for judging interception reachability. The deep neural network for judging interception reachability includes an input layer, a fully connected layer, a ReLU activation function layer, a fully connected layer, a Softmax activation function layer, and an output layer in sequence;

[0148] The input of the deep neural network for judging interception reachability is the state of the interceptor missile and the position of the target point;

[0149] The output of the deep neural network for determining interception reachability is a reachability 0 or 1 label;

[0150] Based on the database obtained in step 2, the constructed deep neural network is trained to obtain a trained deep neural network that can quickly determine interception reachability;

[0151] Use the grid method to sample the target reachable domain in step 1, input the sampled data into a trained deep neural network that can quickly determine interception reachability, and the trained deep neural network that can quickly determine interception reachability outputs 0 or 1;

[0152] 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain;

[0153] The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile.

[0154] Based on the database obtained in step 2, a reachability judgment neural network is designed and trained. With the help of this neural network, the reachability of the interceptor missile to the sampling points in the target reachable domain is quickly judged, and the maneuverable reachable domain of the interceptor missile is represented by a set of reachable points (see Figure 4 ).

[0155] Specifically, according to Figure 5 The structure shown builds a deep neural network, and the input and output are set with reference to the reachability judgment database.

[0156] The activation function is introduced into the network to improve the network's ability to fit nonlinear models. The calculation formula of the ReLU activation function is:

[0157] ReLU(q)=max(0,q) (18)

[0158] Among them, q is the input data.

[0159] Softmax has the characteristics of unilateral inhibition and there is no "gradient disappearance" phenomenon. It is often used in binary classification problems. It can map the input value to the 0-1 interval and construct the neuron output into a probability distribution. Its calculation formula is

[0160]

[0161] Among them, the subscript j represents the jth element of the input and output data, the subscript i represents the i-th element of the input data, and k represents the total number of input quantities.

[0162] During the training process, the loss function selected is the cross-entropy loss function (CE). CE is often used in classification problems. It is a loss function used to evaluate the difference between the probability distribution of the network model output and the actual label. Its calculation formula is:

[0163]

[0164] Among them, L is the loss function value and y is the label value.

[0165] The other steps and parameters are the same as those in Specific Implementation Methods 1 to 5.

[0166] Specific implementation method 7: This implementation method is different from any one of specific implementation methods 1 to 6 in that: in step 4, based on the target reachable domain of step 1 and the maneuverable reachable domain of the interceptor missile in step 3, the coverage capability of the interceptor missile to each target reachable domain is quantitatively evaluated by means of the position coverage theory, and the coverage capability value of the interceptor missile to each target reachable domain is brought into the established many-to-many interception target allocation optimization model, and a two-stage multi-objective integer differential evolution algorithm is used to solve the allocation scheme (which interceptor missile is assigned to which target, an assignment problem);

[0167] The specific process is:

[0168] Firstly, when the target reachable domain and the maneuverable reachable domain of the interceptor missile are known, a weapon target assignment (WTA) model is established based on the position coverage theory. Then, based on the differential evolution (DE) algorithm, an integer differential evolution (IDE) algorithm which is more suitable for solving the WTA problem is proposed.

[0169] Step 41: Calculate the probability density of each target reachable domain sampling point in the corresponding reachable domain ellipsoid;

[0170] The point position error in space is expressed in three-dimensional coordinates, that is, r = (x, y, z) is set as the point position error vector. According to the law of large numbers, each point obeys the normal distribution as a whole and is independent of each other, that is, The probability density function is

[0171]

[0172] Among them, prob is the probability density value, μ x , μ y and μ z are the coordinates of the geometric center of the target reachable domain ellipsoid in the three-axis directions of the observation inertial system, σ x , σ y and σ z are the variances of the three axes of the observed inertial system respectively;

[0173] Step 42: Set optimization indicators;

[0174] Step 43: Use IDE algorithm to optimize and solve the optimization indicators.

[0175] The other steps and parameters are the same as those in Specific Implementations 1 to 6-1.

[0176] Specific implementation eight: This implementation differs from specific implementations one to seven in that: the optimization index is set in step four-two; the specific process is:

[0177] Considering that a single interceptor missile may be able to achieve full coverage of a target's reachable domain, the remaining interceptor missiles may no longer be allocated to the target, resulting in a waste of the defender's combat resources. Therefore, the optimization index setting in this problem adopts a staged approach, that is, the optimization index of the first stage is as shown in formula (22), and the optimization index of the second stage is as shown in formula (23) (the optimization process is set by referring to the index setting method of Pareto optimization). After the first stage converges, the optimization process enters the second stage, and the index is switched accordingly.

[0178] In this WTA problem, the first problem to be solved in decision-making is to allocate the targets to be targeted by each interceptor missile so as to minimize the threat posed by the incoming target. Therefore, the optimization index function of the first stage is as follows:

[0179]

[0180] Among them, J1 is the sum of the remaining threat probabilities, d is the total number of targets, b is the target sequence number, and w b is the total number of uncovered sampling points in the reachable domain of the bth target, o is the sequence number of the uncovered sampling points in the reachable domain of the target; prob b,o is the probability density of the oth uncovered sampling point in the reachable domain of the bth target;

[0181] The optimization index function of the second stage is as follows:

[0182]

[0183] Where J2 is the sum of the probability densities of the covered sampling points, p is the total number of interceptors, the subscript i represents the i-th interceptor, and cover i is the sum of the probability densities of the sampling points that can be covered by the i-th interceptor under the current allocation scheme, J 1,end is the indicator function corresponding to the optimal solution of the first stage.

[0184] The other steps and parameters are the same as those in Specific Implementation 1 to 7-1.

[0185] Specific implementation method nine: This implementation method is different from specific implementation methods one to eight in that: the IDE algorithm is used in step four or three to optimize and solve the optimization index; the DE algorithm has few control parameters, high optimization accuracy, strong robustness and is easy to implement in engineering, and performs well in the field of continuous optimization. However, the target allocation optimization problem in the present invention belongs to an integer optimization problem, so it is considered to improve the classic DE algorithm, and the improvement is mainly from two aspects: (1) The initial population is no longer assigned by randomly generated floating point numbers, but randomly selects integer values ​​within the value range of each variable to assign it; (2) The mutation scaling factor F uses a positive integer. The improved IDE algorithm can ensure that the variable values ​​in all mutant individuals, test individuals, parent and offspring individuals are integers, which is more suitable for the integer assignment problem in the above model. The specific operation of the IDE algorithm is as follows:

[0186] (1) Variation:

[0187] When the population evolves to the Gth generation, Implement the mutation operation to obtain the mutant individual, that is,

[0188]

[0189] Wherein, subscripts r1, r2 and r3 are randomly selected distinct integers (integers that are not the same as each other) between 1 and NP (population size), and the superscript G indicates the Gth generation of the evolutionary process;

[0190] are called basis vectors; is called the difference vector;

[0191] F is a positive integer scaling factor, · is a multiplication sign;

[0192] is the G+1th generation mutant individual, l represents the lth individual in the population;

[0193] (2) Crossover:

[0194] By parent individual and variant individuals The experimental individuals generated by the crossover operation are

[0195]

[0196] Among them, j rand represents the random number calculated for the jth time, CR represents the crossover factor, CR∈(0,1);

[0197] represents the value calculated for the jth time for the lth mutant individual in the Gth generation; rand l,j(0,1) represents the random value in the range of 0 to 1 when performing the j-th calculation on the l-th individual, and j represents the j-th calculation; It represents the value calculated for the jth time for the lth parent individual of the Gth generation; It represents the value calculated for the jth time for the lth test individual of the Gth generation;

[0198] (3) Select:

[0199] By parent individual and experimental individuals The offspring individuals are produced through "greedy competition"

[0200]

[0201] Where f(·) is the fitness value; It represents the value calculated for the jth time by the lth parent individual of the G+1th generation;

[0202] (4) Repeat (1), (2), and (3) until the optimization index value converges and the optimal value of the optimization index is obtained.

[0203] The other steps and parameters are the same as those in Specific Implementation 1 to 8-1.

[0204] The following examples are used to verify the beneficial effects of the present invention:

[0205] Embodiment 1:

[0206] At the initial moment, in the observation inertial coordinate system, the states of targets 1 to 4 are set as shown in Table 1.

[0207] Table 1 The states of each target in the observation inertial system at the initial moment

[0208]

[0209] Assuming the interception time is 115s, the changes of the three-axis overload of target 1 from 0s to 115s are shown in Figures 6 to 8 .

[0210] It is noted that the maximum aerodynamic combined overload of the target is 2.86g during the period from the initial moment to the interception moment. The three-axis overload variation forms of targets 2 to 4 are set similarly to target 1.

[0211] The present invention considers a combat scenario in which 12 interceptor missiles cooperate to intercept 4 targets, so the states of the 12 interceptor missiles at the initial moment in this scenario are set as shown in Table 2.

[0212] Table 2 The status of each interceptor missile in the northeast sky coordinate system at the initial moment

[0213]

[0214]

[0215] First, using the method in step 1, target 1 is filtered and tracked to 35 seconds, and the trajectory and terminal reachable domain of the target are predicted for the next 80 seconds (35 seconds to 115 seconds). The results are shown in Fig. 9 , Fig.10 .

[0216] From the simulation results, it can be seen that the prediction error of the target position after 80 seconds is 1.54 km, and the actual position of the target at the terminal moment is inside the target reachable domain ellipsoid.

[0217] Then, assuming that the maximum aerodynamic overload of the interceptor missile is 4.3g (1.5 times the target), the angle of attack is 20°, and taking interceptor missile 1 as an example, a virtual target point r is assumed in the northeast celestial coordinate system. virtual =[270,75,15]km, the interception time is 80s, and the interception trajectory is solved by using the virtual domain dynamic inverse optimization algorithm. The results are shown in Fig.11 .

[0218] Depend on Fig.11 It can be seen that the method proposed in step 2 can effectively solve the interception trajectory under constraints.

[0219] Furthermore, the above algorithm is used to perform a large number of simulations under different boundary conditions, and the interception reachability judgment problem is abstracted into a binary classification problem (reachable is marked as label 1, unreachable is marked as label 0). A reachability judgment database is established to train a reachability judgment neural network. The loss function in the training iteration process is selected as the cross entropy. The training process is shown in Fig.12 , Fig.13 .

[0220] From the training results, we can see that the loss function converges when the network iterates to the 1254th generation, and the accuracy of the neural network in judging reachability is 95.4%.

[0221] Next, we can use the trained neural network to quickly estimate the maneuverable reachable domain of the interceptor missile and obtain the coverage of the maneuverable reachable domain of each interceptor missile for the reachable domain of each target. Taking interceptor missiles 1, 2, 3 and target 1 as an example, the reachable domain coverage is as follows: Fig.14 .

[0222] Depend on Fig.14 It can be seen that the method proposed in step three can intuitively solve the maneuverable reachable area of ​​the interceptor missile relative to the target.

[0223] Finally, based on the prediction results of the target reachable domain and the solution results of the interceptor missile maneuverable reachable domain, the target allocation algorithm proposed in step 4 is used to optimize the target allocation scheme, and the target allocation scheme can be obtained as shown in Table 3.

[0224] Table 3 Target allocation scheme

[0225] Target Number 1 2 3 4 Assign interceptor missile numbers 1、2、 3、4、5 6、7、8、9 10、11、12

[0226] Under the allocation scheme shown in the table above, the engagement trajectories of 12 interceptor missiles and 4 targets in the northeastern sky coordinate system are as follows: Fig.15 shown.

[0227] In the scenario where 12 interceptors cooperate to intercept 4 targets, the multi-to-multi target allocation method based on reachable domain coverage proposed in the present invention can quickly solve the target allocation scheme based on the target reachable domain and the interceptor missile maneuverable reachable domain. The whole process takes only 0.626 seconds, which is suitable for offensive and defensive confrontation scenarios with rapidly changing battlefield situations. All simulation codes in the embodiments are run on an AMD Ryzen 75800H 3.2GHz processor, the operating system is Windows11, and the simulation software is MATLAB2019b.

[0228] The present invention may also have many other embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art may make various corresponding changes and modifications based on the present invention, but these corresponding changes and modifications should all fall within the scope of protection of the claims attached to the present invention.

Claims

1. A collaborative interception multi-target allocation method based on reachable domain coverage, characterized by: The specific process of the method is: Step 1: Obtain measurement information of the target aircraft based on radar; The measurement information of the target aircraft is the position and speed of the target aircraft; The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained; The state of the target aircraft is the position and speed of the target aircraft; The state estimation result and covariance matrix of the target aircraft are introduced into the maneuver model of the target, and the error propagation is performed with the help of the covariance analysis describing function method to predict the reachable domain of the target aircraft at the interception time. Step 2: For a given interceptor missile and target point, considering the angle of attack constraint and overload constraint, the interception trajectory between the interceptor missile and the target point is solved by using the virtual domain dynamic inverse trajectory optimization method to satisfy the existence of the interception trajectory with the angle of attack constraint and overload constraint, and to characterize the interception accessibility of the interceptor missile to the corresponding target point; Repeat step 2 to build an interceptor missile reachability judgment database; Step 3: construct a deep neural network for judging interception reachability, and train the constructed deep neural network based on the database obtained in step 2 to obtain a trained deep neural network for judging interception reachability; Use the grid method to sample the target reachable domain in step 1, input the sampled data into the trained deep neural network for judging interception reachability, and the trained deep neural network for judging interception reachability outputs 0 or 1; 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain; The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile; Step 4. Based on the target reachable domain of step 1 and the maneuverable reachable domain of the interceptor missile in step 3, the coverage capability of the interceptor missile for each target reachable domain is quantitatively evaluated with the help of position coverage theory. The coverage capability value of the interceptor missile for each target reachable domain is brought into the established many-to-many interception target allocation optimization model, and the two-stage multi-objective integer differential evolution algorithm is used to solve the allocation plan.

2. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 1 is characterized in that: In the step 1, the measurement information of the target aircraft is obtained based on the radar; The measurement information of the target aircraft is the position and speed of the target aircraft; The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained; The state of the target aircraft is the position and speed of the target; The state estimation result and covariance matrix of the target aircraft are introduced into the maneuver model of the target aircraft, and the error propagation is performed with the help of the covariance analysis describing function method to predict the reachable domain of the target aircraft at the interception time. The specific process is: Step 1 The maneuver model of the target aircraft is: Among them, x T ,y T 、z T are the x, y, and z axis coordinates of the target aircraft in the observation inertial system; V Tx 、V Ty 、V Tz are the velocity components of the target aircraft in the x, y, and z axes of the observation inertial system respectively; x , Z y , Z z are the aerodynamic coefficients of the target aircraft under the x, y, and z axes of the observed inertial system, respectively; are the first-order derivatives of each variable with respect to time; ω x ,ω y ,ω z is Gaussian white noise, representing Z x , Z y , Z z The slow time-varying random walk process; ρ is the atmospheric density; R e is the radius of the Earth; μ is the Earth's gravitational constant; θ v and ψ v are the velocity inclination and velocity deviation of the target aircraft in the observed inertial system; V T It indicates the three-axis combined velocity of the target aircraft in the observation inertial system; Step 1 and 2 Acquire measurement information of target aircraft based on radar; The measurement information of the target aircraft is the position and speed of the target aircraft; The nonlinear Kalman filter method is used to track the state of the target aircraft, and the state estimation result and covariance matrix of the target aircraft are obtained; The state of the target aircraft is the position and speed of the target; Step 1-3 The maneuvering model of the target aircraft is linearized based on the covariance describing function method to obtain the maneuvering model of the target aircraft after linearization; Then, the estimated value of the target state at the end of the state estimation of the target aircraft is taken as the initial value, and the covariance is taken as the initial covariance. The error propagation algorithm is used to obtain the position ellipsoid at the specified time, that is, the target reachable domain.

3. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 2 is characterized in that: The aerodynamic coefficient Z x , Z y , Z z The specific form is shown below: Z i =ae bt +c Among them, Z i are the means of the three-axis aerodynamic coefficients, i = x, y, z, a, b, c are the power series coefficients obtained by fitting, and t is the time.

4. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 3 is characterized in that: In the step 2, for a given interceptor missile and a target point, the attack angle constraint and the overload constraint are considered, and the interception trajectory between the interceptor missile and the target point is solved by means of a virtual domain dynamic inverse trajectory optimization method, so as to satisfy the existence of the interception trajectory with the attack angle constraint and the overload constraint, and characterize the interception accessibility of the interceptor missile to the corresponding target point; Repeat step 2 to build an interceptor missile reachability judgment database; The specific process is: Step 21: Set up the optimization model; Step 22: Select the virtual arc length τ f , the ballistic inclination angle θ of the interceptor missile at the terminal moment f , trajectory deviation angle ψ f , normal overload n y and lateral overload n z is the decision variable; The penalty function is designed as follows: C=k1C1+k2C2 Among them, C1 and C2 represent the overload constraint and angle of attack constraint penalty terms respectively, k1 represents the weight coefficient of C1, and k2 represents the weight coefficient of C2; Step 2 and 3 Based on the decision variables and penalty function, the optimization model is solved by genetic algorithm to obtain the penalty function value of the trajectory; If the penalty function value of the trajectory is higher than the threshold C set by the user max , then the trajectory is considered to not satisfy the process constraints, that is, the interceptor missile is unreachable to the target point, otherwise it is reachable; Step 24: Repeat steps 21 to 23 N times, based on the interceptor missile's initial three-axis position e0, k0, m0 in the northeast celestial coordinate system, the interceptor missile's initial flight speed v0, the interceptor missile's initial ballistic inclination θ0, the interceptor missile's initial ballistic deviation ψ0, and the interceptor missile's three-axis terminal position e0 in the northeast celestial coordinate system. f , k f 、m f And the corresponding reachability labels (0 / 1) are obtained to build a reachability judgment sample database; N is a positive integer.

5. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 4 is characterized in that: In the step 21, an optimization model is set; the specific process is: Step 2: Describe the dynamic model of the mid-range guided interceptor missile in the northeast celestial coordinate system as follows: Where: e, k, m are the positions of the interceptor missile in the northeast celestial coordinate system; v is the flight speed of the interceptor missile; θ is the ballistic inclination angle; ψ is the ballistic deviation angle; They are e, The first-order derivatives of v, θ, ψ with respect to time; n x 、n y 、n z are the components of interceptor overload in the axial, normal and lateral directions of the ballistic coordinate system, and g is the gravitational acceleration; Step 212 The trajectory planning problem of guided interception in interceptor missile is described as follows: The state equation is expressed as Among them, X is the 6-dimensional interceptor missile state vector including e, k, m, v, θ, and ψ. is the first-order derivative of X with respect to time; u is the derivative including n y 、n z 2D control quantity; t is time, t0 is the initial time, t f is the terminal moment; f(X(t),u(t),t) represents the function of the first-order derivative of the state quantity calculated by the state quantity, control quantity and time; The boundary conditions at the initial and terminal moments are expressed as Among them, X(t0) represents the state vector of the interceptor at the initial moment; X(t f ) represents the state vector of the interceptor missile at the terminal moment; u(t0) represents the control quantity at the initial moment; u(t f ) represents the terminal moment control quantity; the superscript T represents the transposition; e0, k0, m0, v0, θ0, ψ0, n y0 、n z0 They are e, k, m, v, θ, ψ, and n respectively. y 、n z The value at the initial moment; e f , k f 、m f 、v f ,θ f , f 、n yf 、n zf They are e, k, m, v, θ, ψ, and n respectively. y 、n z The value at the terminal moment; In the guidance phase of interception, the process constraints are mainly considered as the angle of attack constraint and the overload constraint, which can be expressed as: Where α represents the angle of attack, α max represents the maximum value of the angle of attack, represents the combined normal and lateral overload of the interceptor missile, n max Indicates the maximum value of overload; Step 2-3 Design a virtual variable τ, and fit the position of the interceptor missile to τ five times. The expression is as follows In the formula, Indicates the position of the interceptor missile in the northeast celestial coordinate system, represents the fitting coefficient, The fitting coefficients From the initial condition X(t0) and the terminal condition X(t f ) is determined by the algebraic equation (14) Among them, τ f is the virtual arc length, and are the initial and terminal coordinates of the interceptor missile respectively. The initial coordinates of the interceptor missile The first-order derivative in the virtual domain, The initial coordinates of the interceptor missile The second-order derivative in the virtual domain, Terminal coordinates of the interceptor missile The first-order derivative in the virtual domain, Terminal coordinates of the interceptor missile The second-order derivative in the virtual domain; Step 2-4 The mapping relationship between the virtual domain and the time domain is established through the virtual speed λ, and the mapping relationship is defined as So there is Where v′(τ) is the first-order derivative of the interceptor missile velocity with respect to the dummy variable τ, is the first-order derivative of the interceptor missile's flight speed with respect to time t.

6. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 5 is characterized in that: In the step 3, a deep neural network for judging interception reachability is constructed, and the constructed deep neural network is trained based on the database obtained in the step 2 to obtain a trained deep neural network for judging interception reachability; Use the grid method to sample the target reachable domain in step 1, input the sampled data into the trained deep neural network for judging interception reachability, and the trained deep neural network for judging interception reachability outputs 0 or 1; 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain; The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile; The specific process is: Construct a deep neural network for judging interception reachability. The deep neural network for judging interception reachability includes an input layer, a fully connected layer, a ReLU activation function layer, a fully connected layer, a Softmax activation function layer, and an output layer in sequence; The input of the deep neural network for judging interception reachability is the state of the interceptor missile and the position of the target point; The output of the deep neural network for determining interception reachability is a reachability 0 or 1 label; Based on the database obtained in step 2, the constructed deep neural network is trained to obtain a trained deep neural network for judging interception reachability; Use the grid method to sample the target reachable domain in step 1, input the sampled data into the trained deep neural network for judging interception reachability, and the trained deep neural network for judging interception reachability outputs 0 or 1; 1 means that the interceptor missile can reach the sampling point in the target reachable domain, and 0 means that the interceptor missile cannot reach the sampling point in the target reachable domain; The set of interceptor missile reachable sampling points corresponding to 1 represents the maneuverable reachable domain of the interceptor missile.

7. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 6 is characterized in that: In the step 4, based on the target reachable domain of step 1 and the maneuverable reachable domain of the interceptor missile in step 3, the coverage capability of the interceptor missile to each target reachable domain is quantitatively evaluated by means of the position coverage theory, and the coverage capability value of the interceptor missile to each target reachable domain is brought into the established many-to-many interception target allocation optimization model, and a two-stage multi-objective integer differential evolution algorithm is used to solve the allocation scheme; The specific process is: Step 41: Calculate the probability density of each target reachable domain sampling point in the corresponding reachable domain ellipsoid; The probability density function is Among them, prob is the probability density value, μ x , μ y and μ z are the coordinates of the geometric center of the target reachable domain ellipsoid in the three-axis directions of the observation inertial system, σ x , σ y and σ z are the variances of the three axes of the observed inertial system respectively; Step 42: Set optimization indicators; Step 43: Use IDE algorithm to optimize and solve the optimization indicators.

8. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 7 is characterized in that: In step 42, the optimization index is set; the specific process is: The optimization index setting adopts a phased approach, i.e., the optimization index of the first phase and the optimization index of the second phase. After the first phase converges, the optimization process enters the second phase. The optimization index function of the first stage is as follows: Among them, J1 is the sum of the remaining threat probabilities, d is the total number of targets, b is the target sequence number, and w b is the total number of uncovered sampling points in the reachable domain of the bth target, o is the sequence number of the uncovered sampling points in the reachable domain of the target; prob b,o is the probability density of the oth uncovered sampling point in the reachable domain of the bth target; The optimization index function of the second stage is as follows: s.t.J1=J 1,end Where J2 is the sum of the probability densities of the covered sampling points, p is the total number of interceptors, the subscript i represents the i-th interceptor, and cover i is the sum of the probability densities of the sampling points that can be covered by the i-th interceptor under the current allocation scheme, J 1,end is the indicator function corresponding to the optimal solution of the first stage.

9. The method for cooperative interception of multiple targets based on reachable domain coverage according to claim 8, characterized in that: In step 43, the IDE algorithm is used to optimize and solve the optimization index; the specific operation is as follows: (1) Variation: When the population evolves to the Gth generation, Implement the mutation operation to obtain the mutant individual, that is, Wherein, subscripts r1, r2 and r3 are randomly selected distinct integers between 1 and NP, and the superscript G indicates the Gth generation of the evolutionary process; are called basis vectors; is called the difference vector; F is a positive integer scaling factor, · is a multiplication sign; V l G+1 is the G+1th generation mutant individual, l represents the lth individual in the population; (2) Crossover: By parent individual and variant individual V l G The experimental individuals generated by the crossover operation are Among them, j rand represents the random number calculated for the jth time, CR represents the crossover factor, CR∈(0,1); represents the value calculated for the jth time for the lth mutant individual in the Gth generation; rand l,j (0,1) represents the random value in the range of 0 to 1 when performing the j-th calculation on the l-th individual, and j represents the j-th calculation; It represents the value calculated for the jth time for the lth parent individual of the Gth generation; It represents the value calculated for the jth time for the lth test individual of the Gth generation; (3) Select: By parent individual and experimental individuals The offspring individuals produced through "greedy competition" are Where f(·) is the fitness value; It represents the value calculated for the jth time by the lth parent individual of the G+1th generation; (4) Repeat (1), (2), and (3) until the optimization index value converges and the optimal value of the optimization index is obtained.

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