An estimation method for attractive domain of multi-missile simultaneous hitting target under cooperative guidance law

Abstract

The application discloses a kind of attractive domain estimation methods of multiple missiles hitting target simultaneously under cooperative guidance law, belong to multi-agent control technical field, comprising the following steps: S1, according to the relative motion relationship between missile and target point, establish relative kinematics model;S2, estimate the remaining time of each missile to reach target point, and based on each missile itself and neighbor information, design variable navigation ratio for each missile;S3, construct system attractive domain estimation method under Lyapunov function;S4, establish the Lyapunov function with the remaining time error as variable for missile group, and carry out scaling to the expression of its time differential;S5, combined with S3 and S4, obtain the attractive domain estimation of multiple missiles hitting target simultaneously under cooperative guidance law;The method provided by the application innovatively estimates group cooperative consistency task attractive domain, can serve multiple missile simultaneous hit target and other saturation attack tasks, and has wide application prospect.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of multi-agent control, and particularly relates to an attractive domain estimation method for multiple missiles hitting a target simultaneously under a cooperative guidance law. BACKGROUND

[0002] Modern war is complex and changeable, and the enemy's defense system is increasingly strengthened, which makes it particularly crucial to develop the technology of multiple missiles hitting a target simultaneously. When performing saturation attack tasks, precisely coordinating multiple missiles to hit a target in a synchronous manner can not only effectively break through the enemy's defense system, but also significantly improve the damage efficiency of the missiles. The effect of cluster attack is far superior to the traditional mode of single missile hitting a target one by one. The existing researches on the cooperative attack of missile groups focus on the design of cooperative guidance law, lack the analysis of the space-time synchronization capability of missile groups, and the discussion on the initial conditions and safeguards to ensure that the missile groups can hit the target simultaneously is not sufficient and clear enough.

[0003] The existing researches on the design of distributed cooperative guidance law for cooperative attack of missile groups are as follows: (see Zhou J, Yang J, Li Z. Simultaneous attack of a stationary target using multiple missiles: a consensus-based approach [J]. Science China Information Sciences, 2017, 60: 1-14.) discloses a distributed cooperative guidance law based on consensus. This method is established on the framework of the classic proportional guidance law, and through the communication network between missiles to exchange the remaining time estimation value, a variable navigation ratio is designed to solve the problem of multiple missiles hitting a single stationary target simultaneously.

[0004] The existing researches on the estimation of attractive domain of complex network consensus for multi-agent are as follows: (see Zhu S, Zhou J, Chen G, et al. Estimating the region of attraction on a complex dynamical network [J]. SIAM Journal on Control and Optimization, 2019, 57(2): 1189-1208.) discloses the related research on the estimation of the attractive domain of the equilibrium point in the complex dynamic network, and briefly analyzes the stability of the network. Firstly, sufficient conditions and necessary conditions are established for the asymptotic stability of the network equilibrium point, and then by combining the network structure and the node dynamics, a general technique for estimating the attractive domain is developed.

[0005] However, the above research on coordinated strikes of missile groups only focused on the design of coordinated guidance laws, and did not further analyze the spatiotemporal synchronization capabilities of missile groups; and the above research on the estimation of the attraction domain of complex network consistency only focused on fixed equilibrium points, and did not study the estimation of the attraction domain under the coordinated consistency protocol. Summary of the Invention

[0006] The purpose of the present invention is to provide an attraction domain estimation method for multiple missiles to hit the target simultaneously under the cooperative guidance law. By analyzing the estimation of the remaining time of missile hit under the cooperative guidance law and combining the designed Lyapunov function with the remaining time error as the variable, the spatiotemporal synchronization attraction domain of the missile group is constructed to characterize the spatiotemporal synchronization capability of the missile group, and the conditions for the consistency of the remaining time of the missile group are established, which provides a theoretical basis for the feasibility judgment of the subsequent missile group consistency and the target allocation task.

[0007] To achieve the above object, the present invention provides a method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law, comprising the following steps:

[0008] S1. Establish a relative kinematics model based on the relative motion relationship between the missile and the target point;

[0009] S2, estimate the remaining time for each missile to reach the target point, and design a variable navigation ratio for each missile based on its own information and its neighbors;

[0010] S3. Construct a method for estimating the system attraction domain under the Lyapunov function;

[0011] S4. Establish a Lyapunov function for the missile group with the residual time error as the variable, and scale its expression of time differentiation;

[0012] S5. Combine S3 and S4 to obtain the estimation of the attraction domain for multiple missiles to hit the target simultaneously under the cooperative guidance law.

[0013] Preferably, the specific process of S1 is as follows:

[0014] Construct a communication topology graph θ = (υ, ε), where υ is a set of nodes, ε is a set of edges, and edge (i, j) indicates that missile i is a neighbor of missile j and missile j can receive information from missile i. In an undirected graph, for any (i, j) ∈ ε, there is (j, i) ∈ ε. Assume that there are n nodes in the communication topology graph θ, and record the adjacency matrix of the communication topology graph θ as If (i,j)∈ε, then a ij =1, otherwise a ij =0; the Laplacian matrix corresponding to this figure is recorded as in l ij =-a ij , i≠j; denotes a set of real matrices of dimension n x n, l ii denotes a Laplacian matrix denotes the element value of the i-th row and i-th column, l ij denotes a Laplacian matrix denotes the element value of the i-th row and j-th column, a ij denotes the element value of the i-th row and j-th column of the adjacency matrix A;

[0015] Constructing a cluster system composed of n missiles in the same engagement scene

[0016]

[0017] The kinematic relationship of the missile and the target engagement is described as follows:

[0018]

[0019] wherein, r i denotes the relative distance between the missile i and the target point, V i denotes the speed of the missile i, φ i denotes the angle between the speed direction of the missile i and the line-of-sight direction, λ i denotes the angle between the line-of-sight direction of the missile i and the horizontal plane, γ i denotes the angle between the speed direction of the missile i and the horizontal plane, and “·” denotes derivative;

[0020] The proportional guidance method is adopted, denotes as follows:

[0021]

[0022] wherein N i denotes the navigation ratio of the missile i, and the final relative kinematic model is obtained by substituting equation (3) into equation (2), which is specifically denoted as follows:

[0023]

[0024] Preferably, the specific process of S2 is as follows:

[0025] Based on the local information of the missile i and the interaction information of its neighbors, the navigation ratio is designed as shown in the following equation:

[0026] N i = N s (1 + a i r i ξ i -r i |ξ i | 2α-1 sgn(ξ i)), i = 1,..., n (5)

[0027] wherein:

[0028]

[0029] wherein N s , a i and a are constants, a i > 0, 0 < a < 1, denotes the estimate of the remaining time for missile i to hit the target, and i denotes the measure of the remaining time error between missile i and its neighbor nodes under the network topology

[0030] Taking the differential of (6) with respect to time and substituting into (4) gives:

[0031]

[0032] wherein

[0033]

[0034] wherein i is an even function of i and is monotonically increasing in the interval [0, π) and i when |φ | < π,

[0035] Preferably, the specific process of S3 is as follows:

[0036] The attraction domain R0 of the nonlinear system is defined as:

[0037]

[0038] wherein denotes the solution of the nonlinear system under the condition of initial state x = 0 is the equilibrium point of the system, and R n denotes the n-dimensional space;

[0039] In R n , take a region D containing the origin, and a continuously differentiable Lyapunov function V(x): D→R satisfies the following conditions:

[0040] V(0) = 0 (10)

[0041] In the region D-{0}, satisfies:

[0042] V(x) > 0 (11)

[0043]

[0044] Given η>0, choose d∈(0,η] such that:

[0045]

[0046] In area B d The boundary exists ω=min ||x||＝d V(x), from condition (11), we get ω>0, and select β∈(0,ω) so that:

[0047] Ω β ={x∈B r |V(x)≤β} (14)

[0048] Where B d and Ω β Both indicate that R n In the figure, d and β represent a real number selected according to the above requirements.

[0049] Preferably, Ω in S3 β 、B d 、D satisfies The specific content is as follows: Assume Ω β Not in B d Inside, there exists a point p∈Ω β Falls in area B d On the boundary of , for point p, V(p)≥ω>β, but for all x∈Ω β , V(x)≤β holds, which contradicts the condition, so Ω β In B d Internal, satisfied

[0050] Preferably, Ω in S3 β is a subset of the nonlinear system attraction domain R0, which is used as an estimate of the nonlinear system attraction domain. The specific contents are as follows:

[0051] Because Ω β In B d Internally, so Ω β is a compact set, so for any x(0)∈Ω β Under the initial conditions, for all t>0, the system differential equation has a unique solution in Ω β According to conditions (11) and (12), V(x) is monotonically bounded. According to the monotonically bounded theorem, when t→∞, we have:

[0052] V(x(t))→c≥0 (15)

[0053] Assume c>0, then there exists a>0 such that Continuous function on compact set {a≤||x||≤η} There exists a maximum, take According to condition (12), we have -ψ < 0, take x(0) ∈ Ω β , so we have:

[0054]

[0055] So, with the increase of time, V(x(t)) will become negative, however, when t→∞, we have V(x(t))→c≥0, which is contrary to the condition, so when t→∞, we have V(x(t))→0;

[0056] According to condition (11), we have V(x) = 0 if and only if x = 0, combined with the definition of the attractor of nonlinear system, Ω β is a subset of the attractor of nonlinear system R0, which can be used as an estimate of the attractor of nonlinear system.

[0057] Preferably, the specific process of S4 is as follows:

[0058] Let ξ = [ξ1, ξ2,..., ξ n ] T , According to the missile group system under the cooperative guidance law, the following Lyapunov function is obtained:

[0059]

[0060] Differentiate the Lyapunov function with respect to time and substitute it into (5) and (7) and combine and to obtain:

[0061]

[0062] Assume that φ i is a small angle, then:

[0063]

[0064] Define a m = max{a1,...,a n}, since only when the missile hits the target, we have r i = 0, φ i = 0, so we assume that before the time of the missile is consistent, we have r i ≠ 0, φ i ≠ 0, i = 1,...,n, so there exist constants r m and φ m such that r i ≥ r m , |φ i| >= phi m , i = 1,...,n, and the specific expression is as follows:

[0065]

[0066] Preferably, the specific process of S5 is as follows:

[0067] According to the missile group system under the cooperative guidance law, the residual time error region with Lyapunov function less than zero is recorded as D M , and from formula (20), when-a m T T α >= 0, then the expression of B M is as shown in the following formula:

[0068]

[0069] In the formula, For omega M = min ||ξ||＝Δ V(ξ), and beta M is selected from (0, omega M ), and according to S3, it is known that the attractive domain of the multiple missiles hitting the target simultaneously under the cooperative guidance law is estimated as:

[0070] Omega M = {ξ e B M | V(ξ) <= beta M} (22)

[0071] In the formula, omega M and beta M both represent a real number selected according to the above requirements, V(ξ) represents the Lyapunov function under the cooperative residual time error variable, and B M represents a region selected in R n according to the above requirements.

[0072] Therefore, the present application adopts the above-mentioned attractive domain estimation method of multiple missiles hitting the target simultaneously under the cooperative guidance law, and has the following beneficial effects:

[0073] (1) It is used for describing the space-time synchronization ability of the missile group cooperative attack, and provides a theoretical basis for the realizability judgment of the missile group consistency under specific initial conditions and the subsequent target allocation task;

[0074] ​​​(2) For the consistency control under multiple unmanned vehicles and multiple aircrafts, the attraction domain under a specific guidance law can be estimated by the method, so as to characterize the initial condition range of the group for realizing the consistency of multiple intelligent agents, which can serve subsequent task allocation scenarios such as unmanned vehicle group trajectory planning, collaborative transportation of logistics vehicles, unmanned aerial vehicle group waypoint planning and the like.

[0075] The technical solutions of the present application are described in further detail below with the aid of the accompanying drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0076] Figure 1 is the overall flowchart of the attraction domain estimation method for multiple missiles hitting targets simultaneously under a cooperative guidance law;

[0077] Figure 2 is a schematic diagram of the geometric relationship between the missile and the target point of the embodiment of the present application. DETAILED DESCRIPTION

[0078] The detailed description of the embodiments of the present application provided in the accompanying drawings below is not intended to limit the scope of the claimed application, but only represents selected embodiments of the present application. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present application without making creative efforts fall within the scope of the present application.

[0079] Please refer to Figure 1-2 , a method for estimating the attraction domain of multiple missiles hitting targets simultaneously under a cooperative guidance law, comprising the following steps:

[0080] S1, according to the relative motion relationship between the missile and the target point, a relative kinematics model is established; the specific process is as follows:

[0081] A communication topology graph is constructed θ=(υ,ε), where υ is the node set and ε is the edge set, edge (i,j) represents that missile i is a neighbor of missile j and missile j can receive information from missile i, in an undirected graph, for any (i,j) ∈ ε, (j,i) ∈ ε; it is assumed that there are n nodes in the communication topology graph θ, and the adjacency matrix of the communication topology graph θ is denoted as Wherein if (i,j) ∈ ε, then a ij =1, otherwise a ij =0; the Laplacian matrix corresponding to the graph is denoted as Wherein l ij =-a ij , i≠j; denotes a set of real matrices of dimension n x n, l ii denotes the element value of the i-th row and i-th column of the Laplacian matrix ij ​denotes the Laplacian matrix the element value of the i-th row and j-th column of the adjacency matrix A ij denotes the element value of the i-th row and j-th column of the adjacency matrix A

[0082] Constructing a cluster system composed of n missiles in the same engagement scene

[0083]

[0084] The kinematic relationship of the missile and the target engagement is described as follows:

[0085]

[0086] wherein r i denotes the relative distance between the missile i and the target point, V i denotes the speed of the missile i, φ i denotes the angle between the speed direction of the missile i and the line-of-sight direction, λ i denotes the angle between the line-of-sight direction of the missile i and the horizontal plane, γ i denotes the angle between the speed direction of the missile i and the horizontal plane, and “·” denotes derivative;

[0087] The proportional guidance method is adopted, denotes as follows:

[0088] wherein N i denotes the navigation ratio of the missile i, and the final relative kinematic model is obtained by substituting equation (3) into equation (2), and is specifically denoted as follows:

[0089]

[0090] S2, estimate the remaining time for each missile to reach the target point, and design a variable navigation ratio for each missile based on its own and neighbor information; the specific process is as follows:

[0091] Based on the local information of the missile i and the interaction information of its neighbors, the navigation ratio is designed as shown in the following equation:

[0092] N i =N s (1+a i r i ξ i -r i |ξ i | 2α-1 sgn(ξ i )),i=1,...,n (5)

[0093] wherein:

[0094]

[0095] where N s , a i and a are constants, a i > 0, 0 < a < 1, denotes the estimation of the remaining time for missile i to hit the target, and i denotes the measurement of the remaining time error between missile i and its neighbor nodes under the network topology

[0096] Taking the differential of the equation (6) with respect to time and substituting into equation (4), we have:

[0097]

[0098] where

[0099]

[0100] where i is an even function of i and is monotonically increasing in the interval [0, π) when |φ i | < π,

[0101] S3, a method for estimating the attractor domain of a system under a Lyapunov function; the specific process is as follows:

[0102] The attractor domain R0 of a nonlinear system is defined as:

[0103]

[0104] where denotes the solution of the nonlinear system under the condition of the initial state , x = 0 is the equilibrium point of the system, and R n denotes the n-dimensional space.

[0105] In R n , take a region D containing the origin, and a continuously differentiable Lyapunov function V(x): D→R satisfies the following conditions:

[0106] V(0) = 0 (10)

[0107] In the region D-{0}, it satisfies:

[0108] V(x) > 0 (11)

[0109]

[0110] ​Given η>0, choose d∈(0,η] such that:

[0111]

[0112] In area B d The boundary exists ω=min ||x||＝d V(x), from condition (11), we get ω>0, and select β∈(0,ω) so that:

[0113] Ω β ={x∈B r |V(x)≤β} (14)

[0114] Where B d and Ω β Both indicate that R n In the figure, d and β represent a real number selected according to the above requirements.

[0115] Among them, Ω β 、B d 、D satisfies The specific content is as follows: Assume Ω β Not in B d Inside, there exists a point p∈Ω β Falls in area B d On the boundary of , for point p, V(p)≥ω>β, but for all x∈Ω β , V(x)≤β holds, which contradicts the condition, so Ω β In B d Internal, satisfied

[0116] Ω β is a subset of the nonlinear system attraction domain R0, which is used as an estimate of the nonlinear system attraction domain. The specific contents are as follows:

[0117] Because Ω β In B d Internally, so Ω β is a compact set, so for any x(0)∈Ω β Under the initial conditions, for all t>0, the system differential equation has a unique solution in Ω β According to conditions (11) and (12), V(x) is monotonically bounded. According to the monotonically bounded theorem, when t→∞, we have:

[0118] V(x(t))→c≥0 (15)

[0119] Assume c>0, then there exists a>0 such that Continuous function on the compact set {a≤||x||≤η} There is a maximum value, take According to condition (12), we have -ψ < 0, for any x(0) ∈ Ω β Thus, we have:

[0120]

[0121] Therefore, V(x(t)) will become negative as time increases, however, when t→∞, we have V(x(t))→c≥0, which is in contradiction with the condition, thus, when t→∞, we have V(x(t))→0;

[0122] According to condition (11), we have V(x) = 0 if and only if x = 0, combined with the definition of the attractor of nonlinear system, Ω β is a subset of the attractor of nonlinear system R0, which can be used as an estimation of the attractor of nonlinear system.

[0123] S4, establish the Lyapunov function of the missile group facing the time error as a variable, and scale the expression of its time differential; the specific process is as follows:

[0124] Let ξ = [ξ1, ξ2,..., ξ n ] T , According to the cooperative guidance law of the missile group system, the following Lyapunov function is obtained:

[0125]

[0126] Take the time differential of the Lyapunov function and substitute it into (5) and (7) and combine and to obtain:

[0127]

[0128] Assume that φ i is a small angle, then:

[0129]

[0130] Define a m = max{a1,...,a n}, since only when the missile hits the target, we have r i = 0, φ i = 0, therefore, assume that there is r i ≠ 0, φ i ≠ 0, i = 1,...,n, before the time of the missile is consistent, there are constants r m and φ m such that r i ≥ r m , |φ i | ≥ φm , i = 1,..., n, and the specific expression is as follows:

[0131]

[0132] S5, combined with S3 and S4, the cooperative guidance law under the estimation of the attraction domain of multiple missiles hitting the target at the same time; the specific process is as follows:

[0133] According to the missile group system under the cooperative guidance law, the remaining time error area with Lyapunov function less than zero is recorded as D M , from formula (20) when-a m ξ T ξ+(ξ T ξ) α ≥0, then the expression of B M is as follows:

[0134]

[0135] In the formula, ω M = min ||ξ||＝Δ V(ξ), select β M ∈(0, ω M ), combined with S3, the cooperative guidance law under the estimation of the attraction domain of multiple missiles hitting the target at the same time is:

[0136] Ω M ={ξ∈B M |V(ξ)≤β M} (22)

[0137] In the formula, ω M and β M both represent a real number selected according to the above requirements, V(ξ) represents the Lyapunov function under the cooperative remaining time error variable, B M represents a certain region in R n selected according to the above requirements.

[0138] The estimation method of the attraction domain of multiple missiles hitting the target at the same time under the cooperative guidance law is used to describe the space-time synchronization ability of the missile group cooperative attack, and provides a theoretical basis for the realizability judgment of the missile group consistency under certain initial conditions and the subsequent target allocation task. Under the given initial conditions such as the relative distance between the missile and the target, the angle between the line of sight direction of the missile and the target and the speed direction of the missile, the cooperative ability of the missile group hitting the target at the same time under a certain cooperative guidance law can be estimated and judged by the method, thereby serving the missile group saturation attack and other combat scenarios.

[0139] Or for the cooperative search and surveillance task of UAV group, the attraction domain estimation method of the application is used to ensure that the UAV group can maintain spatial synchronization and temporal consistency when performing tasks. By establishing the communication topology between UAVs and the corresponding navigation ratio, the UAV group can perform cooperative search in a wide area while ensuring that each UAV reaches the key monitoring point within a predetermined time. This method can significantly improve search efficiency, reduce search blind area and ensure continuous monitoring of a specific area.

[0140] Or in the unmanned distribution system of the logistics center, the attraction domain estimation method of the application can be used to ensure the efficient cooperation of distribution robots or UAVs during distribution. By using the communication topology of the distribution network and the corresponding navigation strategy, the timing and point distribution efficiency of multiple automated guided vehicles is pre-judged, so that the distribution system can adjust the distribution path in real time to avoid congestion areas. In a wider multi-agent system, the application can be used for consistency control and task allocation. The attraction domain estimation method proposed in the application is used to pre-judge the space-time cooperation ability of the group, so as to dynamically adjust the task allocation to achieve the optimal system performance.

[0141] Therefore, the attraction domain estimation method of multiple missiles hitting targets simultaneously under the above-mentioned cooperative guidance law is used to estimate the attraction domain of group cooperative consistency tasks innovatively, which can serve the saturation attack task of multiple missiles hitting targets simultaneously and has broad application prospects.

[0142] Finally, it should be noted that: the above examples are only used to illustrate the technical solutions of the application and not to limit them, although the application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that: it can still modify or replace the technical solutions of the application, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the application.

Claims

1. A method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law, characterized in that: The following steps are involved: S1. Establish a relative kinematics model based on the relative motion relationship between the missile and the target point; S2, estimate the remaining time for each missile to reach the target point, and design a variable navigation ratio for each missile based on its own information and its neighbors; S3. Construct a method for estimating the system attraction domain under the Lyapunov function; S4. Establish a Lyapunov function for the missile group with the residual time error as the variable, and scale its expression of time differentiation; S5. Combine S3 and S4 to obtain the estimation of the attraction domain for multiple missiles to hit the target simultaneously under the cooperative guidance law; The specific process of S1 is as follows: Build a communication topology diagram ,in is a collection of nodes, is a set of edges, Indicates missile It's a missile A neighbor with missiles Can be launched from missiles Receive information, in an undirected graph, for any ,have ; Assume that in the communication topology diagram There are nodes, the communication topology The adjacency matrix of , among which if ,but ,otherwise ; The Laplacian matrix corresponding to this graph is recorded as ,in , , ; Indicates the dimension A collection of real matrices of dimension , represents the Laplacian matrix No. OK The element value of the column, represents the Laplacian matrix No. OK The element value of the column, Represents the adjacency matrix No. OK The element value of the column; Constructing the same combat scenario missile swarm system : (1) The kinematic relationship between the missile and the target engagement is described as follows: (2) in, Indicates missile The relative distance to the target point, Indicates missile speed, Indicates missile The angle between the velocity direction and the line of sight direction, Indicates missile The angle between the sight line direction and the horizontal plane, Indicates missile The angle between the velocity direction and the horizontal plane, ” indicates derivative; Using proportional guidance method, It is expressed as follows: (3) in Indicates missile The navigation ratio is substituted into (2) to obtain the final relative kinematic model, which is specifically expressed as follows: (4)。 2. The method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law according to claim 1 is characterized in that: The specific process of S2 is as follows: missile-based The local information of and the interactive information of its neighbors, the navigation design is shown as follows: (5) in: (6) Where, , and are all constants, , , , Indicates missile An estimate of the remaining time to hit the target, Indicates missile In the network topology The measure of the remaining time error between the node and its neighbor nodes; For formula (6) Find the time differential and substitute it into equation (4) to obtain: (7) in (8) Where, It's about An even function of Monotonically increasing on the interval, when hour, .

3. The method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law according to claim 2 is characterized in that: The specific process of S3 is as follows: Defining the domain of attraction for a nonlinear system for: (9) Where, Indicates the initial state Nonlinear system under the condition The solution, is the equilibrium point of the system, Represents n-dimensional space; exist Take a region D containing the origin, a continuously differentiable Lyapunov function The following conditions are met: (10) In the area Satisfied: (11) (12) Given , select So that: (13) In the area The boundary exists , obtained from condition (11) , select So that: (14) Where, and All said in The selected area, and represent selected real numbers.

4. The method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law according to claim 3 is characterized in that: S3 、 、 satisfy , the specific content is as follows: Assume Not present Inside, there is a point Falling in the area On the boundary of have , however, for all ,have is established, which is contrary to the condition, so exist Internal, satisfied .

5. The method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law according to claim 4 is characterized in that: S3 The attraction domain of the nonlinear system The subset of , as the estimation of the attraction domain of the nonlinear system, is as follows: because exist Internally, so is a compact set, so in any Under the initial conditions, for all , the system differential equation has a unique solution, According to conditions (11) and (12), Monotone is bounded. According to the monotone bounded theorem, we can get When: (15) Assumptions , then there exists Make , in a tight set Upper continuous function There is a maximum value, take , according to condition (12) we get , any , thus: (16) So, as time goes by, will become negative, but when Sometimes, there are is established, which is contrary to the condition, so when Sometimes, there are ; From condition (11), we can get hour, , combined with the definition of the attraction domain of nonlinear systems, The attraction domain of the nonlinear system The subset of can be used as an estimate of the attraction domain of the nonlinear system.

6. The method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law according to claim 5 is characterized in that: The specific process of S4 is as follows: make , , according to the missile group system under the cooperative guidance law, the following Lyapunov function is obtained: (17) Differentiate the Lyapunov function with respect to time and substitute it into equations (5) and (7) and combine and have to: (18) Assumptions For a small angle, then: (19) definition , since only when the missile hits the target , , so assuming that before the missile remaining time is consistent , , , so there exists a constant and Make , , , the specific expression is as follows: (20)。 7. The method for estimating the attraction domain for simultaneous multi-projectile hits under a cooperative guidance law according to claim 6 is characterized in that: The specific process of S5 is as follows: According to the missile group system under the cooperative guidance law, the residual time error area where the Lyapunov function is less than zero is recorded as , according to formula (20) hour, ,but The expression is as follows: (21) Where, ,right , select , combined with S3, we know that the attraction domain of multiple missiles hitting the target simultaneously under the cooperative guidance law is estimated to be: (22) Where, and represents a real number selected according to the above requirements, represents the Lyapunov function under the cooperative residual time error variable, Indicates An area selected according to the above requirements.

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