Attraction domain estimation method for multiple projectiles to hit target at same time under collaborative guidance law

Abstract

The invention discloses an attraction domain estimation method for simultaneously hitting a target by multiple missiles under a cooperative guidance law, and belongs to the technical field of multi-agent control, and the method comprises the following steps: S1, building a relative kinematics model according to the relative motion relation between missiles and a target point; s2, estimating the remaining time of each missile arriving at the target point, and designing a variable navigation ratio for each missile based on each missile and neighbor information; s3, constructing a system attraction domain estimation method under a Lyapunov function; s4, establishing a Lyapunov function which faces the missile group and takes the remaining time error as a variable, and scaling an expression of time differential by the Lyapunov function; s5, in combination with S3 and S4, obtaining attraction domain estimation that multiple projectiles hit the target at the same time under the collaborative guidance law; the method provided by the invention creatively estimates the group cooperation consistency task attraction domain, can serve saturation strike tasks such as multiple missiles hit a target at the same time, and has a wide application prospect.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-agent control, and in particular relates to an attraction domain estimation method for multiple missiles to hit a target simultaneously under a collaborative guidance law. Background Art

[0002] Modern warfare is complex and changeable, and the enemy's defense system is increasingly strengthened, which makes it particularly important to develop the technology of multiple missiles hitting the target at the same time. When performing saturation strike missions, accurately coordinating multiple missiles to hit the target in a synchronized manner can not only effectively break through the enemy's defense system, but also significantly improve the missile's destructive effectiveness. The effect of cluster strikes far exceeds the traditional mode of a single missile hitting the target in sequence. Existing research on coordinated strikes involving missile groups focuses on the design of coordinated guidance laws, lacks analysis of the space-time synchronization capability of missile groups, and the discussion on the initial conditions and guarantee measures to ensure that missile groups can hit targets at the same time is not sufficient and clear.

[0003] The relevant literature on the design of distributed cooperative guidance laws for coordinated attack by missile groups is 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.) A distributed cooperative guidance law based on consistency is disclosed. This method is based on the framework of the classical proportional guidance law. The remaining time estimates are exchanged through the communication network between missiles, and a variable navigation ratio is designed to solve the problem of multiple missiles hitting a single stationary target at the same time.

[0004] The existing research on the estimation of the region of attraction of complex networks with consistency for multi-agents is 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.) The related research on the problem of estimating the region of attraction of equilibrium points in complex dynamic networks is disclosed, and a brief analysis of network stability is conducted. First, sufficient and necessary conditions are established for the asymptotic stability of network equilibrium points, and then a general technique for estimating the region of attraction is developed by combining network structure and node dynamics.

[0005] However, the above research on coordinated strike of missile groups only focused on the design of coordinated guidance laws, without further analysis of the spatiotemporal synchronization capabilities of missile groups; and the above estimation of the attraction domain of complex network consistency only focused on fixed equilibrium points, without research on 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 a target simultaneously under the cooperative guidance law. By analyzing the estimation of the remaining time of missile hits under the cooperative guidance law, combined with the designed Lyapunov function with the remaining time error as the variable, a missile group spatiotemporal synchronization attraction domain is constructed to characterize the spatiotemporal synchronization capability of the missile group, and the condition of the consistency of the remaining time of the missile group is 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 an attraction domain estimation method for multiple missiles hitting a target simultaneously 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 the information of each missile itself 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 a 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; represents a set of real matrices of dimension n×n, l ii Represents the Laplacian matrix The value of the element in row i and column i, l ij Represents the Laplacian matrix The value of the element in row i and column j, a ij Represents the element value of the i-th row and j-th column of the adjacency matrix A;

[0015] Construct a cluster system consisting of n missiles in the same combat scenario

[0016]

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

[0018]

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

[0020] Using proportional guidance method, It is expressed as follows:

[0021]

[0022] Where N i represents the navigation ratio of missile i. Substituting (3) into (2) yields the final relative kinematic model, which is specifically expressed as follows:

[0023]

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

[0025] Based on the local information of missile i and the interactive information of its neighbors, the navigation equation is designed as shown below:

[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] in:

[0028]

[0029] Where N s , a i and α are both constants, a i >0,0<α<1, represents the estimated remaining time required for missile i to hit the target, ξ i represents the measure of the remaining time error between missile i and its neighbor nodes under the network topology θ;

[0030] For formula (6) Find the time differential and substitute it into equation (4) to obtain:

[0031]

[0032] in

[0033]

[0034] In the formula, δ i It is about φ i is an even function, and is monotonically increasing on the interval [0,π). When |φ i When |<π,

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

[0036] Define the attraction domain R of the nonlinear system 0 for:

[0037]

[0038] In the formula, Indicates the initial state Nonlinear system under the condition The solution is x = 0, which is the equilibrium point of the system. n Represents n-dimensional space;

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

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

[0041] In region D-{0}, the following conditions are satisfied:

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

[0043]

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

[0045]

[0046] In area B d There exists a boundary of 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] In the formula, B d With Ω β Both are expressed in R n In , d and β both represent a real number selected according to the above requirements.

[0049] Preferably, Ω in S3 β , B d , D satisfies The specific contents are as follows: Assume Ω β Not in B d Inside, there exists a point p∈Ω β Falling 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 is contrary to the condition, so Ω β In B d Internal, satisfied

[0050] Preferably, Ω in S3 β The nonlinear system attraction domain R 0 As an estimate of the attraction domain of the nonlinear system, 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 the compact set {a≤||x||≤η} There is a maximum value, take According to condition (12), -ψ<0, any x(0)∈Ω β , thus:

[0054]

[0055] Therefore, as time increases, V(x(t)) will become negative. However, when t→∞, V(x(t))→c≥0 holds, which violates the condition. Therefore, when t→∞, V(x(t))→0.

[0056] From condition (11), we can get V(x) = 0 if and only if x = 0. Combined with the definition of the attraction domain of nonlinear system, Ω β The nonlinear system attraction domain R 0 The subset of can be used as an estimate of the attraction domain of the nonlinear system.

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

[0058] Let ξ=[ξ 1 ,ξ 2 ,...,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 equations (5) and (7) and combine and have to:

[0061]

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

[0063]

[0064] Definition m =max{a 1 ,...,a n}, since only when the missile hits the target, there is r i =0,φ i = 0, so assuming that there are r before the remaining time of the missiles is consistent i ≠0,φ i≠0, i=1,...,n, so there exists a constant r m and φ m Make r i ≥r m ,|φ i |≥φ m , i=1,...,n, 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 remaining time error area where the Lyapunov function is less than zero is recorded as D M , from formula (20) we get -a m ξ T ξ+(ξ T ξ) α ≥0, Then B M The expression is as follows:

[0068]

[0069] In the formula, Yes M =min ||ξ||＝Δ V(ξ), choose β M ∈(0,ω M ), combined with S3, it is known that the attraction domain of multiple missiles hitting the target simultaneously under the cooperative guidance law is estimated to be:

[0070] Ω M ={ξ∈B M |V(ξ)≤β M} (twenty two)

[0071] In the formula, ω M With β M All represent a real number selected according to the above requirements, V(ξ) represents the Lyapunov function under the coordinated residual time error variable, B M Indicates that in R n An area selected according to the above requirements.

[0072] Therefore, the present invention adopts the above-mentioned method for estimating the attraction domain of multiple missiles hitting the target simultaneously under the cooperative guidance law, which has the following beneficial effects:

[0073] (1) It is used to characterize the spatiotemporal synchronization capability of missile group coordinated strikes, and provides a theoretical basis for the feasibility judgment of missile group consistency under specific initial conditions and the subsequent target allocation task;

[0074] (2) When used for consistency control of multiple unmanned vehicles and multiple aerial vehicles, this method can be used to estimate the attraction domain under a specific guidance law, thereby characterizing the range of group initial conditions for achieving multi-agent consistency. This can serve subsequent task allocation scenarios such as unmanned vehicle group trajectory planning, logistics vehicle collaborative transportation, and drone group waypoint planning.

[0075] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 It is an overall flow chart of an attraction domain estimation method for multiple missiles hitting a target simultaneously under a cooperative guidance law of the present invention;

[0077] Figure 2 Schematic diagram of the geometric relationship between a missile and a target point according to an embodiment of the present invention. DETAILED DESCRIPTION

[0078] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the invention claimed for protection, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

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

[0080] S1. Establish a relative kinematics model based on the relative motion relationship between the missile and the target point. The specific process is as follows:

[0081] 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; represents a set of real matrices of dimension n×n, l ii Represents the Laplacian matrix The value of the element in row i and column i, l ijRepresents the Laplacian matrix The value of the element in row i and column j, a ij Represents the element value of the i-th row and j-th column of the adjacency matrix A;

[0082] Construct a cluster system consisting of n missiles in the same combat scenario

[0083]

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

[0085]

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

[0087] Using proportional guidance method, It is expressed as follows:

[0088] Where N i represents the navigation ratio of missile i. Substituting (3) into (2) yields the final relative kinematic model, which is specifically expressed 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 the information of each missile itself and its neighbors. The specific process is as follows:

[0091] Based on the local information of missile i and the interactive information of its neighbors, the navigation equation is designed as shown below:

[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] in:

[0094]

[0095] Where N s , a i and α are both constants, a i >0,0<α<1, represents the estimated remaining time required for missile i to hit the target, ξ i represents the measure of the remaining time error between missile i and its neighbor nodes under the network topology θ;

[0096] For formula (6) Find the time differential and substitute it into equation (4) to obtain:

[0097]

[0098] in

[0099]

[0100] In the formula, δ i It is about φ i is an even function, and is monotonically increasing on the interval [0,π). When |φ i When |<π,

[0101] S3. Construct a method for estimating the system attraction domain under the Lyapunov function. The specific process is as follows:

[0102] Define the attraction domain R of the nonlinear system 0 for:

[0103]

[0104] In the formula, Indicates the initial state Nonlinear system under the condition The solution is x = 0, which is the equilibrium point of the system. n Represents n-dimensional space;

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

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

[0107] In region D-{0}, the following conditions are satisfied:

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

[0109]

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

[0111]

[0112] In area B d There exists a boundary of 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] In the formula, B d With Ω β Both are expressed in R n In , d and β both represent a real number selected according to the above requirements.

[0115] Among them, Ω β , B d , D satisfies The specific contents are as follows: Assume Ω β Not in B d Inside, there exists a point p∈Ω β Falling 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 is contrary to the condition, so Ω β In B d Internal, satisfied

[0116] Ω β The nonlinear system attraction domain R 0 As an estimate of the attraction domain of the nonlinear system, 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), -ψ<0, any x(0)∈Ω β , thus:

[0120]

[0121] Therefore, as time increases, V(x(t)) will become negative. However, when t→∞, V(x(t))→c≥0 holds, which violates the condition. Therefore, when t→∞, V(x(t))→0.

[0122] From condition (11), we can get V(x) = 0 if and only if x = 0. Combined with the definition of the attraction domain of nonlinear system, Ω β The nonlinear system attraction domain R 0 The subset of can be used as an estimate of the attraction domain of the nonlinear system.

[0123] S4. Establish a Lyapunov function for the missile group with the remaining time error as the variable, and scale its expression of time differential; the specific process is as follows:

[0124] Let ξ=[ξ 1 ,ξ 2 ,...,ξ n ] T , According to the missile group system under the cooperative guidance law, the following Lyapunov function is obtained:

[0125]

[0126] Differentiate the Lyapunov function with respect to time and substitute it into equations (5) and (7) and combine and have to:

[0127]

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

[0129]

[0130] Definition m =max{a 1 ,...,a n}, since only when the missile hits the target, there is r i =0,φ i = 0, so assuming that there are r before the remaining time of the missiles is consistent i ≠0,φ i ≠0, i=1,...,n, so there exists a constant r m and φm Make r i ≥r m ,|φ i |≥φ m , i=1,...,n, the specific expression is as follows:

[0131]

[0132] 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 is as follows:

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

[0134]

[0135] In the formula, Yes M =min ||ξ||＝Δ V(ξ), choose β M ∈(0,ω M ), combined with S3, it is known that the attraction domain of multiple missiles hitting the target simultaneously under the cooperative guidance law is estimated to be:

[0136] Ω M ={ξ∈B M |V(ξ)≤β M} (twenty two)

[0137] In the formula, ω M With β M All represent a real number selected according to the above requirements, V(ξ) represents the Lyapunov function under the coordinated residual time error variable, B M Indicates that in R n An area selected according to the above requirements.

[0138] The attraction domain estimation method for multiple missiles to hit the target simultaneously under the cooperative guidance law is used to characterize the spatiotemporal synchronization capability of the missile group's coordinated strike, and provides a theoretical basis for the feasibility judgment of the missile group's consistency under specific initial conditions and the subsequent target allocation task. Given the initial conditions such as the relative distance between the missile and the target, the line of sight direction between the missile and the target, and the angle between the missile's speed direction, this method can be used to estimate the coordinated ability of the missile group to hit the target simultaneously under a certain cooperative guidance law, thereby serving various combat scenarios such as missile group saturation strike.

[0139] Or it can be used for collaborative search and monitoring tasks of drone swarms, and the attraction domain estimation method of the present invention can be used to ensure that the drone swarm can maintain spatial synchronization and temporal consistency when performing tasks. By establishing a communication topology and corresponding navigation ratios between drones, the drone swarm can conduct collaborative searches in a wide area while ensuring that each drone reaches the key monitoring point within a predetermined time. This method can significantly improve search efficiency, reduce search blind spots, and ensure continuous monitoring of specific areas.

[0140] Or it can be used in unmanned delivery systems in logistics centers. The attraction domain estimation method of the present invention can be used to ensure efficient coordination of delivery robots or drones during the delivery process. The communication topology of the distribution network and the corresponding navigation strategy can be used to pre-judge the delivery time efficiency of multiple automatic guided vehicles at fixed times and points, so that the distribution system can adjust the delivery path in real time to avoid congested areas. In a wider range of multi-agent systems, the present invention can be used for consistency control and task allocation. The attraction domain estimation method proposed by the present invention can be used to pre-judge the spatiotemporal coordination ability of the group, thereby dynamically adjusting the task allocation to achieve optimal system performance.

[0141] Therefore, the present invention adopts the above-mentioned method for estimating the attraction domain of multiple missiles hitting the target simultaneously under the collaborative guidance law, and innovatively estimates the attraction domain of group collaborative consistency tasks, which can serve saturation strike tasks such as multiple missiles hitting the target simultaneously, and has broad application prospects.

[0142] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for estimating the attraction domain of multiple missiles hitting a target simultaneously 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 the information of each missile itself 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 a 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.

2. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under a cooperative guidance law according to claim 1 is characterized in that: The specific process of S1 is as follows: 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; represents a set of real matrices of dimension n×n, l ii Represents the Laplacian matrix The value of the element in row i and column i, l ij Represents the Laplacian matrix The value of the element in row i and column j, a ij Represents the element value of the i-th row and j-th column of the adjacency matrix A; Construct a cluster system consisting of n missiles in the same combat scenario The kinematic relationship between the missile and the target engagement is described as follows: Among them, r i represents the relative distance between missile i and the target point, V i represents the speed of missile i, φ i represents the angle between the velocity direction of missile i and the line of sight, λ i represents the angle between the sight line direction of missile i and the horizontal plane, γ i represents the angle between the missile i's velocity direction and the horizontal plane, "·" represents the derivative; Using proportional guidance method, It is expressed as follows: Where N i represents the navigation ratio of missile i. Substituting (3) into (2) yields the final relative kinematic model, which is specifically expressed as follows:

3. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under the cooperative guidance law according to claim 2 is characterized in that: The specific process of S2 is as follows: Based on the local information of missile i and the interactive information of its neighbors, the navigation equation is designed as shown below: N i =N s (1+a i r i x i -r i |x i | 2α-1 sgn(ξ i )),i=1,...,n (5) in: Where N s , a i and α are both constants, a i >0,0<α<1, represents the estimated remaining time required for missile i to hit the target, ξ i represents the measure of the remaining time error between missile i and its neighbor nodes under the network topology θ; For formula (6) Find the time differential and substitute it into equation (4) to obtain: in In the formula, δ i It is about φ i is an even function, and is monotonically increasing on the interval [0,π). i When |<π, 4. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under the cooperative guidance law according to claim 3 is characterized in that: The specific process of S3 is as follows: The attraction domain R0 of the nonlinear system is defined as: In the formula, Indicates the initial state Nonlinear system under the condition The solution is x = 0, which is the equilibrium point of the system. n Represents n-dimensional space; In R n Take a region D containing the origin, a continuously differentiable Lyapunov function V(x): D→R satisfies the following conditions: V(0)＝0 (10) In region D-{0}, the following conditions are satisfied: V(x)>0 (11) Given η>0, choose d∈(0,η] such that: In area B d There exists a boundary of ω=min x＝d V(x), from condition (11), we get ω>0, and select β∈(0,ω) so that: Ω β ={x∈B r |V(x)≤β} (14) In the formula, B d With Ω β Both are expressed in R n In the selected area, d and β both represent the selected real numbers.

5. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under the cooperative guidance law according to claim 4 is characterized in that: S3 medium Ω β , B d , D satisfies The specific contents are as follows: Assume Ω β Not in B d Inside, there is 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 is contrary to the condition, so Ω β In B d Internal, satisfied 6. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under the cooperative guidance law according to claim 5 is characterized in that: S3 medium Ω β is a subset of the nonlinear system attraction domain R0, as an estimate of the nonlinear system attraction domain, the specific contents are as follows: 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: V(x(t))→c≥0 (15) 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), -ψ<0, any x(0)∈Ω β , thus: Therefore, as time increases, V(x(t)) will become negative. However, when t→∞, V(x(t))→c≥0 holds, which violates the condition. Therefore, when t→∞, V(x(t))→0. From condition (11), we can get V(x) = 0 if and only if x = 0. Combined with the definition of the attraction domain of nonlinear system, Ω β It is a subset of the nonlinear system attraction domain R0 and can be used as an estimate of the nonlinear system attraction domain.

7. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under the cooperative guidance law according to claim 6 is characterized in that: The specific process of S4 is as follows: Let ξ=[ξ1,ξ2,...,ξ n ] T , According to the missile group system under the cooperative guidance law, the following Lyapunov function is obtained: Differentiate the Lyapunov function with respect to time and substitute it into equations (5) and (7) and combine and have to: Assume φ i For a small angle, then: Definition m =max{a1,...,a n }, since only when the missile hits the target, there is r i =0,φ i = 0, so assuming that there are r before the remaining time of the missiles is consistent i ≠0,φ i ≠0, i=1,...,n, so there exists a constant r m and φ m Make r i ≥r m ,φ i |≥φ m , i=1,...,n, the specific expression is as follows:

8. The method for estimating the attraction domain of multiple missiles hitting a target simultaneously under the cooperative guidance law according to claim 7 is characterized in that: The specific process of S5 is as follows: According to the missile group system under the cooperative guidance law, the remaining time error area where the Lyapunov function is less than zero is recorded as D M , from formula (20) we get -a m ξ T ξ+(ξ T ξ) α ≥0, Then B M The expression is as follows: In the formula, Yes M =min ξ＝Δ V(ξ), choose β M ∈(0,ω M ), combined with S3, it is known that the attraction domain of multiple missiles hitting the target simultaneously under the cooperative guidance law is estimated to be: Oh M ={ξ∈B M |V(ξ)≤β M } (22) In the formula, ω M With β M All represent a real number selected according to the above requirements, V(ξ) represents the Lyapunov function under the coordinated residual time error variable, B M Indicates that in R n An area selected according to the above requirements.

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