An online generation method for optimal guidance law with fixed attack time

Abstract

The invention discloses an online generation method of an optimal guidance law for a fixed attack time, and belongs to the field of aerospace and trajectory optimization. The method comprises: aiming at the stationary target attack problem under the constraint of attack time, establishing a nonlinear kinematic model of a missile, establishing an optimal control problem related to the guidance problem with a fixed attack time; establishing the first-order necessary conditions of the optimal control problem; using the geometric properties of the optimal trajectory, proposing a new optimality condition; according to the first-order necessary conditions and the newly established optimality condition, establishing a parameterized system to ensure that the solution space of the parameterized system is equivalent to the solution space of the optimal control problem; directly integrating the parameterized system to generate a data set that can reflect the mapping relationship between the missile state and the remaining flight time and the optimal guidance law; using a neural network, the optimal guidance law can be generated in real time according to the missile state and the remaining flight time, so as to control the attack time of the stationary target.

Description

Technical Field

[0001] The invention relates to an online generation method of an optimal guidance law with fixed attack time, and belongs to the technical field of trajectory optimization. Background Art

[0002] With the advent of the intelligent and unmanned era, trajectory optimization is widely used in robot control to improve the robot's work efficiency and reduce energy loss. In the field of aircraft guidance and control, optimizing the flight trajectory of the aircraft during the mission can effectively assist the aircraft design and improve flight performance. In the military field, trajectory optimization has important research significance and application value, and has promoted research directions such as missile guidance and unmanned aircraft trajectory planning.

[0003] Specifically, for the field of missile guidance, the guidance law determines the maneuvering instructions of the missile in the process of approaching the target. With the continuous development of modern warfare, linear optimal guidance law has been widely studied since the 1960s, among which relevant scholars have proved that proportional guidance with a coefficient of 3 has the best performance under certain conditions (see Kreindler, E., "Optimality of Proportional Navigation," AIAA Journal, Vol. 11, No. 6, 1973, pp. 878-880. Bryson, AE, and Ho, YC, Applied Optimal Control, Hemisphere, Washington, DC, 1975, pp. 154-155.). However, proportional guidance only has excellent performance under the small angle assumption; when the small angle assumption is not established, the performance of proportional guidance will be affected under nonlinear conditions. In addition, with the continuous development of modern warfare and the improvement of weapon performance, the risk of using a single missile to destroy a target based on the existing guidance law has gradually increased. Therefore, a strategy of salvo attack to destroy the target has been proposed in the military field to achieve a higher success rate through saturation attack. In this combat plan, there are stricter requirements on the attack time. In order to ensure the secrecy of the combat mission, it is often necessary to require the missiles to arrive at the same time to reduce the possibility of being destroyed by the enemy's defensive weapons. Therefore, the study of the optimal guidance problem with a fixed attack time is of great significance.

[0004] At present, most of the research on guidance law with attack time constraints uses modern control theory. The ITCG (Impact-Time Control Guidance) method proposed by Jeon I.-S. et al. adjusts the gain of proportional guidance according to the difference between the expected remaining flight time and the estimated remaining flight time (see Jeon I.-S., Lee, J.-I., and Tahk, M.-J.,

[0005] “Impact-Time-Control Guidance Law for Anti-Ship Missiles,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 2, 2006, pp. 260–266, but this method is only applicable to the case where the small angle assumption is met. Sliding mode control theory and Lyapunov theory are also used to study the fixed attack time guidance law, but these methods often cannot guarantee the optimality of control efficiency.

[0006] Different from the existing technical methods, the present invention provides a real-time generation scheme of a fixed attack time nonlinear optimal guidance law (FTNOG), which can control the attack time while achieving precise strikes on stationary targets without linearizing the motion model of the missile. Summary of the invention

[0007] In view of the defects of the prior art and the requirements for improvement, the present invention provides an online generation method of the optimal guidance law with fixed attack time, the purpose of which is to utilize the optimality conditions and geometric properties of the solution to the nonlinear optimal interception problem, to establish a parameterized system without directly solving the optimal guidance problem, so that its solution space is equivalent to the solution space of the optimal guidance law, and then to utilize a feedforward neural network to realize the real-time generation of the optimal guidance law.

[0008] To achieve the above object, the basic idea of ​​the present invention includes:

[0009] The present invention provides an online generation method of an optimal guidance law for a fixed attack time, which comprises the following steps:

[0010] S1, for the problem of attacking a stationary target under the constraint of attack time, a nonlinear kinematic model of the missile is established in the Cartesian coordinate system, and the optimal control problem corresponding to the problem of attacking a stationary target is established;

[0011] S2, derive the first-order necessary conditions for the optimal control problem, including determining the Hamiltonian function H according to the Pontryagin maximum principle, determining the co-state equation and the control equation, and giving the transversal condition;

[0012] S3, using the geometric properties of the optimal trajectory, proposes a new optimality condition;

[0013] S4, based on the first-order necessary conditions in S2 and the optimality conditions in S3, a parameterized system is established so that its solution space is equivalent to the solution space of the optimal control problem; using the idea of ​​integration, without directly solving the optimal control problem, a data set that can reflect the mapping between the missile state and the remaining flight time and the optimal guidance law is generated;

[0014] S5, a feedforward neural network is used to train the data set. The resulting neural network can generate the optimal guidance law in real time according to the current state of the missile to control the attack time of the stationary target.

[0015] As a preferred embodiment of the present invention, the S1 process is as follows:

[0016] The target location is defined as the origin of the coordinate system Oxy, with the x-axis pointing east and the y-axis pointing north; (x, y) is the coordinate of the missile in the coordinate system Oxy, θ∈[0,2π] is the angle between the x-axis and the missile velocity V, that is, the heading angle of the missile, and the counterclockwise direction is defined as positive; the missile velocity V is normalized to 1, and the nonlinear kinematic model of the missile is expressed as:

[0017]

[0018] Where t>0 is the time, is the control quantity of the missile, indicating its lateral acceleration;

[0019] For a given initial position and a fixed attack time t f >0, the optimal control problem corresponding to the stationary target attack problem can be described as: by controlling u(·), the missile's initial state (x0, y0, θ0) at time t=0 is transferred to the state at time t=t f The target position at the moment is (0,0), where the terminal heading angle of the missile is free and the control efficiency is minimized, expressed as:

[0020]

[0021] As a preferred solution of the present invention, S2 specifically includes the following steps:

[0022] 3.1) Determine the Hamiltonian function H

[0023]

[0024] where p x 、p y 、p θ are the covariates of x, y, and θ, respectively, and p0 is a nonpositive scalar;

[0025] 3.2) According to the Pontryagin maximum principle, determine the co-state equation

[0026]

[0027] The governing equation is:

[0028]

[0029] According to formula (3), we have:

[0030] u(t)=p θ (t),t∈[0,t f ] (4)

[0031] Since the terminal heading angle is free, the transversal condition is satisfied:

[0032] p θ (t f )=0 (5)

[0033] In addition, the terminal position of the missile must meet the following requirements:

[0034] (x(t f ),y(t f ))=(0,0) (6)

[0035] According to equations (5) and (6), integrating the third equation of equation (2), we have:

[0036] p θ (t) = p x (t)y(t)-p y (t)x(t),t∈[0,t f ] (7).

[0037] As a preferred solution of the present invention, a new optimality condition is proposed in S3, specifically:

[0038] Given in [0,t f ], if there exists any extreme value trajectory (x(t), y(t), θ(t)) on So that the velocity vector Parallel to the Line of Sight, that is, the viewing angle is 0 or π:

[0039]

[0040] Then the corresponding extreme value trajectory is not an optimal trajectory.

[0041] As a preferred solution of the present invention, the establishment of the parameterized system in S4 comprises the following steps:

[0042] Definition about the vector [p xx ,p y ]'s variable α

[0043]

[0044] Define β∈[-π,π]

[0045] p x =αcosβ,p y =αsinβ (8)

[0046] Therefore, formula (7) can be written as:

[0047] p θ (t)=α[y(t)cosβ-x(t)sinβ] (9)

[0048] Define a parameterized system:

[0049]

[0050] in Θ∈[0,2π], for the sake of simplicity of notation, let the three-dimensional matrix:

[0051]

[0052] For the initial state:

[0053] (X(0,α,β),Y(0,α,β),Θ(0,α,β))=(0,0,0)

[0054] Define λ∈[0,2π] as the angle between the line of sight and the x-axis, with counterclockwise being positive, and then obtain the nonlinear kinematic equation in the polar coordinate system:

[0055]

[0056] Where r ≥ 0 is the Euclidean distance between the missile and the target, expressed as:

[0057]

[0058] Define σ∈[-π,π] as the angle between the line of sight and the velocity vector V, which is called the viewing angle, and is positive in the clockwise direction; because σ=π+λ-θ, equation (11) can be written as:

[0059]

[0060] As a preferred solution of the present invention, the generation of the data set reflecting the mapping between the missile state, the remaining flight time and the optimal guidance law in S4 comprises the following steps:

[0061] Use C(r c ,σ c ,t g ) indicates the current state (r c ,σ c ) is the optimal control instruction, where t g >0, indicating the remaining flight time, for t∈[0,t f ], given any optimal trajectory (r(t),σ(t)), if u(t) represents the corresponding optimal control, then C(r(t),σ(t),t f -t)=u(t); According to the definition of optimal control, generating the optimal solution in real time is equivalent to calculating (r c ,σ c ,t g )→C(r c ,σ c ,t g ) mapping;

[0062] Before training a neural network, we need to generate a representation (r c ,σ c ,t g ) and C(r c ,σ c ,t g ) relationship, represented as:

[0063]

[0064] Here

[0065] t(α,β)=min {t>0|cos[Σ(t,α,β)]=1} (15)

[0066] Use the mapping (R(t,α,β),Σ(t,α,β),t)→U(t,α,β) to generate the representation (r c ,σ c ,t g ) and C(r c ,σ c ,t g ), (α, β)∈Λ and t∈[0,T(α,β)]; In order to prevent T(α,β) from being too large and causing difficulty in generating data, let

[0067]

[0068] in is a finite number; all sampled data will eventually be included in the dataset In the dataset The neural network is trained with data from the missile system and can generate the optimal guidance law in real time according to the current state of the missile.

[0069] As a preferred solution of the present invention, the S5 uses a feedforward neural network to train the data set, including:

[0070] Set the parameters for generating the dataset N i , N j , h is 10, 100, 100, 10 and 0.005 respectively, the resulting data set is Contains 4.59×10 6 The neural network used has three hidden layers, each hidden layer contains 30 neurons, and the activation function uses the tanh function.

[0071] Use Pytorch to analyze the dataset After training, the residual (MSE) finally converged to 10 -4 The neural network was tested using the Interi5-9300HCPU@2.4ghz computing platform and it was found that for a given missile state and remaining flight time, the trained neural network can give the corresponding guidance instructions within 0.0935ms. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 A flow chart of an optimal guidance instruction online generation method provided by an example of the present invention;

[0073] Figure 2 A geometric model of a two-dimensional attack problem provided for an example of the present invention;

[0074] Figure 3 A schematic diagram of the optimality condition proof provided for an example of the present invention;

[0075] Figure 4 A schematic diagram of a closed-loop nonlinear optimal guidance provided by an example of the present invention;

[0076] Figure 5A The following is a comparison chart of missile motion trajectories under different attack times and different methods;

[0077] Figure 5B Schematic diagram of trajectories obtained by the method of the present invention at different initial heading angles;

[0078] Figure 5C Schematic diagram of the change of control amount according to the method of the present invention at different initial heading angles;

[0079] Figure 5D A comparison diagram of trajectories obtained by the method of the present invention and the nonlinear programming method;

[0080] Figure 5E Comparison of trajectories obtained by different methods in a salvo attack scenario. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0082] In the present invention, the terms "first", "second", etc. (if any) in the present invention and the accompanying drawings are used to distinguish similar objects but not necessarily to describe a specific order or sequence.

[0083] Figure 1 A flowchart of an optimal guidance method for intercepting a stationary target with a fixed attack time provided by an embodiment of the present invention. Figure 1 , combined with Figure 2-5E , the fixed attack time interception optimal guidance method (FTNOG) for a stationary target in this embodiment is described in detail, and the method includes operations S1 to S5.

[0084] Operation S1: For the stationary target attack problem under the attack time constraint, a nonlinear kinematic model of the missile is established in the Cartesian coordinate system, and the optimal control problem corresponding to the stationary target attack problem is established. In this example, a stationary target on a two-dimensional horizontal plane is considered, and the nonlinear kinematic model of the missile is modeled in the Cartesian coordinate system:

[0085]

[0086] As reference Figure 2 As shown in the figure, the target position is defined as the origin of the coordinate system Oxy, the x-axis points to the east, and the y-axis points to the north. (x, y) is the coordinate of the missile in the coordinate system Oxy, θ∈[0,2π] is the angle between the x-axis and the missile speed V, which is usually called the heading angle of the missile, and is defined as counterclockwise. Normalize the missile speed V, t>0 is the time, and the superscript · represents the differential with respect to time, is the control variable representing the lateral acceleration.

[0087] For a given initial position and a fixed attack time t f>0, the optimal control problem corresponding to the stationary target attack problem can be described as: by controlling u(·), the missile's initial state (x0, y0, θ0) at time t=0 is transferred to the state at time t=t f The target position at the moment is (0,0), where the terminal heading angle of the missile is free and the control efficiency is minimized, expressed as:

[0088]

[0089] Operation S2, based on the nonlinear kinematic model of the missile, derives the first-order necessary conditions of the optimal control problem, including: determining the Hamiltonian function H according to the Pontryagin maximum principle, determining the co-state equation, the control equation, and giving the transversal condition; according to an example of the present invention, operation S2 includes sub-operations S21-S23.

[0090] In sub-operation S21, according to the Pontryagin maximum principle, the Hamiltonian function H is determined:

[0091]

[0092] where p x 、p y 、p θ are the covariates of x, y, and θ, respectively, and p 0 is a non-positive scalar, here we consider p 0 =-1. (See Chen, Z., and Shima, T., "Nonlinear Optimal Guidance for Intercepting a Stationary Target," Journal of Guidance, Control, and Dynamics, Vol. 42, No. 11, 2019, pp. 2418–2431.)

[0093] In sub-operation S22, the co-state equation is determined:

[0094]

[0095] The governing equation is:

[0096]

[0097] Then we have:

[0098] u(t)=p θ (t),t∈[0,t f ]

[0099] In sub-operation S23, the cross-section condition is determined:

[0100] Since the terminal heading angle is free, the cross-section condition satisfies:

[0101] p θ (t f )=0

[0102] In addition, the terminal position of the missile must meet the following requirements:

[0103] (x(t f ),y(t f ))=(0,0)

[0104] According to equations (5) and (6), integrating the third equation of equation (2), we have:

[0105] p θ (|)=p x (t)y(t)-p y (t)x(t),t∈[0,t f ]

[0106] Operation S3 proposes a new optimality condition using the geometric properties of the optimal trajectory.

[0107] Given in [0,t f ], if there exists any extreme value trajectory (x(t), y(t), θ(t)) on So that the velocity vector Parallel to the Line of Sight, that is, the viewing angle is 0 or π:

[0108]

[0109] Then the corresponding extreme value trajectory is not an optimal trajectory. That is, if there is a point in the extreme value trajectory obtained according to the first-order necessary conditions derived from the Pontryagin maximum principle that makes the above equation true, then this extreme value trajectory is not an optimal trajectory, otherwise this extreme value trajectory is an optimal trajectory.

[0110] The proof is as follows:

[0111] As reference Figure 3 As shown in The state quantity at the moment is expressed as make for The control amount that makes the missile move from position A to the origin. Take the time The position state is B. Then there exists an extreme value trajectory γ from B to the origin and a smooth extreme value trajectory from B to A It can be concluded that the control efficiency along the two trajectories is the same, but along the trajectory During the movement, the control quantity will suddenly change at A, which is contrary to the first-order necessary condition based on the Pontryagin maximum principle. Therefore, there is another way that takes time The extreme trajectory from B to the origin makes the control efficiency smaller, the proof is complete.

[0112] Operation S4, such as Figure 4 As shown, based on the first-order necessary conditions in S2 and the optimality conditions in S3, a parameterized system is established so that its solution space is equivalent to the solution of the optimal guidance law; using the idea of ​​integration, without directly solving the optimal control problem, a data set that can reflect the mapping between the missile state and the remaining flight time and the optimal guidance law is generated. Operation S4 includes sub-operations S41-S46.

[0113] In sub-operation S41, define the vector [p x ,p y ] variable α, define β∈[-π,π]

[0114]

[0115] p x =αcosβ,p y =αsinβ

[0116] therefore

[0117] p θ (t) = α[y(t)cosβ-x(t)sinβ]

[0118] In sub-operation S42, a parameterized system is defined:

[0119]

[0120] in Θ∈[0,2π], for the sake of simplicity of notation, let the three-dimensional matrix:

[0121]

[0122] For the initial state:

[0123] (X(0,α,β),Y(0,α,β),Θ(0,α,β))=(0,0,0)

[0124] In sub-operation S43, define λ∈[0,2π] as the angle between the line of sight and the x-axis, where counterclockwise is positive, such as Figure 2 As shown. Then the nonlinear kinematic equation in the polar coordinate system is obtained:

[0125]

[0126] Where r ≥ 0 is the Euclidean distance between the missile and the target, expressed as:

[0127]

[0128] Define σ∈[-π,π] as the angle between the line of sight and the velocity vector V, which is called the viewing angle, and is positive clockwise. Because σ=π+λ-θ, then:

[0129]

[0130] Because this equation is equivalent to equation (1) before r approaches 0, it complies with the optimal control requirements in Cartesian coordinate system and polar coordinate system.

[0131] In sub-operation S44, C(r c ,σ c ,t g ) indicates the current state (r c ,σ c ) is the optimal control instruction, where t g > 0, indicating the remaining flight time. This means that for t∈[0,t f ], given any optimal trajectory (r(t),σ(t)), if u(t) represents the corresponding optimal control, then C(r(t),σ(t),t f -t)=u(t). According to the definition of optimal control, generating the optimal solution in real time is equivalent to calculating (r c ,σ c ,t g )→C(r c ,σ c ,t g ) mapping.

[0132] In sub-operation S45, according to the universal approximation theorem, if a feedforward neural network is used to train a large number of representations (r c ,σ c ,t g ) and C(r c ,σ c ,t g ), then the neural network can be used for any It can be accurately expressed (r c ,σ c ,t g )→C(r c ,σ c ,t g )’s mapping.

[0133] Before training a neural network, we need to generate a representation (r c ,σ c ,t g ) and C(rc ,σ c ,t g ) relationship, represented as:

[0134]

[0135] Here

[0136] T(α,β)=min{t>0|cos[Σ(t,α,β)]=1}

[0137] In sub-operation S46, the mapping (R(t,α,β),Σ(t,α,β),t)→U(t,α,β) can be used to generate the representation (r c ,σ c ,t g ) and C(r c ,σ c ,t g ), (α, β)∈Λ and t∈[0,T(α,β)]. In order to prevent T(α,β) from being too large and causing difficulties in generating data sets, we assume:

[0138]

[0139] in is a finite number. All sampled data will eventually be included in the dataset In the dataset The training data is used to train a neural network that can generate the optimal guidance law in real time according to the current state of the missile.

[0140] Operation S5, using a feedforward neural network to train the data set, the resulting neural network can generate an optimal guidance law in real time according to the current state of the missile, so as to control the attack time of the stationary target.

[0141] In the above steps, the velocity V was normalized, and the velocity was reasonably set in the experimental simulation (see Merkulov, G., Weiss, M., and Shima, T., “Minimum-Effort Impact-Time Control Guidance Using Quadratic Kinematics Approximation,” Journal of Guidance, Control, and Dynamics, Vol. 0, No. 0, 0, pp. 1–14.).

[0142] Specifically, set the parameters for generating the dataset N i , N j , h is 10, 100, 100, 10 and 0.005 respectively, the resulting data set is Contains 4.59×10 6 The neural network used has a total of hidden layers, each hidden layer contains 30 neurons, and the activation function uses the tanh function.

[0143] Use Pytorch to analyze the dataset After training, the residual (MSE) finally converged to 10 -4 The neural network was tested using the Interi5-9300HCPU@2.4ghz computing platform, and it was found that for a given missile state and remaining flight time, the trained neural network can give the corresponding guidance instructions within 0.0935ms. In order to reflect the advantages of the method of the present invention, a comparative verification was carried out with the existing methods.

[0144] In sub-operation S51, the missile motion trajectories of the proposed method under different fixed attack times are compared.

[0145] Consider that the missile has an initial heading angle of 60°, a speed of 500 m / s, is located at the initial point (0,0) m, and the target is located at the coordinate (10000,0) m. Set the attack time to 25, 30, 40, and 50 seconds, and perform modeling and neural network generation according to operations S1-S4.

[0146] See also Figure 5A , the trajectory generated by the method proposed in the present invention is consistent with the optimal solution. However, the trajectory generated by the existing ITCG method has a large deviation from the optimal solution, especially when the attack time is long. When the expected attack time is 40 seconds, it can be found that the trajectory generated by the ITCG method is still far away from the target when the remaining flight time is equal to 0. And when the attack time reaches 50 seconds, the error of the remaining flight time estimation becomes negative, which makes the ITCG method inapplicable. In contrast, the FTNOG method proposed in the present invention can accurately hit the target at any expected attack time. Specifically, the control efficiency of ITCG, FTNOG method and optimal control method at different attack times are expressed by J ITCG , J FTNoOG , J * As shown in Table 1:

[0147]

[0148] Table 1 Control efficiency of different guidance laws at different attack times

[0149] As the remaining flight time increases, the control efficiency deviation between the proposed FTNOG method and the optimal solution is much smaller than the error between the ITCG method and the optimal solution.

[0150] In sub-operation S52, the missile motion trajectories at different initial heading angles obtained by the proposed method are compared.

[0151] Similar to sub-operation S51, the missile speed is also considered to be 500m / s, the initial position is set to (-5000,0)m, the target is located at (0,0)m, different initial heading angles are set, and modeling and neural network generation are performed according to operations S1-S4.

[0152] See also Figure 5B , the trajectory (solid line) of the method proposed in the present invention is almost identical to the optimal trajectory (dashed line) when the initial heading angle is large, and the difference is also small when the initial heading angle is small.

[0153] In sub-operation S53, the ability of the proposed method to generate a global solution is demonstrated.

[0154] Consider the initial state of the missile:

[0155]

[0156] The speed is 600m / s and the fixed attack time is 50 seconds. Next, the neural network is generated according to operations S1-S4, and the optimal guidance instructions are generated online using the proposed method to obtain the trajectory of the missile. Figure 5D ,The trajectories generated by the proposed method are completely different from those generated by the nonlinear programming method (NLP). This is because the NLP method only satisfies the necessary conditions for the optimal solution, and the resulting solution cannot guarantee global optimality. The dataset of the proposed method does not include any non-optimal solutions, so the trajectory generated by the trained network is globally optimal.

[0157] In sub-operation S54, the application of the present invention in a salvo attack scenario is demonstrated.

[0158] In this example, it is assumed that there are four missiles, and the initial position of each missile is set to:

[0159]

[0160]

[0161] The missile speeds are: V1 = 300m / s, V2 = 350m / s, V3 = 400m / s, V4 = 450m / s, and the attack time is 100 seconds. Figure 5E, the trajectory of FTNOG control generated by the trained network is consistent with the trajectory of optimal control, and the control amount and the optimal solution are basically coincident; while the ITCG method is not applicable to the interception of missiles 2 and 3. For missile 4, the trajectory given by it is quite different from the optimal solution. In addition, it can be found that the traditional proportional guidance cannot control the attack time. Therefore, the method proposed in the present invention has excellent performance in the case where the attack time needs to be controlled.

[0162] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. An online generation method of an optimal guidance law for a fixed attack time, characterized in that: The steps include: S1, for the problem of attacking a stationary target under the constraint of attack time, a nonlinear kinematic model of the missile is established in the Cartesian coordinate system, and the optimal control problem corresponding to the problem of attacking a stationary target is established; The S1 process is as follows: The target location is defined as the origin of the coordinate system Oxy, with the x-axis pointing east and the y-axis pointing north; (x, y) is the coordinate of the missile in the coordinate system Oxy, θ∈[0,2π] is the angle between the x-axis and the missile velocity V, that is, the heading angle of the missile, and the counterclockwise direction is defined as positive; the missile velocity V is normalized to 1, and the nonlinear kinematic model of the missile is expressed as: Where t>0 is the time, is the control quantity of the missile, indicating its lateral acceleration; For a given initial position and a fixed attack time t f >0, the optimal control problem corresponding to the stationary target attack problem is described as: by controlling u(·), the missile's initial state (x0, y0, θ0) at time t = 0 is transferred to the state at time t = t f The target position at the moment is (0,0), where the terminal heading angle of the missile is free and the control efficiency is minimized, expressed as: S2, derive the first-order necessary conditions for the optimal control problem, including determining the Hamiltonian function H according to the Pontryagin maximum principle, determining the co-state equation and the control equation, and giving the transversal condition; The S2 specifically includes the following steps: 2.1) Determine the Hamiltonian function H where p x 、p y 、p θ are the covariates of x, y, and θ, respectively, and p 0 is a nonpositive scalar; 2.2) According to the Pontryagin maximum principle, determine the co-state equation The governing equation is: According to formula (3), we have: u(t)=p θ (t),t∈[0,t f ] (4) Since the terminal heading angle is free, the transversal condition is satisfied: p θ (t f )=0 (5) In addition, the terminal position of the missile must meet the following requirements: (x(t f ),y(t f ))=(0,0) (6) According to equations (5) and (6), integrating the third equation of equation (2), we have: p θ (t)=p x (t)y(t)-p y (t)x(t),t∈[0,t f ] (7); S3, using the geometric properties of the optimal trajectory, proposes a new optimality condition; S3 proposes a new optimality condition, specifically: Given in [0, t f ], if there exists any extreme value trajectory (x(t), y(t), θ(t)) on So that the velocity vector Parallel to the line of sight, that is, the viewing angle is 0 or π: Then the corresponding extreme value trajectory is not an optimal trajectory; S4, based on the first-order necessary conditions in S2 and the optimality conditions in S3, a parameterized system is established so that its solution space is equivalent to the solution space of the optimal control problem; using the idea of ​​integration, without directly solving the optimal control problem, a data set reflecting the mapping between the missile state and the remaining flight time and the optimal guidance law is generated; The establishment of the parameterized system in S4 comprises the following steps: Definition about the vector [p x , p y ]'s variable α Define β∈[-π,π] p x =αcosβ,p y =αsinβ (8) Therefore, formula (7) can be written as: p θ (t)=α[y(t)cosβ-x(t)sinβ](9) Define a parameterized system: in Θ∈[0, 2π], for simplicity of notation, let the three-dimensional matrix be: For the initial state: (X(0,α,β),Y(0,α,β),Θ(0,α,β))=(0,0,0) Define λ∈[0, 2π] as the angle between the line of sight and the x-axis, with counterclockwise being positive, and then obtain the nonlinear kinematic equation in the polar coordinate system: Where r ≥ 0 is the Euclidean distance between the missile and the target, expressed as: Define σ∈[-π, π] as the angle between the line of sight and the velocity vector V, which is called the viewing angle, and is positive in the clockwise direction; because σ=π+λ-θ, equation (11) can be written as: The generation of the data set reflecting the mapping between the missile state, the remaining flight time and the optimal guidance law in S4 comprises the following steps: Use C(r c , σ c , t g ) indicates the current state (r c , σ c ) is the optimal control instruction, where t g > 0, indicating the remaining flight time, for t∈[0, t f ], given any optimal trajectory (r(t), σ(t)), if u(t) represents the corresponding optimal control, then C(r(t), σ(t), t f -t)=u(t); According to the definition of optimal control, generating the optimal solution in real time is equivalent to calculating (r c , σ c , t g )→C(r c , σ c , t g ) mapping; Before training a neural network, we need to generate a representation (r c , σ c , t g ) and C(r c , σ c , t g ) relationship, represented as: Here T(α,β)=min{t>0|cos[∑(t,α,β)]=1} (15) Use the mapping (R(t, α, β), ∑(t, α, β), t) → U(t, α, β) to generate the representation (r c , σ c , t g ) and C(r c , σ c , t g ), (α, β)∈Λ and t∈[0,T(α,β)]; In order to prevent T(α,β) from being too large and causing difficulty in generating data, let in is a finite number; all sampled data will eventually be included in the dataset In the data set The neural network that generates the optimal guidance law in real time according to the current state of the missile is trained by the data. S5, using a feedforward neural network to train the data set, the resulting neural network generates an optimal guidance law in real time according to the current state of the missile, so as to control the attack time of the stationary target; The S5 uses a feedforward neural network to train the data set, including: the neural network used has three hidden layers, each hidden layer contains 30 neurons, and the activation function uses the tanh function; the data set is trained using Pytorch Conduct training.

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