Method for on-line generation of nonlinear optimal control commands for aircraft satisfying time constraints

Abstract

The application discloses a kind of aircraft nonlinear optimal control command on-line generation method satisfying time constraint, belong to trajectory optimization field.The application is aimed at the control problem of accurate arrival target position of variable speed aircraft under strict time constraint, establishes the nonlinear motion model considering the time-varying characteristics of speed, constructs optimal control problem;Inequality constraint is converted into regularization equality constraint problem by saturation function processing;Based on the Pontryagin maximum principle, the first-order necessary condition is derived;Parameterized system parameterized by terminal parameter is constructed, and data set is generated by index reduction;Millisecond-level on-line generation of optimal control command is realized by using feedforward neural network training.The application does not need linearization assumption, and simultaneously solves the model mismatch problem caused by time-varying speed and the on-line calculation burden problem of nonlinear optimal control, can satisfy accurate arrival time constraint, and guarantee performance functional optimal.

Description

Technical Field

[0001] This invention belongs to the field of aircraft trajectory optimization and relates to an online method for generating nonlinear optimal control commands for aircraft that satisfy time constraints. Background Technology

[0002] Underactuated systems refer to propulsion systems where the number of independent control inputs is less than the number of system degrees of freedom, such as fixed-wing aircraft, quadcopter drones, and surface vessels. Enabling underactuated systems to reach predetermined positions under precise time constraints in complex mission environments is a key technology for applications such as collaborative operations among various aircraft swarms and robots. The core of such tasks lies in designing control commands for the controlled object that enable precise arrival under strict time constraints, which poses a challenge to traditional control command generation methods.

[0003] Existing methods for generating control commands to achieve aircraft arrival time constraints mainly revolve around improvements to classical proportional guidance laws or designs based on modern control theory. Early representative results, such as ITCG (Impact-Time Control Guidance), achieve time control by introducing a correction term based on the remaining flight time estimation error into a linearized model (see Jeon I.-S., Lee, J.-I., and Tahk, M.-J., “Impact-Time -Control Guidance Law for Anti-Ship Missiles,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 2, 2006, pp. 260–266). However, these methods and their subsequent improvements are generally based on small-angle assumptions and constant-velocity assumptions. The former leads to performance degradation under large initial heading errors; while the latter ignores the time-varying velocity characteristics prevalent in real physical systems, such as velocity variations caused by atmospheric drag, thrust changes, or energy management. Applying control commands designed based on a constant-speed model directly to a variable-speed aircraft (i.e., an aircraft whose speed cannot be considered a constant and whose dynamic speed changes due to thrust, drag, gravity, etc. must be taken into account) will severely affect its terminal time accuracy and trajectory optimality due to model mismatch.

[0004] To overcome the limitations of linearized models, control command generation methods based on nonlinear optimal control theory have been developed. These methods, by solving optimal control problems that satisfy end-point constraints (including position and time), can optimize performance functionals (such as minimizing control energy) under more general nonlinear models. The work of Chen and Shima et al. systematically derived the necessary conditions for such problems using the Pontryagin maximum principle (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). However, to simplify the solution process, most studies on nonlinear optimal control command generation still treat the velocity of the moving body as a constant parameter. For example, although the study by Wang et al. achieved rapid generation of control commands, its core model still did not consider velocity changes (see Wang, K., Chen, Z., Wang, H., Li, J., and Shao, X., “Nonlinear Optimal Guidance for Intercepting Stationary Targets with Impact-Time Constraints,” Journal of Guidance, Control, and Dynamics, Vol. 45, No. 9, 2022, pp. 1614–1626). This greatly limits its application in time-varying velocity systems (such as powered flight phases, reentry gliding phases, or moving bodies subject to varying environmental drag).

[0005] For variable-speed aircraft, existing research has attempted to compensate for the effects of velocity variations by designing time-varying gains or specific feedback structures (such as sliding mode control) (see Zhou, J., Wang, Y., and Zhao, B., “Impact-time-control guidance law for missile with time-varying velocity,” Mathematical Problems in Engineering, Vol. 2016, 2016). However, these methods typically do not systematically address variable-speed models from the perspective of optimal control, and their optimality cannot be strictly guaranteed. On the other hand, although it is theoretically possible to directly handle nonlinear optimal control problems of variable-speed models, their numerical solutions (such as the target-shooting method and pseudospectral method) usually involve complex iterative calculations, which cannot meet the requirements for online real-time generation of control commands in high-dynamic environments, greatly limiting their practicality. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides an online method for generating nonlinear optimal control commands for aircraft that satisfies time constraints. Based on Pontryagin's maximum principle, this invention derives the necessary conditions for the nonlinear optimal control problem; then, it constructs a parameterized system whose solution space is consistent with that of the original optimal control problem; finally, it trains a feedforward neural network using an offline data-driven approach to achieve millisecond-level online generation of optimal control commands.

[0007] To achieve the above objectives, this invention proposes an online method for generating nonlinear optimal control commands for an aircraft that satisfies time constraints. These control commands are used to control a variable-speed aircraft to reach a target position under strict time constraints. The method includes the following steps:

[0008] 1) Establish a nonlinear motion model of a variable-speed aircraft considering time-varying velocity and generalized drag, clarify the constraints of control and state variables, and construct the corresponding optimal control problem;

[0009] 2) By introducing a saturation function to handle the inequality constraints between the state variables and the control variables, the original optimal control problem is transformed into a regularized optimal control problem containing only equality constraints.

[0010] 3) Based on the Pontryagin maximum principle, derive the necessary conditions for the regularized optimal control problem, including constructing the Hamiltonian function and deriving the costate equation and transverse condition;

[0011] 4) Based on the aforementioned necessity conditions, a parameterized system is constructed to ensure that its solution space is consistent with the solution space of the original optimal control problem. The differential-algebraic equations derived from the aforementioned necessity conditions are converted into an ordinary differential equation system through index reduction. The inverse integration method is used to generate a dataset containing the current state of the variable speed aircraft, the remaining arrival time, and the optimal control command.

[0012] 5) The feedforward neural network is trained based on the dataset. Then, the trained neural network is used to generate the optimal control command in real time according to the current state of the variable speed aircraft and the remaining arrival time, so as to control the variable speed aircraft to arrive at the target position at the preset time.

[0013] Further, in step 1), the nonlinear motion model of the variable speed aircraft is established in a Cartesian coordinate system, with the origin of the coordinate system located at the target position; the state variables include position coordinates, velocity, and trajectory tilt angle, the control variable is the angle of attack, and the performance functional is determined to be the maximum terminal velocity; the variable speed aircraft must satisfy the lower limit constraint of velocity and the range constraint of angle of attack (i.e., the constraint conditions of control variables and state variables).

[0014] The optimal control problem includes system dynamics equations, boundary conditions, path constraints, and performance functionals. The system dynamics equations describe the dynamic relationship between the state variables, control variables, and aircraft parameters of the variable-speed aircraft. The boundary conditions include the specific state of the aircraft at the initial moment and the position constraints of the terminal state. The path constraints are inequality constraints between the state variables and the control variables. The performance functionals are the optimization objectives used to evaluate the performance of the control commands.

[0015] Furthermore, in step 2), the constraints of the state variables and control variables refer to the lower limit constraint of the speed and the range constraint of the angle of attack of the variable speed aircraft. For the lower limit constraint of speed, it is constructed as a first-order dynamic system by introducing a first auxiliary control variable. For the range constraint of angle of attack, it is expressed as a function of unconstrained control variables by using a saturation function and introducing a second auxiliary control variable.

[0016] In step 2), an additional regularization term containing the first and second auxiliary control variables is added to the performance functional of the optimal control problem, and the regularization term parameter is set, thereby transforming the original optimal control problem into a regularized optimal control problem containing only equality constraints. When the regularization term parameter approaches zero, the performance functional and optimal trajectory of the regularized optimal control problem converge to the original optimal control problem.

[0017] In step 3), the Hamiltonian function is used to characterize the optimality relationship between the system state, control quantity, and costate variables, and the costate equation is used to describe the dynamic law of the evolution of the costate variables over time. The cross-sectional conditions include: the costate variable corresponding to the terminal trajectory inclination angle being free is zero, the costate variable corresponding to the terminal velocity being free is equal to the partial derivative of the terminal cost, and the costate variable corresponding to the terminal auxiliary state variable being free is zero. The optimal control command can be derived from the minimum condition of the Pontryagin maximum principle.

[0018] In step 4), the parameterization system is constructed using terminal parameters (including terminal velocity, terminal trajectory inclination, and costate variables corresponding to the terminal position) as parameters; the remaining flight time is introduced as an inverse integration variable, and a Lagrange manifold satisfying the terminal cross section condition is defined, thereby generating an extreme trajectory family parameterized by the terminal parameters; the index reduction is achieved by differentiating the differential algebraic equations and using the nonsingularity of the Jacobian matrix to transform them into a system of ordinary differential equations; the dataset is obtained by randomly sampling the terminal parameters and integrating and expanding the system of ordinary differential equations, and is represented as a mapping set from the current state and remaining arrival time of the variable speed aircraft to the optimal control command.

[0019] According to a preferred embodiment of the present invention, in step 5), the feedforward neural network adopts a multi-layer hidden layer structure and is trained using a Bayesian regularization algorithm. After the feedforward neural network is trained, the online generation time of the optimal control command is within milliseconds.

[0020] The beneficial effects of this invention are as follows: This invention eliminates the need for linearization assumptions in the aircraft's motion model. It generates a large dataset of optimal trajectories with guaranteed optimality through Hamiltonian phase space parameterization theory. Then, a simple feedforward neural network is trained using a data-driven approach, enabling real-time online generation of optimal control commands under precise arrival time constraints. Compared to traditional methods, this invention simultaneously solves the model mismatch problem caused by time-varying velocity models, the online computational burden of nonlinear optimal control, and the problem of precise control under arrival time constraints. It has significant theoretical value and promising engineering applications for improving the task adaptability and collaborative operation capabilities of various underactuated systems in complex dynamic environments. Attached Figure Description

[0021] Figure 1 This is a flowchart of the online generation method for nonlinear optimal control commands that satisfy time constraints provided by the present invention;

[0022] Figure 2 This is a geometric model schematic diagram of the arrival time constraint problem of the controlled object (such as a variable speed aircraft) involved in the present invention;

[0023] Figure 3This is a comparative diagram of the flight trajectories of an aircraft obtained using different methods under different expected arrival times;

[0024] Figure 4 This is a diagram comparing the flight speeds of an aircraft obtained using different methods under different expected arrival times. Detailed Implementation

[0025] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0026] In this invention, the terms "first," "second," etc. (if present) in the invention and the accompanying drawings are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence.

[0027] Figure 1 This is a flowchart illustrating an online method for generating optimal control commands for a variable-speed aircraft with arrival time constraints, as illustrated in an embodiment of the present invention. Typically, but not limited to, the variable-speed aircraft described in this invention can be an underactuated system such as a powerless glider, a rocket-powered aircraft, or a reentry vehicle. See also... Figure 1 The following is a detailed description of the online generation method for optimal control commands for variable-speed aircraft with fixed arrival times, based on the accompanying drawings, including steps one through five.

[0028] Step 1: Establish a nonlinear motion model of the variable-speed aircraft considering time-varying velocity and generalized drag, clarify the constraints of control and state variables, and construct the corresponding optimal control problem.

[0029] It should be noted that the nonlinear motion model of the variable-speed aircraft in this invention is established in a Cartesian coordinate system, with the origin of the coordinate system located at the target position. The state variables include position coordinates, velocity, and trajectory inclination angle; the control variable is the angle of attack; and the performance functional is determined by the maximum terminal velocity. The variable-speed aircraft must satisfy both a lower velocity limit constraint and an angle of attack range constraint. The optimal control problem to be constructed in this invention includes system dynamics equations, boundary conditions, path constraints, and a performance functional. The system dynamics equations describe the dynamic relationship between the state variables, control variables, and aircraft parameters of the variable-speed aircraft. The boundary conditions include the specific state of the aircraft at the initial moment and the position constraints of the terminal state; the path constraints are inequality constraints between the state variables and the control variables; and the performance functional is the optimization objective used to evaluate the performance of the control commands.

[0030] In this specific embodiment, step one is performed as follows:

[0031] For reference Figure 2 As shown, the origin of the coordinate system Oxh is located at the target position T. The controlled object (e.g., a variable-speed aircraft) in step one is denoted as P, and its position coordinates are... The speed and trajectory inclination are respectively and The nonlinear kinematic equations of the aircraft can then be expressed as:

[0032]

[0033] in, For time, superscript It represents the derivative of a variable with respect to time. For the mass of the aircraft, For lift, The resistance is given by the following formula:

[0034]

[0035] In the formula, The local atmospheric density, For reference area, The lift coefficient, Let be the drag coefficient. The expressions for the lift coefficient and drag coefficient are as follows:

[0036]

[0037] in, For the angle of attack, The slope of the lift line, Zero-lift drag coefficient, This is the induced drag factor. Simultaneously, the aircraft must satisfy the following constraints during flight:

[0038]

[0039] in, , These are the minimum and maximum permissible angles of attack, respectively. This is the lower limit of speed.

[0040] The aircraft dynamics equations are expressed as: , where the state vector is , This is a function on the right-hand side of the dynamic equations. For given initial conditions... and fixed arrival time The nonlinear optimal control problem under arrival time constraints (fixed arrival time) can be described as: There is controllable above This allows the aircraft to start from its initial position. Move to target location In this embodiment, to ensure the aircraft retains significant energy upon reaching the target position, the performance functional is defined as maximizing the terminal velocity. Therefore, the optimal control problem can be defined as:

[0041]

[0042] Step two: The constraints of the state and control variables refer to the lower limit constraint of the speed and the range constraint of the angle of attack of the variable speed aircraft. For the lower limit constraint of speed, a first auxiliary control variable is introduced to construct it as a first-order dynamic system for processing. For the range constraint of angle of attack, a saturation function is used and a second auxiliary control variable is introduced to express it as a function of unconstrained control variables.

[0043] Furthermore, an additional regularization term containing the first and second auxiliary control variables is added to the performance functional of the optimal control problem, and the regularization term parameter is set, thereby transforming the original optimal control problem into a regularized optimal control problem containing only equality constraints. When the regularization term parameter approaches zero, the performance functional and optimal trajectory of the regularized optimal control problem converge to the original optimal control problem.

[0044] In this embodiment, step two includes the following sub-steps:

[0045] (2.1) Using saturation functions to handle state constraints. Since the velocity constraint is a first-order state constraint (see Graichen, K., Kugi, A., Petit, N., and Chaplais, F., “Handling constraints in optimal control with saturation functions and system extension,” Systems & Control Letters, Vol. 59, No. 11, 2010, pp. 671–679), auxiliary state variables are introduced. and define the saturation function. as follows:

[0046]

[0047] in, It is an exponential saturation function used to convert unconstrained auxiliary states. Mapping to satisfy the lower speed limit actual speed Therefore, we can conclude that:

[0048]

[0049] in This is the first newly introduced auxiliary control variable.

[0050] (2.2) Using saturation functions to handle control constraints. The control quantity is written as (see Graichen, K., and Petit, N., “Constructive methods for initialization and handling mixed state-input constraints in optimal control,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 5, 2008, pp. 1334-1343.):

[0051]

[0052] in, It is a saturation function of the Sigmoid type, used to convert the unconstrained second auxiliary control quantity. Mapping to satisfy angle of attack range constraints Actual angle of attack .

[0053] (2.3) Transform the optimal control problem into a regularized equality constraint problem. Add an additional regularization term to the performance functional. And set the regularization parameter to This leads to a new regularized optimal control problem containing only equality constraints:

[0054]

[0055] in, For the regularization performance functional, the first term Maximizing the terminal speed (equivalent to minimizing its negative value), the second term is a regularization term used to ensure the well-posedness of the optimal control problem. At this time, this regularized optimal control problem The performance functional and the optimal trajectory converge to the original optimal control problem. .

[0056] Step three involves deriving the first-order necessity conditions for optimal control based on the Pontryagin maximum principle. This includes constructing the Hamiltonian function H, deriving the costate equations and transversal conditions. The Hamiltonian function characterizes the optimality relationship between the system state, control variables, and costate variables. The costate equations describe the dynamics of the costate variables' evolution over time. The transversal conditions include: the costate variable corresponding to the terminal trajectory inclination angle being free is zero; the costate variable corresponding to the terminal velocity being free is equal to the partial derivative of the terminal cost (-1 in this embodiment); and the costate variable corresponding to the terminal auxiliary state variable being free is zero. The optimal control command can be derived from the minimum conditions of the Pontryagin maximum principle.

[0057] Step 3 includes the following sub-steps:

[0058] (3.1) Determine the Hamiltonian function :

[0059]

[0060] in, For auxiliary control quantity; State vector The corresponding costate variable, superscript Representing the dual space, It is a state manifold The cotangent bundle, whose elements are pairs of states and costates. This constitutes the phase space of the Hamiltonian system; Auxiliary state variables The corresponding costate variables; For Lagrange multipliers, among which Corresponding constraints , Corresponding constraints Extended control variable is defined as follows: .

[0061] (3.2) Deriving the costate equation. According to Pontryagin's maximum principle (see Lev Semenovich Pontryagin. Mathematical theory of optimal processes. Routledge, 2018), the costate equation is:

[0062]

[0063] (3.3) Determine the necessary conditions. Regularized optimal control problem The optimal solution must satisfy:

[0064]

[0065] Mode ,Mode With formula Jointly determine the optimal control and the multiplier Since the terminal trajectory inclination angle of the aircraft is free, the cross-sectional condition is satisfied:

[0066]

[0067] Since the terminal auxiliary state variable is free, its corresponding costate variable satisfies

[0068]

[0069] In addition, the terminal position of the aircraft must meet the following requirements:

[0070]

[0071] The satisfaction formula -Mode trajectory for The extreme value trajectory.

[0072] Step four: Based on the necessity condition, a parameterized system is constructed to ensure its solution space is consistent with that of the original optimal control problem. The differential-algebraic equations are transformed into a system of ordinary differential equations through index reduction, and a dataset containing the current state of the controlled object, remaining arrival time, and optimal control command is generated using the inverse integration method. Specifically, the parameterized system is constructed using terminal parameters (including terminal velocity, terminal trajectory inclination, and co-state variables corresponding to the terminal position); the remaining flight time is introduced as an inverse integration variable, and a Lagrange multi-manifold satisfying the terminal transverse condition is defined, thereby generating a cluster of extreme trajectories parameterized by the terminal parameters; the index reduction transforms the differential-algebraic equations into a system of ordinary differential equations by differentiating the differential-algebraic equations and utilizing the nonsingularity of the Jacobian matrix; the dataset is obtained by randomly sampling the terminal parameters and integrally expanding the system of ordinary differential equations, representing a mapping set from the current state and remaining arrival time of the variable-speed aircraft to the optimal control command.

[0073] Step four further includes the following sub-steps:

[0074] (4.1) Construct a parameterized system.

[0075] Define the minimum Hamiltonian function :

[0076]

[0077] Let the terminal state and costate be respectively and Terminal auxiliary status is Terminal-assisted comode is In the remaining clusters Define submanifolds :

[0078]

[0079] For the reason A parameterized Lagrange manifold, in which From the formula Confirmed. Remaining flight time will be introduced. For any ,make

[0080]

[0081] Represent the solution trajectory of the following Hamiltonian system:

[0082]

[0083] And the initial conditions of the system are .for extreme value trajectory If satisfied Then there is

[0084]

[0085] Define the parameterized extreme value trajectory family as:

[0086]

[0087] make Let be the projection of the cotangent bundle onto the state space, i.e.

[0088]

[0089] Through parameterized extreme value trajectory clusters Random sampling can be used. A large number of extreme value trajectories are generated (see Wang, K., Chen, Z., Wang, H., Li, J., and Shao, X., “Nonlinear optimal guidance for intercepting stationary targets with impact-time constraints,” Journal of Guidance, Control, and Dynamics, Vol. 45, No. 9, 2022, pp. 1614–1626).

[0090] (4.2) Index Reduction. (See Brenan, KE, Campbell, SL and Petzold, LR, 1995. Numerical solution of initial-value problems in differential-algebraicequations. Society for Industrial and Applied Mathematics.) This is a system of first-order differential-algebraic equations, and direct numerical solutions may encounter difficulties such as numerical drift. Therefore, index reduction is used to transform it into a system of ordinary differential equations (ODEs). Let... Expression ,in Differentiating it, we get:

[0091]

[0092] because If it is not singular, then:

[0093]

[0094] Mode With formula Together, they constitute an extended ODE system. The initial conditions of the parameterized system are determined through... Determined, through solving To obtain the initial optimal control and the multiplier Then, by integrating the extended ODE system, a parameterized extreme value trajectory can be generated.

[0095] (4.3) For any given By integrating the above extended ODE system, a unique extreme trajectory can be obtained. and its corresponding extreme value system instructions By specifying within a certain range It can generate a large number of extreme trajectories and construct a system based on flight state and remaining arrival time. To extreme value control command The dataset consisting of mappings is represented as .

[0096] Step 5: Use the dataset generated in Step 4 to train the feedforward neural network. Then, use the trained neural network to generate the optimal control command in real time based on the current state of the variable speed aircraft and the remaining arrival time, so as to control the variable speed aircraft to accurately reach the target position at the preset time.

[0097] Step five includes the following sub-steps:

[0098] (5.1) Feedforward Neural Network Training. MATLAB was used to train the dataset. Training was performed. The neural network used consisted of two hidden layers, each containing 32 neurons, and was trained using the Bayesian regularization algorithm (trainbr).

[0099] (5.2) Test of the effectiveness of the invention method. The neural network was tested using a 12th Gen Intel(R) Core(TM) i7-1260P (2.10GHz) computing platform. Given the aircraft state and remaining flight time, the neural network can give the corresponding control commands within 0.03ms.

[0100] The relevant parameters and constraints of the aircraft are shown in Table 1:

[0101] Table 1 Aircraft Parameters

[0102]

[0103] Follow steps one through five above to model the optimal control problem, generate the dataset, and train the neural network.

[0104] Considering the initial velocity of the aircraft is The initial flight path inclination angle is ,lie in The target location is Set different arrival times of 25, 27.5, 30, 32.5, and 35 seconds. See also... Figure 3The optimal trajectory generated by the method of this invention is compared with the optimal trajectory generated using nonlinear programming (NLP) and VGPN (see Dong, W., Wang, C., Wang, J. and Xin, M., 2023. Varying-gain proportional navigation guidance for precise impact time control. Journal of Guidance, Control, and Dynamics, 46(3), pp.535-552). The results are shown in Table 2.

[0105] Table 2 Terminal velocity and final arrival time of different control commands under different expected arrival times

[0106]

[0107] As shown in Table 2, the trajectory generated by the method of this invention closely matches the optimal solution, while VGPN deviates significantly from the optimal solution. Further, see... Figure 4 VGPN, this method, and nonlinear programming method are used to determine the final velocity at different arrival times. , and The final arrival time is indicated by... , and express.

[0108] As shown in the figure, the online generation method for nonlinear optimal control commands of this invention can satisfy arrival time constraints, can be calculated online in real time, and can guarantee optimal performance functional. This method has significant theoretical value and promising engineering applications for improving the accurate coordination capabilities of various underactuated systems in complex dynamic environments.

[0109] Those skilled in the art will readily understand that the above description is merely 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 within the scope of protection of the present invention.

Claims

1. An online method for generating nonlinear optimal control commands for an aircraft that satisfy time constraints, wherein the control commands are used to control a variable-speed aircraft to reach a target position under strict time constraints, characterized in that, The method includes the following steps: 1) Establish a nonlinear motion model of a variable-speed aircraft considering time-varying velocity and generalized drag, clarify the constraints of control and state variables, and construct the corresponding optimal control problem; 2) By introducing a saturation function to handle the inequality constraints between the state variables and the control variables, the original optimal control problem is transformed into a regularized optimal control problem containing only equality constraints. In step 2), the constraints on the state and control variables refer to the lower speed limit constraint and angle of attack range constraint of the variable-speed aircraft. For the lower speed limit constraint, a first auxiliary control variable is introduced to construct it as a first-order dynamic system for processing. For the angle of attack range constraint, a saturation function is used and a second auxiliary control variable is introduced to express it as a function of an unconstrained control variable. 3) Based on the Pontryagin maximum principle, derive the necessary conditions for the regularized optimal control problem, including constructing the Hamiltonian function and deriving the costate equation and transverse condition; 4) Based on the aforementioned necessity conditions, a parameterized system is constructed to ensure that its solution space is consistent with the solution space of the original optimal control problem. The differential algebraic equations derived from the aforementioned necessity conditions are converted into ordinary differential equations through index reduction. A dataset containing the current state of the variable speed aircraft, the remaining arrival time, and the optimal control command is generated using the inverse integration method. In step 4), the parameterization system is constructed using terminal parameters, including terminal velocity, terminal trajectory tilt angle, and co-state variables corresponding to the terminal position. The remaining flight time is introduced as a back-integral variable, and a Lagrange manifold satisfying the terminal cross-section condition is defined, thereby generating a cluster of extreme trajectories parameterized by the terminal parameters. The index reduction involves differentiating the differential algebraic equations and utilizing the nonsingularity of the Jacobian matrix to transform them into a system of ordinary differential equations. The dataset is obtained by randomly sampling the terminal parameters and integrally expanding the system of ordinary differential equations, representing a mapping set from the current state and remaining arrival time of the variable-speed aircraft to the optimal control command. 5) Train the feedforward neural network based on the dataset; then use the trained neural network to generate the optimal control command in real time according to the current state and remaining arrival time of the variable speed aircraft, so as to control the variable speed aircraft to arrive at the target position at the preset time.

2. The method according to claim 1, characterized in that, In step 1), the nonlinear motion model of the variable speed aircraft is established in a Cartesian coordinate system, with the origin of the coordinate system located at the target position; the state variables include position coordinates, velocity, and trajectory tilt angle, and the control variable is the angle of attack.

3. The method according to claim 1, characterized in that, The optimal control problem includes system dynamics equations, boundary conditions, path constraints, and performance functionals. The system dynamics equations describe the dynamic relationship between the state variables, control variables, and aircraft parameters of the variable-speed aircraft. The boundary conditions include the specific state of the aircraft at the initial moment and the position constraints of the terminal state. The path constraints are inequality constraints between the state variables and the control variables. The performance functionals are the optimization objectives used to evaluate the performance of the control commands.

4. The method according to claim 3, characterized in that, The lift and drag terms in the system dynamics equations are characterized by the lift coefficient and drag coefficient, respectively, and the lift coefficient and drag coefficient satisfy the following: ； in The lift coefficient, For the angle of attack, The slope of the lift line, The drag coefficient, Zero-lift drag coefficient, It is the induced resistance factor.

5. The method according to claim 1, characterized in that, In step 2), an additional regularization term containing the first and second auxiliary control variables is added to the performance functional of the optimal control problem, and the regularization term parameter is set, thereby transforming the original optimal control problem into a regularized optimal control problem containing only equality constraints. When the regularization term parameter approaches zero, the performance functional and optimal trajectory of the regularized optimal control problem converge to the original optimal control problem.

6. The method according to claim 1, characterized in that, In step 3), the Hamiltonian function is used to characterize the optimality relationship between the system state, control quantity, and costate variables; the costate equation is used to describe the dynamic law of the costate variables evolving over time; the cross-sectional conditions include: the costate variable corresponding to the terminal trajectory inclination angle being free is zero; the costate variable corresponding to the terminal velocity being free is equal to the partial derivative of the terminal cost; and the costate variable corresponding to the terminal auxiliary state variable being free is zero; the optimal control command is derived from the minimum condition of the Pontryagin maximum principle.

7. The method according to claim 1, characterized in that, In step 5), the feedforward neural network adopts a multi-layer hidden layer structure and is trained using the Bayesian regularization algorithm. After the feedforward neural network is trained, the online generation time of the optimal control command is within milliseconds.

8. The method according to claim 1, characterized in that, The variable-speed aircraft include unpowered gliders, rocket-powered aircraft, or reentry vehicles.

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