A multi-constraint closed-loop guidance method for missiles based on Bernstein's method

Abstract

The multi-constraint closed-loop guidance method for missiles based on Bernstein basis belongs to the field of guidance technology. The invention is to realize a real-time and high-computational efficiency multi-constraint guidance method for missiles. The invention establishes a three-dimensional kinematic model of missiles and targets, constructs Bezier curves of state variables of missiles and targets, and control quantities of missiles and targets, and converts the guidance problem of missiles into a nonlinear programming problem under multiple constraints through Bezier curves; solves the constructed nonlinear programming problem under multiple constraints based on sequential quadratic programming algorithm, and constructs a multi-constraint closed-loop guidance method for missiles based on Bernstein basis; takes the terminal velocity of the missile as a performance index, selects N nodes for sampling, converts the optimal control problem into a nonlinear programming problem under multiple constraints through Bezier curves, and then solves it using the SQP algorithm, realizes closed-loop feedback by generating open-loop control online in real time, and obtains a multi-constraint closed-loop guidance method for missiles.

Description

Technical Field

[0001] The invention belongs to the technical field of guidance, and in particular relates to a multi-constraint closed-loop guidance method for a missile based on a Bernstein basis. Background Art

[0002] Guidance law is one of the key technologies for missiles to achieve interception guidance and precision strike. With the complexity and changeability of modern battlefield environments and the continuous improvement of missile defense systems, the requirements for terminal guidance are becoming higher and higher. On the basis of requiring the missile to miss the target as little as possible, it is also hoped that the missile can hit the target with the maximum terminal velocity, thereby achieving the maximum range extension of the missile.

[0003] The design problem of missile guidance methods is essentially a nonlinear optimization problem. Numerical methods for solving optimal control problems are usually divided into indirect methods and direct methods. The former analytically derives the necessary conditions for local optimality, while the latter transforms the original infinite-dimensional problem into a finite-dimensional problem and solves the resulting nonlinear programming (NLP) problem. The transcription process usually uses a set of appropriate polynomials to represent the state, the control, or both. With the improvement of numerical computing capabilities, direct methods have shown increasing advantages in solving complex, highly nonlinear optimal control problems.

[0004] In recent years, pseudo-spectral methods based on Lagrange polynomials have become the mainstream method for nonlinear optimization problems. The advantages of pseudo-spectral methods are wide applicability and the ability to provide high numerical accuracy, especially when solving certain smooth, low-dimensional optimization problems. However, when encountering situations where the state and control quantities change dramatically, the application of pseudo-spectral methods to solve problems may cause the state and control quantities to show oscillations that exceed the expected limits. The open-loop guidance method lacks real-time feedback on environmental changes or system state changes during execution, and cannot adjust the navigation strategy in time to cope with external interference or system errors, which may lead to the accumulation of navigation errors. In addition, the flexibility and adaptability are poor in the face of complex environments, making it difficult to achieve accurate navigation and control. A more flexible and robust closed-loop control method is needed to deal with complex situations. Therefore, how to design a missile multi-constraint guidance method with low solution difficulty, high computational efficiency, and real-time performance is an urgent problem to be solved in the field of missile guidance. Summary of the invention

[0005] The problem to be solved by the present invention is to realize a missile multi-constraint guidance method with real-time performance and high computational efficiency, and a missile multi-constraint closed-loop guidance method based on Bernstein basis is proposed.

[0006] To achieve the above object, the present invention is implemented through the following technical solutions:

[0007] A multi-constraint closed-loop guidance method for a missile based on Bernstein basis comprises the following steps:

[0008] S1. Establish a three-dimensional kinematic model of the missile and the target, set the state variables of the missile and the target, and the control quantities of the missile and the target;

[0009] S2. Construct the Bezier curves of the state variables of the missile and the target, and the control quantities of the missile and the target, and transform the guidance problem of the missile into a nonlinear programming problem under multiple constraints through the Bezier curves;

[0010] S3. Set up a nonlinear programming problem under multiple constraints constructed in step S2 based on a sequential quadratic programming SQP algorithm to construct a missile multi-constraint closed-loop guidance method based on the Bernstein basis;

[0011] S4. Select N nodes for sampling, and use the method of step S2 to transform the optimal control problem into a nonlinear programming problem under multiple constraints through the Bezier curve, and then use the sequential quadratic programming SQP algorithm based on step S3 to solve it. By generating open-loop control online in real time, closed-loop feedback is achieved to obtain the optimal control amount of the missile and the target.

[0012] Furthermore, the specific implementation method of step S1 includes the following steps:

[0013] S1.1. Establish a three-dimensional kinematic model of the missile. The specific expression is as follows:

[0014]

[0015] in, is the rate of change of missile speed with respect to time, g is the acceleration of gravity, V is the missile speed, x m ,y m ,z m are the coordinates of the missile's x-axis, y-axis, and z-axis in the inertial coordinate system, n x ,n y ,n z are the components of the missile's overload vector in the ballistic coordinate system on the x-axis, y-axis, and z-axis, respectively; θ is the ballistic inclination angle, is the rate of change of the trajectory inclination with respect to time, is the ballistic deflection angle, is the rate of change of trajectory angle with time, is the rate of change of missile mass with respect to time, m is the missile mass, m c is the fuel mass flow rate per second, are the rates of change of the x-axis, y-axis, and z-axis coordinates of the missile in the inertial coordinate system with respect to time;

[0016] n x ,n y ,n z The expression is as follows:

[0017]

[0018]

[0019] Among them, P is the missile engine thrust, X is the aerodynamic drag, Y is the lift, Z is the lateral force, α and β are the missile's angle of attack and sideslip angle respectively;

[0020] X, Y, Z are proportional to the dynamic pressure and the characteristic area of ​​the missile. The specific expressions are as follows:

[0021] X=c x Q

[0022] Y=c y Q

[0023] Z=c z Q

[0024]

[0025] Among them, c x ,c y ,c z are the aerodynamic drag coefficient, lift coefficient, and lateral force coefficient, respectively; S is the characteristic area of ​​the missile; q is the aerodynamic force acting on the missile and the dynamic pressure of the incoming flow; and ρ is the air density at the height of the missile;

[0026] S1.2. Establish a three-dimensional kinematic model of the non-maneuvering target. The specific expression is as follows:

[0027]

[0028] in, V is the rate of change of ballistic inclination with respect to time, T is the speed of the target, x T ,y T ,z T are the x-axis, y-axis, and z-axis coordinates of the target in the inertial coordinate system, θ T is the ballistic inclination angle, is the ballistic deflection angle, is the rate of change of trajectory angle with time, is the component of the target's overload in the ballistic coordinate system y, is the component of the target's overload in the ballistic coordinate system z, is the rate of change of the x-axis, y-axis, and z-axis coordinates of the target in the inertial coordinate system with respect to time;

[0029] S1.3. Establish a nonlinear optimal control model for the missile, including constraints, optimization indicators, and optimal problem description;

[0030] S1.3.1. Constraints include missile mass constraints, overload constraints, and terminal constraints;

[0031] Based on the fact that the mass of the missile decreases as the engine fuel is consumed during the initial guidance stage, and the mass of the missile remains unchanged after the engine stops running, the specific expression for setting the mass constraint of the missile is as follows:

[0032]

[0033] in, is the time when the engine stops running, m c0 is the fuel mass flow rate per second during the engine running time;

[0034] The constraints of the missile's overload components in the ballistic coordinate system are set as follows:

[0035] |n x |≤n xmax

[0036] |n y |≤n ymax

[0037] |n z |≤n zmax

[0038] Among them, n xmax ,n ymax ,n zmax are the maximum absolute values ​​of the components of the missile's overload vector in the ballistic coordinate system on the x-axis, y-axis, and z-axis respectively;

[0039] The maximum allowable miss distance of the missile is R max , set the terminal constraints of the missile to:

[0040]

[0041] Among them, t f The final moment when the missile completes guidance;

[0042] S1.3.2. When designing the interception trajectory of a missile, the terminal velocity of the missile is used as the optimization index J, which is expressed as:

[0043] J=min{-[V(t f )] 2};

[0044] S1.3.3. The optimal problem is described as the state variables of the missile and the target, and the control quantities of the missile and the target, expressed as:

[0045]

[0046] d(x,u,t)≤0

[0047] x(t0)=x0

[0048] Among them, x(t) is the state variable of the missile and the target, u(t) is the overload of the missile and the target, t0 represents the initial time, x0 is the initial value of the state variable of the missile and the target; d(x,u,t) is the inequality constraint;

[0049]

[0050]

[0051] Furthermore, the specific implementation method of step S2 includes the following steps:

[0052] S2.1. Setting the Bezier curve is a curve generated by a set of control points and Bernstein basis functions, where the n-order Bernstein basis polynomial B i,n Defined as:

[0053]

[0054] S2.2. Since the quadrature node τ is defined in the interval [-1,1], the Bernstein polynomial is redefined in the interval [-1,1], and the expression is:

[0055]

[0056] S2.3. Select the Bernstein polynomial defined in step S2.2 as the interpolation basis function, and transform the state variables x(t) of the missile and the target and the control variables u(t) of the missile and the target into Bezier curves, expressed as:

[0057]

[0058] Among them, x j (t),u j (t) are respectively the time domain [t j ,t j+1 ] on the state variables and control variables, x sj ,u sj To find the function value at the integration node, X j (τ),U j (τ) is the Bezier curve corresponding to the state variable and control quantity.

[0059] Furthermore, the specific implementation method based on the sequential quadratic programming (SQP) algorithm in step S3 includes the following steps:

[0060] S3.1. Set the state variable x k and the control quantity u kConstruct a new state vector The expression is:

[0061]

[0062] Collect and calculate the initial value of the new state vector And the initial positive definite matrix H1 corresponding to the initial value of the new state vector, let k = 1, set the calculation accuracy value to ε: set 0 < ε < Δ, Δ is the trust radius;

[0063] S3.2. Solve the quadratic programming subproblem:

[0064] S3.2.1. Update the main iterative search direction d according to the solution result. The quadratic programming problem is as follows:

[0065]

[0066] in, for The gradient of is the equality constraint in the optimal control problem, is the inequality constraint in the optimal control problem;

[0067] S3.2.2. Set the approximate solution obtained from the subproblem (d k ,λ k ) must satisfy the KKT condition of the nonlinear programming problem to be solved, where d k is the main search direction, λ k is the step length;

[0068] S3.2.3. Perform calculations based on the settings in steps S3.2.1 and S3.2.2, and then make conditional judgments. When ||d k When ||≤ε, stop calculating and find the optimal solution to the problem Otherwise proceed to the next step;

[0069] S3.2.4. Make a judgment, when set up Use the quasi-Newton formula to calculate the positive definite matrix H k+1 , and set k=k+1, then go to step S3.2.3, otherwise, go to the next step;

[0070] S3.2.5. Solving the one-dimensional optimal problem Get the optimal step size λ k ,make Update the positive definite matrix H according to the quasi-Newton formula k , and obtain the new positive definite matrix H k+1 , let k=k+1, and go to step S3.2.3.

[0071] Furthermore, the specific implementation method of step S4 includes the following steps:

[0072] S4.1. Select N nodes for sampling, set the first sampling time to t1, and calculate the initial value of the state variable obtained from the collected data as x0. Use the method in step S2 to transform the optimal control problem into a nonlinear programming problem under multiple constraints through the Bezier curve, and then use the sequential quadratic programming SQP algorithm in step S3 to solve it and calculate the open-loop optimal control in the interval [t0, t1]

[0073] S4.2. Set i = 1, based on the result obtained in step S4.1 The state variable at time t1 is calculated as x1=x(t1);

[0074] S4.3. Setting t i+1 =t i +Δt, Δt is the optimization calculation time, in the interval [t i ,t i+1 ], with x i As the initial value, Acting on [t i ,t i+1 ],calculate Then based on the obtained Calculate x i+1 , t∈[t i ,t f ];

[0075] S4.4. Repeat iterative step S4.3 until t is the final moment when the missile completes the guidance, thus realizing the multi-constraint closed-loop guidance of the missile.

[0076] Beneficial effects of the present invention:

[0077] The multi-constraint closed-loop guidance method for missiles based on the Bernstein basis described in the present invention solves the problem that traditional open-loop control cannot provide real-time feedback. By implementing closed-loop guidance through continuous open-loop control, feedback can be given on real-time system state changes and other situations, thereby achieving optimal control of each time period. When the external environment changes, the control quantity can be replanned in real time through new information, the ballistic trajectory can be adjusted, the solution accuracy and calculation efficiency can be improved, and the method has strong applicability in actual scenarios. At the same time, the application of the Bernstein basis can eliminate the Gibbs phenomenon when the state quantity and the control quantity change sharply, and achieve the smoothing of the Bezier curve. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 A flowchart of a Bernstein-based missile multi-constraint closed-loop guidance method according to the present invention;

[0079] Figure 2 This is a flow chart of the SQP algorithm based on sequential quadratic programming of the present invention;

[0080] Figure 3 The invention realizes the closed-loop control process curve by continuous online real-time open-loop optimal control. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical solution and advantages of the present invention more clear, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention, that is, the specific embodiments described are only part of the embodiments of the present invention, rather than all of the specific embodiments. The components of the specific embodiments of the present invention described and shown in the drawings herein can be arranged and designed in various different configurations, and the present invention can also have other embodiments.

[0082] Therefore, the following detailed description of the specific 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 the selected specific embodiments of the present invention. Based on the specific embodiments of the present invention, all other specific embodiments obtained by those skilled in the art without making creative work are within the scope of protection of the present invention.

[0083] In order to further understand the content, features and effects of the present invention, the following specific implementation methods are given as examples, and the attached Figure 1 -Attached Figure 3 The detailed instructions are as follows: Specific implementation method one:

[0085] A multi-constraint closed-loop guidance method for a missile based on Bernstein basis comprises the following steps:

[0086] S1. Establish a three-dimensional kinematic model of the missile and the target, set the state variables of the missile and the target, and the control quantities of the missile and the target;

[0087] Furthermore, the specific implementation method of step S1 includes the following steps:

[0088] S1.1. Establish a three-dimensional kinematic model of the missile. The specific expression is as follows:

[0089]

[0090]

[0091] in, is the rate of change of missile speed with respect to time, g is the acceleration of gravity, V is the missile speed, x m ,y m,z m are the coordinates of the missile's x-axis, y-axis, and z-axis in the inertial coordinate system, n x ,n y ,n z are the components of the missile's overload vector in the ballistic coordinate system on the x-axis, y-axis, and z-axis, respectively; θ is the ballistic inclination angle, is the rate of change of the trajectory inclination with respect to time, is the ballistic deflection angle, is the rate of change of trajectory angle with time, is the rate of change of missile mass with respect to time, m is the missile mass, m c is the fuel mass flow rate per second, are the rates of change of the x-axis, y-axis, and z-axis coordinates of the missile in the inertial coordinate system with respect to time;

[0092] n x ,n y ,n z The expression is as follows:

[0093]

[0094] Among them, P is the missile engine thrust, X is the aerodynamic drag, Y is the lift, Z is the lateral force, α and β are the missile's angle of attack and sideslip angle respectively;

[0095] X, Y, Z are proportional to the dynamic pressure and the characteristic area of ​​the missile. The specific expressions are as follows:

[0096] X=c x Q

[0097] Y=c y Q

[0098] Z=c z Q

[0099]

[0100] Among them, c x ,c y ,c z are the aerodynamic drag coefficient, lift coefficient, and lateral force coefficient, respectively; S is the characteristic area of ​​the missile; q is the aerodynamic force acting on the missile and the dynamic pressure of the incoming flow; and ρ is the air density at the height of the missile;

[0101] S1.2. Establish a three-dimensional kinematic model of the non-maneuvering target. The specific expression is as follows:

[0102]

[0103] in, V is the rate of change of ballistic inclination with respect to time, T is the speed of the target, x T ,y T ,z T are the x-axis, y-axis, and z-axis coordinates of the target in the inertial coordinate system, θ T is the ballistic inclination angle, is the ballistic deflection angle, is the rate of change of trajectory angle with time, is the component of the target's overload in the ballistic coordinate system y, is the component of the target's overload in the ballistic coordinate system z, is the rate of change of the x-axis, y-axis, and z-axis coordinates of the target in the inertial coordinate system with respect to time;

[0104] S1.3. Establish a nonlinear optimal control model for the missile, including constraints, optimization indicators, and optimal problem description;

[0105] S1.3.1. Constraints include missile mass constraints, overload constraints, and terminal constraints;

[0106] Based on the fact that the mass of the missile will decrease as the engine fuel is consumed in the initial guidance stage, and the mass of the missile remains unchanged after the engine stops running, the mass constraint m of the missile is set c The specific expression is as follows:

[0107]

[0108] in, is the time when the engine stops running, m c0 is the fuel mass flow rate per second during the engine running time;

[0109] The constraints of the missile's overload components in the ballistic coordinate system are set as follows:

[0110] |n x |≤n xmax

[0111] |n y |≤n ymax

[0112] |n z |≤n zmax

[0113] Among them, n xmax ,n ymax ,n zmax are the maximum absolute values ​​of the components of the missile's overload vector in the ballistic coordinate system on the x-axis, y-axis, and z-axis respectively;

[0114] The maximum allowable miss distance of the missile is Rmax , set the terminal constraints of the missile to:

[0115]

[0116] Among them, t f The final moment when the missile completes guidance;

[0117] S1.3.2. When designing the interception trajectory of a missile, the terminal velocity of the missile is used as the optimization index J, which is expressed as:

[0118] J=min{-[V(t f )] 2};

[0119] S1.3.3. The optimal problem is described as the state variables of the missile and the target, and the control quantities of the missile and the target, expressed as:

[0120]

[0121] d(x,u,t)≤0

[0122] x(t0)=x0

[0123] Among them, x(t) is the state variable of the missile and the target, u(t) is the overload of the missile and the target, t0 represents the initial time, x0 is the initial value of the state variable of the missile and the target; d(x,u,t) is the inequality constraint;

[0124]

[0125] S2. Construct the Bezier curves of the state variables of the missile and the target, and the control quantities of the missile and the target, and transform the guidance problem of the missile into a nonlinear programming problem under multiple constraints through the Bezier curves;

[0126] Furthermore, the specific implementation method of step S2 includes the following steps:

[0127] S2.1. Setting the Bezier curve is a curve generated by a set of control points and Bernstein basis functions, where the n-order Bernstein basis polynomial B i,n Defined as:

[0128]

[0129] S2.2. Since the quadrature node τ is defined in the interval [-1,1], the Bernstein polynomial is redefined in the interval [-1,1], and the expression is:

[0130]

[0131] S2.3. Select the Bernstein polynomial defined in step S2.2 as the interpolation basis function, and transform the state variables x(t) of the missile and the target and the control variables u(t) of the missile and the target into Bezier curves, expressed as:

[0132]

[0133] Among them, x j (t),u j (t) are respectively the time domain [t j ,t j+1 ] on the state variables and control variables, x sj ,u sj To find the function value at the integration node, X j (τ),U j (τ) is the Bezier curve corresponding to the state variable and control quantity;

[0134] S3. Set up a nonlinear programming problem under multiple constraints constructed in step S2 based on a sequential quadratic programming SQP algorithm to construct a missile multi-constraint closed-loop guidance method based on the Bernstein basis;

[0135] Furthermore, the specific implementation method based on the sequential quadratic programming (SQP) algorithm in step S3 includes the following steps:

[0136] S3.1. Set the state variable x k and the control quantity u k Construct a new state vector The expression is:

[0137]

[0138] Collect and calculate the initial value of the new state vector And the initial positive definite matrix H1 corresponding to the initial value of the new state vector, let k = 1, set the calculation accuracy value to ε: set 0 < ε < Δ, Δ is the trust radius;

[0139] S3.2. Solve the quadratic programming subproblem:

[0140] S3.2.1. Update the main iterative search direction d according to the solution result. The quadratic programming problem is as follows:

[0141]

[0142] in, for The gradient of is the equality constraint in the optimal control problem, is the inequality constraint in the optimal control problem;

[0143] S3.2.2. Set the approximate solution obtained from the subproblem (d k ,λ k ) must satisfy the KKT condition of the nonlinear programming problem to be solved, where d k is the main search direction, λ k is the step length;

[0144] S3.2.3. Perform calculations based on the settings in steps S3.2.1 and S3.2.2, and then make conditional judgments. When ||d k When ||≤ε, stop calculating and find the optimal solution to the problem Otherwise proceed to the next step;

[0145] S3.2.4. Make a judgment, when set up Use the quasi-Newton formula to calculate the positive definite matrix H k+1 , and set k=k+1, then go to step S3.2.3, otherwise, go to the next step;

[0146] S3.2.5. Solving the one-dimensional optimal problem Get the optimal step size λ k ,make Update the positive definite matrix H according to the quasi-Newton formula k , and obtain the new positive definite matrix H k+1 , let k=k+1, and go to step S3.2.3.

[0147] S4. Select N nodes for sampling, and use the method of step S2 to transform the optimal control problem into a nonlinear programming problem under multiple constraints through the Bezier curve, and then use the sequential quadratic programming SQP algorithm based on step S3 to solve it. By generating open-loop control online in real time, closed-loop feedback is achieved to obtain the optimal control amount of the missile and the target.

[0148] Furthermore, the specific implementation method of step S4 includes the following steps:

[0149] S4.1. Select N nodes for sampling, set the first sampling time to t1, and calculate the initial value of the state variable obtained from the collected data as x0. Use the method in step S2 to transform the optimal control problem into a nonlinear programming problem under multiple constraints through the Bezier curve, and then use the sequential quadratic programming SQP algorithm in step S3 to solve it and calculate the open-loop optimal control in the interval [t0, t1]

[0150] S4.2. Set i = 1, based on the result obtained in step S4.1 The state variable at time t1 is calculated as x1=x(t1);

[0151] S4.3. Setting t i+1 =t i +Δt, Δt is the optimization calculation time, in the interval [t i ,t i+1 ], with x i As the initial value, Acting on [t i ,t i+1 ],calculate Then based on the obtained Calculate x i+1 , t∈[t i ,t f ];

[0152] S4.4. Repeat iterative step S4.3 until t is the final moment when the missile completes the guidance, thus realizing the multi-constraint closed-loop guidance of the missile.

[0153] It should be noted that relational terms such as "first" and "second" are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, the elements defined by the sentence "comprise a ..." do not exclude the existence of other identical elements in the process, method, article or device including the elements.

[0154] Although the present application has been described above with reference to specific embodiments, various modifications may be made thereto and parts thereof may be replaced with equivalents without departing from the scope of the present application. In particular, as long as there is no structural conflict, the various features in the specific embodiments disclosed in the present application may be used in combination with each other in any manner, and the fact that these combinations are not exhaustively described in this specification is only for the sake of omitting space and saving resources. Therefore, the present application is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims

1. A multi-constraint closed-loop guidance method for a missile based on the Bernstein basis, characterized in that: The steps include: S1. Establish a three-dimensional kinematic model of the missile and the target, set the state variables of the missile and the target, and the control quantities of the missile and the target; S2. Construct the Bezier curves of the state variables of the missile and the target, and the control quantities of the missile and the target, and transform the guidance problem of the missile into a nonlinear programming problem under multiple constraints through the Bezier curves; The specific implementation method of step S2 includes the following steps: S2.

1. Setting the Bezier curve is a curve generated by a set of control points and Bernstein basis functions, where the n-order Bernstein basis polynomial B i,n Defined as: S2.

2. Since the quadrature node τ is defined in the interval [-1,1], the Bernstein polynomial is redefined in the interval [-1,1], and the expression is: S2.

3. Select the Bernstein polynomial defined in step S2.2 as the interpolation basis function, and transform the state variables x(t) of the missile and the target and the control variables u(t) of the missile and the target into Bezier curves, expressed as: Among them, x j (t),u j (t) are respectively the time domain [t j ,t j+1 ] on the state variables and control variables, x sj ,u sj To find the function value at the integration node, X j (τ),U j (τ) is the Bezier curve corresponding to the state variable and control quantity; S3. Set up a nonlinear programming problem under multiple constraints constructed in step S2 based on a sequential quadratic programming SQP algorithm to construct a missile multi-constraint closed-loop guidance method based on the Bernstein basis; The specific implementation method based on the sequential quadratic programming (SQP) algorithm in step S3 includes the following steps: S3.

1. Set the state variable x k and the control quantity u k Construct a new state vector The expression is: Collect and calculate the initial value of the new state vector And the initial positive definite matrix H1 corresponding to the initial value of the new state vector, let k = 1, set the calculation accuracy value to ε: set 0 < ε < Δ, Δ is the trust radius; S3.

2. Solve the quadratic programming subproblem: S3.2.

1. Update the main iterative search direction d according to the solution result. The quadratic programming problem is as follows: in, for The gradient of is the equality constraint in the optimal control problem, is the inequality constraint in the optimal control problem; S3.2.

2. Set the approximate solution obtained from the subproblem (d k ,λ k ) must satisfy the KKT condition of the nonlinear programming problem to be solved, where d k is the main search direction, λ k is the step length; S3.2.

3. Perform calculations based on the settings in steps S3.2.1 and S3.2.2, and then make conditional judgments. When ||d k When ||≤ε, stop calculating and find the optimal solution to the problem Otherwise proceed to the next step; S3.2.

4. Make a judgment, when set up Use the quasi-Newton formula to calculate the positive definite matrix H k+1 , and set k=k+1, then go to step S3.2.3, otherwise, go to the next step; S3.2.

5. Solving the one-dimensional optimal problem Get the optimal step size λ k ,make Update the positive definite matrix H according to the quasi-Newton formula k , and obtain the new positive definite matrix H k+1 , let k=k+1, and go to step S3.2.3; S4. Select N nodes for sampling, and use the method of step S2 to transform the optimal control problem into a nonlinear programming problem under multiple constraints through the Bezier curve, and then use the sequential quadratic programming SQP algorithm based on step S3 to solve it, and realize closed-loop feedback by generating open-loop control online in real time to obtain the optimal control amount of the missile and the target; The specific implementation method of step S4 includes the following steps: S4.

1. Select N nodes for sampling, set the first sampling time to t1, and calculate the initial value of the state variable obtained from the collected data as x0. Use the method in step S2 to transform the optimal control problem into a nonlinear programming problem under multiple constraints through the Bezier curve, and then use the sequential quadratic programming SQP algorithm in step S3 to solve it and calculate the open-loop optimal control in the interval [t0, t1] S4.

2. Set i = 1, based on the result obtained in step S4.1 The state variable at time t1 is calculated as x1=x(t1); S4.

3. Setting t i+1 =t i +Δt, Δt is the optimization calculation time, in the interval [t i ,t i+1 ], with x i As the initial value, Act on [t i ,t i+1 ],calculate Then based on the obtained Calculate x i+1 , t∈[t i ,t f ]; S4.

4. Repeat iterative step S4.3 until t is the final moment when the missile completes the guidance, thus realizing the multi-constraint closed-loop guidance of the missile.

2. A missile multi-constraint closed-loop guidance method based on Bernstein basis according to claim 1, characterized in that: The specific implementation method of step S1 includes the following steps: S1.

1. Establish a three-dimensional kinematic model of the missile. The specific expression is as follows: in, is the rate of change of missile speed with respect to time, g is the acceleration of gravity, V is the missile speed, x m ,y m ,z m are the coordinates of the missile's x-axis, y-axis, and z-axis in the inertial coordinate system, n x ,n y ,n z are the components of the missile's overload vector in the ballistic coordinate system on the x-axis, y-axis, and z-axis, respectively; θ is the ballistic inclination angle, is the rate of change of the trajectory inclination with respect to time, is the ballistic deflection angle, is the rate of change of trajectory angle with time, is the rate of change of missile mass with respect to time, m is the missile mass, m c is the fuel mass flow rate per second, are the rates of change of the x-axis, y-axis, and z-axis coordinates of the missile in the inertial coordinate system with respect to time; n x ,n y ,n z The expression is as follows: Among them, P is the missile engine thrust, X is the aerodynamic drag, Y is the lift, Z is the lateral force, α and β are the missile's angle of attack and sideslip angle respectively; X, Y, Z are proportional to the dynamic pressure and the characteristic area of ​​the missile. The specific expressions are as follows: X=c x qS Y=c y qS Z=c z qS Among them, c x ,c y ,c z are the aerodynamic drag coefficient, lift coefficient, and lateral force coefficient, respectively; S is the characteristic area of ​​the missile; q is the aerodynamic force acting on the missile and the dynamic pressure of the incoming flow; and ρ is the air density at the height of the missile; S1.

2. Establish a three-dimensional kinematic model of the non-maneuvering target. The specific expression is as follows: in, V is the rate of change of ballistic inclination with respect to time, T is the speed of the target, x T ,y T ,z T are the x-axis, y-axis, and z-axis coordinates of the target in the inertial coordinate system, θ T is the ballistic inclination angle, is the ballistic deflection angle, is the rate of change of trajectory angle with time, n yT is the component of the target's overload in the ballistic coordinate system y, is the component of the target's overload in the ballistic coordinate system z, is the rate of change of the x-axis, y-axis, and z-axis coordinates of the target in the inertial coordinate system with respect to time; S1.

3. Establish a nonlinear optimal control model for the missile, including constraints, optimization indicators, and optimal problem description; S1.3.

1. Constraints include missile mass constraints, overload constraints, and terminal constraints; Based on the fact that the mass of the missile will decrease as the engine fuel is consumed in the initial guidance stage, and the mass of the missile remains unchanged after the engine stops running, the mass constraint m of the missile is set c The specific expression is as follows: in, is the time when the engine stops running, m c0 is the fuel mass flow rate per second during the engine running time; The constraints of the missile's overload components in the ballistic coordinate system are set as follows: |n x |≤n xmax |n y |≤n ymax |n z |≤n zmax Among them, n xmax ,n ymax ,n zmax are the maximum absolute values ​​of the components of the missile's overload vector in the ballistic coordinate system on the x-axis, y-axis, and z-axis respectively; The maximum allowable miss distance of the missile is R max , set the terminal constraints of the missile to: Among them, t f The final moment when the missile completes guidance; S1.3.

2. When designing the interception trajectory of a missile, the terminal velocity of the missile is used as the optimization index J, which is expressed as: J=min{-[V(t f )] 2 }; S1.3.

3. The optimal problem is described as the state variables of the missile and the target, and the control quantities of the missile and the target, expressed as: d(x,u,t)≤0 x(t0)=x0 Among them, x(t) is the state variable of the missile and the target, u(t) is the overload of the missile and the target, t0 represents the initial time, x0 is the initial value of the state variable of the missile and the target; d(x,u,t) is the inequality constraint;

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