Online Midcourse Interception Method for High-Speed Maneuvering Targets Based on Convex Programming and Capture Space

Abstract

The present invention discloses an online mid-course interception method for high-speed maneuvering targets based on convex programming and capture space, belonging to the field of spacecraft control technology, including Step 1. transforming the mid-course guidance problem of the mid-course ballistic missile into an interceptor mid-course guidance model; Step 2. transforming the mid-course guidance problem into a discrete convex programming problem; Step 3. using the convex programming method to solve the offline interception trajectory of the mid-course guidance of high-speed maneuvering targets and establishing an online guidance framework; this method relaxes the affine control variables through the weighted relaxation method with terminal constraints, solving the problem of no feasible solution in dynamics due to strong equality constraints; proposes to solve the problem of errors caused by the solution of guidance commands through the online guidance framework; by using the capture space as the criterion for the success of online ballistic correction in mid-course guidance, it provides quantitative support for the success of mid-course guidance, getting rid of the narrow criterion problem of traditional zero-control interception; it can effectively correct the interceptor trajectory online and complete the online mid-course guidance interception of high-speed maneuvering targets.

Description

Technical Field

[0001] The present invention relates to the technical field of spacecraft control, and particularly relates to an online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space. Background Art

[0002] More and more countries are developing space vehicles and weapons in order to gain a dominant position in the space field. Midcourse guidance of an interceptor can provide good interception conditions for terminal guidance. Modifying the end state of midcourse guidance can overcome the influence of target maneuvers and ensure the successful interception of the target during terminal guidance.

[0003] The design ideas of mid-course guidance are mainly divided into two categories. The first is based on the relative motion relationship between the interceptor and the target, and the guidance is completed by controlling the line of sight. By adjusting the guidance direction, an improved three-dimensional pure proportional guidance law is proposed, which improves the guidance performance of the interceptor in the literature "SHIN H S, LI K B. An improvement in three-dimensional pure proportional navigation guidance. IEEE Transactions on Aerospace and Electronic Systems, 2021, 57(5): 3004-3014". A cooperative strategy for the interceptor to cover the target maneuvering distance is proposed, and the guidance parameters are dynamically adjusted to reduce the guidance error under the reference "CHEN ZY, YU J L, DONG XW, et al. Three-dimensional cooperative guidance strategy and guidance law for intercepting highly maneuvering target. Chinese Journal of Aeronautics, 2021, 34(5), 485-495". The literature "LIU S, YAN B, ZHANG T, et al. Three-dimensional coverage-based cooperative guidance law with overload constraints to intercept a hypersonic vehicle. Aerospace Science and Technology, 2022, 130, 107908" designs an offset guidance law based on the standard trajectory based on the coverage of the target escape area, which solves the problem of restricting the maneuvering interceptor from intercepting high-speed maneuvering targets. The literature "GAUDET B, FURFARO R, LINARES R. Reinforcement learning for angle-only intercept guidance of maneuvering targets. Aerospace Science and Technology, 2020, 99, 105746" generates guidance commands only by measuring the line-of-sight angle and its change rate based on reinforcement learning, which is applicable to passive seekers.The literature "XIA, CAIY L. A nonlinear finite-time robust differential game guidance law. Sensors, 2022, 22(17), 6650" proposed a finite-time robust differential guidance law for nonlinear zero-sum games. The zero-control miss distance was used as the terminal constraint, and a critical neural network with a time-varying activation function was used to solve the fixed terminal time constraint. The first design could overcome target maneuvers but might not satisfy some flight restrictions.

[0004] The second is to obtain guidance commands that satisfy flight constraints based on a dynamic model. The literature "ZHOU J, LEI H M, SHAO L, et al. Optimal midcourse trajectory planning considering the capture region. Journal of Systems Engineering and Electronics, 2018, 29(3), 587-600" proposed a midcourse guidance trajectory optimization method based on the Gaussian pseudospectral method considering the capture area of the interceptor, with high precision. Based on the capture region model prediction static planning theory, a multi-stage optimal trajectory planning and guidance method was proposed. However, this method may be difficult to satisfy path constraints during the interception process. The literature "LI H Y, HE S M, WANG J, et al. Near-optimal midcourse guidance for velocity maximization with constrained arrival angle. Journal of Guidance, Control, and Dynamics, 2021, 44(1): 172-180" transformed the nonlinear optimization problem into a linear optimization problem with process constraints and terminal constraints based on the nominal optimal trajectory. A midcourse guidance trajectory with maximum velocity based on angle constraints was generated. The literature "ZHOU H Y, WANG X G, CUI N G. Glide trajectory optimization for hypersonic vehicles via dynamic pressure control. Acta Astronautica, 2019, 164: 376-386" improved the particle swarm optimization algorithm and solved the trajectory planning problem of the hypersonic vehicle glide phase. The disadvantage is that this method cannot guarantee global optimization. The literature "LI W, LI J, LI N, et al. Online Trajectory Planning Method for Midcourse Guidance Phase Based on Deep Reinforcement Learning. Aerospace, 2023, 10(5), 441" used deep reinforcement learning to solve the midcourse guidance trajectory online. By combining process learning with the deep deterministic policy gradient, a training strategy was proposed to improve the convergence of the algorithm.The literature "CHEN W, LI W, SHAO L, et al. Correction Strategy of Online Midcourse Guidance for High-Speed Gliding Target Interceptor. Applied Sciences, 2023, 13(11), 6661" proposed an online midcourse guidance correction algorithm that meets position constraints and energy management. The end constraint of midcourse guidance is provided by a high-probability target presence area.

[0005] In recent years, convex programming has been widely applied due to its unique solutions and computational advantages in solving complex polynomials. The non-linearity and non-convexity of dynamic systems are the main difficulties in applying convex optimization to guidance problems. The literature "LIU, X F, SHEN Z J, LU P. Exact convex relaxation for optimal flight of aerodynamically controlled missiles. IEEE Transactions on Aerospace and Electronic Systems, 2016, 52(4): 1881-1892" introduced a method for dealing with non-linear dynamics. A relaxation method for solving convex non-convex constraints was introduced, and the effectiveness of this method was theoretically proven. The literature "SAGLIANO M. Pseudospectral convex optimization for powered descent and landing. Journal of guidance, control, and dynamics, 2018, 41(2), 320-334" proposed a framework based on pseudospectral optimization and convex programming to solve the guidance problem of non-linear dynamics. The literature "WANG J B, CUI N G, WEI C Z. Rapid trajectory optimization for hypersonic entry using convex optimization and pseudospectral method. Aircraft Engineering and Aerospace Technology, 2019, 91(4): 669-679" constructed a two-stage trajectory optimization framework based on convex programming and pseudospectral non-linear programming methods to generate high-precision reentry trajectories. The literature "SAGLIANO M, ERWIN M. Optimal drag-energy entry guidance via pseudospectral convex optimization. Aerospace Science and Technology, 2021, 117: 106946" proposed an autonomous entry guidance algorithm based on convex programming theory. This method can track and plan trajectories online under disturbances.The literature "ZHANG J L, LI J, ZHOU C J, et al. Fast Trajectory Generation Method for Midcourse Guidance Based on Convex Optimization. International Journal of Aerospace Engineering, 2022" proposed a two-stage convex optimization method for quickly obtaining an initial solution. Relaxation is used to solve problems with strong constraints. Currently, many scholars have studied online midcourse guidance methods and published some research results on midcourse guidance problems; however, the following difficulties still exist:

[0006] (1) There is little research on midcourse guidance correction for continuously maneuvering targets;

[0007] (2) There are strong equality constraints during midcourse guidance and no feasible solutions;

[0008] (3) Considering the algorithm solving time, there are delay errors in the application of guidance commands;

[0009] (4) The criteria for successful midcourse guidance are not clearly defined;

[0010] To address the above problems, it is urgent to design an online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space to solve the problems existing in the above-mentioned prior art. Summary of the Invention

[0011] To address the above existing problems, the present invention aims to provide an online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space. This method first establishes a dynamic model using an affine system. In a continuous-time optimization problem, it transforms the process guidance problem into a process; secondly, through affine control variable relaxation, Gaussian pseudospectral discretization, and constraint linearization, it transforms the problem into a discrete convex programming problem; then it generates an offline trajectory before midcourse guidance, and this trajectory is used as the initial reference trajectory for online correction. Through an online guidance framework, the error caused by the guidance command calculation time is eliminated; at the same time, the design of discrete points decreases as the flight time decreases to improve the solution efficiency; in addition, according to the terminal guidance capture space, it judges the success of online midcourse guidance; it can effectively correct the trajectory of the interceptor online and complete the success of online midcourse guidance for high-speed maneuvering targets.

[0012] To achieve the above object, the technical solutions adopted by the present invention are as follows:

[0013] An online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space, including

[0014] Step 1. When the interceptor intercepts a high-speed maneuvering target, transform the mid-course guidance problem of the interceptor intercepting the high-speed maneuvering target into an interceptor mid-course guidance model;

[0015] The interceptor mid-course guidance model includes an interception dynamics model, a process constraint condition model, a boundary constraint model, a performance index, and a problem description model;

[0016] Step 2. Based on the interceptor mid-course guidance model, transform the mid-course guidance problem into a discrete convex programming problem through the processes of affine control variable relaxation, Gaussian pseudospectral discretization, and constraint linearization;

[0017] Step 3. Use the convex programming method to solve the mid-course guidance offline interception trajectory of the high-speed maneuvering target, establish an online guidance framework, and update the discrete points in the mid-course guidance process.

[0018] Preferably, the establishment process of the interception dynamics model described in Step 1 includes

[0019] Step 111. Based on a fixed ground coordinate system, with the x and z axes pointing east and north respectively, and the h axis forming a right-handed coordinate system, and the interceptor on a flat ground, establish the interceptor dynamics model as

[0020]

[0021] In Equation (1), (h, z, x) is the position coordinate of the interceptor; r = 1 + h is the straight-line distance from the center of the earth to the interceptor; V is the relative velocity of the earth, θ is the ballistic inclination angle; ψ is the ballistic deflection angle; V is scaled by a ratio; g0 is the gravitational acceleration at the surface of the earth at r e ;

[0022] The dimensionless lift L and drag acceleration D are:

[0023]

[0024] In Equations (2) and (3), S is the reference area of the interceptor's force; C L and C D are the lift coefficient and drag coefficient of the interceptor respectively, and are related to the angle of attack α and Mach number Ma; m is the mass of the interceptor, and the atmospheric density

[0025] Step 112. Define the control variable normalization coefficient λ

[0026]

[0027] In Equations (4) and (5), are the lift coefficient and drag coefficient corresponding to the maximum lift-to-drag ratio at a certain Mach number respectively;

[0028] From equations (4) and (5), transform equations (2) and (3) into

[0029]

[0030] In equation (6), the lift acceleration the drag acceleration

[0031] Step 113. Construct affine variables

[0032] u1 = λcosσ, u2 = λsinσ, u3 = λ 2 (7)

[0033] Transform the interception dynamics model through the affine variables into

[0034]

[0035] Step 114. In the affine system, the control variable u must satisfy

[0036]

[0037] Assume the normalization coefficient λ is non - negative, and the upper limit is Get the value range of u3 as

[0038]

[0039] In equation (10),

[0040] Assume the value range of the bank angle σ is within the interval [σ min , σ max , get

[0041] u1tan(σ min ) ≤ u2 ≤ u1tan(σ max ) (11).

[0042] Preferably, the establishment process of the boundary constraint model described in step 1 includes

[0043] Step 131. Assume the initial state of the mid - course guidance of the interceptor is x0, the initial time is t0, and the initial condition constraint is

[0044] x(t0) - x0 = 0 (16)

[0045] Step 132. Assume the ideal terminal state of the interceptor is x p = [h p ; z p ; x p ; ~; θ p; ψ p , the end time of the midcourse guidance of the interceptor is t f , and the terminal constraint is

[0046] x(t f ) = x p (17)

[0047] Equation (17) is a strong equality constraint, representing that the interceptor reaches the terminal constraint state within the specified time. Next, equation (17) is relaxed:

[0048]

[0049] In equation (18), the terminal constraint relaxation coefficient υ = [υ1; υ2; υ3; υ4; υ5], κ1 and κ2 are weight coefficients, and the equation is rewritten as

[0050] x(t f ) - x p = κυ (19).

[0051] Preferably, the relaxation process of the affine control variable described in step 2 includes

[0052] Relaxing equation (9) to obtain:

[0053]

[0054] To ensure the effectiveness of the relaxed affine variable, a term is added to the objective function to ensure that the affine variable satisfies the constraint of equation (9).

[0055] Preferably, the Gaussian pseudospectral discretization process described in step 2 includes

[0056] Step 221. Express the affine dynamics equation (8) as

[0057]

[0058] In equation (23), u = [u1; u2; u3];

[0059]

[0060] Step 222. Transform the time interval t ∈ [t0, t f to τ ∈ [-1, 1], and the transformation formula is

[0061]

[0062] After the transformation, the new time variable τ replaces time t as the independent variable, τ = -1 corresponds to t0, and τ = 1 corresponds to t f ;

[0063] Step 223. The Lagrange-Galerkin collocation method is adopted to approximate the state change by the Lagrange interpolation polynomial basis of order up to N.

[0064]

[0065] In equation (28), P j (τ) is an N-th order Lagrange polynomial.

[0066]

[0067] Taking the time derivative of equation (28), we get

[0068]

[0069] Let and

[0070]

[0071] D ij The coefficient matrix formed is called the differential approximation matrix, D i ∈R n+1 ;

[0072] Step 224. Let Discretize the dynamic equation (24) into the following form:

[0073] D i X - τ c [F(X i ) + Β(X i )U i = 0, i = 1,..., N (32)

[0074] Since τ N < 1, the final state of the midcourse interception of the interceptor cannot be directly represented by the interpolation points. The Gaussian quadrature rule is used to calculate

[0075]

[0076] where w i represents the Gaussian orthogonal weight of each interpolation point;

[0077]

[0078] Discretize the added term The objective function obtained is

[0079]

[0080] Preferably, the constraint linearization process described in step 2 includes

[0081] Step 231. Using the condition Taylor-expand the dynamic equation (32) with respect to the reference trajectory {X * , U *}, omitting the subscript i

[0082] DX - τ c [F(X * ) + A(X * )(X - X * ) + Β(X * )U] = 0 (37)

[0083] where

[0084] Linearize the terminal state constraint equation

[0085] X(τ f ) = X(τ0) + τ c w[F(X * ) + A(X * )(X - X * ) + Β(X * )U] (38)

[0086] Taylor-expand the process constraint equation (15) with respect to the reference trajectory {X * , U *}

[0087]

[0088] In equation (39), h * , V * , u3 * are the state variables in the reference trajectory {X * , U *};

[0089]

[0090] Step 232. Use the variable trust region method to impose the feasible solution constraint

[0091]

[0092] In equation (41), ε x and ε u are the trust region intervals of the state variables, and ξ is the trust region relaxation coefficient;

[0093] After imposing the constraint on the trust region relaxation coefficient in the performance index, the performance index is expressed as

[0094]

[0095] After discretization and linearization, the midcourse guidance problem P1 can be written as a convex programming problem P2, expressed as

[0096]

[0097] In Equation (43), the control variable constraints are Equations (10), (11), and (22), the boundary constraints are Equations (16), (17), and (38), the dynamic equation is Equation (37), the process constraint is Equation (39), and the trust region is Equation (41).

[0098] Preferably, the process of establishing an online guidance framework and updating the discrete points in the midcourse guidance process described in step 3 includes

[0099] Step 31. Solve the offline interception trajectory of the midcourse guidance of a high-speed maneuvering target using a convex programming method;

[0100] Step 32. Establish an online midcourse guidance framework and perform online correction of the ballistic trajectory;

[0101] Step 33. Design a grid update strategy in the online midcourse guidance process;

[0102] Step 34. Determine the success of the midcourse guidance;

[0103] Step 35. Construct a midcourse guidance terminal simulation model.

[0104] Preferably, the process of solving the offline interception trajectory of the midcourse guidance of a high-speed maneuvering target using a convex programming method described in step 31 includes

[0105] Step 311. Set k = 0, determine the discrete points and the initial time domain, and select the initial reference trajectory, i.e., the initial solution (x (0) , u (0) );

[0106] Step 312. In the k-th iteration, solve problem P2 to obtain the results of the optimization variables (x (k) , u (k) );

[0107] where k ≥ 1;

[0108] Step 313. Check whether the results of the optimization variables satisfy the convergence condition of Equation (44). If all discrete points satisfy the convergence condition in Equation (44), go to the next step; otherwise, let k = k + 1 and return to step 312;

[0109]

[0110] Step 314. Terminate the iteration, and the convergent solution is the highly approximate solution of problem P1.

[0111] Preferably, the process of designing the grid update strategy in the midcourse guidance process in step 33 includes

[0112] Establish the grid update strategy N in the midcourse guidance process in step 33 k

[0113]

[0114] In Equation (46), N0 represents the initial grid point number of the offline reference trajectory, N k represents the grid point number in the (k - 1)-th guidance cycle, N l represents the minimum grid number required for the solution accuracy, t k is the start time of the (k - 1)-th guidance cycle, t f is the end time of the midcourse guidance.

[0115] Preferably, the process of determining the success of the midcourse guidance in step 34 includes

[0116] Step 341. Let the actual terminal state of the interceptor's midcourse guidance be x ft =[x ft ; z ft ; h ft ; V ft ; θ ft ; ψ ft , the final ideal correction terminal state and the final target prediction state of the midcourse guidance are x fi =[x fi ; z fi ; h fi ; ~; θ fi ; ψ fi and x fp =[x fp ; z fp ; h fp ; V fp ; θ fp ; ψ fp respectively. The ideal turn-on distance of the interceptor seeker is l. By back-calculating the ideal correction terminal state and the final target prediction state based on the zero-effort miss intercept manifold of the terminal guidance, then x fi and x fp satisfy

[0117]

[0118] Step 342. Determine whether the interceptor after trajectory correction satisfies the capture space constraint according to the midcourse guidance terminal state x ft and the final target prediction state x fp ;

[0119] The relative position vector r between the missile and the targetit In

[0120]

[0121] In Equation (48), e r = r it / r it is the unit vector in the line-of-sight direction;

[0122] The relative velocity vector v between the projectile and the target itf is

[0123]

[0124] The tangential velocity v of the line-of-sight between the projectile and the target θf is

[0125] v θf = v θf e θ = v itf - v rf = v itf - v itf e r = v itf - v rf e r (50)

[0126] In Equation (50), e θ = v θf / v θf represents the unit vector in the tangential direction of the line-of-sight; v θf represents the magnitude of the velocity in the direction of e θ ; v rf represents the end of midcourse guidance, i.e., the velocity of the initial line-of-sight direction of terminal guidance, v rf represents the magnitude of the velocity in the direction of e r ;

[0127] Step 343. In the terminal guidance stage, taking the realistic proportional guidance law as an example to intercept the target, design the allowable collision velocity v rImp according to the capture space given by the target and interceptor information, and solve the boundary value |v rf | of the tangential velocity of the line-of-sight corresponding to the initial line-of-sight velocity v θmax of terminal guidance. The solution is as shown in Equation (51)

[0128]

[0129] In Equation (51), α max represents the saturation maneuver acceleration of the interceptor, and α θ represents the upper limit of the acceleration of the target in the direction of e θ ; α rIndicates the upper limit of the acceleration of the target in the e r direction;

[0130] Step 344. Obtain the success judgment criterion for the midcourse guidance trajectory correction of the interceptor:

[0131] When |v θmax | ≥ |v θf |, it is considered that the correction in the midcourse guidance stage of the interceptor is successful, and the target can be successfully intercepted in the terminal guidance stage;

[0132] When |v θmax | < |v θf |, it is considered that the correction in the midcourse guidance stage of the interceptor is failed, but there is a probability of successfully intercepting the target in the terminal guidance stage.

[0133] The beneficial effects of the present invention are as follows: The present invention discloses an online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space. Compared with the prior art, the improvements of the present invention are as follows:

[0134] The present invention proposes an online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space, including Step 1. When the interceptor intercepts a high-speed maneuvering target, transform the midcourse guidance problem of the interceptor intercepting the high-speed maneuvering target into an interceptor midcourse guidance model including an interception dynamics model, a process constraint condition model, a boundary constraint model, and a performance index and problem description model; Step 2. On the basis of the interceptor midcourse guidance model, transform the midcourse guidance problem into a discrete convex programming problem through an affine control variable relaxation, a Gaussian pseudospectral discretization, and a constraint linearization process; Step 3. Establish an online guidance framework and update the discrete points in the midcourse guidance process; When in use:

[0135] 1. Aiming at the problem of no solution due to strong equality constraints in the dynamics, the present invention proposes a weighted relaxation method with terminal constraints for affine variable relaxation, then transforms the target prediction state into a midcourse guidance terminal constraint, modifies the online trajectory according to the change of the terminal constraint, and transforms the interception problem into an online trajectory solution problem for the midcourse guidance final state;

[0136] 2. Aiming at the error problem caused by solving the guidance command, the present invention proposes an online guidance framework, which solves the delay error caused by solving the guidance command; At the same time, the design of the discrete points decreases with the decrease of the flight time, effectively improving the solution efficiency;

[0137] 3. Aiming at the narrow criterion problem of traditional zero-control interception, the present invention innovatively uses the capture space as the criterion for the successful online trajectory correction in midcourse guidance, providing quantitative support for the success of midcourse guidance and getting rid of the narrow criterion of traditional zero-control interception; the results show that the online trajectory planning method based on convex programming has good optimization performance and can correct the trajectory of the interceptor online. Description of the Drawings

[0138] Figure 1 It is a flowchart of the online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space of the present invention.

[0139] Figure 2 It is a process diagram of the midcourse guidance trajectory correction of the present invention.

[0140] Figure 3 It is an application diagram of the midcourse guidance command in the guidance process of the present invention.

[0141] Figure 4 It is a framework diagram of the online correction of the midcourse guidance trajectory of the present invention.

[0142] Figure 5 It is a convergence process curve diagram of the midcourse guidance offline trajectory in Embodiment 2 of the present invention.

[0143] Figure 6 It is a generation result diagram of the midcourse guidance offline trajectory under different optimization methods in Embodiment 2 of the present invention.

[0144] Figure 7 It is an online midcourse guidance simulation result diagram in Embodiment 2 of the present invention.

[0145] Figure 8 It is a Monte Carlo simulation result diagram in Embodiment 2 of the present invention.

[0146] Among them, in Figure 5 , Fig. (a) is the longitudinal trajectory curve diagram, and Fig. (b) is the lateral trajectory curve diagram;

[0147] In Figure 6 , Fig. (a) is the longitudinal trajectory curve diagram, Fig. (b) is the lateral trajectory curve diagram, Fig. (c) is the angle of attack curve diagram, Fig. (d) is the bank angle curve diagram, Fig. (e) is the trajectory inclination angle curve diagram, and Fig. (f) is the trajectory deflection angle curve diagram;

[0148] In Figure 7 , Fig. (a) is the online midcourse guidance simulation trajectory and ideal terminal curve diagram, Fig. (b) is the online and offline midcourse guidance speed comparison curve diagram, Fig. (c) is the angle of attack curve diagram, Fig. (d) is the bank angle curve diagram, Fig. (e) is the trajectory inclination angle curve diagram, and Fig. (f) is the trajectory deflection angle curve diagram;

[0149] In Figure 8Among them, Figure (a) is the curve graph of the average running time per iteration, and Figure (b) is the curve graph of the end state in the middle section of the capture space during midcourse guidance. Detailed implementation manner

[0150] In order to enable those of ordinary skill in the art to better understand the technical solution of the present invention, the technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0151] Embodiment 1: Refer to the Figures 1 - 4 Online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space as shown, including

[0152] Step 1. When the interceptor intercepts a high-speed maneuvering target, convert the midcourse guidance problem of the interceptor intercepting the high-speed maneuvering target into an interceptor midcourse guidance model including an interception dynamics model, a process constraint condition model, a boundary constraint model, and a performance index and problem description model

[0153] Step 11. Establish an interception dynamics model

[0154] Step 111. The interceptor uses aerodynamic gliding during the midcourse guidance stage. The dynamic model is based on a fixed ground coordinate system, with the x and z axes pointing east and north respectively, and the h axis forming a right-handed coordinate system; ignoring the influence of the earth's rotation, the interceptor is on a flat ground, and the interceptor dynamic model is established as

[0155]

[0156] In Equation (1), (h, z, x) represents the position coordinates of the interceptor; r = 1 + h represents the straight-line distance from the center of the earth to the interceptor; V represents the relative velocity of the earth, θ represents the ballistic inclination angle; ψ represents the ballistic deflection angle; V is scaled by proportion; g0 represents the gravitational acceleration at the surface of the earth where r is e ;

[0157] The dimensionless lift L and drag acceleration D are as follows:

[0158]

[0159] In Equations (2) and (3), S is the reference area of the force on the interceptor; C L and C D are the lift coefficient and drag coefficient of the interceptor respectively, and are related to the angle of attack α and the Mach number Ma; m is the mass of the interceptor, and the atmospheric density ρ0 = 1.225 kg / m 3 , H = 7.25 km;

[0160] Step 112. Define the control variable normalization coefficient λ

[0161]

[0162] In equations (4) and (5), are the lift coefficient and drag coefficient corresponding to the maximum lift-to-drag ratio at a certain Mach number, which can be obtained by interpolation from the aerodynamic data table;

[0163] From equations (4) and (5), equations (2) and (3) can be transformed into

[0164]

[0165] In equation (6), the lift acceleration The drag acceleration

[0166] Step 113. The current interception dynamics model is still not a control-affine system. Now, construct the affine variables

[0167] u1 = λcosσ, u2 = λsinσ, u3 = λ 2 (7)

[0168] Through the affine variables, the interception dynamics model (1) can be transformed into

[0169]

[0170] Step 114. In the affine system, the control variable u must satisfy

[0171]

[0172] In this embodiment, it is assumed that the normalization coefficient λ is non-negative and the upper limit is Then the value range of u3 is

[0173]

[0174] In equation (10),

[0175] Assume that the value range of the bank angle σ is within the interval [σ min , σ max , then

[0176] u1tan(σ min ) ≤ u2 ≤ u1tan(σ max ) (11);

[0177] Step 12. Establish the process constraint condition model

[0178] During the mid-course guidance interception of the interceptor, process constraints need to be satisfied; otherwise, it may cause the deformation of the aerodynamic shape of the interceptor and result in out-of-control phenomena. Among them, the process constraints mainly include overload constraints, heat flux density constraints, dynamic pressure constraints, etc. The functions of the overload constraints, heat flux density constraints, and dynamic pressure constraints are as follows

[0179]

[0180] The process constraints can be written as

[0181]

[0182] In Equation (15), represents the maximum value allowed for the process constraints of the interceptor;

[0183] Step 13. Establish a boundary constraint model

[0184] Step 131. Assume that the initial state of the mid-course guidance of the interceptor is x0 and the initial time is t0, then the initial condition constraint is

[0185] x(t0) - x0 = 0 (16)

[0186] Step 132. Assume that the ideal terminal state of the interceptor is x p =[h p ; z p ; x p ; ~; θ p ; ψ p , where the speed of the interceptor is limited, but in order to effectively damage the target, the greater the speed, the better, and this constraint will be reflected in the objective function; the end time of the mid-course guidance of the interceptor is t f , then the terminal constraint is

[0187] x(t f ) = x p (17)

[0188] Equation (17) is a strong equality constraint, which means that the interceptor reaches the terminal constraint state within the specified time, which may result in no solution within the feasible region; the following is to relax it to ensure that there is a feasible solution

[0189]

[0190] Among them, the terminal constraint relaxation coefficient υ = [υ1; υ2; υ3; υ4; υ5], which is constrained in the objective function; κ1 and κ2 represent weight coefficients to adjust the weights of different state variables;

[0191] The equation can be rewritten as

[0192] x(t f ) - x p=κυ (19);

[0193] Step 14. Establish a performance index and problem description model

[0194] To enable the interceptor to have a greater momentum when colliding with the target, the maximum interceptor velocity is taken as the main performance index, plus the terminal constraint relaxation coefficient

[0195] J0 = -c1V f +c2υ T υ (20)

[0196] where c1 and c1 represent weights;

[0197] Therefore, the midcourse interception guidance problem P1 of the interceptor can be formulated as a continuous-time optimal problem, expressed as follows

[0198]

[0199] Step 2. Based on the midcourse guidance model of the interceptor, transform the midcourse guidance problem into a discrete convex programming problem through the processes of affine control variable relaxation, Gaussian pseudospectral discretization, and constraint linearization

[0200] Problem P1 is still a strongly nonlinear constrained problem. It is necessary to relax the affine variables and discretize the constraints to transform problem P1 into a discrete problem P2, and then convexify problem P2 to transform it into a sequential convex programming problem P3 for easy algorithm solving;

[0201] Step 21. Affine control variable relaxation process

[0202] It can be seen from the affine control variable constraints (9), (10), and (11) that the affine variable constraints are strong equality constraints, which may cause the non-convexity of problem P1. Below, equation (9) is relaxed to solve the non-convex problem of problem P1;

[0203]

[0204] To ensure the effectiveness of the relaxed affine variables, a term is added to the objective function to ensure that the affine variables satisfy the constraint of equation (9);

[0205] Step 22. Gaussian pseudospectral discretization

[0206] Step 221. The affine dynamics equation (8) can be expressed as

[0207]

[0208] In equation (23), u = [u1; u2; u3];

[0209]

[0210] Using the Gauss pseudospectral method, a discrete system with high computational accuracy can be obtained;

[0211] Step 222. Transform the time interval \(t\in[t_0,t]\) to \(\tau\in[-1,1]\), and the transformation formula is f , and the transformation formula is

[0212]

[0213] After the transformation, the new time variable \(\tau\) replaces time \(t\) as the independent variable, \(\tau = -1\) corresponds to \(t_0\), and \(\tau = 1\) corresponds to \(t\); f ;

[0214] Step 223. Adopt the LGR collocation method (Legendre-Gauss-Radau Collocation Method) to approximate the change of the state with the Lagrange interpolation polynomial basis of order up to \(N\).

[0215]

[0216] In equation (28), \(P\) j (\(\tau\)) is an \(N\)-th order Lagrange polynomial;

[0217]

[0218] Since the nodes are independent of time, taking the time derivative of equation (28), we can obtain

[0219]

[0220] Let and

[0221]

[0222] The coefficient matrix composed of \(D\) ij is called the differential approximation matrix, \(D\) i \(\in R\) n+1 ;

[0223] Step 224. Let The dynamic equation (24) can be discretized into the following form:

[0224] \(D\) i \(X-\tau\) c [F(X i )+\(\beta(X\) i )U i =0, \(i = 1,\cdots,N\) (32)

[0225] Since \(\tau\)N <1, the final state of mid-course interception by the interceptor cannot be directly represented by the interpolation points and can be calculated by the Gauss quadrature rule

[0226]

[0227] where w i represents the Gauss orthogonal weight of each interpolation point;

[0228]

[0229] Discretize the added term At this time, the objective function is

[0230]

[0231] Step 23. Process of linearizing the problem constraints

[0232] After the above transformation, the continuous problem is converted into a discrete problem; since the constraints are non-convex, the non-convex constraints can be linearized by Taylor expansion, and the constraints are transformed into convex functions that meet the requirements of the convex programming method;

[0233] Step 231. Utilize the condition Taylor-expand the dynamic equation (32) with respect to the reference trajectory {X * , U *}, and for the sake of simplicity, omit the subscript i

[0234] DX - τ c [F(X * ) + A(X * )(X - X * ) + Β(X * )U] = 0 (37)

[0235] where

[0236] Linearize the terminal state constraint equation

[0237] X(τ f ) = X(τ0) + τ c w[F(X * ) + A(X * )(X - X * ) + Β(X * )U] (38)

[0238] Taylor-expand the process constraint equation (15) with respect to the reference trajectory {X * , U *}

[0239]

[0240] Among them, h * , V * , u3 * are state variables in the reference trajectory {X * , U *};

[0241]

[0242] Step 232. Since the above linearization is achieved through Taylor expansion, the approximate accuracy can only be guaranteed when the solution is close to the reference trajectory {X * , U *}; therefore, a trust region constraint is added; in order to adjust the search range of the feasible solution during the optimization iteration process and improve the optimization speed, a variable trust region method is used to constrain the feasible solution

[0243]

[0244] Among them, ε x and ε u are the trust region intervals of the state variables. ξ is the trust region relaxation coefficient, which is constrained in the objective function;

[0245] After constraining the trust region relaxation coefficient in the performance index, the performance index is expressed as

[0246]

[0247] After discretization and linearization, the midcourse guidance problem P1 can be written as a convex programming problem P2, expressed as

[0248]

[0249] In Equation (43), the control variable constraints are Equations (10), (11), (22), the boundary constraints are Equations (16), (17), (38), the dynamic equation is Equation (37), the process constraint is Equation (39), and the trust region is Equation (41);

[0250] Step 3. Use the convex programming method to solve the midcourse guidance offline interception trajectory of a high-speed maneuvering target, establish an online guidance framework, and update the discrete points during the midcourse guidance

[0251] The process of the interceptor intercepting the target can be divided into initial guidance, midcourse guidance, and terminal guidance; the interceptor can perform offline optimization and solution of the trajectory before midcourse guidance; during the midcourse guidance process, due to prediction information errors and target maneuvers, the predicted hit point of the interceptor will change, which will in turn lead to changes in the midcourse guidance terminal state constraints, so it is necessary to correct the trajectory during the midcourse guidance process;

[0252] Step 31. Solve the midcourse guidance offline intercept trajectory of a high-speed maneuvering target using the convex programming method

[0253] The offline trajectory can be obtained by solving Problem P2. The convex programming problem P2 is linearized by Problem P1, and Problem P1 can be solved by solving Problem P2. Since Problem P1 has strong nonlinearity, the linearization error obtained by solving Problem P2 only once based on the reference trajectory is relatively large, which is generally infeasible for Problem P1.

[0254] Therefore, the convex programming method is used to continuously solve Problem P2 to approximate the solution of Problem P1. The specific steps of the algorithm are as follows:

[0255] Step 311. Set k = 0, determine the discrete points and the initial time domain, and select the initial reference trajectory, that is, the initial solution (x (0) , u (0) );

[0256] Step 312. In the k-th iteration (k ≥ 1), solve Problem P2 to obtain the results of the optimization variables (x (k) , u (k) );

[0257] Step 313. Check whether the results of the optimization variables satisfy the convergence condition of Equation (44). If all discrete points satisfy the convergence condition in Equation (44), go to the next step; otherwise, let k = k + 1 and return to Step 312;

[0258]

[0259] Step 314. The iteration terminates, and the convergent solution is the highly approximate solution of Problem P1;

[0260] When solving the offline trajectory of the above convex programming, there is still a problem, that is, how to obtain the initial solution. The following gives a fast method for generating the initial solution, and the steps are as follows:

[0261] (1) Select the maximum normalized lift coefficient Ensure that the generated initial trajectory can satisfy the trajectory constraints to the greatest extent; take equally spaced values within the bank angle constraint as the control quantity;

[0262] (2) According to the interceptor interception dynamics model, integrate the control quantities at different bank angles respectively to generate a group of initial trajectories;

[0263] (3) From the group of initial trajectories, select two trajectories l1 and l2 that are closest to the terminal desired point. The distances from the terminal desired point are Δl1 and Δl2 respectively, and the bank angles corresponding to the two trajectories are σ1 and σ2 respectively. Select the bank angle σ3 of the initial reference trajectory according to Equation (45),

[0264] σ3 = (σ2Δl1 + σ1Δl2) / (Δl1 + Δl2) (45)

[0265] (4) Calculate the final control quantity based on the tilt angle σ3, and repeat step (2) to generate the initial reference trajectory x (0) , and convert the normalized lift coefficient and the tilt angle σ3 into the initial trajectory control quantity u (0) ;

[0266] Step 32. Establish an online midcourse guidance framework

[0267] During the process of intercepting the target in midcourse guidance, the accuracy of target ballistic prediction will gradually improve; in order to ensure that the interceptor can successfully intercept the target in terminal guidance, it is necessary to correct the trajectory online, and the trajectory correction process is as Figure 2 shown.

[0268] Step 321. The time when the interceptor enters the midcourse guidance stage is t0, the initial predicted impact point is P0, and the guidance command is After a guidance cycle T g , there may be a certain error between the missile flight trajectory and the optimized trajectory. Solve problem P2 to obtain a new guidance command u1, and then continuously repeat this process before the end of a prediction cycle T p ; after the end of the first prediction cycle, the predicted impact point becomes P1, and the guidance command will solve the guidance command within each guidance cycle; then repeat the ballistic correction process within the prediction cycle until the end of midcourse guidance t f ;

[0269] Step 322. Since the guidance command u is obtained by solving the convex programming problem P2, it requires a certain amount of computing time; therefore, in the current guidance cycle, the new guidance command cannot be immediately applied, which will lead to the problem of guidance command delay;

[0270] The application of the midcourse guidance command during the guidance process is as Figure 3 shown. Assume that the start time of the kth guidance cycle of midcourse guidance is t k-1 , the end time of the guidance cycle is t k , the next is the (k + 1)th guidance cycle, the start time is t k , and the end time is t k+1 , and so on; within the kth guidance cycle, the interceptor can, according to the current guidance command u k|k , solve the guidance command u *|k after k guidance cycles; in the (k + 1)th guidance cycle, execute the guidance command u k+1|k solved in the kth guidance cycle, and solve the guidance command u *|k+1 after (k + 1) guidance cycles, and so on. Repeat this process until the midcourse guidance ends;

[0271] Step 33. Design the grid update strategy during online midcourse guidance

[0272] When the trajectory is generated offline, the more grid points of the Gauss pseudospectral method, the higher the solution accuracy. However, the more grid points, the longer the calculation time; during the online correction of the ballistic trajectory, the interceptor continuously approaches the terminal state. During this process, the remaining flight time will gradually decrease, and too many grids will have an adverse effect on the calculation speed; therefore, this embodiment proposes a grid update strategy N during the trajectory correction process k , which not only ensures a certain accuracy but also reduces the solution time required for trajectory correction

[0273]

[0274] In Equation (46), N0 represents the initial number of grid points of the offline reference trajectory, N k represents the number of grid points in the (k - 1)-th guidance cycle, N l represents the minimum number of grid points required for the solution accuracy, t k is the start time of the (k - 1)-th guidance cycle, t f is the end time of midcourse guidance; according to the update strategy, the number of grid points will decrease with the increase of the flight time;

[0275] Step 34. Determine the success of midcourse guidance

[0276] Step 341. Assume that the actual terminal state of the interceptor during midcourse guidance is x ft =[x ft ; z ft ; h ft ; V ft ; θ ft ; ψ ft , the final ideal correction terminal state and the final target prediction state of midcourse guidance are x fi =[x fi ; z fi ; h fi ; ~; θ fi ; ψ fi and x fp =[x fp ; z fp ; h fp ; V fp ; θ fp ; ψ fp , the ideal turn-on distance of the interceptor seeker is l, and the ideal correction terminal state and the final target prediction state are deduced by the zero-effort intercept manifold of terminal guidance, then x fi and x fp satisfy

[0277]

[0278] Step 342. Next, based on the midcourse guidance terminal state x ft and the predicted state x fp of the final target, determine whether the interceptor after trajectory correction meets the capture space constraint;

[0279] The missile-target relative position vector r it is

[0280]

[0281] In Equation (48), e r = r it / r it is the unit vector in the line-of-sight direction;

[0282] The missile-target relative velocity vector v itf is

[0283]

[0284] The tangential velocity v θf of the missile-target line of sight is

[0285] v θf = v θf e θ = v itf - v rf = v itf - v itf e r = v itf - v rf e r (50)

[0286] In Equation (50), e θ = v θf / v θf represents the unit vector in the tangential direction of the line of sight; v θf represents the magnitude of the velocity in the direction of e θ ; v rf represents the end of midcourse guidance, i.e., the initial line-of-sight velocity of terminal guidance, v rf represents the magnitude of the velocity in the direction of e r ;

[0287] Step 343. In the terminal guidance stage, taking the realistic proportional guidance law as an example to intercept the target, based on the capture space given by the target and interceptor information, and combining Equation (51), an appropriate allowable collision velocity v rImp can be designed, and the initial line-of-sight velocity v rfThe corresponding tangential velocity boundary value of the aiming line |v θmax |, and the solution is as shown in Equation (51)

[0288]

[0289] In Equation (51), α max represents the saturation maneuver acceleration of the interceptor, and α θ represents the upper limit of the target's acceleration in the e θ direction, and α r represents the upper limit of the target's acceleration in the e r direction;

[0290] Step 344. In summary, the success judgment criterion for the midcourse guidance trajectory correction of the interceptor is as follows:

[0291] When |v θmax | ≥ |v θf |, it can be considered that the correction in the midcourse guidance stage of the interceptor is successful, and the target can be successfully intercepted in the terminal guidance stage;

[0292] When |v θmax | < |v θf |, it can be considered that the correction in the midcourse guidance stage of the interceptor is failed, but there is a probability of successfully intercepting the target in the terminal guidance stage;

[0293] Step 35. Build a midcourse guidance terminal simulation model

[0294] Through analysis, it can be found that the longer the tracking and filtering time of the high-speed maneuvering target, the smaller the prediction error of the target state; under the same tracking and filtering time, the longer the target state prediction time, the greater the target prediction error; when the target state and the terminal guidance distance are determined, according to the conditions of the zero-control interception manifold, the ideal final state of the midcourse guidance can be obtained. The following is the midcourse guidance final state simulation process;

[0295] Step 351. First, the initial prediction fluctuation error in the midcourse guidance is significantly greater than the later prediction fluctuation error;

[0296] Step 352. Secondly, it decreases as the prediction time decreases. The model is as follows

[0297]

[0298] In Equations (52) and (53), x fc = [x fc ; z fc ; h fc ; ~; θ fc ; ψ fc , and γ represents the weight function before the normal distribution of the prediction error fluctuation between the (i + 1)-th terminal state and the i-th terminal state; The weight representing the fluctuation between different initial states of the interceptor; Φ(i) represents the prediction error fluctuation function, which decreases as i increases; In summary, the online correction framework for the midcourse guidance trajectory of intercepting high-speed maneuvering targets is as Figure 4 shown.

[0299] It can be seen from the above process that this embodiment proposes an online midcourse guidance interception method for high-speed maneuvering targets; This method: First, aiming at the problem of no feasible solution due to strong equality constraints in dynamics, a weighted relaxation method with terminal constraints is proposed; Then, the target prediction state is transformed into the end constraint of midcourse guidance. Modify the online trajectory according to the change of the end constraint. Transform the interception problem into an online trajectory solution problem for the end state of midcourse guidance. In addition, an online guidance framework is proposed to solve the delay error caused by the solution of guidance commands. Finally, innovatively, the capture space is used as the criterion for the success of online midcourse guidance trajectory correction, providing quantitative support for the success of midcourse guidance and getting rid of the narrow criterion of traditional zero-control interception. The simulation results show that the online trajectory planning method based on convex programming has good optimization performance and can correct the trajectory of the interceptor online.

[0300] Embodiment 2: Refer to the attached Figures 5 - 8 shown. To verify the effectiveness of the online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space described in Embodiment 1, this simulation example is designed for verification.

[0301] The interceptor model parameters are m0 = 900 kg, S = 0.4839 m 2 , the boundary of the normalized lift coefficient λ is [0, 4.4016], and the boundary of the bank angle σ is [-80°, 80°]; The path constraint n max = 3.5g0, q max = 150 kPa; The weight coefficients in the objective function are c1 = 0.01, c2 = 0.01, c3 = 100, κ1 = 0.01, κ2 = 1; The trust region constraint is ε x [5000 / r e , 5000 / r e , 10000 / r e , 10π / 180, 10π / 180], and the convergence domain δ x is [100 / r e , 100 / r e , 500 / r e , π / 180, π / 180].

[0302] The initial number of discrete points is N0 = 40, and the minimum number of discrete points is N l= 10; The boundary conditions for the midcourse guidance of the interceptor missile are shown in Table 1; The simulation software is MATLAB R2016a, the computer processor is Intel Core I7-10510 2.30GHZ, and the ECOS

[38] solver is used for programming.

[0303] Table 1: Boundary Conditions for the Midcourse Guidance of the Interceptor

[0304]

[0305] The simulation results of the convex programming are given in Step 1 and compared with the Gaussian pseudospectral optimization method; In Step 2, the effectiveness of the online trajectory correction method is verified;

[0306] Step 1: Verification of Offline Trajectory Generation

[0307] The offline trajectory before midcourse guidance is used as the reference trajectory for the online guidance trajectory; To verify the effectiveness of the method proposed in Example 1 and compare it with the Gaussian pseudospectral optimization method; The end time is 180 seconds.

[0308] Table 2: Simulation Results under Different Methods

[0309]

[0310] It can be seen from Table 2 that both methods can solve the problem of generating the offline trajectory for midcourse guidance. Compared with the Gaussian pseudospectral method, this method has fewer iteration times, shorter total solution time, and higher solution efficiency. The accuracy of this method is slightly lower, but the error can be ignored during midcourse guidance.

[0311] The convergence process of generating the offline trajectory for midcourse guidance using this method is as Figure 5 shown. In Figure 5 (a), the trajectory of the first iteration can quickly converge to the vicinity of the terminal position because of the high solution efficiency of this method and the large influence of the position error on the performance index, which speeds up the trajectory convergence process. In Figure 5 (b), the lateral trajectory after iteration tends to be a straight line due to the influence of the maximum speed performance index, which reduces the energy consumption of the interceptor.

[0312] The results of generating the offline trajectory for midcourse guidance under different optimization methods are as Figure 6 shown. Figure 6 (a) and (b) respectively show the ballistic changes of the midcourse guidance trajectory in the longitudinal and lateral planes. The overall trends of the trajectories generated by the two methods are the same, but the trajectories are not exactly the same, which may be due to the different ways of generating the initial reference trajectory and the different solution algorithms. Figure 6 (c) and Figure 6 (d) respectively give the control variables corresponding to the midcourse guidance trajectory, and all satisfy the constraint conditions.Figure 6 (e) and Figure 6 (f) respectively give the ballistic inclination angle and the midcourse guidance ballistic inclination angle, both of which reach the end constraint. The inclination angle and declination angle of the Gaussian pseudospectral method fluctuate after 100 s, resulting in fluctuations in the ballistic trajectory.

[0313] The simulation results show that both methods can generate the terminal guidance trajectory, and the method proposed in Embodiment 1 of the present invention is more efficient.

[0314] Step 2: Simulation verification

[0315] The online midcourse guidance method proposed in Embodiment 1 is verified as follows; first, assume that the relevant parameters of the ideal end wave iteration error of the interceptor are and Φ(i) = 1 / (i + 10), the distance between the interceptor and the target when the midcourse guidance enters the terminal guidance is 80 km, the guidance cycle time is t z = 1 s, and the predicted final update cycle is t p = 10 s; the terminal time is 180 seconds.

[0316] To conveniently verify whether the state of the interceptor after the midcourse guidance ballistic correction meets the capture space requirements, it is uniformly assumed that the magnitude of the target velocity at the end of the midcourse guidance is 3400 m / s, and the relevant parameters of the terminal guidance capture space are as follows:

[0317] The allowable miss distance r Miss is 0.5 m, the allowable collision velocity v rImp is -4000 m / s, the initial distance r0 between the interceptor and the target is 80 km, the saturation maneuvering acceleration α of the interceptor mmax is 6g, the upper acceleration limit a of the target in e r is 1g, and the upper acceleration limit a of the target in e trmax is 2g, where the gravitational acceleration g is taken as 9.8 m / s θ tθmax 2 .

[0318] Figure 7 The online midcourse guidance simulation results are given and compared with the offline simulation results. In Figure 7 (a), due to the change of the ideal final state, the online terminal guidance trajectory has changed. There is a certain error between the end of the trajectory and the final ideal end, but the error is much smaller than the offline trajectory. The ideal final group 1, ideal final group 2, and ideal final group 3 are the states before 50 s, before 100 s, and at the end of the midcourse guidance respectively; in Figure 7 ​​(b), the terminal velocity of the online trajectory is close to that of the offline trajectory. The velocity curve of the first half of the online trajectory is slightly greater than that of the offline trajectory. This is because the lateral maneuver of the first part of the online trajectory is smaller, less kinetic energy is consumed, and the trajectory is slightly lower than the offline trajectory, converting gravitational potential energy into kinetic energy. From Figure 7 It can be seen from (e) and (f) that the angular fluctuation of the terminal velocity of the online trajectory is greater than that of the offline trajectory, and the kinetic energy loss is larger. The final velocity of the online trajectory is close to that of the offline trajectory.

[0319] Figure 7 (c) and (d) respectively show the changes in the ballistic angle of attack and the bank angle, both of which satisfy the constraints. Because the ideal terminal is constantly changing, both have fluctuations; at the end of the trajectory, due to the power constraints of the interceptor missile, the trajectory cannot reach the ideal terminal, and the state deviation increases, resulting in more severe fluctuations in the angle of attack and the bank angle. Figure 7 (e) and (f) respectively show the changes in the ballistic inclination angle and the declination, both of which meet the constraints. The mid-course ballistic inclination angle is lower than that of the offline trajectory, and higher than that of the offline trajectory at the end, because the altitude at the end of the trajectory is greater than that of the offline end. The trajectory deflection angle continuously increases in the middle and late stages and decreases at the end, because the ideal terminal first deflects in the positive direction of the z-axis and then in the negative direction of the z-axis.

[0320] The tangential velocity v of the offline trajectory terminal in the capture space θon is 1237 m / s, and the tangential velocity boundary |v θmax | on is 259 m / s. Since the terminal state is outside the capture space, this means that the interception target fails. The tangential velocity v of the online trajectory terminal in the capture space θon is 110 m / s, and the tangential velocity boundary |v θmax | on is 287 m / s. Since v θon <|v θmax | on , the terminal state is located within the capture space, which means that the target can be successfully intercepted.

[0321] From Figure 8 (a), it can be seen that the calculation time of the updated discrete point method guidance command decreases with the increase of the flight time. To a certain extent, the calculation efficiency is improved. Figure 8 (b) shows the positions of the online and offline terminal states in the capture space under 100 Monte Carlo simulations. All the online trajectory terminals are located within the capture space, while the offline trajectory terminals are located outside the capture space 65 times. The results show that this method can effectively improve the guidance performance of the interceptor.

[0322] The simulation results show that the method proposed in Embodiment 1 of the present invention can make full use of the maneuverability of the interceptor during midcourse guidance and improve the interception success rate.

[0323] The foregoing has shown and described the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments, and what is described in the above embodiments and the specification is only to illustrate the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will also have various changes and improvements, and these changes and improvements fall within the scope of the present invention claimed. The scope of the present invention claimed is defined by the appended claims and their equivalents.

Claims

1. An online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space, characterized in that: including Step 1. When the interceptor intercepts a high-speed maneuvering target, transform the midcourse guidance problem of the interceptor intercepting the high-speed maneuvering target into an interceptor midcourse guidance model; The interceptor midcourse guidance model includes an interception dynamics model, a process constraint condition model, a boundary constraint model, a performance index, and a problem description model; Step 2. On the basis of the interceptor midcourse guidance model, transform the midcourse guidance problem into a discrete convex programming problem through an affine control variable relaxation, a Gaussian pseudospectral discretization, and a constraint linearization process; Step 3. Use a convex programming method to solve the midcourse guidance offline interception trajectory of the high-speed maneuvering target, establish an online guidance framework, and update the discrete points in the midcourse guidance process; The process of establishing the online guidance framework and updating the discrete points in the midcourse guidance process described in Step 3 includes Step 31. Use a convex programming method to solve the midcourse guidance offline interception trajectory of the high-speed maneuvering target; Step 32. Establish an online midcourse guidance framework and perform online correction on the trajectory; Step 33. Design a grid update strategy in the online midcourse guidance process; Step 34. Determine the success of the midcourse guidance; Step 35. Construct a midcourse guidance terminal simulation model; The process of determining the success of the midcourse guidance described in Step 34 includes Step 341. Set the actual terminal state of the interceptor's midcourse guidance as x ft =[x ft ; z ft ; h ft ; V ft ; θ ft ; ψ ft , the final ideal corrected terminal state and the final target predicted state of the midcourse guidance are x fi =[x fi ; z fi ; h fi ; ~; θ fi ; ψ fi and x fp =[x fp ; z fp ; h fp ; V fp ; θ fp ; ψ fp , the ideal turn-on distance of the interceptor seeker is l, and the ideal corrected terminal state and the final target predicted state are deduced by the zero-effort miss intercept manifold of the terminal guidance, then x fi and x fp satisfy Step 342. Determine whether the interceptor after trajectory correction satisfies the capture space constraint according to the mid-course guidance terminal state x of the interceptor ft and the predicted state x of the final target fp ; The relative position vector r between the projectile and the target it is In Equation (48), e r = r it / r it is the unit vector in the line-of-sight direction; The missile-target relative velocity vector v itf is The tangential velocity v of the line of sight between the projectile and the target θf is v θf = v θf e θ = v itf -v rf = v itf -v itf e r = v itf -v rf e r (50) In Equation (50), e θ = v θf / v θf represents the unit vector in the tangential direction of the line of sight; v θf represents the magnitude of the velocity in the direction of e θ ; v rf represents the end of midcourse guidance, i.e., the velocity of the initial line of sight direction of terminal guidance, v rf represents the magnitude of the velocity in the direction of e r . Step 343. In the terminal guidance stage, taking the realistic proportional guidance law as an example to intercept the target, according to the capture space given by the target and interceptor information, combined with Equation (51), design the allowable collision velocity v rImp , and solve the initial line-of-sight velocity v of the terminal guidance rf corresponding to the tangential velocity boundary value |v θmax |, and the solution is shown in Equation (51) In Equation (51), α max represents the saturation maneuver acceleration of the interceptor, and α θ represents the upper limit of the acceleration of the target in the e θ direction, and α r represents the upper limit of the acceleration of the target in the e r direction; Step 344. Obtain the judgment criterion for the success of the midcourse guidance of the interceptor: When |v θmax | ≥ |v θf |, it is considered that the correction in the mid-course guidance stage of the interceptor is successful and the target can be successfully intercepted in the terminal guidance stage; When |v θmax | < |v θf |, it is considered that the correction in the mid-course guidance stage of the interceptor fails, but there is a probability of successfully intercepting the target in the terminal guidance stage.

2. The online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 1, characterized in that: The establishment process of the interception dynamics model described in Step 1 includes Step 111. Based on a fixed ground coordinate system, with the x and z axes pointing east and north respectively, and the h axis forming a right-handed coordinate system, and the interceptor on a flat ground, establish the interceptor dynamics model as In formula (1), (h, z, x) are the position coordinates of the interceptor; r = 1 + h is the straight-line distance from the earth's center to the interceptor; V is the relative velocity of the earth, θ is the ballistic inclination angle; ψ is the ballistic deflection angle; V is scaled proportionally; g0 is the gravitational acceleration at the earth's surface at r e ; The dimensionless lift L and drag acceleration D are: In formulas (2) and (3), S is the reference area of the force on the interceptor; C L and C D are respectively the lift coefficient and the drag coefficient of the interceptor, which are related to the angle of attack α and the Mach number Ma; m is the mass of the interceptor, and the atmospheric density Step 112. Define the control variable normalization coefficient λ In formulas (4) and (5), are the lift coefficient and drag coefficient corresponding to the maximum lift-drag ratio at a certain Mach number, respectively; From Equation (4) and Equation (5), transform Equation (2) and Equation (3) into In formula (6), the lift acceleration the drag acceleration Step 113. Construct an affine variable u1 = λcosσ, u2 = λsinσ, u3 = λ 2 (7) Transform the interception dynamics model through the affine variable into Step 114. In the affine system, the control quantity u must satisfy Let the normalization coefficient λ be non - negative, with an upper limit of The value range of u3 is obtained as In formula (10), Let the value range of the tilt angle σ be within the interval [σ min , σ max , then u1tan(σ min ) ≤ u2 ≤ u1tan(σ max ) (11).

3. The online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 1, characterized in that: The establishment process of the boundary constraint model described in Step 1 includes Step 131. Let the initial state of the interceptor midcourse guidance be x0, the initial time be t0, and the initial condition constraint be x(t0) - x0 = 0 (16) Step 132. Assume the ideal terminal state of the interceptor is x p = [h p ; z p ; x p ; ~; θ p ; ψ p , and the end time of the mid-course guidance of the interceptor is t f , and the terminal constraint is x(t f ) = x p (17) Equation (17) is a strong equality constraint, representing that the interceptor reaches the terminal constraint state within the specified time. Now, relax Equation (17): In Equation (18), the terminal constraint relaxation coefficient υ = [υ1; υ2; υ3; υ4; υ5], κ1 and κ2 are weight coefficients, and the equation is rewritten as x(t f ) - x p = κυ (19).

4. The online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 2, wherein: The affine control variable relaxation process described in Step 2 includes Relax Equation (9) to obtain: To ensure the validity of the relaxed affine variables, a term is added to the objective function Ensure that the affine variables satisfy the constraints in Equation (9).

5. The online mid-course interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 3, characterized in that: The Gaussian pseudospectral discretization process described in Step 2 includes Step 221. Express the affine dynamics equation (8) as In Equation (23), u = [u1; u2; u3]; Step 222. Transform the time interval \(t\in[t_0,t f \) to \(\tau\in[-1,1]\), and the transformation formula is After the transformation, the new time variable τ replaces time t as the independent variable, where τ = -1 corresponds to t0 and τ = 1 corresponds to t f ; Step 223. Use the LGR collocation method to approximate the change of the state with a Lagrangian interpolation polynomial basis of order at most N In Equation (28), P j (τ) is an Nth-order Lagrangian polynomial, Take the derivative of Equation (28) with respect to time to obtain Let and D ij The coefficient matrix formed is called the differential approximation matrix, D i ∈R n+1 ; Step 224. Let Discretize the kinetic equation (24) into the following form: D i X - τ c [F(X i ) + Β(X i )U i = 0, i = 1,..., N (32) Due to τ N <1, the final state of midcourse interception by the interceptor cannot be directly represented by the interpolation points, and the Gaussian quadrature rule is used to calculate where, w i represents the Gaussian orthogonal weight of each interpolation point; Discretize the added terms The objective function obtained is 6. The online mid-course interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 1, characterized in that: The constraint linearization process described in Step 2 includes Step 231. Using the condition perform Taylor expansion of the dynamic equation (32) with respect to the reference trajectory {X * , U *}, and omit the subscript i DX-τ c [F(X * )+A(X * )(X-X * )+Β(X * )U]=0 (37) Among them, Linearize the terminal state constraint equation X(τ f ) = X(τ0) + τ c w[F(X * ) + A(X * )(X - X * ) + Β(X * )U] (38) The process constraint equation (15) is Taylor-expanded with respect to the reference trajectory {X * , U *} In formula (39), h * , V * , u3 * are the state variables in the reference trajectory {X * , U *}; Step 232. Use the variable trust region method for the feasible solution constraint In formula (41), ε x and ε u are the trust region intervals of the state variables, and ξ is the trust region relaxation coefficient; After constraining the trust region relaxation coefficient in the performance index, the performance index is expressed as After discretization and linearization, the midcourse guidance problem P1 can be written as a convex programming problem P2, expressed as In Equation (43), the control variable constraints are Equations (10), (11), and (22), the boundary constraints are Equations (16), (17), and (38), the dynamic equation is Equation (37), the process constraint is Equation (39), and the trust region is Equation (41).

7. The online mid-course interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 6, characterized in that: The process of using the convex programming method to solve the midcourse guidance offline interception trajectory of a high-speed maneuvering target described in Step 31 includes Step 311. Set \(k = 0\), determine the discrete points and the initial time domain, and select the initial reference trajectory, that is, the initial solution \((x (0) , u (0) )\); Step 312. In the k-th iteration, solve problem P2 to obtain the results of the optimization variables (x (k) , u (k) ); where k ≥ 1; Step 313. Check whether the optimized variable results satisfy Equation (44). Assume that all discrete points satisfy the convergence conditions in Equation (44), then go to the next step; otherwise, let k = k + 1, and return to Step 312; Step 314. The iteration terminates, and the converged solution is the highly approximate solution of Problem P1.

8. The online midcourse interception method for high-speed maneuvering targets based on convex programming and capture space according to claim 1, characterized in that: The process of designing the grid update strategy in the online midcourse guidance process described in Step 33 includes Establish a grid update strategy N during the midcourse guidance process of the design k In Equation (46), N0 represents the initial number of grid points of the offline reference trajectory, N k represents the number of grid points in the (k-1)-th guidance cycle, N l represents the minimum number of grid points required for the solution accuracy, t k is the start time of the (k-1)-th guidance cycle, t f is the end time of the midcourse guidance.

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