Multi-trajectory generation method and system based on pseudospectral method and multi-objective optimization

Abstract

The application provides a multi-trajectory generation method and system based on a pseudospectral method and multi-objective optimization, and the method comprises the following steps: establishing a trajectory model and an atmospheric parameter model; performing trajectory optimization by using the pseudospectral method to obtain a group of optimal control values; converting the trajectory optimization problem into a multi-objective optimization problem; by means of a multi-objective optimization algorithm, using the optimal control values obtained by the pseudospectral method as an initial population, selecting a target function, performing multi-objective optimization, obtaining a plurality of groups of Pareto optimal solutions and a Pareto frontier, and obtaining a plurality of flight trajectories with different combined optimization targets. The method provides a new initialization strategy for multi-objective optimization, can obtain a high-quality initialization population, and then obtain a plurality of trajectory trajectories with different combined optimization targets, can be applied to aircraft simulated trajectory simulation and multi-missile cooperative design, and is a trajectory generation method with wide application prospect.

Description

Technical Field

[0001] This invention relates to the technical field of ballistic planning for boost-glide vehicles, specifically to a method and system for generating multiple ballistic trajectories based on pseudospectral method and multi-objective optimization, and more particularly to a method for generating multiple ballistic trajectories based on pseudospectral method and multi-objective optimization algorithm. Background Technology

[0002] In recent years, boost-glide vehicles have attracted significant attention from major military powers due to their advantages such as long attack range, wide coverage, high accuracy, good maneuverability, and strong penetration capability. Ballistic planning technology for these vehicles is a crucial research area. As future warfare continues to evolve towards informatization, intelligence, and clustering, the requirements for boost-glide vehicle trajectory planning are no longer limited to planning a single trajectory that meets constraints or optimizes a single performance metric. Instead, the focus shifts to planning multiple selectable trajectories or trajectories that satisfy multiple performance requirements. Therefore, it is necessary to solve the multi-target trajectory optimization problem.

[0003] Currently, multi-objective ballistic optimization problems can be solved using algorithms such as NSGA-II, MOPSO, and MOEA / D. However, when multi-objective optimization algorithms are used for ballistic planning, they suffer from problems such as large computational load, slow search speed, and high randomness. Summary of the Invention

[0004] The purpose of this invention is to provide a method and system for generating multiple ballistic trajectories based on pseudospectral method and multi-target optimization.

[0005] According to the present invention, a method for generating multiple ballistic trajectories based on pseudospectral method and multi-target optimization is provided, the method comprising the following steps:

[0006] Step S1: Establish a ballistic motion model and derive the longitudinal plane ballistic equation, as shown in equation (1), where m is the mass of the aircraft, V is the velocity, P is the engine thrust, α is the angle of attack, θ is the ballistic inclination angle, X is the drag, and Y is the lift; establish an atmospheric parameter model.

[0007]

[0008] Step S2: Determine the mathematical description of the ballistic optimization problem;

[0009] Step S3: Select the objective function, set the ballistic constraints, and use the pseudospectral method to solve the ballistic optimization problem to obtain a set of optimal control variables;

[0010] Step S4: Transform the ballistic optimization problem into a multi-objective optimization problem (MOP), discretize the control variables, select a set of optimal control variables obtained by the pseudospectral method as the initial population, and use the multi-objective optimization algorithm to solve the problem, thereby obtaining multiple sets of Pareto optimal solutions and Pareto fronts.

[0011] Step S5: Obtain multiple ballistic trajectories with different combinations of optimized targets.

[0012] Preferably, the ballistic optimization problem in step S2 can be transformed into an optimal control problem for solution, as shown in equations (2)-(5):

[0013]

[0014]

[0015] ψ(x(t0),t0,x(t f ),t f )=0 (4)

[0016] g(x,u,t)≤0 (5)

[0017] In equation (2), J is the objective function, derived from Φ(x(t0),t0,x(t)). f ),t f )and Composition; x(t)∈R n Let u(t) be the state variable of the system, ∈ R. m t is the system's control variable and t is the system's time variable; equations (3)-(5) are the system's state equation, system boundary constraints, and system parameter inequality constraints, respectively; the objective function J must reach its minimum or maximum value under the condition that equations (3)-(5) are satisfied.

[0018] Preferably, the ballistic constraints in step S3 include process constraints, endpoint constraints, and control quantity constraints;

[0019] Step S3 uses the pseudospectral method to solve the ballistic optimization problem, and the specific details are as follows:

[0020] The objective function for the optimal control problem is selected as the maximum range and maximum terminal velocity indices; endpoint constraints include initial state constraints and terminal state constraints, and process constraints include dynamic pressure constraints, normal overload constraints, and flight altitude constraints; the control variable constraint is the angle of attack constraint; the solution uses a time-domain transformation method to transform the [t0,t] corresponding to the optimal control. f The interval transformation is given by the pseudospectral method, which transforms the collocation points into the interval [-1, 1]. The transformation relationship between the two is as follows:

[0021]

[0022] The time domain interval [t0, t] f Discretize the data and divide it into K sub-intervals [t]. k-1 ,t k ], k=1,2,3,…,K and t0 <t1<t2<t K =t f Then, performing the above time-domain transformation in each subinterval yields:

[0023]

[0024] And there are:

[0025]

[0026] In the interval [t] k-1 ,t k Selecting the collocation points to form the required interpolation nodes within the interval [-1, 1], the state variables and control variables are approximated using Lagrange interpolation polynomials, obtaining a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state and control variables, transforming the optimal control problem into a nonlinear programming (NLP) problem. A sequential quadratic programming algorithm is then used to solve this problem, obtaining a set of optimal values ​​for the control variables x* = (x1, x2, ..., x...). N ).

[0027] Preferably, step S4 uses the MOEA / D algorithm to solve the multi-objective optimization problem, as detailed below:

[0028] The original MOP is decomposed into N single-objective optimization subproblems using the Tchebycheff aggregation method, and N uniformly distributed weight vectors λ are selected. 1 ,λ 2 ,λ 3 ,…,λ N Then the objective function of the l-th subproblem is expressed as:

[0029]

[0030] In equation (10), z * Find the global optimum, λ l This is the weight vector corresponding to the l-th subproblem;

[0031] Set t gen Let λ be the maximum number of generations the algorithm can evolve, N be the population size defined by the algorithm, and λ be the maximum number of generations the algorithm can evolve. 1 ,λ 2 ,λ 3 ,…,λ NLet N be the weight vectors corresponding to the population size, and T be the number of neighbor weight vectors corresponding to each weight vector. The algorithm output is denoted as EP, and x is the weight vector of each weight vector. 1 ,x 2 ,x 3 ,…,x N and FV 1 ,FV 2 ,FV 3 ,…,FV N composition.

[0032] Preferably, step S5 uses the Pareto optimal solution obtained by solving the multi-objective optimization problem using the MOEA / D algorithm in step S4 to calculate multiple ballistic trajectories with different combinations of optimized objectives.

[0033] This invention also provides a multi-trajectory generation system based on pseudospectral method and multi-target optimization, the system comprising the following modules:

[0034] Module M1: Establish a ballistic motion model and derive the longitudinal plane ballistic equation, as shown in equation (1), where m is the mass of the aircraft, V is the velocity, P is the engine thrust, α is the angle of attack, θ is the ballistic inclination angle, X is the drag, and Y is the lift; establish an atmospheric parameter model.

[0035]

[0036] Module M2: Determines the mathematical description of the ballistic optimization problem;

[0037] Module M3: Select the objective function, set the ballistic constraints, and use the pseudospectral method to solve the ballistic optimization problem to obtain a set of optimal control variables;

[0038] Module M4: The ballistic optimization problem is transformed into a multi-objective optimization problem (MOP). The control variables are discretized, and a set of optimal control variables obtained by the pseudospectral method is selected as the initial population. The multi-objective optimization algorithm is used to solve the problem and obtain multiple Pareto optimal solutions and Pareto fronts.

[0039] Module M5: Acquires multiple ballistic trajectories with different optimized combinations of targets.

[0040] Preferably, the ballistic optimization problem in module M2 can be transformed into an optimal control problem for solution, as shown in equations (2)-(5):

[0041]

[0042]

[0043] ψ(x(t0),t0,x(t f ),t f)=0 (4)

[0044] g(x,u,t)≤0 (5)

[0045] In equation (2), J is the objective function, derived from Φ(x(t0),t0,x(t)). f ),t f )and Composition; x(t)∈R n Let u(t) be the state variable of the system, ∈ R. m t is the system's control variable and t is the system's time variable; equations (3)-(5) are the system's state equation, system boundary constraints, and system parameter inequality constraints, respectively; the objective function J must reach its minimum or maximum value under the condition that equations (3)-(5) are satisfied.

[0046] Preferably, the ballistic constraints in module M3 include process constraints, endpoint constraints, and control quantity constraints;

[0047] Module M3 uses the pseudospectral method to solve the ballistic optimization problem, as detailed below:

[0048] The objective function for the optimal control problem is selected as the maximum range and maximum terminal velocity indices; endpoint constraints include initial state constraints and terminal state constraints, process constraints include dynamic pressure constraints, normal overload constraints, and flight altitude constraints; the control variable constraint is the angle of attack constraint; during the solution, a time-domain transformation system is used to transform the optimal control [t0,t]... f The interval transformation is given by the pseudospectral method, which transforms the collocation points into the interval [-1, 1]. The transformation relationship between the two is as follows:

[0049]

[0050] The time domain interval [t0, t] f Discretize the data and divide it into K sub-intervals [t]. k-1 ,t k ], k=1,2,3,…,K and t0 <t1<t2<t K =t f Then, performing the above time-domain transformation in each subinterval yields:

[0051]

[0052] And there are:

[0053]

[0054] In the interval [t] k-1 ,t kSelecting the collocation points to form the required interpolation nodes within the interval [-1, 1], the state variables and control variables are approximated using Lagrange interpolation polynomials, obtaining a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state and control variables, transforming the optimal control problem into a nonlinear programming (NLP) problem. A sequential quadratic programming algorithm is then used to solve this problem, obtaining a set of optimal values ​​for the control variables x* = (x1, x2, ..., x...). N ).

[0055] Preferably, module M4 uses the MOEA / D algorithm to solve the multi-objective optimization problem, as detailed below:

[0056] The original MOP is decomposed into N single-objective optimization subproblems using the Tchebycheff aggregation system, with N uniformly distributed weight vectors λ. 1 ,λ 2 ,λ 3 ,…,λ N Then the objective function of the l-th subproblem is expressed as:

[0057]

[0058] In equation (10), z * Find the global optimum, λ l This is the weight vector corresponding to the l-th subproblem;

[0059] Set t gen Let λ be the maximum number of generations the algorithm can evolve, N be the population size defined by the algorithm, and λ be the maximum number of generations the algorithm can evolve. 1 ,λ 2 ,λ 3 ,…,λ N Let N be the weight vectors corresponding to the population size, and T be the number of neighbor weight vectors corresponding to each weight vector. The algorithm output is denoted as EP, and x is the weight vector of each weight vector. 1 ,x 2 ,x 3 ,…,x N and FV 1 ,FV 2 ,FV 3 ,…,FV N composition.

[0060] Preferably, module M5 uses the Pareto optimal solution obtained by solving the multi-objective optimization problem using the MOEA / D algorithm in module M4 to calculate multiple ballistic trajectories with different combinations of optimized objectives.

[0061] Compared with the prior art, the present invention has the following beneficial effects:

[0062] 1. This invention provides a new initialization strategy for multi-objective optimization, which can obtain a high-quality initial population and thus obtain multiple ballistic trajectories with different combinations of optimization objectives. It can be applied to aircraft ballistic simulation and multi-missile collaborative design, and is a ballistic generation method with broad application prospects.

[0063] 2. The present invention uses a set of optimal control variables obtained by the pseudospectral method as the initial population of the multi-objective optimization problem, which can improve the quality of the initial population, speed up the search, and facilitate obtaining the Pareto optimal solution. Attached Figure Description

[0064] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0065] Figure 1 This is a schematic diagram of the process of the present invention;

[0066] Figure 2 The specific embodiments provided for this invention include graphs showing the variation of atmospheric pressure, density, and sound speed with altitude, plotted with reference to the atmospheric parameters of the 1976 US standard.

[0067] Figure 3 A set of curves showing the angle of attack of the optimal control quantity changing with time, obtained by the pseudospectral method in a specific embodiment of the present invention;

[0068] Figure 4 In a specific embodiment of the present invention, the Pareto front plot is obtained by using the maximum range and the maximum terminal velocity as objective functions;

[0069] Figure 5 The range-height variation curve is calculated using the Pareto optimal solution obtained through multi-objective optimization in a specific embodiment of the present invention.

[0070] Figure 6 The velocity-time variation curve is calculated using the Pareto optimal solution obtained through multi-objective optimization in a specific embodiment of the present invention. Detailed Implementation

[0071] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0072] Example 1:

[0073] The present invention provides a method for generating multiple ballistic trajectories based on pseudospectral method and multi-target optimization, the method comprising the following steps:

[0074] Step S1: Establish a ballistic motion model and derive the longitudinal plane ballistic equation, as shown in equation (1), where m is the mass of the aircraft, V is the velocity, P is the engine thrust, α is the angle of attack, θ is the ballistic inclination angle, X is the drag, and Y is the lift; establish an atmospheric parameter model.

[0075]

[0076] Step S2: Determine the mathematical description of the ballistic optimization problem; the ballistic optimization problem can be transformed into an optimal control problem for solution, as shown in equations (2)-(5):

[0077]

[0078]

[0079] ψ(x(t0),t0,x(t f ),t f )=0 (4)

[0080] g(x,u,t)≤0 (5)

[0081] In equation (2), J is the objective function, derived from Φ(x(t0),t0,x(t)). f ),t f )and Composition; x(t)∈R n Let u(t) be the state variable of the system, ∈ R. m t is the system's control variable and t is the system's time variable; equations (3)-(5) are the system's state equation, system boundary constraints, and system parameter inequality constraints, respectively; the objective function J must reach its minimum or maximum value under the condition that equations (3)-(5) are satisfied.

[0082] Step S3: Select the objective function, set the ballistic constraints, and use the pseudospectral method to solve the ballistic optimization problem to obtain a set of optimal control variables. The ballistic constraints include process constraints, endpoint constraints, and control variable constraints. The specific details of solving the ballistic optimization problem using the pseudospectral method are as follows:

[0083] The objective function for the optimal control problem is selected as the maximum range and maximum terminal velocity indices; endpoint constraints include initial state constraints and terminal state constraints, and process constraints include dynamic pressure constraints, normal overload constraints, and flight altitude constraints; the control variable constraint is the angle of attack constraint; the solution uses a time-domain transformation method to transform the [t0,t] corresponding to the optimal control. f The interval transformation is given by the pseudospectral method, which transforms the collocation points into the interval [-1, 1]. The transformation relationship between the two is as follows:

[0084]

[0085] The time domain interval [t0, t] f Discretize the data and divide it into K sub-intervals [t]. k-1 ,t k ], k=1,2,3,…,K and t0 <t1<t2<t K =t f Then, performing the above time-domain transformation in each subinterval yields:

[0086]

[0087] And there are:

[0088]

[0089] In the interval [t] k-1 ,t k Selecting the collocation points to form the required interpolation nodes within the interval [-1, 1], the state variables and control variables are approximated using Lagrange interpolation polynomials, obtaining a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state and control variables, transforming the optimal control problem into a nonlinear programming (NLP) problem. A sequential quadratic programming algorithm is then used to solve this problem, obtaining a set of optimal values ​​for the control variables x* = (x1, x2, ..., x...). N ).

[0090] Step S4: Transform the ballistic optimization problem into a multi-objective optimization problem (MOP). Discretize the control variables, select a set of optimal control variables obtained by the pseudospectral method as the initial population, and use a multi-objective optimization algorithm to solve the problem, obtaining multiple Pareto optimal solutions and Pareto fronts. The MOEA / D algorithm is then used to solve the multi-objective optimization problem, as detailed below:

[0091] The original MOP is decomposed into N single-objective optimization subproblems using the Tchebycheff aggregation method, and N uniformly distributed weight vectors λ are selected. 1 ,λ 2 ,λ 3 ,…,λ N Then the objective function of the l-th subproblem is expressed as:

[0092]

[0093] In equation (10), z * Find the global optimum, λ l This is the weight vector corresponding to the l-th subproblem;

[0094] Set t gen Let λ be the maximum number of generations the algorithm can evolve, N be the population size defined by the algorithm, and λ be the maximum number of generations the algorithm can evolve. 1 ,λ 2 ,λ 3 ,…,λ N Let N be the weight vectors corresponding to the population size, and T be the number of neighbor weight vectors corresponding to each weight vector. The algorithm output is denoted as EP, and x is the weight vector of each weight vector. 1 ,x 2 ,x 3 ,…,x N and FV 1 ,FV 2 ,FV 3 ,…,FV N composition.

[0095] Step S5: Obtain multiple ballistic trajectories with different combinations of optimized targets; calculate multiple ballistic trajectories with different combinations of optimized targets using the Pareto optimal solution obtained by solving the multi-target optimization problem using the MOEA / D algorithm in step S4.

[0096] Example 2:

[0097] Example 2 is a preferred embodiment of Example 1, and is used to illustrate the present invention in more detail.

[0098] This invention also provides a multi-trajectory generation system based on pseudospectral method and multi-target optimization, the system comprising the following modules:

[0099] Module M1: Establish a ballistic motion model and derive the longitudinal plane ballistic equation, as shown in equation (1), where m is the mass of the aircraft, V is the velocity, P is the engine thrust, α is the angle of attack, θ is the ballistic inclination angle, X is the drag, and Y is the lift; establish an atmospheric parameter model.

[0100]

[0101] Module M2: Determines the mathematical description of the ballistic optimization problem; the ballistic optimization problem can be transformed into an optimal control problem for solution, as shown in equations (2)-(5):

[0102]

[0103]

[0104] ψ(x(t0),t0,x(t f ),t f )=0 (4)

[0105] g(x,u,t)≤0 (5)

[0106] In equation (2), J is the objective function, derived from Φ(x(t0),t0,x(t)). f ),t f )and Composition; x(t)∈R n Let u(t) be the state variable of the system, ∈ R. m t is the system's control variable and t is the system's time variable; equations (3)-(5) are the system's state equation, system boundary constraints, and system parameter inequality constraints, respectively; the objective function J must reach its minimum or maximum value under the condition that equations (3)-(5) are satisfied.

[0107] Module M3: Select the objective function, set the ballistic constraints, and use the pseudospectral method to solve the ballistic optimization problem, obtaining a set of optimal control variables. The ballistic constraints include process constraints, endpoint constraints, and control variable constraints. The specific content of solving the ballistic optimization problem using the pseudospectral method is as follows:

[0108] The objective function for the optimal control problem is selected as the maximum range and maximum terminal velocity indices; endpoint constraints include initial state constraints and terminal state constraints, and process constraints include dynamic pressure constraints, normal overload constraints, and flight altitude constraints; the control variable constraint is the angle of attack constraint; the solution uses a time-domain transformation method to transform the [t0,t] corresponding to the optimal control. f The interval transformation is given by the pseudospectral method, which transforms the collocation points into the interval [-1, 1]. The transformation relationship between the two is as follows:

[0109]

[0110] The time domain interval [t0, t] f Discretize the data and divide it into K sub-intervals [t]. k-1 ,t k ], k=1,2,3,…,K and t0 <t1<t2<t K =t f Then, performing the above time-domain transformation in each subinterval yields:

[0111]

[0112] And there are:

[0113]

[0114] In the interval [t] k-1 ,t kSelecting the collocation points to form the required interpolation nodes within the interval [-1, 1], the state variables and control variables are approximated using Lagrange interpolation polynomials, obtaining a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state and control variables, transforming the optimal control problem into a nonlinear programming (NLP) problem. A sequential quadratic programming algorithm is then used to solve this problem, obtaining a set of optimal values ​​for the control variables x* = (x1, x2, ..., x...). N ).

[0115] Module M4: Transforms the ballistic optimization problem into a multi-objective optimization problem (MOP). It discretizes the control variables, selects a set of optimal control variables obtained by the pseudospectral method as the initial population, and uses a multi-objective optimization algorithm to solve the problem, obtaining multiple Pareto optimal solutions and the Pareto front. The MOEA / D algorithm is then used to solve the multi-objective optimization problem. Specific details are as follows:

[0116] The original MOP is decomposed into N single-objective optimization subproblems using the Tchebycheff aggregation method, and N uniformly distributed weight vectors λ are selected. 1 ,λ 2 ,λ 3 ,…,λ N Then the objective function of the l-th subproblem is expressed as:

[0117]

[0118] In equation (10), z * Find the global optimum, λ l This is the weight vector corresponding to the l-th subproblem;

[0119] Set t gen Let λ be the maximum number of generations the algorithm can evolve, N be the population size defined by the algorithm, and λ be the maximum number of generations the algorithm can evolve. 1 ,λ 2 ,λ 3 ,…,λ N Let N be the weight vectors corresponding to the population size, and T be the number of neighbor weight vectors corresponding to each weight vector. The algorithm output is denoted as EP, and x is the weight vector of each weight vector. 1 ,x 2 ,x 3 ,…,x N and FV 1 ,FV 2 ,FV 3 ,…,FV N composition.

[0120] Module M5: Obtain multiple ballistic trajectories with different combinations of optimized targets; calculate multiple ballistic trajectories with different combinations of optimized targets by using the Pareto optimal solution obtained by solving the multi-target optimization problem using the MOEA / D algorithm in Module M4.

[0121] Example 3:

[0122] Example 3 is a preferred example of Example 1, and is used to illustrate the present invention in more detail.

[0123] The purpose of this invention is to provide a method for generating multiple ballistic trajectories based on the pseudospectral method and multi-objective optimization algorithm. By using a set of optimal solutions obtained by the pseudospectral method as the initial population for the multi-objective ballistic optimization problem, the quality of the initial population is improved, the Pareto optimal solution is quickly obtained, and multiple ballistic trajectories with different combinations of optimization objectives are generated.

[0124] The following detailed description, in conjunction with the accompanying drawings and specific embodiments, provides a method for generating multiple ballistic trajectories based on pseudospectral method and multi-objective optimization algorithm. Figure 1 This is a schematic diagram of the workflow of the method, and the specific steps include:

[0125] Step 1 derives the longitudinal plane ballistic equation, as shown in equation (1), where m is the aircraft mass, V is the flight velocity, P is the engine thrust, α is the angle of attack, θ is the trajectory inclination angle, X and Y are the drag and lift obtained by decomposing the aerodynamic forces in the velocity coordinate system, respectively, X = C x qS, Y = C y qS,C x C y These represent the drag coefficient and lift coefficient, respectively; S is the aircraft reference area; q is the incoming dynamic pressure, q = 0.5ρV. 2 ρ represents air density; the atmospheric parameter model adopts the US 1976 standard atmospheric parameters, and the curves of atmospheric pressure, density, and sound speed as a function of altitude are shown in the figure. Figure 2 As shown.

[0126]

[0127] The mathematical description of the ballistic optimization problem in step 2 can be transformed into an optimal control problem for solution, as shown in equations (2)-(5).

[0128]

[0129]

[0130] ψ(x(t0),t0,x(t f ),t f )=0 (4)

[0131] g(x,u,t)≤0 (5)

[0132] In equation (2), J is the objective function, derived from Φ(x(t0),t0,x(t)). f ),t f (Performance index function related to endpoint constraints) and (Integral performance index function) consists of; x(t)∈R n Let u(t) be the state variable of the system, ∈ R. m t is the system's control variable and t is the system's time variable; equations (3)-(5) are the system's state equation, system boundary constraints, and system parameter inequality constraints, respectively; the objective function must reach its minimum (or maximum) value under the condition that equations (3)-(5) are satisfied.

[0133] Step 3 uses the pseudospectral method to solve the trajectory optimization problem, optimizing the glide segment of the boost-glide vehicle's trajectory from the apex (approximately 50 km) to an altitude of 20 km, where the engine thrust is zero. The maximum range is chosen as the objective function; endpoint constraints are shown in Table 1, and process constraints include the dynamic pressure constraint q. max Normal overload constraint n ymax and flight altitude constraints y max As shown in equation (6); the control quantity constraint is the angle of attack constraint.

[0134] Table 1 Endpoint Constraint Parameters

[0135] Parameter name numerical values ballistic peak velocity 1858m / s Ballistic Vertex Height 49.8km Ballistic Apex Ballistic Inclination <![CDATA[-6.2×10 -6 °]]> ballistic terminal height 20km terminal velocity of the ballistic trajectory 500m / s

[0136]

[0137] The solution uses a time-domain transformation method to transform the [t0,t] domain of the optimal control problem. f The interval transformation is given by the pseudospectral method, which transforms the collocation points into the interval [-1, 1]. The transformation relationship between the two is as follows:

[0138]

[0139] Now let the time domain interval [t0, t f Discretize the data and divide it into K sub-intervals [t]. k-1 ,t k ], k=1,2,3,…,K and t0 <t1<t2<t K =t f Then, performing the above time-domain transformation in each subinterval yields:

[0140]

[0141] And there are:

[0142]

[0143] In the interval [t] k-1 ,t k The required interpolation nodes within the interval [-1, 1] are selected by choosing collocation points. The state variables and control variables are approximated using Lagrange interpolation polynomials, thus obtaining a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state variables x(t) and control variables u(t), transforming the optimal control problem into a nonlinear programming problem. A mature sequential quadratic programming algorithm is then used to solve this problem, obtaining a set of optimal values ​​for the control variables x* = (x1, x2, ..., x...). N The simulation results are as follows: Figure 3 As shown.

[0144] Step 4 uses the MOEA / D algorithm to solve the multi-objective optimization problem. Taking the minimization problem as an example, the mathematical description of the multi-objective optimization problem (MOP) can be expressed in the following form:

[0145]

[0146] In equation (10), f1(x), f2(x), ..., f n Let (x) be n objective functions, which together form the objective function vector f(x), g i (x)≤0 and h i (x) = 0 represents the k inequality constraints and l equality constraints of the MOP, respectively, where x = (x1, x2, x3, ..., x...). m Let f represent the decision vector; if there exists f: Ω → Φ, then Ω ∈ R. m ,Φ∈R n Let Ω and Φ be the decision space and objective space of the MOP, respectively.

[0147] The original MOP is decomposed into N single-objective optimization subproblems using the Tchebycheff aggregation method, and N uniformly distributed weight vectors λ are selected. 1 ,λ 2 ,λ 3 ,…,λ N Then the objective function of the l-th subproblem can be expressed as:

[0148]

[0149] In equation (11), z * Generally, the global optimum (also known as the ideal point) is chosen, λ. l This is the weight vector corresponding to the l-th subproblem.

[0150] Before using the MOEA / D algorithm for optimization, the ballistic optimization problem needs to be discretized. This paper adopts the direct firing method to transform it into an NLP problem. The discretization process of the control variables in the direct firing method is described as follows:

[0151]

[0152] The flight time is divided into M time periods, t0 and t... f These represent the start and end times of the flight segment, respectively; u(t) represents the control variable. After discretizing the control variable, the MOEA / D algorithm is used to optimize the control variable u(t), i.e., each individual in the population in the MOEA / D algorithm... The corresponding control variable is u(t).

[0153] When performing multi-objective optimization design for the gliding segment, the maximum range and the maximum terminal velocity are taken as objective functions, as shown in Equation (13). During the optimization process, the angle of attack is used as the control variable, and the inequality constraints are reflected in the objective function in the form of a penalty function.

[0154]

[0155] In equation (13), Indicates range, This indicates the terminal velocity.

[0156] The initial parameters are shown in Table 2:

[0157] Table 2 Initial State Parameters

[0158] Parameter name numerical values ballistic peak velocity 1858m / s Ballistic Vertex Height 49.8km Ballistic Apex Ballistic Inclination <![CDATA[-6.2×10 -6 °]]>

[0159] Set t gen Let λ be the maximum number of generations the algorithm can evolve, N be the population size defined by the algorithm, and λ be the maximum number of generations the algorithm can evolve. 1 ,λ 2 ,λ 3 ,…,λ N Let N be the weight vectors corresponding to the population size, and T be the number of neighbor weight vectors corresponding to each weight vector. The algorithm output is denoted as EP, and x is the weight vector of each weight vector. 1 ,x 2 ,x 3 ,…,x N and FV 1 ,FV 2 ,FV 3 ,…,FV N Composition. During simulation, the multi-objective optimization settings include a population size of N = 400 and an iteration count of It = 20. The MOEA / D algorithm workflow is as follows:

[0160] 1) Initialization operation:

[0161] 1.1) Set the algorithm output EP to an empty set;

[0162] 1.2) Calculate any two weight vectors λ i and λ j Find the weight vector λ by using the Euclidean distance between them. i The weight vector of the T neighbors is denoted as Record the index values ​​of the neighbor weight vector as B(i) = {i1, i2, i3, ..., i T}(i=1,…,N);

[0163] 1.3) Select a set of optimal control variables x* obtained by the pseudospectral method as the initial population x 1 ,x 2 ,x 3 ,…,x N ∈Ω, calculate the corresponding x i The objective function value F(x) i );

[0164] 1.4) Initialize z = (z1, z2, ..., z m ) T Take z i =min{f i (x)|x∈Ω},i=1,2,3,…,m

[0165] 2) Iterative calculation and update:

[0166] For i = 1, ..., N, perform the following operation:

[0167] 2.1) Recombination: From record B i Two indices k and l are randomly selected, and the two individuals corresponding to them are x. k and x l A new individual y is generated from the two individuals using a genetic operator. y is then repaired and improved to obtain y′. If y′ exceeds the range of Ω, a random number within the boundary is used instead.

[0168] 2.2) Update z: For j = 1, ..., m, calculate the function value f corresponding to the new individual y'. j (y'), if z j >f j If (y'), then z needs to be updated, making z j =f j (y');

[0169] 2.3) Update the neighborhood solution: For each j∈B(i), if g te (y'|λ j ,z)≤g te (x j |λj If , z), then it means that the new individual y' is different from x. j Better yet, it should be updated to make x j =y',FV j =F(y').

[0170] 2.4) Update the algorithm output EP: For vectors in the output EP that are dominated by F(y'), remove them. If F(y') is not dominated by vectors in EP, store F(y') in EP.

[0171] 3) Determine if the algorithm has stopped:

[0172] If the algorithm meets the stopping criterion, output EP; otherwise, go to 2.1) to continue iterative updates.

[0173] The simulation was performed according to the above algorithm flow, and the results were as follows: Figure 4 The Pareto front is shown with maximum range and maximum terminal velocity as objective functions.

[0174] Step 5 uses the multiple Pareto optimal solutions obtained in Step 4 as control variables to calculate multiple ballistic trajectories with different optimized combinations. The simulation results are as follows: Figure 5 and Figure 6 As shown.

[0175] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention by utilizing the methods and techniques disclosed above without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

[0176] Those skilled in the art can understand this embodiment as a more specific description of Embodiment 1 and Embodiment 2.

[0177] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0178] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A method for generating multiple ballistic trajectories based on pseudospectral method and multi-objective optimization, characterized in that, The method includes the following steps: Step S1: Establish a ballistic motion model and derive the longitudinal plane ballistic equation, as shown in equation (1), where For the mass of the aircraft, For speed, For engine thrust, For the angle of attack, For the trajectory inclination angle, As resistance, To generate lift, an atmospheric parameter model is established. （1） Step S2: Determine the mathematical description of the ballistic optimization problem; Step S3: Select the objective function, set the ballistic constraints, and use the pseudospectral method to solve the ballistic optimization problem to obtain a set of optimal control variables; Step S4: Transform the ballistic optimization problem into a multi-objective optimization problem (MOP), discretize the control variables, select a set of optimal control variables obtained by the pseudospectral method as the initial population, and use the multi-objective optimization algorithm to solve the problem, thereby obtaining multiple sets of Pareto optimal solutions and Pareto fronts. Step S5: Obtain multiple ballistic trajectories with different combinations of optimized targets; Step S4 uses the MOEA / D algorithm to solve the multi-objective optimization problem, and the specific content is as follows: The original MOP was decomposed using the Tchebycheff polymerization method. There are several single-objective optimization subproblems, taking uniformly distributed... Weight vectors Then the first The objective function for each subproblem is expressed as: （10） In equation (10), Find the global optimum. For the first The weight vectors corresponding to each sub-problem; set up The maximum number of generations the algorithm can evolve. Population size defined for the algorithm, For the corresponding population size A weight vector, The number of neighboring weight vectors corresponding to each weight vector is denoted as EP in the algorithm output. and composition.

2. According to the multi-trajectory generation method based on pseudospectral method and multi-objective optimization as described in claim 1, the trajectory optimization problem in step S2 can be transformed into an optimal control problem for solution, as shown in equations (2)-(5): （2） （3） （4） （5） In equation (2), The objective function is... and composition; For the system's state variables, For the system's control variables, The system time variable is represented by equations (3)-(5), which are the system state equation, system boundary constraints, and inequality constraints of system parameters, respectively; the objective function is... The minimum or maximum value must be reached under the condition that equations (3)-(5) are satisfied.

3. The multi-trajectory generation method based on pseudospectral method and multi-objective optimization according to claim 1, wherein the ballistic constraints in step S3 include process constraints, endpoint constraints and control quantity constraints; Step S3 uses the pseudospectral method to solve the ballistic optimization problem, and the specific details are as follows: The objective function for the optimal control problem is selected as the maximum range and maximum terminal velocity indices; endpoint constraints include initial state constraints and terminal state constraints, process constraints include dynamic pressure constraints, normal overload constraints, and flight altitude constraints; the control variable constraint is the angle of attack constraint; the solution uses a time-domain transformation method to transform the optimal control... The interval transformation is the interval containing the collocation points of the pseudospectral method. The transformation relationship between the two is as follows: （6） Time domain interval Discretize and divide into Sub-intervals And there are Then, performing the above time-domain transformation in each subinterval yields: （7） And there are: （8） In the interval Selecting collocation points to form intervals The required interpolation nodes are used to approximate the state and control variables using Lagrange interpolation polynomials, resulting in a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state and control variables, transforming the optimal control problem into a nonlinear programming (NLP) problem, which is then solved using a sequential quadratic programming algorithm to obtain a set of optimal values ​​for the control variables. .

4. The multi-trajectory generation method based on pseudospectral method and multi-objective optimization according to claim 1, wherein step S5 uses the Pareto optimal solution obtained by solving the multi-objective optimization problem by the MOEA / D algorithm in step S4 to calculate multiple ballistic trajectories with different combinations of optimization objectives.

5. A multi-trajectory generation system based on pseudospectral method and multi-target optimization, characterized in that, The system includes the following modules: Module M1: Establish the ballistic motion model and derive the longitudinal plane ballistic equation, as shown in equation (1), where For the mass of the aircraft, For speed, For engine thrust, For the angle of attack, For the trajectory inclination angle, As resistance, To generate lift, an atmospheric parameter model is established. （1） Module M2: Determines the mathematical description of the ballistic optimization problem; Module M3: Select the objective function, set the ballistic constraints, and use the pseudospectral method to solve the ballistic optimization problem to obtain a set of optimal control variables; Module M4: The ballistic optimization problem is transformed into a multi-objective optimization problem (MOP). The control variables are discretized, and a set of optimal control variables obtained by the pseudospectral method is selected as the initial population. The multi-objective optimization algorithm is used to solve the problem and obtain multiple Pareto optimal solutions and Pareto fronts. Module M5: Acquires multiple ballistic trajectories with different optimized combinations of targets; The module M4 uses the MOEA / D algorithm to solve multi-objective optimization problems, as detailed below: The original MOP was decomposed using the Tchebycheff polymerization system. There are several single-objective optimization subproblems, each with a uniform distribution. Weight vectors Then the first The objective function for each subproblem is expressed as: （10） In equation (10), Find the global optimum. For the first The weight vectors corresponding to each sub-problem; set up The maximum number of generations the algorithm can evolve. Population size defined for the algorithm, For the corresponding population size A weight vector, The number of neighboring weight vectors corresponding to each weight vector is denoted as EP in the algorithm output. and composition.

6. In the multi-trajectory generation system based on pseudospectral method and multi-objective optimization according to claim 5, the trajectory optimization problem in module M2 can be transformed into an optimal control problem for solution, as shown in equations (2)-(5): （2） （3） （4） （5） In equation (2), The objective function is... and composition; For the system's state variables, For the system's control variables, The system time variable is represented by equations (3)-(5), which are the system state equation, system boundary constraints, and inequality constraints of system parameters, respectively; the objective function is... The minimum or maximum value must be reached under the condition that equations (3)-(5) are satisfied.

7. The multi-trajectory generation system based on pseudospectral method and multi-objective optimization according to claim 5, wherein the ballistic constraints in module M3 include process constraints, endpoint constraints and control quantity constraints; Module M3 uses the pseudospectral method to solve the ballistic optimization problem, as detailed below: The objective function for the optimal control problem is selected as the maximum range and maximum terminal velocity indices; endpoint constraints include initial state constraints and terminal state constraints, process constraints include dynamic pressure constraints, normal overload constraints, and flight altitude constraints; the control variable constraint is the angle of attack constraint; during the solution, a time-domain transformation system is used to transform the optimal control... The interval transformation is the interval containing the collocation points of the pseudospectral method. The transformation relationship between the two is as follows: （6） Time domain interval Discretize and divide into Sub-intervals And there are Then, performing the above time-domain transformation in each subinterval yields: （7） And there are: （8） In the interval Selecting collocation points to form intervals The required interpolation nodes are used to approximate the state and control variables using Lagrange interpolation polynomials, resulting in a pseudospectral approximation of the system dynamics equations, constraints, and objective function. The pseudospectral method simultaneously discretizes the system's state and control variables, transforming the optimal control problem into a nonlinear programming (NLP) problem. A sequential quadratic programming algorithm is then used to solve this problem, obtaining a set of optimal values ​​for the control variables. .

8. The multi-trajectory generation system based on pseudospectral method and multi-target optimization according to claim 5, wherein module M5 uses the Pareto optimal solution obtained by solving the multi-target optimization problem using the MOEA / D algorithm in module M4 to calculate multiple ballistic trajectories with different combinations of optimized targets.

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