Aircraft trajectory optimization method combining Birkhoff interpolation and LGL pseudospectral method

Abstract

The present invention discloses an aircraft trajectory optimization method combining Birkhoff interpolation and LGL pseudo-spectral method, and uses the pseudo-spectral method in the numerical solution method to solve the aircraft trajectory optimization problem model. In view of the problem that the differential matrix condition number is high due to the discretization interpolation problem in the process of solving the optimization problem by the pseudo-spectral method, the condition number of the differential matrix in the pseudo-spectral method is greatly reduced by introducing Birkhoff interpolation, thereby reducing the solution time of the aircraft trajectory optimization problem model. The present invention introduces a genetic algorithm in the bionic algorithm to pre-solve the nonlinear programming problem after the discretization transformation of the pseudo-spectral method, and then uses the result of the genetic algorithm solution as the initial value of the sequential quadratic programming algorithm solution, thereby avoiding the divergence of the sequential quadratic programming algorithm due to the sensitivity of the initial value, improving the solution efficiency of the aircraft trajectory optimization problem model by the pseudo-spectral method discretization, and finally improving the trajectory optimization efficiency of the aircraft and the accuracy of problem solving.

Description

Technical Field

[0001] The present invention belongs to the technical field of trajectory optimization, and in particular relates to the design of an aircraft trajectory optimization method combining Birkhoff interpolation and LGL pseudo-spectral method. Background Art

[0002] With the continuous improvement of my country's modernization process, the country's requirements for civil aircraft, rockets, missiles, flying fighters and other aircraft are constantly increasing. As an indispensable tool for national defense and people's travel, the performance requirements of numerous aircraft are constantly increasing. A good aircraft trajectory can not only help the initial modeling design and optimization design of the aircraft, but also help improve the actual flight efficiency of the aircraft, reduce unnecessary flights, and reduce fuel losses.

[0003] The traditional method of aircraft trajectory optimization is to discretize the trajectory optimal control problem through the pseudo-spectral method, directly solve the differential constraints in the problem through the differential matrix constructed based on interpolation, and then solve the discretized nonlinear programming problem through the commonly used sequential quadratic programming problem. Finally, the optimized solution is approximated by interpolation to obtain the solution of the original problem before discretization, thereby realizing the optimization of the aircraft trajectory.

[0004] Since the condition number of the differential matrix constructed based on interpolation is often high (the condition number refers to the measure of the ease of numerical calculation of a problem when solving a problem through numerical methods, that is, the well-posedness of the problem. For a problem with a low condition number, we call it a well-conditioned problem, and for a problem with a higher condition number, we call it an ill-conditioned or non-well-conditioned problem), the existing aircraft trajectory optimization methods usually require a large number of discretized matching points, which in turn leads to an increase in the number of solution iterations when solving the problem, a longer overall solution time for the problem, and insufficient interpolation efficiency. Summary of the invention

[0005] The purpose of the present invention is to solve the problem that the existing aircraft trajectory optimization method based on the pseudo-spectral method has a high differential matrix condition number in the process of solving the trajectory optimization problem, which leads to low trajectory optimization efficiency. A vehicle trajectory optimization method combining Birkhoff interpolation and LGL pseudo-spectral method is proposed.

[0006] The technical solution of the present invention is: a method for optimizing the trajectory of an aircraft combining Birkhoff interpolation and LGL pseudo-spectral method, comprising the following steps:

[0007] S1. Construct a model for the aircraft trajectory optimization problem.

[0008] S2. Select Legendre polynomials as the interpolation basis to initialize and transform the aircraft trajectory optimization problem model in the time domain.

[0009] S3. Select n nodes through the Gauss Lobato numerical quadrature method to discretize the aircraft trajectory optimization problem model, and obtain the discrete time corresponding to the aircraft trajectory optimization problem model, as well as the state variables and control variables corresponding to the discrete time.

[0010] S4. Select LGL collocation points according to Legendre polynomials, and discretize state variables and control variables at the LGL collocation points to obtain discrete state variables and discrete control variables.

[0011] S5. Interpolate the discrete state variables and the discrete control variables through the Lagrange interpolation basis to obtain the state function and the control function.

[0012] S6. Obtain the differential matrix according to the LGL point matching, and combine the differential matrix with the differential expressions of the state function and the control function to obtain the differential expressions of the state variables and the control variables.

[0013] S7. Combining the differential expressions of the state variables and the control variables with the aircraft trajectory optimization problem model, a collocation equation is obtained, and the differential constraints of the aircraft trajectory optimization problem model are obtained based on the collocation equation.

[0014] S8. Generate a new matrix through Birkhoff interpolation, optimize the differential matrix in the differential constraint through the new matrix, and obtain a new differential constraint.

[0015] S9, using a genetic algorithm to pre-solve the discretized aircraft trajectory optimization problem model to obtain a pre-solved result;

[0016] S10, using the pre-solved result as the initial value of the SQP algorithm, and using the SQP algorithm to solve the discretized aircraft trajectory optimization problem model to obtain a discretized solution;

[0017] S11. Perform interpolation approximation and restoration processing on the discretized solution to obtain a solution to the aircraft trajectory optimization problem model, thereby optimizing the aircraft trajectory.

[0018] Furthermore, the aircraft trajectory optimization problem model in step S1 is expressed as:

[0019]

[0020]

[0021]

[0022]

[0023] Where J[·] is the objective function, which represents the maximum flight distance of the aircraft in the gliding phase, x = [H, V, γ] is the trajectory state variable, H, V, γ represent the flight altitude, flight speed and flight track angle of the aircraft respectively, are the differentials of H, V, and γ respectively, u=[α] is the trajectory control variable, α represents the angle of attack, t represents the time, and t 0 represents the starting time of the aircraft trajectory, t f represents the end time of the aircraft trajectory, D represents the resultant force in the direction of resistance, L represents the lift, m represents the mass of the aircraft, g represents the acceleration of gravity, and δ represents the pitch rudder deflection that maintains longitudinal balance under the optimal control variable corresponding to the flight attitude.

[0024] Furthermore, the time domain transformation formula in step S2 is:

[0025]

[0026] Where t represents time, t 0 represents the starting time of the aircraft trajectory, t f represents the termination time of the vehicle trajectory, and τ represents the LGL collocation point.

[0027] Furthermore, the discrete time in step S3 is represented by T n =[t 1 t 2 ...t n t n+1 ], where t p represents the pth discrete time point, and the state variable corresponding to the discrete time is expressed as X n =[H n , V n , γ n ], where H n =[h 1 h 2 ...h n h n+1 ] represents the flight altitude state quantity, h p represents the pth discrete flight height value, V n =[v 1 v 2 ...v n v n+1 ] represents the flight speed state, v p represents the pth discrete flight speed value, γ n =[r 1 r 2 ...r n r n+1 ] represents the flight path angle state, rp It represents the pth discrete flight path angle, and the control variable corresponding to the discrete time is expressed as U n =[α n ], where α n =[a 1 a 2 ...a n a n+1 ] represents the discrete representation of the angle of attack, a p Represents the pth discrete angle of attack, with count value p=1, 2, ..., n, n+1.

[0028] Further, step S4 includes the following sub-steps:

[0029] S41, (1-τ 2 )L n The roots of (τ) are used as LGL collocation points, where L n (τ) is the Legendre polynomial.

[0030] S42, the state variable X n and the control variable U n Discretize at the LGL collocation point to obtain n+1 discrete state variables X i and n+1 discrete control variables U i , i=0, 1, 2, ..., n.

[0031] Furthermore, the formula for interpolating the discrete state variables and the discrete control variables in step S5 is:

[0032]

[0033] Where X(τ) represents the state function, U(τ) represents the control function, represents the n-order Lagrange interpolation basis, which is expressed as:

[0034]

[0035] where τ represents the LGL collocation point, τ i represents the i-th LGL collocation point, τ j represents the jth LGL collocation point.

[0036] Further, step S6 includes the following sub-steps:

[0037] S61, according to LGL point matching τ=(τ 1 , τ 2 ...τ n+1 ) to obtain the differential matrix D ki :

[0038]

[0039] where τ k represents the kth LGL collocation point, L n (τ k ) represents the Legendre polynomial of the kth LGL collocation point.

[0040] S62. Obtain differential expressions of the state function X(τ) and the control function U(τ):

[0041]

[0042] in Represents the differential expression of the state function X(τ), Represents the differential expression of the control function U(τ), represents the differential of the Lagrangian interpolation basis of order n.

[0043] S63. Combining the differential matrix with the differential expressions of the state function and the control function, the differential expressions of the state variable and the control variable are obtained:

[0044]

[0045] in represents the differential expression of the state variable, Represents the differential expression of the control variable.

[0046] Further, step S7 includes the following sub-steps:

[0047] S71. Substitute the differential expressions of state variables and control variables into the aircraft trajectory optimization problem model to obtain the collocation equation:

[0048]

[0049] Where F(X(τ k ), U(τ k )) is the functional expression of state variables and control variables of the constraint function in the aircraft trajectory optimization problem model.

[0050] S72. The differential constraints of the aircraft trajectory optimization problem model are obtained according to the collocation equation:

[0051]

[0052] Where D represents the differential matrix differential matrix D ki , f(x,u) represents F(X(τ k ), U(τ k ))The corresponding state variable and control variable expressions, represents the differential of f(x,u).

[0053] Furthermore, step S8 is specifically as follows: a new matrix B is generated by Birkhoff interpolation, and the differential matrix in the differential constraint is optimized by the new matrix B to obtain a new differential constraint:

[0054]

[0055] Furthermore, step S9 is specifically as follows: using the pre-solved result as the initial value of the SQP algorithm, encapsulating each part of the SQP algorithm into the fmincon function of matlab or the minimize function in the python library, and inputting the discretized aircraft trajectory optimization problem model into the encapsulated fmincon function of matlab or the minimize function in the python library for iterative solution to obtain a discretized solution.

[0056] The beneficial effects of the present invention are:

[0057] (1) The present invention aims at the ever-increasing requirements of the current designs of various types of aircraft and actual flight missions. By improving the aircraft trajectory optimization algorithm, the efficiency of solving the aircraft trajectory optimization problem model can be improved. Due to the improvement of the calculation efficiency of the trajectory optimization algorithm, the time from aircraft development to design to molding can be further shortened, thereby improving the research and development efficiency of the aircraft.

[0058] (2) By introducing Birkhoff interpolation, the present invention can greatly reduce the condition number of the differential matrix in the pseudo-spectral method, thereby reducing the solution time of the aircraft trajectory optimization problem model, improving the solution efficiency of the pseudo-spectral method discretization processing of the aircraft trajectory optimization problem model, and ultimately improving the trajectory optimization efficiency of the aircraft.

[0059] (3) The present invention uses the LGL pseudo-spectral method as the underlying algorithm for aircraft trajectory optimization, and the discretization solution of the optimization problem has a better optimization effect.

[0060] (4) The present invention encapsulates various small parts of the SQP algorithm and can dynamically change the problem settings when initializing and selecting the optimization problem to solve. The algorithm is highly operable and greatly reduces the time used by researchers to test when calling the algorithm.

[0061] (5) The present invention firstly adopts a genetic algorithm to pre-solve the discretized aircraft trajectory optimization problem model, and uses the pre-solved result of the genetic algorithm as the initial value of the subsequent SQP algorithm, thereby solving the problem that the SQP algorithm is sensitive to the initial value. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1The figure shows a flow chart of a method for optimizing an aircraft trajectory by combining Birkhoff interpolation and LGL pseudo-spectral method according to an embodiment of the present invention. DETAILED DESCRIPTION

[0063] Now, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings. It should be understood that the embodiments shown and described in the accompanying drawings are only exemplary and are intended to explain the principles and spirit of the present invention, rather than to limit the scope of the present invention.

[0064] The embodiment of the present invention provides a method for optimizing the trajectory of an aircraft by combining Birkhoff interpolation and LGL (Legendre Gauss Lobato) pseudospectral method, such as Figure 1 As shown, the following steps S1 to S10 are included:

[0065] S1. Construct a model for the aircraft trajectory optimization problem.

[0066] The embodiment of the present invention draws on the modeling idea of ​​the Italian Aerospace Center (CIRA) using the incremental method to realize the PRORA-USV1 hypersonic aircraft wind tunnel test data and general numerical calculation data, and uses the numerical method to simulate the data set of the X-51A similar aircraft to perform modeling. The glide segment trajectory optimization problem of the X-51A similar aircraft is used as the basic problem model for the process description of the present method, and the aircraft trajectory optimization problem model is obtained as follows:

[0067]

[0068]

[0069]

[0070]

[0071] Where J[·] is the objective function, which represents the maximum flight distance of the aircraft in the gliding phase, x = [H, V, γ] is the trajectory state variable, H, V, γ represent the flight altitude, flight speed and flight track angle of the aircraft respectively, are the differentials of H, V, and γ respectively, u = [α] is the trajectory control variable, α represents the angle of attack, t represents the time, and t 0 represents the starting time of the aircraft trajectory, t f represents the end time of the aircraft trajectory, D represents the resultant force in the direction of resistance, L represents the lift, m represents the mass of the aircraft, g represents the acceleration of gravity, and δ represents the pitch rudder deflection that maintains longitudinal balance under the optimal control variable corresponding to the flight attitude.

[0072] S2. Select Legendre polynomials as the interpolation basis to initialize and transform the aircraft trajectory optimization problem model in the time domain.

[0073] The embodiment of the present invention selects Legendre polynomial L according to the idea of ​​solving optimization problems using pseudo-spectral method. n (τ) is used as the interpolation basis. Since the Legendre polynomial acts on τ(-1, 1), it is also necessary to convert the time domain t(t 0 , t f ) is transformed into the region that makes sense for Legendre polynomials.

[0074] In the embodiment of the present invention, the time domain transformation formula is:

[0075]

[0076] Where t represents time, t 0 represents the starting time of the aircraft trajectory, t f represents the termination time of the vehicle trajectory, and τ represents the LGL collocation point.

[0077] S3. Select n nodes to discretize the aircraft trajectory optimization problem model through the Gauss Lobato numerical quadrature method, and obtain the discrete time corresponding to the aircraft trajectory optimization problem model, as well as the state variables and control variables corresponding to the discrete time.

[0078] In the embodiment of the present invention, discrete time is represented by T n =[t 1 t 2 ...t n t n+1 ], where t p represents the pth discrete time point, and the state variable corresponding to the discrete time is expressed as X n =[H n , V n , γ n ], where H n =[h 1 h 2 ...h n h n+1 ] represents the flight altitude state quantity, h p represents the pth discrete flight height value, V n =[v 1 v 2 ...v n v n+1 ] represents the flight speed state, v p represents the pth discrete flight speed value, γ n =[r1 r 2 ...r n r n+1 ] represents the flight path angle state, r p It represents the pth discrete flight path angle, and the control variable corresponding to the discrete time is expressed as U n =[α n ], where α n =[a 1 a 2 ...a n a n+1 ] represents the discrete representation of the angle of attack, a p Represents the pth discrete angle of attack, with count value p=1, 2, ..., n, n+1.

[0079] S4. Select LGL collocation points according to Legendre polynomials, and discretize state variables and control variables at the LGL collocation points to obtain discrete state variables and discrete control variables.

[0080] Step S4 includes the following sub-steps S41-S42:

[0081] S41, (1-τ 2 )L n The root of (τ), that is, the root of the first-order derivative of the Legendre polynomial and the two points τ∈(-1,1) are used as LGL collocation points, where L n (τ) is the Legendre polynomial.

[0082] S42, the state variable X n and the control variable U n Discretize at the LGL collocation point to obtain n+1 discrete state variables X i and n+1 discrete control variables U i , i=0, 1, 2, ..., n.

[0083] S5. Interpolate the discrete state variables and the discrete control variables through the Lagrange interpolation basis to obtain the state function and the control function.

[0084] In the embodiment of the present invention, the formula for interpolating the discrete state variables and the discrete control variables is:

[0085]

[0086] Where X(τ) represents the state function, U(τ) represents the control function, represents the n-order Lagrange interpolation basis, which is expressed as:

[0087]

[0088] where τ represents the LGL collocation point, τ i represents the i-th LGL collocation point, τ j represents the jth LGL collocation point.

[0089] S6. Obtain the differential matrix according to the LGL point matching, and combine the differential matrix with the differential expressions of the state function and the control function to obtain the differential expressions of the state variables and the control variables.

[0090] Step S6 includes the following sub-steps S61 to S63:

[0091] S61, according to LGL point matching τ=(τ 1 , τ 2 ...τ n+1 ) to obtain the differential matrix D ki :

[0092]

[0093] where τ k represents the kth LGL collocation point, L n (τ k ) represents the Legendre polynomial of the kth LGL collocation point.

[0094] S62. Obtain differential expressions of the state function X(τ) and the control function U(τ):

[0095]

[0096] in Represents the differential expression of the state function X(τ), Represents the differential expression of the control function U(τ), represents the differential of the Lagrangian interpolation basis of order n.

[0097] S63. Combining the differential matrix with the differential expressions of the state function and the control function, the differential expressions of the state variable and the control variable are obtained:

[0098]

[0099] in represents the differential expression of the state variable, Represents the differential expression of the control variable.

[0100] S7. Combining the differential expressions of the state variables and the control variables with the aircraft trajectory optimization problem model, a collocation equation is obtained, and the differential constraints of the aircraft trajectory optimization problem model are obtained based on the collocation equation.

[0101] Step S7 includes the following sub-steps S71-S72:

[0102] S71. Substitute the differential expressions of state variables and control variables into the aircraft trajectory optimization problem model to obtain the collocation equation:

[0103]

[0104] Where F(X(τ k ), U( τ k)) is the functional expression of the state variables and control variables of the constraint function in the aircraft trajectory optimization problem model, that is, the expression of the right-hand side function of the constraint differential variable.

[0105] S72. The differential constraints of the aircraft trajectory optimization problem model are obtained according to the collocation equation:

[0106]

[0107] Where D represents the differential matrix differential matrix D ki , f(x,u) represents F(X(τ k ), U(τ k ))The corresponding state variable and control variable expressions, represents the differential of f(x,u).

[0108] S8. Generate a new matrix through Birkhoff interpolation, optimize the differential matrix in the differential constraint through the new matrix, and obtain a new differential constraint.

[0109] In the embodiment of the present invention, a new matrix B is generated by Birkhoff interpolation, and the differential matrix in the differential constraint is optimized by the new matrix B to obtain a new differential constraint:

[0110]

[0111] The idea of ​​Birkhoff interpolation is to abandon the requirement of taking the function value of certain order derivatives at certain points of the original problem. Based on this idea, a new matrix, namely Birkhoff matrix B, is introduced on both sides of the differential constraint equation, so that the condition number of differential matrix BD is significantly smaller than the condition number of differential matrix D. Under the premise of keeping the equations equal, a well-conditioned differential matrix is ​​constructed by reducing the condition number of the differential matrix, thereby improving the efficiency of problem solving.

[0112] S9. Use a genetic algorithm to pre-solve the discretized aircraft trajectory optimization problem model to obtain a pre-solved result.

[0113] In the embodiment of the present invention, the pre-solution result of the genetic algorithm is used as the initial value of the subsequent SQP algorithm, which solves the problem that the SQP algorithm is sensitive to the initial value.

[0114] S10. Using the pre-solved result as the initial value of the SQP algorithm, and using the SQP algorithm to solve the discretized aircraft trajectory optimization problem model to obtain a discretized solution.

[0115] After the discretization processing of the pseudo-spectral method in the above steps, the aircraft trajectory optimization problem model is transformed from a linear programming problem into a nonlinear programming (NLP) problem. Since the SQP (sequential quadratic programming) algorithm is the most effective method for solving strongly constrained NLP problems, in an embodiment of the present invention, each part of the SQP algorithm is encapsulated into the fmincon function of matlab or the minimize function in the python library, and the discretized aircraft trajectory optimization problem model is input into the encapsulated fmincon function of matlab or the minimize function in the python library for iterative solution to obtain a discretized solution.

[0116] S11. Perform interpolation approximation and restoration processing on the discretized solution to obtain a solution to the aircraft trajectory optimization problem model, thereby optimizing the aircraft trajectory.

[0117] The solution obtained through the optimization iteration in step S9 is the discretized solution after the pseudo-spectral processing step. Therefore, in order to obtain the non-discretized solution before the original discretization processing, it is necessary to perform interpolation approximation and restoration processing on the discretized solution. The solution obtained after approximation processing of the discretized solution is the solution of the original aircraft trajectory optimization problem model. The interpolation approximation is also performed through the Lagrange interpolation method for approximation and restoration processing.

[0118] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A method for optimizing aircraft trajectories by combining Birkhoff interpolation and LGL pseudospectral method, characterized in that: The following steps are involved: S1. Construct a model for the aircraft trajectory optimization problem; S2, select Legendre polynomial as interpolation basis, perform initialization processing and time domain transformation on the aircraft trajectory optimization problem model; S3, selecting n nodes to discretize the aircraft trajectory optimization problem model through the Gauss Lobato numerical quadrature method, and obtaining the discrete time corresponding to the aircraft trajectory optimization problem model, as well as the state variables and control variables corresponding to the discrete time; S4, selecting LGL collocation points according to Legendre polynomials, and discretizing state variables and control variables at the LGL collocation points to obtain discrete state variables and discrete control variables; S5. interpolating the discrete state variables and the discrete control variables through the Lagrange interpolation basis to obtain the state function and the control function; S6. Obtain a differential matrix according to the LGL collocation points, and obtain differential expressions of state variables and control variables by combining the differential matrix with differential expressions of state functions and control functions; S7, combining the differential expressions of the state variables and the control variables with the aircraft trajectory optimization problem model to obtain a collocation equation, and obtaining the differential constraints of the aircraft trajectory optimization problem model according to the collocation equation; S8. Generate a new matrix through Birkhoff interpolation, optimize the differential matrix in the differential constraint through the new matrix, and obtain a new differential constraint; S9, using a genetic algorithm to pre-solve the discretized aircraft trajectory optimization problem model to obtain a pre-solved result; S10, using the pre-solved result as the initial value of the SQP algorithm, and using the SQP algorithm to solve the discretized aircraft trajectory optimization problem model to obtain a discretized solution; S11. Perform interpolation approximation and restoration processing on the discretized solution to obtain a solution to the aircraft trajectory optimization problem model, thereby optimizing the aircraft trajectory.

2. The method for optimizing the trajectory of an aircraft according to claim 1, characterized in that: The aircraft trajectory optimization problem model in step S1 is expressed as: Where J[·] is the objective function, which represents the maximum flight distance of the aircraft in the gliding phase, x = [H, V, γ] is the trajectory state variable, H, V, γ represent the flight altitude, flight speed and flight track angle of the aircraft respectively, are the differentials of H, V, and γ respectively, u = [α] is the trajectory control variable, α represents the angle of attack, t represents the time, t0 represents the starting time of the aircraft trajectory, t f represents the end time of the aircraft trajectory, D represents the resultant force in the direction of resistance, L represents the lift, m represents the mass of the aircraft, g represents the acceleration of gravity, and δ represents the pitch rudder deflection that maintains longitudinal balance under the optimal control variable corresponding to the flight attitude.

3. The method for optimizing the trajectory of an aircraft according to claim 1, characterized in that: The time domain transformation formula in step S2 is: Where t represents time, t0 represents the starting time of the aircraft trajectory, and t f represents the termination time of the vehicle trajectory, and τ represents the LGL collocation point.

4. The method for optimizing the trajectory of an aircraft according to claim 1, characterized in that: The discrete time in step S3 is represented by T n =[t1t2...t n t n+1 ], where t p represents the pth discrete time point, and the state variable corresponding to the discrete time is expressed as X n =[H n , V n , γ n ], where H n =[h1h2...h n h n+1 ] represents the flight altitude state quantity, h p represents the pth discrete flight height value, V n =[v1v2...v n v n+1 ] represents the flight speed state, v p represents the pth discrete flight speed value, γ n =[r1r2...r n r n+1 ] represents the flight path angle state, r p It represents the pth discrete flight path angle, and the control variable corresponding to the discrete time is expressed as U n =[α n ], where α n =[a1a2...a n a n+1 ] represents the discrete representation of the angle of attack, a p Represents the pth discrete angle of attack, with count value p=1, 2, ..., n, n+1.

5. The method for optimizing the trajectory of an aircraft according to claim 4, characterized in that: The step S4 comprises the following sub-steps: S41, (1-τ 2 )L n The roots of (τ) are used as LGL collocation points, where L n (τ) is the Legendre polynomial; S42, the state variable X n and the control variable U n Discretize at the LGL collocation point to obtain n+1 discrete state variables X i and n+1 discrete control variables U i , i=0, 1, 2, ..., n.

6. The method for optimizing the trajectory of an aircraft according to claim 5, characterized in that: The formula for interpolating the discrete state variables and the discrete control variables in step S5 is: Where X(τ) represents the state function, U(τ) represents the control function, represents the n-order Lagrange interpolation basis, which is expressed as: where τ represents the LGL collocation point, τ i represents the i-th LGL collocation point, τ j represents the jth LGL collocation point.

7. The method for optimizing the trajectory of an aircraft according to claim 6, characterized in that: The step S6 comprises the following sub-steps: S61, according to LGL point matching τ=(τ1,τ2...τ n+1 ) to obtain the differential matrix D ki : where τ k represents the kth LGL collocation point, L n (τ k ) represents the Legendre polynomial of the kth LGL collocation point; S62. Obtain differential expressions of the state function X(τ) and the control function U(τ): in Represents the differential expression of the state function X(τ), Represents the differential expression of the control function U(τ), represents the differential of the Lagrange interpolation basis of order n; S63. Combining the differential matrix with the differential expressions of the state function and the control function, the differential expressions of the state variable and the control variable are obtained: in represents the differential expression of the state variable, Represents the differential expression of the control variable.

8. The method for optimizing the trajectory of an aircraft according to claim 7, characterized in that: The step S7 comprises the following sub-steps: S71. Substitute the differential expressions of state variables and control variables into the aircraft trajectory optimization problem model to obtain the collocation equation: Where F(X(τ k ), U(τ k )) is the functional expression of state variables and control variables of the constraint function in the aircraft trajectory optimization problem model; S72. The differential constraints of the aircraft trajectory optimization problem model are obtained according to the collocation equation: Where D represents the differential matrix differential matrix D ki , f(x,u) represents F(X(τ k ), U(τ k ))The corresponding state variable and control variable expressions, represents the differential of f(x,u).

9. The method for optimizing the trajectory of an aircraft according to claim 8, characterized in that: The step S8 is specifically as follows: generating a new matrix B by Birkhoff interpolation, optimizing the differential matrix in the differential constraint by the new matrix B, and obtaining a new differential constraint:

10. The method for optimizing the trajectory of an aircraft according to claim 1, characterized in that: The step S9 is specifically as follows: The pre-solved result is used as the initial value of the SQP algorithm, and each part of the SQP algorithm is encapsulated into the fmincon function of matlab or the minimize function in the python library. The discretized aircraft trajectory optimization problem model is input into the encapsulated fmincon function of matlab or the minimize function in the python library for iterative solution to obtain a discretized solution.

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