A Convex Optimization Method for Aircraft Trajectory Sequences Based on High-Dimensional Value Functions

Abstract

The present invention belongs to the technical field of aircraft trajectory planning, and relates to a convex optimization method for aircraft trajectory sequences based on a high-dimensional value function. This method distributes the objective function and the constraint violation degree function in the aircraft trajectory planning problem into a multi-dimensional space to construct a high-dimensional value function. Using the high-dimensional value function as the evaluation function of the iterative solution, a convergence strategy of first feasible and then optimal is adopted to find the iterative solution. The convergence strategy is to first make the iterative solution satisfy the sufficient descent condition of feasibility, and then make the iterative solution satisfy the sufficient descent condition of optimality. The present invention can accurately separately process the objective function and the constraint violation degrees of different types. By constructing a high-dimensional value function and a convergence strategy of first feasible and then optimal, the convergence performance of the sequential convex optimization method for solving aircraft trajectory planning problems with different types of constraints such as equality constraints, inequality constraints, and strongly nonlinear constraints is effectively improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of aircraft trajectory planning, and particularly relates to an aircraft trajectory sequence convex optimization method based on a high-dimensional value function. Background Art

[0002] In the field of aircraft guidance, real-time trajectory planning is a key challenge for achieving high-precision guidance. The sequential convex optimization method exhibits computational efficiency advantages due to its characteristic of linearizing the original non-linear aircraft trajectory planning problem and iteratively solving the linearized convex sub-problems to obtain the optimal solution of the original problem. However, the core problem faced by this method is that it is difficult to guarantee convergence.

[0003] Traditional methods rely on one-dimensional L1 penalty functions to ensure convergence ( Successive Convexification of Non-Convex Optimal Control Problems with State Constraints [J]. IFAC- PapersOnLine, 2017, 50 (1): 4063 - 4069 ). However, this method has obvious defects. Its convergence performance is very sensitive to the penalty parameter. When the penalty parameter is too large, the convergence speed will be extremely slow, consuming a large amount of computing resources and time; when it is too small, the sub-problem may become infeasible, or the algorithm may terminate prematurely and fail to obtain an ideal result. These problems greatly limit the application of the sequential convex optimization method ( SCP ) in complex constraint scenarios.

[0004] The literature ( Sequential Convex Programming without Penalty Function for Reentry Trajectory Optimization Problem [J]. Acta Astronautica, 2024, 225 (1): 402 - 417 ) proposed a filter-based sequential convex optimization method that can avoid introducing penalty parameters, but it cannot group and process the violation degrees of different types of constraints such as equality constraints, inequality constraints, and strongly non-linear constraints. This results in poor convergence performance of the algorithm in complex constraint scenarios and cannot meet the actual application requirements. Summary of the Invention

[0005] The purpose of the present invention is to overcome the deficiencies in the prior art and provide an aircraft trajectory sequence convex optimization method based on a high-dimensional value function.

[0006] To achieve the purpose of the present invention, the following technical solutions are adopted for implementation.

[0007] An aircraft trajectory sequence convex optimization method based on a high-dimensional value function includes the following steps:

[0008] An aircraft trajectory sequence convex optimization method based on a high-dimensional value function is characterized in that it includes the following steps:

[0009] S1. According to the constraint types in the aircraft trajectory planning problem, divide the constraints in the aircraft trajectory planning problem into subscript sets; where: the The sets of subscripts are respectively denoted as ;

[0010] S2. The constraint violation degree functions of the subscript sets obtained in construction step S1:

[0011] (1);

[0012] wherein, is the total optimization variable in the aircraft trajectory planning problem, is the th constraint function in the aircraft trajectory planning problem The violation penalty function of which is defined as follows

[0013] (2);

[0014] wherein, and are respectively the subscript sets of the equality and inequality constraints in the aircraft trajectory planning problem;

[0015] S3. Allocate the objective function in the aircraft trajectory planning problem and the constraint violation degree function , constructed in step S2 into dimensional space to construct a high-dimensional value function:

[0016] (3);

[0017] S4. Use the high-dimensional value function constructed in step S3 as the evaluation function for finding the iterative solution, and adopt a convergence strategy of first feasible and then optimal to find the iterative solution, so as to improve the convergence performance of finding the iterative solution;

[0018] The convergence strategy is to first make the iterative solution satisfy the sufficient descent condition of feasibility, and then make the iterative solution satisfy the sufficient descent condition of optimality;

[0019] The sufficient descent condition of feasibility is:

[0020] (8);

[0021] (9);

[0022] wherein, is an element in , is th component of , is the total constraint violation degree, is the sufficient descent coefficient of feasibility, Denote the \(i\)-th component of the high-dimensional value function as the \(i\)-th component, and \(\mathcal{F}^k\) is the filter set at the \(k\)-th iteration;

[0023] The sufficient decrease condition for optimality is: (13),

[0024] where \(\Delta f^k\) is the decrease in optimality, and \(\gamma\) is the sufficient decrease coefficient for optimality.

[0025] As a preferred embodiment of the present invention, the constraint type is classified according to the degree of non-linearity of the constraint.

[0026] As a preferred embodiment of the present invention, the filter set is used to record the high-dimensional value function values of the iterative solution, where the dimension of each element is the same as the dimension of the high-dimensional value function.

[0027] As a preferred embodiment of the present invention, before the first iteration, the filter set is initialized, and the initial filter set is set as:

[0028] (4);

[0029] where \(\epsilon\), is the maximum tolerance value.

[0030] As a preferred embodiment of the present invention, the specific implementation process of step S4 includes the following steps:

[0031] S41. Solve the convex subproblem at the iterative solution \(\mathbf{x}^k\), where \(k\) represents the number of iterations;

[0032] If the convex subproblem is infeasible, then add \(\mathbf{x}^k\) to the filter set \(\mathcal{F}^k\), i.e.,

[0033] (5);

[0034] where \(\mathcal{F}^{k - 1}\) and \(\mathcal{F}^k\) are the filter sets at the \((k - 1)\)-th and \(k\)-th iterations respectively;

[0035] Subsequently, use feasibility restoration, i.e., iteratively solve the convex subproblem with the objective of minimizing the constraint violation degree to obtain a new iterative solution , make the convex sub-problem at this iteration solution feasible, and then enter the next iteration;

[0036] If the convex sub-problem is feasible, check the optimal solution of the convex sub-problem, that is, the iteration direction , and see if it satisfies the following iteration stop condition:

[0037] (6);

[0038] (7);

[0039] Among them, is the total constraint violation degree at this point, is the absolute value of the objective function of the convex sub-problem, is at the gradient at this point, is the feasibility error tolerance, is the optimality error tolerance;

[0040] If conditions (6) and (7) are satisfied simultaneously, then stop the iterative solution and output as the optimal solution; otherwise, check whether the iterative solution and the iteration direction satisfy the following sufficient feasibility descent condition:

[0041] (8);

[0042] (9);

[0043] Among them, is an element in, is the th component of, is the total constraint violation degree, is the sufficient feasibility descent coefficient, represents the th component of the high-dimensional value function , is the th iteration filter set;

[0044] If and do not satisfy conditions (8) and (9), then shrink the trust region in the convex sub-problem, that is,

[0045] (10);

[0046] Among them, is the scaling ratio;

[0047] Then, continue to solve the convex subproblem until and meet the conditions (8)-(9);

[0048] S42. Denote as the estimated optimality descent; check whether it meets the following conditions:

[0049] (11);

[0050] If it meets the condition (11), then add to the filter set according to formula (5), shrink the trust region in the subproblem according to formula (10) and update the iterative solution according to the following formula (12) and enter the next iteration;

[0051] (12);

[0052] If it does not meet the condition (11), then check whether it meets the following sufficient optimality descent condition:

[0053] (13);

[0054] Among them, is the actual optimality descent, is the sufficient optimality descent coefficient;

[0055] If it meets the condition (13), then update the iterative solution according to formula (12) and enter the next iteration; otherwise, shrink the trust region in the convex subproblem , that is , and then return to step S41.

[0056] Beneficial effects

[0057] Compared with the prior art, the present invention can accurately separately process the objective function and the degrees of constraint violation of different types;

[0058] By constructing a high-dimensional value function and designing a reasonable iterative solution process, the present invention effectively improves the convergence performance of the sequential convex optimization method for solving trajectory planning problems with different types of constraints such as equality constraints, inequality constraints, and strong non-linear constraints, and provides more reliable technical support for the trajectory planning of aircraft in complex environments. Description of the drawings

[0059] 图1 This is the iterative solution flowchart in the present invention;

[0060] 图2 This is a comparison graph of simulation results between the one-dimensional L1 penalty function and the two-dimensional value function and the total constraint violation degree of the three-dimensional value function of the present invention;

[0061] 图3 This is a comparison graph of simulation results between the one-dimensional L1 penalty function and the two-dimensional value function and the absolute value of the convex sub-problem objective function of the three-dimensional value function of the present invention; Specific implementation manners

[0062] The present invention will be further described in conjunction with embodiments and the accompanying drawings.

[0063] As an embodiment of the present invention, as 图1 shown, taking the minimum longitudinal range reentry trajectory planning problem of a certain aircraft as an example, a method for convex optimization of aircraft trajectory sequences based on a high-dimensional value function described in the present invention will be further described. In this embodiment, the aircraft model is selected as the publicly available Crew Exploration Vehicle aircraft ( Reentry guidance with extended range capability for low L / D spacecraft[D]. Massachusetts Institute of Technology, 2006 ); for the setting of the objective function and constraints in the reentry trajectory planning problem, refer to the publicly available literature ( Hybrid-Order Soft Trust Region-Based Sequential Convex Programming for Reentry Trajectory Optimization[J]. Advances in Space Research, 2024, 73 (6): 3195-3208 ).

[0064] The specific implementation steps are as follows:

[0065] S1. According to the degree of constraint non-linearity, the constraints of the trajectory planning problem are divided into two categories, namely . The first category is the motion equation constraint of the geocentric distance, and the second category is other constraints except the motion equation constraint of the geocentric distance. The subscript sets of the corresponding constraints are and respectively;

[0066] S2. Construct the constraint violation degree functions in the and sets:

[0067] (14);

[0068] (15);

[0069] S3. Construct the high-dimensional value function:

[0070] (16);

[0071] Among them, Represents the terminal longitudinal range variable in the minimum longitudinal range reentry trajectory planning problem of the spacecraft, which is included in the total optimization variables , is the objective function for the minimum longitudinal range reentry trajectory planning problem;

[0072] S4. Initialize the filter set to the filter set :

[0073] (17);

[0074] S5. Solve the convex subproblem at the iterative solution , where: represents the number of iterations;

[0075] If the convex subproblem is infeasible, add to the filter set, and then perform feasibility restoration to obtain a new iterative solution such that the convex subproblem at the new iterative solution is feasible, and then enter the next iteration;

[0076] If the convex subproblem is feasible, check the optimal solution of the subproblem, i.e., the iterative direction , to see if it satisfies the following iterative stopping conditions:

[0077] (18);

[0078] (19);

[0079] If both conditions (18) and (19) are satisfied, stop the iterative solution and output as the optimal solution; otherwise, check whether the iterative solution and the iterative direction satisfy the following two sufficient feasibility descent conditions:

[0080] (20);

[0081] (21);

[0082] If and do not satisfy conditions (20) and (21), then reduce the trust region in the convex subproblem; then continue to solve the subproblem until and satisfy conditions (20) and (21);

[0083] S6. Check whether satisfies the following condition:

[0084] (22);

[0085] If condition (22) is satisfied, then is added to the filter set, and the trust region in the reduced convex subproblem is updated, and the iterative solution is updated and the next iteration is entered;

[0086] If condition (22) is not satisfied, then check whether the following conditions are satisfied:

[0087] (23);

[0088] If condition (23) is satisfied, then update the iterative solution and enter the next iteration; otherwise, reduce the trust region in the reduced convex subproblem , that is , and then return to step S5.

[0089] To fully verify the effectiveness of the method of the present invention, the method in the present invention is compared with the sequential convex optimization method based on the one-dimensional penalty function and the sequential convex optimization method based on the two-dimensional value function. From 图2至图3 the simulation results, it can be clearly seen that for the sequential convex optimization method based on the one-dimensional L1 penalty function, after 50 iterations of solution, there is still no sign of convergence; the number of iterative convergences for the method based on the two-dimensional value function is 21 times; while the number of iterative convergences for the method based on the three-dimensional value function of the present invention is only 11 times. Through comparison, it is fully proved that the algorithm proposed in the present invention can significantly improve the convergence speed of the sequential convex optimization method and has important value in practical applications.

[0090] The preferred embodiments of the embodiments of the present application have been described above with reference to the accompanying drawings, and thus do not limit the scope of rights of the embodiments of the present application. Any modifications, equivalent replacements, and improvements made by those skilled in the art without departing from the scope and essence of the embodiments of the present application shall be within the scope of rights of the embodiments of the present application.

Claims

1. A convex optimization method for aircraft trajectory sequences based on high-dimensional value functions, characterized in that: It includes the following steps: S1. According to the constraint types in the aircraft trajectory planning problem, the constraints in the aircraft trajectory planning problem are divided into subscript sets; where: the subscript sets are respectively denoted as ; S2. Construct the constraint violation degree function of each subscript set obtained in step S1: (1)； Among them, is the total optimization variable in the aircraft trajectory planning problem, is the th constraint function in the aircraft trajectory planning problem and its violation penalty function is defined as follows (2)； Among them, and are respectively the subscript sets of the equality and inequality constraints in the aircraft trajectory planning problem; S3. Assign the objective function in the aircraft trajectory planning problem and the constraint violation degree function constructed in step S2 , to a multi-dimensional space to construct a high-dimensional value function: (3)； S4. Use the high-dimensional value function constructed in step S3 as the evaluation function of the iterative solution, and adopt a convergence strategy of first feasible and then optimal to perform the iterative solution, so as to improve the convergence performance of the iterative solution; The convergence strategy is to first make the iterative solution satisfy the sufficient descent condition of feasibility, and then make the iterative solution satisfy the sufficient descent condition of optimality; The sufficient descent condition of feasibility is: (8)； (9)； Among them, is an element in and is the -th component of The total constraint violation degree is The sufficient decrease coefficient of feasibility is denotes the -th component of the high-dimensional value function and is the -th iteration filter set; The sufficient descent condition for optimality is as follows: (13), Among them, is the amount of decrease in optimality, is the sufficient decrease coefficient of optimality.

2. The convex optimization method for the aircraft trajectory sequence based on the high-dimensional value function according to claim 1, wherein: The constraint types are classified according to the non-linearity degree of the constraints.

3. A convex optimization method for aircraft trajectory sequences based on a high-dimensional value function according to claim 1, characterized in that: The filter set is used to record the iterative solution of the high-dimensional value function value, where the dimension of each element is the same as the dimension of the high-dimensional value function.

4. A convex optimization method for aircraft trajectory sequences based on a high-dimensional value function according to claim 3, characterized in that: Before the first iteration, initialize the set of filters , the initial set of filters is set to: (4)； Among them, , is the maximum tolerance value.

5. The convex optimization method for the aircraft trajectory sequence based on the high-dimensional value function according to claim 1, characterized in that: The specific process of step S4 includes the following steps: S41. Solve the iterative solution for the convex sub-problem at, where represents the number of iterations; If the convex subproblem is infeasible, then is added to the filter set i.e., (5)； Among them, and are respectively the filter sets for the and th iterations; Subsequently, feasibility restoration is used, that is, an iterative solution is obtained by minimizing the convex subproblem with the degree of constraint violation as the objective function to obtain a new iterative solution , making the convex subproblem at the new iterative solution feasible, and then entering the next iteration; If the convex subproblem is feasible, check the optimal solution of the convex subproblem, i.e., the iteration direction , to see if it satisfies the following iteration stopping condition: (6)； (7)； wherein, is the total constraint violation degree at the absolute value of the objective function of the convex sub-problem, is at the gradient at is the feasibility error tolerance, and is the optimality error tolerance; If conditions (6) and (7) are both satisfied, stop the iterative solution and output as the optimal solution; otherwise, check whether the iterative solution and the iterative direction satisfy the following sufficient feasibility and descent conditions: (8)； (9)； Among them, is an element in which is the th component of the total constraint violation degree, is the sufficient decrease coefficient of feasibility, represents the th component of the high-dimensional value function is the filter set of the th iteration; If and do not satisfy conditions (8) and (9), then reduce the trust region in the convex subproblem , that is （10）； Among them, is for reducing the scale; Then, continue to solve the convex sub-problem until and satisfy conditions (8) and (9); S42. Denote as the estimated optimal decrease; Check whether the following conditions are satisfied: （11）； If condition (11) is satisfied, then according to formula (5), is added to the filter set, and the trust region in the convex subproblem is reduced according to formula (10) , the iterative solution is updated according to the following formula (12), and the next iteration is entered; （12）； If condition (11) is not satisfied, then check whether the following sufficient descent condition for optimality is satisfied: （13）； Among them, is the actual optimality descent amount, is the sufficient optimality descent coefficient; If the condition (13) is satisfied, update the iterative solution according to formula (12) and enter the next iteration; otherwise, reduce the trust region in the convex subproblem , that is , and then return to step S41.

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