Gradient adaptive finite element orthogonal configuration of hypersonic vehicle trajectory optimal control system

Abstract

The present invention discloses a gradient adaptive finite element orthogonal configuration hypersonic aircraft trajectory optimal control system, which is used to solve the hypersonic aircraft re-entry trajectory optimal control and process constraint full satisfaction. The system includes a data preprocessing module, a dynamic optimization algorithm module and an aircraft control module. The present invention optimizes and controls the important parameter indicators of the hypersonic aircraft re-entry process, overcomes the poor agility of the existing hypersonic aircraft control technology algorithm, and the difficulty in obtaining the optimal parameter control of the system in a short time, which leads to the shortcomings of the hypersonic aircraft control effect. The dynamic optimization method and the gradient-based grid division orthogonal configuration method are introduced to obtain the hypersonic aircraft re-entry trajectory optimization optimal control system, and the aircraft re-entry trajectory OCP optimal control Benchmark scenario test results are better, more efficient, and easy to find the optimal control parameters.

Description

Technical Field

[0001] The present invention relates to the field of hypersonic aircraft flight control, optimal control and dynamic optimization algorithm, and in particular to a hypersonic aircraft trajectory control system combined with an optimal control dynamic optimization method. Background Art

[0002] The predecessor of optimal control theory can be considered as variational calculus, which was born in the 17th century. However, variational calculus has limitations and can only deal with unconstrained or open-set constrained problems. In fact, the admissible control of many control problems belongs to closed sets. Modern optimal control theory began at the end of World War II. The starting point was mainly due to differential games. Because optimal control theory is widely used, it is not limited to differential games and military strategies. It has been applied to the synthesis and design of the fastest control system, the most fuel-saving control system, the minimum energy consumption control system, and linear regulators. Common methods for optimal control problems (OCP) can be divided into two categories: indirect methods and direct methods. Indirect methods transform OCP into first-order optimality conditions by using the Ponderajkin maximum principle. This principle is generally effective in practical problems with strong nonlinearity and complex constraints. In contrast, direct methods transform OCP into a nonlinear programming (NLP) problem, namely control vector parameterization (CVP) and orthogonal collocation method (OC). The main advantage of the direct method is that it can bypass the necessary conditions and directly calculate the optimal performance indicators after discretization. In the past two decades, the OC method has been applied in the numerical solution of OCP. The traditional OC method uses a fixed grid. In order to obtain high-quality convergence results, a fine discretization grid is usually required, but sometimes over-discretization will lead to the Runge phenomenon: on the one hand, high discretization will cause the converted NLP problem to require the optimization of a large number of parameters, which is not a suitable choice in terms of the efficiency of solving the NLP problem; on the other hand, the Runge phenomenon caused by over-discretization makes the obtained results far from the expected optimal results. Therefore, how to refine the grid efficiently and accurately has been a hot topic in the research of OC methods in recent years, prompting the rapid development of optimal control theory research. Summary of the invention

[0003] In order to overcome the deficiency of the traditional OCFE point matching solution in the current optimal control method that cannot take into account both algorithm efficiency and accuracy, the purpose of the present invention is to provide a gradient adaptive finite element orthogonal configuration hypersonic aircraft trajectory optimal control system, which has good test agility and improves the optimal control accuracy while ensuring algorithm efficiency.

[0004] The technical solution adopted by the present invention to solve the technical problem is: a gradient adaptive finite element orthogonal configuration hypersonic aircraft trajectory optimal control system, which is used to solve the hypersonic aircraft re-entry trajectory optimal control and process constraint full satisfaction, including a hypersonic aircraft, a data preprocessing module, a dynamic optimization algorithm module and an aircraft control module, wherein:

[0005] Data preprocessing module: input the data S sent back by the re-entry sensor of the hypersonic vehicle * , and then the data preprocessing module unifies the dimension input to S = {h0, V0, θ0, X0, X f ,X1,X2,X b ,R b}. Wherein, h0 is the initial altitude of the hypersonic aircraft, in km; V0 is the initial velocity of the aircraft, in km / s; θ0 is the initial turning angle of the aircraft; X0 is the initial position of the aircraft and X0 = (x0, y0) is the latitude and longitude of the aircraft; X f Terminal position and X f =(x f ,y f ); X1 and X2 are the waypoint constraints of the aircraft, X1=(x1,y1), X2=(x2,y2); X b is a no-fly zone constraint point and X b =(x b ,y b );R b The radius of the no-fly zone, in km.

[0006] Aircraft control module: satisfies relevant constraints in real time by controlling the aircraft tilt angle σ, acceleration a, heading angle θ, and tilt angle u. Assuming that the earth is a spherical non-rotating sphere, the hypersonic aircraft flight model is

[0007]

[0008] Among them, the path constraints of the hypersonic aircraft control module are fully satisfied including:

[0009] (1) Control constraints

[0010] In order to ensure the stability of the hypersonic vehicle, the normalized tilt angle u constraint of the control variable satisfies

[0011]

[0012] (2) Terminal state constraints

[0013] State equation for terminal hypersonic vehicle

[0014]

[0015] (3) Waypoint constraints

[0016] Assume that the total number of specific waypoints is i end , the position of the i-th waypoint is expressed in longitude and latitude as (x i ,y i ), then the waypoint constraints of the model satisfy

[0017]

[0018] (4) No-fly zone restrictions

[0019] Generally speaking, a no-fly zone is described as a circular restricted area with infinite height. Assume that the center of the j-th no-fly zone is The radius is Then the no-fly zone constraint satisfies

[0020]

[0021] Dynamic optimization algorithm module: By establishing a hypersonic vehicle reentry trajectory optimization system, the system optimization agility and optimization accuracy are improved while ensuring efficiency. The Bolza form of the cost function J of the trajectory optimization problem consists of two parts. One part is the cost function at the terminal time t f The other part is the cost from t0 to t f The cost of the integrand L. The optimal control is to determine the control u(t)∈R m and state x(t)∈R n To minimize the cost function while satisfying the equations of motion, path constraints and boundary constraints.

[0022] The cost function is

[0023]

[0024] At the same time, the constraints are

[0025]

[0026] The general optimal control problem (OCP) has the following form

[0027]

[0028]

[0029] x(t0)=x0

[0030] C i (x(t),u(t),t)≤0

[0031] C e (x(t),u(t),t)=0

[0032] Where J represents the Bolza-type cost function, u:R→R m represents a function of m control variables, x:R→R n represents a function with n state variables; C i :R n ×R m ×R→R p and C ε :R n ×R m ×R→R q They are inequality constraints and equality constraints respectively.

[0033] (1) The orthogonal collocation method of finite elements (OCFE) is introduced to transform the infinite-dimensional OCP problem into a finite-dimensional nonlinear problem (NLP).

[0034] First, the time interval [t0,t f ] is composed of nodes [t1, t2, …, t N-1 ] is divided into N subintervals, and t N =t f . The i-th subinterval [t i-1 ,t i ]Length i =t i -t i-1 At this time, the time interval is projected onto the [-1,1] standard interval through linear interpolation method.

[0035] or

[0036] Therefore, through the above division, the above problem can be transformed into

[0037]

[0038]

[0039] x1(-1)=x0

[0040] x i (-1) = x i-1 (1),(i=2,…,N)

[0041] C i (x i (τ),u i (τ),τ)≤0

[0042] C e (x i (τ),u i (τ),τ)=0

[0043] where x i and u i They correspond to the state variables and control variables on the i-th subinterval respectively.

[0044] Then, the above transformation problem can be solved by orthogonal collocation method. In the i-th subinterval, the state variable can be approximately expressed as

[0045]

[0046] Among them, X i (τ) is the Lagrange interpolation function, K i is the order of the interpolation function; X i,j =X i (τ j ) where τ j (j≠0) is a function node and τ0=-1; K i The Lagrangian basis functions are Then we can get

[0047]

[0048] So we can differentiate the equation to get

[0049]

[0050] Similarly, the control variable can also be approximated by the Lagrange interpolation polynomial

[0051]

[0052] Among them U i,j =U i (τ j ),(j≠0), so the problem can be transformed into

[0053]

[0054]

[0055] X 1,1 =x0

[0056]

[0057] C i (X i,k ,U i,k ,τ k )≤0

[0058] C e (X i,k ,Ui,k ,τ k )=0

[0059] in

[0060] (2) can be solved according to the sequential quadratic programming (SQP) method. The above NLP problem (8) can be expressed as follows:

[0061]

[0062] stc i (z)=0,i∈E,

[0063] c i (z)≤0,i∈I

[0064] The corresponding sub-problem

[0065]

[0066]

[0067]

[0068] (3) The solution process of the sequential quadratic programming (SQP) method is as follows (3.1) Given μ1>0, error ε0>0, initial point z0, and set k=1;

[0069] (3.2) Solve the corresponding sub-problem of the transformed problem and calculate d k ;

[0070] (3.3) Adjustment parameter μ k , so that d k is the iteration point z k Function value ψ(z,μ k ) descends, and the iteration step length α is determined by linear search k ;

[0071] (3.4) Let z k+1 =z k +α k d k , solve the equation Get the parameter λ k+1 ,in

[0072] (3.5) If The iteration terminates and the final result z is output k as the algorithm result; otherwise, let k=k+1 and repeat (3.2).

[0073] (4) The accuracy and efficiency of OCFE are improved through the OCFE adaptive grid refinement strategy based on gradient information (G-OCFE). The proposed gradient-based segmentation strategy analyzes the optimal control strategy under the current segmentation according to the gradient information of the nodes and re-segments according to the given rules.

[0074] Let N T represents the total number of nodes under the current segmentation method, and the gradient of the kth node at the lth iteration, and let σ0 be the preset threshold

[0075]

[0076] in, is the value of the j-th control variable at the l-th iteration at the k-th node.

[0077] When the normalized gradient of a node is too large, The control trajectory of this node usually needs to be subdivided more densely to improve the accuracy of the optimal control strategy. Therefore, choose The midpoint of is used as a segment node to subdivide the entire time interval.

[0078] As a preset parameter, σ0 determines whether the adaptive method tends to global interpolation or local point low-order interpolation. When σ0 is greater than the gradient of all nodes, the entire interval will not be split to generate a global interpolation; on the other hand, when σ0 decreases, more local low-order interpolation will be formed.

[0079] When the feasible initial values ​​of the supersonic aircraft are input into the hypersonic aircraft reentry trajectory optimization system, the aircraft status display control module can control the aircraft in real time to achieve the optimal path and meet the path constraints.

[0080] The beneficial effects of the present invention are mainly manifested in: 1. The trajectory optimal control module and the aircraft state control module perform high-sensitivity optimization and control on the aircraft re-entry trajectory, realizing rapid response and high-precision control of hypersonic aircraft re-entry orbit optimization; 2. The improved gradient adaptive configuration point method can rely on the gradient information of the control trajectory and constraint conditions for adaptive adjustment, ensuring that the control trajectory achieves high robustness and high precision at fast-changing points, avoiding the hypersonic aircraft flight trajectory from exceeding the path constraints, thereby ensuring the optimized aircraft trajectory. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 It is a basic structural diagram of a gradient adaptive finite element orthogonal configuration hypersonic vehicle trajectory optimal control system;

[0082] Figure 2 It is a schematic diagram of the system structure of the dynamic optimization algorithm module. DETAILED DESCRIPTION

[0083] The present invention is described in detail below with reference to the accompanying drawings.

[0084] Reference Figure 1 , a gradient adaptive finite element orthogonal configuration hypersonic aircraft trajectory optimal control system, including a hypersonic aircraft orbit reentry process 1, an aircraft real-time sensor 2 for sensing the aircraft state, a control station 3 for measuring and controlling the aircraft motion state variables, a ground database 4 for receiving and storing data, a trajectory optimal control system 5 and an aircraft state display control module 6. The aircraft real-time sensor 2 and the control station 3 are connected to the hypersonic aircraft orbit reentry process 1, the aircraft real-time sensor 2 and the control station 3 are connected to the ground database 4, the ground database 4 is connected to the input end of the trajectory optimal control system 5, and the output end of the trajectory optimal control system 5 is connected to the aircraft state display control module 6.

[0085] Reference Figure 2 , the trajectory optimal control system 5 also includes:

[0086] Data preprocessing module 7: input the data S sent back by the re-entry sensor of the hypersonic vehicle * , and then the data preprocessing module unifies the dimension input to S = {h0, V0, θ0, X0, X f ,X1,X2,X b ,R b}. Wherein, h0 is the initial altitude of the hypersonic aircraft, in km; V0 is the initial velocity of the aircraft, in km / s; θ0 is the initial turning angle of the aircraft; X0 is the initial position of the aircraft and X0 = (x0, y0) is the latitude and longitude of the aircraft; X f Terminal position and X f =(x f ,y f ); X1 and X2 are the waypoint constraints of the aircraft, X1=(x1,y1), X2=(x2,y2); X b is a no-fly zone constraint point and X b =(x b ,y b );R b The radius of the no-fly zone, in km.

[0087] Aircraft control module 9: satisfies relevant constraints in real time by controlling the aircraft tilt angle σ, acceleration a, heading angle θ, and tilt angle u. Assuming that the earth is a spherical non-rotating sphere, the hypersonic aircraft flight model is

[0088]

[0089] Among them, the path constraints of the hypersonic aircraft control module are fully satisfied including:

[0090] (1) Control constraints

[0091] In order to ensure the stability of the hypersonic vehicle, the normalized tilt angle u constraint of the control variable satisfies

[0092]

[0093] (2) Terminal state constraints

[0094] State equation for terminal hypersonic vehicle

[0095]

[0096] (3) Waypoint constraints

[0097] Assume that the total number of specific waypoints is i end , the position of the i-th waypoint is expressed in longitude and latitude as (x i ,y i ), then the waypoint constraints of the model satisfy

[0098]

[0099] (4) No-fly zone restrictions

[0100] Generally speaking, a no-fly zone is described as a circular restricted area with infinite height. Assume that the center of the j-th no-fly zone is The radius is Then the no-fly zone constraint satisfies

[0101]

[0102] Dynamic optimization algorithm module 8: By establishing a hypersonic vehicle reentry trajectory optimization system, the system optimization agility and optimization accuracy are improved while ensuring efficiency. The Bolza form of the cost function J of the trajectory optimization problem consists of two parts. One part is the cost function at the terminal time t f The other part is the cost from t0 to t f The cost of the integrand L. The optimal control is to determine the control u(t)∈R m and state x(t)∈R n To minimize the cost function while satisfying the equations of motion, path constraints and boundary constraints.

[0103] The cost function is

[0104]

[0105] At the same time, the constraints are

[0106]

[0107] The general optimal control problem (OCP) has the following form

[0108]

[0109]

[0110] x(t0)=x0

[0111] C i (x(t),u(t),t)≤0

[0112] C e (x(t),u(t),t)=0

[0113] Where J represents the Bolza-type cost function, u:R→R m represents a function of m control variables, x:R→R n represents a function with n state variables; C i :R n ×R m ×R→R p and C ε :R n ×R m ×R→R q They are inequality constraints and equality constraints respectively.

[0114] (1) The orthogonal collocation method of finite elements (OCFE) is introduced to transform the infinite-dimensional OCP problem into a finite-dimensional nonlinear problem (NLP).

[0115] First, the time interval [t0,t f ] is composed of nodes [t1,t2,...,t N-1 ] is divided into N subintervals, and t N =t f . The i-th subinterval [t i-1 ,t i ]Length i =t i -t i-1 At this time, the time interval is projected onto the [-1,1] standard interval through linear interpolation method.

[0116] or

[0117] Therefore, through the above division, the above problem can be transformed into

[0118]

[0119]

[0120] x1(-1)=x0

[0121] x i (-1) = x i-1 (1),(i=2,…,N)

[0122] C i (x i (τ),u i (τ),τ)≤0

[0123] C e (x i (τ),u i (τ),τ)=0

[0124] where x i and u i They correspond to the state variables and control variables on the i-th subinterval respectively.

[0125] Then, the above transformation problem can be solved by orthogonal collocation method. In the i-th subinterval, the state variable can be approximately expressed as

[0126]

[0127] Among them, X i (τ) is the Lagrange interpolation function, K i is the order of the interpolation function; X i,j =X i (τ j ) where τ j (j≠0) is a function node and τ0=-1; K i The Lagrangian basis functions are Then we can get

[0128]

[0129] So we can differentiate the equation to get

[0130]

[0131] Similarly, the control variable can also be approximated by the Lagrange interpolation polynomial

[0132]

[0133] Among them U i,j =U i (τ j ),(j≠0), so the problem can be transformed into

[0134]

[0135]

[0136] X 1，1 =x0

[0137]

[0138] C i (X i,k ,U i,k ,τ k )≤0

[0139] C e (X i,k ,U i,k ,τ k )=0

[0140] in

[0141] (2) can be solved according to the sequential quadratic programming (SQP) method. The above NLP problem (8) can be expressed as follows:

[0142]

[0143] stc i (z)=0,i∈E,

[0144] c i (z)≤0,i∈I

[0145] The corresponding sub-problem

[0146]

[0147]

[0148]

[0149] (3) The solution process of the sequential quadratic programming (SQP) method is as follows (3.1) Given μ1>0, error ε0>0, initial point z0, and set k=1;

[0150] (3.2) Solve the corresponding sub-problem of the transformed problem and calculate d k ;

[0151] (3.3) Adjustment parameter μ k , so that d k is the iteration point z k Function value ψ(z,μ k ) descends, and the iteration step length α is determined by linear search k ;

[0152] (3.4) Let z k+1 =z k +α k d k , solve the equation Get the parameter λ k+1 ,in

[0153] (3.5) If The iteration terminates and the final result z is output k as the algorithm result; otherwise, let k=k+1 and repeat (3.2).

[0154] (4) The accuracy and efficiency of OCFE are improved through the OCFE adaptive grid refinement strategy based on gradient information (G-OCFE). The proposed gradient-based segmentation strategy analyzes the optimal control strategy under the current segmentation according to the gradient information of the nodes and re-segments according to the given rules.

[0155] Let N T represents the total number of nodes under the current segmentation method, and the gradient of the kth node at the lth iteration, and let σ0 be the preset threshold

[0156]

[0157] in, is the value of the jth control variable at the kth node at the lth iteration, is the gradient of the j-th control variable at the k-th node at the l-th iteration.

[0158] When the normalized gradient of a node is too large, The control trajectory of this node usually needs to be subdivided more densely to improve the accuracy of the optimal control strategy. Therefore, choose The midpoint of is used as a segment node to subdivide the entire time interval.

[0159] σ0, as a preset parameter, will determine whether the adaptive method tends to global interpolation or local point low-order interpolation. When σ0 is greater than the gradient of all nodes, the entire interval will not be split to generate global interpolation; on the other hand, when σ0 decreases, more local low-order interpolation will be formed. The novel improved gradient adaptive configuration point method of the present invention can be adaptively adjusted based on the gradient information of the control trajectory and the constraint conditions, ensuring that the control trajectory achieves high robustness and high accuracy at fast-changing points.

[0160] When the feasible initial values ​​of the supersonic aircraft are input into the hypersonic aircraft reentry trajectory optimization system, the aircraft status display control module can control the aircraft in real time to achieve the optimal path and meet the path constraints.

[0161] The embodiments of the present invention are used to illustrate the present invention rather than to limit the present invention. Any modifications and changes made to the present invention within the spirit of the present invention and the protection scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A hypersonic vehicle trajectory optimal control system based on gradient adaptive finite element orthogonal configuration, used to solve the hypersonic vehicle reentry trajectory optimal control and process constraint full satisfaction, characterized by: It includes data preprocessing module, dynamic optimization algorithm module and aircraft control module; The input of the data preprocessing module is the data S transmitted by the sensor during the reentry process of the hypersonic vehicle. * , and then the data preprocessing module unifies the dimension input to S = {h0, V0, θ0, X0, X f ,X1,X2,X b ,R b }; Wherein, h0 is the initial altitude of the hypersonic aircraft, in km; V0 is the initial velocity of the aircraft, in km / s; θ0 is the initial turning angle of the aircraft; X0 is the initial position of the aircraft and X0 = (x0, y0) is the latitude and longitude of the aircraft; X f Terminal position and X f =(x f ,y f ); X1 and X2 are the waypoint constraints of the aircraft, X1=(x1,y1), X2=(x2,y2); X b is a no-fly zone constraint point and X b =(x b ,y b );R b is the radius of the no-fly zone, in km; The aircraft control module satisfies relevant constraints in real time by controlling the aircraft tilt angle σ, acceleration a, heading angle θ, and tilt angle u; The flight model of a hypersonic vehicle is: Among them, the path constraints of the hypersonic aircraft control module are fully satisfied including: (1) Control constraints In order to ensure the stability of the hypersonic vehicle, the normalized tilt angle u constraint of the control variable satisfies: (2) Terminal state constraints The state equation for the terminal hypersonic vehicle is: (3) Waypoint constraints: Assume that the total number of specific waypoints is i end , the position of the i-th waypoint is expressed in longitude and latitude as (x i ,y i ), then the waypoint constraints of the model satisfy (4) No-fly zone restrictions: The no-fly zone is described as a circular restricted area with infinite height. Assume that the center of the j-th no-fly zone is The radius is Then the no-fly zone constraint satisfies: The dynamic optimization algorithm module is used to establish a hypersonic vehicle reentry trajectory optimization system; The establishment of the hypersonic vehicle reentry trajectory optimization system is specifically as follows: firstly, the finite element orthogonal collocation method is introduced to transform the infinite-dimensional OCP problem into a finite-dimensional nonlinear problem; then, the problem is solved according to the sequential quadratic programming method SQP; finally, the accuracy and efficiency of OCFE are improved through an OCFE adaptive grid refinement strategy based on gradient information; a gradient-based segmentation strategy is proposed, which analyzes the optimal control strategy under the current segmentation according to the gradient information of the nodes, and re-segments the problem according to the given rules.

2. According to claim 1, the gradient adaptive finite element orthogonal configuration hypersonic vehicle trajectory optimal control system is characterized by: The Bolza form of the cost function J of the trajectory optimization problem consists of two parts. One part is the cost function J at the terminal time t f The other part is the cost from t0 to t f The cost of the integrand L; the optimal control is to determine the control u(t)∈R m and state x(t)∈R n To minimize the cost function while satisfying the equations of motion, path constraints and boundary constraints; The cost function is: At the same time, the constraints are: The general optimal control problem OCP has the following form: Where J represents the Bolza-type cost function, u:R→R m represents a function of m control variables, x:R→R n represents a function with n state variables; C i :R n ×R m ×R→R p and C ε :R n ×R m ×R→R q They are unequal value constraints and equal value constraints respectively; The establishment of the hypersonic vehicle reentry trajectory optimization system comprises the following steps (1) The finite element orthogonal collocation method is introduced to transform the infinite-dimensional OCP problem into a finite-dimensional nonlinear problem; First, the time interval [t0,t f ] is composed of nodes [t1,t2,...,t N-1 ] is divided into N subintervals, and t N =t f ; The i-th subinterval [t i-1 ,t i ]Length i =t i -t i-1 ; At this time, the time interval is projected onto the [-1,1] standard interval through linear interpolation method: Therefore, through the above division, problem (1) is transformed into: where x i and u i They correspond to the state variables and control variables on the i-th subinterval respectively; Then, the above problem (3) is solved by the orthogonal collocation method; in the i-th subinterval, the state variable is approximately expressed as: Among them, X i (τ) is the Lagrange interpolation function, K i is the order of the interpolation function; X i,j =X i (τ j ) where τ j (j≠0) is a function node and τ0=-1; K i Lagrangian basis functions of order and Then get Then, by differentiating equation (4), we get: Similarly, the control variables are approximately obtained by the Lagrange interpolation polynomial: Among them U i,j =U i (τ j ),(j≠0), so the problem is transformed into: in, (2) Solve according to the sequential quadratic programming method SQP; the above NLP problem (8) is expressed as follows: The corresponding sub-problems are: (3) The sequential quadratic programming method is used to solve the problem. The process is as follows: (3.1) Given μ1>0, error ε0>0, initial point z0, and set k=1; (3.2) Solve problem (10) and calculate d k ; (3.3) Adjustment parameter μ k , so that d k is the iteration point z k Function value ψ(z,μ k ) descends, and the iteration step length α is determined by linear search k ; (3.4) Let z k+1 =z k +α k d k , solve the equation Get the parameter λ k+1 ,in (3.5) If The iteration terminates and the final result z is output k as the algorithm result; otherwise, let k = k + 1 and repeat (3.2); (4) The accuracy and efficiency of OCFE are improved through the adaptive mesh refinement strategy of OCFE based on gradient information. The proposed gradient-based segmentation strategy analyzes the optimal control strategy under the current segmentation according to the gradient information of the node and re-segments it according to the given rules. Let N T represents the total number of nodes under the current segmentation method, and the gradient of the kth node at the lth iteration, and let σ0 be the preset threshold: in, is the value of the jth control variable at the lth iteration of the kth node; When the normalized gradient of a node is too large, When The midpoint of is used as a segment node to subdivide the entire time interval.

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