A Cooperative Train Tracking Control Method under Relative Braking Protection Constraints

Abstract

The present invention discloses a multi - train cooperative tracking control method under relative braking protection constraints, including: constructing an optimization model; cooperative tracking control; the construction of the optimization model specifically includes establishing a train dynamic model, train operation constraints, and an optimal control problem; the cooperative tracking control specifically includes model reconstruction based on the augmented Lagrangian method, solving using the trust - region method, and designing a multi - train cooperative tracking control algorithm under relative braking protection constraints. The solution algorithm of the present invention based on the trust - region method realizes the efficient solution of the multi - train cooperative tracking control strategy under relative braking distances. The cooperative control strategy designed by the present invention is solved by the trust - region method, enabling high - speed trains to adjust the control force at each decision moment. This adjustment enables the actual inter - vehicle distance to quickly converge to the desired inter - vehicle distance, ensuring the efficiency and safety of train operation.

Description

Technical Field

[0001] The present invention relates to the field of train operation collaborative tracking operation control, and particularly to a multi-train collaborative tracking control method under relative braking protection constraints. Background Art

[0002] Nowadays, due to the significant advantages of railway transportation in terms of speed, punctuality, and energy efficiency, railways have become one of the most popular passenger transportation modes. To further improve transportation efficiency and flexibility, many emerging train control technologies are being studied, such as the latest train operation control system based on vehicle-to-vehicle communication and the virtual formation train operation control scheme. With the increase in train density, the spacing between trains is getting smaller and smaller. How to ensure the safe spacing between adjacent trains under high train density and tight tracking conditions, and design a tracking controller for multiple trains is of great significance in the theoretical and practical engineering fields.

[0003] In the existing train control system, the absolute braking spacing method is usually used to calculate the safety margin. This method assumes that the train in front is stationary, so the protection interval between adjacent trains is increased accordingly. To further improve efficiency, the spacing protection method based on relative braking has received more and more attention. In this method, both the position information and speed information of the train in front are taken into account. Compared with the absolute braking spacing method, the method based on relative braking spacing significantly reduces the distance between adjacent trains, thereby improving the line capacity. Summary of the Invention

[0004] To solve the problems of the existing technology, the present invention proposes a multi-train collaborative tracking control method under relative braking protection constraints, and the method includes:

[0005] Construct an optimization model;

[0006] Collaborative tracking control;

[0007] The construction of the optimization model specifically includes the establishment of a train dynamic model, train operation constraints, and an optimal control problem;

[0008] The collaborative tracking control specifically includes model reconstruction based on the augmented Lagrangian method, solving using the trust region method, and the design of a multi-train collaborative tracking control algorithm under relative braking protection constraints.

[0009] Furthermore, the establishment of the train dynamic model specifically includes,

[0010] During the operation of the train, it is affected by traction / braking force, basic resistance, external disturbance, and additional resistance composed of ramp and curve resistance. Its dynamic equation is expressed as:

[0011] ;

[0012] in, and Separately for the moment train position, control and speed, For trains Quality, and Train At the moment Basic resistance, additional resistance and external disturbance;

[0013] In order to ensure that the actual motion of the train can track the reference curve, a and Train The reference position and speed; further introduce the deviation variable and Respectively represent the deviation of the actual train position, speed and control force relative to the reference value; Equation (1) is sampled at every time interval Sampling, getting the train The discrete-time state error equation is:

[0014] ;

[0015] in, and is the basic drag coefficient. To simplify the description, the matrix is introduced Refactoring equation (2) rewrites the constraint into the following compact form:

[0016] ;

[0017] in,

[0018] .

[0019] Furthermore, the train operation constraints specifically include:

[0020] To ensure the safe operation of adjacent trains, the train and train The distance between them should meet the minimum spacing constraint; the calculation of the safety spacing is divided into absolute braking distance and relative braking distance;

[0021] The absolute braking distance calculation method ignores the movement of the front train and assumes that the speed is zero. The safety distance constraint is expressed as:

[0022] ;

[0023] in For trains The maximum braking force, represents the length of the train ; is the minimum spacing to ensure the safe operation of the train;

[0024] The relative braking distance calculation method fully considers the position and speed information of the train ahead, and the spacing constraint based on the relative braking distance is expressed as:

[0025] ;

[0026] where is the maximum braking force of the train ;

[0027] The actual control force input is restricted by the actual train actuator, and there are the following control constraints:

[0028] ;

[0029] where and respectively represent the minimum and maximum allowable control inputs of the train ; The actual speed of the train is restricted by the maximum allowable speed, and the expression is as follows:

[0030] ;

[0031] The maximum allowable speed is determined by the line conditions, weather conditions and temporary speed limits.

[0032] Furthermore, the optimal control problem specifically includes

[0033] Constructing an optimal control model for accurately tracking the reference speed curve under the protection of the relative braking distance, as follows:

[0034] ;

[0035] ;

[0036] In the optimal control problem represents the number of in-service trains, and respectively represent the initial time and the termination time, and are the corresponding weight coefficients, aiming to find a balance between punctuality and energy efficiency; specifically, by minimizing the tracking speed deviation and the tracking position deviation in the objective function, the actual position-speed curve of the in-service train is close to the desired speed-position curve; by reducing the third term in the objective function , i.e., the error between the actual input and the reference control force, reduces energy consumption and achieves smooth driving; the train state transition equation is given by Equation (3), while (6) represents the interval constraint with respect to the braking distance; the state and control constraints of the train are given by Equations (7) and (8); the initial speed and the terminal speed are given by Equation (9);

[0037] Based on the model predictive control (MPC) framework, the objective function of the original problem (9) is reconstructed as follows:

[0038] ;

[0039] where, is the finite prediction horizon; thus, the original optimal tracking control problem is formulated as the following optimization problem at each sampling instant :

[0040] ;

[0041] ;

[0042] ;

[0043] ;

[0044] .

[0045] Furthermore, the model reconstruction based on the augmented Lagrangian method specifically includes

[0046] transforming the optimal control problem (11) and the constraint conditions (12)-(15) into an unconstrained optimization problem;

[0047] introducing the augmented Lagrangian method to reformulate the original optimization problem; the constructed optimal control problem includes equality constraints and inequality constraints; for the equality constraint (12), introducing the Lagrangian multiplier and the penalty parameter to obtain the following term:

[0048] ;

[0049] for the relative braking distance inequality constraint (13), the reconstructed expression is as follows:

[0050] ;

[0051] where, is the Lagrangian multiplier;

[0052] the speed constraint (14) is regarded as consisting of two inequalities, and by introducing the Lagrangian multiplier and , and its enhanced form is expressed as follows:

[0053] ;

[0054] Similarly, the enhanced form of the control force constraint (15) is as follows:

[0055] ;

[0056] where and are Lagrange multipliers;

[0057] To simplify the transformation process, the original problem (11) is rewritten in the following compact form:

[0058] ;

[0059] where is the objective function defined in (10), is the vector of control variables, represents the set of equality constraints including the train position and velocity state transition equation (12), represents the set of inequality constraints, including the train headway constraint (13), the train speed constraint (14), and the train control force constraint (15), and are the index sets of equality and inequality constraints respectively;

[0060] By using the augmented Lagrangian method, the constrained optimal control problem (20) is equivalently transformed into the following unconstrained form:

[0061] ;

[0062] where and are the vectors of Lagrange multipliers for equality and inequality constraints respectively, is the penalty parameter, which imposes a penalty when the constraint is violated during the iteration process; is the slack variable, which is equal to zero when the constraint is satisfied and does not affect the control force in the objective function when the constraint is violated; specifically, the constrained optimal control problem (11) with the constraint conditions (12)-(15) is equivalently transformed into the following unconstrained optimization problem:

[0063] ;

[0064] where is the unconstrained problem after transformation, is the decision variable, and is the Lagrange multiplier matrix of equality and inequality constraints.

[0065] Furthermore, the trust region method is used for solving, specifically including:

[0066] After obtaining the unconstrained problem (22) at each sampling moment, the trust region method is used for solving; the basic idea of the trust region method is to construct a trust region sub-problem within a given trust region radius, regarded as the quadratic approximation of the transformed problem (22).

[0067] To facilitate the construction of the trust region sub-problem, first calculate the gradient matrix of the optimization problem (22) and the Hessian matrix The calculation is as follows:

[0068] ;

[0069] ;

[0070] The th iteration trust region sub-problem is expressed as follows:

[0071] ;

[0072] where is the decision variable of the sub-problem (24), is the trust region radius and satisfies , and is an arbitrary vector norm, usually taken as norm or norm;

[0073] By solving the quadratic programming sub-problem (24), the decision variable is obtained, which is also called the trial step size; to check whether the trial step size should be accepted, a new variable is introduced, which represents the ratio of the actual decrease value to the predicted decrease value, and is defined as follows:

[0074] ;

[0075] where and are the actual decrease value and the predicted decrease value of the objective function after applying the trial step size respectively; the actual decrease value is expressed as follows:

[0076] ;

[0077] The predicted decrease value is defined as follows:

[0078] ;

[0079] If the trial step size taken can reduce the value of the objective function (22) to a sufficient extent, i.e., , then accept the trial step size and use it to construct the new initial vector for the next sub-problem; otherwise, the trial step size needs to be rejected; the iterative process is as follows:

[0080] ;

[0081] It should be noted that, as can be seen from (28), the update of the trust region radius also depends on the value of; if , then the radius needs to be reduced in the next iteration because no progress has been made under the current radius; if , it means that the trial step size approximates well, and increase the trust region radius to allow a larger step size; if , then the trust region radius will be adjusted; the adjustment of the trust region radius is summarized as follows:

[0082] ;

[0083] where is the trust region radius at the -th iteration, is the upper bound of the trust region radius, is a positive constant parameter satisfying ; empirically, is set to ;

[0084] After updating the decision vector for a new round of iteration, it is also necessary to determine whether the termination condition is satisfied, and the conditions are set as follows:

[0085] ;

[0086] where is the acceptable termination threshold; if this condition is satisfied, the optimal solution is obtained, otherwise, reconstruct the sub-problem using the updated information;

[0087] Based on the above description, the trust region solution method for the unconstrained cooperative tracking control problem (22) is summarized in Algorithm 1 below;

[0088] Algorithm 1: Trust Region Based Solution Method

[0089] Input:

[0090] Unconstrained problem ;

[0091] Initial point ;

[0092] Initial parameters ;

[0093] Output:

[0094] Optimal solution and optimal value ;

[0095] Steps:

[0096] Set to ;

[0097] Calculate and ;

[0098] If then

[0099] Obtain and and terminate the algorithm

[0100] Otherwise,

[0101] Construct a sub - problem and solve it

[0102] Calculate ;

[0103] Update ;

[0104] If then

[0105] ;

[0106] Otherwise,

[0107] ;

[0108] Update ;

[0109] End the loop

[0110] The input of Algorithm 1 includes the initial vector , the initial trust - region radius , the acceptable termination threshold , and the unconstrained tracking control problem (22); by constructing and solving the trust - region sub - problem at each iteration, the trial step size is obtained and used to update the subsequent iteration variables and the trust - region radius Update according to (28) and (29) respectively; this process is repeated until the iterative termination condition (30) is met; after this condition is satisfied, the solution to the unconstrained optimization problem (22) is obtained, thus providing the optimal control input at each decision moment.

[0111] Furthermore, the design of the multi - train cooperative tracking control algorithm under relative braking protection constraints specifically includes: Based on model reconstruction and the trust - region algorithm, it is proposed that the multi - train cooperative tracking control under relative braking protection constraints is divided into a two - layer structure. The outer layer structure uses the augmented Lagrangian method to transform the original problem into an unconstrained optimal control problem, and the inner layer structure uses the trust - region method to efficiently solve the equivalent problem; the overall framework is shown in Algorithm 2:

[0112] Algorithm 2: Multi - train Cooperative Tracking Control Algorithm under Relative Braking Protection Constraints

[0113] Input:

[0114] The MPC problem (11) at each sampling time

[0115] Initial parameters ;

[0116] Output:

[0117] Optimal control force error ;

[0118] External structure:

[0119] Set to ;

[0120] Reconstruct the unconstrained problem ;

[0121] Internal structure:

[0122] Input into Algorithm 1

[0123] Obtain and values

[0124] End of internal structure

[0125] Calculate value

[0126] If , then

[0127] Set and ;

[0128] Obtain from ;

[0129] Break out of the loop

[0130] Otherwise,

[0131] Update the Lagrange multiplier and penalty factor

[0132] Update

[0133] End the loop

[0134] End the outer structure

[0135] The input of the outer structure is the original optimal control problem (1); by introducing the Lagrange multiplier and penalty parameters, the relative braking margin and other related operation constraints are added to the objective function to obtain the unconstrained problem (22); in each iteration of the outer layer, Algorithm 1 is triggered to solve the unconstrained problem (22), and a new variable is introduced to check whether the termination condition is satisfied, and its calculation is as follows:

[0136] ;

[0137] Then the termination criterion is expressed as:

[0138] ;

[0139] where is the acceptable termination threshold; if the constraint condition (32) holds, the optimal solution of the unconstrained problem is obtained and the iteration is terminated; otherwise, the penalty parameter should be updated according to (33) for the next iteration:

[0140] ;

[0141] where is a constant, which is set empirically; accordingly, the Lagrange multipliers for the equality and inequality constraints (12)-(15) should be updated as follows:

[0142] ;

[0143] ;

[0144] During the iterative interaction process of the inner and outer structures, the outer structure reconstructs the original optimization problem and outputs an unconstrained problem in each iteration. The unconstrained problem is then passed as input to the inner structure, which returns the solution of the unconstrained problem and gives it to the outer structure. The outer structure then uses the solution to check the termination condition and update the unconstrained problem; if the termination condition (32) is satisfied, the current solution is regarded as the global optimal solution, otherwise a new unconstrained problem needs to be reconstructed for the next iteration.

[0145] On the other hand, the present invention also provides an electronic device,

[0146] including a memory, a processor, and computer instructions stored on the memory and running on the processor. When the computer instructions are run by the processor, the above-mentioned method is completed.

[0147] On the other hand, the present invention also provides a computer-readable storage medium for storing computer instructions. When the computer instructions are executed by a processor, the above-mentioned method is completed.

[0148] Beneficial effects of the present invention:

[0149] 1. The present invention proposes a solution algorithm based on the trust region method, realizing an efficient solution for the multi-train cooperative tracking control strategy under the relative braking distance.

[0150] 2. The cooperative control strategy designed by the present invention is solved by the trust region method, enabling the high-speed train to adjust the control force at each decision moment. This adjustment enables the actual inter-vehicle distance to quickly converge to the desired inter-vehicle distance, ensuring the efficiency and safety of train operation. BRIEF DESCRIPTION OF THE DRAWINGS

[0151] The present invention has the following drawings:

[0152] Figure 1 Train spacing based on the absolute braking distance;

[0153] Figure 2 Train spacing based on the relative braking distance;

[0154] Figure 3 Train reference speed curve;

[0155] Figure 4 Train time-space diagram under relative braking control;

[0156] Figure 5 Spacing between adjacent trains. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0157] The following further describes the present invention in detail with reference to the Figures 1-5 accompanying drawings.

[0158] I. Optimization model construction

[0159] In this section, first, the dynamic operation model of the train is described, and the mechanism of the relative braking distance is explained. Then, under the goal of minimizing the tracking error, an optimization model for the cooperative tracking control problem is constructed based on the relative braking distance constraint.

[0160] A. Train dynamic model

[0161] During the operation of the train, it is affected by traction / braking force, basic resistance, external disturbances, and additional resistance composed of gradient and curve resistance. Its dynamic equation can be expressed as:

[0162] ;

[0163] where, are the position, control force, and speed of the train at time respectively, is the mass of the train, are the basic resistance, additional resistance, and external disturbance of the train at time respectively. at time

[0164] To ensure that the actual movement of the train can track the reference curve, and are defined as the reference position and speed of the train respectively. Further, deviation variables are introduced to represent the deviations of the actual position, speed, and control force of the train from the reference values. Sampling the equation (1) at every time interval results in the discrete-time state error equation of the train :

[0165] ;

[0166] where, and are the basic resistance coefficients. To simplify the description, matrix is introduced to reconstruct equation (2), and thus the constraint can be rewritten in the following compact form:

[0167] ;

[0168] where,

[0169] ;

[0170] B. Train operation constraints

[0171] To ensure the safe operation of adjacent trains, the distance between train and train should satisfy the minimum spacing constraint. Generally, the calculation of the safety spacing can be divided into two types: absolute braking distance and relative braking distance.

[0172] For the calculation method of the absolute braking distance, such as Figure 1 ​As shown in the figure, ignoring the movement of the train ahead and assuming its speed is zero, the safety distance constraint can be expressed as:

[0173] ;

[0174] where is the maximum braking force of the train . represents the length of the train , is the minimum distance to ensure the safe operation of the train.

[0175] As Figure 2 shown, the relative braking distance calculation method fully considers the position and speed information of the train ahead. The spacing constraint based on the relative braking distance is expressed as:

[0176] ;

[0177] where is the maximum braking force of the train . By comparing Figure 1 and Figure 2 , it can be clearly seen that the safety distance based on the relative braking distance can shorten the minimum distance between adjacent trains, thus improving the tracking efficiency.

[0178] By comparing Equation (5) and Equation (6), it can be found that the spacing constraint based on the relative braking distance can shorten the tracking interval between trains on the premise of ensuring safety. The present invention adopts Equation (6) to protect the safety of trains.

[0179] In addition, the actual control force input is limited by the actual train actuator, and there are the following control constraints:

[0180] ;

[0181] where and respectively represent the minimum and maximum allowable control inputs of the train . The actual speed of the train is also limited by the maximum allowable speed, and the expression is as follows:

[0182] ;

[0183] The maximum allowable speed is determined by the line conditions, weather conditions and temporary speed limits.

[0184] C. Optimal control problem

[0185] Based on the above analysis, an optimal control model is constructed as shown in Equation (9), aiming to achieve precise tracking of the reference speed curve under the protection of the relative braking distance, as follows:

[0186] ;

[0187] ;

[0188] In the optimization control problem, represents the number of in - service trains, and represent the initial time and the terminal time respectively, and are the corresponding weight coefficients, aiming to find a balance between punctuality and energy efficiency. Specifically, by minimizing the tracking speed deviation and the tracking position deviation in the objective function, the actual position - speed curve of the in - service trains is made to approach the desired speed - position curve as much as possible, thereby improving safety and punctuality. By reducing the third term in the objective function, that is, the error between the actual input and the reference control force, energy consumption is reduced and smooth driving is achieved. The train state transition equation is given by formula (3), and (6) represents the interval constraint with respect to the braking distance. In addition, the state and control constraints of the train are given by formulas (7) and (8). The initial speed and the terminal speed are given by formula (9).

[0189] Furthermore, based on the model predictive control (MPC) framework, the objective function of the original problem (9) is reconstructed as follows:

[0190] ;

[0191] where is the finite prediction horizon. Therefore, the original optimal tracking control problem at each sampling time can be formulated as the following optimization problem:

[0192] ;

[0193] ;

[0194] ;

[0195] ;

[0196] ;

[0197] However, it is noted that formula (13) is a complex constraint that contains quadratic terms and is coupled with adjacent trains, which makes it difficult to directly solve the MPC problem by conventional methods. To solve this difficulty, the present invention proposes a cooperative control strategy solution algorithm based on the trust region method, and the specific content will be described in detail below.

[0198] II. Trust Region Method

[0199] In this section, in order to efficiently handle complex relative braking distance constraints and meet the real-time requirements of train operation, a novel cooperative tracking control strategy is proposed based on the trust region method. First, the augmented Lagrangian method is used to transform the original problem into an unconstrained problem, and then the trust region method is used to efficiently solve the equivalent unconstrained problem.

[0200] A. Model Reconstruction Based on the Augmented Lagrangian Method

[0201] Traditionally, the trust region method is used to handle unconstrained optimization problems. To apply the trust region method to train optimization control problems, the optimization control problem (11) and its constraint conditions (12)-(15) need to be transformed into an unconstrained optimization problem.

[0202] Here, the augmented Lagrangian method is introduced to reformulate the original optimization problem. It can be found that the constructed optimization control problem contains equality constraints and inequality constraints. For the equality constraint (12), the Lagrange multiplier and the penalty parameter are introduced to obtain the following term:

[0203] ;

[0204] For the relative braking distance inequality constraint (13), the reconstructed expression is as follows:

[0205] ;

[0206] where is the Lagrange multiplier.

[0207] The speed constraint (14) can be regarded as consisting of two inequalities. By introducing the Lagrange multipliers and , its augmented form is expressed as follows:

[0208] ;

[0209] Similarly, the augmented form of the control force constraint (15) is as follows:

[0210] ;

[0211] where and are the Lagrange multipliers.

[0212] To simplify the transformation process, the original problem (11) can be rewritten in the following compact form:

[0213] ;

[0214] Among them, is the objective function defined in (10), is the vector of control variables, represents the set of equality constraints including the train position and speed state transition equation (12), represents the set of inequality constraints, including the train headway constraint (13), the train speed constraint (14), and the train control force constraint (15), and are the index sets of the equality and inequality constraints respectively.

[0215] By using the augmented Lagrangian method, the above constrained optimal control problem (20) can be equivalently transformed into the following unconstrained form:

[0216] ;

[0217] Among them, and are the Lagrangian multiplier vectors of the equality and inequality constraints respectively, is the penalty parameter, and during the iteration process, when the constraint is violated, a penalty will be imposed.

[0218] is the slack variable, which is equal to zero when the constraint is satisfied and has no impact on the control force in the objective function when the constraint is violated. Specifically, the constrained optimal control problem (11) and the constraint conditions (12)-(15) can be equivalently transformed into the following unconstrained optimization problem:

[0219] ;

[0220] Among them, is the unconstrained problem after transformation, is the decision variable, and are the Lagrangian multiplier matrices of the equality and inequality constraints.

[0221] B. Trust Region Algorithm

[0222] After obtaining the unconstrained problem (22) at each sampling moment, the trust region method is used to solve it. The basic idea of the trust region method is to construct a trust region subproblem within a given trust region radius, which can be regarded as a quadratic approximation of the transformed problem (22).

[0223] To facilitate the construction of the trust region subproblem, first calculate the gradient matrix and the Hessian matrix The calculation is as follows:

[0224] ;

[0225] ;

[0226] The trust-region subproblem for the th iteration is expressed as follows:

[0227] ;

[0228] where is the decision variable of subproblem (24), is the trust-region radius, and it satisfies , and is an arbitrary vector norm, usually taken as the norm or the norm.

[0229] By solving the quadratic programming subproblem (24), the decision variable is obtained, which is also called the trial step size. To check whether the trial step size should be accepted, a new variable is introduced, which represents the ratio of the actual decrease to the predicted decrease, and is defined as follows:

[0230] ;

[0231] where and are the actual decrease and the predicted decrease of the objective function after applying the trial step size, respectively. The actual decrease is expressed as follows:

[0232] ;

[0233] The predicted decrease is defined as follows:

[0234] ;

[0235] If the trial step size taken can reduce the value of the objective function (22) to a sufficient extent, i.e., , then the trial step size is accepted and used to construct the new initial vector for the next subproblem. Otherwise, the trial step size needs to be rejected. The iterative process can be expressed as follows:

[0236] ;

[0237] It should be noted that, as can be seen from (28), the update of the trust-region radius also depends on the value. If , then the radius needs to be reduced in the next iteration because no progress has been made under the current radius. If , it means that the trial step size is approximated well, and the trust region radius can be increased to allow a larger step size. If , the trust region radius may be slightly adjusted. The adjustment of the trust region radius is summarized as follows:

[0238] ;

[0239] where is the trust region radius at the -th iteration, is the upper bound of the trust region radius, is a positive constant parameter satisfying . Empirically, is set to .

[0240] In addition, after updating the decision vector for a new round of iteration, it is also necessary to determine whether the termination condition is satisfied, and the conditions are set as follows:

[0241] ;

[0242] where is an acceptable termination threshold. If this condition is satisfied, the optimal solution is obtained; otherwise, the sub-problem is reconstructed using the updated information.

[0243] Based on the above description, the trust region solution method for the unconstrained cooperative tracking control problem (22) is summarized in Algorithm 1 below.

[0244] Algorithm 1: Trust Region Based Solution Method

[0245] Input:

[0246] Unconstrained problem ;

[0247] Initial point ;

[0248] Initial parameter ;

[0249] Output:

[0250] Optimal solution and optimal value ;

[0251] Steps:

[0252] Set to .

[0253] Calculate and .

[0254] If , then

[0255] Obtain and and terminate the algorithm.

[0256] Otherwise,

[0257] Construct sub-problems , and solve

[0258] Calculate ;

[0259] Update ;

[0260] If , then

[0261] ;

[0262] Otherwise

[0263] ;

[0264] Update ;

[0265] End the loop

[0266] The input of Algorithm 1 includes the initial vector , the initial trust region radius , the acceptable termination threshold , and the unconstrained tracking control problem (22). By constructing and solving the trust region sub-problem in each iteration, the trial step size is obtained and used to update the subsequent iteration variables and the trust region radius which are updated according to (28) and (29) respectively. This process is repeated until the iteration termination condition (30) is satisfied. After satisfying this condition, we can obtain the solution of the unconstrained optimization problem (22), thus providing the optimal control input at each decision moment.

[0267] C. Design of Multi-Train Cooperative Tracking Control Algorithm under Relative Braking Protection Constraint

[0268] Based on model reconstruction and trust region algorithm, the multi-train cooperative tracking control proposed by the present invention under relative braking protection constraint can be divided into a two-layer structure, where the outer layer structure uses the augmented Lagrangian method to transform the original problem into an unconstrained optimal control problem, and the inner layer structure uses the trust region method to efficiently solve the equivalent problem. The overall framework is shown in Algorithm 2.

[0269] Algorithm 2: Multi-Train Cooperative Tracking Control Algorithm under Relative Braking Protection Constraint

[0270] Input:

[0271] MPC problem (11) at each sampling instant.

[0272] Initial parameters .

[0273] Output:

[0274] Optimal control force error .

[0275] External structure:

[0276] Set to .

[0277] Reconstruct the unconstrained problem .

[0278] Internal structure:

[0279] Input into Algorithm 1.

[0280] Obtain and values.

[0281] End of internal structure

[0282] Calculate value.

[0283] If , then

[0284] Set and ;

[0285] Obtain from .

[0286] Break the loop

[0287] Otherwise,

[0288] Update the Lagrange multipliers and penalty factors.

[0289] Update ;

[0290] End of loop

[0291] End of external structure

[0292] The input of the outer structure is the original optimal control problem (1). By introducing Lagrange multipliers and penalty parameters, the relative braking margin and other related operational constraints are added to the objective function, resulting in the unconstrained problem (22). In each iteration of the outer layer, Algorithm 1 is triggered to solve the unconstrained problem (22), and a new variable is introduced to check whether the termination condition is satisfied, which is calculated as follows:

[0293] ;

[0294] Then the termination criterion is expressed as:

[0295] ;

[0296] where is the acceptable termination threshold. If the constraint condition (32) holds, the optimal solution of the unconstrained problem is obtained and the iteration is terminated. Otherwise, the penalty parameter should be updated according to (33) for the next iteration:

[0297] ;

[0298] where is a constant, which is set empirically. Accordingly, the Lagrange multipliers for the equality and inequality constraints (12)-(15) should be updated as follows:

[0299] ;

[0300] ;

[0301] During the iterative interaction between the inner and outer structures, the outer structure reconstructs the original optimization problem and outputs an unconstrained problem at each iteration, which is then passed as input to the inner structure. The inner structure returns the solution of the unconstrained problem and gives it to the outer structure, and the outer structure then uses this solution to check the termination condition and update the unconstrained problem. If the termination condition (32) is satisfied, the current solution is regarded as the global optimal solution; otherwise, a new unconstrained problem needs to be reconstructed for the next iteration.

[0302] Through Figure 4 and Figure 5 numerical experiments are used to illustrate the performance of the tracking strategy.

[0303] In the numerical example, the real data of the Beijing-Shanghai High-Speed Railway in China is selected to verify the effectiveness and efficiency of the proposed cooperative control strategy. The actual operation data of the high-speed train from Tianjin South Station to Beijing South Station is selected. The distance between the two stations is about 122 kilometers, and the travel time is about 30 minutes. In the simulation experiment, 6 high-speed trains running from Tianjin South Station to Beijing South Station are considered. The length of each train is 210 meters, and the minimum safe distance between trains is set to 1200 meters. In addition, the other parameters of the high-speed train are set as follows: is set to is set to , is set to is set to is set to . The weight coefficients are all uniformly set to 0.1. In our experiment, it is assumed that each train follows the same reference speed curve but departs at different times. The initial positions of the trains are 0, 1310, 2522, 3730, 5010, and 6130 meters respectively, resulting in the initial distances between adjacent trains being 1310, 1212, 1208, 1280, and 1120 meters respectively. The reference speed curve is derived from the actual operation data, as shown in Figure 3 . To verify the performance of the proposed method, a disturbance is introduced to each train at the first sampling moment. Based on the above data, three groups of simulation experiments are designed to illustrate the effectiveness and performance of the proposed cooperative tracking strategy.

[0304] Figure 4 shows the space-time trajectories of these 6 trains under the relative braking distance. By applying the proposed cooperative control strategy, it can be clearly seen from Figure 4 that by appropriately adjusting the distance between trains during operation, all trains follow the train in front more closely while maintaining a safe relative braking distance, demonstrating the effectiveness of the proposed strategy. In addition, Figure 5 shows the variation of the actual distance curve and the reference distance curve between trains with time for each train. As shown in Figure 5 , although the initial distances between adjacent trains due to the initial positions of each train do not reach the expected distance, by implementing the proposed strategy, the trains can dynamically adjust the distance between adjacent trains to reach the expected value. Subsequently, the required safe distance between trains will be maintained until the trains arrive at the station. For each train, the actual tracking curve closely follows the reference curve and is always above it, which confirms that the proposed strategy effectively ensures the safety of train operation.

[0305] As shown in Figure 4 and Figure 5As shown, although the initial position of the train causes a positive or negative deviation of the inter-train distance relative to the expected value, the designed cooperative control strategy is solved by the trust-region method, enabling the high-speed train to adjust the control force at each decision moment. This adjustment enables the actual inter-train distance to quickly converge to the expected inter-train distance, ensuring the efficiency and safety of train operation.

[0306] The above embodiments are only used to illustrate the invention patent, rather than limiting the invention patent. Those of ordinary skill in the relevant technical field can also make various changes and deformations without departing from the essence and scope of the invention patent. Therefore, all equivalent technical solutions also belong to the scope of the invention patent, and the patent protection scope of the invention patent shall be defined by the claims.

[0307] The content not described in detail in this specification belongs to the well-known prior art of those skilled in the art.

Claims

1. A multi - train collaborative tracking control method under relative braking protection constraints, characterized in that, The method includes: Constructing an optimization model; Cooperative tracking control; The construction of the optimization model specifically includes the establishment of a train dynamic model, train operation constraints, and an optimal control problem; The cooperative tracking control specifically includes model reconstruction based on the augmented Lagrangian method, solving using the trust region method, and the design of a multi-train cooperative tracking control algorithm under relative braking protection constraints; Among them, the model reconstruction based on the augmented Lagrangian method transforms the optimal control problem and constraint conditions into an unconstrained optimization problem; The solution using the trust region method constructs a trust region sub-problem within a given trust region radius, regarded as a quadratic approximation of the transformed unconstrained optimization problem; The design of the multi-train cooperative tracking control algorithm under relative braking protection constraints is based on model reconstruction and the trust region algorithm. It is proposed that the multi-train cooperative tracking control under relative braking protection constraints is divided into two-layer structure. The outer layer structure uses the augmented Lagrangian method to transform the original problem into an unconstrained optimal control problem, and the inner layer structure uses the trust region method to efficiently solve the equivalent problem; during the iterative interaction process of the inner and outer structures, the outer structure reconstructs the original optimization problem and outputs an unconstrained problem at each iteration. The unconstrained problem is then passed as input to the inner structure, and the inner structure returns the solution of the unconstrained problem and gives it to the outer structure. The outer structure then uses the solution to check the termination condition and update the unconstrained problem; if the termination condition is satisfied, the current solution is regarded as the global optimal solution, otherwise a new unconstrained problem needs to be reconstructed for the next iteration.

2. The multi - train collaborative tracking control method under relative braking protection constraints as described in claim 1, wherein, The establishment of the train dynamic model specifically includes: During the operation of the train, it is affected by traction / braking force, basic resistance, external disturbance, and additional resistance composed of ramp and curve resistance. Its dynamic equation is expressed as: ; Among them, and are the position, control force and speed of the train at time train respectively, is the mass of the train ; and are the basic resistance, additional resistance and external disturbance of the train at time respectively at time ; To ensure that the actual movement of the train can track the reference curve, , and are defined as the reference position, speed, and control force of the train respectively; further introduce deviation variables and to represent the deviations of the actual position, speed, and control force of the train from the reference values respectively; sample Equation (1) at every time interval to obtain the discrete-time state error equation of the train : ; Among them, and are the basic drag coefficients. For simplicity of description, a matrix is introduced to reconstruct Equation (2), so that the constraints can be rewritten in the following compact form: ; Among them, 。 3. The multi - train cooperative tracking control method under relative braking protection constraints as described in claim 2, wherein, The train operation constraints specifically include: To ensure the safe operation of adjacent trains, the distance between train and train should meet the tracking interval constraint; The calculation of the tracking interval is divided into the tracking interval based on the absolute braking distance and the tracking interval based on the relative braking distance; For the tracking interval based on the absolute braking distance, the movement of the train ahead is ignored during calculation, assuming its speed is zero; the constraint corresponding to this tracking interval can be expressed as: ; wherein is the maximum braking force of the train , represents the length of the train , is the safety margin to ensure the safe operation of the train; For the tracking interval based on the relative braking distance, the position and speed information of the train ahead are fully considered during calculation; the constraint corresponding to this tracking interval can be expressed as: ; Among them is the maximum braking force of the train ; The control force input is restricted by the actual train actuator, and there are the following control constraints: ; wherein and represent the minimum and maximum allowable control inputs of the train, respectively; the actual speed of the train is limited by the maximum allowable speed, and the expression is as follows: ​ ; Maximum allowable speed Determined by track conditions, weather conditions and temporary speed restrictions.

4. The multi-train collaborative tracking control method under relative braking protection constraint according to claim 3, characterized in that The optimal control problem specifically includes: Constructing an optimal control model for precise tracking of the reference speed curve under relative braking protection, as follows: ; ; In the optimization control problem (9), represents the number of trains on the line, and respectively represent the initial time and the termination time, and are the corresponding weight coefficients, aiming to find a balance between punctuality and energy efficiency; based on the model predictive control (MPC) framework, the objective function of the optimization control problem (9) is reconstructed as follows: ; where, is the finite prediction horizon; thus, the optimal control problem (9) at each sampling instant is formulated as the following optimization problem: ; ; ; ; 。 5. The multi - train cooperative tracking control method under relative braking protection constraint according to claim 4, wherein, The model reconstruction based on the augmented Lagrangian method specifically includes: Transforming the optimal control problem (11) and constraint conditions (12)-(15) into an unconstrained optimization problem; The augmented Lagrangian method is introduced to reformulate the optimal control problem (9); the constructed optimal control problem (11) includes equality constraints and inequality constraints; for the equality constraint (12), the Lagrange multiplier and the penalty parameter are introduced to obtain the following terms: ; For the tracking interval constraint (13), the reconstructed expression is as follows: ; wherein, is a Lagrange multiplier; The speed constraint (14) is considered to consist of two inequalities, and by introducing Lagrange multipliers and , its augmented form is expressed as follows: ; Similarly, the augmented form of the control force constraint (15) is as follows: ; wherein, and are Lagrange multipliers; To simplify the transformation process, the optimal control problem (11) is rewritten in the following compact form: ; Among them, is the objective function defined in Equation (10), is the vector of control variables, represents the set of equality constraints including the train position and velocity state transition equation (12), represents the set of inequality constraints, including the tracking interval constraint (13), the train velocity constraint (14), and the train control force constraint (15), and are the index sets of the equality and inequality constraints, respectively; By using the augmented Lagrangian method, the constrained optimal control problem (20) is equivalently transformed into the following unconstrained form: ; where, and are the Lagrange multiplier vectors for equality and inequality constraints respectively, is the penalty parameter, which imposes a penalty when the constraint is violated during the iteration process; is the slack variable, which equals zero when the constraint is satisfied and has no impact on the control force in the objective function when the constraint is violated; specifically, the optimal control problem (11) and the constraint conditions (12)-(15) are equivalently transformed into the following unconstrained optimization problem: ; Among them, is the unconstrained optimization problem after transformation, is the decision variable, and are the Lagrange multiplier matrices of the equality and inequality constraints.

6. The multi - train cooperative tracking control method under relative braking protection constraints according to claim 5, characterized in that, The solution using the trust region method specifically includes: After obtaining the unconstrained optimization problem (22) at each sampling moment, the trust region method is used for solution; the basic idea of the trust region method is to construct a trust region subproblem within a given trust region radius, regarded as a quadratic approximation of the transformed unconstrained optimization problem (22). To facilitate the construction of the trust region subproblem, first calculate the gradient matrix of the unconstrained optimization problem (22) and the Hessian matrix , and are calculated as follows: ; ; The trust-region subproblem for the $k$-th iteration is expressed as follows: ; wherein is the decision variable of the trust region subproblem (24), is the trust region radius, and satisfies , while is an arbitrary vector norm, usually taken as norm or norm; By solving the trust-region subproblem (24), the decision variable , also known as the trial step size, is obtained; to check whether the trial step size should be accepted, a new variable is introduced, which represents the ratio of the actual decrease value to the predicted decrease value and is defined as follows: ; wherein and are respectively the actual decrease value and the predicted decrease value of the objective function after applying the trial step size; the actual decrease value is expressed as follows: ; The predicted decrease value is defined as follows: ; If the trial step size taken can reduce the value of the unconstrained optimization problem (22) to a sufficient extent, i.e., , then accept the trial step size and use it to construct the new initial vector for the next sub-problem; otherwise, the trial step size needs to be rejected; the iterative process is shown as follows: ; It should be noted that, as can be seen from Equation (28), the update of the trust region radius also depends on the value; if , the radius needs to be reduced in the next iteration because no progress has been made with the current radius; if , it means that the trial step size is approximated well, and the trust region radius is increased to allow a larger step size; if , the trust region radius will be adjusted; the adjustment of the trust region radius is summarized as follows: ; where is the trust region radius of the -th iteration, is the upper bound of the trust region radius, is a non - negative constant parameter satisfying ; empirically, is set to ; After updating the decision vector for a new round of iteration, it is also necessary to determine whether the termination condition is satisfied, and the termination condition is set as follows: ; wherein is an acceptable termination threshold; if this condition is met, the optimal solution is obtained, otherwise, the subproblem is reconstructed using the updated information; Based on the above description, the trust region solution method for the unconstrained optimization problem (22) is summarized in Algorithm 1 as follows; The specific content of Algorithm 1 includes: the solution method based on the trust region Input: Unconstrained problem ; Initial point ; Initial parameters ; Output: Optimal solution and optimal value ; Steps: Set to ; Calculation and ; If , then Obtain and and terminate the algorithm Otherwise, Constructor sub-problem , and solve Calculation ; Update ; If , then ; Otherwise, ; Update ; End the loop The input of Algorithm 1 includes the initial vector , the initial trust region radius , the acceptable termination threshold , the maximum number of iterations , and the unconstrained optimization problem (22); By constructing and solving the trust region subproblem at each iteration, the trial step size is obtained. Then, according to Equations (28) and (29) respectively, the trial step size is used to update the subsequent iteration variables and the trust region radius ; This process is repeated until the iteration termination condition (30) is satisfied; After satisfying this condition, the solution of the unconstrained optimization problem (22) is obtained, so as to provide the optimal control input at each decision moment.

7. The multi - train collaborative tracking control method under relative braking protection constraint according to claim 6, wherein, The overall framework of the multi - train cooperative tracking control algorithm design under the relative braking protection constraint is shown in Algorithm 2: The specific content of Algorithm 2 includes: the multi - train cooperative tracking control algorithm under the relative braking protection constraint Input: Under the MPC architecture, the optimal control problem (11) at each sampling moment Initial parameters , maximum number of iterations number the acceptable termination threshold ; Output: Optimal control force error ; External structure: Set to ; Reconstruct the unconstrained problem ; Internal structure: Input into Algorithm 1 Obtain and value End the internal structure Calculate value If , then Let and ; Obtained from ; ; Break out of the loop Otherwise, Update the Lagrange multiplier and the penalty factor Update ; End the loop End the external structure The input of the outer layer structure is the optimal control problem (11); by introducing Lagrange multipliers and penalty parameters, each constraint is added to the objective function to obtain an unconstrained optimization problem (22); in each iteration of the outer layer, Algorithm 1 is triggered to solve the unconstrained optimization problem (22), and a new variable is introduced to check whether the termination condition is satisfied, and its calculation is as follows: ; Then the termination condition is expressed as: ; wherein, is an acceptable termination threshold; if the termination condition (32) holds, the optimal solution of the unconstrained problem is obtained and the iteration is terminated; otherwise, the penalty parameter should be updated according to Equation (33) for the next iteration: ; where is a constant and is set according to experience; correspondingly, the Lagrange multipliers for the equality and inequality constraints (12)-(15) should be updated as follows: ; 。 8. An electronic device, characterized in that it includes a memory, a processor, and computer instructions stored on the memory and running on the processor. When the computer instructions are run by the processor, the method described in any one of claims 1 - 7 is completed.

9. A computer - readable storage medium, characterized in that it is used to store computer instructions. When the computer instructions are executed by the processor, the method described in any one of claims 1 - 7 is completed.

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