A Predictive Cascade Control Method and Device for High-Speed ​​Trains Based on PPC and MPC

Abstract

This application discloses a predictive cascade control method and device for high-speed trains based on PPC and MPC, relating to the field of train control. The method includes obtaining the predicted speed value at time t+1 according to the GS-TCN model; processing the state error at time t using PPC to obtain the initial control force at time t+1 under the initial iteration number; obtaining the predicted speed value at time t+2 based on the predicted speed value at time t+1, the initial control force at time t+1 under the current iteration number, and the GS-TCN model, thereby obtaining the speed error at time t+2 under the current iteration number; inputting the speed error at time t+2 under the current iteration number and the initial control force at time t+1 into the MPC to obtain the initial control force at time t+1 under the next iteration number, and finally obtaining the optimal control force at time t+1. This application can efficiently complete rolling optimization within the effective cycle, effectively alleviating the control delay problem caused by computation time consumption.

Description

Technical Field

[0001] This application relates to the field of train control, and in particular to a predictive cascade control method and device for high-speed trains based on PPC and MPC. Background Technology

[0002] Automatic driving systems for high-speed trains (HSTs) play a crucial role in control, monitoring, optimizing operation, improving safety, and ensuring accuracy. With increasing HST operating speeds and more complex operating environments, the operational performance of automatic driving systems faces new challenges. Therefore, there is an urgent need to develop advanced and reliable modeling and control technologies to support automatic driving operations.

[0003] In the research process of high-speed train modeling, the early mechanism models based on Newtonian mechanics could roughly describe the basic dynamic characteristics of train operation. However, in the face of the inherent strong nonlinearity, time-varying characteristics and multi-source composite disturbances (including environmental changes, track irregularities and load fluctuations) of the HST system, accurate modeling became extremely difficult. The model parameter identification error was often further amplified under complex operating conditions, which seriously restricted the performance of the controller.

[0004] With the development of artificial intelligence technology, data-driven modeling methods based on neural networks have gradually become a research hotspot. Among them, feedforward neural networks, such as radial basis function networks and fuzzy neural networks, can handle nonlinear problems, but they cannot effectively capture temporal dependencies and perform poorly in predicting the dynamic behavior of long sequences. Subsequently, recurrent neural networks and their variants LSTM and GRU were introduced into the field of HST modeling due to their ability to process sequential data. These neural networks can effectively capture long-term dependencies through gating mechanisms, but their sequential computation characteristics make them difficult to parallelize, resulting in low training and inference efficiency. Furthermore, they require stacking a large number of layers to cover sufficient historical information, which not only increases the computational burden but also exacerbates the gradient problem, making them unsuitable for the field of high-speed train control. Summary of the Invention

[0005] The purpose of this application is to provide a high-speed train predictive cascade control method and device based on PPC and MPC, which can efficiently complete rolling optimization within the effective cycle and effectively alleviate the high-speed train control delay problem caused by computation time consumption.

[0006] To achieve the above objectives, this application provides the following solution: In a first aspect, this application provides a high-speed train predictive cascade control method based on PPC and MPC, including: acquiring the train operation information and state error of the high-speed train at time t; t is the current time; the train operation information at time t includes the actual speed at time t, the optimal control force at time t, and the track data at time t.

[0007] The train operation information of the high-speed train at time t is input into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+1. The GS-TCN model is a TCN model with the GELU activation function, the number of convolutional units in each residual block is greater than 2, and the input of the first causal dilated convolution in each residual block is connected to the output of the compressed excitation module.

[0008] A PPC-based controller is used to process the state error of the high-speed train at time t to obtain the initial control force of the high-speed train at time t+1 under the initial iteration number.

[0009] At the current iteration number, the train operation information of the high-speed train at time t+1 at the current iteration number is input into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+2 at the current iteration number; the train operation information at time t+1 includes the initial control force at time t+1, the track data at time t+1, and the speed prediction value at time t+1.

[0010] The speed error of the high-speed train at time t+2 is obtained based on the predicted speed of the high-speed train at the current iteration number and the target speed at time t+2.

[0011] The speed error of the high-speed train at time t+2 under the current iteration number, the initial control force of the high-speed train at time t+1 under the current iteration number, and the optimal control force at time t are input into the MPC-based controller for one iteration to obtain the initial control force of the high-speed train at time t+1 under the next iteration number. Then, the iteration number is updated and the next iteration begins until the iteration stops. The initial control force of the high-speed train at time t+1 under the last iteration number is determined as the optimal control force at time t+1. The high-speed train at time t is controlled according to the optimal control force at time t+1.

[0012] Secondly, this application provides a high-speed train predictive cascade control device based on PPC and MPC, including: an acquisition module for acquiring train operation information and state error of the high-speed train at time t; t is the current time; the train operation information at time t includes the actual speed at time t, the optimal control force at time t, and the track data at time t.

[0013] The first speed prediction module is used to input the train operation information of the high-speed train at time t into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+1. The GS-TCN model is a TCN model with the GELU activation function, the number of convolutional units in each residual block is greater than 2, and the input of the first causal dilated convolution in each residual block is connected to the output of the compressed excitation module.

[0014] The initial control force determination module is used to process the state error of the high-speed train at time t using a PPC-based controller to obtain the initial control force of the high-speed train at time t+1 under the initial iteration number.

[0015] The second speed prediction module is used to input the train operation information of the high-speed train at time t+1 at the current iteration number into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+2 at the current iteration number; the train operation information at time t+1 includes the initial control force at time t+1, the track data at time t+1, and the speed prediction value at time t+1.

[0016] The speed error calculation module is used to obtain the speed error of the high-speed train at time t+2 based on the predicted speed value of the high-speed train at the current iteration number and the target speed at time t+2.

[0017] The control module is used to input the speed error of the high-speed train at time t+2 under the current iteration number, the initial control force of the high-speed train at time t+1 under the current iteration number, and the optimal control force at time t into the MPC-based controller for one iteration to obtain the initial control force of the high-speed train at time t+1 under the next iteration number. Then, the iteration number is updated to enter the next iteration until the iteration stop condition is reached. The initial control force of the high-speed train at time t+1 under the last iteration number is determined as the optimal control force at time t+1. The high-speed train at time t is controlled according to the optimal control force at time t+1.

[0018] According to the specific embodiments provided in this application, this application has the following technical effects: This application provides a high-speed train prediction cascade control method and device based on PPC and MPC. Temporal Convolutional Network (TCN) provides a superior solution through causal dilated convolution mechanism: its causal convolution ensures that the model input depends only on historical data, meeting the causal requirements of time-series prediction; its dilated convolution expands the receptive field exponentially, requiring only a small number of convolutional layers to cover long-term sequence data; and TCN's unique parallel computing capability significantly improves training and inference efficiency; and the residual connection structure can effectively alleviate the gradient vanishing problem. These characteristics make TCN more suitable for high-speed train control scenarios, not only efficiently processing long-sequence data, but also meeting real-time requirements through parallel computing, while its convolutional kernel can naturally learn the periodic dynamic characteristics of train operation.

[0019] This application makes three improvements to the traditional TCN structure: (1) It introduces a squeeze-and-excitation block (SEBlock) as a channel attention mechanism, which adaptively learns the weights of each input channel through the compression-excitation process, so that the model can dynamically focus on key physical features, effectively overcoming the shortcomings of traditional TCN in treating all channel features equally and being susceptible to noise interference; (2) It adopts a Gaussian error linear unit (GELU) to replace the traditional ReLU activation function, avoiding the problem of information loss in negative value regions; (3) It optimizes the residual structure of TCN into a convolutional unit with more than 2 layers, extracting more refined temporal patterns through deeper layers with less memory and computational overhead, making up for the shortcomings of insufficient receptive field and expressive ability of traditional two-layer residual structure, and achieving a better balance between accuracy and efficiency. Secondly, in terms of control architecture design, Prescribed Performance Control (PPC) and Model Predictive Control (MPC) are cascaded together: PPC generates initial control commands instantaneously based on the current state error. These commands serve as the initial input to the MPC controller, providing a high-quality starting point for MPC optimization. If used as the initial value for the "warm start" of MPC, it not only significantly accelerates the subsequent optimization convergence process of MPC and alleviates the real-time bottleneck, but also effectively improves the solution accuracy in the transient stage, guiding MPC to obtain a higher-precision control solution within a finite number of iterations, effectively alleviating the control delay problem caused by computation time. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a flowchart illustrating a high-speed train predictive cascade control method based on PPC and MPC, provided as an embodiment of this application.

[0022] Figure 2 This is a structural diagram of a high-speed train model.

[0023] Figure 3 This is a structural diagram of the GS-TCN model provided in an embodiment of this application.

[0024] Figure 4A schematic diagram of a high-speed train predictive cascade control method based on PPC and MPC provided in an embodiment of this application.

[0025] Figure 5 A trend comparison chart of different evaluation criteria for convolutional units provided in an embodiment of this application.

[0026] Figure 6 A comparison chart of single-step identification errors for four models provided in an embodiment of this application.

[0027] Figure 7 A prediction time-domain rolling prediction error diagram provided in an embodiment of this application.

[0028] Figure 8 The diagram shows the global tracking effect of three control algorithms provided in one embodiment of this application.

[0029] Figure 9 A comparison chart of the global tracking errors of three control algorithms provided in an embodiment of this application.

[0030] Figure 10 A comparison diagram of the global control force curves of three control algorithms provided in an embodiment of this application. Detailed Implementation

[0031] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0032] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0033] In one exemplary embodiment, such as Figure 1 As shown, a predictive cascade control method for high-speed trains based on PPC and MPC is provided, including: Step 201: Obtain the train operation information and state error of the high-speed train at time t; t is the current time; the train operation information at time t includes the actual speed at time t, the optimal control force at time t, and the track data at time t.

[0034] Step 202: Input the train operation information of the high-speed train at time t into the GS-TCN model to obtain the predicted speed value of the high-speed train at time t+1; for example... Figure 3As shown, the GS-TCN model is a TCN model with the GELU activation function, the number of convolutional units in each residual block being greater than 2, and the input of the first causal dilated convolution in each residual block being connected to the output of the compressed excitation module.

[0035] Step 203: Use a PPC-based controller to process the state error of the high-speed train at time t to obtain the initial control force of the high-speed train at time t+1 under the initial iteration number.

[0036] Step 204: At the current iteration number, input the train operation information of the high-speed train at time t+1 at the current iteration number into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+2 at the current iteration number; the train operation information at time t+1 includes the initial control force at time t+1, the track data at time t+1, and the speed prediction value at time t+1.

[0037] Step 205: Based on the predicted speed of the high-speed train at time t+2 under the current iteration number and the target speed at time t+2, obtain the speed error of the high-speed train at time t+2 under the current iteration number.

[0038] Step 206: Input the speed error of the high-speed train at time t+2 under the current iteration number, the initial control force of the high-speed train at time t+1 under the current iteration number, and the optimal control force at time t into the MPC-based controller for one iteration to obtain the initial control force of the high-speed train at time t+1 under the next iteration number. Then update the iteration number and enter the next iteration until the iteration stop condition is reached. Determine the initial control force of the high-speed train at time t+1 under the last iteration number as the optimal control force at time t+1. Control the high-speed train at time t based on the optimal control force at time t+1.

[0039] In another exemplary embodiment of this application, the number of convolutional units in each residual block is 3.

[0040] against Figure 2 The high-speed train model shown Representing the current moment, Let g represent the mass of the train, and g represent the acceleration due to gravity. The train control force (traction or braking force) at time t. and These represent the basic resistance and additional resistance of the train at time t, respectively. Let be the train's speed at time t. According to Newton's laws, after force analysis of the high-speed train model, its dynamic relationship can be expressed as: .

[0041] in, to Used to simulate mass disturbances or equivalent random disturbances in the movement of people. Sampling period. Basic resistance. With additional resistance They can be represented as follows: and .

[0042] in, , , This is the basic drag coefficient, based on empirical parameters for different vehicle models (CRH380B). , , . , , These are additional resistance on ramps, additional resistance in tunnels, and additional resistance on curves, respectively; the unit of measurement is N.

[0043] The calculation formulas are as follows: .

[0044] in, This represents the track gradient corresponding to the train's current position. Since the gradient is generally small, the tangent value can be used to approximate the sine value. The length of the tunnel; A constant of 450 to 800 is generally used (600 is used in the simulation of this application). The radius of the curve is denoted as .

[0045] Due to the complex dynamic characteristics of high-speed trains, their basic and additional resistance parameters are influenced by multiple coupled factors, and the train mass also dynamically changes with the number of passengers, resulting in strong time-varying and nonlinear characteristics in the model. Furthermore, the train operation generates massive amounts of data and various random disturbances are difficult to avoid. Therefore, this application reconstructs the actual dynamic relationships of high-speed trains into a nonlinear mapping model. .

[0046] in, For an unknown nonlinear function, and These are two positive parameters, representing the historical order of the input characteristic variables of the dynamic system, i.e., the number of historical time steps. This represents the control force data at time t. This represents Gaussian white noise with a mean of 0. For velocity time series, For the control force time series, This application presents time-series data covering three types of road conditions (slopes, tunnels, and curves). Based on this, a GS-TCN model is designed. Unlike CNNs, TCNs use dilated causal convolutions to exponentially expand the receptive field, requiring only a small number of convolutional units to cover long sequences. Therefore, TCNs are well-suited for processing sequential data. Furthermore, the inclusion of residual blocks makes the model more practical in terms of storage and gradient mitigation. Simultaneously, TCN's parallel processing capabilities effectively reduce memory consumption and accelerate model training and testing. The proposed GS-TCN model differs from traditional TCN models in the following three aspects.

[0047] (1) A Squeeze-and-Excitation Block (SEBlock) is introduced as a channel attention mechanism. This module adaptively learns and generates weight scores for each input channel through a compression-excitation process. By multiplying these weights by the original features, the model dynamically enhances the channel features that contribute more to the prediction task while suppressing redundant or interfering information. This improvement allows the model to more accurately focus on the changing patterns of key physical features. Especially during the acceleration and deceleration phases of high-speed trains, it effectively captures the dynamic physical features that fluctuate dramatically, thereby improving the model's prediction robustness under unsteady conditions.

[0048] (2) The Gaussian error linear unit activation function is used instead of the traditional ReLU to avoid information loss caused by ReLU being completely zeroed in the negative region and gradient oscillations that may be caused by non-differentiability at zero. At the same time, the smooth differentiability of GELU ensures the stability of the training process and helps the model converge, thereby enhancing the ability to capture complex nonlinear control-velocity dynamic relationships and improving the accuracy and robustness of the prediction results.

[0049] (3) In order to obtain better prediction results, this application improves the residual structure of traditional TCN by modifying the original two-layer residual structure into a three-layer residual structure (convolutional unit), so that the model can extract more detailed temporal feature information.

[0050] In practical applications, under the condition of input sequence length alignment, it can be set as follows: equal , and by The unified representation of the GS-TCN model's internal computational logic includes: Step 1: processing the control force time series data... orbital data Speed ​​data and other derived feature data By merging, we get: , It represents the rate of change of control force or velocity.

[0051] in, , The matrix is ​​formed by concatenating the first dimension. Time series characteristic data. , This represents the number of features, which will be expressed as the number of channels later.

[0052] Step 2: Introduce SEBlock before the first convolutional unit of each residual block. The core idea of ​​SEBlock is to extract statistical information of the feature input sequence features through global average pooling: .

[0053] in, , indicating channel The global mean, express No. 1 in the sequence Channel at each time step The data was then concatenated from all the average pooled channels to obtain... Then, two fully connected layers are used to perform dimensionality reduction and dimensionality increase operations respectively, and combined with nonlinear activation to generate dynamic weights: .

[0054] in, Represents the matrix after dimensionality reduction. This represents the matrix after channel average pooling concatenation. This represents the dimension-reduced matrix after activation by the nonlinear error unit GELU. This represents the GELU activation function. The GELU setting enables the entire activation module to learn complex nonlinear functions, not just a linear transformation. Represents the matrix after dimensionality increase. Indicates to The matrix after normalization, For learnable matrix weight parameters, For the bias vector term; As a dimensionality reduction factor, for The function makes the original output Normalized to the [0,1] interval, and broadcast via a broadcast mechanism. Expanding to the same spatial dimension as the input feature map, the final channel attention weights are generated and multiplied element-wise with the input features to achieve channel-level weighting: Output features The original dimensions were preserved, but the feature intensity of each channel was recalibrated.

[0055] Step 3: Let represent the attention-weighted input sequence, k represent the kernel size, and d represent the dilation factor. To ensure causality, the input... Fill the left side along the time dimension, with a fill size of [value missing]. Then, using a strict causal expansion mechanism, causal convolution is performed on the weighted exogenous sequence data to obtain new feature values ​​for the current time step. This is the process of establishing a nonlinear mapping relationship between the input data and the predicted output by the neural network. The following analyzes the single-sample time series convolution calculation process: For time step... and its corresponding output channel The essence of convolutional computation is to transform the input signal into a new feature space through extraction, learning, and combination. (Number of channels) .

[0056] in, Representing the Each convolutional kernel in Multi-channel convolution output at different times. For the first The convolutional kernel at the _th ... The first input channel Each weight (hereinafter referred to as a weight) Indicates by The matrix formed ), For bias parameters (hereinafter referred to as bias parameters) Indicates by (the bias matrix formed) It is after filling The input sequence at time step ,aisle The corresponding eigenvalues, This represents the total number of input channels. It can be observed that the sequence length remains unchanged in the new time series data obtained after convolution, but the feature quantities at each time step change. These new feature quantities are no longer the original physical quantities, but rather abstract features learned by the convolution kernel that can better represent the local patterns of the input sequence.

[0057] Step 4: The probability distribution of the causal output is weighted and preserved by using the non-linear activation function GELU on the output of the convolutional unit, so that the model has stronger expressive power and a more stable training process; and avoids the negative value truncation problem that is easy to cause when using the ReLU activation function in the traditional TCN structure.

[0058] The approximate expression for GELU is: .

[0059] in, It is the output sequence, a time series output by multi-channel convolution of all convolutional kernels at all times. It was pieced together. It is the hyperbolic tangent function.

[0060] Step 5: Adaptation Layers, the core idea of ​​which is to dynamically adjust the probability of discarding data without changing the data dimensions. and according to probability Set some elements in the input tensor to zero; during training, this probability will be adjusted according to changes in the training loss.

[0061] Step 6: After multiple residual blocks of convolutional units calculate and perform residual connections on the input time series feature data, a mapping relationship between the input features of the entire sequence is finally established. A fully connected regression layer is then used to process the information at the last time step. All new features are subjected to regression fitting, mapping the high-dimensional features back to one-dimensional values ​​to obtain the predicted value for the next time step: .

[0062] in, This represents the predicted velocity value at time t+1. , These are the output layer weight parameters and bias parameters, respectively. This represents the number of output channels of the last convolutional unit in the last residual block. It is the output sequence The last time step information.

[0063] In summary, if we consider the GS-TCN model as a nonlinear function... Then the predicted speed value can be expressed as: , This indicates that the velocity at time t+1 is predicted at time t.

[0064] This application also provides an embodiment, taking a single batch of samples as an example, to illustrate the update rules of the GS-TCN model: setting the corresponding loss function for the time series of a single batch of samples during training. for: .in, Let be the target velocity value at time t. The velocity prediction value of the GS-TCN model at time t is... It represents the set of all learnable parameters of the model, including the parameters. , , , , , , , The model can update its parameters. During model training, each training iteration accesses all sample data in the training set in batches (in random order) and updates the model parameters using the AdamW optimization algorithm. Since this example uses a single batch of samples, the number of parameter updates in each training iteration is equal to the total number of samples. Therefore, we can let... express Given the set of model parameters at time step 1, first calculate the gradient of the loss function with respect to the model parameters. : .

[0065] in, Indicates the parameter The gradient operator, i.e., the gradient operator with respect to all parameters of the model. Calculate the partial derivatives for each step. Then calculate the first-order moment estimate at time t. and second-order moment estimation . . .in, This represents the first moment estimate at time t. This represents the second moment estimate at time t. and The momentum decay coefficient, This represents element-wise multiplication. To eliminate bias caused by initial values, according to the formula... and Perform deviation correction. Indicates the deviation correction , After deviation correction , and They represent the momentum decay coefficients, respectively. t power and momentum decay coefficient t to the power of t.

[0066] Finally, the parameters of each GS-TCN model are updated according to the following rules: .in, This represents the set of model parameters at time t. Let be the learning rate at time t. To prevent small constants from being divided by zero, This represents the weight decay coefficient. Through iterative training and validation, the model parameters are corrected to better fit the distribution of the training set data. In the above parameter update algorithm, the learning rate... A smaller value will result in slow convergence of the GS-TCN model, while an excessively large value will cause convergence oscillations. This application addresses this by adopting... Adaptive learning rate algorithm ( When the gradient is large, the learning rate is reduced to prevent excessively large update step sizes from causing oscillations; when the gradient is small, the learning rate is increased to accelerate convergence, and the GS-TCN parameters are adaptively learned. This algorithm has been proven to accelerate model convergence.

[0067] In another exemplary embodiment of this application, the state error at time t includes the speed error at time t and the position error at time t; the calculation process of the state error of the high-speed train at time t specifically includes: obtaining the position error of the high-speed train at time t based on the actual position of the high-speed train at time t and the target position at time t+1. . .

[0068] The speed error of the high-speed train at time t is obtained based on the actual speed of the high-speed train at time t and the target speed at time t+1. . .

[0069] in, , These represent the actual position and actual speed of the train at time t. , These represent the target position and target speed of the train at time t+1, respectively.

[0070] In practical applications, to better illustrate the combination of the MPC-based controller and the GS-TCN model, the following diagrams are drawn: Figure 4 The schematic diagram is shown. The diagram assumes... This is MPC rolling optimization in the prediction time domain. Special cases at that time, For example, time The input features of the collected GS-TCN model do not include the control force to be applied. Instead, it includes the optimal control force obtained by the MPC method at time t-1. , Compared to It includes the initial control force output by the PPC at time t, or the initial control force obtained by the MPC-based controller. The sampling period is determined by the data collection process. Train schedule information Predict using the trained GS-TCN model The train speed at a given time is then processed by the feature data calculation unit to obtain... Train operation information at all times The initial control force at the initial iteration count is generated by a PPC-based controller. The role of the PPC-based controller here is to quickly generate the initial force to be applied to the train based on the current train state error. This adjustment process, unlike MPC control which requires continuous iterative optimization, is very fast and robust. Furthermore, because of the initial solution obtained by the PPC-based controller, the subsequent iterative optimization process of the MPC-based controller for the control force is significantly shortened. This design fully considers the time-varying and uniform speed-changing problems of train operation. At this point, the initial control force is generated by a PPC-based controller. Then, by combining the GS-TCN model and using the same method to continuously predict the train speed in the next time step, the optimal control command in the prediction time domain is optimized and solved using the adaptive gradient descent method. and immediately transmit to the actuator It is the first and only control command in the system.

[0071] against Figure 4 The described rolling optimization, this constrained MPC loss function can be defined as follows: This loss function is updated in each iteration of optimization. The initial control force at time t+j, obtained from the PPC-based controller, is input into this formula for the initial number of iterations. In subsequent iterations, the initial control force at time t+j obtained from the MPC-based controller will be input into this formula. ,right Through iterative optimization, the optimal solution is finally obtained. The optimal control force at time t+j is the final output of MPC.

[0072] Constraints: .

[0073] in, Energy consumption weighting parameter, To predict the target velocity at time t+j+1 in the time domain, The velocity prediction value at time t+j+1 output by the GS-TCN model. Indicates the lower limit of speed. Indicates the upper limit of speed. These are the corresponding constraint limits. It equals the increment of control force at time t+j.

[0074] The loss function can be simplified to: ,in , express The transpose of the matrix, This represents the velocity error matrix at time t. express The transpose of the matrix, The control input increment at time t, Represents t+N p The velocity error at time +1 Represents t+N u Incremental control input at any given time.

[0075] In another exemplary embodiment of this application, the MPC-based controller includes a loss function: ,in, Indicates the time domain of position control. This represents the velocity error at time t+j+1. This represents the energy consumption weighting parameter. Represents the time domain of speed control. For control input, let represent the control force at time t+j. The initial control force at time t+j is obtained using a PPC-based controller for the initial number of iterations, while the initial control force at time t+j is obtained using an MPC-based controller for subsequent iterations. This represents the optimal control force at time t+j-1.

[0076] In another exemplary embodiment of this application, the MPC-based controller further includes a second constraint: Where | represents absolute value, This indicates the upper limit of the control increment.

[0077] In another exemplary embodiment of this application, the speed error of the high-speed train at time t+2 under the current iteration number, the initial control force of the high-speed train at time t+1 under the current iteration number, and the optimal control force at time t are input into the MPC-based controller for one iteration to obtain the initial control force of the high-speed train at time t+1 under the next iteration number. Specifically, the speed error of the high-speed train at time t+2 under the current iteration number, the initial control force of the high-speed train at time t+1 under the current iteration number, and the optimal control force at time t are input into the MPC-based controller, and then the adaptive gradient descent algorithm is used for one iteration to obtain the initial control force of the high-speed train at time t+1 under the next iteration number.

[0078] In practical applications, the solution obtained by performing one iteration using the adaptive gradient descent algorithm in the p-th iteration is as follows: .

[0079] In the formula, This represents the control force obtained after the (p+1)th iteration optimization, i.e., the initial control force of the high-speed train at the (p+1)th iteration number. For learning rate, To optimize the number of iterations, To prevent division by zero, a small constant is generally taken as... ; and First-order moment estimates and second-order moment estimation The deviation correction and the calculation methods for both are as follows: and .

[0080] In the formula, Indicates the momentum decay coefficient power of p+1 Indicates the momentum decay coefficient power of p+1 and The calculation formula is: and .

[0081] In the formula, and The momentum decay coefficient, , Let represent the first-order moment estimate obtained from the p-th calculation and the second-order moment estimate obtained from the p-th calculation, respectively. This represents element-wise multiplication; The gradient of the loss function with respect to the model parameters is calculated using the following formula: .

[0082] In the formula, Indicates the control force parameters gradient operator, This represents the control force obtained after the p-th optimization. This control force optimization method shares some similarities with the parameter update rule of the GS-TCN model, such as the momentum decay coefficient. and However, there are also significant differences, and the adaptive aspect refers to the method automatically adjusting the actual update step size for each parameter, rather than referring to the base learning rate. It will change over time.

[0083] In another exemplary embodiment of this application, the PPC-based controller includes: a nonlinear gain function: To map bounded errors to an unbounded space, facilitating subsequent feedback controller design, this nonlinear gain function is used for error transformation. This function maps bounded errors to an unbounded space. Mapped to unbounded space This inherently implies that the error must satisfy certain conditions. This is a prerequisite. It allows the controller to amplify the negative feedback, forcing the train's state error to remain within a preset boundary.

[0084] Control law calculation function: ,in, Represents the position error transformation function at time t; Represents the velocity error transformation function at time t; This indicates the preset performance boundary at time t. The velocity preset performance boundary at time t represents the velocity, and both are first-order differentiable and strictly positive decreasing functions or fixed values; Indicates position control gain. Indicates speed control gain. , Requires a value greater than 0; This represents the position error at time t; This represents the velocity error at time t; Represents the virtual control force at time t+1; This represents the initial control force at time t+1;

[0085] In another exemplary embodiment of this application, the PPC-based controller further includes: a performance boundary function. ,in, This indicates the upper limit of the initial position error. Indicates the upper limit of the initial velocity error. This indicates the upper limit of the steady-state error. Indicates the upper limit of the steady-state error of the velocity; Indicates the position convergence parameter. This represents the velocity convergence rate parameter. , The value must be greater than 0.

[0086] In another exemplary embodiment of this application, the PPC-based controller further includes: a first constraint: The train's state error must always meet this constraint.

[0087] The MPC-based controller uses the GS-TCN model to perform rolling predictions of the future operating state of the high-speed train and solves for the optimal control force. The PPC-based controller, on the other hand, rapidly generates an initial control command that meets preset performance boundaries based on the current instantaneous state error. This command serves as the input starting point for the MPC-based controller, providing a high-quality starting point for MPC optimization. By introducing a PPC-based controller, the optimization process of the MPC-based controller can be guided to begin with a high-performance initial solution, thereby greatly accelerating the convergence speed and ensuring that the method combining PPC and MPC consistently meets the preset performance requirements under transient error dynamics.

[0088] This application also provides a simulation experiment to verify the above method. The steps are as follows: Step 4.1: Set the simulation parameters according to Table 1 and Table 2.

[0089] Table 1 Simulation parameters of CRH380B train

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[0091] Table 2 Simulation parameters of the method provided in this application

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[0093] Step 4.2: Model evaluation criteria.

[0094] Before deploying the GS-TCN model to a closed-loop system, its optimal structure and parameters need to be determined through offline training. To improve prediction accuracy and stability while reducing runtime, this application adjusts the number of convolutional units in the residual structure and conducts experiments for different configurations. The prediction performance is evaluated using three commonly used metrics: mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). Figure 5 Part (a) shows the trend of the mean absolute percentage error during the training process over 150 durations. Figure 5 Part (b) shows the trend of the root mean square error during the training process over 150 durations. Figure 5 Section (c) shows the trend of the mean absolute error during training over 150 time periods. To highlight the subtle fluctuations after model convergence, Figure 5 The rapid decline in metrics during the first 20 durations is omitted, and only the training results from durations 21 to 150 are presented. It can be seen that among various convolutional layer configurations, the three-layer convolutional unit structure exhibits the best stability while maintaining high prediction accuracy.

[0095] To verify the predictive performance of the GS-TCN model, this application compares it with three other neural network models: Transformer, Gated Recurrent Unit (GRU), and traditional TCN. GRU dynamically manages information flow through a gating mechanism to learn long-term dependencies in a sequence; however, such recursive structures may struggle to fully capture and retain all key temporal features when dealing with long sequences or complex patterns due to challenges in gradient propagation or information transfer efficiency. While the Transformer model effectively overcomes some of the shortcomings of recursive structures with its powerful global attention mechanism, its computational complexity and sensitivity to hyperparameter tuning may pose challenges in practical engineering applications. In contrast, the GS-TCN model, based on the TCN structure, provides a more stable and efficiently parallelizable gradient propagation path through causal convolution and layer expansion design. Figure 6This experiment showcases a speed prediction comparison between the GS-TCN model and three other models: TCN, GRU, and Transformer. The single-step identification error on the test set refers to the difference between each model's predicted speed and the target speed value. The results show that the GS-TCN model performs best in speed prediction on the test set after multiple training runs on the same training set, further confirming its advantage in train dynamics modeling. In this experiment, a new dataset was constructed by randomly sampling segments and shuffling the original dataset to create a significantly expanded dataset. This dataset breaks the temporal patterns and operational logic of speed changes in the original data, randomly combining different motion segments to enhance data diversity and challenge. The new dataset is divided into training and test sets in a 7:3 ratio to evaluate the generalization ability and control performance of neural networks under complex, non-stationary sequences.

[0096] Figure 7 This demonstrates the self-circulating multi-step prediction and identification errors of the GS-TCN model and three other models under the same conditions. Figure 7 As can be seen, the prediction errors of all models increase with the number of prediction steps, which is consistent with the basic law of error accumulation in time series forecasting. Overall, the GS-TCN model exhibits the lowest error level across the entire prediction interval, with a relatively gentle growth slope. In contrast, the other three models show faster error growth, especially the GRU model, whose performance gap widens further after more than four prediction steps. This experimental result demonstrates that the GS-TCN model has a stronger ability to capture the complex dynamic characteristics of high-speed trains and can more effectively suppress the cumulative amplification effect of historical prediction errors, thus maintaining higher accuracy in long-term prediction tasks.

[0097] Step 4.3: Comparative simulation of model predictive dynamic cascade control methods.

[0098] This application introduces random disturbances into the train dynamics model to simulate common environmental and load fluctuations encountered in actual operation. Under this setting, the proposed high-speed train predictive cascade control method based on PPC and MPC was simulated and verified. Its performance was compared with two benchmark schemes: Full Format Dynamic Linearization Model-Free Adaptive Control (FFDL-MFAC) and Transformer-based Model Predictive Control (Transformer-MPC), aiming to evaluate the comprehensive performance of different data-driven models and control architectures in high-speed train automatic driving scenarios.

[0099] like Figure 8 As shown, all three control schemes can basically complete the task of tracking the target's velocity trajectory. From a global perspective, the overall tracking performance of the three is highly similar, all able to follow the general trend of the target trajectory, indicating that each scheme is feasible at the macro level. However, a detailed comparison reveals identifiable differences between the different schemes in local details and tracking accuracy.

[0100] A detailed comparison of the tracking errors and the global control force output curves of the three control algorithms are shown below. Figure 9 and Figure 10 As shown. From Figure 9 From the error curves, the high-speed train predictive cascade control method based on PPC and MPC provided in this application exhibits relatively smaller instantaneous tracking error amplitudes and more convergent error fluctuations in most operating segments, demonstrating superior local tracking accuracy and stability. In contrast, the FFDL-MFAC algorithm has a relatively larger error fluctuation range, while the performance of Transformer-MPC falls between the two. Regarding global control force output, such as... Figure 10 As shown, the control commands generated by the high-speed train predictive cascade control method based on PPC and MPC provided in this application exhibit good smoothness under traction, braking, and cruise conditions. Its output range is stable between -156kN and 180kN, and the transition during condition switching is natural with no obvious jitter. The control force fluctuation range of the FFDL-MFAC algorithm is slightly wider (-156kN to 200kN). The control force range of the Transformer-MPC algorithm (-157kN to 179kN) is similar to that of the high-speed train predictive cascade control method based on PPC and MPC provided in this application, but its smoothness in the transient process is slightly inferior, and its control force change curve has more spikes and exhibits high-frequency jitter. Furthermore, to accurately evaluate the tracking performance and comfort of each control scheme, the following formula is used... The evaluation indicators shown quantify the tracking and control effectiveness and operational smoothness and comfort of the train throughout its overall operation period.

[0101] in, This represents the integral of the total absolute error (a tracking performance metric). The integral representing the absolute value of acceleration (a comfort index). Total running time, This represents the target velocity at time t. Let be the acceleration at time t. The evaluation results are shown in Table 3.

[0102] Table 3 Comparison of IASE and IAAV data

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[0104] As shown in Table 3, the quantitative evaluation results demonstrate that the high-speed train predictive cascade control method based on PPC and MPC provided in this application exhibits significant advantages in overall performance. In terms of speed tracking accuracy, its IASE index is 245.32, far lower than Transformer-MPC and FFDL-MFAC, indicating that this strategy can track the target speed trajectory more accurately and has the smallest cumulative tracking error. Regarding operational stability and ride comfort, the IAAV indices of the three control algorithms are similar. Although the high-speed train predictive cascade control method based on PPC and MPC provided in this application is slightly better than the other two algorithms, considering the extremely small numerical difference, it can be considered that all three have achieved a high level in acceleration fluctuation suppression. It is worth noting that while achieving good tracking accuracy (IASE index), the high-speed train predictive cascade control method based on PPC and MPC provided in this application has not sacrificed comfort; its IAAV index remains within a range comparable to the best. This demonstrates that the algorithm achieves an excellent balance between "high precision" and "high comfort," and verifies the effectiveness of the PPC-based controller in terms of transient response of the constrained system and smooth control output. It fully verifies that the high-speed train predictive cascade control method based on PPC and MPC provided in this application has significant improvements in tracking performance, running stability, and model accuracy compared to existing methods.

[0105] This application significantly improves the prediction accuracy and computational efficiency of the model by introducing a channel attention mechanism (SEBlock), a Gaussian error linear unit (GELU), and optimizing the residual structure to a three-layer dilated causal convolutional unit on the basis of traditional TCN. In terms of control architecture design, it innovatively integrates Preset Performance Control (PPC) and Model Predictive Control (MPC) in a cascaded manner. The PPC-based controller rapidly generates initial control commands that meet performance constraints as the starting point for MPC optimization, greatly accelerating the convergence speed of MPC. Simultaneously, relying on the high-precision prediction capability of the GS-TCN model, the entire control strategy can complete rolling optimization in real time and efficiently within the effective cycle, effectively alleviating the latency problem of the control system caused by computational time consumption.

[0106] The core advantage of MPC lies in its ability to explicitly embed system inputs and state constraints, dynamically correcting the control strategy through time-domain recursive optimization. However, this algorithm is highly dependent on model accuracy, facing a bottleneck of command execution lag in high-speed train control. Since MPC needs to solve a finite-time optimal control problem within each sampling period, its computation time increases significantly with the length of the prediction time domain. This results in a deviation in the actual train operating state within the time window from state sampling to the actual application of control commands to the system. The deviation caused by the timing misalignment between state observation and command execution accumulates rapidly with the computation time, potentially leading to control performance degradation or even system instability. Existing research has attempted to alleviate this problem by reducing the control update frequency, optimizing the control algorithm, and using distributed parallel solutions, but the effects have been limited. PPC, as a control strategy that constrains the dynamic behavior of system errors through a pre-defined performance function, has received widespread attention in recent years due to its ability to explicitly guarantee the transient response and steady-state accuracy of the system. The core idea of ​​PPC (Programmable Control Programming) is to limit the system's output error within a predefined dynamic range by designing performance boundaries, such as error convergence rate, overshoot, and steady-state error range. This enhances the system's robustness and disturbance rejection capability while ensuring control accuracy. This characteristic allows PPC to be flexibly combined with other control algorithms to form a cascaded control framework with complementary advantages. This application integrates PPC and MPC (Multi-Process Control), enabling PPC to quickly generate initial control commands based on the state error, which serve as the initial input for MPC. This retains PPC's ability to accurately guide the system's transient response while fully leveraging the optimization characteristics of MPC, effectively shortening the MPC optimization iteration process and significantly improving the dynamic response characteristics and computational efficiency in high-speed train control systems.

[0107] This application's GS-TCN model extracts more detailed train feature information by establishing a three-layer convolutional unit, thereby establishing a nonlinear mapping between input and output at each time step and predicting the train speed at the next time step. Secondly, leveraging the high prediction accuracy of the GS-TCN model, a reliable prediction foundation is built for the subsequent MPC optimization problem. Furthermore, by introducing PPC, control commands are pre-set based on MPC control, limiting the train's state tracking error within a pre-selected performance envelope, ensuring that the error converges at a predetermined convergence rate, with no overshoot exceeding a specified amount, and with a set steady-state accuracy.

[0108] Based on the characteristics of measurable data sources from high-speed trains, this application designs a GS-TCN model with dynamic modeling capabilities. The GS-TCN model can perform fast and effective identification in scenarios with complex disturbances and large datasets. In addition, compared with other deep learning models, the GS-TCN model has a lower parameter scale and an explicit computational architecture.

[0109] Leveraging the superior performance of the GS-TCN model in single-step prediction and combined with the high-quality initial solution provided by PPC, this application can efficiently complete the MPC rolling optimization process, thereby achieving high-precision, low-latency high-speed train control.

[0110] This application introduces PPC on the basis of MPC. By presetting the performance boundary, a better control force command is initialized before the optimal control force command is obtained by MPC control, thereby accelerating the iterative optimization process of MPC control and improving the dynamic response, tracking accuracy and robustness of high-speed train automatic driving.

[0111] Based on the same inventive concept, this application also provides a PPC and MPC-based high-speed train predictive cascade control device for implementing the aforementioned PPC and MPC-based high-speed train predictive cascade control method. The solution provided by this device is similar to the implementation described in the above method. Therefore, the specific limitations of one or more PPC and MPC-based high-speed train predictive cascade control device embodiments provided below can be found in the above-described limitations of the PPC and MPC-based high-speed train predictive cascade control method, and will not be repeated here.

[0112] In an exemplary embodiment, a high-speed train predictive cascade control device based on PPC and MPC is provided, comprising: an acquisition module for acquiring train operation information and state error of a high-speed train at time t; t is the current time; the train operation information at time t includes the actual speed at time t, the optimal control force at time t, and the track data at time t.

[0113] The first speed prediction module is used to input the train operation information of the high-speed train at time t into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+1. The GS-TCN model is a TCN model with the GELU activation function, the number of convolutional units in each residual block is greater than 2, and the input of the first causal dilated convolution in each residual block is connected to the output of the compressed excitation module.

[0114] The initial control force determination module is used to process the state error of the high-speed train at time t using a PPC-based controller to obtain the initial control force of the high-speed train at time t+1 under the initial iteration number.

[0115] The second speed prediction module is used to input the train operation information of the high-speed train at time t+1 at the current iteration number into the GS-TCN model to obtain the speed prediction value of the high-speed train at time t+2 at the current iteration number; the train operation information at time t+1 includes the initial control force at time t+1, the track data at time t+1, and the speed prediction value at time t+1.

[0116] The speed error calculation module is used to obtain the speed error of the high-speed train at time t+2 based on the predicted speed value of the high-speed train at the current iteration number and the target speed at time t+2.

[0117] The control module is used to input the speed error of the high-speed train at time t+2 under the current iteration number, the initial control force of the high-speed train at time t+1 under the current iteration number, and the optimal control force at time t into the MPC-based controller for one iteration to obtain the initial control force of the high-speed train at time t+1 under the next iteration number. Then, the iteration number is updated to enter the next iteration until the iteration stop condition is reached. The initial control force of the high-speed train at time t+1 under the last iteration number is determined as the optimal control force at time t+1. The high-speed train at time t is controlled according to the optimal control force at time t+1.

[0118] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0119] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A high-speed train prediction cascade control method based on PPC and MPC, characterized in that, The high-speed train prediction cascade control method based on the PPC and the MPC comprises the following steps: acquiring train operation information and state errors of a high-speed train at a time t, wherein t is a current time, the train operation information at the time t comprises an actual speed at the time t, an optimal control force at the time t and track data at the time t; inputting the train operation information of the high-speed train at the time t into a GS-TCN model to obtain a speed prediction value of the high-speed train at a time t+1, wherein the GS-TCN model is a TCN model with a GELU activation function as an activation function, a number of convolution units in each residual block being greater than 2 and an input end of a first causal dilated convolution in each residual block being connected to an output end of a compression excitation module; processing the state errors of the high-speed train at the time t by using a controller based on the PPC to obtain an initial control force of the high-speed train at the time t+1 under an initial iteration number; inputting train operation information of the high-speed train at the time t+1 under a current iteration number into the GS-TCN model to obtain a speed prediction value of the high-speed train at a time t+2 under the current iteration number, wherein the train operation information at the time t+1 comprises the initial control force at the time t+1, the track data at the time t+1 and the speed prediction value at the time t+1; obtaining a speed error of the high-speed train at the time t+2 under the current iteration number according to the speed prediction value of the high-speed train at the time t+2 under the current iteration number and a target speed at the time t+2; inputting the speed error of the high-speed train at the time t+2 under the current iteration number, the initial control force of the high-speed train at the time t+1 under the current iteration number and the optimal control force at the time t into a controller based on the MPC to perform one iteration to obtain an initial control force of the high-speed train at the time t+1 under a next iteration number, then updating the iteration number to enter the next iteration, until an iteration stop condition is reached, the initial control force of the high-speed train at the time t+1 under the last iteration number is determined as an optimal control force at the time t+1, and the high-speed train at the time t is controlled according to the optimal control force at the time t+1.

2. The PPC and MPC based high-speed train predictive cascade control method according to claim 1, characterized in that, The state errors of the high-speed train at the time t comprise a speed error at the time t and a position error at the time t, and a state error calculation process of the high-speed train at the time t comprises the following steps: obtaining the position error of the high-speed train at the time t according to an actual position of the high-speed train at the time t and a target position of the high-speed train at the time t+1; obtaining the speed error of the high-speed train at the time t according to an actual speed of the high-speed train at the time t and a target speed of the high-speed train at the time t+1.

3. The PPC and MPC based high-speed train predictive cascade control method according to claim 2, characterized in that, The controller based on the PPC comprises: Nonlinear gain function: ; Control law computation function: wherein, represents a position error transformation function at time t, represents a velocity error transformation function at time t, represents a position preset performance boundary at time t, represents a velocity preset performance boundary at time t, represents a position control gain, represents a velocity control gain, represents a position error at time t, represents a velocity error at time t, represents a virtual control force at time t, represents a virtual control force at time t+1, represents an initial control force at time t+1.

4. The PPC and MPC based high-speed train predictive cascade control method according to claim 3, characterized in that, The PPC-based controller further comprises a performance boundary function: wherein, represents an initial position error upper limit, represents an initial velocity error upper limit, represents a position steady state error upper limit, represents a velocity steady state error upper limit; represents a position convergence parameter, represents a velocity convergence rate parameter.

5. The PPC and MPC based high-speed train predictive cascade control method according to claim 4, characterized in that, The PPC-based controller further comprises a first constraint condition: .

6. The PPC and MPC based high-speed train predictive cascade control method according to claim 1, wherein, The speed error of the high-speed train at t+2 time under the current iteration number, the initial control force of the high-speed train at t+1 time under the current iteration number and the optimal control force at t time are input into the MPC-based controller to perform one iteration to obtain the initial control force of the high-speed train at t+1 time under the next iteration number, specifically, the speed error of the high-speed train at t+2 time under the current iteration number, the initial control force of the high-speed train at t+1 time under the current iteration number and the optimal control force at t time are input into the MPC-based controller, and then one iteration is performed by using the adaptive gradient descent algorithm to obtain the initial control force of the high-speed train at t+1 time under the next iteration number.

7. The PPC and MPC based high-speed train predictive cascade control method according to claim 1, wherein, The MPC-based controller includes a loss function : wherein, denotes the position control time domain, denotes the speed error at time t+j+1, denotes the energy consumption weight parameter, denotes the speed control time domain, denotes the control force at time t+j, denotes the optimal control force at time t+j-1.

8. The PPC and MPC based high-speed train predictive cascade control method according to claim 7, characterized in that, The MPC-based controller further comprises a second constraint condition: wherein || denotes an absolute value, denotes an upper limit of the control force increment. 9.The PPC and MPC based high-speed train predictive cascade control method according to claim 1, wherein, The number of convolution units in each residual block is 3.

10. A high-speed train predictive cascade control device based on PPC and MPC, characterized in that, The high-speed train prediction cascade control device based on the PPC and the MPC comprises: An acquisition module is configured to acquire train operation information and a state error of the high-speed train at t time; t is a current time; the train operation information at t time comprises an actual speed at t time, an optimal control force at t time and track data at t time; A first speed prediction module is configured to input the train operation information of the high-speed train at t time into a GS-TCN model to obtain a speed prediction value of the high-speed train at t+1 time; the GS-TCN model is a TCN model with a GELU activation function as an activation function, a number of convolution units in each residual block being greater than 2 and an input end of a first causal dilated convolution in each residual block being connected to an output end of a compression excitation module; An initial control force determination module is configured to process the state error of the high-speed train at t time by using a PPC-based controller to obtain an initial control force of the high-speed train at t+1 time under an initial iteration number; A second speed prediction module is configured to input train operation information of the high-speed train at t+1 time under the current iteration number into the GS-TCN model to obtain a speed prediction value of the high-speed train at t+2 time under the current iteration number; the train operation information at t+1 time comprises the initial control force at t+1 time, track data at t+1 time and the speed prediction value at t+1 time; A speed error calculation module is configured to obtain a speed error of the high-speed train at t+2 time under the current iteration number according to the speed prediction value of the high-speed train at t+2 time under the current iteration number and a target speed at t+2 time; A control module is configured to input the speed error of the high-speed train at t+2 time under the current iteration number, the initial control force of the high-speed train at t+1 time under the current iteration number and the optimal control force at t time into the MPC-based controller to perform one iteration to obtain the initial control force of the high-speed train at t+1 time under the next iteration number, then update the iteration number to enter the next iteration, until an iteration stop condition is reached, the initial control force of the high-speed train at t+1 time under the last iteration number is determined as the optimal control force at t+1 time, and the high-speed train at t time is controlled according to the optimal control force at t+1 time.

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