A Model Predictive Control Method for High-Speed ​​Trains Based on the Koopman Operator

Abstract

This invention discloses a high-speed train model predictive control method based on the Koopman operator. Specifically, it involves: acquiring input and output data using control inputs that can stimulate the dynamic characteristics of the high-speed train; establishing a dataset based on the generated output data and control input data; offline establishing a high-dimensional linear model corresponding to the high-speed train using the Extended Dynamic Mode Decomposition (EDMD) method according to Koopman operator theory; designing a model predictive controller based on the reference speed and the high-dimensional linearized model of the train at the current time; updating the high-dimensional linear model of the train online; using the updated model to calculate the optimal control sequence satisfying constraints at time k+1; and applying the first solution of the optimal control sequence as the control quantity to the high-speed train to achieve speed tracking control. This invention significantly improves control efficiency and accuracy, while also facilitating constraints on speed and acceleration, ensuring the safety of the control quantity.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of automatic control in the process of high-speed train operation, and particularly relates to a high-speed train model predictive control method based on a Koopman operator. BACKGROUND

[0002] In recent years, automatic driving technology has become a hot spot, and the driving mode of high-speed trains has developed towards automatic driving. The automatic driving (ATO) system of high-speed trains is a highly complex integrated control system, and is the key to ensuring operation safety, comfort and parking accuracy. The speed tracking control of the train is a key control unit of ATO, and the design and research of the control algorithm are related to the safety and efficiency of the entire ATO. As known, with the change of the running speed of high-speed trains, due to the high-speed air dynamic effect, complex operating environment and uncertain disturbances (such as wind, rain, etc.), the dynamic model of high-speed trains will present the characteristics of parameter time-varying, non-parametric uncertainty and strong nonlinearity, thus making the speed tracking control of high-speed trains more complex and challenging. In order to achieve this purpose, it is necessary to study the system model identification and tracking control strategy of high-speed trains.

[0003] In the research of the dynamic model of high-speed trains in the process of operation: some people solve the strong nonlinearity problem of the dynamic model of high-speed trains through Takagi-Sugen (T-S) model, and the establishment of T-S model depends on fuzzy rules and membership functions, and the determination process is relatively complex, so the practical application of this method will have limitations; some people regard high-speed trains as a whole and adopt single-particle modeling, which is a macro description of the overall operating state, and cannot accurately describe the control force distribution of each car, especially when high-speed trains run on complex road conditions (such as curves and slopes), which will produce large modeling errors; on this basis, some people later proposed a multi-particle modeling method, which regards each car as a particle to improve the accuracy of the model. Although the accuracy of the model is improved, too many nonlinear characteristics are considered, and the calculation is too complex, which is not conducive to the design of subsequent controllers, and even less conducive to real-time online control.

[0004] In the speed tracking control research of high-speed train operation: some people use pole placement method to design train speed controller according to optimal preview algorithm, however, the pole placement method and other analytical solving methods are aimed at the simplified train dynamics model, on the one hand, the simplified model cannot accurately reflect the real state of the train, on the other hand, the solving process needs strong mathematical foundation, which limits its application; some people combine traditional PID control method with some intelligent control methods, such as fuzzy-PID, genetic algorithm-PID, neural network-PID and the like, which realizes the optimization control of train speed to a certain extent, but may fall into local optimal solution; some people solve the problem of time-varying parameters of train dynamics model through adaptive control, but have no effect on non-parametric uncertainty; some people learn the control law of the train through deep learning, which lacks safety, and the speed control of the train needs to ensure safety, which limits its application to a certain extent.

[0005] In summary, the current train modeling strategy is mainly based on physical modeling, on the one hand, the train dynamic model established by such modeling method is a strong nonlinear system, which is not conducive to the design of subsequent control algorithm from the real-time and computational efficiency, on the other hand, the train operation environment is complex, there are many time-varying parameters and non-parametric uncertainties in the model, it is difficult to obtain an accurate dynamics model. The current train speed tracking control strategy mainly includes PID control, adaptive control, LQR control, neural network and other algorithms, on the one hand, these algorithms are difficult to handle control and state constraints, and the safety of the obtained control law cannot be guaranteed, which limits the further application in practical occasions. SUMMARY

[0006] In order to overcome the shortcomings of the prior art, the present application provides a high-speed train model predictive control method based on Koopman operator.

[0007] The high-speed train model predictive control method based on Koopman operator provided by the present application comprises the following steps:

[0008] Step 1: input and output data acquisition is carried out by using control input capable of exciting the dynamic characteristics of high-speed train, and a data set is established according to the generated output data and control input data.

[0009] Step 2: according to the Koopman operator theory, the high-dimensional linear model corresponding to the high-speed train is established offline by using the method of extended dynamic mode decomposition (EDMD).

[0010] Step 3: a model predictive controller is designed according to the reference speed and the high-dimensional linear model of the train at the current time; the model used at the initial time is the high-dimensional linear model designed offline in step 2, and the model used at the subsequent time is the high-dimensional linear model updated iteratively.

[0011] Step 4: update the high-dimensional linear model of the train according to the high-dimensional linear model of the train at time k, the train control input and output data at time k and the tracking error at time k, use the updated high-dimensional linear model of the train to calculate the optimal control sequence under the constraints at time k+1 in step 3, and use the first solution of the optimal control sequence as the control quantity acting on the high-speed train; repeat steps 3 and 4 to realize the speed tracking control of the train.

[0012] Further, step 1 is specifically:

[0013] The general longitudinal dynamics model of the train is described as:

[0014]

[0015] In the formula, p, v and m are the position, speed and mass of the train respectively; u is the traction force or braking force; r r is the running resistance per unit mass; r p is the additional resistance per unit mass; d(t) is the unmodeled disturbance.

[0016] The running resistance r r is composed of the rolling mechanical resistance r m and the aerodynamic resistance r a , and is described as:

[0017]

[0018] In the formula, c0(t) is the rolling resistance coefficient, c1(t) is the resistance coefficient caused by friction and train vibration, and c2(t) is the aerodynamic resistance coefficient.

[0019] The additional resistance r p is caused by track slope, curvature and tunnel, and is described as:

[0020]

[0021] In the formula, r g is the slope resistance from the track slope, θ(t) is the slope angle of the current track; r c is the curvature force from the track curvature, and D(t) is the curvature degree.

[0022] Define x(t) = [p(t) v(t)] T , and the train model (1) is described as The discrete-time train model is obtained by using the fourth-order Runge-Kutta method as follows:

[0023]

[0024] Wherein, k1, k2, k3, k4 are expressed as:

[0025]

[0026] In the formula, k is the sampling time, T s is the sampling period.

[0027] The control signal is applied to the high-speed train to obtain the speed and position data at the current time and the next time, which are stored as matrix X and X + , based on X, X + , U data set can be solved by high-dimensional linear model, the data set collected as follows:

[0028] X=[x1 x2…x k ],X+=[x2 x3…x k+1 ],U=[u1 u2…u k ] (5)

[0029] In the formula, k is the number of recorded data points, x(k)=[p(k) v(k)] T .

[0030] Further, step 2 is specifically:

[0031] According to the Koopman operator theory, the method of extended dynamic mode decomposition EDMD is used to establish the Koopman high-dimensional linear model corresponding to the high-speed train offline as follows:

[0032]

[0033] In the formula, z(x)=ψ(x)=[x T ,ψ1(x),…,ψ N (x)] is the dimension function, which evolves in a linear way in high-dimensional space, which contains the state is the self-defined basis function.

[0034] Since the position and speed state of the train itself are the first two elements of the dimension function, the matrix C does not need to be solved by the least square method, and C=[0 1 0 0…0] N+2 , that is, y represents the speed of the train.

[0035] The method of extended dynamic mode decomposition EDMD is used to solve the matrix A, B as follows:

[0036] The data set is dimensioned:

[0037]

[0038] The solved matrix A, B is converted into the following least square problem:

[0039]

[0040] The analytical solution of equation (8) is:

[0041] Rewrite the result in equation (8) as the result of equation (9):

[0042]

[0043]

[0044] The analytical solution of equation (9) is:

[0045] In order to update the high-dimensional linear model of high-speed trains in subsequent iterations, define a variable P as the initial value of subsequent iterative updates:

[0046] Further, step 3 is specifically:

[0047] Write the prediction equation of the future state:

[0048]

[0049] In the formula, N p ,N c are the lengths of the prediction time domain and the control time domain, respectively, and N p ≥N c .

[0050] Further, write the prediction equation in the following compact form:

[0051]

[0052] Solve the control quantity:

[0053] Define the cost function as J = (Y(k) - Y r (k)) T Q(Y(k) - Y r (k)) + U T (k)RU(k), where Q and R are constant positive definite diagonal matrices, then solving the optimal control sequence is equivalent to solving the following optimization problem:

[0054]

[0055] Further, step 4 is specifically:

[0056] Assume that the high-dimensional linear model of the train at time k is M k = [A B], and the state of the train before the control signal u at time k is x k, the train state is measured as x k+1 , define φ(k) = [ψ(x k ); u], ξ(k) = ψ(x k+1 ), the velocity error is defined as e_v = |y r (k+1) - [0 1]*x k+1 |, then the updated train high-dimensional linear model at time k is:

[0057]

[0058] wherein the calculation of P(k+1) is as follows:

[0059]

[0060]

[0061] The iterative update of the high-dimensional linear model is proved as follows:

[0062] The train high-dimensional linearization model at time k is:

[0063]

[0064] wherein:

[0065] X = [x1 x2…x k ], X+ = [x2 x3…x k+1 ], U = [u1 u2…u k ]

[0066] Φ k = [φ(0), φ(1), …, φ(k-1)] T

[0067] Y k = [y(0), y(1), …, y(k-1)]

[0068] The latest data of the train at time k+1, the new data set is:

[0069]

[0070] Y k+1 = [y(0), y(1), …, y(k-1), y(k)] = [Y k y(k)]

[0071] According to formula (9), the high-dimensional linearization model of the train at time k+1 is represented as:

[0072]

[0073] The matrix inversion lemma is known as follows:

[0074] (A+BCD) -1 =A -1 -A -1 B(C -1 +DA -1 B) -1 DA -1

[0075] Therefore, equation (14) has the following meaning:

[0076]

[0077]

[0078] The above is the proof of the iterative update algorithm without introducing the forgetting factor λ. When the forgetting factor λ is introduced, we have... Y k+1 =[ρY k y(k)], let λ=ρ 2 This is the result of expression (13).

[0079] The beneficial technical effects of this invention are as follows:

[0080] (1) High modeling accuracy: Unlike previous modeling based on complex mechanistic models, the data-driven modeling based on the Koopman operator described in this invention only relies on input and output data, eliminating the need for model analysis and using a high-dimensional linearized model to describe the original strongly nonlinear system. Simultaneously, to address the issues of time-varying train dynamic model parameters, nonparametric uncertainties, and unmodeled dynamics, this invention proposes online iterative updates to the train model (instead of iteratively solving matrices A and B using the EDMD method, thus avoiding excessive computational burden), thereby improving modeling accuracy. Furthermore, as the model iterates and updates, the rectification capability of new data decreases. To prevent this data saturation phenomenon, a forgetting factor λ is introduced. λ is a quantity related to the train speed tracking error rather than a constant, which further improves modeling accuracy and tracking control accuracy.

[0081] (2) Simple controller solution: When designing the model predictive controller, a high-dimensional linear model is used as the dynamic model of the high-speed train, which avoids the problem of non-convex optimization solution and greatly improves the computational efficiency.

[0082] (3) Good safety: Train speed control has extremely high safety requirements. Previous control algorithms were not convenient to add state constraints and control constraints. In the process of solving the model predictive controller, state constraints and control quantity constraints were considered to ensure the safety of the obtained control quantity. Attached Figure Description

[0083] Figure 1 The flow chart of collecting train data set offline provided by the present application.

[0084] Figure 2 The schematic diagram for the action of Koopman operator.

[0085] Figure 3 The online control flow chart of the high-speed train model predictive control method based on Koopman operator provided by the present application.

[0086] Figure 4 The closed-loop principle block diagram of the high-speed train model predictive control method based on Koopman operator provided by the present application.

[0087] Figure 5 The prediction effect diagram of Koopman high-dimensional linear model on the future state of high-speed train.

[0088] Figure 6 The tracking curve diagram of high-speed train on target speed.

[0089] Figure 7 The tracking curve diagram of high-speed train on target position. DETAILED DESCRIPTION

[0090] The present application will be further described below in combination with the drawings and embodiments.

[0091] The high-speed train model predictive control method based on Koopman operator of the present application: firstly, the high-dimensional linear representation of the strong nonlinear system of high-speed train is obtained by using the characteristic that Koopman operator can identify nonlinear system as high-dimensional linear model. In order to solve the problems of time-varying of train dynamic model parameters, non-parametric uncertainty and unmodeled dynamics, the high-dimensional linearization model is iteratively updated online according to the real-time running state data of train, current tracking error and high-dimensional linearization model of last time, so as to ensure the accuracy of modeling. Secondly, the speed tracking controller is designed by combining model predictive control based on high-dimensional linear model. Model predictive control has the characteristics of multi-step prediction, rolling optimization and feedback control, which greatly improves the control efficiency and accuracy, and also facilitates the constraint of speed and acceleration, and ensures the safety of control quantity.

[0092] The high-speed train model predictive control method based on Koopman operator of the present application comprises the following steps:

[0093] Step 1: input-output data collection is carried out by using control input capable of exciting the dynamic characteristics of high-speed train, and a data set is established according to the generated output data and control input data.

[0094] Taking into account mechanical resistance, aerodynamic resistance, and road conditions, the general longitudinal dynamics model of a train can be described as follows:

[0095]

[0096] In the formula, p, v, and m represent the position, speed, and mass of the train, respectively; u is the traction force or braking force; r r The running resistance per unit mass; r p d(t) represents the additional drag per unit mass; d(t) represents the unmodeled disturbance.

[0097] Operating resistance r r Due to rolling mechanical resistance r m and aerodynamic drag r a Composition, described as:

[0098]

[0099] In the formula, c0(t) is the rolling resistance coefficient, c1(t) is the resistance coefficient caused by friction and train vibration, and c2(t) is the aerodynamic drag coefficient. Here, considering some physical characteristics of the train, such as the train's length, cross-sectional area, and widely varying aerodynamics, the drag coefficients c0(t), c1(t), and c2(t) are expressed as time-varying to describe the actual dynamics of the train (1).

[0100] Additional resistance r p It is caused by track gradient, curvature, and tunnels, and is described as follows:

[0101]

[0102] In the formula, r g The slope resistance is the slope angle from the track gradient, where θ(t) is the current track slope angle; r c Let D(t) be the curvature force derived from the track curvature, and D(t) be the curvature degree. Due to the complex and dynamically changing operating conditions of high-speed trains, it is difficult to obtain resistance and disturbances. In particular, the disturbance d(t) varies with changes in the train's operating environment, such as strong winds and different wheel-rail contact conditions, making it difficult for equation (1) to describe the actual train dynamics. Therefore, using Koopman operator theory, a data-driven train model that can fully reflect the characteristics of train dynamics is proposed.

[0103] Combination Figure 1 It can be seen that, in order to collect data, the continuous-time series dynamics (1) is transformed into discrete-time series dynamics. Define x(t) = [p(t) v(t)] T The train model (1) is redescribed as The discrete-time train model obtained using the fourth-order Runge-Kutta method is as follows:

[0104]

[0105] where k1, k2, k3, k4 are expressed as:

[0106]

[0107] where k is the sampling time, T s is the sampling period.

[0108] When collecting data, sufficient control input data that can stimulate the dynamic characteristics of the system should be selected according to the dynamic characteristics of the system, and then the input and output data of the system are collected to form a data set. After the control signal u (traction force or braking force) is applied to the high-speed train, the speed and position data at the current time and the next time are obtained, and are stored as matrices X and X + , respectively, based on the X, X + , U data set, a high-dimensional linearization model can be solved, and the collected data set is as follows:

[0109] X = [x1 x2…x k ], X + = [x2 x3…x k+1 ], U = [u1 u2…u k ] (5)

[0110] where k is the number of recorded data points, x(k) = [p(k)v(k)] T .

[0111] It is worth noting that the data in (5) does not need to be time-ordered, nor does it need to be from the same trajectory of (4). However, the same sampling time needs to be used to obtain the data set.

[0112] Step 2: According to the Koopman operator theory, an extended dynamic mode decomposition (EDMD) method is used to establish an offline Koopman high-dimensional linear model corresponding to the high-speed train.

[0113] As shown in Fig. Figure 2 , the Koopman operator uses an up-dimensional function to lift the original nonlinear system to a high-dimensional space, and evolves in a linear manner in the high-dimensional space. Therefore, the biggest feature of the Koopman is that it can learn the dynamic model of the unknown system from the data, and represents the original nonlinear system in a linear high-dimensional space, which is beneficial to the design of the subsequent controller. At the same time, in order to solve the problems of time-varying parameters, non-parametric uncertainties and unmodeled dynamics of the train dynamic model, the present application will update the high-dimensional linearization model online according to the real-time running data of the train, the speed tracking error and the high-dimensional linearization model at the last time, to ensure the accuracy of the modeling.

[0114] The selection of the dimension-lifting function is crucial, and is usually selected as a set of nonlinear functions, such as a vector composed of Gaussian functions, radial basis functions, polynomial functions, sinusoidal curves, etc. The dimension-lifting function is defined as z(x) = ψ(x) = [x T , ψ1(x), …, ψ N (x)] here, where x(k) = [p(k)v(k)] T , and the remaining components are selected as Thinplate radial basis functions:

[0115] ψ3 = ||x-a1|| 2 log(||x-a1||)

[0116] ψ4 = ||x-a5|| 2 log(||x-a2||)

[0117] ψ5 = ||x-a3|| 2 log(||x-a3||)

[0118]

[0119] ψ N = ||x-a N-2 || 2 log(||x-a N-2 ||)

[0120] where a1, a2, … a N-2 are random numbers in [-1, 1].

[0121] According to the Koopman operator theory, the high-dimensional linear model corresponding to the high-speed train is established offline using the method of extended dynamic mode decomposition (EDMD) as follows:

[0122]

[0123] In the formula, z(x) = ψ(x) = [x T , ψ1(x), …, ψ N (x)] is a dimension-lifting function that evolves in a linear manner in a high-dimensional space, where x(k) = [p(k)v(k)] T , is a self-defined basis function.

[0124] Since the position and speed state of the train itself are the first two elements of the dimension-lifting function, the matrix C does not need to be solved by the least square method, and C = [0 1 0 0…0] N+2 can be obtained, where y represents the speed of the train.

[0125] Solve the matrix A, B by the method of extended dynamic mode decomposition (EDMD) as follows:

[0126] Upgrade the dimension of the data set:

[0127]

[0128] Convert the solution of the matrix A, B into the following least squares problem:

[0129]

[0130] The analytical solution of equation (8) is: Up to now, the Koopman high-dimensional linear model of high-speed train has been solved.

[0131] Step 4 will update the high-dimensional linear model of the train at time k according to the high-dimensional linear model of the train at time k, the train control input and output data at time k, and the speed tracking error at time k, that is, update the matrix A, B. In order to facilitate subsequent formula derivation and improve the calculation efficiency of model updating (avoid solving a least squares problem every time), the result in equation (8) is rewritten as the result of equation (9) (both are equivalent):

[0132]

[0133]

[0134] The analytical solution of equation (9) is:

[0135] In order to update the high-dimensional linear model of high-speed train in subsequent iterations, define a variable P as the initial value of subsequent iterative update:

[0136] The online control flow chart of high-speed train model predictive control method based on Koopman operator is shown in Figure 3 The closed-loop principle diagram of high-speed train model predictive control method based on Koopman operator is shown in Figure 4 , which involves related algorithms as shown in steps 3 and 4.

[0137] Step 3: Design a model predictive controller according to the reference speed and the high-dimensional linear model of the train at the current time. The model used at the initial time is the high-dimensional linear model designed offline in step 2, and the model used at the subsequent time is the iteratively updated high-dimensional linear model.

[0138] Write the prediction equation of the future state:

[0139]

[0140] where N p c are the prediction horizon and control horizon length, respectively, and N p ≥ N c .

[0141] The prediction equation is further written in the following compact form:

[0142]

[0143] Solving the control variable:

[0144] Define the cost function as J = (Y(k) - Y r (k)) T Q(Y(k) - Y r (k)) + U T (k)RU(k), where Q and R are constant positive definite diagonal matrices, then solving the optimal control sequence is equivalent to solving the following optimization problem:

[0145]

[0146] The solution of the above optimization problem is the optimal control sequence that satisfies the constraints at the current time. The first solution of the optimal control sequence is used as the control variable to act on the high-speed train, thereby realizing the speed tracking control of the train.

[0147] Step 4: Update the high-dimensional linear model of the train at time k based on the high-dimensional linear model of the train at time k, the train control input and output data at time k, and the tracking error at time k. The updated high-dimensional linear model of the train is used to calculate the optimal control sequence that satisfies the constraints at time k+1 in step 3. The first solution of the optimal control sequence is used as the control variable to act on the high-speed train. Steps 3 and 4 are repeated to realize the speed tracking control of the train.

[0148] The specific updating method is as follows:

[0149] Assume that the high-dimensional linear model of the train at time k is M k = [A B], the state of the train before the control signal u at time k is x k , and the measured state of the train after the control signal u at time k is x k+1 . Define φ(k) = [ψ(x k ); u], ξ(k) = ψ(x k+1 ), and the speed error is defined as e_v = |y r (k+1) - [0 1]*x k+1 |. Then the updated high-dimensional linear model of the train at time k is:

[0150]

[0151] ​where P(k+1) is calculated as follows:

[0152]

[0153]

[0154] The iteration initial value P(0) in the above equation is calculated in step 2.

[0155] The iteration update of the high-dimensional linear model is proved as follows:

[0156] The high-dimensional linear model of the train at time k is:

[0157]

[0158] where:

[0159] X = [x1 x2...x k ], X+ = [x2 x3...x k+1 ], U = [u1 u2...u k ]

[0160] Ф k = [φ(0), φ(1),..., φ(k-1)] T

[0161] Y k = [y(0), y(1),..., y(k-1)]

[0162] The latest data of the train at time k+1, the new data set is:

[0163]

[0164] Y k+1 = [y(0), y(1),..., y(k-1), y(k)] = [Y k y(k)]

[0165] According to equation (9), the high-dimensional linear model of the train at time k+1 is represented as:

[0166]

[0167] The matrix inversion lemma is known as follows:

[0168] (A+BCD) -1 = A -1 -A -1 B(C -1 +DA -1 B) -1 DA -1

[0169] Therefore, equation (14) has the following meaning:

[0170]

[0171]

[0172] The above is the proof of the iterative update algorithm without introducing the forgetting factor λ. When the forgetting factor λ is introduced, we have... Y k+1 =[ρY k y(k)], let λ=ρ 2 This is the result of expression (13).

[0173] Clearly, this method updates the train model online in real time without repeatedly solving the least squares problem; it only requires the train data at the current moment. It solves the problems of time-varying train dynamic model parameters, non-parametric uncertainties, and unmodeled dynamics with almost no increase in computational load. Furthermore, as the model iterates, the matrix p(k) decreases rapidly, indicating a decline in the rectification capability of the new data—a phenomenon known as data saturation. Therefore, a forgetting factor λ is introduced to maintain the correction capability of the new data, allowing us to obtain an accurate estimate. The forgetting factor λ determines its tracking capability; convergence is only possible if λ ≤ 1. A smaller λ indicates that old data is forgotten faster, making the algorithm more sensitive to noise. To balance interference suppression and fast tracking, λ ∈ [0.95, 0.99] is typically chosen. To obtain a more accurate model, λ in this invention is not a constant but a function of the train speed tracking error, making train modeling more accurate and achieving better tracking results.

[0174] Example:

[0175] To verify the effectiveness of the control algorithm designed in this invention, a simulation experiment on the tracking control of high-speed train speed and position was conducted using MATLAB as the simulation platform and a train operation system similar to CRH2-A as the controlled object. The following detailed explanation of the high-speed train model predictive control method based on the Koopman operator proposed in this invention, in conjunction with the simulation experiment and accompanying figures, provides a detailed description.

[0176] The railway line selected in this embodiment is approximately 72km from the departure station to the terminal station, with one curve and one ramp along the route, and the train's operating time is 2000s. To verify the control effect of this invention under complex road conditions, time-varying resistance parameters and non-parametric uncertainty disturbances are introduced. The parameters and constraints of train dynamics are shown in Table 1.

[0177] Table 1. Parameters and constraints of train dynamics

[0178]

[0179] The speed reference curve of the high-speed train is a function of time t, and the integral of the speed reference signal is the position reference curve, the unit of t is s, and the unit of speed is m / s. The speed reference curve is as follows:

[0180]

[0181] Before designing the controller, the input and output data set of the train needs to be established, and this verification contains fifty thousand data points (50 trajectories, 1000 input and output data points are collected for each trajectory), and the dimension increasing function is z(x) = ψ(x) = [x T , ψ1(x), …, ψ N (x)], wherein N = 6, x(k) = [p(k)v(k)] T , and the remaining components are selected as Thinplate radial basis functions: ψ i = ||x-a|| 2 log(||x-a||), a is a random constant vector in [-1, 1].

[0182] On the basis of the above data set, according to the Koopman operator theory, the method of extended dynamic mode decomposition (EDMD) is used to establish the Koopman high-dimensional linear model corresponding to the high-speed train offline.

[0183] Before designing the controller, the accuracy of the modeling is verified first, the initial state x0 = [0; 49.5] is selected, the state of the train is predicted based on the high-dimensional linear model, and the prediction result is as shown in Figure 5 , wherein True represents the true state, Koopman represents the prediction result based on the fixed high-dimensional linear model, and Koopmanrls represents the prediction result based on the iteratively updated high-dimensional linear model. The results show that when facing parameter time-varying and unknown uncertainty interference, the prediction result of the iteratively updated high-dimensional linear model proposed by the application is closer to the true state, the model can be more accurately established, and the subsequent control effect is improved.

[0184] Design of the controller: the prediction time domain length and the control time domain length of MPC are both taken as 30, the sampling interval time is 0.01 s, the simulation time is 2000 s, Figure 6 and Figure 7 are the speed tracking simulation results and the position tracking simulation results (since all tests do not violate the speed and acceleration constraints, they are not displayed), it can be seen from the figure that the controller designed based on the iteratively updated high-dimensional linear model can obtain better tracking effect. The positive and negative maximum speed tracking error and the position tracking error range are as shown in Table 2:

[0185] Table 2 Positive and negative maximum velocity tracking error and position tracking error range

[0186] Control method Velocity tracking error range (m / s) Position tracking error range (m) Kmpc [-0.152,0.064] [-31.262,20.958] Krlsmpc [-0.091,0.026] [-19.103,8.652]

[0187] As can be seen from Table 2, although the controller designed based on the fixed high-dimensional linear model and the controller designed based on the iteratively updated high-dimensional linear model can both achieve tracking effects and meet the speed error requirement of the Chinese Train Control System Level 3 (CTCS-3) (the speed error range is not more than 0.56 m / s when the speed v is less than or equal to 8.33 m / s, and the speed error should be within 2% of the speed when the speed v is greater than 8.33 m / s), it is obvious that the method provided by the present application has smaller speed and position tracking errors and higher control precision when facing parameter time-varying and non-parameter uncertainty disturbances.

Claims

1. A high-speed train model predictive control method based on a Koopman operator, characterized in that, The method comprises the following steps: Step 1: input and output data are collected by using control input capable of exciting the dynamic characteristics of the high-speed train, and a data set is established according to the generated output data and the control input data; Step 2: according to the Koopman operator theory, a high-dimensional linear model corresponding to the high-speed train is established offline by using the method of extended dynamic mode decomposition (EDMD); Step 3: a model predictive controller is designed according to the reference speed and the high-dimensional linear model of the train at the current time; the model used at the initial time is the high-dimensional linear model designed offline in step 2, and the model used at the subsequent time is the high-dimensional linear model updated iteratively; Step 4: the high-dimensional linear model of the train at the k time, the train control input and output data at the k time and the tracking error at the k time are used to update the high-dimensional linear model of the train online, the updated high-dimensional linear model of the train is used to calculate the optimal control sequence under the constraint at the k+1 time in step 3, and the first solution of the optimal control sequence is used as the control quantity acting on the high-speed train; steps 3 and 4 are repeated, so that the speed tracking control of the train is realized; Assume the high-dimensional linear model of the train at time k is , the train state before the control signal u at time k acts is , the measured train state after the control signal u at time k acts is , define , the speed error is defined as , and the updated high-dimensional linear model of the train at time k is （13） wherein The calculation is as follows: ； The iterative update of the high-dimensional linear model is proved as follows: The high-dimensional linearization model of the train at the k time is: ； Wherein: ； The latest data of the train at the k+1 time, and the new data set is: ； According to formula (9), the high-dimensional linearization model of the train at the k+1 time is represented as: （14）； The matrix inversion lemma is known as follows: ； Therefore, formula (14) has the following representation: ； The above is the result of the expression (13) without introducing the forgetting factor The proof of the iterative update algorithm when the forgetting factor is introduced The above is the result of the expression (13) without introducing the forgetting factor , let The above is the result of the expression (13) without introducing the forgetting factor 2. The high-speed train model predictive control method based on a Koopman operator according to claim 1, characterized in that, The step 1 is specifically: The general longitudinal dynamic model of the train is described as: （1） where p, v, m are the position, velocity and mass of the train, respectively; u is the traction or braking force; is the running resistance per unit mass; is the additional resistance per unit mass; is the unmodeled disturbance; Running resistance consists of rolling mechanical resistance and aerodynamic resistance is described as: （2） wherein is the rolling resistance coefficient, is the resistance coefficient caused by friction and train vibrations, is the aerodynamic resistance coefficient; Additional resistance is caused by track grade, curvature, and tunnel, described as: （3） wherein is the slope resistance from the track slope, is the slope angle of the current track; is the curvature force from the track curvature, is the curvature degree; Definitions , the train model (1) is re-described as ; the discrete time train model is obtained by using the fourth-order Runge-Kutta method as follows: （4） Wherein, k1, k2, k3 and k4 are represented as: ； In the formula, is the sampling time, is the sampling period; The control signal is applied to the high-speed train to obtain the speed and position data at the current time and the next time, which are stored as matrices With , based on , , The data set can be solved by high-dimensional linearization model. The collected data set is as follows: （5） where k is the number of data points recorded, .

3. The high-speed train model predictive control method based on a Koopman operator according to claim 2, characterized in that, The step 2 is specifically: According to the Koopman operator theory, the high-dimensional linear model corresponding to the high-speed train is established offline by using the method of extended dynamic mode decomposition (EDMD) as follows: （6） wherein is an elevation function that evolves in a linear fashion in a high-dimensional space that contains the state , is a user-defined basis function; Since the position and velocity of the train are the first two elements of the lift function, the matrix C does not need to be solved by least squares, but rather i.e., where denotes the velocity of the train. The matrices A and B are solved by using the method of extended dynamic mode decomposition (EDMD) as follows: The data set is upgraded: （7） The solved matrices A and B are converted into the following least square problem: （8） The analytical solution of equation (8) is: ; The result in formula (8) is rewritten as the result in formula (9): ； （9） The analytical solution of equation (9) is: ; In order to update the high-dimensional linearization model of the high-speed train in the subsequent iteration, a variable P is defined as the initial value of the subsequent iteration update: .

4. The high-speed train model predictive control method based on a Koopman operator according to claim 3, characterized in that, The step 3 is specifically: The prediction equation of the future state is written as: （10） In the formula, are the prediction and control horizon lengths, respectively, and ; The prediction equation is further written in the following compact form: （11） The control quantity is solved: The cost function is defined as where Q, R are constant positive definite diagonal matrices, then solving the optimal control sequence is equivalent to solving the following optimization problem: （12）。

Images