Distributed speed cooperative sliding mode control method for high-speed train

Abstract

The invention provides a high-speed train distributed sliding-mode control method based on an extended sliding-mode disturbance observer, and compared with a distributed control method without an extended sliding-mode disturbance observer, the high-speed train distributed sliding-mode control method based on the extended sliding-mode disturbance observer has the advantages that the distributed sliding-mode control method based on the extended sliding-mode disturbance observer is high in reliability; the method can eliminate the influence of unknown composite interference on the speed synchronization of each compartment when a high-speed train runs, is more suitable for nonlinear and strong-coupling systems such as the high-speed train, adopts an extended sliding mode disturbance observer to estimate the unknown composite interference of each compartment, and enhances the robustness and anti-interference capability of the method. The control method disclosed by the invention has the advantages of high response speed, high control precision and better robustness for unknown composite interference, and can enable each compartment to quickly and accurately track a target speed curve, so that the high-speed train can efficiently, stably and reliably run under uncertain factors and external interference.

Description

Technical Field

[0001] The present invention relates to the technical field of high - speed train control systems, and particularly to a distributed speed cooperative sliding - mode control method for high - speed trains based on an extended sliding - mode disturbance observer. Background Art

[0002] High - speed trains have many advantages such as safety, environmental protection, and strong passenger - carrying capacity, and are an important representative of China's high - end equipment manufacturing industry. The construction speed and scale of China's high - speed railways are in the leading position in the world. High - speed trains have the advantages of high speed, strong transportation capacity, high safety, and large passenger capacity, and have become an indispensable means of transportation in China. However, when high - speed trains operate under actual working conditions, affected by complex and variable operating environments, such as factors like temperature, operating duration, and external disturbances, the speeds between carriages are asynchronous, leading to problems such as longitudinal impulses of the train, and further making the train operation unstable, and even causing safety accidents. Therefore, the safety and stability issues of high - speed train operation have become increasingly prominent.

[0003] The coordinated control of the speeds of the carriages of high - speed trains is the key to train operation control. Studying the speed - coordinated control algorithm for the carriages of high - speed train multiple units can not only improve the synchronization accuracy and operation efficiency, but also have a significant impact on the safe operation of high - speed trains and is of great significance for the development of China's high - speed railways. Therefore, in order to ensure the stable operation of the high - speed train system under parameter perturbation and external disturbance conditions, new control methods need to be sought to achieve the efficient, safe, and reliable operation of high - speed trains under actual working conditions. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a distributed speed cooperative sliding - mode control method for high - speed trains based on an extended sliding - mode disturbance observer in view of the deficiencies and defects of the prior art.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] A distributed speed cooperative sliding - mode control method for high - speed trains based on an extended sliding - mode disturbance observer, characterized by comprising the following steps:

[0007] Step 1, establish a multi - agent mathematical model of a high - speed train:

[0008] Step 1.1, the high - speed train system can be regarded as a multi - agent system composed of a virtual leader carriage 0 and follower carriages. The dynamic equation of the virtual leader carriage 0 is , where is the speed of the virtual leader carriage 0, is the displacement of the virtual leader carriage 0, Control input for the virtual leader car 0

[0009] Step 1.2, the dynamic equation of the nth follower car is , where is the displacement of the nth follower car, is the speed of the nth follower car, is the mass of the nth follower car, is the control input of the nth follower car, ( ) is the non - linear coupling force of the nth follower car, ( ) is the unknown composite disturbance of the nth follower car; is the connection weight between adjacent follower cars, indicates that there is information transfer between car i and car j, indicates that there is no information transfer between car i and car j;

[0010] Step 1.3, the dynamic model of the nth follower car can be simplified as: , where , ;

[0011] Step 2, design the sliding - mode controller for the virtual leader car 0

[0012] Step 2.1, define the state error of the virtual leader car 0 as: , where is the actual speed of the virtual leader car 0, is the given speed of the virtual leader car 0;

[0013] Step 2.2, select the sliding - mode surface as , where , is the parameter to be designed;

[0014] Step 2.3, select the exponential reaching law , where, , is the parameter to be designed;

[0015] Step 2.4, design the control law of the virtual leader car 0 as: ; where is the sign function;

[0016] Step 3, design the extended sliding mode disturbance observer of the th follower car to observe the disturbance:

[0017] Step 3.1, when the high-speed train is subject to unknown combined disturbances during actual operation, its state equation is: ; where , represents the state variable of the th follower car; represents the input signal of the th follower car; represents the unknown combined disturbance signal received by the th follower car, which is an unknown bounded function, i.e., , D is a bounded quantity; , and are the coefficient matrices of the th follower car;

[0018] Step 3.2, expand the unknown combined disturbance signal into a state variable, and the extended state equation of the high-speed train can be obtained as: ; where is the change rate of the unknown combined disturbance signal ;

[0019] Step 3.3, select the sliding mode surface as , where is the observation error of the system state variable; design the extended sliding mode disturbance observer of the th follower car as ; where is the estimated value of ; is the estimated value of the unknown combined disturbance ; represents the change rate of the estimated value of the unknown combined disturbance ; is the sliding mode term of the observer to be designed; is the observer gain to be designed;

[0020] Step 3.4, the dynamic error equation of the extended sliding mode disturbance observer can be obtained as: ; is the observation error of the unknown composite disturbance signal;

[0021] Step 3.5, select the reaching law as: ; where, and are the designed coefficients respectively; is the sign function;

[0022] Step 3.6, design the sliding mode control law of the extended sliding mode disturbance observer as: ;

[0023] Step 4, design the distributed velocity cooperative sliding mode controller for the th follower carriage:

[0024] Step 4.1, let the displacement error between the th follower carriage and the virtual leader carriage 0 be ; let the velocity error between the th follower carriage and the virtual leader carriage 0 be ;

[0025] Step 4.2, select the displacement sliding mode surface of the th follower carriage as ; select the sliding mode surface of the velocity of the th follower carriage ;

[0026] Step 4.3, feedback the estimated value of the unknown composite disturbance observed by the extended sliding mode disturbance observer to the distributed sliding mode controller;

[0027] Step 4.4, design the distributed velocity cooperative sliding mode controller for the th follower carriage as: , where, is the distributed velocity cooperative control algorithm term, is the sliding mode control term; they are respectively

[0028] ; where, is the mass of the th follower carriage; is the connection weight between adjacent carriages, indicates that there is information transfer between carriage i and carriage j, indicates that there is no information transfer between carriage i and carriage j; and and are parameters to be designed; indicates the State variables of the follower carriages Denote the state variables of the th follower carriage; Denote the speed of the th follower carriage; is the sign function; wherein,

[0029] Furthermore, for the extended sliding mode disturbance observer, select the gains , , and . The state error will converge within a finite time, and at this time, the estimated value of the unknown disturbance signal can be obtained as: wherein, , denote the parameters to be designed for the

[0030] Furthermore, for the distributed speed cooperative sliding mode controller of the high-speed train, when , and and satisfy , , that is, the speed of the th follower carriage is synchronized with the speed of the virtual leader carriage 0.

[0031] In order to effectively solve the problem of speed asynchrony of each carriage of the high-speed train caused by unknown composite disturbances during operation, the present invention adopts a distributed speed cooperative sliding mode control strategy for high-speed trains based on an extended sliding mode disturbance observer (ESMD0), so that the speed of each carriage of the high-speed train EMU is consistent with the speed of the virtual leader carriage. This strategy is based on the multi-particle model of the high-speed train, combines multi-agent distributed cooperative control with sliding mode variable structure control to design a distributed sliding mode controller; at the same time, introduces ESMDO into the control algorithm, accurately estimates the unknown composite disturbances suffered during the operation of the high-speed train through the implementation of ESMDO, eliminates the influence of the unknown composite disturbances on the distributed sliding mode controller, and effectively improves the robustness of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 is the system structure block diagram of an embodiment of the present invention;

[0033] In the figure, is the given speed of the virtual leader car 0; is the speed of the virtual leader car 0, is the displacement of the virtual leader car 0; is the displacement of the is the speed of the is the control input of the section follower car; ESMDO1, ESMDO2, …… ESMDOn are the extended sliding mode disturbance observers (ESMDOs) of the respectively the estimated values of the unknown composite disturbances of the

[0034] Figure 2 This is an embodiment of the present invention: the communication topology diagram between multiple intelligent agents of the carriages.

[0035] Figure 3 This is an embodiment of the present invention: the speed tracking diagram of the follower carriages without ESMDO.

[0036] Figure 4 This is an embodiment of the present invention: the displacement error diagram of the follower carriages without ESMDO

[0037] Figure 5 This is an embodiment of the present invention: the displacement curve diagram of the follower carriages with ESMDO.

[0038] Figure 6 This is an embodiment of the present invention: the speed tracking diagram of the follower carriages with ESMDO.

[0039] Figure 7 This is an embodiment of the present invention: the speed error diagram of the follower carriages with ESMDO.

[0040] Figure 8 This is an embodiment of the present invention: the uniform disturbance observation effect of follower carriage 1.

[0041] Figure 9 This is an embodiment of the present invention: the slowly varying disturbance observation effect of follower carriage 2.

[0042] Figure 10 This is an embodiment of the present invention: the high-frequency disturbance observation effect of follower carriage 3.

[0043] Figure 11For an embodiment of the present invention: the mutation disturbance observation effect of the follower car body 4. Detailed implementation manners

[0044] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments.

[0045] Figure 1 It is a block diagram of an embodiment of the distributed speed cooperative sliding mode control method for a high-speed train based on ESMDO proposed by the present invention. Figure 2 It is a communication interaction topology diagram between multiple intelligent agents of the car bodies of the high-speed train proposed by the present invention.

[0046] First, assume that the high-speed train consists of n car bodies. Each car body of the high-speed train is regarded as a particle. By analyzing the force conditions during the operation of the high-speed train, a multi-particle model of the high-speed train is established:

[0047] (1)

[0048] Where , , ( ) are the mass, displacement, and velocity of the -th car body respectively; ( ) is the unknown composite disturbance; ( ) is the non-linear coupling force of the -th car body.

[0049] The high-speed train system consists of a leader car body 0 and multiple follower car bodies . During its movement, each of its car bodies is regarded as a particle, and multiple car bodies form a multi-particle multi-intelligent agent system. Assume that the -th follower car body is a node in the multi-intelligent agent system, then the virtual leader car body 0 is the root node in graph theory. The information transfer between every two car bodies interacts through the edge set in graph theory to form an adjacency graph.

[0050] Definition: The information interaction between multi-intelligent agents is an undirected graph composed of a node set , an edge set , and a weighted adjacency matrix . Each node in the node set represents a follower car body. In the edge set , if the follower car body can send information to the car body , then It is the edge set between these two follower carriages, and . The weight connection matrix In, if there is information exchange between follower carriages and , then , otherwise . Therefore, the matrix is a symmetric matrix.

[0051] Laplacian matrix , where D n = [ d ij ] ∈ R n × n is the in-degree matrix. When , is the in-degree of node , where . If there is information exchange between any two nodes, then the undirected graph is connected, and its corresponding is a symmetric positive semi-definite matrix, and its eigenvalues satisfy . If the matrix is defined, then is a symmetric positive definite matrix.

[0052] The traditional high-speed train controller is a PI controller, which cannot well adapt to the application scenarios of high-speed train control systems facing complex working conditions, especially when there are unknown combined disturbances such as electrical parameter perturbations, mechanical parameter perturbations, external disturbances, and unmodeled dynamics. This embodiment proposes a distributed sliding mode control algorithm for high-speed trains based on an extended sliding mode disturbance observer (ESMD0). The virtual leader carriage transmits information to the controller, and then the speed and displacement information are transmitted to each follower carriage through the distributed sliding mode controller. Then, the ESMDO feeds back the compensation value of the disturbance observation to the controller, so as to offset the influence of the unknown combined disturbance on the follower carriage, and make the speed of each follower carriage of the high-speed train multiple units consistent with that of the virtual leader carriage.

[0053] S1 Establish a mathematical model of high-speed train multi-agent

[0054] The high-speed train system can be regarded as a multi-agent system composed of a virtual leader carriage 0 and follower carriages. The dynamic equation of the virtual leader carriage 0 is:

[0055] (2)

[0056] Among them, is the speed of the virtual leader carriage, is the displacement of the virtual leader car body. is the control input of the virtual leader car body.

[0057] From Equation (1), the th follower car body has the following dynamic model:

[0058] (3)

[0059] where is the displacement of the th follower car body, is the velocity of the th follower car body, is the mass of the th follower car body, is the control input of the th follower car body, ( ) is the non - linear coupling force of the th follower car body, ( ) is the unknown composite disturbance of the th follower car body; is the connection weight between adjacent car bodies, indicates that there is information transfer between car body i and car body j, indicates that there is no information transfer between car body i and car body j;

[0060] Equation (3) can be further simplified as:

[0061] (4)

[0062] In the formula, , .

[0063] S2 Design the sliding - mode controller of the virtual leader car body

[0064] Design the sliding - mode controller of the virtual leader car body 0 to make the velocity of the virtual leader car body consistent with the target velocity function . The traditional virtual leader car body 0 generally uses a PID controller. To improve the robustness of the virtual leader car body controller, design the sliding - mode controller of the virtual leader car body 0.

[0065] Define the velocity error of the virtual leader car body 0 as

[0066] (5)

[0067] Among them, is the speed of the virtual leader car 0, is the displacement of the virtual leader car 0. is the control input of the virtual leader car 0.

[0068] It can be obtained from equation (8):

[0069] (6)

[0070] Select the sliding mode surface as

[0071] (7)

[0072] In the formula , are parameters to be designed.

[0073] Select the exponential reaching law

[0074] (8)

[0075] In the formula, , are parameters to be designed.

[0076] Therefore, the control law of the virtual leader car 0 can be obtained as:

[0077] (9)

[0078] When selecting the exponential reaching law,

[0079] (10)

[0080] the reachability condition of the sliding mode can be satisfied.

[0081] S3 Design the extended sliding mode disturbance observer of the th follower car to observe the disturbance

[0082] When the high-speed train is affected by unknown composite disturbances during actual operation, its state equation is:

[0083] (11)

[0084] Among them, , represents the state variable of the th follower car; represents the input signal of the th follower car; denotes the unknown composite disturbance signal received by the -th follower car, which is an unknown bounded function, i.e., , where D is a bounded quantity; , and are the coefficient matrices of the -th follower car;

[0085] Expand the unknown composite disturbance into a state variable. From Equation (11), the extended state equation of the high-speed train can be obtained as:

[0086] (12)

[0087] where is the rate of change of the unknown composite disturbance.

[0088] Select the sliding mode surface as , where is the observation error of the system state variable; for the system in (14), design the extended sliding mode disturbance observer for the -th follower car as

[0089] (13)

[0090] where is the estimated value of ; is the estimated value of the unknown composite disturbance ; represents the rate of change of the estimated value of the unknown composite disturbance ; is the sliding mode term to be designed; is the observer gain to be designed;

[0091] From Equations (12) and (13), the dynamic error equation of the extended sliding mode disturbance observer can be obtained as:

[0092] (14)

[0093] where: is the observation error of the system state variable; is the observation error of the unknown composite disturbance signal.

[0094] Select the reaching law as

[0095] (15)

[0096] where and are the designed switching gain and exponential term coefficient, respectively; is the sign function.

[0097] Combining equations (14) and (15), the sliding mode control law for designing the extended sliding mode disturbance observer is:

[0098] (16)

[0099] When the sliding mode surface is selected, the control law in (16) is selected, and the gains , and are selected, the state error equation will converge to 0 in finite time.

[0100] Select the following Lyapunov function :

[0101] (17)

[0102] And take the derivative of , and substitute the control law in (16) to get:

[0103] (18)

[0104] To ensure , the parameter should satisfy:

[0105] (19)

[0106] It can be seen from equation (19) that when appropriate gains are selected, , the extended sliding mode disturbance observer is asymptotically stable, and the system will reach the sliding mode surface in finite time, and there is .

[0107] Therefore, when appropriate parameters , and are selected, the designed observer is asymptotically stable and convergent. Therefore, the state error of the extended sliding mode disturbance observer designed in this embodiment will converge in finite time.

[0108] From this, the estimated value of the unknown composite disturbance of the follower car body can be obtained as:

[0109] (20)

[0110] where , represent the parameters to be designed for the follower car body.

[0111] S4 Design of the Distributed Speed Cooperative Sliding Mode Controller for the Follower Carriage Section

[0112] Let the displacement error between the -th follower carriage and the virtual leader carriage be ; Let the speed error between the -th follower carriage and the virtual leader carriage be ;

[0113] Select the displacement sliding mode surface of the -th follower carriage as ; Select the sliding mode surface of the speed of the -th follower carriage ;

[0114] Feed the estimated value of the unknown composite disturbance observed by the extended sliding mode disturbance observer back to the distributed sliding mode controller;

[0115] Design the distributed speed cooperative sliding mode controller for the -th follower carriage as follows:

[0116] (21)

[0117] (22)

[0118] (23)

[0119] In equations (21)-(23), is the distributed consensus control algorithm, is the sliding mode control term. is the mass of the -th follower carriage; is the connection weight between adjacent carriages, indicates that there is information transfer between carriage i and carriage j, indicates that there is no information transfer between carriage i and carriage j; , , are parameters to be designed; represents the state variable of the -th follower carriage; represents the state variable of the -th follower carriage; is the speed of the -th follower carriage, is the The speed of the follower carriages; is the sign function; is a parameter to be designed, .

[0120] Introducing a sliding mode term into the distributed sliding mode control algorithm of the follower carriages can enhance the robustness of the system, and the introduction of the observed value of the unknown composite disturbance eliminates its impact on the system.

[0121] Assume that there is information exchange between any two follower carriages, its communication topology graph is an undirected graph, and each follower carriage can obtain the output signal of the virtual leader carriage. Then the Laplacian matrix is a symmetric positive semi - definite matrix.

[0122] Let the control input of the distributed speed coordination sliding mode controller of the follower carriages in the high - speed train be in Equation (21). If the inequality is satisfied, and and satisfy , , When the speed of the

[0123] Let , , . Substituting Equation (24) into Equation (6) gives

[0124] (24)

[0125] Rewriting Equation (24) into matrix form gives:

[0126] (25)

[0127] where, x ˜ = [ x ˜ 1 , x ˜ 2 , … , x n ] , v ˜ = [ v ˜ 1 , v ˜ 2 , … , v ˜ n ] , ε = [ d ˜ 1 m 1 − v ˙ 0 , d ˜ 2 m 2 − v ˙ 0 , … d ˜ n m n − v ˙ 0 ] ; , is a diagonal matrix.

[0128] Define the error vector X e = [ x ˜ T , v ˜ T ] , and select the following Lyapunov function:

[0129] (26)

[0130] Taking the derivative of Equation (26) gives:

[0131] (27)

[0132] because , so when When , we can get:

[0133] (28)

[0134] in, For the matrix The minimum eigenvalue of . Because is a symmetric positive definite matrix, so , so , then the system is globally stable. X = [ x ˜ v ˜ ] T = 0 , follower carriage Speed ​​with Virtual Navigator Synchronous, that is .

[0135] Next, the distributed sliding mode controller system of high-speed train based on extended sliding mode disturbance observer (ESMDO) was modeled and simulated. Figure 1 The synchronization of speed and displacement in two cases, distributed sliding mode controller without ESMDO and distributed sliding mode controller with ESMDO, was analyzed by MATLAB simulation to verify the control performance of the proposed algorithm. The carriage parameters in the high-speed train EMU shown in Table 1 meet the requirements of CRH high-speed train. Four carriages were selected to verify the effectiveness of the algorithm.

[0136] Table 1 High-speed train compartment parameters

[0137] parameter numerical value unit <![CDATA[m i , i = 1, 2, 3, 4]]> 8000 kg <![CDATA[k 0i > 8000 N / m <![CDATA[a 3i > 0.01176 N / kg <![CDATA[a 2i > 0.00077616 Ns / mkg <![CDATA[a 1i > 0.00016 <![CDATA[Ns 2 / m 2 kg]]>

[0138] In addition, in order to simulate the complexity of high-speed trains during operation, four types of signals, namely uniform signal, sudden change signal, slowly changing signal and high-frequency signal, are selected as unknown composite disturbances. Carriage 1 is subject to uniform disturbance, car 2 is subject to slowly changing disturbance, car 3 is subject to high-frequency disturbance and car 4 is subject to sudden change disturbance.

[0139] In order to better simulate the actual operation of high-speed trains, three different stages are set: acceleration, constant speed, and braking. Set the reference speed signal of virtual navigator 0 for:

[0140]

[0141] Reference speed signal for high-speed trains It is only transmitted to the virtual leader car 0, and the follower cars 1, 2, 3, and 4 track the speed of the virtual leader car 0. Select the follower cars for information exchange, and at this time, the connection weights between adjacent follower cars , and the communication topology diagram between multiple agents is as Figure 2 shown.

[0142] From Figure 3 and Figure 4 , it can be seen that when the ESMDO is not introduced to estimate the unknown composite disturbance during the acceleration, constant speed, and braking processes, there are relatively large tracking errors between each follower car and the virtual leader car 0 during the acceleration stage, constant speed stage, and braking stage.

[0143] From Figure 5 , it can be seen that after the ESMDO is introduced, the synchronization rate of the displacement of the follower cars of the high-speed train reaches 99.9%. From Figure 6 and Figure 7 , it can be seen that during the acceleration stage of the high-speed train, the speed of the follower cars starts to accelerate from 0, and the speed errors between the follower cars and the leader car 0 are all kept within 0.0021%. During the constant speed stage of the high-speed train, the speed errors between the follower cars and the leader car 0 are all kept within 0.0025%. During the deceleration braking stage of the high-speed train, the speed error between the follower cars and the leader car 0 is kept within 0.0029%, meeting the speed requirements for the actual operation of the high-speed train.

[0144] In order to test the accuracy of the extended sliding mode disturbance observer, Figure 8 for the observation effect of the uniform disturbance received by follower car 1, Figure 9 for the observation effect of the slow-varying disturbance of follower car 2, Figure 10 for the observation effect of the high-frequency disturbance of follower car 3, Figure 11 for the observation effect of the mutation disturbance of follower car 4.

[0145] From Figures 8 to 11 , it can be seen that for the above different types of unknown composite disturbances, the designed ESMDO reaches the stable value at 0.002s, 0.003s, 0.053s, and 0.0066s respectively; after the observation effect of the ESMDO reaches stability, its steady-state errors are 0.301%, 0.649%, 0.532%, and 0.752% respectively, meeting the designed performance requirements.

[0146] In summary, the distributed speed cooperative sliding mode control of high-speed trains based on the extended sliding mode disturbance observer can effectively solve the problem of speed asynchrony between train carriages caused by unknown composite disturbances in the high-speed train system. By using the ESMDO to provide real-time feedback on the unknown composite disturbances experienced during the operation of high-speed trains, the impact of unknown composite disturbances on high-speed trains is effectively eliminated, the convergence time of speed errors between carriages is shortened, and the anti-interference ability of the system is improved. By replacing the traditional PID controller of the virtual leader carriage with a sliding mode controller and adopting a new distributed speed cooperative sliding mode control algorithm for the follower carriages, the stability of train operation is enhanced.

[0147] The above-described embodiments are merely preferred embodiments cited to fully illustrate the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or transformations made by those skilled in the art on the basis of the present invention are within the protection scope of the present invention. A distributed speed cooperative sliding mode control method for high-speed trains.

Claims

1. A distributed speed cooperative sliding mode control method for high-speed trains based on an extended sliding mode disturbance observer, characterized in that, it includes the following steps: Step 1, establish a multi-agent mathematical model of high-speed trains: Step 1.1, the high-speed train system can be regarded as a multi-agent system composed of a virtual leader car 0 and follower cars. The dynamic equation of the virtual leader car 0 is , where is the speed of the virtual leader car 0, is the displacement of the virtual leader car 0, is the control input of the virtual leader car 0; Step 1.2, the dynamic equation of the follower carriages is , where is the displacement of the -th follower carriage, is the velocity of the -th follower carriage, is the mass of the -th follower carriage, is the control input of the -th follower carriage, ( ) is the non - linear coupling force of the -th follower carriage, ( ) is the unknown composite disturbance of the -th follower carriage; is the connection weight between adjacent follower carriages, indicates that there is information transfer between carriage i and carriage j, indicates that there is no information transfer between carriage i and carriage j; Step 1.3, the dynamic model of the follower carriages can be simplified as: , where , ; Step 2, design a sliding mode controller for the virtual leader car 0 Step 2.1, define the state error of the virtual leader car 0 as: , where is the actual speed of the virtual leader car 0, is the given speed of the virtual leader car 0; Step 2.2, select the sliding mode surface as , where is the parameter to be designed; Step 2.3, select the exponential reaching law , where , is the parameter to be designed; Step 2.4, design the control law of the virtual leader car 0 as: ; where is the sign function; Step 3, design the extended sliding mode disturbance observer of the th follower car to observe the disturbance:​​​ Step 3.1, when the high-speed train is subject to unknown composite disturbances during actual operation, its state equation is: ; where , represents the state variable of the th follower car; represents the input signal of the th follower car; represents the unknown composite disturbance received by the th follower car, which is an unknown bounded function, that is , and D is a bounded quantity; , and are the coefficient matrices of the th follower car; Step 3.2, expand the unknown composite disturbance into state variables, and the extended state equation of the high-speed train can be obtained as follows: ; where is the unknown composite disturbance is the change rate of Step 3.3, select the sliding mode surface as , where is the observation error of the system state variable; design the extended sliding mode disturbance observer for the th follower car body as ; where is the estimated value of ; is the estimated value of the unknown composite disturbance ; represents the change rate of the estimated value of the unknown composite disturbance ; is the sliding mode term of the observer to be designed; is the observer gain to be designed; Step 3.4, from which the dynamic error equation of the extended sliding mode disturbance observer can be obtained as follows: ; is the observation error of the unknown composite disturbance signal; Step 3.5, select the reaching law as: ; where , are the designed coefficients respectively; is the sign function. Step 3.6, the sliding mode control law of the extended sliding mode disturbance observer is designed as: ; Step 4, design the distributed speed cooperative sliding mode controller for the section follower carriages: Step 4.1, let the displacement error between the th follower car body and the virtual leader car body 0 be ; let the speed error between the th follower car body and the virtual leader car body 0 be ; Step 4.2, select the sliding mode surface of the displacement of the th follower car body as ; select the sliding mode surface of the speed of the th follower car body ; Step 4.3, feed the estimated value of the unknown composite disturbance observed by the extended sliding mode disturbance observer back to the distributed sliding mode controller; Step 4.4, the designed distributed speed cooperative sliding mode controller for the section follower carriages is as follows: , where is the distributed speed cooperative control algorithm term, is the sliding mode control term; they are respectively ; where is the mass of the th follower car; is the connection weight between adjacent cars, indicating that there is information transfer between car i and car j, indicating that there is no information transfer between car i and car j; , , are parameters to be designed; represents the state variable of the th follower car; represents the state variable of the th follower car; is the speed of the th follower car, is the speed of the th follower car; is the sign function; , where is a parameter to be designed, .

2. The distributed speed cooperative sliding mode control method for high-speed trains based on an extended sliding mode disturbance observer according to claim 1, characterized in that, For the extended sliding mode disturbance observer, the gains , , and are selected. The state error will converge within a finite time. At this time, the estimated value of the unknown composite disturbance of the follower car body is: , where , represent the parameters to be designed for the follower car body.

3. The distributed speed cooperative sliding mode control method for high-speed trains based on an extended sliding mode disturbance observer according to claim 1, characterized in that, The distributed speed cooperative sliding mode controller of the high-speed train, when and and satisfy , , that is, the speed of the th follower car is synchronized with the speed of the virtual leader car 0.

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