A control method for small-radius steering of a high-speed maglev train

Abstract

The present invention relates to a control method for small-radius steering of a high-speed maglev train, belonging to the field of maglev trains. This method adopts adaptive and backstepping techniques to perform real-time smooth control on the train's guiding system, establishes a dynamic mathematical model of the high-speed maglev train's guiding system under small-radius steering; constructs the state-space equation of the high-speed maglev train's guiding system under small-radius steering, and transforms the system state-space equation into a matching lateral centrifugal interference control state equation of the high-speed maglev train's guiding system; designs a lateral displacement adaptive backstepping controller for the high-speed maglev train under small-radius steering to achieve real-time smooth control of the high-speed maglev train for small-radius steering at high speed. The present invention ensures that the guiding coil of the high-speed maglev train does not rub against the track guiding surface, while ensuring that the train can pass through small-radius curves at high speed, ensuring the smooth and reliable operation of the high-speed maglev train during small-radius steering.

Description

Technical Field

[0001] The present invention relates to the field of maglev trains, and particularly to a control method for small-radius turning of high-speed maglev trains. Background Art

[0002] High-speed maglev trains have attracted extensive attention from researchers around the world due to their advantages such as high transmission efficiency, good riding comfort, and low running noise, and have now become an indispensable part of the modern transportation system. Among them, the guidance control system of high-speed maglev trains is one of its most important subsystems, which can keep the train stable during running. The speed of high-speed maglev trains reaches 600 km / h, and the extremely high running speed greatly increases the safety requirements for high-speed maglev trains.

[0003] To ensure the safe operation of the train, the guidance control of high-speed maglev trains based on the guidance air gap is essential, which can maintain the coincidence of the central axis of the high-speed maglev train and the central axis of the track during operation, and avoid the friction between the guidance coil and the track guidance surface. Especially during the turning process, in order to ensure the safe operation of the high-speed maglev train, it is necessary to reduce the traveling speed to reduce the lateral centrifugal interference generated during the turning process. Therefore, the high-speed maglev train will extend the running time and reduce the running efficiency when decelerating through curves.

[0004] Due to the high nonlinearity, strong coupling, and essentially unstable characteristics of the guidance system, it is extremely challenging to achieve its stable control. In the research on the guidance control of maglev trains, traditional control methods such as PID control and state feedback control can maintain the guidance stability of the train, but as a system with extremely demanding transient performance requirements, traditional control methods cannot guarantee the frictionless control between the train and the track.

[0005] Therefore, a control method for small-radius turning of high-speed maglev trains according to the present invention can ensure that the train passes through a curve with a relatively small radius of the horizontal curve at a speed of 600 km / h. At the same time, it can ensure that there is no friction between the guidance coil of the high-speed maglev train and the track guidance surface during operation. Summary of the Invention

[0006] The main purpose of the present invention is: aiming at the existing technical deficiencies and gaps, the present invention provides a control method for small-radius turning of high-speed maglev trains. By using the guidance adaptive backstepping control method of high-speed maglev trains, a Lyapunov Function is constructed, and the virtual control law of each order of the dynamic mathematical model of high-speed maglev trains is designed to ensure that there is no friction between the guidance coil of the high-speed maglev train and the track guidance surface during operation. The lateral centrifugal interference adaptive estimation compensation of high-speed maglev trains is designed, and the lateral centrifugal interference received by the high-speed maglev train during high-speed turning is offset through the design of the control law, ensuring the smooth operation of the high-speed maglev train.

[0007] To achieve the above object, a control method for small-radius turning of a high-speed maglev train according to the present invention includes the following steps:

[0008] Step 1, construct a dynamic mathematical model of the guiding system of a high-speed maglev train under small-radius turning;

[0009] Step 2, construct a state-space equation of the guiding system of a high-speed maglev train under small-radius turning;

[0010] Step 3, construct a matching lateral centrifugal interference control state equation of the guiding system of a high-speed maglev train under small-radius turning;

[0011] Step 4, design a lateral displacement adaptive backstepping controller for a high-speed maglev train under small-radius turning;

[0012] The dynamic mathematical model of the guiding system of a high-speed maglev train under small-radius turning in the said Step 1 is:

[0013]

[0014] In the formula, x is the lateral displacement of the high-speed maglev train, is the second derivative of the lateral displacement, i = l, r represents the left and right sides of the high-speed maglev train, F d is the lateral centrifugal force received by the high-speed maglev train during the turning process, μ 0 is the vacuum magnetic permeability, N is the number of turns of the guiding coil of the high-speed maglev train, A is the effective area of the magnetic pole of the guiding coil of the high-speed maglev train facing the guiding surface of the track, m is the mass of the high-speed maglev train carried by a single pair of guiding coils, i r is the input current of the right guiding coil of the high-speed maglev train, i l is the input current of the left guiding coil of the high-speed maglev train, i i is the first derivative of the input current, x 0 is the air gap between the guiding coil and the guiding surface of the track when the guiding system of the high-speed maglev train is stable, μ is the PWM signal of the driving circuit of the guiding coil of the high-speed maglev train, U dc is the power supply voltage of the driving circuit of the guiding coil of the high-speed maglev train, L is the equivalent inductance of the guiding coil of the high-speed maglev train, R is the equivalent resistance of the guiding coil of the high-speed maglev train;

[0015] The state-space equation of the guiding system of a high-speed maglev train under small-radius turning in the said Step 2 is:

[0016]

[0017] In the formula, x 1 is the lateral displacement of the high-speed maglev train, x 2is the lateral displacement speed of the high-speed maglev train, x 3 is the bias current of the guide coil of the high-speed maglev train, is the first derivative of the lateral displacement, is the first derivative of the lateral displacement speed, is the first derivative of the bias current, y is the output of the guide system of the high-speed maglev train, i 0 is the reference current of the guide coil of the high-speed maglev train.

[0018] The matching centrifugal interference control state equation of the guide system of the high-speed maglev train under small-radius turning in step 3 is:

[0019]

[0020] In the formula, are respectively the first derivatives of z 1 , z 2 , z 3 , F d , K z1 = Kx 1 R / L, K x1 = 4K c i 02 / x 0 3 , K z3 = -R / L, K x3 = -4K c i 0 / x 0 2 , f z4 = z 4 , μ o satisfies:

[0021]

[0022] Among them, g(x 3 , μ) = -R / Lx 3 + U dc / Lμ;

[0023] The design steps of the adaptive backstepping controller for the lateral displacement of the high-speed maglev train under small-radius turning in step 4 are:

[0024] A) Set the tracking errors and first derivatives of the lateral displacement, its speed, and acceleration of the guide system of the high-speed maglev train as:

[0025]

[0026] In the formula, z 1dis the lateral displacement tracking target of the high-speed maglev train guidance system, z 2d is the lateral displacement speed tracking target of the high-speed maglev train guidance system, z 3d is the lateral displacement acceleration tracking target of the high-speed maglev train, are respectively, z 1d 、z 2d 、z 3d the first-order derivative of;

[0027] B) Select the Lyapunov Function

[0028]

[0029] Taking the first-order derivative of Equation (5) gives:

[0030]

[0031] Let the virtual control rate of the lateral displacement speed of the high-speed maglev train guidance system be:

[0032]

[0033] where κ 1 > 0 is a constant, which is the gain of the virtual control rate of the lateral displacement speed of the high-speed maglev train guidance system, then Equation (7) is rewritten as:

[0034]

[0035] C) For e 1 e 2 in the subsystem, select the Lyapunov Function

[0036]

[0037] Taking the first-order derivative of Equation (9) gives:

[0038]

[0039] where is the second-order derivative of z 1d Let the virtual control rate of the bias current of the high-speed maglev train guidance coil be:

[0040]

[0041] Then Equation (10) is rewritten as:

[0042]

[0043] where κ 2> 0 is a constant, which is the gain of the virtual control rate of the bias current of the guide coil of the high-speed maglev train.

[0044] D) For e in Equation (12) 2 e 3 For the subsystem, select the Lyapunov Function

[0045]

[0046] In the formula, λ > 0 is a constant, is the estimation error of the lateral centrifugal interference of the high-speed maglev train, is the estimated value of the lateral centrifugal interference of the high-speed maglev train. Take the first derivative of Equation (13) as:

[0047]

[0048] Design the backstepping control rate of the high-speed maglev train guide system as:

[0049]

[0050] In the formula, is the first derivative of, is z 1d the third derivative of, is the second derivative of, κ 3 > 0 is a constant, which is the gain of the backstepping control rate of the high-speed maglev train guide system; design the adaptive rate of the lateral centrifugal interference of the high-speed maglev train as:

[0051]

[0052] Then Equation (14) is rewritten as:

[0053]

[0054] Equations (15) and (16) constitute the backstepping adaptive controller for the lateral displacement of the high-speed maglev train under small-radius turning. Then, the system composed of Equations (2), (15), and (16) is a closed-loop system, and the closed-loop system is asymptotically stable.

[0055] In the previous formulas, in the field of mathematics, a dot above a letter represents the first derivative of the letter, two dots represent the second derivative of the letter, and three dots represent the third derivative of the letter. Description of the Drawings

[0056] Appendix Figure 1 is a schematic diagram of the high-speed maglev train guide system of the present invention.

[0057] Appendix Figure 2It is the simulation waveform of the lateral centrifugal interference and the output of the lateral centrifugal interference adaptability rate.

[0058] Appendix Figure 3 It is the simulation waveform of the lateral horizontal displacement and the maximum allowable limit of the high-speed maglev train.

[0059] Appendix Figure 4 It is the simulation curve of the output of the guidance adaptive backstepping controller and the triangular carrier wave of the high-speed maglev train.

[0060] Appendix Figure 5 It is the simulation curve of the virtual control rate of the lateral displacement speed of the high-speed maglev train.

[0061] Appendix Figure 6 It is the simulation curve of the virtual control rate of the bias current of the guidance coil of the high-speed maglev train.

[0062] Appendix Figure 7 It is the simulation curve of the input current of the guidance coil of the high-speed maglev train.

[0063] Among them, 1 is the train car body, 2 is the track, 3 is the left guidance coil, 4 is the right guidance coil, 5 is the left guidance surface of the track, and 6 is the right guidance surface of the track. Specific implementation manner

[0064] The present invention will be further described in detail below with reference to the accompanying drawings.

[0065] The guidance system of the high-speed maglev train is as shown in the appendix Figure 1 As shown, 3 and 4 in 1 are respectively facing 5 and 6 in 2. After passing the reference current i0 into 3 and 4, 3 and 4 respectively generate equal and opposite electromagnetic suction forces on 5 and 6. At this time, the axial center line of 1 coincides with the center line of 2; when 1 is subjected to lateral centrifugal force interference during the small-radius turning process, the control input μ changes accordingly, generating a bias current, so that the currents i l , i r change accordingly, the electromagnetic suction forces on 5 and 6 change accordingly, generating a differential force to suppress the lateral centrifugal force interference during the turning process, and maintaining the axial center line of the train near the center line of the track.

[0066] A method for controlling the small-radius turning of a high-speed maglev train according to the present invention, in order to realize passing through a small-radius curve during the high-speed travel of the high-speed maglev train, specifically includes the following steps:

[0067] Step 1, establish a dynamic mathematical model of the guidance system of the high-speed maglev train under small-radius turning:

[0068] The modeling process is as follows:

[0069] As shown in the appendix Figure 1 As shown, the guiding electromagnetic suction force generated after the guiding coils on both sides of the train are energized is:

[0070] F i (i i , δ i ) = Ki i 2 / δ i 2

[0071] As shown in the appendix Figure 1 During the steering process of the high-speed maglev train, it is vertically affected by gravity mg, and laterally affected by the guiding electromagnetic suction force F i (i i , δ) and the lateral centrifugal force F 1 generated during the traveling process. F+ is the positive direction specified for modeling. During the traveling process, the mechanical equation in the guiding direction of the high-speed maglev train is:

[0072]

[0073] The voltage balance equation of the guiding coil is:

[0074] d(Li i ) / dt = μU dc - Ri i

[0075] In summary, the dynamic mathematical model of the high-speed maglev train guiding system is:

[0076]

[0077] Introduce the coordinate transformation of the guiding air gap and horizontal displacement of the high-speed maglev train:

[0078]

[0079] In summary, the dynamic mathematical model of the high-speed maglev train guiding system under small-radius steering is:

[0080]

[0081] where x is the lateral displacement of the high-speed maglev train, is the second derivative of the lateral displacement, i = l, r represent the left and right sides of the high-speed maglev train respectively, Fd is the lateral centrifugal force received by the high-speed maglev train during the small-radius steering process, μ 0 is the air magnetic permeability, N is the number of turns of the guiding coil of the high-speed maglev train, A is the effective area of the magnetic pole of the guiding coil of the high-speed maglev train facing the guiding surface of the track, m is the mass of the high-speed maglev train carried by a single pair of guiding coils, i r is the input current of the right guiding coil of the high-speed maglev train, i l is the input current of the left guiding coil of the high-speed maglev train, is the first derivative of the input current, x 0 is the air gap between the guide coil and the track guide surface when the high-speed maglev train guide system is stable, μ is the PWM signal of the drive circuit of the high-speed maglev train guide coil, U dc is the power supply voltage of the drive circuit of the high-speed maglev train guide coil, L is the equivalent inductance of the high-speed maglev train guide coil, R is the equivalent resistance of the high-speed maglev train guide coil;

[0082] Step 2, construct the state space equation of the high-speed maglev train guide system under small-radius turning:

[0083] Express the current of the high-speed maglev train guide coil as a linear superposition of the reference current and the bias current:

[0084]

[0085] Based on the above transformation and Equation (1), we get:

[0086]

[0087] Respectively with x, i as the state x 1 ~x 3 and with x as the output y, the state space equation of the high-speed maglev train guide system under small-radius turning is:

[0088]

[0089] In the formula, x 1 is the lateral displacement of the high-speed maglev train, x 2 is the lateral displacement speed of the high-speed maglev train, x 3 is the bias current of the high-speed maglev train guide coil, is the first derivative of the lateral displacement, is the first derivative of the lateral displacement speed, is the first derivative of the bias current, y is the output of the high-speed maglev train guide system, i 0 is the reference current of the high-speed maglev train guide coil.

[0090] Step 3, construct the matching centrifugal interference control state equation of the high-speed maglev train guide system under small-radius turning:

[0091] Linearize the above state space equation (2) of the high-speed maglev train guide system under small-radius turning near the equilibrium point (x 1 = 0, x 3 = 0) as:

[0092]

[0093] In the formula, They are respectively z 1 , z 2 , z 3 , F d 's first-order derivatives, K x1 = 4K c i 0 2 / x 0 3 , K x3 = -4K c i 0 / x 0 2 , μ o satisfies

[0094]

[0095] wherein, g(x 3 , μ) = -R / Lx 3 + U dc / Lμ.

[0096] Introduce the lateral centrifugal interference matching transformation:

[0097]

[0098] The matching centrifugal interference control state equation of the high-speed maglev train guidance system under small-radius turning is obtained as:

[0099]

[0100] In the formula, K z1 = K x1 R / L, K z3 = -R / L.

[0101] Step 4, design the lateral displacement adaptive backstepping controller of the high-speed maglev train under small-radius turning:

[0102] A) Set the lateral displacement, its speed, acceleration tracking errors and first-order derivatives of the high-speed maglev train guidance system as:

[0103]

[0104] In the formula, z 1d is the lateral displacement tracking target of the high-speed maglev train guidance system, z 2d is the lateral displacement speed tracking target of the high-speed maglev train guidance system, z 3d is the lateral displacement acceleration tracking target of the high-speed maglev train, They are respectively, z 1d , z 2d , z3d The first derivative of

[0105] B) Select the Lyapunov Function

[0106]

[0107] Taking the first derivative of Equation (5) gives:

[0108]

[0109] Let the virtual control rate of the lateral displacement velocity of the high-speed maglev train guidance system be:

[0110]

[0111] where κ 1 > 0 is a constant, which is the gain of the virtual control rate of the lateral displacement velocity of the high-speed maglev train guidance system. Then Equation (7) is rewritten as:

[0112]

[0113] C) For e in Equation (8) 1 e 2 subsystem, select the Lyapunov Function

[0114]

[0115] Taking the first derivative of Equation (9) gives:

[0116]

[0117] where is the second derivative of z 1d Let the virtual control rate of the bias current of the high-speed maglev train guidance coil be:

[0118]

[0119] Then Equation (10) is rewritten as:

[0120]

[0121] where κ 2 > 0 is a constant, which is the gain of the virtual control rate of the bias current of the high-speed maglev train guidance coil.

[0122] D) For e in Equation (12) 2 e 3 subsystem, select the Lyapunov Function

[0123]

[0124] where λ > 0 is a constant, is the estimation error of the lateral centrifugal interference of the high-speed maglev train, is the estimated value of the lateral centrifugal interference of the high-speed maglev train. The first derivative of Equation (13) is:

[0125]

[0126] The backstepping control law of the high-speed maglev train guidance system is designed as:

[0127]

[0128] where is the first derivative of, is the third derivative of z 1d , is the second derivative of, κ 3 > 0 is a constant, which is the gain of the backstepping control law of the high-speed maglev train guidance system; the adaptive rate of the lateral centrifugal interference of the high-speed maglev train is designed as:

[0129]

[0130] Then Equation (14) is rewritten as:

[0131]

[0132] Equations (15) and (16) constitute a backstepping adaptive controller for the lateral displacement of the high-speed maglev train under small-radius turning. Then, the system composed of Equations (2), (15), and (16) is a closed-loop system, and the closed-loop system is asymptotically stable.

[0133] The following uses a preferred embodiment to further illustrate the present invention.

[0134] Embodiment 1:

[0135] The parameters of the high-speed maglev train guidance system are as follows: the equivalent inductance value of the guidance coil is L = 0.1387 H, the equivalent resistance value of the guidance coil is R = 2.77 Ω, the effective area of the guidance coil magnetic pole is A = 0.0552 m 2 , the number of turns of the guidance coil is N = 200 turns, the parameter specifications of the guidance coils on both sides of the train are the same, the total mass of the guide body is m = 1995 kg, the power supply voltage of the drive circuit is U dc = 200 V, the vacuum permeability is μ 0 = 4π × 10-7, the steady-state air gap between the guidance coil and the track guidance surface is δ 0 = 10 mm, the steady-state current i 0= 10A, the control target of the train's horizontal displacement is set to 0.

[0136] The track curve radius R = 8000m, the train running speed v = 600km / h, the track cross slope angle θ = 10.02°, and the transition curve length is 666.67m are adopted for the track and operation data. According to the Figure 1 force analysis shown, the lateral centrifugal interference on the train after entering the R8000 curve should be:

[0137] F 1 / m = gsin(θ) - v 2 cos(θ) / R = -1.71m / s 2 ≈ -1.72m / s 2

[0138] Combined with the conditions of the transition curve and the train running speed, etc., the simulation time is set to 70s. As shown by the Figure 2 dashed line, the lateral centrifugal interference on the train during the turning process is:

[0139]

[0140] Among them, else represents other moments not involved in the lateral centrifugal interference F d during the simulation time. That is, from the 10th s to the 25th s, the high-speed maglev train enters the first R8000 curve: from the 10th s to the 14th s, the train enters the entry transition curve; from the 14th s to the 21st s, the train enters the R8000 curve; from the 21st s to the 25th s, the train enters the exit transition curve; from the 40th s to the 55th s, the train enters the second R8000 curve, and the extension direction of this curve is opposite to the previous one: from the 40th s to the 44th s, the train enters the entry transition curve; from the 44th s to the 51st s, the train enters the R8000 curve; from the 51st s to the 55th s, the train enters the exit transition curve.

[0141] According to the above simulation conditions, the system is simulated to verify the stable control ability of the high-speed maglev train guidance adaptive backstepping control system under lateral centrifugal interference during small-radius turning. As shown in Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 、 Figure 6 、 Figure 7 .

[0142] As shown by the Figure 2 solid line, the adaptive estimation output of the high-speed maglev train guidance adaptive backstepping controller under small-radius turning is shown, as shown in Figure 3The solid line shows the lateral displacement of the high-speed maglev train's guidance adaptive backstepping control system during a small-radius turn, and the dashed line is the maximum allowable displacement between the effective area of the magnetic poles of the high-speed maglev train's guidance coil and the track guidance surface. Attached Figure 4 The dashed line is the output signal of the high-speed maglev train's guidance adaptive backstepping controller, and the solid line is the triangular carrier wave of the drive circuit. Attached Figure 5 It is the waveform of the virtual control rate of the speed of the high-speed maglev train's lateral displacement. Attached Figure 6 It is the waveform of the virtual control rate of the bias current of the high-speed maglev train's guidance coil. Attached Figure 7 It is the waveform of the current of the high-speed maglev train's guidance coil. Among them, the solid line is the current waveform of the right-side guidance coil of the train, and the dashed line is the current waveform of the left-side guidance coil of the train.

[0143] Combined with the attached drawings, it can be seen that when the high-speed maglev train enters the first transition curve of the bend, due to the action of the lateral centrifugal force, as attached Figure 3 shown, the waveform of the train's lateral displacement begins to decline, indicating that the train begins to shift to the right. At the same time, as attached Figure 2 , Figure 4 , Figure 5 , Figure 6 shown, the waveform of the disturbance adaptive estimation in the high-speed maglev train's guidance adaptive backstepping controller begins to decline to estimate the lateral centrifugal disturbance. The trend of the waveform of the virtual control rate of the train's lateral displacement speed is opposite to the direction of the lateral displacement waveform, slowing down the influence of the lateral centrifugal disturbance on the train. At the same time, the bias current tracking signal fluctuates according to the lateral displacement of the train, increasing the current of the right-side guidance coil while decreasing the current of the left-side guidance coil to offset the influence brought by the lateral centrifugal disturbance force when the high-speed maglev train turns at a speed of 600 km / h.

[0144] The above results show that the method for controlling the small-radius turn of a high-speed maglev train in the present invention ensures that the guidance coil of the high-speed maglev train does not rub against the track guidance surface, while ensuring that the train can pass through small-radius curves at high speed, ensuring the reliable operation of the high-speed maglev train when passing through small-radius curves at high speed. The system has good anti-interference ability and improves the operation efficiency of the train.

Claims

1. A control method for small-radius turning of a high-speed maglev train, characterized in that: The method comprises: S1. Establish a dynamic mathematical model of the high-speed maglev train guidance system under small radius turning Where x is the lateral displacement of the high-speed maglev train, is the second-order derivative of the lateral displacement, i=l, r are the left and right sides of the high-speed maglev train respectively, F d is the lateral centrifugal force on the high-speed maglev train during the small radius turning process, μ0 is the air magnetic permeability, N is the number of turns of the high-speed maglev train guide coil, A is the effective area of ​​the magnetic pole of the high-speed maglev train guide coil facing the track guide surface, m is the mass of the high-speed maglev train carried by a single pair of guide coils, i r is the input current of the right guide coil of the high-speed maglev train, i l is the input current of the left guide coil of the high-speed maglev train, is the first-order derivative of the input current, x0 is the air gap between the guide coil and the track guide surface when the high-speed maglev train guide system is stable, μ is the PWM signal of the high-speed maglev train guide coil drive circuit, U dc is the power supply voltage of the high-speed maglev train guide coil driving circuit, L is the equivalent inductance of the high-speed maglev train guide coil, and R is the equivalent resistance of the high-speed maglev train guide coil; S2. Constructing the state space equation of the high-speed maglev train guidance system under small radius turning Where x1 is the lateral displacement of the high-speed maglev train, x2 is the lateral displacement speed of the high-speed maglev train, x3 is the bias current of the guide coil of the high-speed maglev train, is the first-order derivative of the lateral displacement, is the first-order derivative of the lateral displacement velocity, is the first-order derivative of the bias current, y is the output of the high-speed maglev train guidance system, and i0 is the reference current of the high-speed maglev train guidance coil; S3. Based on the state equation of formula (2), introduce the transformation: z1 = x1, z2 = x2, z3 = K x1 x1+K x3 x3+F d , Constructing the state equation of centrifugal disturbance control for high-speed maglev train guidance system under small radius turning In the formula, They are z1, z2, z3, and F. d The first derivative of K z1 =K x1 R / L, K x1 =4K c i0 2 / x0 3 , K z3 = -R / L, f z4 =z4,K x3 =-4K c i0 / x0 2 , μ o satisfy: where \(g(x_3,\mu)=-\frac{R}{L}x_3 + \frac{U}{L}\mu\); dc / L\mu; S4. Design an adaptive backstepping controller for the lateral displacement of a high-speed maglev train under small-radius turns.

2. According to claim 1, a control method for small-radius turning of a high-speed maglev train is characterized in that: The specific design steps of S4 are: S41. Set the lateral displacement of the high-speed maglev train guidance system and its velocity, acceleration tracking error and first-order derivative as: In the formula, z 1d is the lateral displacement tracking target of the high-speed maglev train guidance system, z 2d is the lateral displacement velocity tracking target of the high-speed maglev train guidance system, z 3d The lateral displacement acceleration tracking target of the high-speed maglev train. They are respectively 1d 、z 2d 、z 3d The first derivative of ; S42. Select Lyapunov Function V1: The first-order derivative of formula (5) is: Assume that the virtual control rate of the lateral displacement velocity of the high-speed maglev train guidance system is: Where κ1>0 is a constant, which is the virtual control rate gain of the lateral displacement velocity of the high-speed maglev train guidance system. Then, equation (7) can be rewritten as: S43. For the e1e2 subsystem in equation (8), select Lyapunov Function V2: The first-order derivative of equation (9) is: In the formula, Yes 1d The second-order derivative of , assuming that the virtual control rate of the high-speed maglev train guide coil bias current is: Then formula (10) can be rewritten as: Where, κ2>0 is a constant, which is the virtual control rate gain of the bias current of the high-speed maglev train guide coil; S44. For the e2e3 subsystem in equation (12), select Lyapunov Function V3: In the formula, λ>0 is a constant, is the estimation error of the lateral centrifugal disturbance of the high-speed maglev train, is the estimated value of the lateral centrifugal disturbance of the high-speed maglev train, and the first-order derivative of equation (13) is: Design the backstepping control rate of the high-speed maglev train guidance system: In the formula, for The first derivative of Yes 1d The third-order derivative of for The second-order derivative of κ3>0 is a constant, which is the gain of the backstepping control rate of the high-speed maglev train guidance system; the adaptive rate of the lateral centrifugal disturbance of the high-speed maglev train is designed as: Then formula (14) can be rewritten as: Equations (15) and (16) constitute the backstepping adaptive controller for the lateral displacement of a high-speed maglev train under small radius turning. The system composed of equations (2), (15) and (16) is a closed-loop system, and the closed-loop system is asymptotically stable.

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