Control method for small-radius steering of high-speed maglev train

Abstract

The invention relates to a control method for small-radius steering of a high-speed maglev train, and belongs to the field of maglev trains. According to the method, self-adaption and backstepping technologies are adopted, real-time stable control is conducted on a train guiding system, and a dynamic mathematical model of the high-speed maglev train guiding system under small-radius steering is established; constructing a state-space equation of the high-speed maglev train guide system under small-radius steering, and changing the state-space equation of the system into a matched lateral centrifugal interference control state equation of the high-speed maglev train guide system; and a high-speed maglev train lateral displacement self-adaptive backstepping controller under small-radius steering is designed, and real-time stable control over high-speed small-radius steering of the high-speed maglev train is achieved. According to the invention, the high-speed maglev train can pass through a small-radius curve at a high speed while the guide coil of the high-speed maglev train is ensured not to generate friction with the guide surface of the track, and the stable and reliable operation of the high-speed maglev train during small-radius steering is ensured.

Description

Technical Field

[0001] The invention relates to the field of maglev trains, and in particular to a control method for small-radius steering of a high-speed maglev train. Background Art

[0002] High-speed maglev trains have attracted extensive attention from researchers around the world due to their high transmission efficiency, good riding comfort, and low operating noise. They 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 maintain the stability of the train during driving. High-speed maglev trains reach a speed of 600km / h, and the extremely high operating speed greatly increases the safety requirements of high-speed maglev trains.

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

[0004] Due to the high nonlinearity, strong coupling and inherent instability of the guidance system, it is extremely challenging to achieve stable control. In the research of maglev train guidance control, 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 frictionless control between the train and the track.

[0005] To this end, a control method for small-radius steering of a high-speed maglev train of the present invention can ensure that the train passes through a flat curve with a small radius at a speed of 600 km / h, and at the same time, ensure that no friction between the guide coil and the guide surface of the track occurs during the operation of the high-speed maglev train. Summary of the invention

[0006] The main purpose of the present invention is to provide a control method for small-radius steering of a high-speed maglev train in view of the shortcomings and blanks of the existing technology. The present invention constructs a Lyapunov Function through a high-speed maglev train guidance adaptive backstepping control method, designs a virtual control rate of each order of a high-speed maglev train dynamic mathematical model, ensures that the high-speed maglev train will not have friction between the guide coil and the track guide surface during operation, designs an adaptive estimation compensation for the lateral centrifugal interference of the high-speed maglev train, offsets the lateral centrifugal interference suffered by the high-speed maglev train when turning at high speed through the design of the control rate, and ensures the smooth operation of the high-speed maglev train.

[0007] In order to achieve the above object, the present invention provides a control method for small-radius turning of a high-speed maglev train, comprising the following steps:

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

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

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

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

[0012] The dynamic mathematical model of the high-speed maglev train guidance system under small radius turning in step 1 is:

[0013]

[0014] Where x is the lateral displacement of the high-speed maglev train, is the second-order derivative of lateral displacement, i=l, r is the left and right sides of the high-speed maglev train, F d is the lateral centrifugal force on the high-speed maglev train during the turning process, μ0 is the vacuum 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, i i 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;

[0015] The state space equation of the high-speed maglev train guidance system under the small radius turning in step 2 is:

[0016]

[0017] 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.

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

[0019]

[0020] In the formula, They are z1, z2, z3, and F. d The first derivative of K z1 =Kx1R / L, K x1 =4K c i 02 / x0 3 , K z3 =-R / L, K x3 =-4K c i0 / x0 2 , f z4 =z4,μ o satisfy:

[0021]

[0022] Where g(x3,μ)=-R / Lx3+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 as follows:

[0024] A) The lateral displacement of the high-speed maglev train guidance system and its velocity, acceleration tracking error and first-order derivative are set as:

[0025]

[0026] 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 ;

[0027] B) Select Lyapunov Function

[0028]

[0029] The first-order derivative of formula (5) is:

[0030]

[0031] Assume that the virtual control rate of the lateral displacement velocity of the high-speed maglev train guidance system is:

[0032]

[0033] 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:

[0034]

[0035] C) For the e1e2 subsystem in equation (8), select the Lyapunov Function

[0036]

[0037] The first-order derivative of equation (9) is:

[0038]

[0039] 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:

[0040]

[0041] Then formula (10) can be rewritten as:

[0042]

[0043] Where κ2>0 is a constant and is the virtual control rate gain of the bias current of the high-speed maglev train guide coil.

[0044] D) For the e2e3 subsystem in equation (12), select the Lyapunov Function

[0045]

[0046] 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. The first-order derivative of equation (13) is:

[0047]

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

[0049]

[0050] 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:

[0051]

[0052] Then formula (14) can be rewritten as:

[0053]

[0054] 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.

[0055] In the previous formula, in the field of mathematics, adding one dot to a letter represents the first-order derivative of the letter, adding two dots represents the second-order derivative of the letter, and adding three dots represents the third-order derivative of the letter. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0057] Attached Figure 2 Output simulation waveforms for lateral centrifugal disturbance and lateral centrifugal disturbance adaptive rate.

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

[0059] Attached Figure 4 The output of adaptive backstepping controller and triangular carrier simulation curve for high-speed maglev train guidance.

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

[0061] Attached Figure 6 This is the simulation curve of the virtual control rate of the bias current of the high-speed maglev train guide coil.

[0062] Attached Figure 7 This is the simulation curve of the input current of the guide coil of the high-speed maglev train.

[0063] Among them, 1 is a train carriage, 2 is a track, 3 is a left guide coil, 4 is a right guide coil, 5 is a left guide surface of the track, and 6 is a right guide surface of the track. DETAILED DESCRIPTION

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

[0065] High-speed maglev train guidance system as attached Figure 1 As shown, 3 and 4 in 1 are facing 5 and 6 in 2 respectively. After the reference current i0 is passed through 3 and 4, 3 and 4 generate equal and opposite electromagnetic attraction forces on 5 and 6 respectively. At this time, the axial center line of 1 coincides with the center line of 2. When 1 is disturbed by the lateral centrifugal force during the small radius turning process, the control input μ changes accordingly, generating a bias current, so that the current i0 in 3 and 4 l ,i r Corresponding changes occur, and the electromagnetic attraction of 5 and 6 changes accordingly, generating a differential force, suppressing the lateral centrifugal force interference during the turning process, and maintaining the axial centerline of the train near the centerline of the track.

[0066] The present invention provides a method for controlling a small radius turning of a high-speed maglev train, which specifically comprises the following steps in order to enable the high-speed maglev train to pass through a small radius curve during high-speed travel:

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

[0068] The modeling process is as follows:

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

[0070] F i (i i ,δ i )=Ki i 2 / δ i 2

[0071] As attached Figure 1 As shown in the figure, during the turning process, the high-speed maglev train is subjected to gravity mg in the vertical direction and the guiding electromagnetic attraction F in the lateral direction. i (i i ,δ) and the lateral centrifugal force F1 generated during the journey, F+ is the positive direction specified by the modeling. During the journey, the mechanical equation of the high-speed maglev train in the guiding direction is:

[0072]

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

[0074] d(Li i ) / dt=μU dc -Ri i

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

[0076]

[0077] Introducing the high-speed maglev train guide air gap and horizontal displacement coordinate transformation:

[0078]

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

[0080]

[0081] 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, Fd is the lateral centrifugal force received by 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;

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

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

[0084]

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

[0086]

[0087] x, i is state x1~x3, x is output y, and the state space equation of the high-speed maglev train guidance system under small radius turning is:

[0088]

[0089] 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.

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

[0091] The state space equation (2) of the high-speed maglev train guidance system under the above small radius turning is linearized near the equilibrium point (x1 = 0, x3 = 0) as follows:

[0092]

[0093] In the formula, They are z1, z2, z3, and F. d The first derivative of K x1 =4K c i0 2 / x0 3 , K x3 =-4K c i0 / x0 2 , μ o satisfy

[0094]

[0095] Where g(x3,μ)=-R / Lx3+U dc / Lμ.

[0096] Introduce lateral centrifugal interference matching transformation:

[0097]

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

[0099]

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

[0101] Step 4: Design an adaptive backstepping controller for the lateral displacement of a high-speed maglev train under small-radius turning:

[0102] A) The lateral displacement of the high-speed maglev train guidance system and its velocity, acceleration tracking error and first-order derivative are set 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 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 ;

[0105] B) Select Lyapunov Function

[0106]

[0107] The first-order derivative of formula (5) is:

[0108]

[0109] Assume that the virtual control rate of the lateral displacement velocity of the high-speed maglev train guidance system is:

[0110]

[0111] 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:

[0112]

[0113] C) For the e1e2 subsystem in equation (8), select the Lyapunov Function

[0114]

[0115] The first-order derivative of equation (9) is:

[0116]

[0117] 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:

[0118]

[0119] Then formula (10) can be rewritten as:

[0120]

[0121] Where κ2>0 is a constant and is the virtual control rate gain of the bias current of the high-speed maglev train guide coil.

[0122] D) For the e2e3 subsystem in equation (12), select the Lyapunov Function

[0123]

[0124] 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. The first-order derivative of equation (13) is:

[0125]

[0126] Design the backstepping control rate of the high-speed maglev train guidance system:

[0127]

[0128] 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:

[0129]

[0130] Then formula (14) can be rewritten as:

[0131]

[0132] 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.

[0133] The present invention is further described below using a preferred embodiment.

[0134] Embodiment 1:

[0135] The parameters of the high-speed maglev train guidance system are as follows: the equivalent inductance of the guidance coil is L = 0.1387H, the equivalent resistance of the guidance coil is R = 2.77Ω, and the effective area of ​​the guidance coil magnetic pole is A = 0.0552m 2 The number of turns of the guide coil is N = 200 turns. The parameters and specifications of the guide 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 =200V, the vacuum magnetic permeability is μ0=4π×10-7, the steady-state air gap between the guide coil and the track guide surface is δ0=10mm, the steady-state current i0=10A, and the control target of the horizontal displacement of the train is set to 0.

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

[0137] F1 / 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 train speed, the simulation time is set to 70s. Figure 2 As shown by the dotted line, the lateral centrifugal interference that the train experiences during turning is:

[0139]

[0140] Among them, else represents the lateral centrifugal disturbance F during the simulation time. dOther moments not covered. That is, from 10s to 25s, the high-speed maglev train enters the first R8000 curve: from 10s to 14s, the train enters the curve entry transition curve, from 14s to 21s, the train enters the R8000 curve, from 21s to 25s, the train enters the curve exit transition curve; from 40s to 55s, the train enters the second R8000 curve, the extension direction of which is opposite to the previous one: from 40s to 44s, the train enters the curve entry transition curve, from 44s to 51s, the train enters the R8000 curve, and from 51s to 55s, the train enters the curve 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 small radius turning under lateral centrifugal interference. Figure 2 , Figure 3 , Figure 4 , Figure 5 , Figure 6 , Figure 7 shown.

[0142] Attached Figure 2 The solid line shows the adaptive estimated output of the adaptive backstepping controller for high-speed maglev train guidance under small radius turning. Figure 3 The solid line shows the lateral displacement of the high-speed maglev train guidance adaptive backstepping control system under small radius turning, and the dotted line shows the maximum displacement allowed between the effective area of ​​the magnetic pole of the high-speed maglev train guidance coil and the track guidance surface. Figure 4 The dashed line is the output signal of the adaptive backstepping controller for high-speed maglev train guidance, and the solid line is the triangular carrier of the drive circuit. Figure 5 The speed virtual control rate waveform of the lateral displacement of the high-speed maglev train is shown in the figure. Figure 6 This is the virtual control rate waveform of the bias current of the high-speed maglev train guide coil. Figure 7 It is the current waveform of the guide coil of the high-speed maglev train, where the solid line is the current waveform of the guide coil on the right side of the train, and the dotted line is the current waveform of the guide coil on the left side of the train.

[0143] It can be seen from the attached figure that when the high-speed maglev train enters the first transition curve, due to the lateral centrifugal force, Figure 3 As shown in the figure, the train lateral displacement waveform begins to decrease, indicating that the train begins to deviate to the right. Figure 2 , Figure 4 , Figure 5 , Figure 6As shown in the figure, the interference adaptive estimation waveform in the high-speed maglev train guidance adaptive backstepping controller begins to decrease, and the lateral centrifugal interference is estimated. The trend of the train lateral displacement speed virtual control rate waveform is opposite to the lateral displacement waveform, which reduces the influence of lateral centrifugal interference on the train. At the same time, the bias current tracking signal fluctuates according to the lateral deviation of the train, increasing the current of the right guide coil while reducing the current of the left guide coil to offset the influence of lateral centrifugal interference force when the high-speed maglev train turns at a speed of 600km / h.

[0144] The above results show that the method for small-radius steering control of a high-speed maglev train of the present invention ensures that the guide coil of the high-speed maglev train does not rub against the guide surface of the track while ensuring that the train can pass through a small-radius curve at high speed, ensuring that the high-speed maglev train can operate reliably when passing through a small-radius curve at high speed. The system has good anti-interference ability and improves the operating 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 z3 = -R / L, μ o satisfy: where \(g(x_3,\mu)=-\frac{R}{L}x_3 + U\) 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. 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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