Vibration analysis and multi-state feedback control of maglev vehicle-guideway coupling system

1.   Introduction

Maglev train has many advantages such as high speed, low noise, small vibration and low maintenance cost, which is the development direction of modern rail transit ^[1]. Different from the other wheel rail transit, maglev train relies on the electromagnet installed on the vehicle to suspend the car body above the track, so as to realize the non-contact support between the vehicle and the track. However, due to the need for active control to suspend the car body in the air, when the track is elastic, or the controller performance can't cooperate with the track well, the vehicle-track coupling vibration will occur randomly ^[2]. Vehicle-track coupled vibration is an important technical problem in maglev train suspension system.

In the test, the vehicle-track coupling vibration often occurs on different tracks. When TR04 of Germany and HSST04 of Japan are suspended on the steel frame bridge, the coupling vibration phenomenon of vehicle and track has been found ^[3]. AMT maglev train in the United States has good levitation performance in Edgewater test center, but after moving to Old Dominion University, vehicle and track coupling vibration occurs ^[4]. When TR08 of Germany is suspended on the steel structure track in the depot of the maintenance base, or running at a lower speed, it is prone to severe vehicle -track coupling vibration. However, when TR08 is running on the main line with a density of 7 ton/ m, the coupling vibration rarely occurs ^[5,6]. In engineering, although the possibility of coupling vibration can be reduced by adding up the mass and stiffness of the track, the cost of the system is significantly increased.

At present, many scholars have studied the mechanism and characteristics of vehicle-track coupling vibration. Kim et al. ^[7] studied the dynamic behavior of the maglev vehicle with elastic track when it was stationary or running at low speed through numerical simulation. Zhang et al. ^[2] studied the dynamic characteristics of magnetic levitation system with elastic guideway and velocity signal delay, and pointed out that the track elasticity is an important factor to induce the coupling vibration of the system. Wang et al. ^[8] proposed an on-line optimization method of control structure to suppress vibration aiming at the problem of track deformation and irregularity. Zhou et al. ^[9] proposed an adaptive vibration compensation method, which significantly reduces the vibration amplitude. Xu et al. ^[10] established the mathematical model of maglev system, and studied the dynamic behavior of maglev train on elastic track considering the state delay of position signal and speed signal of the electromagnet.

It is not difficult to find that in previous studies, some studies ignored the role of suspension controller and focused on the internal relationship between track elasticity and coupled vibration. While, some researchers only regard the track elasticity as an external disturbance, ignoring the coupling effect of the track structure and of the suspension control ^{[11,12,13,14]}. In fact, the vehicle-track coupling vibration of maglev system is closely related to the structure parameters of the track and control parameters of the controllers ^[11,15,16].

In order to suppress the vehicle-track coupling vibration of maglev system with flexible track, this paper propose a nonlinear stability analysis method and a multi-state feedback control method. The contributions of this paper are summarized in the following two aspects.

1) A stability analysis method for the vehicle-track coupling vibration of maglev system is presented. This method transforms the stability problem of the nonlinear maglev system with flexible track into an equivalent linear system stability problem, and the coupling vibration frequency of the system can be calculated.

2) A multi-state feedback control method with gap differential signal against the vehicle-track coupling vibration of maglev system is proposed. This method introduces the flexible track information to the control system, the vehicle-track coupling vibration of maglev train can be effectively avoided.

The remainder of this paper is organized as follows. In Section 2, the vehicle-track coupling system model considering the flexible guideway is built. In Section 3, a four-state feedback control algorithm with gap differential signal is proposed. In Section 4, the stability analysis method for the nonlinear vehicle-track coupling system is presented. In Section 5, the effectiveness of the proposed vibration control method is verified by some simulation experiments. In this paper, by adjusting some key control parameters, the dynamic evolution behavior between the maglev train and the elastic track is comprehensively investigated, which can provide a reference for suppressing the vehicle-track coupling vibration.

2.   Vehicle-track coupling suspension system modeling considering elastic track

The maglev train with electromagnetic attraction relies on the electromagnetic force provided by some electromagnets to suspend and support the carriage. Due to the distributed control of each electromagnet, the mechanical decoupling can be realized to a great extent, so the coupling effect between the electromagnets is greatly reduced. In order to simplify the research and clarify the basic characteristics of suspension system of maglev train, the model of the suspension control system with a single electromagnet is established ^[17,18,19].

Considering the elastic mode of track, the suspension control system of single electromagnet mainly includes a suspension electromagnet group, an elastic track and a suspension controller. It is the smallest unit of the suspension control system of maglev train, and can reflect the basic dynamic characteristics of the interaction between the vehicle and track ^[20,21].

Figure 1 is the schematic diagram of the vehicle-track coupling system of maglev train with a single electromagnet. In Figure 1, $O$ is the left fulcrum of the track, $OX$ is the running direction of the train; $OZ$ is the direction of gravity acceleration, ${z}_{1}$ is the vertical displacement of the electromagnet, ${z}_{g}$ is the vertical displacement of the track; $m$ is the mass of the electromagnet, $M$ is the mass of the carriage and passengers, $l$ is the distance between the center of the electromagnet and the origin $O$ in the ox direction $OX$. The displacement sensor is installed on the electromagnet, which measure the displacement between the track and the electromagnet as ${z}_{1}-{z}_{g}$. ${F}_{e}$ is the attractive force of the electromagnet.

Due to the influence of track joint, track irregularity or communication error code, the displacement signal is often interrupted. Because of the discontinuity of the displacement signal, the control difficulty of suspension system of the maglev train increases greatly, and it is easy to cause the failure of the control system or the vehicle-rail coupling vibration.

2.1.   Model of the elastic track

When the elastic mode of the track is considered, any point ${z}_{g}(l, t)$ of the beam satisfies the Bernoulli-Euler dynamic equation ^[3,11,17].

where, the parameters of the beam are given as below. ${E}_{g}$ is the modulus of elasticity, ${I}_{g}$ is the section inertia, ${T}_{g}$ is the tension, ${\rho }_{g}$ is the mass linear density, ${b}_{g}$ is the equivalent damping coefficient, ${k}_{g}$ is the equivalent elastic coefficient of the beam, and $p(l, t)$ is the distributed load density on the beam.

Combined with the test phenomenon, when the vehicle-track coupling vibration occurs, the vibration frequency of the track is in a certain frequency band. Because the excitation of the high-order vibration of the track needs a lot of energy, it is generally considered that the vehicle-track coupling vibration is mainly related to the first-order mode of the track.

The modal analysis method is adopted, in which only the first mode of the track is considered. Let

where, ${\phi }_{1}\left(l\right)$ is the first mode function, ${q}_{1}\left(t\right)$ is the generalized coordinates of the beam.

Here, ${l}_{g}$ is the span of the beam, ${m}_{g}$ is the mass of the track. It is known that the normalized mode function ${\phi }_{1}\left(l\right)$ is

The natural frequency of the first mode of the track is

The first order generalized force exerted on the track is

where, ${\eta }_{1}$ is the damping ratio of the first mode.

Assuming that the length of electromagnet is far less than that of track, the electromagnetic force on the track can be regarded as a concentrated force. At this time, there is

By combining the above Eqs (1)–(6), the coupling vibration model considering only the first mode of the track is obtained as follows

2.2.   Model of the maglev system

The attractive force of the electromagnet is

where, ${\mu }_{0}$ is the vacuum permeability, N is the number of turns of the coil, i is the coil current, and A is the area of magnetic pole.

Let ${C}_{1} = \frac{{\mu }_{0}{N}^{2}A}{4}$. The voltage of the electromagnet is

where, $R$ is the coil resistance.

Because the carriage and electromagnet rely on the air spring to realize the elastic support, when the train is stationary and suspended, only the vertical movement of the electromagnet is considered. Then

2.3.   Vehicle-track coupling system model

Let ${C}_{2} = \frac{2{C}_{1}}{{\rho }_{g}{l}_{g}}{\mathit{sin}}^{2}(\frac{\pi x}{{l}_{g}})$. By combining the above Eqs (7)–(10), the vehicle-track coupling system model considering only the first elastic mode of the track is obtained ^[3,11,17],

Select $x = ({x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}{)}^{T} = ({z}_{1}, {\dot{z}}_{1}, {z}_{g}, {\dot{z}}_{g}, i{)}^{T}$. The open-loop state space equation of the suspension system is obtained

Obviously, the system (12) is a nonlinear system, and its stability should be analyzed according to the nonlinear system method.

3.   Design of the multi-state feedback control law

The suspension control system (12) is a typical multi-variable system. In engineering, the PID control and multi-variable feedback control is usually used ^[5].

3.1.   Design of multi-state feedback control law

In the past, the three-state combined feedback control law was often used as follows ^[2,15,17],

where, ${u}_{ec}$is the initial voltage of electromagnet, ${k}_{s}$is the gap feedback coefficient, ${k}_{c}$is the current feedback coefficient, and ${k}_{bi}$is the coefficient of the electromagnet acceleration signal after integration.

The above control law does not contain the track information, so it has some limitations in suppressing vehicle-track coupling vibration ^[2,5]. In this paper, the gap differential signal is introduced into the control algorithm, and the feedback coefficient is ${k}_{sd}$, which is equivalent to adding the track information in the feedback control. The four-state feedback control law is

It can be seen from (14) that when the above control law is applied to the suspension control system, it needs to be able to obtain the five state variables${x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}$.

In the actual suspension system, the displacement sensor ${S}_{1}$, current sensor ${S}_{2}$and accelerometer ${S}_{3}$are installed on the electromagnet, as shown in Figure 2. Here, the displacement sensor ${S}_{1}$can measure the displacement ${x}_{1}-{x}_{3}$ between the electromagnet and the track. The current sensor ${S}_{2}$ can directly measure the current ${x}_{5}$ of the electromagnet by a low pass filter. For the accelerometer ${S}_{3}$, the velocity signal ${x}_{2}$ of the electromagnet is obtained by a high pass filter and an integrator. Although the signals ${x}_{1}, {x}_{3}$ can not be obtained directly by the above sensors, we can acquire the differential signals ${x}_{2}-{x}_{4} = {\dot{x}}_{1}-{\dot{x}}_{3}$ indirectly by a tracking differentiator in Figure 2.

3.2.   Design of the tracking differentiator

In engineering, it is very important to extract the differential signal from the original signal. For the magnetic suspension system, whether using PID control or multi-state feedback control, we both need to use the appropriate differential signal.

In the past magnetic levitation system, the differential signal extracted from the relative displacement of electromagnet and track can not be used directly. The main reason is the obvious increase of noise.

Assume that the original signal is $v$. The traditional differential signal is obtained by the following differential link

where, T is a relatively small time constant.

In order to reduce the noise amplification effect, in Figure 2, an improved tracking differentiator is used to extract the differential signal ^[15,16].

where, ${\tau }_{1}, {\tau }_{2}$ are relatively small time constant, and $0 < {\tau }_{1} < {\tau }_{2}$.

Here, $T = 0.001, {\tau }_{1} = 0.01, {\tau }_{2} = 0.02, N = 2000, v = \mathit{sin}t+0.0002rand(1, N)$. Using (15) and (16), the corresponding differential signals are obtained respectively, as shown in Figures 3.

4.   Stability analysis

4.1.   Stability analysis of closed loop system

By substituting the state feedback (14) into the open-loop state system (12), the following closed-loop state space equation of the system is obtained ^[11,17]:

For the system (17), let $\dot{x} = ({x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}{)}^{T} = 0$, and define ${C}_{3} = \sqrt{\frac{(m+M)g}{{C}_{1}}}$, the singular point of the system can be obtained as follows

It is not difficult to find that the equilibrium point of the four-state feedback system with gap differential information is the same as that of the three-state feedback system without gap differential information. Therefore, the four-state feedback with gap differential information does not change the static operating point of the system.

At the equilibrium point, we have ${x}_{1}-{x}_{3} = {z}_{0}$. Further, the initial static voltage ${u}_{ec0}$ of the electromagnet is obtained as

In fact, its size determines the size of the suspension gap in the stable state.

At the equilibrium point ${x}_{0}$, the system (17) is linearized and the Jacobian matrix of the system is

In this way, the nonlinear system (17) at the equilibrium point can be expressed as

where, $O(x-{x}_{0})$ is a higher order infinitesimal.

Lemma 1. There is a nonlinear system (17) and its corresponding linear system (21) at the equilibrium point. If ${x}_{0}$ is an isolated singular point of system (17) and all eigenvalues of $J\left({x}_{0}\right)$ have non-zero real parts, it is called ${x}_{0}$ is a hyperbolic singular point of system (17). While, if some eigenvalues of $J\left({x}_{0}\right)$ have zero real parts, it is called ${x}_{0}$ is a non-hyperbolic singular point of system (17).

Theorem 1 ^[11,17]. (Hartman Grobman Theory) if ${x}_{0}$ is a hyperbolic singular point of system (17) and satisfies the condition.

Then the system (17) has the same topological structure as the corresponding linear system (21) at the isolated singular point ${x}_{0}$.

The above theorem is also called the first approximation theorem. According to the theorem, for the nonlinear vehicle-track coupling control system (17) of maglev train, if its equilibrium point is hyperbolic singularity, we only need to analyze its linear system at the equilibrium point, and then we can get the stability of nonlinear system (17) near the equilibrium point.

The characteristic polynomial of Eq (20) is

where, ${a}_{1} = 2{\eta }_{1}{\omega }_{1}-\frac{{z}_{0}({k}_{c}-R)}{2{C}_{1}}$, ${a}_{2} = {\omega }_{1}^{2}+\frac{{C}_{3}{k}_{bi}}{m}-\frac{{\eta }_{1}{\omega }_{1}{z}_{0}({k}_{c}-R)}{{C}_{1}}+\frac{{C}_{3}{k}_{sd}}{m}+\frac{{C}_{2}{C}_{3}{k}_{sd}}{{C}_{1}}$, ${a}_{3} = \frac{{C}_{3}{k}_{s}}{m}+\frac{{C}_{2}{C}_{3}^{2}({k}_{c}-R)}{{C}_{1}}+\frac{{C}_{2}{C}_{3}{k}_{s}}{{C}_{1}}+\frac{2{C}_{3}{\eta }_{1}{\omega }_{1}{k}_{bi}}{m}$$-\frac{{\omega }_{1}^{2}{z}_{0}({k}_{c}-R)}{2{C}_{1}}+\frac{{C}_{3}^{2}{k}_{s}}{m}+\frac{2{C}_{3}{\eta }_{1}{\omega }_{1}{k}_{sd}}{m}$, ${a}_{4} = \frac{2{C}_{3}^{2}{\eta }_{1}{\omega }_{1}({k}_{c}-R)}{m}+\frac{2{C}_{3}{\eta }_{1}{\omega }_{1}{k}_{s}}{m}+\frac{{C}_{3}{\omega }_{1}^{2}{k}_{bi}}{m}+\frac{{C}_{3}{\omega }_{1}^{2}{k}_{sd}}{m}$, ${a}_{5} = \frac{{C}_{3}^{2}{\omega }_{1}^{2}({k}_{c}-R)}{m}+\frac{{C}_{3}{\omega }_{1}^{2}{k}_{s}}{m}$.

Because of the complexity of the characteristic polynomial, it is difficult to obtain the analytical solution of the system directly. The stability of the system can be analyzed by Hurwitz criterion.

The Hurwitz determinant is constructed as follows

Lemma 2. (Hurwitz criterion) the linear system (21) corresponding to matrix (20) is asymptotically stable, that is, all eigenvalues of characteristic polynomial (23) have negative real parts, if and only if ${a}_{1} > 0, {a}_{2} > 0, {a}_{3} > 0, {a}_{4} > 0, {a}_{5} > 0$, and ${{\mathit{\Delta}} }_{2} = {a}_{1}{a}_{2}-{a}_{3} > 0$,

Lemma 2 shows that the linear system (23) corresponding to the nonlinear system (17) at the equilibrium point is asymptotically stable, which also means that the nonlinear system (17) is asymptotically stable near the equilibrium point.

4.2.   Analysis of vehicle-track coupling vibration

It is generally believed that when the vehicle-track coupling vibration occurs, the system is in a critical stable state, and the corresponding linear system may have a Hopf bifurcation point.

Theorem 2 ^[11,17]. The linear system (21) corresponding to the matrix (20) is in a critical stable state, that is, the characteristic polynomial (20) has a pair of pure imaginary roots $\pm j\omega$, and the other three characteristic roots all have negative real parts, if and only if ${a}_{1} > 0, {a}_{2} > 0, {a}_{3} > 0, {a}_{4} > 0, {a}_{5} > 0$, and ${{\mathit{\Delta}} }_{2} = {a}_{1}{a}_{2}-{a}_{3} > 0$, ${{\mathit{\Delta}} }_{4} = \left|\begin{array}{cccc}{a}_{1} & 1 & 0 & 0\\ {a}_{3} & {a}_{2} & {a}_{1} & 1\\ {a}_{5} & {a}_{4} & {a}_{3} & {a}_{2}\\ 0 & 0 & {a}_{5} & {a}_{4}\end{array}\right| = 0$. Here, $\omega = \sqrt{\frac{{a}_{2}{a}_{5}}{{a}_{3}}}$.

From Theorem 2, the possible Hopf bifurcation point of the system can be obtained.

5.   System simulation

The parameters of the suspension system of a maglev train are as follows.

$m = 500kg, M = 1000kg, R = 4\mathrm{\Omega }, {\eta }_{1} = 0.005$, ${\omega }_{1} = 50\pi, {C}_{1} = 0.0026, {C}_{2} = 4e-8, l_{g} = 25m$, ${z}_{0} = 0.010m, {l}_{e} = 3m.$ The initial value is $x = (\mathrm{0.020, 0}, \mathrm{0.0002, 0}, 0)$.

1) The system is in critical stable state without gap differential signal.

At this time, select the control parameter as ${k}_{c} = -40, {k}_{bi} = 800, {k}_{s} = \mathrm{172,321}, {k}_{sd} = 0$, and get the response curve of the system, as shown in Figure 4. Figure 4(a) is the displacement curve of electromagnet, Figure 4(b) is the displacement curve of track, Figure 4(c) is the relative displacement curve of electromagnet and track, and Figure 4(d) is the phase trajectory diagram of relative displacement of electromagnet and track.

It can be seen from Figure 4 that when ${k}_{s} = \mathrm{172,321}, {k}_{sd} = 0$, the suspension system was in a critical stable state at that time. It can be seen from Figure 4(c) that the relative displacement between the electromagnet and the track is in a state of constant amplitude oscillation. If the control is not adopted effectively in time, the coupled vibration will occur continuously. From Theorem 2, we can get ${a}_{1} > 0, {a}_{2} > 0, {a}_{3} > 0, {a}_{4} > 0, {a}_{5} > 0$, and ${{\mathit{\Delta}} }_{2} = {a}_{1}{a}_{2}-{a}_{3} > 0$. At that time, when ${{\mathit{\Delta}} }_{4} = 0$, we get ${\omega }_{0} = 50.3\pi$. The value is very close to the first-order frequency of the track and the simulation result in Figure 4(b). It is not difficult to find that the coupling vibration frequency is ${\omega }_{0} = 50.1\pi$.

2) The system is in a stable state without gap differential signal.

At this time, the control parameter is selected as ${k}_{c} = -40, {k}_{bi} = 800, {k}_{s} = \mathrm{179,035}, {k}_{sd} = 0$, and the response curve of the system is obtained, as shown in Figure 5. Figure 5(a) is the displacement curve of electromagnet, Figure 5(b) is the displacement curve of track, Figure 5(c) is the relative displacement curve of electromagnet and track, and Figure 5(d) is the phase trajectory diagram of relative displacement of electromagnet and track.

It can be seen from Figure 5 that, when ${k}_{s} = \mathrm{179,035}, {k}_{sd} = 0$, the suspension system can reach a stable working state at that time. From Figure 5(c) that after 1.4 seconds, the relative displacement fluctuation range between the electromagnet and the track is less than 2 mm, realizing stable suspension.

3) The system is in a stable state with gap differential signal.

At this time, the control parameter is selected as ${k}_{c} = -40, {k}_{bi} = 800, {k}_{s} = \mathrm{172,321}, {k}_{sd} = 200$, and the response curve of the system is obtained, as shown in Figure 6. Figure 6(a) is the displacement curve of electromagnet, Figure 6(b) is the displacement curve of track, Figure 6(c) is the relative displacement curve of electromagnet and track, and Figure 6(d) is the phase trajectory diagram of relative displacement of electromagnet and track.

It can be seen from Figure 6 that the suspension system was still in a stable working state at that time. As can be seen from Figure 6(c), after 0.5 seconds, the relative displacement fluctuation range between the electromagnet and the track is less than 1 mm, realizing stable suspension.

Notation: For the magnetic levitation system, the performance differences of different control methods are mainly reflected in the convergence speed and steady-state fluctuation range of the vertical displacement ${x}_{1}$ of the electromagnet. In Figure 5(a), with the three-state combined feedback control method, the convergence speed of ${x}_{1}$ is 1.6 s and steady-state fluctuation range of ${x}_{1}$ is 2 mm. While, in Figure 6(a), with the four-state improved feedback control method, the convergence speed of ${x}_{1}$ is 0.7 s and steady-state fluctuation range of ${x}_{1}$ is 1 mm.

Through the above analysis, it can be seen that the system may be in a critical stable state when there is no gap differential signal feedback, and the vehicle and track will continue to vibrate. After introducing the gap differential signal to the feedback control law, the regulation time of the system is shortened and the stability is improved.

6.   Conclusions and discussion

Aiming at the problem of vehicle-track coupling vibration in maglev train suspension system, a four-state feedback control algorithm is proposed in this paper. By investigating the dynamic evolution behavior of the interaction between structure parameters and control parameters of the suspension system, the vehicle-track coupling vibration of maglev train has been effectively avoided.

In order to fully describe the real dynamic characteristics of the system, this paper fully considers the elastic modal factors of the track, and establishes a coupling system model with elastic track. Furthermore, by using the first approximation theorem, the stability problem of the nonlinear maglev system is transformed into an equivalent linear system stability problem. Then, the gap differential signal is introduced to the feedback system, and a four-state feedback control strategy is adopted to the magnetic suspension system. According to the Hurwitz algebraic criterion, the coupling vibration frequency of the system can be calculated, and the coupling vibration of the system can be judged. The above research provides a useful reference for effectively suppressing the coupling vibration between vehicle and elastic track.

In this paper, only the first order vibration mode is considered. In fact, the results will be closer to the actual situation when the higher order vibration mode is considered. Further, the above four-state feedback control algorithm method will be applied to the practical maglev system to verify its feasibility.

Acknowledgments

The authors would like to thank the referees for their comments and suggestions which helped to improve the manuscript. This work was supported by the Scientific Research Key Project of Hunan Provincial Education Department (20A267).

Conflict of interest

The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.