Effect of levitation gap feedback time delay on the EMS maglev vehicle system dynamic response

EMS maglev train uses the active control system to maintain the levitation stability, time delay is widespread in control system. However, the existing maglev dynamics studies rarely consider the effect of time delay, so these analysis results cannot directly guide the engineering design. This paper starts from a theoretical analysis of the levitation stability of a single electromagnet levitation system to obtain the theoretical critical value for the time delay. Then the model is gradually extended to a complete vehicle model and a vehicle-girder coupling dynamics model to find the time delay engineering critical values for the complex coupling system. In order to seek ways to reduce the influence of time delay on the dynamic responses, this paper analyzes the influence regularities of the running speed and control parameters on the dynamic response under the effect of time delay. The result shows that the theoretical critical value of the time delay is equal to the ratio of the differential coefficient and proportional coefficient of the PID controller. For a complex maglev system, the engineering critical value is less than the theoretical critical value. Higher running speeds lead to time delay having a more obvious effect on the maglev system’s dynamic responses. Selecting the smaller proportional coefficient and appropriate differential coefficient for the levitation control system can expand the stability region and reduce the influence of time delay on the maglev system. This analysis is helpful and meaningful to the understanding of the EMS vehicle system stability, and helpful to explore the reason of violent coupled vibration in actual engineering.


Introduction
EMS maglev vehicles use electromagnetic attraction force between electromagnet and rail to achieve levitation and guidance, and use linear motors to achieve longitudinal traction. The EMS maglev vehicle has advantages of high operational safety, good ride comfort, low vibration and noise, better curve negotiation and climbing ability. In recent years, the world has made rapid developments with respect to these technologies [1][2][3]. But in almost every lowspeed maglev line operation, there are violent vehicleguideway (switch) coupling vibration phenomena, and in serious cases, these can lead to electromagnet and rail collisions and may even lead to levitation instability [4,5]. Therefore, the levitation stability and coupling vibration problems have been the focus of EMS maglev transportation research and a key but difficult problem for practical engineering.
Because EMS maglev systems are inherently unstable, a large number of early studies conducted stability analyses based on single levitation control systems, and these studies typically linearized the system at the rated operation point [4,[6][7][8]. Shi and She (2005) analyzed the bifurcation condition and stability of the cascade PID controller from the perspective of nonlinear dynamics, which shows that a small-amplitude vibration may occur in a wide range of control parameters and may induce vehicle-bridge coupling resonance, which also fully indicates that the levitation control system has strong nonlinear characteristics, and the PID control parameters must be carefully selected to avoid maglev vehicle levitation instability [9]. There are also many scholars working on adopting better levitation control strategies to improve the system's stability [10][11][12]. All these studies analyzed the nonlinear characteristics of the levitation control system itself but ignored the nonlinear characteristics caused by the time delays that are prevalent in each feedback channel. In reality, during the development and commissioning of maglev vehicles, there have indeed been violent vehicle-bridge coupling vibrations due to time delays in the control system [13,14].
In recent years, many scholars have analyzed the influence of time delay in the feedback channel of the levitation gap, gap variation speed, and gap variation acceleration on the system levitation stability from the perspective of nonlinear dynamics, based on more complex nonlinear dynamics theories such as Hopf bifurcation theory, central manifold theory, and the Poincare normal form. The results of these studies show that when the time delay is greater than a critical value, the levitation system will generate the Hopf bifurcation phenomenon, generating unstable vibrations or even levitation instability, and this critical value is closely related to the control algorithm and control parameters [14][15][16][17][18]. Li (2015) analyzed the influence of the time delay of each feedback channel on the stability of the system from the energy perspective, and the results pointed out that the levitation gap feedback (LGF) time delay is the most detrimental to the stability of the maglev vehiclebridge coupling system [19]. The above studies analyzed the stability of a single electromagnet levitation (SEL) system at the theoretical level but cannot reflect the influence of feedback channel time delay on the vehicle system dynamics under the combined action of multiple levitation control points, and the complex theoretical results cannot be directly applied to engineering practice. Wu and Zeng (2019) also focused on the effect of LGF time delay on the maglev vehicle-bridge coupling system. This study established a simplified 10-degrees-of-freedom maglev vehicle-girder dynamics model and conducted a numerical simulation, demonstrating that the LGF gain coefficient and velocity feedback gain coefficient have a large effect on the critical time delay value, while the secondary suspension stiffness, damping, and guideway stiffness have relatively small effects on the critical time delay value. This study only analyzed the effect of gap feedback time delay on the dynamic responses of the maglev vehicle system, however, without conducting an in-depth analysis of the causes of the effect [20].
Therefore, this paper first analyzes the causes underlying the impact of gap feedback time delay on the stability of the system, based on a third-order SEL system that considers the levitation gap feedback time delay and PD controller. Then a fifth-order SEL model taking into account levitation gap feedback time delay, current loop feedback, and the PID levitation controller is developed using Simulink software to further analyze the stability of a more complete SEL system, as well as to determine the necessary conditions for the stability of an SEL system under effect of LGF time delay. Then, based on the previously established levitation control system, the vehicle system dynamics model taking into account the levitation gap feedback time delay is established in the form of a Simulink and Simpack co-simulation, and the influence of the levitation gap feedback time delay on vehicle system dynamics is analyzed. On this basis, this paper analyzes the influence regularity of the running speed and levitation control parameters when considering the gap feedback time delay in order to seek ways to reduce the influence of the levitation gap feedback time delay on the system's dynamic responses.
In summary, the main contribution of this manuscript is three aspects. (1) Uses single electromagnet levitation system model to explain the reason of levitation instability caused by the time delay from theoretical perspective. Gives the theoretical critical value of time delay s c , and the relationship between the theoretical critical value control system parameters. (2) The time delay module is added into the EMS vehicle system dynamics simulation, the influence law of time delay on the vehicle system dynamics responses is analyzed, and the engineering critical value s e is given. (3) Propose some suggestions for reducing the detrimental influence of time delay on the system dynamics responses. This analysis is helpful and meaningful to the understanding of the EMS vehicle system stability, and helpful to explore the reason of violent coupled vibration in actual engineering, so as to reduce the violent coupled vibration from the perspective of levitation control system optimization designing.
2 Transfer function of a single electromagnet system considering time delay EMS maglev vehicles usually use several electromagnets to obtain sufficient levitation carrying capacity; the electromagnet windings in each levitation module are generally divided into two groups, corresponding to a set of independent levitation controllers, so an electromagnet system controlled by a single levitation controller is the basic unit of maglev vehicle systems. A schematic of this system, called a single electromagnet levitation (SEL) system in this paper, is shown in Fig. 1. The gap sensor first measures the levitation gap between the electromagnet and the rail, the deviation between measured levitation gap and rated value is used as the input signal to the controller [21]. The controller adjusts the electromagnet coil current through the control algorithm, and to change the electromagnetic force magnitude and maintain a stable levitation gap. According to Newton's theorem, the kinetic equations of the SEL system can be described as follows: where m is the levitation mass, z is the levitation gap between the electromagnet and rail, i is the current of coil winding, and F m (i,z) is the electromagnetic force. Based on the assumption as below: Assumption 1 The permeability of the ferromagnetic material in the magnetic circuit is infinite.
Assumption 2 Ignoring the magnetoresistance of the ferromagnetic core and rail.
The electromagnetic force F m (i,z) can be calculated as: where l 0 is the air permeability, N is the number of coil windings, and A is the area magnetic pole. The current is generated by the voltage, and the voltage of the electromagnet can be calculated using the equation: where U is the voltage of the coil winding, R c is the resistance of the coil winding, L indicates the inductance of the coil winding, and the inductance at the rated operating position is L 0 ¼ l 0 N 2 A 2z 0 . In the ideal case without considering the external excitation, the single electromagnet can be maintained near the rated operation position under the active adjusting of the levitation control system, thus the linearization of single electromagnet levitation system Fig. 1 Schematic diagram of a single electromagnet levitation system at the rated operation position is conducted [8,18]. Linearization of the electromagnetic force at the rated operation position (z 0 ; i 0 ) is substituted into Eq. (2) to obtain Eq. (4): The coil voltage shown in Eq. (3) can be linearized to obtain Eq. (5): Using the classical PID control algorithm combined with the current loop feedback, the voltage control law of the electromagnet can be expressed as: where k p , k i , k d , and k c are the proportional, integral, differential, and current feedback gain coefficients, respectively. The control flow of the SEL system is shown in Fig. 2. Following Fig. 2, the closed-loop transfer function of this system can be written as shown in Eq. (7): where . Equation (7) shows that the linearized SEL system is a fourth-order system, when not accounting for the LGF time delay. Most maglev vehicles use eddy current gap sensors to measure the levitation gap, and in engineering practice, there may be time delays in each segment, such as measurement, sampling, signal transmission, and control law calculation. The transfer function of a pure time delay is shown in Eq. (8): where s 0 is the time delay. Because the time delay value is always tiny, e s 0 s can be linearized and expanded to 1 þ s 0 s, according to the Taylor Series, which means the time delay can be approximated as a one-order filter. This transfer function can therefore be simplified as in Eq. (9): If the LGF time delay is integrated into Eq. (7), the whole system is upgraded to a fifth-order system with the transfer function shown in Eq. (10): where The characteristic equation of the transfer function shown in Eq. (10) is also upgraded to a quintuple parameter equation, and the exact analytical solution of the characteristic root cannot be obtained directly, it is difficult to theoretically analyze the stability of a fifth-order SEL system. Because the single electromagnet levitation system is the smallest levitation unit in the EMS maglev vehicle system, and also is the most important part, it can fully reflect the properties of EMS system, such as inherent instability characteristic and the dynamics performance under the adjusting from control system. In addition, the time delay mainly exists in the process of signal measurement, signal transmission, control law calculation, electromagnet response, these factors are all considered in the single electromagnet levitation system model. Therefore, the theoretical analysis is carried out by using the single electromagnet levitation system model. In this section, the fifth-order SEL system is simplified as a third-order system by the assumptions as below: Assumption 3 The SEL system is in the rated operation conditions and doesn't need the integration part in PID controller to adjust the static deviation.

Assumption 4
The current loop function provides optimal adjustment function to ensure the coil current can track the voltage variation completely.
Based on these assumptions, this study analyzes the stability of the SEL system that considers the LGF time delay from a theoretical perspective to obtain its theoretical critical value. Then a complete fifth-order SEL system model is built in Simulink software, further analyzing the stability of the complete SEL system to obtain the necessary condition for ensuring the system stability. Table 1 shows the primary parameters of the SEL system.

Stability analysis based on a simplified thirdorder SEL system
The main function of the integral term of the PID controller is to eliminate the static deviation by integrating the difference between the levitation gap and the initial gap to lead it to gradually approach the initial levitation gap value. This function of integral term can be considered as a static process, has only a slight influence on the stability of the levitation system with the determined levitation mass and the stability can be estimated without including this integral term. The simplified third-order system is shown in Fig. 3.
The transfer function of this third-order system is shown in Eq. (11): where The characteristic equation corresponding to Eq. (11) i s s 0 s 3 þ s 2 þ ðk d k 1 À k 2 s 0 Þs þ ðk p k 1 Àk 2 Þ ¼ 0. Because of the prerequisites of the Routh Criterion, it is clear that the stability of the system requires that all coefficients in the characteristic equation are positive, so that Eq. (12) can be obtained as follows: On this basis, the conditions for the stability of the system based on the Routh stability criterion can be expressed as follows: Magnet pole area(A) m 2 0.028 Fig. 3 Control flow diagram of the simplified third-order system Combining Eqs. (12) and (13), it can be seen that 0\s 0 \ k d k p \ k d k 3 =k 2 ; and s c ¼ k d k p is the critical value of the LGF time delay for ensuring the system stability, which can be described as the theoretical critical value s c of the LGF time delay. In other words, this simplified third-order system can remain stable when the LGF time delay s 0 \s c .
Solving the characteristic equation corresponding to the transfer function, the characteristic roots include one real root and two conjugate imaginary roots, and the system is stable only when all the roots have negative real parts.
Three sets of control parameters are selected as examples: k p 10000 k d 70, k p 7000 k d 70, and k p 7000 k d 120. Calculating the characteristic roots for these three sets of control parameters at different LGF time delays, the root locus is shown in Fig. 4a. It can be seen that the real part of the conjugate imaginary root approaches the x-axis faster than the real root, changing from negative to positive when the time delay is greater than the theoretical critical value s c , at which point the system changes from stable to unstable. This indicates that the positive and negative of the real part of the conjugate imaginary root plays a decisive role in the stability of the system. The real part of the conjugate imaginary root is 0 around time delay is 7 ms, 10 ms, and 17 ms, at which point the system is in a critical stable state, and the critical time delay is consistent with the theoretical critical value calculated by the Routh Criterion. Furthermore, Fig. 4b shows the critical values of the LGF time delay corresponding to different values of k p and k d , where it can be seen that the critical value of the LGF time delay increases as k p decreases and k d increases, which is also consistent with the theoretical critical value calculated by the Routh Criterion.
The damping ratio of this system can be determined by the real part of the conjugate imaginary root and the imaginary root modulus, as shown in Theorem 1: Figure 5 shows the relationship between the system damping ratio and the time delay. It can be seen that asthe time delay of the LGF increases, the damping ratio of the system decreases, and when the time delay isgreater than 10 ms, the damping ratio of the system is less than 0 and the system is unstable. It should also benoted that the system damping ratio is already less than 5% when the time delay is greater than 8.6 ms,and for a complex system, there is a greater risk of instability under the action of external excitation.
When the LGF time delay exceeds the critical value and the system becomes unstable, a fixed frequency exists for each response of the system. The value of this frequency can be determined by the imaginary part of the conjugate imaginary root, as shown in Theorem (2): Based on the previously described SEL system, the conjugate imaginary root of the system characteristic equation is 0 ± 55i when the control parameter is set to k p 7000 k d 70, from which the inherent frequency of the system can be calculated to be about 8.7 Hz. The imaginary part of the conjugate imaginary root is related to the constant term k p 9 k 2 -k 3 of the characteristic equation.

Stability analysis based on the complete fifthorder SEL system
The above results of this analysis are based on a simplified third-order system, ignoring the integral term in the PID controller, and assuming an ideal current loop. In this section, a fifth-order SEL system model with an LGF time delay consistent with that shown in Eq. (10) is developed in Simulink. At a time step of 3 s in the simulation, the time delay begins. Gaussian white noise with an amplitude of ± 1 mm is integrated into the levitation gap signal as an external excitation. The integration step is set to 1E -4 s. Figure 6 shows the calculation results of the levitation gap when the time delay is varied from 0 to 11 ms. It can be seen that when the LGF time delay is increased from 0 to 9 ms, there is basically no change in the fluctuation range of the levitation gap; when the time delay is increased to 10 ms, the fluctuation range of the gap signal after 3 s increases significantly compared with that without the time delay effect, but the system is still stable; when the time delay is increased to 10.1 ms, the levitation gap response reaches the critical stable state, and the fluctuation amount of the gap signal is approximately ± 4 mm, close to the ideal working boundary of the levitation control system. When the time delay is further increased to 10.5 ms or even 11 ms, the levitation gap reaches 0 after 2.8 s and 1.6 s of delayed action, respectively, and the system is completely destabilized, the collision between electromagnet and rail occurs.
In summary, when the control parameters are set to k p = 7000, k i = 1000, k d = 70, k c = 300, the LGF critical value of the time delay is around 10 ms, and the system is unstable once the time delay increases beyond the critical value. The results of the simulation of the LGF critical value obtained from the fifth-order system model are consistent with the theoretical critical value calculated from the theoretical calculation. Figure 7 shows the coil current and electromagnet acceleration, which shows that when there is a 10.1 ms time delay in LGF, the coil current fluctuation amplitude is about 15-35A, the maximum electromagnet acceleration is close to 5 m/s 2 , which is significantly larger than without a time delay, and there are obvious periodic fluctuations and a divergent trend in the response curve. The Fourier transfer of the acceleration response at different time delays shows that the amplitude of the periodic vibration becomes larger and larger as the time delay increases, and when the time delay is near the critical value of 10 ms, the frequency of periodic vibration is prominently manifested at 8 Hz, which is close to the fixed frequency result obtained from the third-order system theory analysis.
In the research of Zhang and Zhang (2013), a single electromagnet levitation system model is also used to analyze the stability of the system from the perspective of nonlinear dynamics. It is pointed out that the system is stable when the time delay is less than 11 ms, and when the time delay is larger than 15 ms, the system becomes unstable. And it also indicates that when the time delay is 11.4 ms, the levitation system will occur the supercritical Hopf bifurcation phenomenon [13].  The theoretical critical value of time delay calculated in this paper is 10 ms, which is close to the result given in reference [13], and that means the theoretical analysis model is correct and the results are credible.
4 The engineering critical value of the time delay and its effect on vehicle system dynamics Based on the previous analysis of the single electromagnet levitation system stability, the theoretical critical value of LGF time delay s c was then calculated through theoretical analysis. However, the mediumlow speed maglev system is a complex coupling system, so the s c can only be used as a necessary condition to ensure the stability of the more complex coupling system. Therefore, this section simulates and analyzes the stability of the complex medium-low speed maglev vehicle system under the influence of the LGF time delay. The medium-low speed maglev vehicle system dynamics model is constructed using Simpack software as shown in Fig. 8, including the carbody and the five levitation bogies arranged longitudinally under the carbody. Each levitation bogie consists of two levitation modules on the left and right and two antirolling beams on the front and rear. Each levitation module contains the a side beams and four electromagnet coils, where the four electromagnets are divided into two groups at the front and rear of the module, independently controlled by two levitation controllers. The carbody is connected to the levitation frame by a secondary suspension system (air springs); ignoring the connection between the side beams and the electromagnet coils, the levitation module can be considered as a whole rigid body. The electromagnetic force generated by each coil winding is equivalent to a concentrated force; that means the interaction between the levitation module and the rail is generated by four concentrated forces with a longitudinal interval of 0.68 m. The levitation electromagnetic force in the model is controlled by the control system model constructed in Simulink, and the other force elements are simulated using stiffness-damping units [22]. The medium-low speed maglev vehicle system dynamics model in this paper is verified by comparing the simulation results with the measurement data obtained from the field dynamics experiment on Changsha Express Line, its calculation results are in good agreement with the measured results, thus ensuring the accuracy of the vehicle system dynamics model [23].
Based on the previous analysis, the PID levitation control model accounting for the LGF time delay (2) is constructed in Simulink; the levitation gap and levitation electromagnet force are used as the input and output of Simpack and Simulink; and the Simat interface is used to realize the co-simulation analysis. The German lowdisturbance track irregularity is used to model external excitation on the maglev vehicle system [1]. The analysis focuses on the coil current and levitation gap at the first control point on the right first levitation module, the vertical acceleration responses of the first levitation frame, and the carbody.

Vehicle system dynamic responses with different time delays in levitation gap feedback
The effect of LGF time delay on the dynamic response of the maglev vehicle system is analyzed using the dynamics model established in the previous content.
The levitation control parameters are still set to k p 7000, k i 1000, k d 70, and k c 300, which correspond to the levitation control system adjustment frequency of about 7.5 Hz [22], and the corresponding theoretical critical value for the time delay of 10 ms. The simulation speed is 80 km/h, and the LGF time delay is set to 0 ms, 3 ms, 6 ms, 8 ms, and 9 ms. As shown in Fig. 9a, when the vehicle is running at 80 km/h and the LGF time delay increases from 0 to 6 ms, the fluctuation range of coil current, levitation gap, levitation frame acceleration, and carbody acceleration all increase significantly, but the vehicle system can still maintain stability under the vibration-reducing effect of the levitation control and suspension system. When the LGF time delay is further increased to 8 ms, each dynamic response showing the divergent trend at a fixed frequency of 16 Hz, the maximum coil current can reach 80A, the levitation gap fluctuation amplitude is more than ± 4 mm, and the acceleration amplitude of the levitation frame and carbody is 25 m/s 2 and 2 m/s 2 , respectively. At this moment, the system is at the critical point of stability, but finally the system becomes unstable after about 4 s in the simulation. When the complete vehicle system is considered, the whole system has more degrees of freedom and is more complex, so the critical value of the time delay for the whole system is smaller than the theoretical critical value s c . When the time delay is further increased to 9 ms, the coil current fluctuation range has reached the set threshold range of 0-120A, and the system is completely unstable. This means that the critical value of the LGF time delay to ensure the stability of the vehicle system is between 8 and 9 ms, which also shows that for the complex vehicle system, it is necessary to have a critical value used for engineering purposes, s e , that is less than the theoretical critical value obtained from the theoretical calculation s c . In other words, when the actual time delay in vehicle system is close to the engineering critical value s e , the dynamic responses such as levitation gap, coil current, levitation module acceleration and carbody acceleration exhibit periodic response in a short period of time, and these responses tend to diverge over a longer time range gradually.
Taking the LGF time delay of 8 ms as an example, comparing the response of the coil current at 40 km/h, 80 km/h, and 120 km/h, Fig. 10 shows that the coil current shows a significant divergent trend with a time Fig. 9 Effect of different time delays on vehicle dynamic response delay of 8 ms at speeds of 80 km/h and 120 km/h. However, when the speed is reduced to 40 km/h, the coil current response does not show a tendency to divergence, which is a preliminary indication that the running speed also has an effect on the stability of the vehicle system given an LGF time delay.
From the previous analysis, it can be concluded that the LGF time delay has a notable effect on the dynamic response of medium-low speed maglev vehicle systems. But in engineering practice, the LGF time delay is a widespread phenomenon in levitation control systems, so the next step is to take the time delay of 6 ms as an example, analyze the influence of running speed, the proportional coefficient k p , and differential coefficient k d on the vehicle system dynamic response, in order to seek methods to reduce the influence of LGF time delay on system stability.

Vehicle system dynamic responses
with different running speeds (time delay 6 ms) When the time delay is set as 6 ms, the vehicle system dynamic response results can be obtained completely, so this section uses the 6 ms time delay condition as an example to further analyze the influence of running speed on system stability under effect of the LGF time delay. Figure 11 compares the trend of the vehicle dynamics response when the running speed increases from 20 to 160 km/h with a 6 ms time delay versus no time delay. From (a) and (b), it can be seen that the LGF time delay has a notable effect on the coil current and levitation gap, and as the running speed continues to increase, the fluctuation range of the coil current and levitation gap widens, making the effect of the LGF time delay even more evident. From (c) and (d), it can be seen that with the increase in running speed, the vertical acceleration of the levitation frame and the carbody increases, and the LGF time delay also has a significant effect on the acceleration response of the vehicle system; the higher the running speed, the more pronounced this effect is. When the speed reaches 160 km/h, the levitation frame's vertical acceleration approaches 15 m/s 2 and the carbody's vertical acceleration approaches 0.55 m/s 2 , both of which are more than twice the acceleration response when the LGF time delay is not considered.
The results of the above analysis show that the effect of the LGF time delay on the vehicle dynamics response is more obvious when the running speed of the maglev vehicle is higher.

Vehicle system dynamic responses
with different control parameters (time delay 6 ms) As the LGF time delay is an objective reality of maglev systems, engineers can only seek to reduce its impact on the maglev vehicle dynamics system, but they cannot completely eliminate it. Therefore, this section maintains the 6 ms time delay as the example, analyzing the influence of two key parameters of the vehicle system dynamics: proportional coefficient k p and differential coefficient k d . The other control parameters k i , k d , and k c are set to 1000, 70, and 300, respectively, and the running speed is set to 80 km/h, and maintain k p as the unique variable. The six values 5500, 6000, 7000, 8000, 9000, and 10,000 are selected as values of k p , corresponding to the critical values of the LGF time delay, s c , of 12.7 ms, 11.6 ms, 10 ms, 8.7 ms, 7.7 ms, and 7 ms, respectively. When the k p is 10,000, the calculation is divergence. The effect of the proportional coefficient k p on the vehicle dynamic response is shown in Fig. 12. As shown in (a) and (b), the coil current fluctuates slightly, increasing with k p , when the LGF time delay is not considered, the levitation gap fluctuation decreases as k p increases. However, Fig. 10 Coil current response at three running speeds (with time delay of 8 ms) assuming a 6 ms LGF time delay, when k p is set to 5500, 6000, 7000, and 8000, the fluctuation range of coil current and levitation gap is significantly larger than the result without considering the time delay, but it is not significantly affected by the change in k p . Only when the k p value increases to 9000 does the fluctuation range of coil current and levitation gap increase significantly. From (c) and (d), it can be seen that with no time delay, the vertical acceleration response of the levitation frame and the carbody does not change much; however, given a 6 ms time delay, the vertical acceleration of the levitation frame and the carbody shows a nonlinear increasing trend as k p increases. When the k p value increases from 5500 to 8000, the acceleration of the levitation frame and carbody increases gradually, and as the k p value approaches 9000, the acceleration response increases sharply. This analysis shows that for complete maglev vehicle dynamics systems under effect of time delay, an excessive proportional coefficient k p may lead to system instability. However, the k p can't be infinitesimal, the reason is that the selection of k p needs to meet the basic stability requirements of the PID controller firstly. The k p should be larger than k 3 /k 2 , as shown in Eq. (12), where the k 2 ¼ 2g i 0 and k 3 ¼ 2g z 0 , thus the minimum value of k p = i 0 /z 0 = 30/0.008 is 3750 for the single electromagnet levitation system. On the premise of meeting the PID controller basic requirements, a smaller proportional coefficient k p may reduce the impact of LGF time delay on the vehicle system.
The control parameters k p , k i , and k c are set to 7000, 1000, and 300, respectively. The LGF time delay and  Figure 13 shows the effect of different values for the differential coefficient k d on the dynamic response of the vehicle system. From (a) and (b), it can be seen that the fluctuation range of coil current and levitation gap decreases with the increase of k d where there is no LGF time delay. Assuming a 6 ms time delay, the coil current and levitation gap fluctuations still decrease as k d increases, but the fluctuation range is significantly larger. From (c) and (d), it can be seen that from the perspective of levitation frame and carbody acceleration, when the LGF time delay is ignored, the levitation frame acceleration is relatively stable and the carbody acceleration decrease as k d increases. When the LGF time delay is considered, the acceleration response is significantly larger. When k d is increased from 50 to 110, the acceleration response tends to decrease and then increase, and both larger (k d = 50) and smaller (k d = 110) values of k d may cause a relatively large acceleration response.
A comprehensive analysis of Figs. 12 and 13 shows that the responses are significantly larger given an LGF time delay, which means that the LGF time delay makes the levitation control system more sensitive to the parameter choices of k p and k d . The dynamics responses are more significantly influenced by the value of the proportional coefficient k p , this indicates that the influence of the LGF time delay is primarily reflected in the proportional term of the PID controller, and the proportional coefficient k p must be selected with more care in comparison with the differential coefficient k d .

Effect of time delay on vehicle-girder coupling system dynamics
When a medium-low speed maglev vehicle runs on a guideway girder, the girder is usually considered to be a periodic harmonic excitation of the vehicle system with a wavelength equal to the girder span, but the actual vehicle-girder coupling interaction may make the variation of the levitation gap more complicated. The vehicle dynamics simulation above simulates the vehicle running on a rigid guideway girder without considering the vehicle-girder coupling interaction. Therefore, this section maintains the dynamics model of the medium-low speed maglev vehicle model established in the previous section; This section adopts the Bernoulli-Euler beam model to simulate the guideway girder in vertical direction, the boundary condition of each Bernoulli-Euler beam is consistent with the simply support condition. In other words, the translation motion in x and z direction is constrained at one end of the guideway girder model, and at the other end, only the translation motion in z direction is constrained, the rotation motion at both ends is not constrained. The response of the girder is calculated using the modal superposition method [24] to further analyze the effect of the LGF time delay on the maglev vehicle-girder coupling system. The same control parameters are used (k p 7000, k i 1000, k d 70, and k c 300), and the simulation speed is 80 km/h. It should be noted that the vehicle runs on the rigid guideway before 2.5 s and the vehicle enters the flexible girder after 2.5 s. Figure 14 compares three different LGF timed delays for the carbody vertical acceleration and levitation frame vertical acceleration responses when the maglev vehicle is running on the flexible girder at 80 km/h. As the LGF time delay becomes larger, the acceleration response of the carbody and levitation frame similarly increase. Without the time delay, the the carbody acceleration amplitude increases to 7.35 m/s 2 , far above the limit value of 2.5 m/s 2 , and the levitation frame acceleration amplitude also increases significantly, to 30.17 m/s 2 . Figure 15 shows the levitation gap and coil current response. Without the LGF time delay, the levitation gap fluctuates ± 0.4 mm around the rated levitation LGF time delay of 6 ms, the levitation gap fluctuation increases slightly, but when the LGF time delay increases to 7.8 ms, the levitation gap response also shows periodic fluctuation, and the fluctuation range increases significantly, to ± 3.4 mm, which is close to the ideal fluctuation range ± 4 mm of the levitation controller. As for the coil current, with a time delay of 7 ms, the fluctuation range is 29-37A, and when the time delay increases to 7.8 ms, the fluctuation range of coil current increases significantly to 0-67A. Figure 16 shows the deflection and vertical acceleration response in the middle span of the girder when the LGF time delay is considered. When a 6 ms LGF time delay is compared to no delay, the deflection is not much different, and the acceleration of the girder increases only slightly, from 0.1 (0) to 0.14 m/s 2 (6 ms). When the time delay increases to 7.8 ms, the deflection of the girder fluctuates, and the dynamic fluctuation can reach 0.6 mm. The acceleration of the girder also increases significantly, and the acceleration amplitude can reach 3.6 m/s 2 .
The time delay engineering critical value in this paper is slightly larger than that indicated by Wu (2019), whose analysis indicated that the system becomes unstable when the time delay is 6.1 ms [20]. The difference is mainly caused by the different control system parameters; the k p and k d are 7000 and 50 in Wu's research. The theoretical critical value can be calculated by the conclusion in this paper is 7.1 ms, which is smaller than the theoretical critical value of 10 ms calculated by the control parameters of this paper (k p and k d are 7000 and 70, respectively). Therefore, it is reasonable that the engineering critical Fig. 16 Effect of LGF time delay on girder dynamic response value in this paper is larger, and this comparison also indicates that the control system parameters have significant influence on the critical value of time delay. Figure 17 compares the effect of three different girder flexural stiffnesses on the dynamic response of the maglev vehicle-girder coupling system given a 6 ms LGF time delay. Taking the vertical acceleration response of the levitation frame as an example (only the acceleration response after 2.5 s is given), it can be seen that the increase of the girder flexural stiffness can appropriately reduce the acceleration of the levitation frame. However, as the girder flexural stiffness increases, the mid-span acceleration continues to increase. That means increasing the girder flexural stiffness can effectively reduce the effect of LGF time delay on the vehicle system, but also makes the girder acceleration response increase, which means that the girder flexural stiffness can only be increased to a certain degree to reduce the effect of LGF time delay on the system while meeting the bridge design criteria and construction cost.
In summary, the LGF time delay has a significant effect on maglev vehicle-girder coupling systems. As the LGF time delay increases, the dynamic response of the coupling system increases, especially when the time delay reaches 7.8 ms, at which point the maglev vehicle-girder coupling system reaches a critical stable state, the responses in vehicle system, levitation control system, and guideway girder all increase significantly and show periodic fluctuations, which indicates that the LGF engineering critical value for maglev vehicle-girder coupling systems is 7.8 ms.

Conclusion
1. From the perspective of a single electromagnet levitation system, the levitation gap feedback time delay raises the fourth-order system to the fifthorder, making the system more prone to instability. The theoretical critical value for this levitation gap feedback time delay is s c ¼ k d =k p (based on the parameters in this paper, s c ¼ 10 ms). 2. Levitation gap feedback time delay has a significant effect on the dynamic response of EMS maglev systems, and to ensure the stability of complex maglev systems, the actual system time delay must be smaller than the engineering critical value s e (based on the parameters in this paper s e ¼ 7:8 ms). The system dynamic responses increase as time delay increases, and when the time delay approaches the critical value, the system exhibits a periodic response with a fixed Fig. 17 Effect of girder flexural stiffness and LGF time delays on system dynamic responses frequency which should be carefully considered in the design process of controllers and vehicle systems. 3. The higher the running speed, the more obvious the effect of the levitation gap feedback time delay will be on the operating safety and ride comfort of the maglev vehicle, and the smaller the region of system stability. When the running speed reaches 160 km/h, the acceleration response of the vehicle system nearly doubles. 4. Selecting a smaller proportional coefficient k p and appropriate differential coefficient k d can expand the region of system stability and reduce the impact of levitation gap feedback time delay. Based on the parameters in this paper, k p should be less than 8000 and a k d of 80 can be selected. 5. This analysis is helpful and meaningful to the understanding of the EMS vehicle system stability, and helpful to explore the reason of violent coupled vibration in actual engineering, so as to reduce the violent coupled vibration from the perspective of levitation control system optimization designing.

Outlook
This paper is only a preliminary analysis about the influence law of time delay on the maglev vehicle system. In next stage, the simulation results should be compared with the actual operation data of mediumlow speed maglev system to validate the correctness and feasibility. In particularly, the validation of engineering critical value s e calculation result has significant meaning for the design and optimization of EMS maglev system. And more attention should be paid to propose method which can reduce the effect of time delay on the maglev system dynamic performance. Data availability Enquiries about data availability should be directed to the authors.

Declarations
Conflict of interest The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.
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