A New Strategy for Improving the Tracking Performance of Magnetic Levitation System in Maglev Train

S S symmetry

Article A New Strategy for Improving the Tracking Performance of Magnetic Levitation System in Maglev Train

Mingda Zhai , Zhiqiang Long * and Xiaolong Li

College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China * Correspondence: [email protected]

Received: 18 July 2019; Accepted: 13 August 2019; Published: 16 August 2019

Abstract: The maglev train is a whole new method of transportation without wheels, consisting of 20 groups of symmetry suspension units. The magnetic levitation system plays a major role in suspending the maglev train stably and following the track quickly with the desired gap. However, vertical track irregularity in the maglev train line has a dreadful effect on the tracking performance of the magnetic levitation system. The investigations carried out by our team have revealed that the fluctuation of the suspension gap becomes more and more serious with increases in running speed. In this paper, a mathematical model with consideration of vertical track irregularity is established. In order to overcome and suppress the fluctuation of the suspension gap, we propose a new strategy which includes installing an accelerometer on the electromagnet to address this problem. This strategy has already been successfully implemented and applied to the suspension controller for a magnetic levitation system in the Changsha maglev express. Real operation data indicates the tracking performance of the magnetic levitation system was obviously improved.

Keywords: maglev train; symmetry; magnetic levitation system; vertical track irregularity; tracking performance; strategy

1. Introduction In 1814, the first steam locomotive was invented by George Stephenson, renowned as the “Father of Railways”. The Stockton and Darlington Railway was opened to carry passengers in 1825. Rail transport was one of the most important technological inventions of the industrial revolution. Ever since then, the wheel of time has never stopped. Time goes by as fast as a high-speed train, and we have already entered into an age without wheels. The maglev train is a whole new method of transportation without wheels. Plenty of scholars and engineers in Japan, the United States, China and South Korea are carrying out research in related fields and have already achieved many successes [1–3]. It is well known that the Shanghai high-speed maglev train went into operation in 2002. The Incheon airport maglev line of South Korea officially began to carry passengers on 3 February 2016. A superconducting maglev train with a high speed of 603 km/h will come out in 2027. As shown in Figure1, the Changsha maglev express is the first domestic medium-low speed maglev line in China and completely owns the proprietary intellectual property rights. This line stretches over 18.55 km and started trial operations on 6 May 2016. Beijing’s 1st maglev Line S1 is the second domestic medium-low speed maglev line in China and began operating on 31 December 2017. The Qingyuan maglev travel line is under construction, and will go into operation on September 2019. In recent years, the maglev train increasingly began to receive considerable attention and has become an active traffic field.

Symmetry 2019, 11, 1053; doi:10.3390/sym11081053 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1053 2 of 18 Symmetry 2019, 11, x FOR PEER REVIEW 2 of 19

Figure 1. A photograph of the medium-low speed maglev train in Changsha.

Compared with conventional railway systems, the medium-low speed maglev train makes full use of electromagnetic electromagnetic force force to to achieve achieve active active suspen suspensionsion and and runs runs the the train train using using a linear a linear motor. motor. It completelyIt completely abandons abandons the the wheel-rail wheel-rail relationship relationship and and achieves achieves non-contact non-contact motion, motion, which which not not only eliminates friction to reduce noise and vibration, but also allows the maglev train to travel through buildings on viaducts smoothly and quietly. What’s more, there are zero emissions to the atmosphere during running. Therefore, Therefore, the the maglev maglev train train is is sa safe,fe, sound, sound, low-carbon low-carbon and and environmentally environmentally friendly. friendly. This paper isis concernedconcerned chiefly chiefly with with the the study study of of the the magnetic magnetic levitation levitation system system in ain medium-low a medium- lowspeed speed maglev maglev train. train. The The magnetic magnetic levitation levitation system system plays plays a major a major role inrole suspending in suspending the maglev the maglev train trainstably stably and following and following the track the quickly. track quickly. Most scholars Most scholars concentrate concentrate on the research on the onresearch the levitation on the levitationcontrol algorithm control underalgorithm the conditionsunder the ofconditions a straight of track. a straight However, track. vertical However, track irregularityvertical track in irregularitythe maglev trainin the line maglev has a banefultrain line influence has a onba theneful tracking influence performance. on the tracking Unfortunately, performance. many Unfortunately,experts are not concernedmany experts about are this not issue. concerned In order about to address this issue. this issue,In order the to mathematical address this model issue, with the mathematicalconsideration ofmodel vertical with track consideration irregularity of is vertical established track in irregularity this paper. Foris established the benefit in of this improving paper. For the thetracking benefit performance, of improving we proposethe tracking a new performance, strategy of installing we propose an accelerometer a new strate ongy the of electromagnetinstalling an accelerometerto address this on problem. the electromagnet to address this problem.

2. Nominal Nominal Controller of Magnetic Levitation System Magnets of the same same polarity polarity repel repel each each other, other, whil whilee those those of of different different polarity polarity attract attract each each other. other. The medium-low speed maglev train takes advantageadvantage of the attraction between the electromagnet and the F-type F-type track track to to realize realize suspension suspension at at a adefin defineded height. height. A Acomplete complete magl maglevev train train has has 20 20sets sets of electromagnets,of electromagnets, installed installed on on the the bottom bottom of of the the ca carriage.rriage. When When the the current current is is switched switched on, upward electromagnetic force is generated. The F-type track wi willll suck up the base of the vehicle, and the base and the carriagecarriage areare connectedconnected by by air air springs. springs. Thus, Thus, the th magleve maglev train train is ableis able to suspendto suspend itself itself in thein the air, air,against against gravity. gravity. The interactionThe interaction between between electromagnetic electromagnetic force and force gravity and gravity will always will keepalways a dynamic keep a dynamicbalance, levitatingbalance, levitating the maglev the train maglev above train the rail.above No the matter rail. howNo matter the speed how and the railway speed lineand changes,railway linethe gapchanges, between the thegap trackbetween and the the track electromagnet and the electromagnet has to remain has unchanged to remain under unchanged the action under of the actionsuspension of the controller suspension for controller the magnetic for the levitation magnetic system. levitation system. As shown in Figure 22,, thethe magneticmagnetic levitationlevitation controlcontrol systemsystem consists of a suspension controller, a gap sensorsensor andand anan actuator, actuator, which which is is the the electromagnet. electromagnet. The The gap gap sensor sensor measures measures the the gap gap between between the theelectromagnet electromagnet and theand F-type the F-type track constantly.track constantly. The suspension The suspension controller controller calculates calculates the control the quantity control quantityaccording according to the measured to the measured gap signal gap to adjustsignal theto ad currentjust the of current the electromagnet of the electromagnet in real time. in Whenreal time. the Whencurrent the varies, current the intensityvaries, the of theintensity closed-loop of the magneticclosed-loop field magnetic established field between established the electromagnet between the electromagnetand the F-type trackand the will changeF-type accordingly.track will change The electromagnetic accordingly. forceThe electromagnetic will consequently force vary will and consequentlymake the electromagnet vary and make fall or the float electromagnet to levitate the fall maglev or float train to levitate in the target the maglev gap. train in the target gap.

Symmetry 2019, 11, x FOR PEER REVIEW 3 of 19 Symmetry 2019, 11, 1053 3 of 18 Symmetry 2019, 11, x FOR PEER REVIEW 3 of 19

Figure 2. The structure of the magnetic levitation system in the medium-low speed maglev train. Figure 2. TheThe structure structure of the magnetic levitation syst systemem in the medium-low speed maglev train. Because of the symmetry of the magnetic levitation system in medium-low speed maglev trains, the mathematical model of single magnetic levitation system is easily established. If all factors are Because of the symmetry of the the magnetic magnetic levitati levitationon system system in in medium-low medium-low speed speed maglev maglev trains, trains, taken into account, the order of the whole model will be very high, which is not conducive to the mathematical model of single magnetic levitation system is easily established. If If all all factors are resolving the characteristics of the magnetic levitation system and designing the suspension control taken intointo account,account, the the order order of theof the whole whole model model will bewill very be high,very whichhigh, iswhich not conducive is not conducive to resolving to algorithm [4]. Therefore, some minor factors are neglected in practice: resolvingthe characteristics the characteristics of the magnetic of the levitation magnetic system levitation and designingsystem and the designing suspension the control suspension algorithm control [4]. algorithmTherefore,• The magnetic [4]. some Therefore, minor flux factorsleakage some areminor and neglected edge factors effect inare practice: are neglected neglected, in practice: and it is determined that the magnetic flux is uniformly distributed in the air gap. • The magnetic flux flux leakage and edge effect effect are neglected,neglected, and it is determined that the magnetic • It is considered that the electromagnetic force provided by the electromagnet concentrates on fluxflux is uniformly distributed in the air gap. the geometric center, and the geometric center of the electromagnet coincides with the center of • It isis consideredconsidered thatthat the the electromagnetic electromagnetic force force provided provided by by the the electromagnet electromagnet concentrates concentrates on theon • mass. geometricthe geometric center, center, and and the geometricthe geometric center center of the of electromagnet the electromagnet coincides coincides with thewith center the center of mass. of • There is no dislocation between the magnetic pole surface and the F-type track, namely, there is Theremass. is no dislocation between the magnetic pole surface and the F-type track, namely, there is • only vertical motion of the electromagnet relative to the track. onlyThere vertical is no dislocation motion of between the electromagnet the magnetic relative pole to surface the track. and the F-type track, namely, there is only vertical motion of the electromagnet relative to the track. 2.1. Reference Coordinate Equation 2.1. Reference Coordinate Equation 2.1. ReferenceThe reference Coordinate coordinate Equation system XOY is established in Figure 3. zt()is the absolute The reference coordinate system XOY is established in Figure3. z(t) is the absolute displacement ofdisplacement theThe electromagnet reference of the relativecoordinateelectromagnet to the system reference relative XOY plane to theis and establishedreferencec(t) is the plane suspensionin Figure and ct gap()3. is betweenzt ()the issuspension the the trackabsolute andgap displacementthebetween electromagnet. the track of the and When electromagnet the theelectromagnet. track does relative not When haveto thethe vertical trreferenceack does motion, planenot have the and following vertical ct()is motion, formulathe suspension the is satisfied:following gap formula is satisfied: between the track and the electromagnet. When the track does not have vertical motion, the following c(t) = z(t) (1) formula is satisfied: ct()= zt () (1) ct()= zt () (1)

Figure 3. The model of magnetic levitation system in a medium-low speed maglev train. Figure 3. The model of magnetic levitation system in a medium-low speed maglev train. Figure 3. The model of magnetic levitation system in a medium-low speed maglev train.

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2.2. Electrical Equation Since the electromagnet is an inductive component, the electrical equation is:

. 2 2 . µ0N A i(t) µ0N Ai(t) z(t) u(t) = Ri(t) + (2) 2 · z(t) − 2 [z(t)]2

In this equation, u(t) is the control voltage applied to both ends of the electromagnet coil, and R is the resistance. The number of turns, the polar area and the permeability of vacuum are N, A and µ0 respectively.

2.3. Electromagnetic Force Equation

2 2 µ0N A i(t) F(i, z) = [ ] (3) 4 z(t) In this equation, F(i, z) is the electromagnetic force, and i(t) is the current of the electromagnet coil.

2.4. Kinetic Equation

.. mz(t) = (m + M)g F(i, z) (4) − In this equation, m is the equivalent mass of electromagnet, and M is the equivalent mass of carriage. Thus, the state variables are speciﬁed, and the mathematical model of magnetic levitation system is established: T . T x = x1 x2 x3 = z z i  .  x1 = x2  2  . + x  x = M m g K 3 (5)  2 m − m x2  . 1  x2x3 Rx1x3 x1  x3 = + u x1 − 2K 2K 2 where, K = µ0AN /4 The parameters of the magnetic levitation system are indicated in Table1.

Table 1. Parameter list of magnetic levitation system.

Symbol Meaning Quantity Unit m Mass of electromagnet 400 kg M Mass of carriage 1000 kg R Resistance 1 Ω N Number of turns 360 A Polar area 0.038 m2 7 µ0 vacuum permeability 4π 10 × − z0 set gap 10 mm

From the above discussion, it is found that the magnetic levitation system is typically nonlinear. It is very formidable work to analyze the performance of the system by resolving the non-linear diﬀerential equation. The linear analysis is often adopted as a solution to address this problem. Hartman-Grobman theory: If x0 is hyperbolic singular point of a nonlinear system, and the following formula is satisﬁed: X(x x0) lim ( − ) = 0 (6) x x0 x x → | − 0| Symmetry 2019, 11, 1053 5 of 18

Therefore, the nonlinear system is determined to have the same topological structure as the corresponding linear system at the isolated singularity x0. What’s more, if x0 is the equilibrium point of the nonlinear system, the stability near the equilibrium point also depends on the corresponding linear system. For the magnetic levitation system above, the equation at equilibrium position is presented:

2 i0 (m + M)g = F(i0, z0) = K [ ] (7) · z0

In this equation, i0 is the current of electromagnet coil at equilibrium position, while z0 is the suspension gap at equilibrium position. The linearization is accomplished near the equilibrium point, and the linearization model is obtained [5].

 .. 2  2Ki0 2Ki0  mz(t) = 3 ∆z(t) 2 ∆i(t)  z0 − z0  . . (8)  2K 2Ki0  ∆u(t) = R∆i(t) + ∆i(t) 2 ∆z(t) z0 − z0 Thus, the state variables of linearization system are speciﬁed, and the state space equation of linearization system is deﬁned:

T . T x = x1 x2 x3 = ∆z ∆z ∆i

   0 1 0       0   .  2Ki 2 2Ki     =  0 0 0  +    x  z 3m z 2m x  0 u  0 − 0   z  (9)   i0 Rz0   0   0 z K 2K  0 −2  y = 1 0 0 0 x The characteristic polynomial is as follows:

λ 1 0 2 − = 2Ki0 λ 2Ki0 λI A z 3m z 2m | − | − 0 0 (10) 0 i0 λ + Rz0 z0 2K − 2 3 Rz0 2 Ri0 = λ + λ 2 2K − z0 m According to Equation (10), the coeﬃcients of characteristic polynomial fail to satisfy the Hurwitz stability criterion. This linearization system is unstable. Therefore, the magnetic levitation system is unstable according to the Hartman-Grobman theory. The controllability of the magnetic levitation system is determined by achieving and analyzing the rank of the controllability matrix.

rank([B AB A2B]) = 3 (11)

Therefore, the magnetic levitation system is completely controllable. By designing a perfect controller, the magnetic levitation system is able to work steadily in the desired state. In the ﬁeld of industrial and traﬃc control, the most successful and mature control algorithm is the PID algorithm. In our system, the fundamental features of the PID algorithm are reproduced. Firstly, the PID algorithm is less dependent on the accuracy of the model, and is ideal for a nonlinear system. Moreover, the PID algorithm facilitates engineering realization. Consequently, the PID algorithm is the conventional control algorithm in magnetic levitation systems of maglev trains. Previous work has also proved that the PID controller is nominal. A PID controller is not only easy to design, but also conducive to realization in engineering. Symmetry 2019, 11, 1053 6 of 18

Based on the idea of cascade control, our control algorithm consists of two parts. The inner loop is the current loop and the outer loop is the position loop. When the magnetic levitation system works in the equilibrium position, the inductance of the electromagnet varies in an extremely small range. We refer to it as a constant consequently. Near the equilibrium point, the voltage-current equation of the electromagnet is described as: 2K . u = R i + i (12) · z0 · The voltage and current transfer function of the electromagnet is as follows:

I(s) 1/R Gi(s) = = (13) U(s) 2K s + 1 Rz0 · The above formula serves to illustrate that the electromagnet is a typical inertia link, and Symmetryaccordingly 2019, the11, x currentFOR PEER response REVIEW seriously lags behind the voltage change. The response time was7 of not19 able to meet the demand of the maglev train. Therefore, the current feedback method was adopted to

reduceSymmetry the 2019 response, 11, x FORtime PEER inREVIEW Figure 4. 7 of 19 uˆ z k u i i c 2K 

uˆ R z k u i i c 2K 

Figure 4. The schematic diagram of current loop.

Figure 4. The schematic diagram of current loop. The input voltage before introducing the current loop was uˆ , and the voltage acting on the Figure 4. The schematic diagram of current loop. electromagnetThe input is voltagedescribed before as: introducing the current loop was uˆ, and the voltage acting on the electromagnet is described as: =⋅− The input voltage before introducinguk the currentc () uiˆ loop was uˆ , and the voltage acting on(14) the u = kc (uˆ i) (14) electromagnet is described as: · − The function of negative feedback is to make the output equal to the input. The purpose of The function of negative feedback is to make the output equal to the input. The purpose of uk=⋅−() uiˆ (14) preamplifier kc is to speed up response. As shownc in Figure5, the response time of the electromagnet preampliﬁer kc is to speed up response. As shown in Figure5, the response time of the electromagnet waswas reduced reducedThe function down down ofto to negative10 milliseconds. feedback The The is torelationsh relationship make theip betweenoutput between equal the the voltage voltageto the andinput. and the the The current current purpose of of the the of electromagnetelectromagnet was was settled settled to to be be a a proportional proportional link link in in the the frequency frequency band band of of the magnetic levitation preamplifier kc is to speed up response. As shown in Figure5, the response time of the electromagnet system.system. In In addition, addition, the the current current loop loop also also decreased decreased the the order order of of the the magnetic magnetic levitation levitation system system from from was reduced down to 10 milliseconds. The relationship between the voltage and the current of the threethree to to two. two. electromagnet was settled to be a proportional link in the frequency band of the magnetic levitation system. In addition, the current loop also decreased the order of the magnetic levitation system from 1.2 three to two. voltage current

1.2 0.8 voltage current

0.8 0.4

0.4 0 0 2 4 6 8 10 T (ms)

FigureFigure 5. 5. TheThe0 response response of of the the electrom electromagnetagnet using using a a current current loop. loop. 0 2 4 6 8 10 T (ms) The next step was to perform the corresponding position loop. Figure 5. The response of the electromagnet using a current loop. =−++t − ukzzˆ pdi1011() kzk ()dt zz 0 (15) The next step was to perform the corresponding position0 loop. Among, k is proportional gain, k is differential gaint and k is integral gain. The outer p1 =−++d1 −i1 ukzzˆ pdi1011() kzk ()dt zz 0 (15) position loop was designed completely. 0 Equation (15) was brought into Equation (14), and the nominal controller for the magnetic Among, k p1 is proportional gain, kd1 is differential gain and ki1 is integral gain. The outer levitation system was designed completely. In order to correspond with the actual engineering signal, position loop was designed completely. the final control law is presented here: Equation (15) was brought into Equation (14), and the nominal controller for the magnetic t levitation system was designeduk=⋅ completely.(( kzz −+ In ) order kzk + to correspond ( zz − )dt) with − ki the actual engineering signal, cp11 0 d 1 i 10 0 c 2 the final control law is presented here: t (16) =−++ −t − kzzpdic()00 kzk ()dt zz ki uk=⋅(( kzz −+ ) kzk0 + ( zz − )dt) − ki cp11 0 d 1 i 10 0 c 2

In order to examine the performance of the desit gned controller, the designed control algorithm(16) =−++kzz() kzk ()dt zz − − ki was directly substituted into thepdic nonlinear00 model of0 the magnetic levitation system in Figure 6. The dynamic characteristics were verified. In order to examine the performance of the designed controller, the designed control algorithm was directly substituted into the nonlinear model of the magnetic levitation system in Figure 6. The dynamic characteristics were verified.

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The next step was to perform the corresponding position loop.

Z t . uˆ = kp1(z z0) + kd1z + ki1 (z z0)dt (15) − 0 −

Among, kp1 is proportional gain, kd1 is diﬀerential gain and ki1 is integral gain. The outer position loop was designed completely. Equation (15) was brought into Equation (14), and the nominal controller for the magnetic levitation system was designed completely. In order to correspond with the actual engineering signal, the ﬁnal control law is presented here:

. R t u = k (k (z z ) + k z + k (z z )dt k i) Symmetry 2019, 11, x FOR PEER REVIEWc1 p1 0 d1 i1 0 0 c2 8 of 19 · − . R t − − (16) = kp(z z ) + k z + k (z z )dt kci − 0 d i 0 − 0 − 22Kdi Kidz() z uRi=+ − In order to examine the performanceu zdt of the()z 2 designed dt controller, the designed control algorithm was directly substituted into the nonlinear model of the magnetic levitation system in Figure6. The dynamic i 2 () z characteristicsSymmetry 2019, 11 were, x FOR veriﬁed. PEER REVIEW i Fiz, 1 z z 8 of 19 K  −   z m ()M + mg 22Kdi Kidz() z uRi=+ − u zdt()z 2 dt − kc

2 i i Fiz(), 1 z z z K u  − m   + kdz ()M + mg 

+ k − i − kc z0

+ k p u + kd  Figure 6. The nominal control block diagram of magnetic levitation system. + ki − z The initial gap for the electromagnet was set to 25 mm and the0 desired gap was set to 10 mm. + k The suspension gap of the electromagnetp when the floating instruction was issued is presented in Figure 7. As shownFigure Figurein Figure 6. 6.The The 7, nominalthe nominal electromagnet controlcontrol block began diagram to rise of of slowlymagnetic magnetic after levitation levitation receiving system. system. the command signal. The suspension gap became smaller little by little, and eventually was fixed at the set gap of 10 mm. ThereTheThe initialwas initial no gap obviousgap for for the fluctuationthe electromagnet electromagnet in the whole was set proc to to ess,25 25 mm and mm andthe and systemthe the desired desired overshoot gap gap was and was set response set to to10 10mm. time mm. TheThewere suspension suspension within a gapreasonable gap of of the the range. electromagnet electromagnet The simulation whenwhen result the floating ﬂoating shows instructionthe instruction designed was wascontroller issued issued is able ispresented presented to levitate in in FigureFigurethe7 maglev. 7. train with the desired gap. As shown in Figure 7, the electromagnet began to rise slowly after receiving the command signal. The suspension gap became smaller26 little by little, and eventually was fixed at the set gap of 10 mm. There was no obvious fluctuation in the whole process, and the system overshoot and response time were within a reasonable range.22 The simulation result shows the designed controller is able to levitate the maglev train with the desired gap. 18 26

Gap(mm) 14 22 10 18

0 1 2 3 4 5

Gap(mm) 14 T(s) FigureFigure 7. 7.The The suspension suspension gap gap of of thethe electromagnet.electromagnet. 10

3. The Tracking Performance of Magnetic Levitation System on the Vertical Track Irregularity Condition 0 1 2 3 4 5 T(s) However, due to topographic fluctuations or other spatial constraints, the profile of a maglev train line will change [6–8].Figure The prof7. Theile suspension of a line is gap composed of the electromagnet. of several gradient lines. If the angles of two adjacent slopes exceed the allowable range, a vertical curve is needed to connect. The vertical 3. The Tracking Performance of Magnetic Levitation System on the Vertical Track Irregularity curve of a maglev line is usually a circular curve, and there is no transition curve between the straight Condition line and the vertical curve [9–12]. The appearance of a vertical curve is not ideal and is consequently disadvantageousHowever, due to to the topographic magnetic levitationfluctuations system or othe [13–15].r spatial The constraints, tracking performance the profile ofof aa magneticmaglev trainlevitation line will system change with [6–8]. consideration The profile ofof thea line vertical is composed curve in of a several maglev gradient train line lines. needs If the to anglesbe further of twodiscussed adjacent and slopes analyzed, exceed which the allowable plays an importantrange, a vertical role in thecurve safety is needed and comfort to connect. of maglev The verticaltrains. curve of a maglev line is usually a circular curve, and there is no transition curve between the straight line and the vertical curve [9–12]. The appearance of a vertical curve is not ideal and is consequently

disadvantageous to the magnetic levitation system [13–15]. The tracking performance of a magnetic levitation system with consideration of the vertical curve in a maglev train line needs to be further discussed and analyzed, which plays an important role in the safety and comfort of maglev trains.

Symmetry 2019, 11, 1053 8 of 18

As shown in Figure7, the electromagnet began to rise slowly after receiving the command signal. The suspension gap became smaller little by little, and eventually was ﬁxed at the set gap of 10 mm. There was no obvious ﬂuctuation in the whole process, and the system overshoot and response time were within a reasonable range. The simulation result shows the designed controller is able to levitate the maglev train with the desired gap.

3. The Tracking Performance of Magnetic Levitation System on the Vertical Track Irregularity Condition However, due to topographic fluctuations or other spatial constraints, the profile of a maglev train line will change [6–8]. The profile of a line is composed of several gradient lines. If the angles of two adjacent slopes exceed the allowable range, a vertical curve is needed to connect. The vertical curve of a maglev line is usually a circular curve, and there is no transition curve between the straight line and the vertical curve [9–12]. The appearance of a vertical curve is not ideal and is consequently disadvantageous to the magnetic levitation system [13–15]. The tracking performance of a magnetic Symmetrylevitation 2019 system, 11, x FOR with PEER consideration REVIEW of the vertical curve in a maglev train line needs to be further9 of 19 discussed and analyzed, which plays an important role in the safety and comfort of maglev trains. AsAs described described in in Figure Figure 88,, thethe profileprofile ofof aa maglevmaglev traintrain lineline isis composedcomposed ofof fivefive parts,parts, thethe straitstrait sectionsection ABAB,, the the concave concave curvecurve ofof sectionsection BCBC, the, the oblique oblique curve curve of of section sectionCD CD, the, the convex convex curve curve of ofsection sectionDE DandE and the the strait strait section sectionEF .E TheF .The radius radius of of circular circular curve curve is isr andr and the the angle angle of of slope slope is isα . α The. The displacement displacement of of the the track track relative relative to to the the absolute absolute reference reference plane plane is iszzGG.. ItIt isis assumedassumed that the the tangenttangent velocity velocity along along the the track track is is VV and and remains remains constant. constant.

θ β

zG α

Figure 8. TheThe illustration illustration of of the the vertical vertical trac trackk irregularity irregularity in in maglev maglev train train line. line.

TheThe strait strait section section ABAB:: ( ) = zG t = 0 z(t)0G The concave curve of section BC: The concave curve of section BC : β 2 β2 V2 t2 2 22 BM zG(t) = r (1 cos β) = r 2(sinββ) r =Vt⋅ · z(t)=⋅·rrr (1cos)− −β =⋅· 2(sin)2 2 ≈⋅≈ · 2 = BM2r G 222r In this equation, M is any point on the concave curve of section BC. β is very small, so the In this equation, M is any point on the concave curve of section BC . β is very small, so the approximate operation is performed in above formula. tBM is the time spent through the BM segment. approximateThe oblique operation curve is of pe sectionrformedCD in: above formula. tBM is the time spent through the BM segment. The oblique curve of section CD : z (t) = r (1 cos α) + V t sin α G =⋅· −− αα + ⋅· CN ⋅· z(t)GCNrVt (1cos) sin In this equation, N is any point on the oblique curve of section CD. tCN is the time spent through the CNIn this segment. equation, N is any point on the oblique curve of section CD . tCN is the time spent throughThe the convex CN segment. curve of section DE: The convex curve of section DE : π Vt ( ) = ( ) + + ( + DK ) zG t r 1 cos α CD sin α r sinπ Vtα r cos α ·z (t)−=⋅rCDr (1 − cosαα ) +· ⋅ sin +⋅ sin(· − α2 +− DK ) −⋅ rr cos α − · G 2 r

In this equation, K is any point on the convex curve of section DE . tDK is the time spent through the DK segment. The strait section EF : =⋅ −αα + ⋅ +⋅ − α z(t)G rCDr (1cos) sin (1cos)

When the track displacement changes, the suspension gap is no longer equal to the displacement of the electromagnet. In this case, the suspension gap is equal to the difference value between the displacement of the electromagnet and the displacement of the track. Namely: =− ct() zt () zG (t) (17)

Therefore, the model of the magnetic levitation system will change depending on the vertical track irregularity conditions. The electrical equation is defined again:

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In this equation, K is any point on the convex curve of section DE. tDK is the time spent through the DK segment. The strait section EF:

z (t) = r (1 cos α) + CD sin α + r (1 cos α) G · − · · − When the track displacement changes, the suspension gap is no longer equal to the displacement of the electromagnet. In this case, the suspension gap is equal to the diﬀerence value between the displacement of the electromagnet and the displacement of the track. Namely:

c(t) = z(t) z (t) (17) − G

SymmetryTherefore, 2019, 11, x the FOR model PEER REVIEW of the magnetic levitation system will change depending on the vertical10 of 19 track irregularity conditions. The electrical equation is deﬁned again: 22Kdi Ki dz()− z =+2K di − 2Ki d(z zG)G uRiu = Ri + − (18)(18) −z z dt 22 dt z zGG dt−()(zzz− z ) dt − − G The electromagnetic force equation is become:

!2 2 i i FicF()(i,, c) = K (19)(19) z −zG zz− G The kinetic equation is presented under these circumstances: The kinetic equation is presented under these circumstances: .. mz(t)=+ =()()(m + M)g −F(i, c) (20) mtz() m Mg− Fic , (20)

However, the controller is unchanged. The function of the controller is to levitate the magnetic levitation system at the defined deﬁned height.height. In other words, the suspension gap is controlled so that it will remain invariable under various line conditions. As shown shown in in Figure Figure 99,, thethe aboveabove nominalnominal controllercontroller was brought into the new nonlinear model with consideration of track irregularity in order to achieve an overall evaluation of tracking performance on the vertical tracktrack irregularityirregularity condition.condition.

()− − 22Kdi Ki dz z zzG uRi=+ − G z− z dt()− 2 dt u G zzG zG

2 i i Fic(), 1 z z z K − − m   zzG ()M + mg

− kc

u + kd 

+ ki −

+ k p

Figure 9.9.TThehe controlcontrol block block diagram diagram of magnetic of magnetic levitation levitation system system with consideration with consideration of track irregularity of track irregularityin maglev train in maglev line. train line.

Symmetry 2019, 11, 1053 10 of 18

According to the above analysis, the new model of the magnetic levitation system is closely related to the radius of circular curve r and the angle of slope α. Their settings are clearly stipulated in the regulations for track construction of maglev lines, with a minimum radius of 1500 m and a maximum slope angle of 7%. What’s more, the model of the magnetic levitation system also includes the running speed V. However, the radius of circular curve and the slope angle are fixed once the maglev train line is built; the investigation of the relationship of tracking performance and speed makes more sense. When the maglev train passes through the track shown in Figure8 at a certain speed, the magnetic levitation system will go through four different stages: entering the concave curve stage, withdrawing from the concave curve stage, entering the convex curve stage and withdrawing from the convex curve stage. In order to discuss the tracking performance of the magnetic levitation system at different speeds, three representative speeds were selected to observe the tracking performance on the above four stages. As described in Figure 10, the tracking performance of a magnetic levitation system when passing through the vertical track irregularity in a maglev train line was investigated and achieved. SymmetryThe suspension 2019, 11, x FOR gap PEER will fluctuateREVIEW upward or downward, and stabilize in the set gap after a period11 of 19 of time. It is apparent that the fluctuation of the suspension gap becomes more and more serious with of time. It is apparent that the fluctuation of the suspension gap becomes more and more serious with increases in speed. At some stages, the tracking performance of the magnetic levitation system is increases in speed. At some stages, the tracking performance of the magnetic levitation system is extremely awful and unacceptable, and the electromagnet is prone to collide with the F-type track. extremely awful and unacceptable, and the electromagnet is prone to collide with the F-type track.

11 10.5 20 km/h 10 10.5 9.5 10 20 km/h 9 9.5 23.5 24 24.5 25 25.5 26 4.5 5 5.5 6 6.5 7 11 10.5 60 km/h 10.5 10 9.5 10 60 km/h 9 Gap(mm) 9.5 Gap(mm) 11.5 12 12.5 13 13.5 14 4.5 5 5.5 6 6.5 7 12 11 100 km/h 11 10 10 9 100 km/h 9 8 4.5 5 5.5 6 6.5 7 8.5 9 9.5 10 10.5 11 T(s) T(s) (a) (b)

12 14 10 20 km/h 12 8 20 km/h 10 6 37.5 38 38.5 39 39.5 40 8 57.5 58 58.5 59 59.5 60 12 14 10 60 km/h 12 8 60 km/h 10 6

Gap(mm) 15.5 16 16.5 17 17.5 18

Gap(mm) 8 22.5 23 23.5 24 24.5 25 12 14 10 100 km/h 12 8 100 km/h 10 6 11.5 12 12.5 13 13.5 14 8 T(s) 15.5 16 16.5 17 17.5 18 T(s) (c) (d)

FigureFigure 10. 10. TheThe tracking tracking performance performance of of magnetic magnetic levitati levitationon system system when when passing passing through through the the vertical vertical tracktrack irregularity irregularity in in a amaglev maglev train train line. line. (a ()a )The The suspension suspension gap gap when when entering entering the the concave concave curve curve stage;stage; (b (b) )the the suspension suspension gap gap when when withdrawin withdrawingg from from the the concave concave curve curve stage; stage; (c ()c )the the suspension suspension gap gap whenwhen entering entering the the convex convex curve curve stage; stage; (d (d) )the the suspension suspension gap gap when when wi withdrawingthdrawing from from the the convex convex curvecurve stage. stage.

In order to verify the rationality of the above analysis, we conduct a series of simulations and investigations to achieve the relationship of the maximum fluctuations of suspension gap and speeds. The maximum fluctuation of the suspension gap at five different speeds is shown in Figure 11.

(a) (b)

Symmetry 2019, 11, x FOR PEER REVIEW 11 of 19

of time. It is apparent that the fluctuation of the suspension gap becomes more and more serious with increases in speed. At some stages, the tracking performance of the magnetic levitation system is extremely awful and unacceptable, and the electromagnet is prone to collide with the F-type track.

12 14 10 20 km/h 12 8 20 km/h 10 6 37.5 38 38.5 39 39.5 40 8 57.5 58 58.5 59 59.5 60 12 14 10 60 km/h 12 8 60 km/h 10 6

Gap(mm) 15.5 16 16.5 17 17.5 18

Gap(mm) 8 22.5 23 23.5 24 24.5 25 12 14 10 100 km/h 12 8 100 km/h 10 6 11.5 12 12.5 13 13.5 14 8 T(s) 15.5 16 16.5 17 17.5 18 T(s) (c) (d)

Figure 10. The tracking performance of magnetic levitation system when passing through the vertical track irregularity in a maglev train line. (a) The suspension gap when entering the concave curve stage; (b) the suspension gap when withdrawing from the concave curve stage; (c) the suspension gap Symmetrywhen2019 entering, 11, 1053 the convex curve stage; (d) the suspension gap when withdrawing from the convex11 of 18 curve stage.

InIn orderorder toto verifyverify thethe rationalityrationality ofof thethe aboveabove analysis,analysis, wewe conductconduct aa seriesseries ofof simulationssimulations andand investigationsinvestigations to to achieve achieve thethe relationshiprelationship ofof thethe maximummaximum fluctuationsfluctuations ofof suspensionsuspension gapgap andand speeds. TheThe maximummaximum fluctuationfluctuation of of the the suspension suspension gap gap at at five five di differentfferent speeds speeds is is shown shown in in Figure Figure 11 11..

Symmetry 2019, 11, x FOR PEER REVIEW 12 of 19 (a) (b)

FigureFigure 11. 11. TheThe maximum maximum fluctuation fluctuation of of the the suspension suspension ga gapp when when passing passing through through the the vertical vertical track track irregularityirregularity in in maglev maglev train train line. line. (a ()a )Entering Entering the the concave concave curve curve stage; stage; (b (b) )withdrawing withdrawing from from the the concaveconcave curve curve stage; stage; ((cc)) enteringentering the the convex convex curve curve stage; stage; (d )( withdrawingd) withdrawing from from the convexthe convex curve curve stage. stage. Taking into account the above analysis and discussion, a number of significant conclusions wereTaking acquired. into Itaccount is clear the that above the fluctuationanalysis and of discussion, the suspension a number gap becomes of significant more conclusions and more serious were acquired.with the increaseIt is clear of that speed the influctuation the above of four the stages.suspension Under gap high-speed becomes more conditions, and more the fluctuationserious with of thethe increase suspension of speed gap isin too the violent. above four When stages. withdrawing Under high-speed from the convexconditions curve, the stage, fluctuation the maximum of the suspensionfluctuation gap at a speedis too ofviolent. 100 km When/h is up withdraw to 3.7532ing mm, from and the electromagnetconvex curve isstage, prone the to collidemaximum with fluctuationthe F-type at track. a speed In other of 100 words, km/h theis up tracking to 3.7532 performance mm, and the of theelectromagnet nominal controller is prone isto unpleasant. collide with the F-type track. In other words, the tracking performance of the nominal controller is unpleasant. 4. A New Strategy for Improving the Tracking Performance of the Magnetic Levitation System 4. A NewAlthough Strategy the for designed Improving nominal the Tracking controller Pe isrformance able to make of the the Magnetic maglev train Levitation stay within System in the target gap, the tracking performance is not ideal for the vertical track irregularity condition. What’s Although the designed nominal controller is able to make the maglev train stay within in the more, the fluctuation of the suspension gap is too violent, and the electromagnet is prone to collide target gap, the tracking performance is not ideal for the vertical track irregularity condition. What’s with the F-type track at some stages. Therefore, the existing control strategy is not satisfied on the more, the fluctuation of the suspension gap is too violent, and the electromagnet is prone to collide vertical track irregularity condition. We propose a new strategy by installing an accelerometer on the with the F-type track at some stages. Therefore, the existing control strategy is not satisfied on the electromagnet as a solution to address this problem. vertical track irregularity condition. We propose a new strategy by installing an accelerometer on the In the magnetic levitation system, the existing gap sensor can only measure and supply gap electromagnet as a solution to address this problem. signals. The suspension controller is not enough, especially when the magnetic levitation system In the magnetic levitation system, the existing gap sensor can only measure and supply gap passes through the vertical track irregularly in a maglev train line. At the same time, the vertical signals. The suspension controller is not enough, especially when the magnetic levitation system acceleration of the electromagnet will change above four stages. Therefore, we propose to improve the passes through the vertical track irregularly in a maglev train line. At the same time, the vertical tracking performance of the magnetic levitation system by obtaining and utilizing the acceleration of acceleration of the electromagnet will change above four stages. Therefore, we propose to improve the electromagnet. the tracking performance of the magnetic levitation system by obtaining and utilizing the acceleration of the electromagnet. However, when the maglev train passes through the vertical curve, the data measured by the accelerometer includes not only the gravity acceleration component, but also the centrifugal acceleration component. Therefore, for the vertical curve, it is necessary to analyze the measurements of the accelerometer at() separately. The strait section AB : at()=− g zt () The concave curve of section BC : v2 at()=+− g cosθ  zt () r The oblique curve of section CD : =−θ at() g cos0  zt ()

Symmetry 2019, 11, 1053 12 of 18

However, when the maglev train passes through the vertical curve, the data measured by the accelerometer includes not only the gravity acceleration component, but also the centrifugal acceleration component. Therefore, for the vertical curve, it is necessary to analyze the measurements of the accelerometer a(t) separately. The strait section AB: .. a(t) = g z(t) − The concave curve of section BC:

v2 .. a(t) = g cos θ + z(t) Symmetry 2019, 11, x FOR PEER REVIEW r − 13 of 19

The convexThe oblique curve curve of section of section DECD: : .. a(t) = g cos θ 2z(t) 0 −v at()=−− g cosθ  zt () The convex curve of section DE: r

v2 .. The strait section EF : a(t) = g cos θ z(t) − r − =− The strait section EF: at() g zt () .. a(t) = g z(t) Therefore, the measured acceleration signals were− not able to be directly used in the design of the controller.Therefore, The acceleration the measured signal acceleration must be signals pre-processed. were not able The to be low-frequency directly used in signals, the design such of as the gravity theacceleration controller. Thecomponent acceleration and signal centrifugal must be pre-processed. acceleration, The are low-frequency filtered, and signals, the suchdynamic as and the gravity acceleration component and centrifugal acceleration, are filtered, and the dynamic and vertical high-frequency signals of electromagnet are retained. The high-pass filter is used to filter and vertical high-frequency signals of electromagnet are retained. The high-pass filter is used to filter and extract signals.extract signals. In summary,In summary, the new the newcontrol control block block diagram diagram of thethe magnetic magnetic levitation levitation system system was achieved.was achieved. ConsiderationConsideration of vertical of vertical curves curves in a in maglev a maglev train train line isis presented presented in Figurein Figure 12. 12.

()− − 22Kdi Ki dz z zzG uRi=+ − G zzdt− ()− 2 dt u G zzG zG

2 i i Fic(), 1 z z z K − − m   zzG ()M + mg

− kc

− k u a  + kd

+ k i − z0

+ k p

Figure 12.Figure The 12. newThe control new control block block diagram diagram of of magnetic magnetic levitation system system with with consideration consideration of vertical of vertical curve incurve maglev in maglev train trainline. line.

When the maglev train passes through the track shown in Figure 8 at a certain speed, the magnetic levitation system by utilizing the new strategy will also go through four different stages: entering the concave curve stage, withdrawing from the concave curve stage, entering the convex curve stage and withdrawing from the convex curve stage. In order to achieve and investigate the tracking performance of the magnetic levitation system at different speeds, three representative speeds were also selected to observe the tracking performance on the above four stages (Figure 13).

11 10.5 20 km/h 10.5 10 10 9.5 20 km/h 9.5 9 4.5 5 5.5 6 6.5 7 23.5 24 24.5 25 25.5 26 11 10.5 60 km/h 10.5 10 10 9.5 60 km/h

Gap(mm) 9.5 9

4.5 5 5.5 6 6.5 7 Gap(mm) 11.5 12 12.5 13 13.5 14 12 11 100 km/h 11 10 10 9 100 km/h 9 8 4.5 5 5.5 6 6.5 7 8.5 9 9.5 10 10.5 11 T(s) T(s) (a) (b)

Symmetry 2019, 11, x FOR PEER REVIEW 13 of 19

The convex curve of section DE : v2 at()=−− g cosθ  zt () r The strait section EF : at()=− g zt () Therefore, the measured acceleration signals were not able to be directly used in the design of the controller. The acceleration signal must be pre-processed. The low-frequency signals, such as the gravity acceleration component and centrifugal acceleration, are filtered, and the dynamic and vertical high-frequency signals of electromagnet are retained. The high-pass filter is used to filter and extract signals. In summary, the new control block diagram of the magnetic levitation system was achieved. Consideration of vertical curves in a maglev train line is presented in Figure 12.

()− − 22Kdi Ki dz z zzG uRi=+ − G zzdt− ()− 2 dt u G zzG zG

2 i i Fic(), 1 z z z K − − m   zzG ()M + mg

− kc

− k u a  + kd

+ k i − z0

+ k p

SymmetryFigure2019 12., 11 The, 1053 new control block diagram of magnetic levitation system with consideration of vertical13 of 18 curve in maglev train line.

WhenWhen the the maglev maglev train train passes passes through through the the track track show shownn in Figure in Figure 8 at8 a at certain a certain speed, speed, the magnetic the magnetic levitationlevitation system system by by utilizing utilizing the the new new strategy strategy will will also also go go through through four four different diﬀerent stages: stages: entering entering the the concaveconcave curve curve stage, stage, withdrawing withdrawing from from thethe concave curve curve stage, stage, entering entering the theconvex convex curve curve stage stage and and withdrawingwithdrawing from from the convexthe convex curve curve stage. stage. In order In toorder achieve to achieve and investigate and investigate the tracking the tracking performance of theperformance magnetic levitationof the magnetic system levitation at diﬀerent system speeds, at different three speeds, representative three representative speeds were speeds also selectedwere to observealso theselected tracking to observe performance the tracking on theperformance above four on the stages above (Figure four stages 13). (Figure 13).

11 10.5 20 km/h 10.5 10 10 9.5 20 km/h 9.5 9 4.5 5 5.5 6 6.5 7 23.5 24 24.5 25 25.5 26 11 10.5 60 km/h 10.5 10 10 9.5 60 km/h

Gap(mm) 9.5 9

4.5 5 5.5 6 6.5 7 Gap(mm) 11.5 12 12.5 13 13.5 14 12 11 100 km/h 11 10 10 9 100 km/h 9 8 4.5 5 5.5 6 6.5 7 8.5 9 9.5 10 10.5 11 Symmetry 2019, 11, x FOR PEERT(s) REVIEW T(s) 14 of 19 (a) (b)

12 14 20 km/h 10 12 8 20 km/h 10 6 8 37.5 38 38.5 39 39.5 40 57.5 58 58.5 59 59.5 60

12 14 60 km/h 10 12 8 10 60 km/h

6 Gap(mm) 8 Gap(mm) 15.5 16 16.5 17 17.5 18 22.5 23 23.5 24 24.5 25 14 12 100 km/h 12 10 8 10 100 km/h 6 8 11.5 12 12.5 13 13.5 14 15.5 16 16.5 17 17.5 18 T(s) T(s) (c) (d)

Figure 13. TheThe tracking tracking performance performance of magnetic levitati levitationon system by utilizing the new strategy when passing through the verticalvertical tracktrack irregularityirregularity inin maglevmaglev traintrain line.line. ( a) The suspension gap when entering the concave curve stage; (b) the suspension gap when withdrawingwithdrawing from the concave curve stage; (c) the suspension gap when enteringentering the convex curve stage; (d) the suspensionsuspension gap whenwhen withdrawing from thethe convexconvex curvecurve stage.stage.

To illustrateillustrate thethe foregoingforegoing analysisanalysis andand discussion,discussion, a number of significantsignificant simulations and investigations were performed to check the trackingtracking performanceperformance of the magneticmagnetic levitationlevitation system when thethe vertical vertical curve curve is passed.is passed. In order In order to test to the test validity, the validity, we performed we performed several sets several of comparative sets of analyses.comparative Compared analyses. with Compared a nominal with controller, a nominal the maximum controller, fluctuations the maximum of the fluctuations suspension gapof the at disuspensionfferent speeds gap at by different utilizing speeds the new by strategy utilizing are the described new strategy in Figure are described 14. in Figure14. Our investigation has reached plenty of conclusions. The fluctuation of the suspension gap is obviously suppressed by utilizing the new strategy. The maximum reduction is nearly 1 mm. Moreover, the maximum fluctuations of the suspension gap are all limited to less than 3 mm. Therefore, the tracking performance of the magnetic levitation system by utilizing the new strategy improved the vertical track irregularity condition.

(a) (b)

Figure 14. The contrastive maximum fluctuation of suspension gap when passing through the vertical track irregularity in maglev train line. (a) Entering the concave curve stage; (b) withdrawing from the concave curve stage; (c) entering the convex curve stage; (d) withdrawing from the convex curve stage.

Symmetry 2019, 11, x FOR PEER REVIEW 14 of 19

12 14 20 km/h 10 12 8 20 km/h 10 6 8 37.5 38 38.5 39 39.5 40 57.5 58 58.5 59 59.5 60

12 14 60 km/h 10 12 8 10 60 km/h

6 Gap(mm) 8 Gap(mm) 15.5 16 16.5 17 17.5 18 22.5 23 23.5 24 24.5 25 14 12 100 km/h 12 10 8 10 100 km/h 6 8 11.5 12 12.5 13 13.5 14 15.5 16 16.5 17 17.5 18 T(s) T(s) (c) (d)

Figure 13. The tracking performance of magnetic levitation system by utilizing the new strategy when passing through the vertical track irregularity in maglev train line. (a) The suspension gap when entering the concave curve stage; (b) the suspension gap when withdrawing from the concave curve stage; (c) the suspension gap when entering the convex curve stage; (d) the suspension gap when withdrawing from the convex curve stage.

To illustrate the foregoing analysis and discussion, a number of significant simulations and investigations were performed to check the tracking performance of the magnetic levitation system when the vertical curve is passed. In order to test the validity, we performed several sets of comparative analyses. Compared with a nominal controller, the maximum fluctuations of the Symmetrysuspension2019, gap11, 1053 at different speeds by utilizing the new strategy are described in Figure14. 14 of 18

(a) (b)

Symmetry 2019, 11, x FOR PEER REVIEW 15 of 19

Our investigation has reached plenty of conclusi ons. The fluctuation of the suspension gap is obviously suppressed by (utilizingc) the new strategy. The maximum reduction(d) is nearly 1 mm. Moreover, the maximum fluctuations of the suspension gap are all limited to less than 3 mm. Figure 14. The contrastive maximum fluctuation of suspension gap when passing through the vertical Therefore,Figure the 14. trackingThe contrastive performance maximum of the ﬂuctuation magnetic of suspensionlevitation gapsystem when by passing utilizing through the new the vertical strategy track irregularity in maglev train line. (a) Entering the concave curve stage; (b) withdrawing from the improvedtrack the irregularity vertical track in maglev irregularity train line. condition. (a) Entering the concave curve stage; (b) withdrawing from the concaveconcave curve curve stage; stage; (c )(c entering) entering the the convex convex curve curve stage; stage; (d) withdrawing (d) withdrawing from thefrom convex the convex curve stage.curve stage. 5. 5.Implementation Implementation and and Experiment Experiment WeWe employed employed a model a model 4610-010 4610-010 MEMS MEMS accelerome accelerometerter with with exceptional exceptional performance performance over over a full a full operatingoperating temperature temperature range range of of −55°C55 toC to+125°C.+125 AsC. Asshown shown in Figure in Figure 15, 15 this, this accelerometer accelerometer has has small small − ◦ ◦ sizesize and and low low weight. weight. The The static static and and dynamic dynamic measurem measurementsents are are all all feasible. feasible. In Inaddition, addition, the the frequency frequency responseresponse is istaken taken from from DC DC to to2700 2700 Hz. Hz.

FigureFigure 15. 15. TheThe photograph photograph of ofmodel model 4610-010 4610-010 accelerometer. accelerometer.

TheThe parameters parameters of ofthe the model model 4610-010 4610-010 accelerometer accelerometer are are indicated indicated in inTable2. Table2 .

Table 2.Parameter list of model 4610-010 accelerometer.

Meaning Quantity Unit Range ±10 g Sensitivity 200 mV/g Frequency Response 0–2000 Hz Residual Noise 400 μV RMS Non-Linearity (%FSO) ±0.1 / Transverse Sensitivity <3 % Damping Ratio 0.7 / Shock Limit 6000 g

What’s more, this strategy has already been successfully implemented and applied to the suspension controller for the magnetic levitation system in the Changsha maglev express. Figure 16 is a photograph of the suspension controller which was designed and manufactured by our team.

Symmetry 2019, 11, 1053 15 of 18

Table 2. Parameter list of model 4610-010 accelerometer.

Meaning Quantity Unit Range 10 g ± Sensitivity 200 mV/g Frequency Response 0–2000 Hz Residual Noise 400 µV RMS Non-Linearity (%FSO) 0.1 / ± Transverse Sensitivity <3 % Damping Ratio 0.7 / Shock Limit 6000 g

Figure 16. The photograph of designed and manufactured controller. Figure 16. TheThe photograph photograph of of designed designed and and manufactured manufactured controller. Since the Shanghai-Kunming high-speed railway passes through the middle of the maglev line, Since the Shanghai-Kunming high-speed railway pa passessses through through the the middle middle of of the the maglev maglev line, line, the maglev line has to cross under the high-speed railway line. As presented in Figure 17, the vertical the maglev line has to cross under the high-speed railwayrailway line. As presented presented in Figure 17,17, the vertical track irregularity is similar to in Figure 8. When the maglev train passes through this section, the track irregularity is similar to in FigureFigure8 8.. WhenWhen thethe maglevmaglev traintrain passespasses throughthrough thisthis section,section, thethe magnetic levitation system will also go through the above four different stages. The radius of the magnetic levitation system will also go through the above four didifferentﬀerent stages.stages. The radius of the circular curve is 2000 m and the angle of slope is 4.1%. circular curve is 2000 m and the angle of slope is 4.1%.

Figure 17. A photograph of the maglev train passing through the vertical section in Changsha maglev Figure 17. A photograph of the maglev train passing through the vertical section in Changsha maglev Figureexpress. 17. A photograph of the maglev train passing through the vertical section in Changsha maglev express. express. In order to observe the tracking performance of the maglev train passing through the above In order to observe the tracking performance of the maglev train passing through the above sections, the suspension gap and running speed need to be measured and recorded. The magnetic sections, the suspension gap and running speed need to be measured and recorded. The magnetic levitation system sends data to the on-board computer every 100 ms through the CAN network, and levitation system sends data to the on-board computer every 100 ms through the CAN network, and these data are stored in the on-board computer. In the Changsha maglev express line, the Mc1, M and these data are stored in the on-board computer. In the Changsha maglev express line, the Mc1, M and Mc2 maglev trains are organized into groups. Each maglev train has 20 sets of magnetic levitation Mc2 maglev trains are organized into groups. Each maglev train has 20 sets of magnetic levitation systems. The suspension gap and speed of the tenth suspension point in each maglev train were systems. The suspension gap and speed of the tenth suspension point in each maglev train were selected and observed in Figure 18. selected and observed in Figure 18.

Symmetry 2019, 11, 1053 16 of 18

In order to observe the tracking performance of the maglev train passing through the above sections, the suspension gap and running speed need to be measured and recorded. The magnetic levitation system sends data to the on-board computer every 100 ms through the CAN network, and these data are stored in the on-board computer. In the Changsha maglev express line, the Mc1, M and Mc2 maglev trains are organized into groups. Each maglev train has 20 sets of magnetic levitation systems. The suspension gap and speed of the tenth suspension point in each maglev train were Symmetry 2019, 11, x FOR PEER REVIEW 17 of 19 selected and observed in Figure 18.

15 14 12

10 10 8 Gap(mm) Gap(mm) 5 6 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 T(s) T(s)

45 45

40 40

35 35 V(km/h) V(km/h) 30 30

25 25 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 T(s) T(s)

(a) (b)

8 Gap(mm) 6 0 10 20 30 40 50 60 70 80 90 100 T(s)

V(km/h) 30

25 0 10 20 30 40 50 60 70 80 90 100 T(s)

(c)

FigureFigure 18. 18. TheThe suspension suspension gap gap and and speed speed of of suspension suspension point point in in the the magl maglevev train train when when the the vertical vertical sectionsection is is passed. passed. (a (a) )The The tenth tenth suspension suspension point point in in the the Mc1 Mc1 maglev maglev train; train; (b (b) )The The tenth tenth suspension suspension pointpoint in in the the M M maglev maglev train; train; ( (cc)) The The tenth tenth suspension suspension point point in in the the Mc2 Mc2 maglev maglev train. train.

BasedBased on analysisanalysis of of the the operation operation data, data, it was it foundwas found that the that performance the performance of the actual of the suspension actual suspensiongap curve isgap diff curveerent fromis different that of from the simulation that of th curve.e simulation This is curve. because This the is on-board because computerthe on-board only computercollects data only from collects the magneticdata from levitation the magnetic system levi everytation 100system ms inevery actual 100 operation. ms in actual The operation. sampling Thefrequency sampling is too frequency low to fully is reflecttoo low the to real-time fully reflect dynamic the characteristics real-time dynamic of the suspensioncharacteristics gap. of What’s the suspensionmore, the maglev gap. What’s train does more, not the pass maglev through train the do abovees not section pass atthrough a fixed speedthe above in actual section line at operation, a fixed speedand the in speedactual is line constantly operation, changing. and the However, speed is itconstantly is obviously changing. presented However, that the suspensionit is obviously gap presentedalways fluctuates that the up suspension and down gap near always the desired fluctuat gap,es and up theand maximum down near fluctuations the desired of thegap, suspension and the maximumgap are almost fluctuations all limited of the to suspension less than 3 mm,gap are except almost the all impression limited to of less noise. than The 3 mm, fluctuations except the of impressionthe suspension of noise. gap The are allfluctuations in the reasonable of the susp rangeension and gap the are electromagnet all in the reasonable doesn’t collide range withand the the electromagnettrack. In other doesn’t words, thecollide tracking with performance the track. ofIn theother magnetic words, levitation the tracking system performance by utilizing theof newthe magneticstrategy waslevitation improved system and by qualified utilizing the the vertical new trackstrategy condition. was improved and qualified the vertical track condition.

6. Discussions In this paper, it is emphasized that the tracking performance of the magnetic levitation system is closely related to the running speed in the vertical track condition. An important and significant conclusion was acquired, that the fluctuation of suspension gap becomes more and more serious with the increase of speed. However, when the maglev train passes through the vertical curve, the tracking performance of the magnetic levitation system is also dictated by the radius of circular curve r and the angle of slope α .Therefore, the tracking performance of the magnetic levitation system in the

Symmetry 2019, 11, 1053 17 of 18

6. Discussion In this paper, it is emphasized that the tracking performance of the magnetic levitation system is closely related to the running speed in the vertical track condition. An important and significant conclusion was acquired, that the fluctuation of suspension gap becomes more and more serious with the increase of speed. However, when the maglev train passes through the vertical curve, the tracking performance of the magnetic levitation system is also dictated by the radius of circular curve r and the angle of slope α.Therefore, the tracking performance of the magnetic levitation system in the different radiuses of circular curve and angles of slope need to be further analysis, calculation and verification. This will become a study emphasis in our further research.

7. Conclusions In this paper, we established a mathematical model with consideration of vertical track irregularity to discuss and analyze the impact on the tracking performance of the magnetic levitation system. The tracking performance of the magnetic levitation system on the vertical track irregularity condition was investigated and achieved at different speeds. The investigation carried out by us has revealed that the fluctuation of suspension gap becomes more and more serious with the increase of running speed. For the benefit of improving the tracking performance of the magnetic levitation system, we proposed a new strategy by installing an accelerometer on the electromagnet as a solution to address this problem. What’s more, this strategy has already been successfully implemented and applied to the suspension controller for a magnetic levitation system in the Changsha maglev express. The actual operation data demonstrates that the tracking performance of the magnetic levitation system was evidently improved. This fruitful work has produced an enormous application value.

Author Contributions: M.Z. is responsible for paper writing, modeling, algorithm design, simulation and experimental verification. X.L. is responsible for development and manufacture of controller. Z.L. is responsible for overall planning. Funding: This research was funded by National Key R & D Program of China, grant number 2016YFB1200601. Acknowledgments: This work was supported by “National Key R & D Program of China” under Grant 2016YFB1200601. Conflicts of Interest: The authors declare no conflict of interest.

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