Hopf bifurcation analysis of maglev vehicle–guideway interaction vibration system and stability control based on fuzzy adaptive theory

Computers in Industry 108 (2019) 197–209

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Computers in Industry journal homepage: www.elsevier.com/locate/compind

Hopf bifurcation analysis of maglev vehicle–guideway interaction vibration system and stability control based on fuzzy adaptive theory Yougang Suna , Junqi Xua,* , Haiyan Qiangb , Wenjia Wangb , Guobin Lina a

College of Transportation Engineering, National Maglev Transportation Engineering R&D Center, Tongji University, 201804, China Key Laboratory of Road and Traffic Engineering of the State Ministry of Education, Shanghai Key Laboratory of Rail Infrastructure Durability and System Safety, 201804, China b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 29 November 2018 Received in revised form 26 January 2019 Accepted 4 March 2019 Available online xxx

The vehicle–guideway interaction vibrations often occur when the parameters of the maglev system change. This phenomenon corresponds to the Hopf bifurcation in nonlinear dynamics. In order to solve the problem of maglev vehicle–guideway interaction vibration, the vehicle–guideway coupling dynamics model of maglev system considering the elasticity of the guideway is established firstly. Then, according to nonlinear dynamics theory and numerical simulations, the Hopf bifurcation rules of the maglev system are studied. Next, based on the Hopf bifurcation rules and the influence of control parameters on system vibration, the fuzzy inference method is used to establish the fuzzy control rules. A fuzzy adaptive tuning PID controller with variable parameter is designed for the vehicle–guideway interaction system. By identifying the disturbance or the change of the system parameters, the control parameters are adjusted automatically to keep the closed loop system away from the Hopf bifurcation point, which can restrain the vehicle–guideway interaction vibration. The simulation results show that the proposed fuzzy controller can adjust the levitation control proportional gain parameter K p ðtÞ online, which can improve the dynamic performance of the system and make the maglev system obtain a large state stability range, thus restrain the vehicle–guideway interaction vibration effectively. © 2019 Elsevier B.V. All rights reserved.

Keywords: Vehicle–guideway system Hopf bifurcation Fuzzy control Adaptive adjust Maglev

1. Introduction Maglev train is a new type of transportation, which has advantages that traditional wheel-rail transportation does not have [1,2], such as non-contact supports, low noise and strong climbing ability. At the same time, the maglev system is strong nonlinear and sensitive to the rail quality. The maglev vehicle– guideway interaction vibrations often happen in medium-low speed maglev line, which brings us many new challenges in engineering application and theoretical research. Interaction vibrations between vehicle and guideway have appeared in the test line of maglev vehicles in various countries [5– 7]. In the application of maglev train, how to solve the maglev vehicle–guideway interaction vibration is always the key and difficult points. Large vehicle–guideway interaction vibration not only affects the ride comfort of passengers seriously, but also causes damage to vehicle and guideway structure in the long operation. In recent years, this problem has been studied and discussed by scholars all over the world. In reference [8,9], the

* Corresponding author. E-mail address: [email protected] (J. Xu). https://doi.org/10.1016/j.compind.2019.03.001 0166-3615/© 2019 Elsevier B.V. All rights reserved.

vehicle–guideway coupling vibration analysis module, numerical integration method and PSD were utilized to study the influence of track irregularities on coupled vibration. Sun et al. [10] proposed a non-linear control method to identify the external disturbance of maglev train on-line, which effectively restrained the influence of external disturbance. However, the guideway was simplified to rigid body. In reference [11], the elasticity of guideway has been considered, the dynamic behavior of maglev vehicle is studied by numerical simulation when standing still or moving at low speeds. In reference [12], the guideway resonance problem of maglev train is solved from the point of view of levitation control system design by an adaptive vibration control scheme is proposed. In reference [13], the relationship between control parameters, guideway parameters and vibration characteristics of maglev vehicle– guideway interaction system under static levitation is studied from the point of nonlinear characteristics. Based on the flexible guideway, Xu et al. [14] analyzed the influence of double delay on coupled vibration of train and guideway, and proposed a method to estimate Hopf bifurcation of maglev system. The theoretical analysis was presented, but the method to suppress the vibration has not been proposed. In recent years, intelligent control algorithms such as nonlinear control [15–17], fuzzy control [18– 21], sliding mode control [22] have been applied to magnetic

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levitation system [23,24]. Sun et al. [25] presented an adaptive neural-fuzzy robust position control scheme for maglev train system. Xu et al. [26] proposed a magnetic flux observer to develop an adaptive sliding mode control for a maglev system. Sun et al. [27] presented a fuzzy H1 robust controller for magnetic levitation system based on T-S model, the simulation and experiment results showed the novel control strategy can solve the problems of model uncertainty and exogenous disturbances simultaneously. However the guideway was simplified as a rigid body. In references [28–30], the applications of nonlinear control law to the single suspension module system of maglev vehicles were discussed. Unfortunately, these studies all ignored the deformation of the guideway. In these studies above-mentioned, the structural experts often neglect the role of the levitation controller, meanwhile the control experts often simplify other factors beyond vehicles as external disturbances. So, the influence rule and control method of maglev vehicle–guideway interaction vibration did not solved completely. Previous studies on vehicle–guideway coupled vibration focus the structure of vehicle bogies and track characteristics generally. A simplified vehicle model is used to study the vehicle resonance problem caused by different track fundamental frequencies and irregularities [3,4]. In research, the maglev system is often simplified to a spring support system. Obviously, the traditional method has not studied the interaction vibration of maglev system systematically and comprehensively. Besides, the problem of coupling resonance between vehicle and guideway has not been studied and improved from the viewpoint of control system. A large number of studies have shown the coupled vibration of maglev vehicle–guideway corresponds to Hopf bifurcation in nonlinear dynamics [34–38]. In our paper, we study this problem from the novel viewpoints of coupling the guideway structure with levitation control. We proposed an effective vehicle–guideway interaction model for maglev system and illuminated the rule of the coupling vibration by Hopf bifurcation theory. An adaptive fuzzy control law was designed to remove the maglev coupled vibration, which has great significance in theory and practical value in engineering.

In thispaper, considering levitation control and guideway structure together, the vehicle–guideway interaction model of maglev system is constructed. Based on the nonlinear theory and simulation, the Hopf bifurcation law of maglev system is studied. Then, the influence of the key control parameters of the maglev system on the vibration characteristics of the guideway is studied. An adaptive control method based on fuzzy theory is proposed to suppress the coupled vibration between vehicle and guideway of maglev system. In our paper, the vehicle–guideway interaction vibration system is established in Section 2. Section 3 analyzes the maglev vehicle–guideway interaction system based onHopf bifurcation theory. Section 4 designs an adaptive controller to remove the vehicle–guideway interaction vibration based on fuzzy logic and inference. Simulation results are shown in Section 5. Finally, the conclusions and future work directions are drawn in Section 6. 2. Vehicle–guideway interaction system As shown in Fig. 1, maglev vehicles are levitated by multiple levitation points. According to the decoupling analysis of maglev bogie [31], the maglev system of train can be decomposed into the control problem of single electromagnet-guideway system. It is more universal to analyze and study the stability of single electromagnet-guideway system than to study multi electromagnet-guideway system. Without loss of generality, the following assumptions should be made before constructing the dynamic model of maglev vehicle–guideway interaction system: 1. The single electromagnet units (levitation points) have been fully decoupled mechanically through the maglev bogie and the levitation points do not affect each other. 2. In the analysis, the secondary suspension of the carriage is neglected, because the stiffness of the air spring supporting the carriage is less than the levitation stiffness. So the load variation of the electromagnet caused by the micro-vibration of the carriage can be neglected, that is to say, the air spring isolates the vibration of the levitation electromagnet and the carriage.

Fig. 1. Illustration of vehicle–guideway interaction system.

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3. Guideways on maglev line are generally simply supported. The length of guideway is much longer than the height of crosssection, and the deformation of guideway is much smaller than the length of guideway, so the mathematical model of guideway can be built according to Bernoulli–Euler beams. 4. Compared with the span length of the guideway, the length of the electromagnet is very small, so the length of the electromagnet can be ignored in the modeling. The illustration of vehicle–guideway interaction system is shown in Fig. 1. The origin of coordinates is O point in Fig. 1. Electromagnets and loads are simplified to an equivalent mass, which is denoted by m. z1 and z3 denote absolute displacement of electromagnet and guideway respectively. d ¼ z1  z3 denotes air gap. The current and voltage of the electromagnet are iðtÞ and uðtÞ respectively. Number of turns of coil is N. Coil resistance is R and area of coil is A. m0 denotes the permeability of air. According to Maxwell's equation andBiot–Savar's theorem: Rt cðiðtÞ; xðtÞÞdt ð1Þ F m ðiðtÞ; xðtÞÞ ¼ 0 @dðtÞ Ignore the leakage flux of the electromagnet winding, then

fL ¼ 0. According to the Kirchhoff law of the magnetic circuit: NiðtÞ cm ðiðtÞ; xðtÞÞ ¼ N RðdÞ

ð2Þ

2dðtÞ RðdÞ ¼ m0 A

ð3Þ

Substituting (2) and (3) into (1) can obtain the electromagnetic force equation of the magnetic levitation system as: F m ðiðtÞ; xðtÞÞ ¼ 

m0 AN iðtÞ

2 ð4Þ

dðtÞ

4

The relationship between voltage and current across the solenoid coil is: uðtÞ ¼ iðtÞR þ Nfm_

ð5Þ

where Lm is the instantaneous inductance of the electromagnet, which can be expressed as: Lm ¼

m0 N 2 A 2dðtÞ

;

fm ¼

Lm iðtÞ N

ð6Þ

Take a derivative of (6) with respect to time:

fm_ ¼

m0 NAdiðtÞ 2dðtÞ dt



m0 NAiðtÞddðtÞ 2

2d ðtÞ

ð7Þ

dt

By the formulas (5) and (7) can get electrical equation of electromagnet: um ðtÞ ¼ im ðtÞRm þ

m0 N2m Am dim ðtÞ 2xm ðtÞ

dt



be described by the Bernoulli–Euler beam equation as follows: EIg

@4 z3 ðy; tÞ @z ðy; tÞ @2 z3 ðy; tÞ þ rg þ Cg 3 ¼ f ðy; tÞ @t @y4 @t2

m0 N2m Am im ðtÞdxm ðtÞ 2x2m ðtÞ

dt

ð8Þ

Assuming the disturbance force of the suspension electromagnet (such as: wind force, force generated by sudden changes in the power grid, etc.), the dynamic equation of the suspension system in the vertical direction is described as follows:   2 d dðtÞ m AN2 iðtÞ 2 ¼ 0 þ mg þ f d m 2 dðtÞ 4 dt

ð9Þ

The unit length mass and bending stiffness of the guideway are rg and EIg , respectively; Cg is the viscous damping coefficient, and the span of the beam is L. The motion of the maglev guideway can

ð10Þ

where f ðy; tÞ is the external force acting on the guideway and can be approximated as: f ðy; tÞ ¼ F m ðtÞdðy  y0 Þ

ð11Þ

where y0 is the position of the electromagnet, F m ðtÞ is the electromagnetic force, and dðÞ is the Dirac function. According to the vibration theory of continuous beams, it is known that high-order mode of vibration can occur only when the energy contained in the excitation of the orbit is high [32,33]. Therefore, maglev guideway is only discussed for first-order mode generally. The modal analysis method is used to research the problem expressed by Eq. (10). For the simple supported beam, the firstorder modal frequency v1 and the first-order modal shape function f1 ðyÞ are: sffiffiffiffiffiffiffi EIg v1 ¼ l1 2 ð12Þ ; f1 ðyÞ ¼ sinðl1 yÞ

rg

where l1 ¼ p=L.According to the modal superposition theory, the solution of Eq. (10) can be expressed as: z3 ðy; tÞ ¼ f1 ðyÞq1 ðtÞ

where Rðxm Þ can be expressed as:

2

199

ð13Þ

where q1 ðtÞ is the amplitude of the first-order modal displacement as a function of time. Substitute (13) into (10), multiply both sides by the previously mentioned equation f1 ðyÞ, and then integrate both from 0 to L simultaneously. €1 ðtÞ þ 2j1 v1 q_ 1 ðtÞ þ v21 q1 ðtÞ ¼ q

2f1 ðy0 Þ F ðtÞ rg L m

ð14Þ

where v1 and j1 are the first-order modal frequency and damping 1 is defined. Both sides of ratio respectively, and A1 ¼ 2f1 ðy0 Þr1 g L 2

Eq. (14) are multiplied by f1 ðyÞ to obtain: €z3 ðtÞ þ 2j1 v1 z_ 3 ðtÞ þ v21 z3 ðtÞ ¼ A1 F m ðtÞ

ð15Þ

Additionally the remaining system state variables are selected as follows: z2 is the vertical speed of the electromagnet, z4 is the vertical speed of the guideway, and z5 ¼ i is the electromagnet current. According to (8), (9) and (15), the first-order mode of maglev vehicle–guideway interaction system is: 8 dðtÞ ¼ z1 ðtÞ  z3 ðtÞ > > > > A2 z25 ðtÞ > > > F m ðtÞ ¼ > > m d2 ðtÞ < ð16Þ m€z1  ¼ mg F m ðtÞ   > > > > U ¼ 2A2 z_  2A2 z5  ðz_  z_ Þ þ Rz > > 5 5 1 3 2 > d > d > :€ z3 þ 2j1 v1 z_ 3 þ v21 z3 ¼ A1 F m ðtÞ 2

2

ðl1 yÞ where A1 ¼ 2 sinrgL and A2 ¼ m0 4N A.

At present, the practical application of the maglev train system utilizes the PID controller basically, that is, two states of air gap error z1 ðtÞ  z3 ðtÞ  dref and electromagnet speed z2 are used as feedback control loop. It is of practical significance to research the dynamics of the vehicle–guideway interaction under the existing controller. The feedback controller is: U ¼ U ec þ K p ðz1  z3  dref Þ þ K d z2

ð17Þ

where U ec is the voltage at the equilibrium point and dref is the reference air gap. K p and K d are the feedback gains for each loop.

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Substituting Eq. (17) into (16), the state equation of maglev vehicle–guideway interaction system can be obtained as follows: 8 z_ > > > 1 > > > > > z_ 2 > > > <_ z3 > > > z_ 4 > > > > > > > > : z_ 5

¼ z2 ¼g ¼ z4

The characteristic equation of matrix AðK p ; K d Þ can be expressed as follows: 5

4

3

2

f ðlÞ ¼ jlI  Jj ¼ l þ Al þ Bl þ C l þ Dl þ E

ð22Þ

where

 2 A2 z5 m z1  z3

dref R ; A ¼ 2j1 v1 þ 2A2



2 z5  2j1 v1 z4  v21 z3 z1  z3 z5 ðz2  z4 Þ Rðz1  z3 Þz5 z1  z3 ¼  þ ðU ec þ K p ðz1  z3  dref Þ þ K d z2 Þ 2A2 2A2 z1  z3

B ¼ v1 2 þ

Kd

pffiffiffiffiffiffiffiffiffiffiffiffi mgA2 j1 v1 dref þ R A2 m A2

¼ A2 A1

ð18Þ 2

rffiffiffiffiffiffiffiffiffiffi   2g þ 2A1 mg  v1 2dref g  R C ¼ ð2j1 v1 K d þ K p þ A1 K p mÞ mA2 2A2

2

ðl1 yÞ where A1 ¼ 2 sinrgL and A2 ¼ m0 4N A. K p and K d are parameters that

can be adjusted in the control system.

2gj1 v1 R þ 2j1 v1 K p A2 rffiffiffiffiffiffiffiffiffiffi 2 gv g ¼  1 R þ v21 K d A2 m A2

D¼

3. Analysis of maglev vehicle–guideway interaction system 3.1. Hopf bifurcation criterion of maglev system Let z_ 1 ¼ 0, z_ 2 ¼ 0, z_ 3 ¼ 0, z_ 4 ¼ 0, z_ 5 ¼ 0, then the unique singularity of the system can be obtained within the working interval as follows: rffiffiffiffiffiffiffi A1 mg A1 mg mg þ dref ; z02 ¼ 0; z03 ¼ ; z04 ¼ 0; z05 ¼ dref z01 ¼ A2 v1 2 v1 2 The Jacobian matrix of the system in the singularity can be expressed as follows: 2

0 6 6 2g 6 6 dref 6 6 6 0 6 AðK p ; K d Þ ¼ 6 6 2A mg 1 6 6 6 dref 6 6 4 kp dref 2A2

1

0

0

2g dref

0

0

0

1

0

2A1 mg  v1 2 dref

2j1 v1

0

rffiffiffiffiffiffiffi mg dref kd þ A2 2A2





kp dref 2A2



rffiffiffiffiffiffiffi mg A2

0 pffiffiffiffiffiffiffiffiffiffiffiffi 2 mgA2  mdref

3

7 7 7 7 7 7 7 0 7 pffiffiffiffiffiffiffiffiffiffiffiffi 7 2A1 mgA2 7 7 7 dref 7 7 7 dref R 5  2A2

Theorem 2. Classical Hopf bifurcation criterionAccording to Hopf bifurcation theory of nonlinear systems [39–42], when:

The equivalent system of the nonlinear system (18) is: z 2 Rn

E

According to the definition of Routh criterion, the Routh table of system (18) is written as shown in Table 1. According to the above Routh table, the following conclusions can be drawn: when A, B, C, D and E are positive and R3;1 , R4;1 and R5;1 are greater than zero, the symbols in the first column of the Routh table are all positive. In this case, all eigenvalues of matrix AðK p ; K d Þ have negative real parts, so the approximate linear system corresponding to system (18) at the working point is stable. Since there is no pure imaginary root in matrix AðK p ; K d Þ, the singularity of system (18) is hyperbolic singularity. At the same time, the partial derivative of the system (18) exists everywhere near the singular point. According to Theorem 1, the system (18) has the same topological structure at the working point as its corresponding approximate linear system, so the system (18) is also stable near that point. When R5;1 ðmÞ ¼ 0, there may be Hopf bifurcation in the system. A large number of studies have shown that the coupled vibration of Maglev vehicle-rail corresponds to Hopf bifurcation in non-linear dynamics [34–38]. Next, the Hopf bifurcation of maglev vehicle–guideway interaction system is judged.

ð19Þ

_ ¼ AðK p ; K d Þðz  z0 Þ þ Oðz  z0 Þ; z

rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi g g þ v21 K d ; mA2 mA2

ð20Þ

where Oðz  z0 Þ is infinitesimal of higher order.

1) The Jacobian matrix J of the maglev vehicle-rail coupling system at the equilibrium point has a pair of complex roots l and l.

Theorem 1. Hartman–Grobman theorem If z0 is the hyperbolic singularity of a nonlinear system (18), and satisfies the condition:   Oðz  z0 Þ ð21Þ ¼0 lim z!z0 jz  z0 j Then the nonlinear system (18) at the isolated singular point x0 has the same topological structure of the linearized system (20) at its equilibrium point. The Hartman–Grobman theorem is also known as the first approximation theorem. By analyzing this theorem, the following can be easily known: Note 1: The vector field of the linearized hyperbolic nonlinear system at the equilibrium point of hyperbolic nonlinear system is topologically equivalent to that of the original system near the equilibrium point. Note 2: If the nonlinear system and the linearized system are topologically equivalent at the equilibrium point, then the stability of the nonlinear system near the equilibrium point can be determined by the linearized system.

lðmÞ ¼ aðmÞ þ ivðmÞ where vðm0 Þ ¼ v0 > 0; aðm0 Þ ¼ 0; a0 ðm0 Þ 6¼ 0. 2) The other n  2 characteristic roots of Jðm0 Þ have negative real parts Then the Hopf bifurcation occurs at parameter m ¼ m0, that is, the resonance of the maglev system occurs near m ¼ m0. m is the parameter of the vehicle-rail coupling system. Table 1 Routh table of vehicle–guideway interaction system.

l 5

R

l l4 l3 l2 l1

R1;1 ¼ 1 R2;1 ¼ A

l0

R6;1 ¼ E

R1;2 ¼ B R2;2 ¼ C

R3;1 ¼ ABC A 2

2

A DþAE R4;1 ¼ ABCCABC 2

R5;1 ¼ ABCDA

D2 AB2 Eþ2ADEC 2 DE2 þBCE ABCC 2 A2 DþAE

R3;2 ¼ ADE A R4;2 ¼ E

R1;3 ¼ D R2;3 ¼ E R3;3 ¼ 0 R4;3 ¼ 0

R5;2 ¼ 0

R5;3 ¼ 0

R6;2 ¼ 0

R6;3 ¼ 0

Y. Sun et al. / Computers in Industry 108 (2019) 197–209 Table 2 System parameters when Hopf bifurcation occurs. m

Kp

dref

A1

j1

v1

750 750 750 750

140,000 160,000 120,000 140,000

0.01 0.01 0.01 0.01

0.0005 0.0005 0.0005 0.0005

0.0005 0.0005 0.0005 0.0005

154.37 177.48 122.24 126.99

mc can be solved from equation R5;1 ðmÞ ¼ 0, and the Hopf bifurcation can be judged by ReðUðmc ÞBðmc ÞVðmc ÞÞ 6¼ 0 and R3;1 ðmc Þ > 0 when m ¼ mc . When the parameters of the maglev change, the Hopf bifurcation point of parameter v1 can be calculated. Parts of the data are listed in Table 2. In Table 2, the system parameters are given when the Hopf bifurcation occurs in the vehicle–guideway coupling system. If we want to find the parameters range of the system stability, the type of the Hopf bifurcation needed to be further analyzed. For the fifthorder system such as maglev vehicle–guideway coupling, the related analytical expressions are very complex, and it is very difficult to use the analytical method to judge the type of Hopf bifurcation. The numerical simulation method can be utilized to study the relationship between these system parameters and the system stability or the stability range [43–45].

Fig. 3. Phase trajectory for system when v1 ¼ 175 Hz.

3.2. Numerical simulation Taking the data of first row in Table 2 as an example, the effectiveness of the above Hopf bifurcation point calculation is verified by numerical simulation, and the direction and stability range of Hopf bifurcation are studied. Because the calculation and simulation steps of different system parameters are the same. In this section, the numerical simulation of the guideway frequency v1 is carried out only. The data of first row in Table 2 shows that v1 of Hopf bifurcation point of the vehicle–guideway system is 154:37 Hz at this time. When v1 ¼ 175 Hz, the air gap response of the vehicle– guideway system is shown in Fig. 2. The phase trajectory of the air gap is shown in Fig. 3. The guideway and current response are shown in Figs. 4 and 5. We can learn from Fig. 2, the air gap reaches the reference air gap and remains stable after 1 s. Fig. 3 of phase trajectory indicates the system is asymptotically stable. It can be concluded that the system is asymptotically stable when the

Fig. 2. The air gap response of system when v1 ¼ 175 Hz.

Fig. 4. The guideway response when v1 ¼ 175 Hz.

Fig. 5. The current response when v1 ¼ 175 Hz.

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Y. Sun et al. / Computers in Industry 108 (2019) 197–209

parameter v1 is larger than the Hopf bifurcation point, that is, on the right side of the bifurcation point. When v1 ¼ 154:37 Hz, the air gap response of the vehicle– guideway system is shown in Fig. 6. The phase trajectory of the air gap is shown in Fig. 7. As can be seen from Fig. 6, at the Hopf bifurcation point, the system has a continuous self-excited oscillation, and the air gap has a periodic motion solution, which is a coupled oscillation. In Fig. 7, the air gap phase trajectory shows a limit cycle. In fact, the phase trajectory of the air gap always has a limit cycle when w fluctuates about 10% upward at 154.37, but the amplitude of the guideway decreases gradually and converges with time, while the vibration frequency increases. We can learn from Figs. 2–9 that the limit cycle exists in the range of parameter v1 less than bifurcation points, so it is subcritical Hopf bifurcation. Similarly, you can do a bunch of simulations. Only simulation results of Hopf bifurcation points of v1 are given here. Combining with the numerical and simulation results in Table 2, it can be concluded that if the natural frequency v1 of the track is taken as the bifurcation parameter, the corresponding v1 of the coupled vibration of the system is the calculated critical value. The critical value of Hopf bifurcation point is the lower bound in the range of v1 corresponding to system stability. That is to say, when the control parameters and system parameters are completely determined, the track parameters v1 can make the coupled system stable in the open interval with the Hopf bifurcation point as the lower bound, which is called sub-

Fig. 8. The guideway response.

Fig. 9. The current response.

Fig. 6. The air gap response of system v1 ¼ 154:37.

critical bifurcation [46–48]. By using the same method, the influence rule and the stability range of the system can be obtained when other parameters (such as other parameters m, j1 and A1 , control parameters K p ; K d ) are taken as Hopf bifurcation parameters. 4. Maglev vehicle–guideway coupling vibration control method 4.1. Influence of control parameters on vibration

Fig. 7. Phase trajectory for system when v1 ¼ 154:37.

System stability is only the basic requirement of control. In order to achieve fast and stable control, the system also needs good dynamic performance. In this section, in addition to studying the stable region of the control parameter degree to the system, it will also analyze the regulation of the dynamic characteristics of the system by adjusting the control parameters on the basis of system stability. In order to study the effect of K p on the stability and dynamic characteristics of the system, based on the first line data in Table 2, keep K d ¼ 100 and the values of K p are respectively: K p ¼ 6290, K p ¼ 6310, K p ¼ 6800, kp ¼ 7200. The simulation

Y. Sun et al. / Computers in Industry 108 (2019) 197–209

results of the air gap response of the maglev vehicle–guideway interaction system are shown in Fig. 10. Meanwhile, in order to study the effect of kd on the stability and dynamic characteristics of the system, kp ¼ 6900 is chosen and the values of kd are respectively: kd ¼ 100, kd ¼ 200, kd ¼ 300, kd ¼ 500. The simulation results of the air gap response of the maglev vehicle–guideway interaction system are shown in Fig. 11. As shown in Fig. 10, the value of K p has a great relationship with the type of Hopf bifurcation. With the increase of K p , that is, K p is less than 6300, the levitation air gap is divergent and unstable. When K p changes from 6290 to 6310, the levitation air gap of the system is obviously convergent, but the response speed is high and there is a large static error. When K p changes to 6800, the response speed of levitation air gap becomes faster and the static error becomes smaller. But K p cannot be too large. When K p changes to 7200, the levitation air gap begins to oscillate, although the response speed is faster and the static error is smaller. So K p can’t be too large. From the simulation analysis, under the first line of system parameters in Table 2, it can be concluded that the range of K p is ð6300; 7500Þ which can make the system stable. From Fig. 11, it can be seen that the levitation air gap of the system is still convergent and the static error remains unchanged with the increase and decrease of the K p . However, the increase of K p can suppress the oscillations to a certain extent, but will slow down the system response speed. Therefore, it can be concluded

that the value of K p cannot change the type of Hopf bifurcation, but it has an important influence on the dynamic characteristics of the system. In order to study the dynamic response of the system under external interference, K p ¼ 6800, K d ¼ 100 is selected. The external interference is shown in Fig. 12. Under the interference, the air gap response of maglev system is shown in Fig. 13. Where, the blue solid line is air gap response, the dotted line is error e, and the red solid line is error change rate ec. From Fig. 13, during the rising time of the system from 0 to t1, we can see that the maximum e gradually decreases to 0, and the ec first changes from 0 to the maximum, then decreases gradually. When the system is in steady state, the positive and negative impulse interference force is applied to the system at t2 and t5. As shown in Fig. 12, the time periods of t2–t3 and t5–t6 are the process of the system deviating from steady state when positive and negative impulse interference forces are applied, e and ec are the same signs. The time periods of t3–t4 and t6–t7 are respectively the process of the system returning to steady state after being disturbed by positive and negative impulse, e and ec are different signs. Different system parameters correspond to different control parameters. A method to improve the dynamic performance of the system is proposed based on the magnetic levitation control. kd is changed to variable parameter kd ðtÞ. The system focuses on rapid response and stable regulation. It can be seen from Fig. 13 that changing kd ðtÞ through the control system can shorten the rise time of the system and improve the steady-state characteristics. When the system deviates from the steady state, e and ec are the

Fig. 10. The dynamic response of the system when Kp changes. Fig. 12. External interference force.

Fig. 11. The dynamic response of the system when Kd changes.

203

Fig. 13. Effect of disturbance on dynamic response of maglev system.

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same signs. kd ðtÞ can be adjusted by adjusting the system, which can effectively reduce or suppress the deviation phenomenon and improve the anti-interference ability of the system. At the same time, it can effectively reduce or avoid overshoot in the regulation stage of the system, shorten the regulation time of the system and improve the stability of the system. 4.2. Fuzzy adaptive controller for maglev system As is described in Figs. 10–13, it is difficult to determine the quantitative relationship between e, ec and gain K p ðtÞ according to the actual output response in the rising and stable stages of the maglev vehicle–guideway system, and to establish an accurate mathematical model which aims to realize the variable parameter control of the system. Conventional PID controllers are generally only suitable for linear systems and cannot adjust fixed proportional gain kp , integral gain ki and differential gain kd on-line and simultaneously according to the actual situation of the system. In order to adapt to the uncertainties, nonlinearity, time-varying and strong disturbance of the control plant, it is necessary to adjust the kp , ki and kd adaptively on-line. Fuzzy control [49,50] is a nonlinear control strategy based on fuzzy set theory, fuzzy linguistic variables and fuzzy logic inference. Fuzzy control system realizes the control of the system according to the system error and error change rate in the control process. Fuzzy control has the advantages of good robustness and high control performance for the control of nonlinear and complex objects. Fuzzy adaptive PID control uses the method of fuzzy mathematics to express the control rules as rules base by fuzzy sets in the computer knowledge base. According to the actual response of the control system, the computer carries out fuzzy inference and decision-making, which can automatically realize the most optimal adjustment of the PID parameters and restrain the coupled vibration of the train and guideway. Combining the advantages of PID control and fuzzy control, according to the e and ec changing rules of the output response of the system, the fuzzy adaptive tuning method of kp , ki and kd is used to change the Kp(t) of the system on-line. When the parameters of maglev system are changed, the PID parameters are required to be adjusted automatically online based on fuzzy inference. The control schematic diagram of the proposed fuzzy adaptive controller is shown in Fig. 14. As shown in Fig. 14, the fuzzy parameter adaptive PID consists of a fuzzy control part and a variable parameter PID part. The e and de/

dt in Fig. 14 represent the error and the error change rate respectively. The relationship between output variables and PID parameters is as follows: 8 < kp ¼ kp0 þ Dkp ð23Þ k ¼ ki0 þ Dki : i kd ¼ kd0 þ Dkd where kp0 , ki0 and kd0 are the initial value of the parameter kp, ki and kd . Dkp , Dki and Dkd are the output of the fuzzy controller. The fuzzy adaptive PID controller designed in this section has two inputs (air gap error e and change rate of air gap error ec), and three outputs (proportional coefficient ½ Emin Emax , integral coefficient Dki and differential coefficient Dkd ). The fuzzy relationship between PID parameters and E, EC is calculated continuously by changing the fuzzification into fuzzy linguistic variables E and EC. According to the rules of fuzzy control, Dkp , Dki and Dkd are revised on-line to realize the control parameters adaptive adjusting. Consequently, the maglev system will have good dynamic and static characteristics, and restrain the coupled vibration of vehicle and guideway. 4.3. Fuzzification of control parameters and membership function The actual range of input (e, ec) and output (Dkp , Dki , Dkd ) of the two-dimensional fuzzy PID controller is called the basic universe, which is recorded separately as ½ emin emax , ½ demin demax , ½ kpmin kpmax , ½ kimin kimax , Dkp . After the fuzzification, the basic domains correspond with the fuzzy domains, which is respectively ½ Emin Emax , ½ DEmin DEmax , ½ KP min KPmax , ½ KImin KImax  and ½ KDmin KDmax . The linguistic variables of (e, ec) and (Dkp , Dki , Dkd ) are E, DE and KP, KI, KD separately. The whole fuzzy universe is divided into seven fuzzy subsets: positive big (PB), positive medium (PM), positive small (PS), zero (ZO), negative small (NS), negative medium (NM), negative big (NB). Therefore, the fuzzy domains of e and ec are f NB; NM; NS; ZO; PS; PM; PB g, and similarly, the fuzzy universes of DK p , DK i and K d are fNB; NM; NS; ZO; PS; PM; PBg. The level of fuzzy domains are E ¼ f6; 4; 2; 0; 2; 4; 6g, DE ¼ f3; 2; 1; 0; 1; 2; 3g, KP ¼ KI ¼ KD ¼ f6; 4; 2; 0; 2; 4; 6g. In the process of fuzzify deviation e and deviation change rate ec, the definition of quantization factor is introduced. When dealing with input, take deviation e as an example: assuming the change range of actual input e is ½emin ; emax , and the

Fig. 14. Schematic diagram of proposed fuzzy adaptive controller.

Y. Sun et al. / Computers in Industry 108 (2019) 197–209

range of change after fuzzy processing is ½ Emin quantization factor can be defined as: ke ¼

Emax  Emin emax  emin

205

Emax , so the

ð24Þ

When dealing with the control output, take the proportional link as an example. Assuming the range of change of proportional link K p is ½ kpmin kpmax , and the range of change after fuzzy treatment is ½ KPmin KPmax , the quantitative factor can be defined as: kp ¼

kpmax  kpmin KPmax  KPmin

ð25Þ

According to formulas (24) and (25), we can define the quantization factors of the transformation between the actual domain and the fuzzy domain in the control system. If the basic domain of deviation e is [8,8] mm and the fuzzy domain is f6; 4; 2; 0; 2; 4; 6g mm, the quantitative factor mapping the deviation e to the fuzzified domain is ke ¼ 12=16 ¼ 0:75 mm. Similarly, the fuzzy domain of the deviation change rate ec is f3; 2; 1; 0; 1; 2; 3g, and the quantization factor is 0.375. The fuzzy domain of DK p , kp and DK d is [6 4  2 0 2 4 6]. The basic domain of DK p is [0 12000]; the basic domain of DK i is [0 1200]; the basic domain of DK d is [0 2100]. Similarly, the fuzzy domain can be mapped to the basic domain according to formula (25) when the output of control variables is controlled. The quantization factors of defuzzification are respectively defined as 1000, 100 and 175. Considering that the control action of suspension system is more precise near the equilibrium point, on the basis of dividing the input-output domain into seven levels, NM, NS, ZO, PS and PM are used triangular membership functions with strong sensitivity, while NB and PB are used S-type membership functions to soften the mutation of input. The selection of membership functions is shown in Figs. 15–17. 4.4. Fuzzy control rules The key to realize the fuzzy controller is to determine the fuzzy control rule table. Fuzzy control rule table is a series of control rules based on expert experience and referring to a large number of manual control practices. These rules are essentially parameters kp , ki and kd . The effects of parameters kp , ki and kd on the dynamic characteristics, response time and steady-state characteristics are as follows: the main function of kp is to adjust the response speed; the main function of ki is to eliminate the steady-state error of the system; and the main function of kd is to improve the dynamic characteristics of the system. According to the influence of the above parameters kp , ki and kd on the system control performance, the fuzzy parameter adaptive

Fig. 16. Membership function curve of ec.

Fig. 17. Membership function curve of DK p , DK i , DK d .

control principle of the relationship between input and output variables is obtained as follows: (1) When the e is large, in order to improve the response speed and reduce the error rapidly, kp should take a larger value. While in order to prevent differential supersaturation, kd should take a smaller value. In order to avoid overshoot caused by integral saturation, the integral effect should be limited by taking ki ¼ 0. (2) When (e  de/dt) < 0, the output of the system tends to change toward steady-state value. kd should take a larger value to eliminate the deviation as soon as possible, while ki and kd should take a smaller value. (3) When (e  de/dt) > 0, the output of the system begins to deviate from the steady state value, and the deviation changes in the direction of increasing. In order to control overshoot, kd should take a larger value, while kp and ki should take a smaller value. (4) The de/dt reflect the change rate of error. The larger the value of de/dt is, the larger the value of ki should be, and the smaller the value of kp should change. Concerning the controlling effects of parameters kp , ki and kd on various error and change rate of error and engineering experience, the fuzzy control rules table of maglev system is obtained as shown in Tables 3–5. 4.5. Fuzzy inference and defuzzification The process of fuzzy inference is the process of obtaining the fuzzy subset of the output variables by fuzzy inference or solving the fuzzy relation equation according to the fuzzy control rules for the input fuzzy variables. Fuzzy inference algorithms directly related to system performance can be expressed as follows: IfðPerform index is Aij Þ then ðDkp is Aik Þ and ðDki is Ail Þ and ðDkd is Aim Þ

Fig. 15. Membership function curve of e.

ð26Þ

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Table 3 Fuzzy inference rules of DKp.

DKp ec

e

NB NM NS ZO PS PM PB

NB

NM

NS

ZO

PS

PM

PB

PB PB PB PB PM PS ZO

PB PB PB PB PM ZO ZO

PM PM PM PM PS ZO NS

PM PM PS ZO PS NS NM

PS ZO NS NM NM NM NM

ZO ZO NM NB NB NB NB

ZO PS NM NB NB NB NB

ec

e

NB NM NS ZO PS PM PB

NB

NM

NS

ZO

PS

PM

PB

NB NM NM NS NS ZO ZO

NM NM NS NS ZO ZO ZO

NM NM NS ZO ZO ZO NS

PS PS ZO ZO ZO NS NS

PS ZO ZO ZO PS PM PM

ZO ZO ZO PS PS PM PM

ZO ZO PS PS PM PM PB

Table 5 Fuzzy inference rules of DKd.

DKd ec

NB

NM

NS

ZO

PS

PM

PB

PS NB NB NS NB NB PS

PS NB NB NS NB NB PS

ZO NM NM NS NM NM ZO

NS NS NS NS NS NS ZO

ZO PM PS ZO PS PM ZO

PS PB PS PS PB PB PS

PS PB PM PM PB PB PS

Perform index can be any observable dynamic or static parameters, such as steady-state error, overshoot, etc. Its advantage is that the adjustment precision is high. However, due to the need for several complete control processes to obtain performance parameters, the real-time performance of the actual control is poor. The error-driven fuzzy inference algorithm designed in this paper is as follows: IfðE is A11 Þ and ðEC is A12 Þ then ðDkp is A21 Þ and ðDki is A22 Þ and ðDkd is A23 Þ

where Dkx is the decision result of weighted average algorithm on 0 the fuzzy set Dk x ; W xj is the weighted coefficient, the value is taken from each element in the domain; x 2 ðp; i; dÞ; uj represents the membership function. The weighted averaging algorithm not only highlights the main information but also takes into account the rest information. Practice shows that the weighted averaging algorithm can meet the control requirements of the system.

According to the test prototype of the national maglev center, a set of parameters values is given in Table 6. The value range of K p ðtÞ, which keeps the system stable, is defined as follows:

G ¼ ðK PL K PR Þ

ð29Þ

where K PL and K PR are left poles and right poles of proportional feedback coefficient K p ðtÞ, respectively. Based on the simulation results of Fig. 10, we can get that: G ¼ ð6300 7500Þ. 5.1. Simulation 1

e

NB NM NS ZO PS PM PB

ð28Þ

5. Numerical simulation

Table 4 Fuzzy inference rules of DKi.

DKi

control object is as follows: PL 0 j¼1 W xj uj ðDk x Þ Dkx ¼ P 0 L j¼1 uj ðDk x Þ

ð27Þ

The value of Aij in (26) is a fuzzy range fNB; NM; NS; ZO; PS; PM; PBg. If each of the fuzzy variables E and EC has L fuzzy subsets, the output variables accordingly have L2 membership rules. According to Tables 3–5 and (27), the adjusted membership degree of three output variables DK p , DK i , DK d on the fuzzy rules for all different E and EC can be obtained. In the formulation of fuzzy rules, the relationship between input e, ec and output variable DK p , DK i , DK d can be observed more intuitively by surf view observer, as shown in Fig. 18. The result of the fuzzy inference is a set of fuzzy vectors, which cannot be directly used as control variables. The precise quantity obtained by the fuzzy inference can be used as the output control quantity only after the defuzzification. The defuzzification is a mapping from the action space of fuzzy control to the precise control action space. There are many algorithms for defuzzification. The weighted average algorithm designed according to the

Let K p0 ¼ 6800, and obviously K p 2 G. After the system runs for 10 s, the system step response is shown in Fig. 19 when the suspension mass suddenly decreases by 20%. As can be seen from Fig. 19, after 10 s of operation, the system undergoes a transient adjustment process due to the sudden decrease of suspension mass. After returning to the steady state, the system produced an obvious self-excited vibration. As the suspended mass decreases, it can be seen from the previous simulation that the interval G shifts to the left. Because the value of K p0 is fixed, the distance between K p0 and K PR decreases. Because K p0 is close to K PR , the system generates selfexcited vibration. 5.2. Simulation 2 Let the proportional feedback coefficient be adjusted according to Hopf bifurcation law with the change of suspension mass. The step response of the system and the change curve of control parameter K p ðtÞ under the action of fuzzy PID are shown in Figs. 20 and 21, respectively. As can be seen from Fig. 20, after the system runs for 10 s, the suspension mass suddenly decreases. When the system reaches steady state again, the suspension clearance remains near the ideal suspension clearance, and there is no obvious self-excited vibration in the system. However, in Fig. 21, after the suspension mass decreases, the proportional feedback coefficient K p ðtÞ decreases correspondingly under the regulation of fuzzy PID. According to the calculation in simulation 1, G shifts to the left after the suspension mass decreases by 20%. According to the designed fuzzy reasoning system, K p also moves to the left. Moreover, the relative position between K p ðtÞ and G is guaranteed to remain unchanged. Therefore, the dynamic characteristics of the system are not changed, and there is no self-excited vibration. This is the reason why the fuzzy adaptive control system can suppress the vehicle–guideway interaction vibration system. It can be seen from Figs. 19–21 that when fuzzy adaptive tuning PID variable parameter control is adopted, the dynamic

Y. Sun et al. / Computers in Industry 108 (2019) 197–209

207

Fig. 18. The relation between e, ec and DK p , DK i , DK d .

Table 6 Parameters values of the system. Physical quantity

Value

Physical quantity

Value

Mass, m/kg Number of turns of coil, Nm Area of coil, Am/m2 Coil resistance, Rm/V

750 450 0.024 1.2

Controls parameter, Kd Permeability of air, m0/(H m1) Guideway frequency, v1 Damping ratio, z1

100 4p  107 185 0.008

Fig. 19. System dynamic response while the mass decreases 20% and Kp is fixed after running 10 s.

208

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Fig. 20. System dynamic response while the mass decreases 20% and Kp is adjusted adaptively after running 10 s.

References

Fig. 21. The variance curve of control parameter Kp(t) under fuzzy PID.

performance and anti-disturbance performance of the system are significantly improved. Because the proportional feedback coefficient K p ðtÞ can be automatically adjusted according to fuzzy reasoning, it has a significant suppress effect on the vehicle– guideway interaction vibration caused by parameter perturbation. 6. Conclusion The maglev vehicle–guideway interaction vibration problem is a significant problem which greatly influences the stability of the maglev system during operation. In this paper, based on the methods of electromagnetics, structural dynamics and modal analysis, the mathematical model of vehicle–guideway interaction system is established. Based on Hopf bifurcation theory, the vibration rule of maglev interaction system is analyzed. The effect of the control parameters on the vehicle–guideway coupling system is discussed. Then, according to analysis results and the strong adaptive ability of fuzzy control, fuzzy adaptive tuning PID control technology is utilized to tune the three gain parameters of the PID on-line, and the proportional gain parameter K p ðtÞ of the system can be changed at different stages of the system output response to avoid self-excited vibration. The simulation results show that the proposed adaptive fuzzy control method is effective. Our future works will study the influence rule between the track irregularities and velocity of the maglev train. Acknowledgements This research is supported by the National Key Technology R&D Program of the 13th Five-year Plan, Research on Key Technologies of Medium Speed Maglev Transportation System (2016YB1200601).

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