Control algorithm to reject the effect of iron sleepers for magnetic levitation vehicle using LSM with PM Halbach and ironless coil/ Geleziniu pabegiu poveikio pasalinimo transporto priemonei su magnetine pagalve valdymo algoritmas naudojant linijinius sinchroninius variklius su nuolatiniais Halbacho magnetais ir gelezies neturinciomis ritemis.

Zhang, Xiao ; Cheng, Hu ; Li, Yungangm 等

1. Introduction

In magnetic levitation (maglev) system, suspension, guidance and propulsion are provided by magnetic forces. Basically, there are two kinds of linear motors utilized to provide propulsion force for maglev vehicles [1, 2]. One design is called linear synchronous motor (LSM), and the other is linear induction motor (LIM). LSM possesses the dominant advantage of high efficiency, and is widely applied in high-speed maglev vehicles, such as Germany TR maglev vehicle and Japan MLU/MLX superconductive maglev vehicle. LIM is widely used in low-speed maglev systems, such as Japan HSST maglev vehicle. The reason is that LIM suffers from the influence of eddy current [3], which in return generates acceleration resistance for the maglev vehicle. As the increase of people's demands for rapidity and energy-saving, it is appreciable that the tendency of propulsion system for maglev vehicles is toward LSM.

Typically, LSM in the maglev system is composed of iron coils on the guideway supplying primary magnetic field and electromagnets on the vehicle offering secondary magnetic field. There are energy-consuming in both the electromagnets and the iron coils in this kind of LSM. One method to reduce the energy-consuming on the vehicle is to replace electromagnets with Permanent Magnets (PMs) [4, 5], which can provide field without power supply. The improvement of the performance of PM materials [6] and the present of Halbach array [7] greatly promote the application of PM in LSM. Currently the maximum magnetic energy product of permanent magnets applicable to commercial manufacture has achieved at 397.9 kJ/[m.sup.3]. And Halbach array is an innovative combination of PMs arranged with different magnetization directions and it can enhance the magnetic field on one side while weaken the magnetic field on the other side. Iron coils should not be utilized if the magnetic field on the vehicle is provided by PM Halbach, or there will be strong attractive force (normal force) between the PMs on the vehicle and the iron on the guideway. Thus LSM with PM Halbach and ironless coil is a good candidate to maglev propulsion system, and it has the characteristics of zero power supply on the vehicle and zero normal force. This innovative structure of LSM has been successfully utilized in American GA maglev vehicle.

The magnetic field and force generated by LSM with PM Halbach and ironless coil are studied in previous literatures. H. Bergh et al. presented the approximate analytical solution to magnetic field of Halbach near the surface and deduced the propulsion and normal force generated by the LSM [8]. To investigate the magnetic field far away from the surface, J.F. Hoburg analyzed the static magnetic field of dual Halbach array for GA maglev vehicle by magnetization charge theory, but the results are quite complicated [9]. C.S. Li et al. investigated the optimization technique of PM Halbach array for electrodynamic suspension (EDS) maglev [10]. Magnetic field and optimization design of rotary motor using PM Halbach are conducted by Z.P. Xia et al. [11] and M. Markovic et al [12]. X. Zhang et al. investigated the magnetic field and the magnetic force of LSM with PM Halbach and ironless coil by introducing surface current theory [13].

However, no attention has been paid to the iron sleepers on the guideway, with which attractive force can be generated between the PM Halbach and the guideway. The direction of this attractive force is opposite to the suspension force, which will increase the suspension load and affect the suspension performance.

This paper focuses on analysis of the effect of the iron sleepers on the suspension system using LSM with PM Halbach and ironless coil and the design of the control strategy to reject this effect. The analytical model of the attractive force is proposed by introducing the approximate analytical expression for the field of PM Halbach. And then the suspension performance under consideration of the iron sleepers on the guideway is testified by simulation, which demonstrates that the effect of iron sleepers on the suspension system cannot be ignored. The control strategy to reject this effect is designed, and its effectiveness is illustrated by simulation.

This paper is organized as follows. Section 2 presents the structure of LSM with PM Halbach and ironless coil, and the iron sleepers on the guideway. Section 3 introduces the analytical description of the magnetic field of PM Halbach, and develops analytical solution to attractive force generated by PM Halbach and iron sleepers. Section 4 investigates the effect of iron sleepers on the suspension system by adding the proposed force to suspension system, and simulation result shows that the attractive force brings resonance to suspension system with typical state feedback controller. In section 5, a nonlinear controller is designed using a feedback linearization technique, and simulation result shows that it is quite effective to reject the resonance resulting from sleepers. Finally, a brief summary are discussed in Section 6.

2. System description

2.1. Structure of the LSM with PM Halbach and the ironless coil

The basic structure of LSM with PM Halbach and ironless coil is shown in Fig. 1 [14]. The PM Halbach is located under the maglev vehicle, and the ironless coil is installed on the guideway. The magnetization directions of the magnetic cubes in the array among a period are different from each other, and Fig. 1 shows an example that the magnetization directions change by [pi]/4 orderly. When three-phase current is added into the ironless coil on the guideway, there will be magnetic forces generated between the vehicle and the guideway. The vertical force acts as normal force, and the horizontal force acts as propulsion force. Under synchronous control strategy, the propulsion force can reach its maximum value while the normal force is zero. Thus the LSM has the characteristics of zero power supply on the vehicle because of PM Halbach array and zero normal force because of ironless coil on the guideway.

[FIGURE 1 OMITTED]

2.2. Structure of the iron sleepers on the guideway

[FIGURE 2 OMITTED]

The simple structure of the iron sleepers on the guideway is shown in Fig. 2 [15, 16]. The iron sleepers are utilized to connect the guide rail and the girder, and they are periodically distributed along the guideway. As the sleepers are made of iron, attractive force will be generated between the PM Halbach on the vehicle and the iron when the vehicle passes through the guideway. Obviously, the attractive force acting on the vehicle points downwards and it accordingly adds extra suspension load on the vehicle. Moreover, the attractive force acts periodically pulsate, which will continuously affect the performance of the suspension system.

3. Calculation of the force between PM Halbach and the ironless sleepers

The attractive force between the PM Halbach and the iron sleepers acts as magnetic force between PM and ferromagnetic material. The calculation starts from the magnetic field generated by PM Halbach.

3.1. Expression of the magnetic field of the pm halbach

Motors in the maglev vehicle are all linear. The PM Halbach utilized to the LSM is linear Halbach, as shown in Fig. 1. In Fig. 1, the magnetic field under the array is enhanced, while the magnetic field above the array is greatly weakened. Here only the magnetic field on the enhanced side is concerned.

As shown in [8], the peak magnetic field at the surface of Halbach array [B.sub.0] can be calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [B.sub.r] is the residual PM flux density, d is the thickness of the array, n is the number of magnets in one period of array, and

k = 2[pi]/[lambda] (2)

with [lambda] being the wavelength of the Halbach array. The structural parameters [lambda] and d are shown in Fig. 1. From Eq. (1) it can be seen that [B.sub.0] is a constant if the structure of the array is established.

Build coordinate Y-O-Z for the array as shown in Fig. 1. The axis OY points rightwards, the axis OZ points downwards, and the origin O locates on the surface of one cube with the magnetization direction pointing upwards. It is known that, the peak of the magnetic field of the array decreases exponentially as the distance from the surface increases. And then the peak magnetic field when the vertical coordinate equals to z is [9]

[B.sub.m] = [B.sub.0][e.sup.-kz] (3)

Also it is known that the magnetic field changes sinusoidally along the axis OY and cosinusoidally along the axis OZ. And then the expression of the components of the magnetic field at point (y,z) can be described as [9]

[B.sub.y] = [B.sub.m] sin (ky)} [B.sub.z] = [B.sub.m] cos (ky)} (4)

3.2. Calculation of the attractive force

The attractive force F between the PM Halbach and the iron sleepers can be approximately established by the following equation [17, 18]

F = [B.sup.2] S/2[[mu].sub.0] (5)

where B is the magnetic induction, [[mu].sub.0] is permeability in the vacuum, and S is the area of the ferromagnetic material vertically to the direction of the magnetic field.

The result presented in Eq. (5) is only applicable to the cases that the magnetic field is evenly distributed. For Eq. (4), the magnetic field changes to the position y. Then the attractive force between the PM Halbach and the iron sleepers should be calculated by implementing integral of y.

[FIGURE 3 OMITTED]

As shown in Fig. 3, only the magnetic field along z axis [B.sub.z] contributes to the attractive force. Suppose the length of the Halbach is infinite along Y axis. Consider one piece of iron sleeper, and denote the left of the sleeper locates at (y,z). Choosing an arbitrary point on the surface of the sleeper as (y', z), then from Eq. (5) the forces acts on the area with width dy' is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

Supposing the width (the length vertical to the surface of the paper) of the LSM is l, the width of the sleeper is w, and substituting Eqs. (1)-(4) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Conducting integral of Eq. (7) with respect to y' yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

It can be seen that [F.sub.z] relates to y and z.

4. Effect of sleepers on the suspension system

4.1. Model of a single-point suspension system

A single-point system is the basic unit for the maglev suspension system, and its structure is shown in Fig. 4, where [delta] is the suspension gap between the guideway and the magnet, u is the voltage added to the coil, i is the current in the coil, F is the suspension force generated by the electromagnet, m is the mass of the electromagnet, and [F.sub.z] is the attractive force generated by Halbach array and the iron sleepers.

Choosing the state as x = [[[delta] [??] i].sup.T], the model of the suspension system can be described as [19, 20]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

where g is the gravity acceleration, R is the resistance of the coil and [k.sub.a] is a parameter determined by the experimental setup. To investigate the effect of the sleepers, following a full state feedback controller is designed and the control performance is testified for the suspension system Eq. (9) under both [F.sub.z] = 0 and [F.sub.z] [not equal to] 0.

[FIGURE 4 OMITTED]

4.2. Full state feedback controller design

In experimental setup, the operation point is selected as [[delta].sub.0] = 0.01m. Then according to Eq. (9), the equilibrium is [20]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

Linearizing Eq. (9) at the equilibrium Eq. (10) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Obviously system (11) is controllable. Thus its poles can be arbitrarily placed with the following control law [21]

[DELTA]u = r - [k.sub.1][DELTA][x.sub.1] - [k.sub.2][DELTA][x.sub.2] - [k.sub.3][DELTA][x.sub.3] (12)

Choose the parameters as shown in Table for the system. Under the performance requirements proposed in [19], the closed-loop poles can be chosen as

p = [[-40 -45 -200].sup.T] (13)

Then the following control parameters can be obtained

[k.sub.1] = -436208, [k.sub.2] = -8817, [k.sub.3] = 122 (14)

4.3. Performance testification under [F.sub.z] = 0

Utilizing controller Eq. (12) with parameters Eq. (14) in system (9) with [F.sub.z] = 0 yields the response of the suspension gap shown in Fig. 5.

[FIGURE 5 OMITTED]

It can be seen from Fig. 5 that the performance is good and the response of the suspension gap is satisfactory even in enlarged view under the condition that the attraction force [F.sub.z] generated by the sleepers is ignored.

4.4. Performance testification under [F.sub.z] [not equal to] 0

In experiment setup, z in Eq. (8) and [x.sub.1] in Eq. (9) satisfy the following equation

z + [x.sub.1] = a, a = 0.12 (15)

Moreover the variable y in Eq. (8) denotes the position of the vehicle on the guideway. For simplicity, let the speed of the vehicle is 1m/s, then

y = t (16)

Substituting parameters in Table, Eqs. (15) and (16) to (8) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

Utilizing controller Eq. (12) with parameters Eq. (13) in system Eq. (9) with [F.sub.z] described in Eq. (17) yields the response of the suspension gap shown in Fig. 6.

[FIGURE 6 OMITTED]

It can be seen from Fig. 6 that the system can also be stabilized around [[delta].sub.0] = 10 mm using full state feedback controller when [F.sub.z] is considered. But the performance is not satisfactory. In the enlarged view, resonance can be seen. And the amplitude of the resonance is about 0.05 mm. Although the resonance is small and passengers cannot feel it, it is bad for the suspension system. The resonance can be amplified under some unexpected situation, such as the elasticity of the guideway, disturbance from the wind and so on. Also, the frequency of the resonance is proportion to the speed of the vehicle, which increases the possibility of unsatisfactory performance.

5. Control design to reject the effect of the sleepers on the guideway

Substituting Eq. (8) into Eq. (9), the model of a single-point suspension system can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

In the expression [k.sub.s], y is the only variable. In experimental setup, there is position and speed detection system, from which y can be directly measured. Then [k.sub.s] can be regarded as a known parameter.

It is surmised that the resonance shown in Fig. 6 results from the strong nonlinearity shown in Eq. (18). To reject the resonance, a nonlinear controller will be designed by feedback linearization. One reason to use feedback linearization is that it provides a technique for designing and synthesizing nonlinear systems by linear control theory, and the other reason is that it is simple to implement, which is quite important for the current experimental setup.

The first step is to check if the system (18) is feedback linearizable or not. Considering the following nonlinear SISO system

{[??] = f(x) + g (x)u (20) [y = h (x)

where x [member of] [R.sup.n], f (x) and g (x) are sufficiently smooth on a domain D [subset] [R.sup.n], and u [member of] [R.sup.l] is a control function. The nonlinear system could be feedback linearized by input-state if and only if there exists a region [OMEGA] [member of] [R.sup.n], in which the following two conditions hold [22-25].

The vector fields {g, [ad.sub.f] g, ..., [ad.sup.n-1.sub.f] g} linearly independent in [OMEGA].

The set {g, [ad.sub.f]g, ..., [ad.sup.n-2.sub.f] g} involved in [OMEGA].

It is easy to check that system (18) satisfies the above conditions i and ii. And the transformation matrix T can be obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

Substituting Eq. (21) into system (18) yields the following linear model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

where u = [alpha](x) + [beta] (x) v, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

It is obvious that linear system (22) is controllable. Then the following controller can be obtained choosing the same closed-loop poles described in Eq. (13)

v = r - 359999 [z.sub.1] -18799 [z.sub.2] - 285 [z.sub.3] (24)

Combining Eq. (23) and Eq. (24) yields the resultant controller

u = [alpha](x) + [beta](x)(r -359999[z.sub.1] - 18799[z.sub.2] - 285[z.sub.3]) (25)

where Z is determined by Eq. (21).

Utilizing controller Eq. (25) in system Eq. (18), the response of the suspension gap under the nonlinear controller for the suspension system (18) with consideration of the sleepers on the guideway can be obtained as shown in Fig. 7.

[FIGURE 7 OMITTED]

It can be seen from Fig. 7 that the suspension performance is satisfactory using the proposed feedback linearization controller with consideration of the effect of the sleepers on the guideway. Comparing Fig. 5 with Fig. 7, it can be concluded that the proposed control algorithm is better than typical state feedback control strategy. And the obtained nonlinear controller can entirely reject the resonance caused by the effect of the sleepers.

6. Conclusion

In this work, an engineering problem is proposed and solved for LSM with PM Halbach located on the vehicle and ironless coils on the guideway. This kind of LSM has been utilized in GA maglev vehicle. It is a good candidate for propulsion system. Above all, the effect of the sleepers on the guideway is investigated. And the following results are obtained. First, the closed-form of the attractive force is deduced from the expression of the magnetic field of Halbach array. Second, the analytical result is added to a suspension system, the result of which shows the phenomenon of resonance. That is to say the typical state feedback control strategy can not guarantee the suspension performance while there are iron sleepers on the guideway. The proposed analytical model of the attractive force generated between the sleepers and the Halbach on the vehicle is simple, and suitable for analysis and controller design. Thus a nonlinear controller is designed using a feedback linearization technique. Simulation result shows the proposed control algorithm can effectively reject the resonance resulting from the effect of the sleepers on the guideway.

Received March 21, 2011

Accepted December 15, 2011

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Xiao Zhang, Hu Cheng, Yungang Li College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha 410073, China, E-mail: [email protected]

Table

Parameters of the LSM and the sleeper

Parameter   Value     Parameter   Value

[lambda]    400(mm)   l           500(mm)
d           50(mm)    w           100(mm)
n           8         [B.sub.r]   1.085(T)
R           3(Q)      m           1000(kg)
[k.sub.a]   0.0025    g           10(m/[s.sup.2])