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Air Gap Control of the Novel Cross (+) Type 4-Pole

MAGLEV Carrier System

Enes Mahmut GÖKER

Mechatronics Engineering, Yıldız Technical University, Istanbul, TURKEY

enesgkr@hotmail.com

Ahmet Fevzi BOZKURT

Mechatronics Engineering, Yıldız Technical University, Istanbul, TURKEY

afbozkurt90@gmail.com

Bora BAYKAL

Uskudar American Academy, Istanbul, TURKEY

bbaykal23@my.uaa.k12.tr

Kadir ERKAN

Mechatronics Engineering, Yıldız Technical University, Istanbul, TURKEY

kerkan@yildiz.edu.tr

Abstract – Mechanical non-contact carrier systems

based on magnetic levitation (MAGLEV) are used in

special transportation areas (clean rooms, chemical

areas, etc.). Among these types of carriers, 4-pole

hybrid electromagnetic systems (containing permanent

magnets and electromagnets) stand out with their low

energy consumption. The main problem of maglev

carrier systems is their non-linear characteristics and

unstable open loop response. In this study, PID and I-

PD controllers are designed for the air gap control of

the new cross-type 4-pole mechanical contactless

carrier system.

Thus, the instability problem was overcome and the

desired reference tracking for each degree of freedom

was successfully carried out in simulation

environments, and the results were compared.

Keywords – Magnetic Levitation, Maglev Carrier

System, PID I-PD Controller Design

1 INTRODUCTION

With the developing technology, differences in the

technological needs of people have begun to occur. These

differences have led to innovations in the fields of

production and transportation. Transport of manufactured

materials is of great importance. In particular, the material

produced in clean rooms, chemical areas and high

technology production areas should be away from factors

such as vibration, noise and dust. Existing technologies

are insufficient for such transport environments (Kim et

al. 2011; Jiangheng and Koseki 2001). In the solution of

these problems, the use of electromagnetic forces allowed

the carrier platform to be levitation, enabling contactless

transportation (Atherton, 1980; Han, Kim, 2016; Erkan et

al. 2016). The carrier platforms in the literature are

divided into two types as active rail and passive rail, but

the levitation method is similar in systems where

electromagnetic levitation topology is used (Bozkurt et al.

2018; Ertuğrul 2014). U-type, E-type and 4-pole U-type

hybrid electromagnets are used for levitation. (Tzeng, Y.

and Wang 1998; Lee et al. 2013; Ertuğrul 2014). The

proposed cross-type hybrid electromagnet carrier system

has a multi-degrees-of-freedom control structure. Each

pole produces the electromagnetic force required for

magnetic levitation. Pole terminals energized by closed-

loop controllers keep the carrier platform levitation and

provide the axial or radial movement with internal or

external thrust components. Thus, it can be used in

multiple engineering applications such as transportation

systems, frictionless bearings, and spacecraft design

(Jiangheng L and Koseki T 2001).

The cross-type hybrid electromagnet structure contains

permanent magnets and electromagnets as in the 4-pole U

type electromagnets available in the literature. But due to

its structure, it consists of fewer parts compared to the

convectional 4-pole. While the convectional 4-pole is

formed by the combination of 4 silica sheet metals, the

novel cross-type 4-pole is formed by combining two U-

types. This has been beneficial for the system to reduce

leakage fluxes (Göker E.M. and Erkan K. 2022).

In this article, the model of the novel cross-type 4-pole

maglev carrier system has been developed. Since the

system has an unstable structure, it needs an active

controller. For this, PID and I-PD controllers have been

designed by using canonical structure in analytical model

and analytical model was linearized, simulation studies

have been made. Therefore, by using PID and I-PD type

controller, the carrier system can levitate without steady

state error. When comparing PID and I-PD controllers for

air gap position control, the I-PD controller provides the

best results.

2 MAIN CONTENT

2.1 Levitation Model of Cross Type Hybrid

Electromagnet

Figure 1. The cross-type hybrid electromagnet

The core and coil windings of the cross-pole hybrid

electromagnet are shown in Figure 1.

The dimensions of the cross-type 4-pole hybrid maglev

carrier system and the centralized control geometric

transformation matrices are given (Göker E.M. and

Erkan K 2022).

Controllers are designed on this model by linearizing the

system in the predicted operating range. Ferromagnetic

body resistance, magnetic saturation, hysteresis fuco

losses and flux leakages are neglected in the modelling.

The operating point where the linearization is made is

the point where the axes of rotation are absent. If

balancing is done at this point;

0

0

( , , ) ( ,0,0)

( , , ) ( ,0,0)

z z

z z

i i i i

α β

α β

=

=

(1)

should be. The dynamics of movement in the Z-axis

direction at this point;

( ) ( ( ), ( )) ( )

z z d

m z t f z t i t mg F t

∆ = − −

&&

(2)

If

z

f

, which we express as the gravitational force,

2

2

0

2

0

2 ( ( ))

( ( ), ( )) . .4

2 ( ( ) )

PM z

z z

PM

S E Ni t

B

f z t i t S z t l

µ

µ

+

= = +

(3)

is expressed as.

denotes the AT value of permanent

magnets, N stands for winding count, i stands for pole

current,

PM

l

magnet thickness, S stands for magnet area,

µ

0

stands for magnetic permeability constant. In Figure 2,

the air gap and the force graph formed by the system

against the currents are given.

Figure 2. Electromagnetic Attractive Force

Characteristic in Z Axis

The linear model for the levitation system of the

electromagnet is found by linearizing the

electromagnetic attraction force by choosing as in the

1rd equation. Minor variations around the linearized

model of the system;

0

0

0

0

( ( ), ( )) ( ( ), ( ))

( , ) ( ) ( )

z z z z

z z A B z

f z t i t f z z t i i t

f z i K z t K i t

= − ∆ + ∆

= + ∆ + ∆

(4)

expression is made. Here;

0

00

2

0

( , ) 3

0

4 ( )

( )

z

PM z

z

A z i

PM

S E Ni

f

Kz z l

µ

+

∂

= − =

∂ +

(5)

0

00

0

( , ) 2

0

4 ( )

( )

z

PM z

z

B z i

z PM

SN E Ni

f

Ki z l

µ

+

∂

= =

∂ +

(6)

formula is obtained. If Equation 2 is rearranged;

( ) ( ) ( ) ( )

A B z d

m z t K z t K i t F t

∆ = ∆ + ∆ −

&&

(7)

obtained. m; mass (kg), F

d

; is expressed as the

disturbance input force (N). If the laplace transform of

Equation 7 is done according to the air gap;

2 2

1

( ) ( ) ( )

B

z d

A A

K

Z s I s F s

ms K ms K

= −

− −

(8)

available in the form. Angular displacement axis

dynamics;

0

2

2

0

( ) ( ( ), ( ))

( ( ), ( )) ( , ( ),0, , ( ),0) ( ) ( )

z C D

d t

J T t i t

dt

T t i t T z t i i t K t K i t

α α α

α α α α α

θθ

θ θ θ

=

≅ ≅ ∆ − ∆

(9)

{ }

0

0 0

( ,0)

lim

lim lim ( , )

C

Di

T

K

K T i

i

α

α

θ

α α

θ

α

θ

θ

θ

→

→ →

∂

=∂

∂

=∂

(10)

is expressed as. Since the cross-pole maglev carrier has a

symmetrical structure, the β axis dynamics shows the

same characteristics as the α axis dynamics. For this

reason, the β-axis dynamic equations are not included.

In Equation 8, the control signal is given in the form of a

current source. The coils used in levitation system can be

energized by using a voltage source instead of a current

source. From this point of view, if the dynamic equations

are expressed again;

0

( )

( ) ( ) ( ) ( )

z PM

z z z z

PM

Ni t E

d d

e t R i t N R i t NS

dt dt z t l

µ

+

= + Φ= + +

(11)

is expressed as and this equation is linearized;

. .

( ) ( ) ( ) ( )

A z

z z z z z

B

K L

e t R i t L i t z t

K

∆ = ∆ + ∆ + ∆

(12)

form is obtained. Taking the laplace transform according

to the air gap of the equation;

1

( ) ( ) ( )

A z

z z

z z B

K L

I s E s sZ s

L s R K

= −

−

(13)

is found. The linearized system dynamics for the Z-axis

is given in Figure 3 with block diagrams.

( )

z

E s

1

z z

L s R

−

( )

d

F s

( )

Z s

A z

B

K L

s

K

2

B

A

K

ms K

−

1

B

K

( )

z

I s

Figure 3. Z-Axis Linearized System Dynamics

In Table 1, the model parameters of the cross-pole

electromagnet are extracted.

Table 1. Cross Pole Electromagnet Model Parameters

Size/Unit

Value

Size/Unit

Value

Size/Unit

Value

m

[kg]

10.00 z

0

[mm]

19.30 α

0

,β

0

[rad]

0.00

Jα,β

[kg.m

2

]

0.30 i

z0

[A]

0.00 i

α0

,i

β0

[A]

0.00

S

[A

2

]

12*10

-4

K

A

[N/m]

12473

K

C

[Nm/rd]

79.28

β

[AT/N]

19.85 K

B

[N/A]

9.88

K

D

[Nm/A]

2.38

R

z

,

α,β[Ω]

1.00 L

z,α,β

[H]

0.016 E

pm

[AT]

3970

2.2 Simulations and Results

There are 4 control inputs to control the levitation of the

cross-pole hybrid electromagnet. These control inputs can

be modelled in 3 axes (z and α, β axis) by transforming

them into central control axis matrices. The Z axis model

is similar to the α, β axis models, and the difference only

appears in the relevant parameters. The model of the

voltage input system is given in Equation 14.

2 2

( ) ( ) ( )

( )( ) ( )( )

B z z

d

z z A A z z z A A z

K R L

Z s E s F s

z

R Ls ms K K Ls R Ls m K K Ls

+

= −

+ − + + − +

(14)

2.2.1

PID Controller Design

PID controllers are the most used controllers in the

literature. PID controllers are used in many fields due to

their simple structure, low number of control variables

and easy physical implementation. The PID structure of

the z-axis controller of the cross-pole hybrid

electromagnet is given in Figure 4.

p

K s

s

*

( )

Z s

2

( )( )

B

z z A A z

K

R L s ms K K L s

+ − +

( )

Z s

z z

B

L s R

K

+

( )

d

F s

( )

z

E s

2

d

K s

s

i

K

s

Figure 4. Structure of The System's z Axis PID Controller

When the transfer function of the block diagram is found;

*

2

2 2

( ) ( )

1 ( ) ( )

( )

( ) [ ( ) ]

B d P i

B d P i z z A A z

G s Z s

G s Z s

K K s K s K

K K s K s K s R L s ms K K Ls

= =

+

+ +

+ + + + − +

(15)

obtained. The canonical polynomial approach was used to

calculate the controller coefficients in the closed-loop

transfer function (Mochizuki and Ichihara 2013; S.

Manabe. 1998).

Table 3. PID and I-PD Controller Coefficients

Kp

Kd

Ki

1756

32

3088

The Matlab-Simulink model is given in Figure 5. In this

simulation, the response of the system against the entered

air gap reference value is observed.

Figure 5. PID Controller Simulink Model

Looking at Figure 6, it is seen that the PID controller

creates an overshoot by exceeding 1.5 millimetres against

the 0.5 mm reference signal given in the 1st second. In

the real system, this overshoot causes the system to

become instability (mechanical constraints and non-linear

characterism). This problem can be solved by using the I-

PD controller instead of the PID controller.

Figure 6. Step Reference Input PID Controller z Axis

Response

2.2.2

I-PD Controller Design

Significant changes in the reference signal input in the

PID controller produce large input signals as a result of

proportional (P) and derivative (D) coefficients. This

causes overshoots in the system, that is, saturation. Since

the integral (I) block is used first in the I-PD controller, it

integrates the first effect on the error signal. Thus, it

limits the input signal to be applied to the system.

Proportional and derivative expressions are integrated

into the system as feedback from the closed loop.

(Mochizuki and Ichihara 2013).

When the transfer function of the block diagram is found;

* 4 3 2

( )

( ) ( ) ( ) ( ) ( )

B i

z z B d P B A z B i

K K

Z s

Z s s mL s mR s K K s K K K R K K

=+ + + − +

(16)

is expressed.

Figure 7. I-PD Controller Simulink Model

Figure 7 shows the simulation study of the I-PD

controller. In this simulation study, a reference air gap

value was entered and the system was controlled in this

air gap with the I-PD controller.

The model was carried out using the parameters used in

PID. Looking at Figure 8, it is seen that the I-PD

controller responds more smoothly to the 0.5 mm

reference signal given at 1 second, without creating an

overshoot.

Figure 8. Step Reference Input I-PD Controller z Axis

Response

Figure 9 shows the comparison of PID and I-PD

controllers. Looking at this comparison, it is seen that

using the I-PD controller instead of the PID controller is

better in terms of control.

Figure 9. Comparison of Step Reference Input I-PD

Controller with PID Controller's z Axis Response

3 CONCLUSION

In this study, the air gap control of the maglev carrier

system, which contains a novel cross-shaped 4 pole

hybrid electromagnet, was carried out. Analytical model

was linearized, PID and I-PD controllers were designed

by using canonical structure in the analytical model and

controllers were compared in simulation environment.

The successful realization of simulation provides

opportunities for future studies.

For future studies, it is aimed to perform the experimental

setup and to improve the system performance by studying

different controllers.

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