THE MINISTRY OF EDUCATION AND TRAINING
THAI NGUYEN UNIVERSITY
NGUYEN THI THANH BINH

Title:
IMPROVING THE CONTROL QUALITY
OF MAGNETIC BEARINGS Speciality: Control Engineeering and Automation.
Research code: 62.52.02.16
DOCTORAL DISSERTATION ABSTRACT
Supervisors
1. Prof. Dr Do Khac Duc
2. Assoc.Prof. Dr Nguyen Nhu Hien

Thai Nguyen - 2013

Training Institution:
Thai Nguyen University of technology – Thai Nguyen University.


min to 10,000 r/min), requiring great precision, working in
environments not using lubricants (temperature, pressure very high
or very low).
Currently, research is focusing on the following four directions:
1.1 Research and apply in industrial and biomedical devices.
1.2. Research to reduce the size of the magnetic bearing.
1.3. Research to work in special environments such as vacuum,
temperature, pressure very high or very low
1.4. Research to apply modern controllers. Dissertation’s content
will research following the 4
th
trend.
Motivation of topic
Unlike mechanical bearings, magnetic bearings have no contact between
the stationary and moving elements. Thus, magnetic bearings used in
electric motors are considered as an advanced and environmentally friendly
technology. They strongly contribute in raising the rotational speed to the
engine and help the engine be applied in the special environment that
common engine with ball mechanical bearing can not work or work with
very high maintainance cost. The studies about magnetic bearings typically
are in the developed countries such as Japan, America, France, Germany
and Switzerland. Currently, the applicability of electric motor using
magnetic bearings is necessary in many areas; the study of manufacturing
electric motors using magnetic bearings and applications is also being
promoted in developing countries such as China, Korea, Brazil 2

Magnetic bearings used in electric motors currently classify as the


3

Research Methodology
- Based on the mechanical, electrical and magnetic laws to
calculate dynamics for a radial magnetic bearing 4 DOFs.
- The mathematical description of the radial magnetic bearings
4DOFs in the form of EL.
- Design the controller of nonlinear channel separation for the
magnetic bearings.
- Research magnetic bearing by simulation.
- Empirical study on the magnetic bearing.
The content and scope of the problem will be studied in depth,
resolution and prospects for achieving results
- Develop mathematical description of magnetic bearing 4 DOFs,
under the radial force.
- Design to control nonlinear channel separation for system under
the effects of noise and uncertainty parameter.
- Simulations and experiments on the control system in laboratory
equipment.
The theoretical and practical meaning of topics
- This is a new problem to be studied in Vietnam and is one of the
first scientific studies of the magnetic bearings announcing in Viet
Nam and is a new research direction having highly promising
application in scientific research…
- The results of the research will be the reference to students and
graduate students. From the results of the research will serve as a
basis for many subsequent studies, in order to widespread application
of the electric magnetic bearing to drive system in Vietnam,
especially in systems with special requirements (speed high, high or
a)
Displacement
sensor
Controller

Power amplifier
b)
Figure 1.6: a) Shape and b) the basic elements of magnetic bearing

Contro
ller

Electromagnetic actuator
Power
amplifier
Rotor

Rotor
Rotor weight mg
Magnetic force
f
m

magnetic bearings 4 DOFs.
Chapter 2: The mathematical description of the active
magnetic bearing
2.1. Introduction
Principles of magnetic bearing are a electric magnet instead of
creating shaft torque, it generates displacement forces in the x and y
axis, these forces are automatically adjusted to maintain the gap
between stator and rotor around the nominal value 6

2.2. Mathematical basis of magnetic levitating system
- According to [1] magnetic levitating system was studied on the
basis of approximations based on assumptions such as: flux is closed
completely in the ferromagnetic core, not consider at the saturation
mode etc
- The calculation is based on the average length of the path of
magnetic field and the length of the air gap to the nominal value x
0
.
2.2.1. Magnetic flux density of the magnetic circuit
With these assumptions have not leakage flux, we have:
0
0 I
0
0 0 0
2
1
2

core of steel
Electromagnetic force including the magnetic saturation of core of
steel is determined [1]:
2 2 2 2
a 0 a 0 a
2 2
0
C I C I
0 0
r r r r
W 2 S N i S N i
1
F
x 2
l l l l
2x 2x
 
   

   

   
   
   
   
   
(2.16)
2.2.4. Electromagnetic force not including the magnetic
saturation of core of steel


 

(2.17)
2.2.5. The relationship between electromagnetic forces and
currents in the AMB
When changing the current i compared to the nominal polarized
current i
0
, it will alter the electromagnetic force and after
transforming, we have the basic equation to describe the motion of
an object is levitated by the electromagnetic force in a fixed direction
[23]:a n
mx K i K x
 

(2.24)
The model built for magnetic bearing from the above will be
generalized for magnetic bearings 4DOFs.
2.3. Building a mathematical model of magnetic levitating system
using 4 poles
2.3.1. The structural form of the current magnetic bearing and
researching trend.
4-pole active magnetic bearing is actively promoted to research
[5, 6], [14, 15] and is attracting the attention of many manufacturers
(Synchrony, Mutecs). This is the trend that author will research.
1
1 1 0
2 2 2
0
0 1 2
0 1
1
1 1 0
2 2 2
0
0 1 2
0 2
2
2 2 0
2 2 2
0
0 1 2
2 2
( )
( )
( )
(2 )
( )
(2 )
p
x
x xc bx
p
x
x xc bx




   





 

 
 





   




 

 
 


























 









9

And the levitating force in the x direction:

2
2
2
0 0 0
1
1
1 1 1 1 1
2 2 2 2 2 2 2
0 0
0 1 2 0 1 2
2
2
2
0 0 0
2
2
2 2 2 2 2
2 2 2 2 2 2 2
0 0
0 1 2 0 1 2
2 2
2 2
(2 ) (2 )
(2.31 )
2 2



 




     


   





     


   



The magnetic flux and levitating force in the y direction are
similar to x.
2.3.4. The dynamic characteristics of magnetic bearing 4 DOFs
This section presents the relationship of the parameters of the
magnetic bearing through properties:
- Characteristic of magnetomotive force and stator current (λ =
ψ.N or λ = Li).

 
 




   







   





We can move to the center of the rotor shaft:
1 2
2 2
1 2
( ) 2 2
(2.32 )
( ) 2( ) 2( )
2 2 2 2
x a x x n a xs n
a a a a

2
2 2
2
2
2 2
x a xs n
a a
x a xd n x x x
y a ys n
a a
y a yd n y y y
F K i K x mx
h h
T K i K J
F K i K y my
h h
T K i K J
 
 


  



 






 







(2.34)

2.5. General model of AMB system 4 DOF as Euler - Lagrange
(EL) system
2.5.1. Introducing the Euler-Lagrange system
This section presents the basic principles for a mathematical
model to satisfy the EL system.
2.5.2. General model of AMB 4 DOF system as EL
With the linearized model of (2.34ab), not fully reflect the
nonlinearity of the magnetic bearing. So we put (2.34ab) on general
mathematical model of AMB 4 DOF as follows:

2 2
2 2
a a
y y rm k x n r yd
a a
x x rm k y n r xd
n xs
n ys
h h

 
 



(2.41b)
where

,
r r
x y
: The decline of the rotor to the x-axis and y-axis
Jx =Jy =Ji: Inertia moment on the x axis and y axis
J
k
: Inertia moment on the z axis
K
a
: Proportional coefficient to the current
K
n
: Proportional coefficient to the displacement


x
,

y
: The deflection angle in the x and y direction
F

: Jumping component when levitating shaft
Transforming the equations to matrix form:

u M(q)q+C(q,q)q+g(q)

  
(2.50)
Then, prove that




0 (*)
T
M q M q
 

and






(**)
T
M q C q C q
 

. Derive equations describing

working point
From the linearized model transforming to the center of the rotor
shaft (2.34), building control systems with PID controllers with
structure as Figure 3.1.
With controlling in the x, y directions in 2 magnetic bearings, we
need to have 4 controllers corresponding to control channels x
1
, x
2
,
y
1
, y
2
. Each channel has two control loops: the inner control loop is
the current control loop (enhanced load carrying capacity of the
magnetic bearing) and outer control loop is position control loop.
From the above analysis, the control structure diagram of x
1
channel
is in Figure 3.2.

directions x, y so that the control structure diagrams of magnetic
bearing at x
1
and y
1
as shown in Figure 3.3.

Due to the capacity variation with time constant is very small, the
PID controller of current control loop (inner loop) is set according to
optimal modularity standard [50]. But the position control loop is
designed according to the symmetrical standard [50].
3.2.2 Simulation of system in Matlab-Simulink
- Simulation Data:
The number of turns in a coil 200 turns
Resistance of coil 3.1 Ω
Inductance of coil 12.61 mH
Polarization flux 0.058 T
Rotor mass 1 Kg
Inertial moment in z direction 0.00241 kgm
2


+ With disturbance Figure 3.5ab: Displacement response of magnetic bearing 1 in (y,x)

Figure 3.6a,b: Displacement response of magnetic bearing 1 in (y,x)
15

Figure 3.9ab: Displacement response
of magnetic bearing 1 in (y,x)
Figure 3.10ab: Displacement
response of magnetic bearing 1 in (y,x)
Figure 3.11: Displacement response in directions x, y with the impact
of channel alternating with PID controller: a) Magnetic bearing 1; b)
Magnetic bearing 2.
16

q
Figure 3.14: Control structure PD to compensate gravity (3.15) AMB 4
DOF

∑
C(q,dq)+g
(q)

K
1

K
2


∑
∑
∑
The EL full system is a system class having the model as follows:

( ) ( , ) ( )
M q q C q q q g q u
  
  
(3.11)
Using the controller introduced in references [50, 53]:

( ) ( , ) ( )
u M q v C q q q g q
  
 

(3.12)
Becomes linear system (Figure 3.12) and the linear form of the
second order integral (unstable):
( ) ( , ) ( ) ( ) ( , ) ( )
M q q C q q q g q u M q v C q q q g q q v
       
     
(3.13)
To stabilize by designing additional PD controller in the outer loop:

1 2
ref
v q K e K e


Figure 3.13: PD controller to compensate gravity for magnetic bearing Controller

(3.12)

EL –based
plant (3.11)

v
u

qOuter
controller
(3.14)
q
ref
e
17

2
1

J
h
K c
k
J
k
h
u
K c
k
k
m
K
m
K
y y
x x


 
 
 
 
 


 





 







 







 





  

 





y y k y y y
x x k x x x
       

   
   
   
   
   
   
   
   

   
   
   
    
   
     
 
  
   
  
   
  
   
  
   
  
 



1 1
2 2
1 2
0 0 0 0
0 0 0 0
;
0 0 0 0 0 0
0 0 0 0
p d
p d
p n dn
k k
k k
K K
k k
   
 
 
 
 
 
 
 
 
 
 
 
 

( )
q t
,

( )
q t
of system (3.11)
asymptotically stable tracking on the desired set trajectory


ref
q t
,


ref
q t

. From above, we have detailed diagram of control method as
figure 3.14
3.3.1 Building control system
According to expression (3.15), we have accurately linearized
controller - channel separation for 4 DOF AMB system is
represented as follows:

J J c
y y
h h
K c K
J c
J K
x x
h h
K K c
  
 
 

 
 
 
 
 
   
 
 
 
 
 
 
 
 
 
 
  

2 2
2 2
0 0 0
0 0 0
0 0 0
0 0 0
a
n
a
n
a
n
a
n
K
K
y y
K
x x
K
y K y
K
x x
K
K
 





 


 
 

 

 

 

 

 

 

 


 
 
 
 
 
 

   



+ With disturbance

Figure 3.21: Displacement
response in directions x anf y of
magnetic bearing 1

Figure 3.22: Displacement
response in directions x anf y of


3.4. Quality assessment of PID controller and nonlinear
controller by squared error integral standard
To assess the quality of the 2 controllers for control system of
magnetic bearings four degrees of freedom, we use the squared error
integral standard.
3.5. Conclusion of Chapter 3
This nonlinear control method has solved the fundamental
problem of the 4 DOF AMB system:
(1) Stable tracking control for system with the change of the input
signal with high precision, according th the squared error integral
standard, the error is smaller when the system works with accurately
linearized controller . This demonstrates the quality of the system has
been enhanced.
(2) Ensuring the channel separation for control system.
Figure 3.26a: The Impact of channel
alternating of magnetic bearing 1 in
the x and y

directions is avoided

Figure 3.26b: The Impact of channel
alternating of magnetic bearing 2 in
the x and y directions is avoided
20

(3) The controller is designed in this chapter will be empirically


21

4.2 Experimental results
- System responses:
- Decoupling action in x and y axes:
Remarks:
This experiment is implemented at rotating speed of 5.000 RPM
of rotor shaft. Obtained experimental results indicate the similarity to
those of aforementioned simulation.
4.3. Conclusion
According to the simulation and experimental results, it can be
seen that:
- Control performance of the magnetic bearings is considerably
improved on nonlinear decoupling control regime compared to the
performance of linearization at the operating point control regime
(position error and vibration are reduced).
- However, as working at steady-state, AMB system is still
affected by disturbances. This indicates that the difference between
simulation results and experimental results is partly caused by the
imperfectness of mathematical model of the system.
Figure 4.15: Position displacement of rotor shaft in x and y axes
compared to the balanced position of bearings 1 and 2
23

CONCLUSION AND FUTURE WORKS
Conclusion:
Automated electric drive systems applied magnetic bearings to