Dynamic modeling and control for conical magnetic bearing systems = Điều khiển hệ thống nâng từ trường hụt cơ cấu chấp hành.pdf

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
MASTER THESIS
Dynamic modeling and Control for
Conical Magnetic Bearing systems
VU LE MINH
Control Engineering and Automation
Supervisor: Nguyen Danh Huy
Dr.
School: School of Electrical and Electronic Engineering
HA NOI, 2022

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
MASTER THESIS
Dynamic modeling and Control for
Conical Magnetic Bearing systems
VU LE MINH
Control Engineering and Automation
Supervisor: Nguyen Danh Huy
Dr.
School: School of Electrical and Electronic Engineering
HA NOI, 2022
Supervisor’s Signature

C NG HÒA XÃ H I CH T NAM
Ộ Ộ Ủ NGHĨA VIỆ
Độ ậ – ự – ạ
c l p T do H nh phúc
B N XÁC NH NH S A
Ả ẬN CHỈ Ử LUẬN VĂN THẠC SĨ
H và tên tác gi
ọ ả luận văn: Vũ Lê Minh
Đề ận văn:
tài lu Điề ể ệ ố ừ ờ ụt cơ cấ ấ
u khi n h th ng nâng t trư ng h u ch p hành
Chuyên ngành: K thu u khi n và t ng hóa
ỹ ậ ề
t đi ể ự độ
Mã số SV: 20202856M
Tác giả, Người hướng d n khoa h c và H
ẫ ọ ội đồ ấ
ng ch m luận văn xác nhận tác
gi a ch a, b sung lu n h p H ng ngày 04/05/2022
ả đã sử ữ ổ ận văn theo biên bả ọ ội đồ
v i dung sau:
ới các nộ
˗ ẫ ẽ ộ ận văn (Hình 1.1 ử
Trích d n hình v trong n i dung lu - Hình 1.6). S a
Fig. 1.8 thành Fig. 1.7 trong trang 7, m c 1.1.2.
ụ
˗ ỉ ử ứ ận văn, đưa các công thứ
Ch nh s a các công th c trong lu c vào trong
d u ngo
ấ ặc đơn và thể hi n công th c (2.9) -(2.13) b ng h
ệ ứ ằ ệ phương trình.
˗ ả ế ả ỏ ị ả ỏ
Mô t thêm các k t qu mô ph ng trong k ch b n mô ph ng 2 (Hình
4.7- Hình 4.12).
Ngày tháng năm
Giáo viên hướ ẫ ả ận văn
ng d n Tác gi lu
T
CHỦ ỊCH HỘ Ồ
I Đ NG

Acknowledgement
I would like to thank Hanoi University of Science and Technology for
building, maintaining, and developing a leading research and studying
environment. Also, thanks to the School of Electrical Engineering and the
Department of Industrial Automation teachers for teaching and imparting
necessary knowledge from fundamental to in-depth. In particular, many thanks to
my supervisor, Nguyen Danh Huy for his guidance and support through each
Dr.
stage of the process, and also for giving me this great opportunity. I would like to
thank Assoc. Prof. Nguyen Tung Lam his constructive criticism, and inspiring
for
advice throughout this course of project. is the most interesting advisor I
the He
have ever known. The knowledge, challenges, and experiences in studying and
r firm foundation and significant experience
esearching at the university will be a
for me as I pursue my research and development orientation.

TABLE OF CONTENT
CHAPTER 1. INTRODUCTION ....................................................................... 1
1.1 State of the art ............................................................................................ 1
1.1.1 ............................................. 1
Introduction of Magnetic Bearings
1.1.2 Conical Magnetic Bearings modelling and control .................... 7
1.1.3 Fundamental of ADRC ............................................................... 8
1.1.4 Fractional Order Calculus ........................................................... 9
1.2 .................................................................................................. 9
Motivation
1.3 ............................................................................................ 10
Contributions
1.4 ........................................................................................... 10
Thesis outline
CHAPTER 2. MATHEMATICAL MODELLING OF CONICAL AMBs.. 12
2.1 .................................... 12
General schema and theoretical model of AMBs
2.1.1 Structure of AMBs.................................................................... 12
2.1.2 .................................................... 13
Theoretical models of AMBs
2.2 .................................................................... 15
Modelling of Conical AMBs
2.2.1 .......................... 15
Overview of the modelling of Conical AMBs
2.2.2 modelling of Conical AMBs ..................................... 16
Dynamic
2.3 Conclusion................................................................................................ 20
CHAPTER 3. CONTROL SYSTEM DESIGN ............................................... 21
3.1 ............................... 21
Decoupling the coupling components of the model.
3.1.1 ....................... 21
The control structure "Different driving mode".
3.1.2 ................. 22
Decoupling the coupling components of the model
3.2 Fractional order active disturbance rejection control method.................. 23
3.2.1 ....................... 23
Active Disturbance Rejection Control (ADRC)
3.2.2 Fractional Order Control (FOC) ............................................... 26
3.2.3 Fractional order active disturbance rejection control for second-
order system ................................................................................................. 28
3.3 Fractional order active disturbance rejection control for Conical AMBs
system 30
3.4 Conclusion................................................................................................ 32
CHAPTER 4. SIMULATION RESULTS........................................................ 33
4.1 Conical AMBs model parameters............................................................ 33

4.2 Simulation results......................................................................................33
CHAPTER 5. CONCLUSIONS AND FUTURE WORKS.............................47
5.1 Results of the thesis ..................................................................................47
5.2 Future works .............................................................................................47
REFERENCE......................................................................................................48

LIST OF TABLES
Table 1.1 Advantages and disadvantages of PMBs ............................................... 2
Table 1.2 Advantages and disadvantages of AMBs .............................................. 3
Table 3.1 ADRC Controller parameters .............................................................. 25
Table 4.1 System parameters ............................................................................... 33

LIST OF FIGURES
Figure 1.1 Passive Magnetic Bearing PMBs [2]................................................. 2
–
Figure 1.2 Siemens steam turbine SST-600 with active magnetic bearings [3].... 2
Figure 1.3 Hybrid bearing development for high-speed turbomachinery in
distributed energy systems [4] ............................................................................... 3
Figure 1.4 AMBs in the artificial heart .......................................................... 6
[11]
Figure 1.5 AMBs in Vacuum pump [11]............................................................... 6
Figure 1.6 AMBs in flywheel energy storage systems- FESS [12]....................... 6
Figure 1.7 System with cylindrical AMB (a) and Conical AMB (b) [18]............. 7
Figure 2.1 AMBs structure with single-DOF ...................................................... 12
Figure 2.2 Simple electromagnet structure.......................................................... 13
Figure 2.3 Model of the rotor in a cone magnetic bearings system..................... 16
Figure 2.4 Simple model of Conical AMBs ........................................................ 17
Figure 3.1 Conceptual control loop of the cone- ...................................... 21
shaped
Figure 3.2 Control loop structure with active disturbance rejection control (ADRC)
.............................................................................................................................. 24
Figure 3.3 Fractional order state observer ........................................................... 27
Figure 3.4 FOADRC with a second-order system............................................... 28
Figure 3.5 Desired response and system outputs with different
{50,100,200,500,600}................................................................................... 30
Figure 3.6 System outputs with different ...................................................... 31

Figure 4.1 Response to the position of the x, y, z axes........................................ 34
Figure 4.2 The position of the axis angle y
q , z
q .................................................. 34
Figure 4.3 Control current response .................................................................... 34
Figure 4.4 Impact force of electromagnets .......................................................... 35
Figure 4.5 Velocity deviation of x, y, z axes according to the observer ............ 35
Figure 4.6 Velocity deviation of y
q , z
q .............. 35
axes according to an observer
Figure 4.7 Response to the position of the x, y, z axes........................................ 36
q , z
q .................................................. 36
Figure 4.9 Control current response .................................................................... 37
Figure 4.10 Impact force of electromagnets ........................................................ 37
Figure 4.11 Velocity deviation of x, y, z axes according to the observer .......... 37
q , z
q ......... 38
axes according to the observer
Figure 4.13 Response to the position of the x, y, z axes...................................... 38
q , z
q ................................................ 39
Figure 4.15 Control current response .................................................................. 39

Figure 4.17 Velocity deviation of x, y, z axes according to the observer........... 40
q , z
q ........... 40
Figure 4.19 Response to the position of the x, y, z axes...................................... 41
q , z
q ................................................ 41
Figure 4.21 Control current response................................................................... 42
Figure 4.23 Velocity deviation of x, y, z axes according to the observer........... 42
q , z
q ........... 43
Figure 4.25 Comparison of response to the position of x- ............................ 44
axis
Figure 4.26 Comparison of response to the position of y- ............................ 44
axis
Figure 4.27 Comparison of response to the axis angle y
q ................................... 44
Figure 4.28 FO ADRC with control current response ......................................... 45
Figure 4.29 ADRC with control current response .............................................. 45
Figure 4.30 FO ADRC with the impact force of electromagn ........................ 45
ets
Figure 4.31 ADRC with the impact force of electromagnets .............................. 46

1
CHAPTER 1. INTRODUCTION
1.1 State of the art
1.1.1 Introduction of Magnetic Bearings
a. Development of Magnetic Bearings
The concept of designing magnetic bearings and implementing in practical
them
applications has been around for a long time. The theory behind these was
proposed more than a century ago and passive systems based on permanent
magnets have been around for over 150 years. Beginning in 1842, Samuel
Earnshaw's famous paper titled " On the Nature of the Modecular Forces which
regulate the Constitution of the Luminiferous Ether" was related to the force
equilibrium in static fields was published. In 1939, Werner Braunbek interprets the
theorem in terms of magnetic levitation, demonstrating that purely permanent
magnetic stabilization of an object is only possible with diamagnetic materials. In
one of Beams' well-known experiments, published in 1950 , he idled a small
[1]
propeller (1/64 inch in diameter) in a magnetic field and was able to demonstrate
speeds of 800000 rpm. Despite their long history, magnetic bearings have only
been used in practice since the 1980s, thanks to advancements in control
technology, both hardware, and software, which allow for smaller bearings
significantly reducing the size of the controller, power supply, and power
converter. Professor Schweitzer (ETHZ), Prof. Allaire (University of Virginia),
and Professor Okada founded the International Magnetic Bearing Association at
the first International Symposium on Magnetic Bearings (ISMB) in Zurich in 1988.
(University of Ibaraki). Every two years since then, the conference has attracted
scientific and industrial contributions.
Magnetic bearings are now being researched and developed for a wider range of
applications, with advancements in areas such as rotor and stator design materials
to optimize flux, minimize energy loss, high-speed processor for advanced
a
controller designs and enhanced precision.
, ….
b. Classification of Magnetic Bearings
Magnetic bearings can be classified in several ways:
- Based on the magnet force's controllability: A magnetic bearings
ctive
(AMBs), Pass magnetic bearings (PMBs), or magnetic bearings
ive Hybrid
(HMBs).
- Based on type of force: Magnetic repulsion bearings and
the magnetic
Magnetic attraction bearings.
- Based on the movement of the rotor: Radial magnetic bearings Axial
and
magnetic bearings.
Among the above classifications, there are several common types of magnetic
bearings:

2
• Passive Magnetic Bearing (PMBs):
Figure 1.1 Passive Magnetic Bearing PMBs
– [2]
With PMBs (Fig. 1.1), the bearing is only made of permanent magnets and
ferromagnetic materials to conduct magnetic flux, the passive magnetic cell has no
active component, which is a copper coil.
Because it is made entirely from permanent magnets, PMBs have the
following advantages and disadvantages:
Table 1.1 Advantages and disadvantages of PMBs
Advantages of PMBs Disadvantages of PMBs
- Compact size.
- No energy consumption.
- need for a closed-
Don’t loop
controller to maintain
stability since the levitation
force is fixed.
- D y manufactur and
ifficult in ing
process permanent magnets on
ing
demand, resulting in a higher cost
price.
- Because the levitation force is
fixed and cannot be controlled,
keeping the object stable for
variable weights is difficult.
- Inadequate vibration damping.
• Active Magnetic Bearings – Bs:
AM
Figure 1.2 Siemens steam turbine SST-600 with active magnetic bearings [3]

3
The AMBs (Fig. 1.2), which works on the principle of adjusting
system
magnetic force through an electric current, is made up of many parts, including an
electromagnet, amplifiers, and a position sensor. AMBs enable control of the
bearing's stiffness and damping factors, which can affect the dynamic state of the
object during operation.
- Electromagnet: produces magnetic force to lift the rotor.
- .
Position sensor: provides feedback on the rotor position
- Controller: regulates the current supplied to the electromagnet via the power
amplifier, thereby regulating the magnetic force acting on the rotor and
keeping it balanced.
AMBs have the following advantages isadvantages:
and d
Table 1.2 Advantages and disadvantages of AMBs
Advantages of AMBs Disadvantages of AMBs
- Has excellent dynamic properties.
- The force of levitation can be
adjusted.
- It is possible to actively control it.
- Large size, complex
structure.
- .
Difficult to manage
- .
High cost
• HMBs (Hybrid Magnetic Bearings)
The hybrid magnetic bearing known as a
is magnetic bearing that uses an
electromagnet and one or more permanent magnet rings (with the effect of
supporting objects or reducing the load on conventional bearings). On the other
hand, permanent magnets can be integrated into active magnetic bearings to
provide bias flux for the bearing's linearization characteristic while consuming
no power. turbomachinery is shown in Fig. 1.3.
HMBs in
Figure 1.3 Hybrid bearing development for high-speed turbomachinery in distributed
energy systems [4]

4
c. Advantages and disadvantages of AMBs
As AMBs are contactless and friction- , they are gradually replacing
free
traditional mechanical bearings. Mechanical bearings have many limitations,
including poor high-speed operating ability, noise, poor dampening ability, contact
movement, and the need for oil. the
Therefore, moving towards the use of "non-
contact bearing" which brings many advantages to electric motor users, the
mechanical bearings will be replaced by a circular bearing, inside with
electromagnet coils, and rotor will be suspended in the space between the magnetic
bearing by the magnetic force generated by the electromagnets, allowing the motor
to operate without friction.
The magnetic bearings use the electromagnetic force of attraction and
p the rotor
ropulsion generated by the magnetic field of the electromagnets to lift
shaft so that allows rotate in the bearing (stator), even though the distance
it to
between the rotor shaft and the stator is very small (only 0.5 2mm). Magnetic
to
bearings have the potential to bring many breakthroughs to manufacturing
industries because of the outstanding advantages that mechanical bearings lack:
- Due to the contact-free structure of the electromagnetic attraction and
repulsion generated by the magnetic field, the magnetic drive does not cause
friction and has a higher operating speed [5].
- The magnetic bearing does not require a lubrication system, it is virtually
maintenance- , operating, and maintenance costs.
free, lowering both initial
- The shaft is stiffer and less sensitive to vibrations due to the lack of grease
seals and the ability to withstand a larger shaft diameter on the bearing side.
- Magnetic bearings can be used in harsh environments such as gravity,
corrosive environments, extremely low temperatures, and high
temperatures .
[6]
- The vibration-free and friction-free structure may extend the machine's
working life, which primarily ages due to mechanical wear.
- The control electronics include features such as rotor status monitoring,
operation monitoring, and data logging. As a result, this data can be used to
evaluate and inspect the magnetic bearings operating conditions and
quality.
- chieve higher running
Magnetic bearing motors with low power loss can a
speeds, higher efficiency, and a longer machine life than conventional
bearings.
- The rotor position accuracy is controllable and is determined by the quality
of the measurement signal.
- keep the rotor out of
Sliding bearings or ball bearings can be added to
contact with the stator the event of a malfunction. Under normal
in
operating conditions, these additional bearings do not come into contact
with the rotor.

5
- The load capacity of the magnetic bearing depends on the magnetic material
and the design of the bearing.
- The air gap can be adjusted: depending on the size of the actuator, the air
gap can be adjusted from 0 to several millimeters (up to 20mm in special
cases).
However, AMBs also have some drawbacks.
- The price of AMBs is much higher than traditional bearings due to the time-
consuming design, mechanical processing, control design, etc.
- AMBs take up more space and are heavier than traditional bearings.
- Backup bearings are still required in many systems in the event of an AMB
system breakdown.
- Environmental conditions need to be ensured to avoid magnetic force
attracting materials such as iron, and steel billet outside.
d. Applications of AMBs
In recent decades, active magnetic bearings s) has been of increasing
the (AMB
interest to the manufacturing industry due to its properties of being contactless,
lubrication-free, no mechanical wear, and high-speed capability , , . The
[7] [8] [9]
motion resolution of the suspended object in translation or high-speed rotation is
restricted solely by the actuators, sensors, and servo system utilized due to the
noncontact nature of a magnetic suspension. The characteristics mentioned above
allow AMBs to be used in a variety of applications such as:
- AMBs can be utilized in almost any environment as long as the
electromagnetic coils are suitably shielded, for example, in the air at temperatures
ranging from 235°C to 450°C [10].
- In medical devices: a very specific application area of AMBs is in the
pumping of blood within the artificial heart (Fig. 1.4) , which helps to
[11]
maintain the amount of blood being ejected at the desired rate to meet blood
circulation requirements in the human body.
- Due to the non-contact nature of magnetic levitation, AMBs have no
friction loss and a higher operating speed. As a result, magnetic bearings are
increasingly being used in industrial applications such as compressors, pumps
(Fig. 1.5), turbine generators, and flywheel energy storage systems (Fig. 1.6)
[12] [13]
, .
- Magnetic bearings' main advantage is their extremely high positioning
accuracy, which makes them ideal for metalworking machines such as milling
machines and precision grinding machines for small objects.
- AMBs are used to work in hazardous environments in contact with
and
corrosive substances...
- AMBs are used in systems where vibration suppression is required due to
their outstanding advantages of being able to control and eliminate vibrations
while also achieving a predefined dynamic response.

6
Figure 1.4 AMBs in the artificial heart [11]
Figure 1.5 AMBs in Vacuum pump [11]
Figure 1.6 AMBs in flywheel energy storage systems- FESS [12]

7
1.1.2 Conical Magnetic Bearings modelling and control
In recent years, many researchers, in particular, have endeavored to design a
range of AMBs that are compact and simple-structured while still performing well.
Because of the advantages of a cone-shaped active magnetic bearings s
(AMB )
system, such as its simple structure, low heating, and high dependability, there is
an increasing number of studies on it [14], [15]. The structure of a conical magnetic
bearing is identical to that of a regular radial magnetic bearing, with the exception
that both the stator and rotor working surfaces are conical, allowing force to be
applied in both axial and radial directions [16] .
, [17]
To control the rotor in a regular magnetic bearings system by degrees of
five
freedom (DOF), two systems of electromagnets are required to keep the rotor
balanced in the radial direction, as well as one system of electromagnets with a
shaft disc to keep the rotor balanced in the axial direction in Fig. 1.7. The shaft
disc causes an imbalance when the rotor is running at high speeds. The conical
form saves axial space, which can be used to install gears and other components
for added mechanical benefit. It also conserves energy for optimal load support.
Figure 1.7 System with cylindrical AMB (a) and Conical AMB (b) [18]

8
Conical electromagnetic bearings feature two coupled properties as
compared to ordinary radial electromagnetic bearings: current-coupled effect and
geometry-coupled effect, making dynamic modelling and control of these systems
particularly difficult. The current-coupled effect exists because the axial and radial
control currents flow in the bearing coils at the same time. Furthermore, the
inclined angle of the magnet core causes a geometry-coupled effect. Coupled
dynamic characteristics of the rotor conical magnetic bearing system became
known due to the existence of the two coupled effects. conical AMBs systems
As
are inherently unstable, a controller is required to keep the rotor in the desired
stable position. Furthermore, fabricating the conical stator is a challenge that
necessitates high precision mechanical engineering.
So far, several researchers have discussed the modelling and control of cone-
shaped AMBs , , . Lee CW and Jeong HS presented a control method
[8] [19] [20]
for conical magnetic bearings in [17], which allows the conical rotor to float in the
air stably. They proposed a completely connected linearized dynamic model for
the cone-shaped magnet coil that covers the relationships between the input voltage
and output current. The connected controller uses a linear quadratic regulator with
integral action to stabilize the cone-shaped AMB system, while the decoupled
controller is used to stabilize the five DOF systems. Abdelfatah M. Mohamed et
al. proposed the Q-parameterization control method for designing system
[16]
stabilization in terms of two free parameters. The proposed technique is validated
using digital simulation. As a result, plant parameters such as transient and forced
response are good, and stiffness characteristics are obtained at p = 15000 rpm, with
oscillation amplitudes ranging from 7.05-7.1296 % of total airgap length.
Recently, in , E. E. Ovsyannikova and A. M. Gus'kov created a mathematical
[21]
model of a rigid rotor suspended in a blood flow and supported by conical active
magnetic bearings. They used the proportional integral differential (PID) control,
-
which takes into account the influence of hydrodynamic moments, which affect
the rotor from the side of blood flow, as well as external influences on the person.
The experimental findings are reported, with a rotor speed range of 5000 to 12000
rpm and a placement error of less than 0.2 mm. In [22], modelling of conical AMB
structure for complete support of the five DOF rotor system was reported by
Arvind Katyayn and Praveen Kumar Agarwal, who improved the system
performance by creating the Interval type-2 fuzzy logic controller (IT2FLC) with
an uncertain bound algorithm. This controller reduces the need for precise system
modelling while also allowing for the handling of parameter uncertainty. The
simulation results show that the proposed controller outperforms the type-1 fuzzy
logic controller in terms of rising time, overshoot, and settling time.
1.1.3 Fundamental of ADRC
The Active Disturbance Rejection Control (ADRC) is a new control
technique proposed by Prof. Han that aims to bridge the gap between control
[23]

9
theory and practice. A survey paper [24] recently summarized the methodology of
ADRC and the progress of its theoretical analysis. Following [25], ADRC is
primarily based on the ability to estimate unknown disturbance inputs impacting
plant behavior live using adequate observers and then cancel them using an
appropriate feedback control rule based on the obtained disturbance estimate. It is
an appealing choice for practitioners because it rejects in real-time the unknown,
unmeasurable disparities between the actual system (including plant modelling
errors and external perturbations) and its assumed mathematical representation,
which promises good robustness against process variations. There are several
successful implementations of this framework in various fields such as space
ob [26], motion control
servatory antenna [27], attitude tracking of rigid spacecraft
[28], robotic system [29] technique is
,…But on the other hand, the ADRC
recognized to be a model-free controller. As a result, several concepts have been
proposed to improve the controller in order to increase the robustness of
ADRC
the ADRC approach, with one of them being the incorporation of ADRC with the
fractional order controller [30]
.
1.1.4 Fractional Order Calculus
Fractional calculus is a more than 300 years old topic. roots of
The early this
theory were discovered in 1695 [31]. Since then the concept of fractional calculus
has drawn the attention of many famous mathematicians, including Euler, Laplace,
Fourier, Liouville, Riemann, Abel, and Laurent. These mathematical phenomena
enable a more accurate description of real-world objects compared to traditional
"Integer O - " techniques. The voltage- -infinite
rder IO current relation of a semi
lossy transmission line is an example of a non-integer (fractional) order system
[32]. The absence of solution methods for fractional differential equations was the
primary reason for using integer-order models. The theory and calculation of the
Fractional-Order (FO) Fractional-Order
calculus are the fundamental basis for
Control (FOC). The mathematical complexity of fractional controllers limited their
practical application, but with the discovery of fractional calculus, this has
changed. calculus was not widely used until recently, when the benefits of
FO
applying its principles to a variety of scientific domains, such as system modelling
and automated control, became evident. It is also clear that the increased interest
is tied to the availability of more efficient and powerful computing tools made
possible by technological advancement [33],[34]. With the benefits listed above,
fractional calculus can be useful in a variety of industrial and scientific areas,
including the study of electrical circuits [35], signal processing , robotic
[36] and s
[37].
1.2 Motivation
As aforementioned, AMBs have piqued the interest of many people due to
their distinct properties and wide range of applications. -shaped active
Cone
magnetic bearings are an upgraded version of traditional active magnetic bearings

10
that provides several benefits to the manufacturing industry [38], .
[39] However, in
addition to their benefits, Conical AMBs include features such as current
the -
coupled effect and geometry-coupled effect, which make dynamic modelling and
control of these systems particularly complex. The control of Conical AMBs is
predicted to be a topic of frequent discussion in the coming years. Disturbances
are common in Conical system operations, such as external system
AMBs
vibration, exogenous noises, and measurement uncertainties. Disturbances caused
by these sources might have a significant impact on system performance and could
potentially damage mechanical parts when a shaft rotates at high speed.
As a result, it is important for AMBs to reduce the impact of external
d After exploring model design taking into account aspects of AMBs
isturbances.
systems such as gyroscopes, and coupling phenomena, the way to further improve
control performance through a new controller what the thesis aims. The thesis
is
proposes to use a control called Fractional-order active disturbance rejection
control ( -ADRC). This method is based on a control method called Active
FO
Disturbance Rejection Control (ADRC) , [40]
[23] combined with Fractional Order
Control (FOC) [41] [42] above- .
for solving the mentioned
1.3 Contributions
- The major original contribution in this work are listed as follows:
- The principle of operation and design of Conical AMBs are studied. The
thesis presents the concept of conical magnetic bearings for both radial and axial
control.
- The electromagnetic equations governing the relationship between
magnetic forces, air gaps, gyroscopic force, and control currents are used to build
the nonlinear model of a conical magnetic bearing.
- The fundamentals of ADRC C are presented.
and FO
- To increase the efficiency and improve the control performance of Conical
AMBs system, a control method that combines ADRC and C used. It is
FO is
demonstrated that the system shows a better control performance.
1.4 Thesis outline
This thesis is structured as follows:
Chapter 1. Introduction. A detailed overview of the AMBs, including its
development, classification, applications, advantages, and disadvantages,
applications are discussed. Then thesis presents a discussion about Conical
the the
type of AMBs, modelling, and some control requirements. Then it briefly discusses
ADRC and F . The motivations of the thesis are provided, as well as the thesi
OC s's
main contributions. A thesis outline and conc given at the end of the
lusion are
chap.
Chapter 2. Dynamic modelling of conical magnetic bearing. The construction
and working principles of AMBs and Conical AMBs are proposed. Then, based on

11
the mechanical and electromagnetic analyses of the system, a ve DOF
fi
mathematical description of the model is presented.
Chapter 3. Control system design. At first virtual current controls are
identified to decouple the electrical sub-system, then the ADRC and FOC are
discussed and combined. The FO-ADRC is calculated and applied to a Conical
AMBs system.
Chapter 4. Simulation results. A control system for the Conical AMBs model
is completed that includ gyro force. Some simulation scenarios evaluate the
es
performance of ADRC, and FO-ADRC controllers, as well as the system response
to these controllers.
Chapter 5. The conclusions, challenges, and future works are summarized.

12
CHAPTER 2. MATHEMATICAL MODELLING OF CONICAL AMBs
In this chapter, the main components of active magnetic bearings as well as
the basic working principle are presented. Then, the electromagnetic force of an
electromagnet is calculated using basic physics rules and equations. In addition,
the mathematical model of the cone- s is created in a
shaped magnetic bearing
linearized form. The final model created will serve as the basis for creating the
methods of control discussed in the next chapters.
2.1 General schema and theoretical model of AMBs
AMBs use electromagnets to exert forces on the rotor without making direct
physical contact. The electromagnets attract the ferromagnetic rotor, generating
forces. The strength of these forces can then be adjusted by varying the currents in
the magnetic coils.
2.1.1 Structure of AMBs
Electromagnet
Rotor
AMB
Control system
Controller
Position
Power
amplifier
Sensor
Reference signal
2
F
1
F
mg
Figure 2.1 AMBs structure with single-DOF
AMBs ve a structure similar to an electric motor. However, instead of
ha
creating torque to rotate the rotor, it creates an axial force to lift the rotor in the
bearing. The structure of basic magnetic bearings is shown in Fig. 2.1, where
s components of the AMB system can be seen, including controller,
everal major
rotor, electromagnet, power amplifier, and position sensor.
For clarity, consider examining a basic structure, such as the AMBs structure
with single- illustrated in Fig. 2.1. The analysis and control plan for the AMBs
DOF
system with more than -DOF will thus be easier and more convenient to
single
implement. The working principle of the magnetic drive is similar to that of an
electromagnet, that is, a mechanical displacement in a certain direction can be
made by electromagnetic (attracting or repulsive) forces. A position sensor
measures the deviation between the desired position and the actual rotor position

13
and provides this information to the controller. A controller (microprocessor)
generates a control signal from the measuring device. A power amplifier converts
this control signal into a control current, and this control current is then applied to
the magnet coil leading to the variation of attractive force, which maintain the rotor
in the original position. This means the balance between the attraction of the two
electromagnets with the gravity P = m.g of the rotor shaft at the stationary working
point.
When the rotor moves away from its equilibrium position as a result of an
external disturbance, the position sensor detects the movement and sends it to the
microprocessor. The controller will send a control signal to the power amplifier to
open and close the power valves and change the voltage value applied to the
magnet coil 1 and 2. Then the current in the coils will be changed and generate the
appropriate electromagnetic force 1 2
F, F to bring the rotor shaft to the desired
equilibrium position.
2.1.2 dels of AMBs
Theoretical mo
The physical structure of the AMBs system must be analyzed to establish its
dynamic interactions. The object to be analyzed includes the following basic
physical components: voltage applied to the coil, current flowing in the coil,
dynamic force, magnetic flux, inductance (magnetic flux density), magnetic field,
energy stored in the air gap, magnetic force, magnetic field strength.
A dynamic model of the system is constructed based on the balancing
equations around these physical facts. calculating the force from an
To make
electromagnet in the x-direction easier, consider using a basic electromagnet
construction like Fig. 2.2 as an example to compute the magnetic forces. As a
result, apply the same formula to the object of the thesis which is the five DOF
conical AMBs model.
U
N, I
g
l
g
A
Figure 2.2 Simple electromagnet structure

14
Where I is the current flowing in the coil [A], gis is
the air gap ,
[m] N the
number of coil turns, g
A is the cross-section of a steel core [ 2
m ], and l is the length
of the area surrounding the flux's surface [m]
The current I flowing through the coil will generate a dynamic magnetic
force, resulting in magnetic flux. This magnetic flux loops through the steel core,
the air gap, and the rotor, creating an electromagnetic attraction that pulls the rotor
towards the electromagnet's steel core.
Ampère’s circuital law, which states that “the line integral of the magnetic
field surrounding closed-loop equals to the number of times the algebraic sum of
currents passing through the loop.”, presents the relation in Eq (2.1) between the
magnetic field and the current sum enclosed by the closed integration path.
Ampère’s circuital law is expressed as the following formula:
i
i
l
I
= 
 (2.1)
g g s s
i i
i i
i i
i g s
B l B l
Bl
N I ,
= + =
  
  (2.2)
where H is the magnetic field strength [A / m ], B the [
is magnetic flux density
2
Wb / m ], is the permeability of magnetic material,
B
H
 = [ /
H m], s
l is the
average length of the iron core loop, and g
l the length of magnetic flux through
is
the air gap.
For the system in Fig. 2.1, because there are 2 air gaps and the air
permeability coefficient is much smaller than the iron permeability coefficient (
g s
 
), the number s
s
Bl

in Eq (2.2) can be ignored.
Then Eq (2.2) can be simplified:
g
g
2B g
NI.
=

(2.3)
or written in the form:
g
g
NI
B .
2g

= (2.4)
The total magnetic flux 0
 generated by the magnetic flux force F consists
of two components: g
 passes through the air gap, creating an electromagnetic
attraction that attracts the rotor to the magnet, and r
 is the magnetic flux loops
through the steel core called magnetic leakage
flux . Ignoring magnetic flux
leakage, from (2.4), the expression from the magnetic flux through the air gap
Eq
is shown as follows:

15
g g
g g g
NIA
B .A ,
2g

 =  = = ( 2.5)
where g
A is the cross area of the air
-sectional gap.
Ideally, the magnetic field is well distributed, thus energy stored in the
volume of air gap can be calculated as:
g g g
V
1 1
E H.BdV B H A 2g.
2 2
= =
 ( 2.6)
Assuming the magnetic force is F, the previous derivation can be used to
determine the mathematical relationship between magnetic force and rotor
position. As a result, the magnetic force F can be obtained by considering the
energy stored in the air gap as follows:
2
g g g g g
g
dE 1
F B H A B A .
dg
= = =

( 2.7)
Substituting the formula in Eq (2.4) to Eq (2.6), the expression for calculating
the electromagnetic force for the electromagnet mechanism Fig. 2.2 is shown as
follows:
2 2
g g
2
N I A
F .
4g

= ( 2.8)
Eq (2.8) shows the quadratic dependence of the force on the
electromagnetic
current and the inversely quadratic dependence on the air
flowing through the coil
gap. As a result, the electromagnetic force can be changed by varying the value of
the current flowing through the coil while also taking the air gap distance into
account.
In the next section, the above-proven theories will be applied to build
descriptive kinematics equations for the conical five DOF.
AMBs with
system
2.2 Modelling of Conical AMBs
In this section, the electromagnetic system model for magnetic bearings with
conical air gaps on two sides of the rotor (refer to Fig. 2.3) will be developed in
order to build the mathematical model of the AMB system. To fully position a
rotating shaft in a magnetic field, force must be applied along five axes. When
using a cylindrical gap magnetic bearing, five pairs of electromagnets are required.
However, if the gap has a conical form, four pairs are adequate.
2.2.1 Overview of the modelling of Conical AMBs
The figure below depicts the rotor model of conical magnetic bearings,
which has a cylindrical shape in the middle like other regular rotors. Especially at
the two ends of the rotor is beveled cone so that the force generated by the
electromagnet can be separated into two components axial and radial. From there,
the electromagnet system at both ends of the shaft can be used to control both axial

16
and radial movements of the rotor. A shaft that is supposed to be levitated with
Conical AMB has five degrees of freedom, two (y z-direction)
radials and at each
end of the rotor, and one axial (x-direction).
Figure 2.3 Model of the rotor in a cone magnetic bearings system
2.2.2 modelling of Conical AMBs
Dynamic
Consider the simplified model of a conical magnetic bearing system shown
in Fig. 2.3. For simplicity, the rotor is assumed rigid and its center of mass and
geometric center are consistent, i.e, with no eccentricity. Here, m
R and are the
effective radius and inclined angle of the magnetic core, 
 and 
 are the distances
between the two radical magnetic bearings and the center gravity point of the rotor;
1
F (j =1 to 8) are the magnetic forces produced by the stator and exerted on the
rotor; (x,y,z) and x y z
( , , )
   are the displacement and angular coordinates defined
with respect to the mass center. The cone-shaped system can be modeled
AMBs
by using Newton’s law of motion and Euler’s motions equations as follows:
1 2 5 6
3 4 7 8
1 2 3 4
5 6 7 8
d y 6 5 1 7 8 2
5 6 8 7 m x z
d z 1 2 1 4 3 2
2 1 3 4 m
m F F F )sin
(F F F F )sin mg.
m F F F )cos .
m F F F )cos .
J F )b (F F )b ]cos
(F F F F )R sin J
J F )b (F F )b ]cos
(F F F F )R s
= + + + 
− + + + −
= − + − 
= − + − 
− + − 
+ − + − + 
− + − 
+ − + − x y
in J













− 


(2.9)

17
where J the moment of inertia of the rotor along the axis of rotation. The mass
is
and diametrical mass moment of inertia of the rotor, respectively, are m and d
J .
W - other two axes.
e consider also the effect of the x axis rotation on the
Figure 2.4 Simple model of Conical AMBs
Here, the first three equations in Eq (2.9) are the kinematics of the rotor’s
reciprocating motion, while the last two equations represent the rotor’s rotational
dynamics. In addition, in the two rotational kine s equations, there is an
matic
additional component of the feedback force. Suppose that when the rotor rotates
rapidly if a force is applied to the y-axis (z-axis) that is sufficiently large to deflect
the rotor from the axis of motion by a small angle, the rotor itself will also react
back to a torque of the corresponding magnitude equal to x z
J . Similarly, the
component of gyro force along the z-axis is computed.
In order to linearize, the dynamic Eq (2.9 small motions of the rotor are
)
considered. Fig. 2.4 shows the change of air gap of the cone-shaped magnet, which
is written as:
1,2
3,4
1,2
3,4
y o 1 z
y o 2 z
z o 1 y
z o 2 y
g g xsin (y b )cos .
g g xsin (y b )cos .
g g xsin (z b )cos .
g g xsin (z b )cos .
= −   +  


= +  − 


= −   +  

 = +  + 

(2.10)
where o
g is the steady-state nominal air gap. In terms of the actual air gap and the
current, the change in magnetic force can be written as:

18
1 1,2
1,2
2 3,4
3,4
1 1,2
1,2
2 3,4
3,4
2 2
o p o y
1,2 2
y
2 2
o p o y
3,4 2
y
2 2
o p o z
5,6 2
z
2 2
o p o z
7,8 2
z
A N (I i )
F .
4g
A N (I i )
F .
4g
A N (I i )
F .
4g
A N (I i )
F .
4g
  +
=



 +
 =



 +

=


  +
 =


(2.11)
where -7
o ( 4 10 H / m).
 =  stands for the magnetic permeability of a vacuum;
p
A
A
cos
=

, A is the cross-sectional area, N is the number of coil turns;
j
q
i (j 1,4&q y,z)
= = is the control current of each magnet; 1
o
I and 2
o
I are the bias
current in the upper and lower bearing.
Assume that the current change and the displacement of the rotor are small
relative to the bias current o
I and the nominal air gap. Apply Eq (2.10) to Eq (2.11)
and use the Taylor expansion series to obtain the magnetic force, which is
linearized as:
1 1 1,2 1 1
2 2 3,4 2 2
1 1 1,2 1 1
2 2 3,4 2 2
1,2 o i y q q 1 z
3,4 o i y q q 2 z
5,6 o i z q q 1 y
7,8 o i z q q 2 y
F F K i K xsin K (y b )cos .
F F K i K xsin K (y b )cos .
F F K i K xsin K (z b )cos .
F F K i K xsin K (z b )cos ,
= + +  +  


= + −  − 


= + +   −  

 = + −  +  

(2.12)
where
2 2
o p oj
oj 2
o
A N I
F , j 1,2
4g

= = are the steady-state magnetic forces and are the
position and current stiffnesses, respectively.
Apply Eq (2.12) to Eq (2.9) to obtain linear differential kinematics equations:

19
1 2 1 1 2 1 2 2 3 4 3 4
1 2
1 1 2 2 3 4 1 2
1 2
1 1 2 2 3 4 1
o o i y y z z i y y z z
q q
i y y i y y q q
z q 1 q 2
i z z i z z q
m 4F K (i i i i ) K (i i i i )
4x(K K )sin ]sin mg.
m (i i ) K (i i ) 2ycos (K K )
2 cos (K b K b )]cos .
m (i i ) K (i i ) 2zcos (K K
= − + + + + − + + +
+ +   −
= − + − +  +
+   − 
= − + − +  + 2
1 2
1 1 2 2 3 4
1 2 1 2
1 2 1 2
q
y q 1 q 2
d y i 1 m z z i 2 m z z
2 2 2
y q 1 q 2 m q 1 q 2
2
q 1 q 2 m q q x z
d z
)
2 cos (K b K b )]cos .
J ( b cos R sin )(i i ) K (b cos R sin )(i i )
[2cos (K b K b ) sin(2 )R (K b K b )]
z[ 2cos (K b K b ) sin(2 )R (K K )] J
J
−   − 
−  +  − + −  −
+  + −  +
+ −  − +  − + 
1 1 2 2 3 4
1 2 1 2
1 2 1 2
i 1 m y y i 2 m y y
2 2 2
z q 1 q 2 m q 1 q 2
2
q 1 q 2 m q q x y
K (b cos R sin )(i i ) K (b cos R sin )(i i )
[2cos (K b K b ) sin(2 )R (K b K b )]
y[2cos (K b K b ) sin(2 )R (K K )] J

















= −  − − −  −


+  + −  +


+  − −  − − 


(2.13)
The linear differential equation showing the kinematics of the five DOF conical
AMB systems can be briefly rewritten from Eq (2.13) as follows:
b b b b ibm b
M (2.14)
where b
q is the displacement vector defined in the mass center coordinates; m
i is
the control current vector and b
M , b
K and ibm
K are the mass, position stiffness,
and current stiffness matrices, respectively.
1 2 3 4 1 2 3 4
T
y z
T
y y y y z z z z
b
m
{x,y,z, , }
{i ,i ,i ,i ,i ,i ,i ,i }
=  
=
q
i
1 1 1
1 1 1
2 2 2
2 2 2
1 1 1
1 1 1
2 2 2
2 2 2
i i i
i i i
i i i
i i i
ibm
i i i
i i i
i i i
i i i
K sin K cos 0 0 K
K sin K cos 0 0 K
K sin K cos 0 0 K
K sin K cos 0 0 K
K sin 0 K cos K 0
K sin 0 K cos K 0
K sin 0 K cos K 0
K sin 0 K cos K 0
  
 
 
 −  − 
 
 
−   − 
 
−  −  
 
=
 
  
 
 −  − 
 

−   

−  −  − 

 
K
T
b
d
d
m 0 0 0 0
0 m 0 0 0
0 0 m 0 0
0 0 0 J 0
0 0 0 0 J
 
 
 
 
=
 
 
 
 



M
z
y
z y y
y
z z z
xx
yy y
zz z
b
.
x
.
y
x
0 0 0 0 0 K 0 0 0 0
0 0 0 0 0 0 K 0 0 K
0 0 0 0 0 0 0 K K 0
G
0 0 K K 0
0 0 0 0 J
0 K 0 K 0
0 0 0 J 0
K



 
  
  −
 
   
− −
   
   
− −
= =
   
  − −
  
   
− −
   
−   
 

20
1 2
1 2
z 1 2
y 1 2
y y z z 1 2 1 2
2
xx q q
2
yy zz q q
2
y q 1 q 2
2
z q 1 q 2
2 2 2
q 1 q 2 m q 1 q 2
K 4(K K )sin
K K 2cos (K K )
K 2cos (K b K b )
K 2cos (K b K b )
K K 2cos (K b K b ) sin(2 )R (K b K b )


   
= + 
= =  +
=  −
= −  −
= =  + −  +
y 1 2 1 2
z 1 2 1 2
2
z q 1 q 2 m q q
2
y q 1 q 2 m q q
1 m
1 m
2 m
K 2cos (K b K b ) sin(2 )R (K K )
K 2cos (K b K b ) sin(2 )R (K K )
b cos R sin
b cos R sin
b cos R sin


= −  − +  −
=  − −  −
 = − + 
 =  − 
 =  − 
2.3 Conclusion
In this chapter, the dynamic model of Conical has been analyzed. A
AMBs
five model has been proposed based on the electromagnetic analysis. The
DOF
system's equation is normally more complicated and non-linear, but after b
K and
ibm
K of Eq (2.14) have been linearized, the system's equation becomes simpler.
However, (2.14) is still interleaved because the
the system’s equation Eq
components outside the main diagonal of the matrices, b
K , ibm
K and non-
G are
zero. So, the linear control rules cannot be applied directly. As a result, the
Fractional-Order Active Disturbance Rejection Control (FO-ADRC)
– algorithm
is proposed to solve the problems.

21
CHAPTER 3. CONTROL SYSTEM DESIGN
In this chapter, virtual current controls are identified to decouple the electrical
sub-system, then the ADRC and FO are discussed and used to develop a FO-
ADRC controller. This controller is calculated and applied to a Conical AMBs
system.
3.1 coupling components of the model.
Decoupling the
3.1.1 The control structure "Different driving mode".
The conical AMB system is unstable, a closed-loop control is required to
stabilize the rotor position. The control current of the system can be calculated
through the control structure “ ”, which is shown in Fig. 3.1.
different driving mode
Figure 3.1 Conceptual control loop of the cone-shaped
The main principle of the aforementioned structure: where controlling the
position of the rotor according to the x-axis and y-axis, the magnet pair are in the
poles that are opposite each other. For example, 1
y
i and 2
y
i magnets, as well as, 3
y
i
and 4
y
i , 1
z
i and 2
z
i , 3
z
i and 4
z
i are similarly controlled by this structure. Here, the
magnet in each pair is controlled by the sum of the bias current and control current,
and the other with the difference between the bias current and control current. This
means that when the rotor is displaced from its equilibrium position, the different
“
driving mode” controls the pairs of magnets, whereas when the rotor is in its
equilibrium position, only the bias current is present on each pair of magnets.
When the rotor deviates from the equilibrium position, the current through the pairs
of magnets is written as:

22
1 1
2 1
3 2
4 2
1 1
2 1
3 2
4 2
y o
y o
yt
y o
yd
y o
zt
z o
zd
z o
x
z o
z o
i I 1 0 0 0 1
i I 1 0 0 0 1
I
i I 0 1 0 0 1
I
i I 0 1 0 0 1
I
i I 0 0 1 0 1
I
i I 0 0 0 1 1
I
i I 0 0 0 1 1
i I 0 0 0 1 1
    −
 
   
 
− −
   
 
 
   
 

   
 

−
   
 

= +
   
 
− 
   
 

− −
   
 

 
   
 
   
 
−
     
 
   
.






(3.1)
where 1 1 2 2 1 1 2 2
T
o o o o o o o o
o [I ,I ,I ,I ,I ,I ,I ,I ]
=
I is the bias current. At steady-state,
consider o 0
=
I .
T
yt yd zt z x
r d
[I ,I ,I ,I ,I ]
=
i is the x,y, and z axes’ virtual control
current. x
I is the virtual control current of x-axes. The virtual control current in the
upper half of the y and z axes is yt zt
(I ,I ) , whereas the virtual control current in the
bottom half is yd zd
(I ,I ).
T
1 1 0 0 0 0 0 0
0 0 1 1 0 0 0 0
.
0 0 0 0 1 0 0 0
0 0 0 0 0 1 1 1
1 1 1 1 1 1 1 1
−
 
 
−
 
 
=
 
− −
 
 
− − − −
 
H
In this case, Eq (2.14) can be rewritten as’
r
b b b b b ibm
.
=
M K Hi (3.2)
3.1.2 Decoupling the coupling components of the model
Although Eq (2.14) depicted the object model in linear form, there is still
interleaving between the control variables in the matrices. The decoupling
technique will be utilized in this part to remove the interstitial components between
the control variables in order to reduce the amount of computation while increasing
the controller's accuracy.
Since the control is performed in the bearing coordinates, rewriting the
equations of motion in bearing coordinates utilizing the relationship between the
mass center coordinates y z
(x,y,z, , )
  and the bearing coordinates ( 1 2 1 2
x,y ,y ,z ,z
), given by:
T
y z
b {x,y,z, , }
 
=
q and T
1 2 1
se 2
{x,y ,y ,z ,z }
=
q
se b
=
q Tq is the coordinate transfer matrix.
with T

23
1
2
1
2
1 0 0 0 0
0 1 0 0 b
.
0 1 0 0 b
0 0 1 b 0
0 0 1 b 0
 
 
 
 
= −
 
−
 
 
 
T
(3.2) shows that the inter-channel effect occurs at
Eq b
K and
ibm
K H because the major non-diagonal components are not zero. The b
K and ibm
K
H are clearly inverse. The following control structure is used to eliminate the
interstitial component:
1 1
r ibm b se
( ) ( ).
− −
= +
i K H v K T q (3.3)
Eq (3.2) can be rewritten as:
b b b
M (3.4)
where v is the new control signal’s vector. The interstitial component has
been removed in the control channels (x, y, z) leaving just the interstitial
component in the control channel ( y z
,
 ) owing to the gyroscope force.
The original model of the magnetic bearing is a complex, multivariable
nonlinear system, through the process of linearity and decoupling, we have the
linear form of the system shown in Eq (3.4) with 5 inputs and 5 outputs. The full
form of Eq (3.4) is shown as follows:
..
1
..
2
..
x
3
..
d y
..
d z
. .
z 4
. .
x y 5
m
y
J
x v
m v
mz v
J
J
v
J v
 =


=


 =

  −

  
+


 =
 =
(3.5)
Then, the FO-ADRC controller will be used to remove the remaining
interstitial components as well as stabilize the control object. FO-ADRC controller
is used for each input and output pair 1 2 3 y 4 z 5
(x,v ) (y,v ),(z,v ),( ,v ),( ,v )
,   .
3.2 thod.
Fractional order active disturbance rejection control me
3.2.1 Active Disturbance Rejection Control (ADRC)
The thesis introduces a new control design method called Active disturbance
rejection control (ADRC) [23], [40] to stabilize the cone-shaped AMB system.
ADRC has developed as an option that combines the easy applicability of
conventional PID control methods with the strength of modern model
-type -based

24
approaches. The foundation of ADRC is an observer that treats actual disturbances
and modelling uncertainty together, using only a very coarse process model to
create a control loop.
Process
Observer
P
K
0
1
b
D
K
1
ˆ ˆ
y x
=
=
=
=
=
2
ˆ ˆ
y x
=
=
=
=
=
3
ˆ ˆ
f x
=
=
=
=
=
0
u u y
r
−
−
−
−
− −
−
−
−
− −
−
−
−
−
Figure 3.2 Control loop structure with active disturbance rejection control (ADRC)
where f(t) is the sum of the unknown components of system, and b0 is the
knowledge component of the object’s model. The according structure of the
control loop with ADRC is presented in Fig. 3.2. The fundamental idea of ADRC
is to implement an extended state observer (ESO) that provides an estimate, f(t),
such that we can compensate for the impact of f(t) on our process (model) by means
of disturbance rejection. The equation for the extended state observer is given as:
1
1 1
1 2 0 2 1
3 3
1
1
1
2 2 0
3 3
x̂ (t)
x̂ (t) l
0
0 1 0
ˆ ˆ ˆ
x (t) 0 0 1 x (t) b .u(t) l .(y(t) x (t))
ˆ
0 0 0 x (t) l
0
x̂ (t)
x̂ (t) 0
l 1 0
ˆ
l 0 1 x (t) b
ˆ
l 0 0 x (t) 0
 
     
 
 
     
 
 
= + + −
     
 
 
   
   
 
   
   
 
 
−   
 
 
 
− +
 
 
  
 
−
   
=
 
1
2
3
1 1
2 2
3 3
l
.u(t) l .y(t)
l
x̂ (t) l
x̂ (t) B.u(t) l .y(t).
A
x̂ (t)
( LC)
l
 
 
 +  

  
 
   
   
+ +
   
   

−

=
 
(3.6)
where 1 2 3
ˆ
(t) (t) y(t); (t) f(t)
ˆ ˆ ˆ
x y(t);x = =
= . Removing the unknown components
is done through the following control law:
P D
0 0
(t)
K .((r(t) y(t)) K .y
ˆ
(t) f(t)) u u (t)
 − −
= − + 
(3.7)
Where r is the setpoint. In order to work properly, observer parameters, 1 2 3
l ,l ,l ,
in Eq (3.6) still has to be determined. According to [40], the ADRC’s parameters
can be chosen to tune the closed-loop to a critically damped behavior and a desired
2% settling time settle
T . The tuning procedure is summarized as follows:

25
CL 2 CL
p d
ESO ESO 2 ESO 3
1 2 2
K (s ) ,K 2.s
l 3.s ,l 3.(s ) ,l (s )
= = −
= − = =
with CL
settle
6
s
T
= − negative-real double closed-loop pole, and
is the
ESO CL
s (3...10).s
 observer pole.
is the
Using the ADRC controller to calculate the variables x; y and z are calculated
similarly:
..
01 1 01 1
f (t)
1 P1 D1
1
x ( .d(t) b.u(t)) b v f(t) b .v (t).
m
v (t) K ((r(t) x(t)) K .x(t)).
= +  + = +
= − −
For equations containing the two variables ( y
 and z
 ), which have an interleaved
component between the two equations. Because the interleaved component is
unknown, the extended observer can be used to estimate and analyze it. Using the
ADRC controller with variable y
 , as follows:
..
y 4 x 04 4 04 4 04
4 P4 y D4 y
.
1 1
( d(t) b.v (t) J z) b v f(t) b .v (t);b .
J J
v (t) K ((r(t) (t)) K . (t)).
ˆ
 = +  +   + = + =
= − − 
The ADRC controller parameters are calculated as follows:
Table 3.1 ADRC Controller parameters
Name Symbol
01 02 03
b b b
= = 1/m
04 05
b b
= 1/J
settle
T 0.1 (s)
CL
s -60
Pi
K (i 1,...,5)
= 3600
Di
K (i 1,...,5)
= 120
ESO
s -420
1i
l (i 1,...,5)
= 1260
2i
l (i 1,...,5)
= 529200
3i
l (i 1,...,5)
= 74088000

26
3.2.2 Fractional Order Control (FOC)
Fractional order calculus has been around for more than 300 years and has
recently been used [43] [44]
in a variety of areas , . The use of Fractional-order
calculus in system modelling and control has grown significantly. Fractional
calculus is a preferred tool for describing complex natural objects and dynamic
processes such as electrical disturbances, and [45]. On the other
chaotic systems
hand, the fractional-order controller has the potential to provide greater and more
robust control performance than the integer [46]
-order controller .
The typical integer order single input single output (SISO) transfer function
can be extended to the case of the Linear fractional-order system (FOS). Over the
years, several definitions of fractional-order operators have been presented,
including those of Grünwald– –
Letnikov, Riemann Liouville, and Caputo [47].
Caputo is probably the more common in engineering applications. The fractional
operator functions as a non-local operator in all three formulations, which means
fractional derivatives have a memory of previous values.
Using conventional definitions, it is impossible to directly construct the
fractional ractional
-order operator in domain for Linear f
the time the -order plant.
To remedy this difficulty, the normative integer-order operators are used to
approximate fractional-order operators. A great deal of effort and study has been
done in this field. W. Krajewskia and U. Viaro use the Oustaloup approach to
approximate a fractional–order integrator with a rational filter [48], [49]. Based on
numerical quadrature, Piché provides discrete-time approximations of fractional
order operators [50]. This thesis uses the approximation stated in [51], which is
based on network theory approximations and may provide required accuracy over
any frequency band.
a. Fractional order system (FOS)
A SISO linear FO system is defined as :
[52]
j
i
0 0
n m
it t jt t
i 0 j 0
y(t) a D y(t) b D u(t),


= =
+ =
  (3.8)
where u is the input, y is the output,
(t) (t) i
(1 ≤ i ≤n) and j
 (1 are real
≤ i ≤m)
positive numbers, and 1 2 n
...
   , 1 2 m n m
... ( )
      
. Model
coefficients i(1 i n)
   and j
b (1 i m)
  are constants. 0
t t
D
i j
( or )
 =   = 
is a fractional-order differential operator. The upper and lower limits of the integral
interval are denoted by 0
t t, respectively.
and
The fractional derivative of order  defined by Caputo with variable t and
is
starting point 0
t 0
= llows:
as fo
( ) ( )
t (m 1)
0 t
0
1 y ( )
D y(t) d( ),
1 t
+



= 
 −  −
 (3.9)

27
where (z) is Euler's gamma function, and m
 = + 
, m  
, and (0). If the
order differentiations in Eq (3.8) are integer multiples of a single based order: ie
i j
i , j
 =   = , the system is called commensurate order and has the following
form:
j
n m
i
i0 t j0 t
i 0 j 0
y(t) a D y(t) b D u(t).


= =
+ =
  (3.10)
When the initial conditions are set to zero, the Laplace transform of Eq (3.10) is as
follows:
j
i
m
j
j 0
m
i
j 0
b s
Y(s)
G(s) .
U(s) 1 a s


=
=
= =
+


(3.11)
Fractional order state observer
A corresponding FOS allows the following state-space representation:
0 t
D x(t) Ax(t) Bu(t)
,
y(t) Cx(t)

 = +

=

(3.12)
where matrix A, B, and C are constants. By generalizing the classical Luenberger
state observer, a novel fractional-order state observer with full-dimensionality for
the corresponding linear FOS is developed [53]. Fig 3.3 depicts the structure of the
state observer.
1
s




Y
X
U
B C
L
A
X̂
B
A LC
−
−
−
−
−
1
s




+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Figure 3.3 Fractional order state observer
where L is the matrix of unknown coefficients and
1
s
-
represents the fractional
order integer operator. The observer error is given by E X X
= − , where X is the
actual state and X is the predicted state.

28
0 t
D E (A LC)E.

= − (3.13)
Eq (3.13) will be asymptotically stable if the eigenvalues of matrix (A LC) remain
–
in the stable zone.
3.2.3Fractional order active disturbance rejection control for second-order
system
a) -ADRC extended states observer (FOESO)
FO
Assume the following for a second-order linear FO of comparable order:
S
2
2 1
01
2 2
Y(s) b
U(s) s a s a
b
Y(s) 1/ m
U(s) s s
 
 
=
+ +
= =
(3.14)
where 1 2
a ,a ,b < 1) It can be rewritten as follows:
and α (0 < α are constants.
(2 ) ( ) ( )
2 1
y ( a y a y) bu f(y ,y,t) bu.
  
= − − + = + (3.15)
Let 1
x y
= , ( )
2
x y 
= and ( )
3
x f(y ,y,t)

= where 1
x , and 2
x represent system
states and 3
x -space version of
represents the external state. The augmented state
Eq (3.14) can be written as:
( )
x Ax Bu Eh
y Cx

 = + +

=

(3.16)
where ( )
1 1
( ) ( )
2 2
( )
3 3
( )
x x 0 1 0 0
x x ,x x ,A 0 0 1 ,B b ,
x x 0 0 0 0
0
C (1 0 0),E 0 ,h f .
1

 


      
      
= = = =
      
    
 
    
 
 
 
= = =
 
 
 
Linear Fractional
PD controller
Process
Fractional Extended
State Observer
1
b
1
( )
z t
0
( )
u t ( )
u t ( )
y t
( )
r t
−
−
−
−
− −
−
−
−
−
2 ( )
z t
3
( )
z t
Figure 3.4 FOADRC with a second-order system

29
Fig. 3.4 shows the configuration of a second-order FO-ADRC. A linear Fractional
Extend State Observer (FOESO) is designed to estimate the state 1 2
x ,x , and 3
x as
follows:
( )
z Az Bu L(y y)
y Cz

 = + + −


=


(3.17)
( ) ( ) ( ) ( ) T T
1 2 3 1 2 3
T
1 2 3
z (z z z ) , z (z z z ) , B (0 b 0),
L ( ) ,
   
= = =
=   
where L are observer gains, b is the estimated value of b. 1 2
z , z and 3
z are outputs
of FOESO: 1
z is the estimation of the state 1
x , 2
z is the estimation of the state 2
x
, and 3
z is the estimation of the total disturbance 3
x . The bandwidth-
parameterization approach is used to simplify the tuning procedure. The FOS
[54]
is converted into an integer-order system in w-plane via w-plane mapping. The
following is generated by putting the poles of the translated characteristic equation
(w)
 , as follows:
in the same place
3 2 3
1 2 3 0
(w) w w w (w w ) ,
 = + + + = + (3.18)
where the observer gains are linearized as:
1 0
2
2 0
3
3 0
3w
3w
w
 =


 =


 =

(3.19)
Variable 0
w is known as the ESO bandwidth for the integer order system. 0
w has
the bandwidth characteristics when it comes to S. The variable
the FO 0
w in Eq
(3.18) represents the w-plane bandwidth of FOESO. The primary goal of
for
FOESO is to estimate total disturbance in real-time, and larger w-plane bandwidth
results in faster reaction. However, in fact, the upper limit of the bandwidth is
related to the sampling ratio, and exceeding the limit magnifies sensor noises and
dynamic uncertainties. As a result, a well-tuned 0
w must achieve a balance
between rapidity and stability.
b) Design the PD controller for the FO-ADRC structure
W -ADRC structure, the control law can be designed as:
ith the FO
3
z u
u
b
− +
= (3.20)
In order to get the desired response, where 0
u is a conventional linear fractional
PD control:
0 p 0 1 d 2
u k (v z ) k ( z ),
= − + − (3.21)

30
where p
k and d
k . Using the approach in [54], the
represent controller gains
parameter adjustment is further simplified.
d c
2
p c
k 2w
k w
=



=


(3.22)
where c
w is the w-plane bandwidth of the controller. Then Eq (3.14) takes on a
comparable cascade fractional order integrator form, as follows:
(2 )
0
y u

 (3.23)
Using Eq (3.21) and Eq (3.23), the required response of FO-ADRC may be derived
as -
the following closed loop transfer function:
P
2
D P
Y(s) k
.
U(s) s k s k
 

+ +
(3.24)
In practice, the design method is divided into two sections. A linear FOESO is
developed in the first step, and a well-tuned 0
w is chosen to ensure accurate
predictions. In the second step, a fractional-order PD controller is created on the
premise that the total disturbance is well approximated, and any known methods
for developing linear or nonlinear controllers may be employed in this stage.
3.3 Fractional order active disturbance rejection control for Conical AMBs
system
FO-ADRC controller is used for each input and output pair of Conical AMBs
system, shown in Eq (3.5), including 1 2 3 y 4 z 5
(x,v ) (y,v ), (z,v ), ( ,v ), ( ,v )
,   .
Using the FO-ADRC controller to calculate the variables x; y and z are calculated
similarly. With 1
m , the following Eq (3.25) is used to test the FOESO single
parameter 0
w .
01
2 2
b
Y(s) 1/ m
U(s) s s
 
= = (3.25)
With 0.98
 = , is set as {50, 100, 200, 500, 600}. The tracking responses are
shown in Fig. 3.5.
Figure 3.5 Desired response and system outputs with different 
{50,100,200,500,600}

31
As seen in Fig. 3.5, increasing 0
w closes the gap between the output results and
the anticipated response. 0
w 600
= shows that the system achieves good response.
This is due to the fact that a greater observer w-plane bandwidth provides exact
disturbance estimate, and a compensator may subsequently be used to reject the
entire disturbance.
Similarly, tuning C
w in order to make the system stable and the response of the
system.
Fig. 3.6 shows that mall C
w makes the system response slower, while large C
w
makes it faster. In Fig 3.6, C
w 60
= shows that the system achieves significant
efficiency.
Figure 3.6 System outputs with different 
In practice, the design method is divided into two sections. A linear FOESO is
developed in the first step, and a well-tuned 0
w is chosen to ensure accurate
predictions. In the second step, a fractional-order PD controller is created on the
premise that the total disturbance is well approximated, and any known methods
for developing linear or nonlinear controllers may be employed in this stage.
As s result, -ADRC the variables x, y, and z
FO controller parameters for are
calculated as:
2
0 C D C P C 01 02 03
1
0.98, w 600, w 60, K 2w 120, K w 3600, and b b b .
m
 = = = = = = = = = =
S , using the FO-ADRC controller to
imilarly calculate the two variables ( y
 and
z
), which have an interleaved component between the two equations. Because
the interleaved component is unknown, the FOESO can be used to estimate and
analyze it.
With x
..
y
.
d
.
z 4
J J v

 +  = ,
. .
x z
J  will be considered as disturbance and included in
the total system disturbance. The following Eq (3.26) is used to test the FOESO
single parameter 0
w .

32
04
2 2
b
Y(s) 1/ J
U(s) s s
 
= = (3.26)
Similar to the calculation of the variables x, y, and z, -ADRC controller
FO
parameters for the two variables y
 , and z
 are calculated as:
2
0 C D C P C 04 05
1
0.99, w 600, w 60, K 2w 120, K w 3600, and b b .
J
 = = = = = = = = =
3.4 Conclusion
At first, this chapter presented how virtual current controls are identified to
decouple the electrical sub-system. Then the ADRC and FO are discussed and
combined. The FO-ADRC is calculated and applied to a Conical AMBs system.
This controller's response efficiency will be examined in the next chapter.

33
CHAPTER 4. SIMULATION RESULTS
To verify the correctness as well as efficiency and quality of the proposed
Fractional-order active disturbance rejection control, Conical AMBs system
the is
constructed in MATLAB/Simulink environment. The simulation scenario is
displayed and compares the -ADRC
response of the system with 2 ADRC and FO
controllers. The control quality is analyzed, evaluated, and compared to the above
2 controllers that have been performed on the conical AMBs model to demonstrate
the efficiency as well as the superiority of the method proposed in the thesis.
4.1 Conical AMBs model parameters
Table 4.1 System parameters
4.2 Simulation results
Consider 3 scenarios to compare and evaluate the efficiency of using FO-ADRC
and ADRC controller in case of variable rotation speed and rotor load disturbance.
A Simulation scenario 1:
The thesis introduces the -ADRC controller with a rotor rotation speed
FO
of 3000 rpm. The initial values of the rotor center of mass position are:
3 3 3 3 3
0 0 0 y z
x 0.25.10 ;y 0.2.10 ;z 0.15.10 ; 0.1.10 ; 0.2.10 .
− − − − −
= = =  =  =
Select the coefficients of the -ADRC as follows:
FO
2
1 01 C1 D1 C1 P1 C1
0.98, w 600, w 60, K 2w 120, K w 3600
 = = = = = = = , and
Bearing design parameters Symbol Value
Radial air gap 0
g 0.5 mm
Cross-sectional area A 18*10 mm
Inclined angle  10•
Magnetic coils N 300 turns
Resistance of wire R 2
Inductance of wire 0
L 20 mH
Rotor mass M 1.86 Kg
Diametrical moment of inertia d
J 0.00647 kg. 2
m
Polar moment of inertia p
J 0.00121 kg. 2
m
Bias current 01 02
I , I 1.6 A,1 A
Bearing span 1 2
b , b 81.7 mm,71.6mm

34
01 02 03
1
b b b .
m
= = = 2 02 C2 D2 C2
0.99, w 600, w 60, K 2w 120
 = = = = = ,
2
P2 C2
K w 3600
= = , and 04 05
1
b b .
J
= =
Figure 4.1 Response to the position of the x, y, z axes
q , z
q
Figure 4.3 Control current response

35
Figure 4.4 Impact force of electromagnets
Figure 4.5 Velocity deviation of x, y, z axes according to the observer
q , z
q axes according to an observer
The position of the center of mass and the deflection angle of the rotor return
to the equilibrium position after a time interval of 0.02 seconds and there is no
overshoot in Fig. 4.1 and Fig. 4.2. From Fig. 4.3 , n the rotor position
initially whe
deviates from the equilibrium position, a control current is generated to bring the
rotor back to the equilibrium position. After the rotor is in the equilibrium position,
the control current is zero so that the bias currents 01
I and 02
I keep the rotor in this

36
equilibrium state. The impact force of the magnet is shown in Fig.4.4 as having a
significant value at first to bring the rotor to equilibrium, but once the rotor returns
to equilibrium, the force is kept stable at the values 01
F and 02
F . From the above
results, the controller is designed to completely satisfy the requirements. Based on
F 4.5 and Fig. 4.6, the observer satisfied the requirements, and the
ig. estimated
velocity values were near to the real velocity value after 0.02 s.
The rotor speed will be changed to 12000 rpm to the controllability
evaluate
of the controller when the rotor is in the high-speed , the initial values of the
region
rotor center of mass position are:
3 3 3 3 3
0 0 0 y z
x 0.25.10 ;y 0.2.10 ;z 0.15.10 ; 0.1.10 ; 0.2.10 .
− − − − −
= = =  =  =
FO
2
1 01 C1 D1 C1 P1 C1
0.98, w 600, w 60, K 2w 120, K w 3600
 = = = = = = = , and
01 02 03
1
b b b .
m
= = = 2 02 C2 D2 C2
0.99, w 600, w 60, K 2w 120
 = = = = = ,
2
P2 C2
K w 3600
= = , and 04 05
1
b b .
J
= =
q , z
q

37

38
q , z
q axes according to the observer
The simulation results on the x, y, z, y
 and z
 axes are identical to the first
simulation scenario. When the rotor rotates at high speeds, no axis of motion is
significantly influenced. According to Fig. 4.11 and Fig. 4.12, the observer met the
conditions, and the predicted velocity values were close to the true velocity value
after 0.02s. The suggested controller takes into account the rotor speed factor and
demonstrates its capacity to function well in the high-speed region.
To test the controller's ability to resist load disturbance, the ADRC
controller is developed with load disturbance operating on the rotor. Assuming a
force F=5 is applied to the x-axis at time t = 0.05s, the initial values of the rotor
N
center of mass position are:
3 3 3 3 3
0 0 0 y z
x 0.25.10 ;y 0.2.10 ;z 0.15.10 ; 0.1.10 ; 0.2.10 .
− − − − −
= = =  =  =
FO
2
1 01 C1 D1 C1 P1 C1
0.98, w 600, w 60, K 2w 120, K w 3600
 = = = = = = = , and
01 02 03
1
b b b .
m
= = = 2 02 C2 D2 C2
0.99, w 600, w 60, K 2w 120
 = = = = = ,
2
P2 C2
K w 3600
= = , and 04 05
1
b b .
J
= =

39
q , z
q

40
q , z
The rotor position along the axes all reached the equilibrium value of 0 after
0.02 . 4.13 and Fig. 4.14, only the x-axis position reached
seconds as shown in Fig
the new -0.25mm after 0.02s since the disturbance impact
equilibrium position of
force. Because at 0.2 seconds, a force of magnitude 5N has been applied to the
rotor along the x-axis from top to bottom, causing the new equilibrium position of
the rotor to drop by a distance of 0.25 mm. This new equilibrium position has not
yet exceeded air gap, therefore it is still within acceptable limits; but, if the
the
force is increased, the rotor will collide with the stator, chipping and damaging the
rotor. The subsequent figures demonstrate that the current and force have also
changed, resulting in a new equilibrium position that corresponds to the rotor's new
equilibrium position. As shown in Fig. 4.17 and Fig. 4.18, the observer has met the
set requirements, the estimated velocity values have closely followed the actual
velocity value after 0. s. The controller reacts effectively, and the reaction time
02
is very short.

41
This thesis designs an ADRC controller with load disturbance acting on the
rotor to evaluate the controller's ability to resist load noise. Suppose a force
F=5*sin(400 *t) N is applied to the x-axis at time t=0.03s.
π
3 3 3 3 3
0 0 0 y z
x 0.25.10 ;y 0.2.10 ;z 0.15.10 ; 0.1.10 ; 0.2.10 .
− − − − −
= = =  =  =
FO
2
1 01 C1 D1 C1 P1 C1
0.98, w 600, w 60, K 2w 120, K w 3600
 = = = = = = = , and
01 02 03
1
b b b .
m
= = = 2 02 C2 D2 C2
0.99, w 600, w 60, K 2w 120
 = = = = = ,
2
P2 C2
K w 3600
= = , and 04 05
1
b b .
J
= =
q , z
q

42

43
q , z
Figure 4.19 shows that after being exposed to disturbance, the x-axis
load
position value varies around the equilibrium point with an amplitude of 7.

m.
Figures 4.21 and 4.22 agnet force
indicate that the control current and electrom
fluctuate around the equilibrium position with amplitudes of 0.05A and 1.5N,
respectively. The x-axis velocity estimator remains effective and varies by very
small amplitude. The controller still achieves good results with the given
parameters
To compare the efficiency between two ADRC and FO-ADRC controllers,
test the system with noise equivalent to 20% of the initial value. From the running
results of the system, it is possible to evaluate the performance of the controller.
3 3 3 3 3
0 0 0 y z
x 0.25.10 ;y 0.2.10 ;z 0.15.10 ; 0.1.10 ; 0.2.10 .
− − − − −
= = =  =  =
Select the coefficients of the - controller as follows:
FO ADRC
2
1 01 C1 D1 C1 P1 C1
0.98, w 600, w 60, K 2w 120, K w 3600
 = = = = = = = , and
01 02 03
1
b b b .
m
= = = 2 02 C2 D2 C2
0.99, w 600, w 60, K 2w 120
 = = = = = ,
2
P2 C2
K w 3600
= = , and 04 05
1
b b .
J
= =
And the coefficients of the ADRC are as follows:
CL ESO CL CL 2 CL
p d
ESO ESO 2 ESO 3
1 2 3
6
s , s 7s , K (s ) , K 2.s .
0.1
l 3.s , l 3.(s ) , l (s ) .
−
= = = = −
= − = =

44
Figure 4.25 Comparison of response to the position of x-axis
Figure 4.26 Comparison of response to the position of y-axis
Figure 4.27 Comparison of response to the axis angle y
q

45
Figure 4.28 FO ADRC with control current response
Figure 4.29 ADRC with control current response
Figure 4.30 FO ADRC with the impact force of electromagnets

46
Figure 4.31 ADRC with the impact force of electromagnets
The position of the center of mass and the deflection angle of the rotor return
to the equilibrium position after 0.01s with FO-ADRC and 0.03s with ADRC.
Similar to the control current and the magnetic force acting on these two
controllers. FO-ADRC controller shows superior response time compared to
ADRC controller. However, because the response time of the FO-ADRC controller
is quite short, the magnetic force and control current when using this controller
reach a larger value than the ADRC controller with load disturbance. However,
these values are still within the allowable range.

47
CHAPTER 5. CONCLUSIONS AND FUTURE WORKS
Conical Magnetic Bearings are increasingly widespread in applications in
various fields, including those in industry or the medical and energy sectors.
However, these mechatronic systems considered underactuated and strongl
are y
coupled systems. Therefore, problems of dealing with coupling components or the
impact of external disturbances are the content of interest. Moreover, the design of
controllers able to respond well to disturbances to improve control quality has also
been a potential research field for this Conical AMBs model.
5.1 Results of the thesis
In this thesis, conical magnetic bearings are considered as characterized by a
class of underactuated and strongly coupled systems. Base control current
d on the
distribution, coupling mechanism in the -system is solved.
the electrical sub
Subsequently, Fractional Order Linear ADRC-Based Controller is adopted to
a
tackle rotational motion-induced disturbance acting on the system. The simulations
are carried out to prove oposed control can effectively bring the rotor to
that the pr
equilibrium. The results also indicate that coupling effects from low to high
rotational speed do not have noticeable impacts on translational motions of the
rotor.
5.2 Future works
The cone-shaped magnetic bearing not only affected by coupling
is
phenomena, and exogenous disturbance but there are many such as the
construction of the electromagnet and the rotor case
. Furthermore, in of a relevant
axial perturbation, coil current saturation may occur, with adverse consequences
for system stability. As a result, taking these events into account while calculating
the control law is essential for improving control quality and calculating an
acceptable control signal for the system.

48
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52
SUMMARY
Topic: Dynamic modeling and Control for Conical Magnetic Bearing systems
Author: Vu Le Minh Academic year: CH2020B
Supervisor: Dr. Nguyen Danh Huy
Keywords: Conical active magnetic bearings, over-actuated systems, ADRC,
coupling mechanism, - , Fractional calculus.
FO ADRC
Summary of content:
a) ational
The r e of the study
The conical- active magnetic bearings advantage obtained is the
shape with
elimination of a pair of axially compare to
-control electromagnetic coils
cylindrical active magnetic bearings systems. Conical AMBs are increasingly
replacing regular magnetic bearings in practical applications due to their high
dependability and axial space savings for mounting other rotor applications.
However, due to the underactuated and coupling phenomena, the kinematic and
control model of these bearings is very complicated. As a result, the requirements
for control design as well as linearization for a conical magnetic bearing are
becoming challenging and playing a vital role in the design of high precision
industrial machines.
b) , research subjects, research scope.
Objective
Objective:
- Research and propose a magnetic bearings mathematical model that
conical
takes into account the strong coupling effects between movements.
- Research and propose methods to decouple the .
coupling component
- Research and propose FO-ADRC tackle rotational motion-
algorithm to
induced disturbance acting on conical AMBs system based on fractional-order
control control
(FOC) and active disturbance jection
re (ADRC) methods.
- Research and apply a fractional extended state observer capable of
ing is
reducing the influence of model errors, combined with the proposed -
FO ADRC
controller, and verifying the quality of the whole system.
Research subject: Conical active magnetic bearings system. Primarily focus
on research, construction, and propose anti- with Fractional
an coupling method
Order ADRC-Based Controller and improves the control quality of the system.
Research scope: Research and design an a d rejection
ctive isturbance
controller combined with fractional derivatives in the case of taking into account
the coupling external disturbance
factor due to the moving axes and acting on the
conical AMBs system.
c) Thesis Content and Contribution
Scientific significance and contribution:
- The thesis focuses on solving the problem of designing a controller for a
conical active magnetic bearings system, which is an upgraded version of
traditional active magnetic bearings (simple structure, allowing force to be

53
applied in both axial and radial directions). The approach is based on the
de hod to reduce the coupling effect of the system.
coupling met
- The proposed controller is based on the fractional derivative technique
combined with an active disturbance rejection control method, so the control law
takes advantage of these techniques in eliminating the uncertainties, and nonlinear
components as well as improv system stability.
ing
- The controller is designed with better disturbance rejection and increases
the robustness and adaptability of the entire system when compared with a regular
active disturbance rejection controller. The simulation results show the availability
and correctness of the theoretical analysis and the effectiveness of the proposed
controller.
Thesis layout:
- A detailed overview of the AMBs, including its
Chapter 1: Introduction.
development, classification, applications, advantages, and disadvantages,
applications are discussed. The thesis then discusses Conical AMBs, modeling,
and some control requirements. It then goes through ADRC and FOC briefly. The
motivations of the thesis are provided, as well as the thesis's main contributions.
From there, analyze, synthesize, and give research directions for the thesis.
- The main
Chapter 2: Mathematical modelling of conical AMBs.
components of conical active magnetic bearings as well as the basic working
principle are presented. Then the kinematic and dynamic models of conical AMBs
with 5 DOF are presented and analyzed.
- The chapter content is the m
Chapter 3: Control system design. ain
contribution of the thesis. First, virtual current controls are found in order to
decouple the electrical subsystem, and then the Active disturbance rejection
control and
(ADRC) Fractional order control ( ) are explained and integrated.
FOC
The FO-ADRC is calculated and applied to a Conical AMBs system.
- Chapter 4: Simulation results. The whole system together with the proposed
controller is simulated to verify the correctness of the method. Simulation
scenarios are built to demonstrate the effectiveness of the control law and show the
quality of the system. The results show that the proposed method has solved the
desired problems.
- : . Present the thesis's conclusions
Chapter 5 Conclusions and future works
and key contributions, as well as the issues encountered during the research and
completion process, and propose directions for future research.
d) R .
esearch method
- Statistical and synthesis
- Theoretical research
- Professional solution
- Simulation and verification

54
e) Conclusion
The thesis has achieved the research objective and proposed an active
disturbance rejection control algorithm with fractional derivatives for cone-shaped
active magnetic bearings affected by exogenous disturbance, coupling components
between the axes, and model deviation.
In addition, in the process of completing the thesis, the research on the active
disturbance rejection -shaped active magnetic
control algorithm for the cone
bearings object has also been published in an international journal.

Dynamic modeling and control for conical magnetic bearing systems = Điều khiển hệ thống nâng từ trường hụt cơ cấu chấp hành.pdf

Dynamic modeling and control for conical magnetic bearing systems = Điều khiển hệ thống nâng từ trường hụt cơ cấu chấp hành.pdf

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Dynamic modeling and control for conical magnetic bearing systems = Điều khiển hệ thống nâng từ trường hụt cơ cấu chấp hành.pdf