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

The magnetic bearings use the electromagnetic force of attraction and propulsion generated by the magnetic field of the electromagnets to lift the rotor shaft so that allows it to rotate

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: Dr Nguyen Danh Huy

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: Dr Nguyen Danh Huy

School: School of Electrical and Electronic Engineering

HA NOI, 2022

Supervisor’s Signature

CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM

Độc lập – Tự do – Hạnh phúc

BẢN XÁC NHẬN CHỈNH SỬA LUẬN VĂN THẠC SĨ

Họ và tên tác giả luận văn: Vũ Lê Minh

Đề tài luận văn: Điều khiển hệ thống nâng từ trường hụt cơ cấu chấp hành Chuyên ngành: Kỹ thuật điều khiển và tự động hóa

Mã số SV: 20202856M

Tác giả, Người hướng dẫn khoa học và Hội đồng chấm luận văn xác nhận tác giả đã sửa chữa, bổ sung luận văn theo biên bản họp Hội đồng ngày 04/05/2022 với các nội dung sau:

˗ Trích dẫn hình vẽ trong nội dung luận văn (Hình 1.1- Hình 1.6) Sửa Fig 1.8 thành Fig 1.7 trong trang 7, mục 1.1.2

˗ Chỉnh sửa các công thức trong luận văn, đưa các công thức vào trong dấu ngoặc đơn và thể hiện công thức (2.9) -(2.13) bằng hệ phương trình

˗ Mô tả thêm các kết quả mô phỏng trong kịch bản mô phỏng 2 (Hình 4.7- Hình 4.12)

Ngày tháng năm

CHỦ TỊCH HỘI ĐỒNG

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, Dr Nguyen Danh Huy for his guidance and support through each stage of the process, and also for giving me this great opportunity I would like to thank Assoc Prof Nguyen Tung Lam for his constructive criticism, and inspiring advice throughout this course of the project He is the most interesting advisor I have ever known The knowledge, challenges, and experiences in studying and researching at the university will be a firm foundation and significant experience 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 Introduction of Magnetic Bearings 1

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 Motivation 9

1.3 Contributions 10

1.4 Thesis outline 10

CHAPTER 2 MATHEMATICAL MODELLING OF CONICAL AMBs 12

2.1 General schema and theoretical model of AMBs 12

2.1.1 Structure of AMBs 12

2.1.2 Theoretical models of AMBs 13

2.2 Modelling of Conical AMBs 15

2.2.1 Overview of the modelling of Conical AMBs 15

2.2.2 Dynamic modelling of Conical AMBs 16

2.3 Conclusion 20

CHAPTER 3 CONTROL SYSTEM DESIGN 21

3.1 Decoupling the coupling components of the model 21

3.1.1 The control structure "Different driving mode" 21

3.1.2 Decoupling the coupling components of the model 22

3.2 Fractional order active disturbance rejection control method 23

3.2.1 Active Disturbance Rejection Control (ADRC) 23

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 [11] 6

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-shaped 21

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 𝑤0 {50,100,200,500,600} 30

Figure 3.6 System outputs with different 𝑤𝑐 31

Figure 4.1 Response to the position of the x, y, z axes 34

Figure 4.2 The position of the axis angle qy,q z 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 qy,q z axes according to an observer 35

Figure 4.7 Response to the position of the x, y, z axes 36

Figure 4.8 The position of the axis angle qy,q 36 z 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

Figure 4.12 Velocity deviation of qy,q axes according to the observer 38 z Figure 4.13 Response to the position of the x, y, z axes 38 Figure 4.14 The position of the axis angle qy,q 39 z

Figure 4.16 Impact force of electromagnets 39

Figure 4.17 Velocity deviation of x, y, z axes according to the observer 40

Figure 4.18 Velocity deviation of qy,q z axes according to the observer 40

Figure 4.19 Response to the position of the x, y, z axes 41

Figure 4.20 The position of the axis angle qy,q 41 z Figure 4.21 Control current response 42

Figure 4.22 Impact force of electromagnets 42

Figure 4.23 Velocity deviation of x, y, z axes according to the observer 42

Figure 4.24 Velocity deviation of qy,q z axes according to the observer 43

Figure 4.25 Comparison of response to the position of x-axis 44

Figure 4.26 Comparison of response to the position of y-axis 44

Figure 4.27 Comparison of response to the axis angle qy 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 electromagnets 45

Figure 4.31 ADRC with the impact force of electromagnets 46

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 them in practical 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 [1], he idled a small 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, a high-speed processor for advanced 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: Active magnetic bearings (AMBs), Passive magnetic bearings (PMBs), or Hybrid magnetic bearings (HMBs)

- Based on the type of magnetic force: Magnetic repulsion bearings and Magnetic attraction bearings

- Based on the movement of the rotor: Radial magnetic bearings and Axial magnetic bearings

Among the above classifications, there are several common types of magnetic bearings:

• 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

- 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 – AMBs:

The AMBs system (Fig 1.2), which works on the principle of adjusting 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 and disadvantages:

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 is known as a 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 HMBs in turbomachinery is shown in Fig 1.3

Figure 1.3 Hybrid bearing development for high-speed turbomachinery in distributed

c Advantages and disadvantages of AMBs

As AMBs are contactless and friction-free, they are gradually replacing 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 Therefore, moving towards the use of the "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 propulsion generated by the magnetic field of the electromagnets to lift the rotor shaft so that allows it to rotate in the bearing (stator), even though the distance between the rotor shaft and the stator is very small (only 0.5 to 2mm) Magnetic 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-free, lowering both initial, operating, and maintenance costs

- 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

- Magnetic bearing motors with low power loss can achieve higher running 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

- Sliding bearings or ball bearings can be added to keep the rotor out of contact with the stator in the event of a malfunction Under normal operating conditions, these additional bearings do not come into contact with the rotor

- 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 consuming design, mechanical processing, control design, etc

time 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, the active magnetic bearings (AMBs) has been of increasing interest to the manufacturing industry due to its properties of being contactless, lubrication-free, no mechanical wear, and high-speed capability [7], [8], [9] The 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) [11], which helps to 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 and in contact with 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

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]

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 (AMBs) 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 five degrees of 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]

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 As conical AMBs systems 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 shaped AMBs [8], [19], [20] Lee CW and Jeong HS presented a control method 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

cone-al [16] proposed the Q-parameterization control method for designing system 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 [21], E E Ovsyannikova and A M Gus'kov created a mathematical 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 [23] that aims to bridge the gap between control

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 observatory antenna [26], motion control [27], attitude tracking of rigid spacecraft [28], robotic system [29],…But on the other hand, the ADRC technique is recognized to be a model-free controller As a result, several concepts have been proposed to improve the ADRC controller in order to increase the robustness of 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 The early roots of 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 Order - IO" techniques The voltage-current relation of a semi-infinite 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) calculus are the fundamental basis for Fractional-Order Control (FOC) The mathematical complexity of fractional controllers limited their practical application, but with the discovery of fractional calculus, this has changed FO calculus was not widely used until recently, when the benefits of 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 [36], and robotics [37]

1.2 Motivation

As aforementioned, AMBs have piqued the interest of many people due to their distinct properties and wide range of applications Cone-shaped active magnetic bearings are an upgraded version of traditional active magnetic bearings

that provides several benefits to the manufacturing industry [38],[39] However, in addition to their benefits, Conical AMBs include features such as the current-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 AMBs system operations, such as external system 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 disturbances After exploring model design taking into account aspects of AMBs systems such as gyroscopes, and coupling phenomena, the way to further improve control performance through a new controller is what the thesis aims The thesis proposes to use a control called Fractional-order active disturbance rejection control (FO-ADRC) This method is based on a control method called Active Disturbance Rejection Control (ADRC) [23], [40] combined with Fractional Order Control (FOC) [41] [42] for solving the above-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 and FOC are presented

- To increase the efficiency and improve the control performance of Conical AMBs system, a control method that combines ADRC and FOC is used It 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 the thesis presents a discussion about the Conical type of AMBs, modelling, and some control requirements Then it briefly discusses ADRC and FOC The motivations of the thesis are provided, as well as the thesis's main contributions A thesis outline and conclusion are given at the end of the chap

Chapter 2 Dynamic modelling of conical magnetic bearing The construction and working principles of AMBs and Conical AMBs are proposed Then, based on

the mechanical and electromagnetic analyses of the system, a five DOF 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 includes gyro force Some simulation scenarios evaluate the 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

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-shaped magnetic bearings is created in a 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

Figure 2.1 AMBs structure with single-DOF

AMBs have a structure similar to an electric motor However, instead of 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 several major components of the AMB system can be seen, including controller, rotor, electromagnet, power amplifier, and position sensor

For clarity, consider examining a basic structure, such as the AMBs structure with single-DOF illustrated in Fig 2.1 The analysis and control plan for the AMBs system with more than single-DOF will thus be easier and more convenient to 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

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 F , F1 2 to bring the rotor shaft to the desired equilibrium position

2.1.2 Theoretical models of AMBs

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 To make calculating the force from an 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

Where I is the current flowing in the coil [A], gis the air gap [m], N is the number of coil turns, A is the cross-section of a steel core [g m2], 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 i l

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 (

The total magnetic flux 0 generated by the magnetic flux force F consists

of two components:  passes through the air gap, creating an electromagnetic gattraction that attracts the rotor to the magnet, and r is the magnetic flux loops through the steel core called magnetic flux leakage Ignoring magnetic flux leakage, from Eq (2.4), the expression from the magnetic flux through the air gap

is shown as follows:

where A is the cross-sectional area of the air gap g

Ideally, the magnetic field is well distributed, thus energy stored in the volume of air gap can be calculated as:

In the next section, the above-proven theories will be applied to build descriptive kinematics equations for the conical AMBs system with five DOF

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

and radial movements of the rotor A shaft that is supposed to be levitated with Conical AMB has five degrees of freedom, two radials (y and z-direction) 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 Dynamic modelling of Conical AMBs

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, Rm and  are the effective radius and inclined angle of the magnetic core, 𝑏1 and 𝑏2 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 ( ,   are the displacement and angular coordinates defined x y, z)with respect to the mass center The cone-shaped AMBs system can be modeled

by using Newton’s law of motion and Euler’s motions equations as follows:

where J is the moment of inertia of the rotor along the axis of rotation The mass and diametrical mass moment of inertia of the rotor, respectively, are m and Jd

We consider also the effect of the x-axis rotation on the other two axes

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 kinematics equations, there is an 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 J x z 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

The linear differential equation showing the kinematics of the five DOF conical AMB systems can be briefly rewritten from Eq (2.13) as follows:

x

y x

K of Eq (2.14) have been linearized, the system's equation becomes simpler However, the system’s equation Eq (2.14) is still interleaved because the components outside the main diagonal of the matrices, Kb,Kibm and G are non-

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

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 Decoupling the coupling components of the model

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 “different driving mode”, which is shown in Fig 3.1

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,

of magnets is written as:

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 (x, y,z,  and the bearing coordinates (y, z) x, y , y ,z ,z1 2 1 2), given by:

1 2 1 2

K H because the major non-diagonal components are not zero The Kb and Kibm

H are clearly inverse The following control structure is used to eliminate the

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 (  ) owing to the gyroscope force y, z

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

J

m z vJ

is used for each input and output pair (x, v ) (y, v ),(z, v ),( , v ),( , v )1 , 2 3 y 4 z 5

3.2 Fractional order active disturbance rejection control method

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-type control methods with the strength of modern model-based