Luận văn thạc sĩ nonlinear control of conical magnetic bearing systems Điều khiển hệ thống Ổ từ chủ Động hình nón

Based on extended state observer, a fractional order sliding mode control FOSMC is designed to achieve fast response and minimize tracking errors as well as better control quantity with

TIANOI UNIVERSITY OF SCIENCE AND TECIINOLOGY

MASTER THESIS

Nonlinear control of conical magnetic

bearing systems

TA THE TAI TALTT212587m@sis.hust.eduyn

Control Engineering and Automation

Supervisor: Dr Nguyen Danh Huy

Superviser’s signature

School School of Flectrical and Electronie Bnginteering

Hanoi, 03/2023

CONG HOA XÃ HOI CHU NGHIA VIRT NAM

Độc lập - Tự do— Hạnh phúc

BẢN XÁC NHẬN CHỈNH SỬA LUẬN VĂN THẠC SĨ

Tọ và tên tác giá luận văn: Tạ Thế Tải

Đề

Chuyên ngành: Kỹ thuật điều khiến và tự động hóa

Mã số SV: 20212587M

luận văn: Điều khiển hệ thống ô từ chủ động hình nón

‘Tac gia, Người hướng dẫn khoa học vả Liệt đồng chấm luận vẫn xác

nhận tác giả đã sửa chữa, bổ sung luận văn theo biên bản hợp Hội đồng ngày 28/04/2023 với các nội dung sau:

-_ Sữa số thứ tự các phương trình trong chương 2 và chương 3

Mô tâ thêm các kết quã mỗ phỏng trong các kịch bản mô phỏng

THESIS TOPIC

Nonlinear control of conical magnetic bearing systems

Supervisor

Acknowledgment

I would like to thank Tanoi University of Science and Technology for building, maintaining, and developing a leading research and studying envirorment Also, thanks to the School of Electrical Enginccring 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 las oriented, guided, encouraged, and helped me throughout the process

of studying, researching, and completing the thesis ‘The knowledge, challenges, and experiences in studying and researching at the university will be a solid foundation and valuable experierwe for me to continue with iny research and

development orientation

Abstract

Aclive magnetic bearmgs (AMBs) are clectromagnetic mechanism systems in which non-contact bearings support a rotating shaft using atwactive forces

generated by electromagnets through closed-loop control For complete support of

a five-degree of freedom (DOF) rotor system, most AMB structures imclude two

radial actuators and one for the axial direction Conical active magnetic bearing (CAMB) is one of the development directions of conventional magnetic bearings

in which the requirement of the axial bearing can be elimmated Due to the

nonlinearities and inherett coupling properlics of cornaal aetrve magnetic bearing system, it is essential to accomplish an appropriate mathematical model as well as

design a high accuracy control scheme In this thesis, extended state observer

(FSO) is applied to deal with the lumped disturbances of CAMB system which

come from extemal disturbances, uncertain electromagnetic forces and parametric

uncertainties The convergence properties of the tracking error are analytically

proven using Lyapunov’s theory Based on extended state observer, a fractional

order sliding mode control (FOSMC) is designed to achieve fast response and minimize tracking errors as well as better control quantity without chattering The

control performance of the proposed FOSMC-ESO is illustrated in Ierrns of very

good disturbance rejection capabilily that is demonstrated through

MATLAB/Simulink simulation results In addition, comparative simulations

combine wilh three performance indices are performed to quantitatively evaluate

the tracking performance of proposed controllers against SMC and ADRC

controllers

Author

TABLE OF CONTENT CHAPTER 1 INIRODUCTION

11 State of the aFt ào

1.11 IaodueHonof Active Maenetic Hearing

Í

1.12 Principles of Magnetic Bearing Function

1.13 Advantages and disadvantages of AMBs

118 — Extended state cbserver wd

Linearized bearing forces sec TỔ

Extcrnal đisturbaiiccs c cocociecccecree ¬ 4

Magnetic bearings actuation - - 22

CHAPTER 3 EXTENDED STATE OBSERVER BASED CONTROL

eu 24

3.2 Kracuonal Order Bliding Mode Contol

3.21 Principle of Sliding Mode Conuol and Chattering Problem

33 Conclusion 28 CHAPTER 4 NUMERICAL SIMULATION STUDY

CHAPTER § CONCLUSIONS AND FUTURE WORKS

Table 4.2 The coniroi performanee benchmark in sccnari

Table 4.3 The control performance benohmiark _ - - 53

LIST OF FIGURES Figure 1.1 Active magnetic bearings in compressor [6j

Figure 1.2 Munction principle of an active electromagnetic bearing [3]

Figure 1.3 Applications of AMBs

Figure 1.4 System with cylindrical AMEs [14]

Figure 1.5 System conical AMBs [15] (1) impeller, (2) centering tip; (3) conical

geometry: (1) rotor; (5) electric motor, (6) magnetic actuators

Figure 2.1 AMBs structure with single-DOF

Figure 2.2 Simple electromagnet structure

Tigure 2.3 Sehematie of conical active rnagnetic bearing Íorcs

Figure 2.4 Ihustration of unbalanee rotor

Figure 4.2 Response to the position z, x, y

Figure 4.3 Response (o the position of the axis angle 6, ,Ay

Figure 4.4 Upper control currents response

Figure 4.5 Under control currents response .jsssssssiessstsoneeineen

Figure 4.6 Upper impact forces of electromagnsts

Figure 4.7 Under impact forces of electromagnets

Figure 4.8 Comparison betwoon observer and response position of z

Figure 4.9 Comparison between observer and response position af x

Figure 4.10 Comparison between observer and response position of y

Figure 4.11 Comparison between observer aud response position of x

Figure 4.12 Comparison between observer and respanse position of 0y

Figure 4.13 Comparison between observer and response velocity of z

Figure 4.14 Comparison between observer and response velocity of x

Figure 4.15 Comparison between observer and response velocity of y

Figure 4.16 Comparison between observer and response velocity of 8x

Figure 4.17 Comparison between observer and response velocity of Gy

Figure 4.18 Comparison between sign and sigmoid function

Figure 4.19 7, axis transionl response under the parameter uncerlaintics

Figure 4.20 X axis transient response under the parameter uncertairilics

Figure 4.21 Y axis transient response under the parameter uncertainties

Figure 4.22 Ox axis transient response under the parameter uncertainties

5 11

12

„Ö l$ 1d

20 31

Figure 4.23 @y axis transient response under the parameter uncertainties

Figure 4.24 Upper current response under the parameter uncertainties

Figure 4.25 Under current response under the parameter uncertainties

Figure 4.26 Lixternal force Fa=30N oo ee cceccseteens cere ieeesiees

Figure 4.27 ADRC controller position response under Tu=30N

Figure 4.28 SMC controller position response under Fa=30N

Figure 4.29 FOSMC controller position response under Fa-30N

Figure 4.30 Current response with ADRC controller under Fa=30N

Figure 4.31 Current response with SMC controller under Fy 30N-

Figure 4.32 Current response with FOSMC controller under Fg-30N 43

Figtue 4.33 Comparison betweơn obsorver and posilion of x axis with Fa 30N 44 Figure 4.34 Comparison between observer and velocity of x axis with Fg=30N 44 Figure 4.35 Comparison between observer and disturbance of x axis with Fa=30N

Figure 4.37 ADRC controller position response under #a-10UN 0

Figure 4.38 SMC controller position respouse under Fa 100N

Figure 4.39 FOSMC controller position response under Fa—100N

Figure 4.40 Current response with ADRC controller under Fa TOON

Figure 4.41 Current response with SMC controller under Fa=100N

Figure 4.12 Current response with FOSMC controller under Pa-L00N

Figure 4.43 Comparison between observer and position of x axis with Fa-100N

44 -Öò độ

5S]

Figure 4.53 Comparison between observer and response position of x axis with

Figure 4.54 Comparison between observer and response velocity of x axis with

Figure 4.55 Comparison belween observer and real disturbance of x axis with

CHAPTER 1 INTRODUCTION 1.1 State of the art

1.1.1 Introduction of Active Magnetic Bearing

The study of the mechanism and development of bearings, one of the most crucial parts of any rotating machinery, is becoming increasingly important as the demand

for more high-precision, high-speed devices grows By conventional definition, a bearing is the stationary component of a machine that supports a system's rotating

component Active magnetic bearings are alternative to the traditional types of

bearings such as ball or fluid bearings, in which non-contact bearing support a rotating shaft using attractive forces generated by electromagnets through closed-

loop control The history of AMB is briefly addressed: Wemer Braunbek interprets

the theorem in terms of magnetic levitation in 1939 and the first applications of the electromagnetic suspension principle have been in experimental physics and

suggestions to use this principle for uranium centrifuges in the 1940s AMBs are

commonly referred to as magnetic suspension systems for one degree of freedom

(DOF) systems, which are utilized in ground transportation systems to float the vehicle using a mix of regulated electromagnetic and permanent magnetic forces

A suspended cylindrical rotor in the AMBs system rotates at different speeds

depending on the applications for systems with higher DOF Thus, the noncontact

suspension mechanism of the AMBs system is its most distinguishing feature This mechanism offers many benefits over conventional bearings, including lower

rotating losses, higher operating speeds, the elimination of expensive lubrication

systems, the ability to operate in extreme temperatures and vacuum, as well as a

longer lifespan [1]-{3] Due to these significant reasons, AMBs have been

successfully used in various applications for several decades such as industrial

machinery and medical equipment, power and vacuum technologies [4]-[6], and

Auxiliary bearing,

Radial AMB

Figure 1.1 Active magnetic bearings in compressor [6]

1.1.2 Principles of Magnetic Bearing Function

Generating contact-free magnetic field forces by actively controlling the dynamics

of an electromagnet is the principle that is actually used most often among

magnetic suspensions The schematic arrangement of the system's rotor and

magnetic coil (stator) is shown in Fig 1.2, along with the typical construction of the AMBs system

Gap Sensor

Electro- Magnet

Rotor

Power Amplifier

Figure 1.2 Function principle of an active electromagnetic bearing [3]

AnAMB-Rotor system's fundamental parts comprise the rotor, magnetic actuators,

position sensors, power amplifier, and controller The controller determines the

appropriate commands to the power amplifiers based on the position sensors’

measurements of the rotor locations so that the output currents to the magnetic actuators produce the necessary electromagnetic force to levitate the rotor The

rotor is stabilized at the equilibrium position of the air gap as a result of the closed

loop created by this

1.1.3 Advantages and disadvantages of AMBs

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

- Because magnetic bearings can operate at high temperatures, the overall

system can be vastly improved and there are no contacting parts in magnetic

bearings so the lubrication system is unnecessary Therefore, a system with AMBs weighs up to 5% less than systems with conventional bearings

= Cilemissions are reduced by removing lubrication from bearings, providing

direct environmental benefits The removal of oil from the system also

makes it more fire-resistant

Friction losses are eliminated because AMBs are non-contact bearings Furthermore, non-contact system eliminates the fatigue and wear associated with ball bearings The operating speed and efficiency can also

be increased

Bevause magnetic bearing is an active system, it has several advantages aver a passive one At critical speeds, the controller can compensate for unbalance and actively control rotor behavior The AMB can then be used

as a sensor lo provide information aboul changes in shall dynamics, allowing for system monitoring ‘This system diagnosis allows for lower maintenance costs by extending the time between engine services

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, ete

Backup bearings are still required in many systems inthe event of an AMBs system breakdown

Environmental conditions necd to be ensured to avoid magnetic force attracting materials such as iron, and steel billet outside

1.1.4 Applications of AMBs Active magnetic bearings (AMBs) have gained popularity in recent decades They are necessary to suspend shafts that can spin at high speeds without mechanical conlact or lubrication AMBs are curenily employed globally in a variety of industrial, space, and laboratory applications The widespread adoption of this technology is primarily due to its numerous advantages over traditional bearing lechnology The characlerislics mentioned above allow AMBs to be used in a

variety of applications such as:

Tn medical equipment: a very specific application arca of AMBs is in the pumping of blood within the artificial heart [7], which helps to maintain the amount of blood being ejected at the desired rate to meet blood circulation requirements in the human body AMBs can minialurize the geometry of the rotor suspension structure, which is an important factor when designing

an auxiliary mechanical blood circulation apparatus

Due to no contacting parts in magnetic bearings, AMBs climinate friction and lubrication, reduce rotor vibration, and are more advantageous to the development of motors towards high-speed and high-power-density As a result, magnetic bearings are increasingly being used in rotating machinery apphealions such as compressors [8], pumps |9|, wind turbines [10], and flywheel energy storage systems (FESS) [11]

al Vacuum Vessel Blood Pump ` Maglev Trains ` ss

Figure 1.3 Applications of AMBs

- AMBs are used to work in hazardous and high temperatures environments

such as nuclear power plants [12], gas turbine applications and aircraft

turbo-machinery to eliminate the need for mechanical bearings and shaft

seals

accuracy, which makes them ideal for metalworking machines such as

milling machines and precision grinding machines for small objects

1.1.5 Conical Magnetic Bearings: An overview

The present trend in AMBs focuses on the development of various geometrical

bearing designs to save axial space for mounting additional mechanical

components such as gearboxes A potential development path is to use a conical shape for active magnetic bearings (CAMB)

For complete support of five degrees of freedom (DOF) rotor system, two radial AMBs and an axial AMB in Fig 1.4 are required which results in increased

complexity of the system The shaft disc causes an imbalance when the rotor is

running at high speeds In order to save the axial space for mounting additional

mechanical components such as gears etc., a conical structure of magnetic bearing

in Fig 1.5 where the rotor surface at the bearing end has a small angle, resulting ina slanted airgap between the rotor and bearing The electromagnetic coils supply

both the axial and radial forces to the system in this configuration, and the most

obvious benefit obtained is the elimination of a pair of axially-controlled

electromagnetic coils In addition, it also saves energy for supporting the maximum

load and reduces copper loss [13]

Figure 1.5 System conical AMBs [15] (1) impeller; (2) centering tip; (3) conical

geometry; (4) rotor; (5) electric motor; (6) magnetic actuators

Nonetheless, CAMB highlights two coupling properties: current-coupled and

geometry-coupled effects, making dynamic modelling and control of these frameworks especially troublesome In addition, the nonlinear nature of the

dynamics, small natural damping in the process, the strict positioning specifications often required by the application, and the unstable open-loop system

dynamics make the controller design for the CAMB system a challenging task In

most cases, a proportional-integral-derivative (PID) controller is chosen due to its

simplicity and intuitiveness in the tuning of the controller parameters However,

there are times when a conventional PID controller is unable to meet the industry

performance standards for CAMB systems Many previous researchers have proposed some control methods of a CAMB

Lee and Jeong presented an optimal control method for conical magnetic bearings

in [16], which allows the conical rotor to float in the air stably ‘They proposed a completely connected linearized dynamic model that included the linkages

between the input vollage and output current in the conical magnel coi The

connected controller uses a lincar quadratic regulator with integral action to stabilize the cone-shaped AMB system, while the decoupled controller is used to

stabllize the five DOF systems The study in [17], Mohamed and Emad have

provided a model of the conical bearing derived in state variable form with airgap flux, airgap displacement, and velocity used as state variables ‘'his description converts the equations of rotor dynamics into a simple form However, considering the dynamic of systom, the study ignores the variation in parametric and gyroscopic effects which is a significant nonlinearity ‘The (-parameterization control method was then proposed for designing system stabilization in terms of

two [ree paramoters A recent publication on modeling for CAMB system, where

the mismatched disturbances are determined using a disturbance estimator, is presented in [18] Lhe observer is built on the basis of the Kalman filter, which is

designed lo ellectively handle the trade-off belweert noise Tejection and margin recovery However, when considering the technological characteristics of the CAMB, the above model ignores the match disturbances effect on the control input

channel Based on the estimate of mismatched disturbances, Molina used offset-

free model predichye control (OF-MPC) to effectively handle coil current

saturation

The key point lo guarantec the system output quality is to properly handle the coil current saturation and lumped disturbances of CAMB, Conventional decentralized

PID controls still have significant limitations when combined with magnetic

bearings, making i difficult Lo fully realize the active potentials thal might allow

a much higher degree of control of rotor vibration, positioning, and alignment control Due to the high nonlinearity in a CAMB system, nonlinear control leclmiques are a logical choice [hal can allow for a more thorough analysis of the nonlincarities and underlying coupling features as well as allow for greater use of available clearance during operation Controllers designed by using the fuzzy techmque have been recently studied, and [19}{21 have shown for the CAMB

system that the quahty and stability of the syslem are guaranteed and the

displacement over-shoot is small In [21], Arvind Katyayn and Praveen Kumar Agarwal published modeling of the conical AME structure for full support of the [ive DOF rotor system They erihanced the syslem performance by developing the Interval type-2 fuzzy logic controller (IT2FLC) using an uncertain bound method

‘this controller enables the managing of parameter uncertainty while reducing the requirement for exact system modeling The simulation results demonstrate that in

6

terms of rising time, overshoot, and settling time, the suggested contraller performs better than the type-l fuzzy logic controller Nonlinear adaptive control is proposed in [22] to improve performance with uncertainty in the rotor angular velocity and stabilize the system over a greater range of error than the nonlinear

controller without adaptation

1.1.6 Fundamental of Sliding Made Control

In control systems, Sliding Mode Control (SMC) is a notable robust nonlinear control method that alters the dynamics of a nonlinear system by applying a sct-

valued control signal thal forces the system lo "slide" along a cross-section of the

system’s normal behavior Sliding mode control is a variable structure control

method, where trajectories will slide along the boundaries of the control structures

This will result ina new class of system dynamics that slide on a so-called sliding,

surface The main of this control method is its robustness, so it has many

applications in electric drives, robotics and other nonlinear systems

SMC controllers have been utilized ina variety of ways to control the AMB system

in order to either achieve the necessary system performance or solve a particular application-oriented issuc Many academics have suggested combining SMC with the feedback linearization approach for AMB management

The main difficulties in controlling the AMB system are rotor variations from the

nominal position and unbalance brought on by mass imbalance Because of its advantages in terms of robustness, faster convergence, case of design, and

improved Wansient and steady-stale responses, SMC has been employed in a

variety of research studies to address these issues [23]

To accomplish reliable regulation of the rotor to the center when the system is subjected to external disturbance, parameter uncertainty, and unmodeled

dynamics, Smith and Weldon [24] have worked on the nonlinear formulation of

cascaded SMC and feedback Imeanvalion controller The second-order nonlinear

coupling effect is eliminated by the feedback linearized control rule in this study's voltage control, which demanded that the system be linearized at an equilibrium point established al a predelormined bias euent Because (he umeerlaimly is still

present, due (o the parametric changes, the SMC control technique is developed

such that the tacking of the rotor position can be performed

Secondary electromagnetic effects like flux leakage, fringing flux, and finite core permeance are some of the contributing variables that degrade the system

performance for some AMBs syslems Yeh [25] have researched the effeels of this

nonlinearity effect in AMBs system and proposed the employment of the SMC approach to stabilize the system

A few research groups are considering the control approach specifically created to seduce the vibration effect caused by rotor imbalance in the study of AMBs system This is because vibration brought on by imbalance is proporlional lo toler spocd squared and is certainly becoming more important for high-speed applications

‘Therefore, it appears that adaptive vibration control is the best control strategy to

achieve this design goal In [26] an integral sliding mode controller (SMC) is designed for stable levitation, with the assumption of known mass imbalance

SMC has proven to be able to provide reliable rotor positioning in a variety of

system conditions, even in the presence of parametric uncertainties, nonlinearities, and disturbances, m the family of nonknear robust control algorithms for AMBs

systems The evalution of the SMC control algorithm over the last few years has made it possible for this controller type to systematically accommodate varied systems and design needs This has been inade possible by dhe adaption of numerous linear and nonlinear system design tools, This has made a promising contribution to research, particularly in the area of AMB's control system

1.1.7 Fractional Order Calculus

Fractional-order calculus (FOC) has been around over 300 years, and it is applied

lo different fields in recent years The utilization of FOC in system modeling and control has been raised a lot In tenus of system modeling, many phenomena demonstrate fractional-order characteristics These mathematical phenomena allow describing a real object more accurately than the classical “integer-order” incthods Fractional calculus provides a preferable method to describe complicated natural objects and dynamical processes such as electrical noises, chaotic systems, and organic dielectric materials Permanent magnet synchronous motor [27], gas lurbines [28], supercapacitors, baiteries and fuel cells [29] can be identified as

fractional-order models

Since fractional-order caloulus wns created, iL las been widely used in dilTerenl fields of science Researchers have found that fractional-order controllers have the potential to provide higher and more robust control performance than the integer- order controller [30], which has made them particularly significant and fascinating

in the field of control engineering Many [raclional-order controllers have beer proposed, including fractional-order sliding mode controller [31], [32], fractional- order PID controller [33], [34] fractional-order intelligent PID controller [35], and

so on, With the benefits listed above, fractional caleulus can be useful in a variely

of industrial and scientific areas, including the study of electrical circuits [36], signal processing [37], and robotics [38]

Recently, a fractional-order controller was employed to control active magnetic bearings systems In [39], to achieve this trade-off between the simplicity of the controller siructure and Ihe performance of the closed-loop system in control design for AMBs systems, fractional-order control has been applied A relevant work that studied and implemented fractional-order control on AMB systems was demonstrated in [40], where a fractional-order PID (FOPID) controller was applied

lo the rotor susperision by using radial and thrust AMBs A study in [41 J, describes the design of the FO controller for a simple construction of a magnetic levitation system As the overshoot requirements were wsed to select the parameters, rather than using intricate frequency domain [atures or artificial intelligence, the process

of tuning parameters was made simpler Lastly as is customary in studies of this kind, this paper contrasted the performances of PID and KO ‘The work in [15] had shown the benefit of utilizing FOADRC controllers for improving the transient and steady performances and robustness with respect to parameter uncertainty and disturbance applied 10 conical active magnelic bearing system

1.1.8 Extended state observer

A class of high-gain observers knows as extended slate observers (FSO) are effective for controlling the output feedback of nonlinear systems The

fundamental idea behind an L/SO is to simultaneously estimate the state and the

defined extended slate while viewing the system's enitire uncertainty as an extended

state of the system The system total uncertainty can be directly corrected in the

contro! action in real-time based on the output of an ESO The most advantage of ESO is only relative to dogree of the system, sc it requires relatively less information about a dyramic system Varions ESO-based control methods have

been implemented in industrial systems [42]-[44]

In the study [45], based on the accurate model of the AMB-rotor system with imbalance vibration, the extended state observer is applied to estimate the imbalance mass information ESO is also introduced to improve the adaptive oontrol algorithm for a magnetic bearing syslom in [46] Tn he work im [47], ESO

is used to estimate the unknown disturbances of the active magnetic bearing system, such as model inaccuracy and external disturbance ‘hen, the controller continuously updates its contrat law through the kumped disturbance estimated by

the ESO in real-time, thus, reducing the impact of uncertainties on the system

1.2 Motivation

As previously noted CAMB is now a possible candidate for a variety of magnetic

force-supporled applications because of ils distinctive goometrie design, which

lowers the number of active magnets needed However, in addition to their

‘benefits, the nonlinearities and inherent coupling properties of CAMB system

make the controller design is a challenging (ask The control of CAMB is predicted

to be a topic of frequent discussion in the coming years In CAMB model, velocity and current dynamics have parameter uncertainties and disturbance Besides

disturbances come from external such as gyroscopic effect and rotor mass

unbalance In the worst-case scenario, disturbances caused by these sources might have a significant impact on system performance, causing the rotor to partially or

even annularly mb the stator, resulting in permanent bearing system damage

As a result, it is important for CAML to reduce the impact of disturbances In combination with the disturbance compensation capacity of the ESO with the advantages of the FOSMC, the thesis proposed a type of FSO based fractional order sliding mode control (FOSMC-ESO) strategy to achieve fast response and minimize tracking errors as well as better control quantity without chattering In

addition, ESO estimates other states of systems such as the position and velocity, which helps to lessen sensor and sensor measurement noise,

13 Contributions

‘The major original contributions in this work are listed as follows:

- The principle of operation and design of CAMB is presented

- The electromagnetic equations governing the relationship between magnetic forces, air gaps, pyroscopic force, and control currents are used to build the nonlinear model of a conical magnetic bearing

Proposed using ESO to deal with the hưnged dishubancos of CAMB system The LO is designed with new state variable to avoid the amplification of disturbances that come from extemal disturbances, uncertam clectromagnetic forecs and parametric uncertaintics Moreover, LSO estimates the position and velocity of systems, which helps to lessen sensor and sensor measurement noise

- Based on extended stale observer, a fractional order shding mode control

(FOSMC) is designed to increase the efficiency and improve the control performance of CAMB system It is demonstrated that the system shows a

toeller control performance

1.4 Thesis outline

This thesis is structured as follows:

Chapter 1 Introduction A detailed overview of the AMBs, including its

development, applications, advantages, and disadvantages are discussed Then the thesis presents a disoussion about the CAMB, modeling, and some control requirements Then it briefly discusses SMC, FSO and FOC The motivations of

the thesis are provided, as well as the dhesis's main conlbutions A thesis oullme

and conclusion are given at the end of the chap

Chapter 2 Dynamic modshng of conical magnelic bearing The eonstruction and working principles of AMBs and CAMB are presented ‘then, based on the mechanical and electromagnetic analyses of the system, a five DOI mathematical description of the madel is presented

Chapter 3 Control system design At first, BSO is identified to estimate lumped

disturbances, Ilen the FOSMC is discussed and combmed The FOSMC is

calculated and applied to a CAMB systom

Chapter 4 Some simulation scenarios are shown including a comparison of the

performance between ADRC, SMC-ESO and FOSMC-FSO controllers

Chapter 5 The conclusions, challenges, and future works are summarized

10

CHAPTER 2 SYSTEM DESCRIPTION AND MATHEMATICAL

MODEL

The main components of active magnetic bearings, as well as the basic operating

principle, are discussed in this chapter Then, the electromagnetic force of an

electromagnet is calculated using basic physics rules and equations In addition,

the mathematical model of the CAMB is created The final model created will

serve as the basis for creating the methods of control discussed in the next chapters 2.1 AMBs general schema

This section provides an introduction to active magnetic bearings The primary components as well as the fundamental operating principle are presented

Fig 2.1 depicts the basic AMB components Electromagnets are made up of a soft

magnetic core and electrical coils They resemble the stator of an electric motor in

certain ways

iron core Both its magnetic saturation and magnetic permeability must be

high The core usually consists of insulated lamination sheets to reduce

eddy current losses

- Windings: The magnetic field is created by the current flowing through the

winding An insulated conductor is used to create the winding, which is

wound around a soft magnetic core The conductor must have low electrical

resistance and be wound with a high fill-factor in order to increase the

AMB's efficiency

has been shrunk onto a non-magnetic shaft To avoid unbalances,

manufacturing tolerances must be extremely tight To overcome the

11

centrifugal stress caused by high-speed rotation, the mechanical properties

of the rotor lamination must be good

Position Sensors: Position sensors are present in AMBs in the majority of

applications AMBs are actively controlled in relation to the sensor signal,

hence sensor performance has a significant impact on control performance

Inductive, eddy current, capacity, and optical displacement sensors are

among the sensor types utilized in AMBs

Controller: AMBs are closed-loop regulated by the controller Several

techniques are used, including PD, PID, optimal output feedback, and

observer-based state feedback

Power Amplifiers: The power amplifiers transform the control signals into the control currents Since switching amplifiers have low losses, they are

frequently utilized In an AMB system, the amplifier frequently acts as a

limiter Although they are still uncommon, amplifiers with voltage or flux density control may occasionally enhance AMB performance

2.1.2 Theoretical models

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, cwrent flowing in the coil,

dynamic force, magnetic flux, inductance (magnetic flux density), magnetic field, energy stored in the air gap, magnetic force and magnetic field strength

LN

Figure 2.2 Simple electromagnet structure

Where J is the current flowing in the coil [A], gis the air gap [m], Nis the

number of coil tums, 4, is the cross-section of a steel core [m°], and / is the

length of the area surrounding the flux's surface [m]

The current 7 flowing through the coil will generate a dynamic magnetic force,

resulting in magnetic flux This magnetic flux loops through the steel core, the air

12

gap, and the rotor, creating an electromagnetic attraction that pulls the rotor towards the electromagnet's steel core

Ampére’s circuttal 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 (Hg, 2.1) between the magnetic field and the current sum enclosed by the closed integration path

AAmpére’s circuital law is expressed as the following formula:

Dota Set Sta NE, T AC âm 22)

where 77 ís the magnetic field strength [A/m ], # is the magnetic flux density

[Wh/m'], jis the permeability of magnetic material, = 2 [#/m], 1, is the average length of the iron core loop, and 7, is the length of magnetic flux through the air gap

For the system in Fig 2.1, because there are two air gaps and the air permeability

coefficienL is much smaller than (he iron permeability coeflicienl (4, <u, ), the

xumber 2.2 can be ignored

'Then Liq, 2.2 can be simplified

‘The total magnetic flux 4 generated by the magnetic flux foree H consists of two

components: g, passes (through the air gap, crealing art clectromagnetic allraction

that attracts the rotor to the magnet, and ý, is the magnetic flux loops through the stecl core called magnetic flux leakage Ignoring magnetic tlux leakage, from Eq 2.4, the expression from the magnetic flux through the air gap is shown as follows:

ca

ø~#,—B,„4,— (2

where 4, is the cross-sectional arca of the air gap

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

3 2 Assuming the magnetic force is /2, the previous derivation can be used to determine the mathematical relationship between magnetic force and rotor position As a result, the magnetic force # can be obtained by considering the energy stored in the air gap as follows:

Substituting the formula in Eq 2.4 lo Eq, 2.6, the expression for calculating the electromagnetic force for the electromagnet mechanism Mig 2.2 is shown as follows:

order fo build the mathematical mods] of the AMBs system To fully position a

rolaling shafl ina magnetic Geld, 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 modeling of CAMB

‘The figure below depicts the rotor model of conical magnetic bearings, which has a cylindrical shape in the middle like ather regular rotors Especially at

the two ends of the rotor is a beveled cone so that the force generated by the

clovlromagncl can be separated into two componenls axial and radial Froin (here,

the electromagnet system at both ends of the shaft can be used to control both axial

amd radial movements of the rotor A shaft that is supposed to be levitated with

CAMB has five degree of freedom, two radials (y and z-direction) in cach end of the rotor and one axial (x-direction)

14

Figure 2.3 Schematic of conical active magnetic bearing forces

2.2.2 Electromechanical interaction

The forces (F, ,F,) generated by the active magnetic bearings are integrated

into the rotor dynamics They enter the rotor equations of motion as the forces

magnetic bearings Table 2.1 lists the parameters of the system

Table 2.1 System parameters

8o radial air gap 0.45 mm

A cross-sectional area 118 mm2

8 inclined angle 0.98 rad

N magnetic coils 82 tums

The equations of motion can be written as following Newton’s second law and

Euler’s second equations:

where F,F,,F.,M,,.M, are the external disturbances acting, on the rotor

In the two rotational kinematics equations, gyroscopic effect [48] which is a significant nonlinearity associated with the rotor dynamics is considered The coupling between pitch (rotation around the x-axis) and yaw (rotation around the y-axis) motions caused by the gyroscopic effect in the AML system is proportional

to the speed of the rotar As a result, stabilizing the system for high-speed application becomes a more difficult process

2.2.3 Linearized bearing forces

Assumptions, the reluctance of the iron is neglected concerning gap reluctance In addition, all magnets have identical structure and the fringing effect can be neglected, the electromagnetic forces are given by:

control purposes, il is much mare desirable 1o have a Hnear relation between the

control current and force The bearing air gaps presented in Eq 2.10 can be referred

to the center of gravity (COG) coordinates involving the geometrical quantities introduced in Fig 2.3 as

2 —e,—s8inB = yb, cos8 Bie = Bo sin TE y HA, cosh

85:7 Be F8iN BT «—BB, cos 8 (2.42)

gn = 8, —Z8in dF ¥=B9, cos

where g, is the steady-state nominal air gap In Lerms of the actual air gap aned the

current, the change in magnetic force can he written as

16

2.13 to arrive al the magnelic force, which is lnearized as:

F-F,+K,i,-K,zsin B+ K, (y+ 6,0,)o0s 8 F,-F,+K,i,+K,ssinB—K, (y+b,0,)oos 8 P-K Ki, K,zsinft KU 68,)oos #

Tà T1 + 6=, ssín 8— K, (y—b8,)cos 8 F,-F,+K,i,+K,zsin f+ K,(x—b,0.)o0s 8

Kak 1K, 1K, csing K(x 6,6,)cosf

~12 1K K,rsinfik,(a &8,)c0s 6 Kaw 1K K,2sinf K(x b6,)oos 8

are the position and current stifthesses,

respectively

Apply Eq 2.14 to Eq, 2.9 the magnetic foree can be rewritten as

Whore q is displaccmont matrix, iis current matrix, K, is displacoment stiffness

matrix of the AMR and K, is stiffness current matrix,

q-[s x Ox 9y Ï

Â~ E,á,ñ; 0k; 8,8 61”

Kyoosp Ka 0

038 —Ế œ 0 Kop Ky 0

Q

Kook Ky 0 |

0 Ky a— b, cos B—K, sinB

-K, 0 o~—b, cos B—R, sini f

0 -K,

Ku TK, — (h + ŒK, —K, 3s đoos 8

=Ku=2e08' 8Q +.)

— Ky- KuTĂcos) ,Œ, 1K, Xb bà)

in ACK, —K,, [B+ 2, Joos 82K, sin /)]

Ky — Ky — 008 AK, 1 K, (1 Bp oos 8 R,(4, + 6,)eos Ø) |

‘the linear magnetic field is assumed above, however in practice there exist

hysteresis and saturation in the magnetic field so that the magnetic field is not

linear Hence, the exists uricerlainties are called urmmodeled magnelic force include

fringing ellect, Mux leakage, high-order components are omitted in Taylor expansion,

2.2.4 External disturbances 2.2.4.1 Rotor mass imbalance

Tn addilion to gyroscopic effecis and paramclor coupling, imbalance mass can produce system vibrations in active magnetic bearing rotor systems The co- channel disruption brought on by the mass imbalance is the most crucial element Rolor mass imbalance of magnetically levitated rotors causes deteriorations af

dynamic and static characteristics The eccentricily of the rotor causes unbalanced

vibration, which is a common problem in rotating machinery The centrifugal force

produced by the aceclcration of the inertia center when the rotor is "foreed" to

rotate around its center of inertia, G, , rather than its center of geometry, G, also manifests as synchronous rotor displacement In the worst situation, because the imbalance effect is proportional to the rotor rotational speed, high-speed operation

resulls in the rotor whirling beyond the permitted airgap, which causes the rotor to

18

partially or, worse still, annularly rub the stator and cause long-term damage to the

bearing system

Figure 2.4 Illustration of unbalance rotor

The imbalance is modelled as a residual mass displaced from the rotating axis

which combined into one variable called imbalance moment and phase The static and dynamic imbalances, which is formulation is presented as:

where e is eccentricity, ¢ is the is inclination angle in the shaft and the

equilibrium position when the rotor rotates, @ is a static and dynamic unbalanced

phase angle

2.2.4.2 Hydrodynamic effect

Another type of external disturbance exist in canned motor pumps which utilizes

AMBs to eliminate the need for mechanical bearings and shaft seals [49] is

hydrodynamic effect As an absolutely leak-proof safety pump, a canned motor

pump requires a more complicated and clever design The rotor is completely surrounded by a hermetically sealed can with a thin fluid layer between the rotor

19

can and the stator can However, this causes whirling of the rotor, which can

become more serious at high speed In addition, hydrodynamic interactions

generate cross-coupling forces between the radial axes of significant magnitude

This phenomenon, which is the main cause of system instability, is far more

complex and cannot be neglected

The hydrodynamic force amplitude is not significant during normal operation and

is only brought on by the velocity difference between the cans of the rotor and stator However, forces on the impeller, mass imbalances in the motor, and external disturbances all cause the rotor to wander slightly away from its geometric center,

which immediately causes radial hydrodynamic forces to act on the rotor These

forces depend on the rotor’s rotational speed and movement The hydrodynamic

forces can be described by the dimensionalized stiffness and damping matrices

coefficients K,, and C,,, respectively [50] Their interaction with the axial force on

the rotor is given by:

to as the bearing Summerfield number; F, is the static force, e=eg,=Jx +y" ,

and 0<¢<1 is the eccentricity ratio, ,« is the absolute viscosity; 7 is the length

of rotor; R, is the radius of the rotor, @ is the angular velocity

The values of the non-dimensional stiffness matrix are given by [50]

4

ye

‘The hydrodynamic forces only act in the radial motion of the rotor and have no

effeot on the rotation of the rotor axis about the z-axis Henwe, the hydrodynamic

force equations in CAMB coordinate system can be written as follows:

1000 0

0100 4 T-|001 4% ©

0100 +

0914 0

2.2.5 Magnetic bearings actuation Every magnetic pole pair of the CAMB can provide both the axial force and radial

forge to the roler, constitutes an mbhorently unstable syslers The stabilizing

control of the electromagnet current is necessary to find a solution for this issue

‘The coil current of the system operating in differential driving mode is employed

to control the rolor The general principle is one clevtromagnel is driven wilh the sum of a bias an] a control current, whereas the opposile one is driven with their

difference By increasing the forces on the lower bearings and reducing the forces

on the upper bearings in Fig 2.3, any positive movement along y away from the

equilibrium point ean be compensated for The goal of this control stralegy is to

sustain levitation and maintain the rotor position at the center of the stator with

only the bias current present on each magnetic pole pair Five control currents are used to controt (he five-DOF plant The following is how the currents passing through the coils are expressed:

AMBs is a group of a nonlinear, and unstable dynamical system with lumped

disturbances [51], [52] “The model of lumped disturbance combined effect of

extemal disturbance, the uncertain clectromagnetic force and parametric

imeertainties The parametric uncertainties come from manufacturing and assembly errors in the nominal air gap go, inclined angle f, rotor mass m or the other parameter of Rotor- CAMB

Combine Eq, 2.9, Bq 2.15 and Faq 2.16 we have

Consider the parametric uncertainties in the Eq 2.27 and use Eq, 2.26 we have

(M- AMG +(G= AG) Q+(K, +AK,)q— (K, +AK Hi, +F, (2.28)

where M,G,K,,K_ is the nominal value, the variations of time-varying system

parameters are symbolized as AM,AG,AK,,.AK, The lumped disturbances can

‘be denoted as

The K, and KI are invertible The interstitial component ean be climinated by using the following control struclure

Rg 3.28 can be rewrillen as

where w is he control signal From here, the FOSMC-ESO is applied to eliminate

the interstitial components as well as stabilize the control object

Assumption 1 The desired trajectory and its derivatives are bounded, suggesting

that the positions of the rotor are also bounded

Assumption 2 ‘he unknown disturbance L and its derivatives are bounded, ie., [L sa, L|<a,, where aa, are positive constants

2.3 Conclusion

To this chapler, the dynamic model of CAMB has been analyzed The five DOF qaodel has been proposed based on the electromagnetic analysis, The equation for the system is typically more complicated and nonlinear ‘hus, it is impossible to vse linear control rules directly The FOSMC-ES0 algorithm is consequently suggested as a solution to the issues,

bà a

CHAPTER 3 EXTENDED STATE OBSERVER BASED CONTROL

DESIGN

This chapter identifies extended state observers to estimate the hunped disturbance

before discussing and utilizing the SMC and FOC to create a FOSMC controller

This controller is calculated and applicd to a CAMB system

3.1 Extended state observer

Using above differential equations, nonlinear state space model for Rotor-CAMB system is developed by rotor position and rotor velocity as two state variables as

X, = qx, = q with rotor displacement q as output variable and b= M_”

‘The idea of LISO is to estimate the total disturbance through an extended state and compensated them For the active magnetic bearing system, HSO can estimate the sum disturbing forces acting on the magnetic hearing system by measuring the rotor output displacement and the controller output

Define x,—M L is the extended state, Eq 3.1 can be rewritten as:

and « denotes positive observer gain to he determined

Denote the estimation errors as:

310 0 0

A-|-3 0 1], #E-|1|, B.-|0

‘The A, is Llurwitz matrix, thus, there are positive define matrix P satisfying these

functions where I denotes an identify matrix ATP | PA, = L

‘The definition of a Lyapunov function 77 is

TLis evident that F< 0if the observer gaits «1 is selected satisfying a> 0 and

g's >2y, As a result, the estimation errors ¢ is bounded, indicating that ¥, is

finally bounded The aforementioned ESO provides estimates for the entire system states, including the overall disturbance and unmeasured velocity ‘the next

comroller design will imcorporale the estimaled slales

3.2 Fractional Order Sliding Mode Control

3.2.1 Principle of Sliding Mode Control and Chattering Problem

Sliding mode control is a kind of variable structure control algorithm The fundamental difference between sliding mode control and conventional control

strategy is the discontinuity of its control; that is, the output of the controller changes with time and presenis switching charactoristics Smee sliding mode is independent of system parameters and disturbances, the system in sliding mode has strong robustness in the sliding mode controller, the control law usually

consists of (he equivalent control u,, aud the switching control u,, The equivatent

iy

control keeps the state of system on the sliding surface, while the switching control forces the system sliding on the sliding surface [53]

A second-order nonlinear system can be described as:

#- fQ,Ð+bu0)+ đÐ (48)

where 6 > 0, wis the control output d denotes external disturbance and uncertainty while we assune |đ(2| <Ð

The design of a SMC system comprises two steps:

- First, the design of a suitable sliding surface which depends on system variables

A sliding surface can be designed as follows

where ¢> Oand # is the tracking error

- Second, the SMC's control law wis built to push the system variable toward the sliding surface while keeping the state variable stationary

‘To find the equivalent term of the control law, choose §— 0, we get

The switching control and equivalent control arc both inchuded in the sliding mode

controller, then we have

Chattering can lead to the wear of moving mechanical parts and high heat losses

in the power circuit The problems of reducing or climinating this challering is solved with many procedures have been designed One of them, in the simplest

case, is replacing of discontinuous signum function sgn(s) by a continuous

approximation, a smooth function, At this thesis, we choose the sigmoid function

26