Abstract Active magnetic bearings AMBs are electromagnetic mechanism systems in which non-contact bearings support a rotating shaft using attractive forces generated by electromagnets t
INTRODUCTION
State of the art
1.1.1 Introduction of Active Magnetic Bearing
The increasing demand for high-precision, high-speed devices has made the study of bearing mechanisms and development essential, as bearings are critical components in rotating machinery Traditionally, bearings support a machine's rotating parts, but active magnetic bearings (AMBs) offer a non-contact alternative to conventional ball or fluid bearings, utilizing attractive forces from electromagnets through closed-loop control The history of AMBs dates back to 1939 when Werner Braunbek explored magnetic levitation, with early applications in experimental physics and suggestions for uranium centrifuges in the 1940s AMBs, often called magnetic suspension systems, are employed in ground transportation to levitate vehicles using a combination of regulated electromagnetic and permanent magnetic forces.
The Active Magnetic Bearings (AMBs) system features a suspended cylindrical rotor that rotates at varying speeds based on the application's requirements for higher degrees of freedom (DOF) Its most notable characteristic is the noncontact suspension mechanism, which provides numerous advantages over traditional bearings, such as reduced rotating losses, increased operating speeds, the removal of costly lubrication systems, and the capability to function in extreme temperatures and vacuum conditions, all contributing to an extended lifespan Consequently, AMBs have been effectively utilized in a wide range of applications for decades, including industrial machinery, medical equipment, power and vacuum technologies, and artificial hearts.
Figure 1.1 Active magnetic bearings in compressor [6]
1.1.2 Principles of Magnetic Bearing Function
The principle of generating contact-free magnetic field forces through the active control of electromagnet dynamics is commonly utilized in magnetic suspensions Figure 1.2 illustrates the schematic arrangement of the system's rotor and magnetic coil (stator), highlighting the typical construction of Active Magnetic Bearings (AMBs) systems.
Figure 1.2 Function principle of an active electromagnetic bearing [3]
The essential components of an AMB-Rotor system include the rotor, magnetic actuators, position sensors, power amplifier, and controller The controller issues commands to the power amplifiers based on the rotor's position measurements from the sensors, ensuring that the output currents to the magnetic actuators generate the required electromagnetic force for rotor levitation This process establishes a closed loop that stabilizes the rotor at the equilibrium position of the air gap.
1.1.3 Advantages and disadvantages of AMBs
Magnetic bearings utilize the electromagnetic force generated by electromagnets to lift the rotor shaft, enabling it to rotate within the stator while maintaining a minimal gap of just 0.5 to 2mm These innovative bearings offer significant advantages over traditional mechanical bearings, positioning them as a transformative solution for the manufacturing industry.
Magnetic bearings enable operation at elevated temperatures, significantly enhancing system performance With no contacting parts, magnetic bearings eliminate the need for a lubrication system, resulting in a weight reduction of up to 5% compared to conventional bearing systems.
- Oil emissions are reduced by removing lubrication from bearings, providing direct environmental benefits The removal of oil from the system also makes it more fire-resistant
Active Magnetic Bearings (AMBs) eliminate friction losses due to their non-contact design, which also reduces fatigue and wear commonly associated with traditional ball bearings This non-contact system enhances operating speed and overall efficiency.
Magnetic bearings, as an active system, offer significant advantages over passive systems by compensating for unbalance and actively controlling rotor behavior at critical speeds Additionally, active magnetic bearings (AMB) serve as sensors that provide valuable insights into changes in shaft dynamics, facilitating effective system monitoring This capability enhances system diagnosis, ultimately leading to reduced maintenance costs and extended intervals 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, etc
- Backup bearings are still required in many systems in the event of an AMBs system breakdown
- Environmental conditions need to be ensured to avoid magnetic force attracting materials such as iron, and steel billet outside
Active magnetic bearings (AMBs) have become increasingly popular in recent decades due to their ability to suspend high-speed shafts without mechanical contact or lubrication This technology is utilized worldwide across various industrial, space, and laboratory applications The growing adoption of AMBs is largely attributed to their numerous advantages over traditional bearing systems, making them suitable for a wide range of applications.
A significant application of artificial magnetic bearings (AMBs) in medical equipment is their role in pumping blood within artificial hearts, ensuring that blood is ejected at the required rate for proper circulation in the human body Additionally, AMBs enable the miniaturization of the rotor suspension structure, which is crucial for the design of auxiliary mechanical blood circulation devices.
Active magnetic bearings (AMBs) eliminate friction and lubrication by having no contacting parts, which reduces rotor vibration and supports the advancement of motors towards higher speeds and power densities Consequently, the use of magnetic bearings is on the rise in various rotating machinery applications, including compressors, pumps, wind turbines, and flywheel energy storage systems (FESS).
AMBs are utilized in hazardous and high-temperature environments, including nuclear power plants, gas turbine applications, and aircraft turbo-machinery, to replace mechanical bearings and shaft seals.
- Magnetic bearing's 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
1.1.5 Conical Magnetic Bearings: An overview
Current advancements in active magnetic bearings (AMBs) emphasize the creation of diverse geometrical bearing designs aimed at optimizing axial space for the installation of supplementary mechanical components like gearboxes A promising direction for future development is the implementation of a conical shape for active magnetic bearings (CAMB).
To achieve full support for a five degrees of freedom (DOF) rotor system, two radial active magnetic bearings (AMBs) and one axial AMB are necessary, which adds complexity to the system High-speed operation of the rotor can lead to shaft disc imbalance To optimize axial space for additional mechanical components like gears, a conical magnetic bearing design is proposed, featuring a slanted airgap between the rotor and bearing This configuration allows electromagnetic coils to provide both axial and radial forces, eliminating the need for a pair of axially-controlled coils, thereby conserving energy for maximum load support and reducing copper losses.
Figure 1.4 System with cylindrical AMBs [14]
Figure 1.5 System conical AMBs [15] (1) impeller; (2) centering tip; (3) conical geometry; (4) rotor; (5) electric motor; (6) magnetic actuators
CAMB presents two significant coupling properties: current-coupled and geometry-coupled effects, which complicate dynamic modeling and control The challenges in controller design for CAMB systems arise from the nonlinear dynamics, minimal natural damping, stringent positioning requirements, and unstable open-loop system dynamics Typically, a proportional-integral-derivative (PID) controller is preferred for its simplicity and ease of parameter tuning.
6 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
Motivation
CAMB's unique geometric design positions it as a promising candidate for various magnetic force-supported applications by reducing the number of required active magnets However, the nonlinearities and inherent coupling properties of the CAMB system present significant challenges in controller design As a result, the control of CAMB is expected to be a prominent topic in future discussions The CAMB model faces parameter uncertainties and disturbances in velocity and current dynamics, with external factors such as gyroscopic effects and rotor mass unbalance contributing to these disturbances In severe cases, these disturbances can severely impact system performance, potentially causing the rotor to rub against the stator, which may lead to permanent damage to the bearing system.
To minimize the impact of disturbances, CAMB must implement effective strategies This article proposes a novel approach that combines the disturbance compensation capabilities of the Extended State Observer (ESO) with the benefits of Fractional Order Sliding Mode Control (FOSMC) The proposed FOSMC-ESO strategy aims to achieve rapid response times, reduce tracking errors, and enhance control performance while eliminating chattering effects.
10 addition, ESO estimates other states of systems such as the position and velocity, which helps to lessen sensor and sensor measurement noise.
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, gyroscopic force, and control currents are used to build the nonlinear model of a conical magnetic bearing
The article proposes the use of an Extended State Observer (ESO) to effectively manage lumped disturbances in the CAMB system The ESO is engineered with a novel state variable to prevent the amplification of disturbances caused by external factors, uncertain electromagnetic forces, and parametric uncertainties Additionally, the ESO provides estimates of the system's position and velocity, which significantly reduces sensor noise and measurement inaccuracies.
A fractional order sliding mode control (FOSMC) is developed using an extended state observer to enhance the efficiency and control performance of the CAMB system The results indicate a significant improvement in the system's control performance.
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 discussion about the CAMB, modeling, and some control requirements Then it briefly discusses SMC, ESO 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 modeling of conical magnetic bearing The construction and working principles of AMBs and CAMB are presented 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, ESO is identified to estimate lumped disturbances, then the FOSMC is discussed and combined The FOSMC is calculated and applied to a CAMB system
Chapter 4 Some simulation scenarios are shown including a comparison of the performance between ADRC, SMC-ESO and FOSMC-ESO controllers
Chapter 5 The conclusions, challenges, and future works are summarized
SYSTEM DESCRIPTION AND MATHEMATICAL MODEL
AMBs general schema
This section provides an introduction to active magnetic bearings The primary components as well as the fundamental operating principle are presented
Figure 2.1 AMBs structure with single-DOF
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
The iron core is essential for conducting the magnetic field across the air gap, requiring high magnetic saturation and permeability Typically made from insulated lamination sheets, the core is designed to minimize eddy current losses.
The magnetic field in an AMB is generated by the current passing through the winding, which consists of an insulated conductor wrapped around a soft magnetic core To enhance efficiency, the conductor should possess low electrical resistance and be wound with a high fill-factor.
In traditional designs, the rotor consists of a laminated packet that is tightly fitted onto a non-magnetic shaft, requiring precise manufacturing tolerances to prevent imbalances.
12 centrifugal stress caused by high-speed rotation, the mechanical properties of the rotor lamination must be good
Position sensors play a crucial role in active magnetic bearings (AMBs) across various applications, as their performance directly influences control effectiveness The types of sensors commonly used in AMBs include inductive, eddy current, capacitive, and optical displacement sensors.
- 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 convert control signals into control currents and are commonly used due to the low losses associated with switching amplifiers In an Active Magnetic Bearing (AMB) system, these amplifiers often function as limiters While still relatively rare, amplifiers that utilize voltage or flux density control can sometimes improve the performance of AMB systems.
Analyzing the physical structure of the Active Magnetic Bearings (AMBs) system is essential for understanding its dynamic interactions Key components to consider include the voltage applied to the coil, the current flowing through it, dynamic force, magnetic flux, inductance, magnetic field, energy stored in the air gap, magnetic force, and magnetic field strength.
Where I is the current flowing in the coil [A], g is the air gap [m], N is the number of coil turns, A g is the cross-section of a steel core [m 2 ], 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
13 gap, and the rotor, creating an electromagnetic attraction that pulls the rotor towards the electromagnet's steel core
Ampère’s circuital law establishes that the line integral of the magnetic field around a closed loop is equal to the algebraic sum of the currents passing through that loop This relationship is mathematically represented in the formula: \$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}\$, where \$\mathbf{B}\$ is the magnetic field, \$d\mathbf{l}\$ is the differential length element along the loop, and \$I_{enc}\$ is the total current enclosed by the integration path.
(2.2) where H is the magnetic field strength [A / m], B is the magnetic flux density
[Wb / m 2 ],is the permeability of magnetic material, B
= H [H m/ ], l s is the average length of the iron core loop, and l g 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 coefficient is much smaller than the iron permeability coefficient ( g s ), the number s s
in Eq 2.2 can be ignored
Then Eq 2.2 can be simplified:
= (2.3) or written in the form:
The total magnetic flux \$\Phi_0\$ produced by the magnetic flux force \$F\$ comprises two components: \$\Phi_g\$, which traverses the air gap to generate electromagnetic attraction that pulls the rotor towards the magnet, and \$\Phi_r\$, which represents the magnetic flux that loops through the steel core, known as magnetic flux leakage By disregarding magnetic flux leakage, the expression for the magnetic flux through the air gap can be derived from Equation 2.4.
= = = ( 2.5) where A g is the cross-sectional area of the air gap
Ideally, the magnetic field is well distributed, thus energy stored in the volume of air gap can be calculated as:
The magnetic force \( F \) can be derived from the mathematical relationship between magnetic force and rotor position, utilizing the energy stored in the air gap.
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:
The electromagnetic force exhibits a quadratic relationship with the current flowing through the coil, as indicated by Eq 2.8, and an inversely quadratic relationship with the air gap Consequently, by adjusting the current in the coil and considering the air gap distance, the electromagnetic force can be effectively modified.
In the next section, the above-proven theories will be applied to build descriptive kinematics equations for the CAMB system with five DOF
This section develops the electromagnetic system model for magnetic bearings featuring conical air gaps on both sides of the rotor, as illustrated in Fig 2.3, to establish the mathematical model of the Active Magnetic Bearings (AMBs) system To accurately position a rotating shaft within a magnetic field, forces must be applied along five axes While a cylindrical gap magnetic bearing necessitates five pairs of electromagnets, a conical gap design only requires four pairs.
2.2.1 Overview of the modeling of CAMB
The rotor model of conical magnetic bearings features a cylindrical center, similar to standard rotors, with beveled cones at both ends This design allows the electromagnet to generate forces that are divided into axial and radial components, enabling precise control of the rotor's movements The shaft, designed for levitation with conical active magnetic bearings (CAMB), offers five degrees of freedom: two radial movements in the y and z directions at each end, along with one axial movement in the x direction.
Figure 2.3 Schematic of conical active magnetic bearing forces
The active magnetic bearings produce forces (F₁, ,F₈) that are incorporated into the rotor dynamics, influencing the rotor's equations of motion A simple model of conical active magnetic bearings is illustrated in Fig 2.3, while Table 2.1 provides a detailed list of the system parameters.
Symbol Description Value g 0 radial air gap 0.45 mm
A cross-sectional area 118 mm2 m rotor mass 0.755 kg β inclined angle 0.98 rad
I 02 bias current 1 A b 1 bearing span 55 mm b 2 bearing span 55 mm
The equations of motion can be written as following Newton’s second law and Euler’s second equations:
5 6 8 7 sin sin cos cos cos sin cos sin z x y d x p y m x d y p x m mz F F F F F F F F F mx F F F F F my F F F F mg F
(2.9) where F F F M M z , x , y , x , y are the external disturbances acting on the rotor
The gyroscopic effect introduces significant nonlinearity in rotor dynamics, impacting the two rotational kinematics equations In the Active Magnetic Bearing (AMB) system, the coupling between pitch (rotation around the x-axis) and yaw (rotation around the y-axis) motions is directly proportional to rotor speed Consequently, stabilizing the system for high-speed applications presents increased challenges.
Conclusion
This chapter analyzes the dynamic model of CAMB, proposing a five degrees of freedom (DOF) model based on electromagnetic analysis The system's equations are complex and nonlinear, making it impractical to apply linear control rules directly Therefore, the FOSMC-ESO algorithm is recommended as a solution to these challenges.
EXTENDED STATE OBSERVER BASED CONTROL
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
1 = ; 2 x q x q with rotor displacement q as output variable and b=M −
The Extended State Observer (ESO) aims to estimate and compensate for total disturbances in systems In the context of active magnetic bearing systems, the ESO effectively calculates the cumulative disturbing forces by analyzing the rotor's output displacement alongside the controller's output.
Define x 3 =M L − is the extended state, Eq 3.1 can be rewritten as:
(3.2) where β 1 =3 ; 3 ; 3 ; 3 ; 31 1 1 1 1 β 2 = 3 1 2 ; 3 1 2 ; 3 1 2 ; 3 1 2 ; 3 1 2 β 3 = 1 3 ; 1 3 ; 1 3 ; 1 3 ; 1 3 and 1 denotes positive observer gain to be determined
Denote the estimation errors as:
The definition of the scaled estimating errors is provided below to aid in the stability analysisε= ε 1 ε 2 ε 3 T = x 1 x 2 /1 x 3 /1 2 T and the dynamics (3.3) can be rewritten as
The A 1 is Hurwitz matrix, thus, there are positive define matrix P 1 satisfying these functions where I denotes an identify matrix A P 1 T 1 +P A 1 1 = −I
The definition of a Lyapunov function V 1 is
The derivative of V 1 can be defined as
The inequality theorem states that it is possible to determine the boundness of the aforementioned function by
It is evident that V 1 0if the observer gains ω1 is selected satisfying 1 0 and
The estimation errors, denoted as \$\epsilon\$, are bounded, which implies that the state \$x_i\$ is also ultimately bounded The Extended State Observer (ESO) offers estimates for all system states, encompassing both the total disturbance and the unmeasured velocity The subsequent controller design will utilize these estimated states.
Fractional Order Sliding Mode Control
3.2.1 Principle of Sliding Mode Control and Chattering Problem
Sliding mode control is a variable structure control algorithm characterized by its discontinuous control output, which changes over time and exhibits switching behavior Unlike conventional control strategies, sliding mode control is robust against system parameters and disturbances, ensuring strong performance in varying conditions The control law in a sliding mode controller typically comprises two components: the equivalent control \$u_{eq}\$ and the switching control \$u_{sw}\$.
26 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:
( , ) ( ) ( ) x= f x t +bu t +d t (3.8) where b0, u is the control output d denotes external disturbance and uncertainty while we assume d t( ) D
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: s= +ce e (3.9) where c0and e is the tracking error
- Second, the SMC's control law u is 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 s=0, we get
The control law is designed as
To ensure reaching conditions of sliding mode control, the reaching condition must be satisfied
Switching control must be chosen whose control law is
The switching control and equivalent control are both included in the sliding mode controller, then we have eq sw u=u +u (3.14)
The chattering phenomenon refers to oscillatory motion around the sliding surface, often caused by switching imperfections like time delays or discontinuities in the switching function These factors contribute to the dynamic behavior observed near the sliding surface.
Chattering can cause wear on moving mechanical parts and result in significant heat losses within the power circuit To address the issues associated with chattering, various procedures have been developed One effective solution involves replacing the discontinuous signum function, denoted as sgn( ), with a continuous approximation using a smooth function In this thesis, we have selected the sigmoid function for this purpose.
Where is a small positive scalar It can be observed that pointwise: lim s sgn( ) s s
The classical integer calculus is extended to non-integer orders via fractional calculus Riemann-Liouville defines the fractional derivative of the fractional order [34]
= − − (3.17) where Γ(x) is Euler's gamma function and n is an integer satisfying n-1 ≤ α ≤ n Under the zero initial condition, the Laplace transform of fractional calculus based on Riemann– Liouville definition is
The displacement error can be defined as:
= = 1 e q - r x - r (3.19) where r is the reference input position
A fractional sliding surface is created using the concepts of fractional calculus, where c 1 0,c 2 0 are controller parameters and (0,1]is the fractional order
In addition, the observing sliding mode variable is ˆs=c e e c 1 ˆ ˆ+ + 2 D − eˆ, where ˆ ˆ= 1 e x - r, eˆ ˆ=x 2 −r The system dynamic along the sliding surface is
The fractional order SMC law can be chosen as:
In a fully actuated system with complete state feedback, under the conditions of assumption 1 and assumption 2 for the CAMB system, it has been demonstrated that the sliding control method ensures the system response attains the sliding surface, which is also asymptotically stable.
The asymptotic stability of the system amidst uncertainties is ensured by the negative definiteness of the derivative of the Lyapunov function By differentiating Equation 3.25 with respect to time, we derive a crucial equation that supports this stability analysis.
Substituting Eq 3.23 into the Eq 3.26, one can obtain
After applying the control input value from Eq 3.24, the result is:
Because the extended observer has converged,
is bounded and sufficiently small, and then we have V 0.
Conclusion
At first, extended state observer are identified to estimate the lumped disturbance, and other states of system, then the SMC and FOC are discussed and used to
29 develop a FOSMC controller This controller is calculated and applied to a CAMB system This controller's response efficiency will be examined in the next chapter
NUMERICAL SIMULATION STUDY
Simulation settings
This section presents a comparative analysis of the efficacy of FOSMC, SMC, and PID controllers using the MATLAB/Simulink environment, focusing on scenarios involving variable rotation speed, rotor load disturbances, measurement noise, and external forces The controllers are employed to adjust the rotor's five axes to achieve equilibrium, starting from the initial positions of the rotor's center of mass The coefficients for the three controllers are detailed in Table 4.1.
Controller Symbol Description Value b 1, b 2, b 3 1/m b 4, b 5 1/J d ω 0 observer bandwidth 600 β 1 Observer gain 1800 β 2 Observer gain 1080000 β 3 Observer gain 216000000
FOSMC c 1 Sliding surface parameter 500 c2 Sliding surface parameter 10
The control strategy of the CAMB is illustrated in Figure 4.1, where position sensors assess the discrepancy between the desired and actual rotor positions These measurements are relayed to the ESO and the "different driving mode" structure, which generates the necessary control signal.
The control signal, along with available information, is utilized by the current controller to adjust the attraction force of the electromagnets, thereby altering the rotor's position To assess the effectiveness of the proposed control system against traditional Active Disturbance Rejection Control (ADRC) and Sliding Mode Control (SMC) for the Controlled Active Magnetic Bearing (CAMB), four distinct scenarios are analyzed.
Scenario 1: Response of the FOSMC controller and ESO with a rotor rotation speed of 3000 rpm and the initial values of the rotor center of mass position are:
Scenario 2: The initial values of the rotor same with scenario 1 under the parameter uncertainties including M, G, K d and K i (i.e., the actual parameters of CAMB deviate about 20% from the nominal value)
Scenario 3: The rotor is brought to the equilibrium position at the initial time
The conical active magnetic bearing operates at a speed of 9000 rpm and is influenced by parameter uncertainties, with actual parameters deviating approximately 20% from their nominal values Additionally, external disturbances, such as unbalanced mass vibrations, are discussed in subsection 2.3.4, particularly when an external force is applied.
Fd 0Nacting on rotor Then increase external force F d 0Nand consider the effect on the system
Scenario 4: The rotor speed is initialized to 9,000 rpm to evaluate the performance of controllers when the rotor is in the high-speed region under hydrodynamic force The rotor is brought to the equilibrium position at the initial time Then, a damped oscillation contact the rotor at 0,04 s This simulation scenario includes parameter uncertainties.
Results and Discussion
Figure 4.2 Response to the position z, x, y
Figure 4.3 Response to the position of the axis angle x , y
Figure 4.4 Upper control currents response
Figure 4.5 Under control currents response
Figure 4.6 Upper impact forces of electromagnets
Figure 4.7 Under impact forces of electromagnets
The center of gravity and the rotor's deflection angle return to their equilibrium position within 0.02 seconds, as shown in Figures 4.2 and 4.3, without any overshoot.
When the rotor position deviates from equilibrium, a control current is generated to restore it to the equilibrium state Once the rotor is stabilized, the control current becomes zero, allowing the bias currents I₀₁ and I₀₂ to maintain this equilibrium The initial impact force of the magnet is significant for bringing the rotor back to equilibrium, as illustrated in the figures, but stabilizes at a consistent value thereafter These findings indicate that the controller is effectively designed to meet all operational requirements.
Figure 4.8 Comparison between observer and response position of z
Figure 4.9 Comparison between observer and response position of x
Figure 4.10 Comparison between observer and response position of y
Figure 4.11 Comparison between observer and response position of θx
Figure 4.12 Comparison between observer and response position of θy
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 θx
Figure 4.17 Comparison between observer and response velocity of θy
Figure 4.18 Comparison between sign and sigmoid function
Figures 4.8 to 4.17 illustrate the position and velocity tracking performance of the proposed method, demonstrating that the observer meets the required standards with no discrepancies between the estimated and actual position values.
The analysis of the switching function's output reveals a chattering phenomenon, akin to vibrations that cause mechanical wear, as illustrated in Fig 4.18 However, when the sigmoid function is applied to the switching function, it significantly enhances performance and effectively reduces chattering.
Figure 4.19 Z axis transient response under the parameter uncertainties
Figure 4.20 X axis transient response under the parameter uncertainties
Figure 4.21 Y axis transient response under the parameter uncertainties
Figure 4.22 θx axis transient response under the parameter uncertainties
Figure 4.23 θy axis transient response under the parameter uncertainties
Table 4.2 The control performance benchmark in scenario 2
Axis Controller Overshoot Undershoot Settling time z ADRC 3.26% 13.07% 84 (ms)
FOSMC-ESO 1.6% 0% 9(ms) x ADRC 1.4% 18.1% 28(ms)
FOSMC-ESO 0% 0% 12(ms) y ADRC 0% 28.21% 60(ms)
The performance of the FOSMC controller is evaluated against SMC and ADRC controllers, as detailed in Table 4.2 The FOSMC system demonstrates a stable position response with no fluctuations, reduced settling time, and minimal overshoot, illustrated in Figures 4.19 to 4.23 Consequently, the FOSMC controller exhibits enhanced controllability compared to both ADRC and SMC controllers, particularly when the rotor deviates from its equilibrium position amid parameter uncertainties.
Figure 4.24 Upper current response under the parameter uncertainties
Figure 4.25 Under current response under the parameter uncertainties
The levitation control currents for the upper and lower coils, calculated using FOSMC, ADRC, and SMC controllers, are illustrated in Figures 4.24 and 4.25 Notably, the FOSMC controller demonstrates a swift stabilization of the rotor with minimal peak current, outperforming both the ADRC and SMC controllers in this regard.
During normal operation, the control system aligns the geometric centers of the rotor and stator, ensuring that unbalance forces remain tangential to the rotor's surface due to the velocity differential between the rotor and stator cans However, any slight displacement of the rotor from its geometric center, caused by external forces or disturbances, can lead to the immediate development of radial unbalance forces These forces are influenced by the rotor's movement and its rotational speed.
Figure 4.27 ADRC controller position response under F d 0N
Figure 4.28 SMC controller position response under F d 0N
Figure 4.29 FOSMC controller position response under F d 0N
Figure 4.30 Current response with ADRC controller under F d 0N
Figure 4.31 Current response with SMC controller under F d 0N
Figure 4.32 Current response with FOSMC controller under F d 0N
Figure 4.33 Comparison between observer and position of x axis with F d 0N
Figure 4.34 Comparison between observer and velocity of x axis with F d 0N
Figure 4.35 Comparison between observer and disturbance of x axis with F d 0N
At 0.05 s, external forces Fd0(N) in Fig 4.26 make contact with the rotor With the external forces applied to the rotor, the FOSMC system achieves lower overshoot, fluctuation, and a faster settling time, as shown in Fig 4.27 to 4.29 The position of the rotor oscillates around the equilibrium point with the FOSMC controller and has a stability state of 0.01 s These results show that the FOSMC controller has faster response times and better vibration-damping ability than ADRC and SMC controllers when the rotor is operated under external forces Fig 4.30 to 4.32 shows the responses of typical direct control currents in systems controlled by FOSMC, ADRC, and SMC As can be seen, when the system is under the influence of external force, the control currents generated by the FOSMC controller are quite large but still within the allowable limit The FOSMC control current quickly brings the rotor into equilibrium and remains stable around the bias current, whereas the control currents generated by ADRC controllers fluctuate around the bias current Based on Fig 4.33 to Fig 4.35, the observer satisfied the requirements as well as has no different between position estimated and real values and has small different between velocity and disturbance observer and real value
Figure 4.37 ADRC controller position response under F d 0N
Figure 4.38 SMC controller position response under F d 0N
Figure 4.39 FOSMC controller position response under F d 0N
Figure 4.40 Current response with ADRC controller under F d 0N
Figure 4.41 Current response with SMC controller under F d 0N
Figure 4.42 Current response with FOSMC controller under F d 0N
Figure 4.43 Comparison between observer and position of x axis with F d 0N
Figure 4.44 Comparison between observer and velocity of x axis with F d 0N
Figure 4.45 Comparison between observer and real disturbance of x axis with F d 0N
The analysis reveals that with a larger external force (Fd0N), the controllers exhibit increased overshoot, fluctuation, and slower settling times Notably, the FOSMC controller demonstrates superior response times and enhanced vibration-damping capabilities compared to ADRC and SMC The control currents in systems managed by FOSMC, ADRC, and SMC show that while FOSMC generates substantial control currents, they remain within permissible limits, effectively stabilizing the rotor around the bias current In contrast, ADRC control currents exhibit fluctuations around the bias current Additionally, the observer meets the necessary requirements, showing negligible differences between estimated and actual positions, as well as minimal discrepancies between velocity and disturbance observations and their real values.
Figure 4.46 External force in scenario 4
Figure 4.47 ADRC controller position response under hydrodynamic force
Figure 4.48 SMC controller position response under hydrodynamic force
Figure 4.49 FOSMC controller position response under hydrodynamic force
Figure 4.50 Current response with ADRC controller under hydrodynamic force
Figure 4.51 Current response with SMC controller under hydrodynamic force
Figure 4.52 Current response with FOSMC controller under hydrodynamic force
Figure 4.53 Comparison between observer and response position of x axis with hydrodynamic force
Figure 4.54 Comparison between observer and response velocity of x axis with hydrodynamic force
Figure 4.55 Comparison between observer and real disturbance of x axis with hydrodynamic force
The FOSMC controller demonstrates a rapid return of the rotor's eccentricity ratio to nearly 0 within 0.1 seconds, as illustrated in Figures 4.47 to 4.49 In contrast, the SMC and ADRC controllers take significantly longer to achieve a similar result This indicates that the FOSMC controller offers superior response times and enhanced vibration-damping capabilities compared to the ADRC and SMC controllers when the rotor operates under hydrodynamic forces Additionally, Figures 4.50 to 4.52 present the direct control current responses for systems managed by FOSMC, ADRC, and SMC.
The FOSMC controller generates substantial control currents under hydrodynamic force, yet these remain within acceptable limits due to its sensitivity to uncertainties and external disturbances This controller effectively stabilizes the rotor around the bias current with minimal oscillation, in contrast to the significant fluctuations observed with ADRC controllers Figures 4.53 to 4.55 demonstrate that the observer meets the required standards, showing negligible differences between estimated and actual position values, as well as minor discrepancies between the velocity and disturbance observer compared to real values, despite the impact of large disturbances.
The closed-loop responses of ADRC, SMC, and FOSMC control actions regarding rotor state tracking errors are evaluated using three performance indices: Integral Squared Errors (ISE), Integral Absolute Errors (IAE), and Integral Time-Multiplied Absolute Errors (ITAE) As shown in Table 4.2, both FOSMC and SMC exhibit performance degradation when faced with disturbances, which is anticipated since these controls are not inherently robust to such noise Nevertheless, FOSMC demonstrates a slight advantage over SMC in performance.
Integral time absolute error (ITAE):
Table 4.3 The control performance benchmark
CONCLUSIONS AND FUTURE WORKS
Results of the thesis
This thesis explores CAMB as a class of underactuated and strongly coupled systems, proposing a control strategy utilizing the FOSMC-ESO controller to enhance transient and steady-state performance while ensuring robustness against parameter uncertainties and disturbances The convergence of the extended state observer and the stability of the proposed control strategy are rigorously demonstrated Simulation results indicate that the proposed controller outperforms traditional ADRC and SMC controllers, exhibiting superior performance and reduced sensitivity to measurement noise Future experiments will focus on the real-time implementation of the fractional system to validate the designed control structure for practical applications.
Future works
The CAMB's performance is influenced by various factors, including coupling phenomena, external disturbances, the design of the electromagnet, and the rotor Notably, axial perturbations can lead to coil current saturation, negatively impacting system stability Therefore, it is crucial to consider these factors when calculating the control law to enhance control quality and generate an acceptable control signal Upcoming experiments will validate the designed control structure for practical applications.
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TÓM TẮT LUẬN VĂN THẠC SĨ Đề tài: Điều khiển hệ thống ổ từ chủ động hình nón
Tác giả luận văn: Tạ Thế Tài Khóa: CH2021B
Người hướng dẫn: TS Nguyễn Danh Huy
Từ khóa (Keyword): Ổ từ chủ động hình nón, điều khiển trượt, đạo hàm phân số, bộ quan sát trạng thái mở rộng
Nội dung tóm tắt: a) Lý do chọn đề tài
Hiện nay, thiết kế hệ thống AMB chú trọng vào sự nhỏ gọn và khả năng chịu tải cao Để hỗ trợ hệ thống rotor năm bậc tự do, cần hai AMB ngang trục và một AMB dọc trục, dẫn đến sự phức tạp của hệ thống Ổ từ chủ động hình nón có thể hỗ trợ tải trọng ngang và dọc trục với chỉ hai hệ thống nam châm điện, tiết kiệm không gian và vật liệu So với ổ từ cơ bản, ổ từ hình nón cho phép điều khiển chuyển động với số lượng cực từ tối thiểu Tuy nhiên, mô hình động học và điều khiển của ổ đỡ từ hình nón là phi tuyến và phức tạp, đòi hỏi khả năng chế tạo cơ khí chính xác cao Do đó, thiết kế điều khiển phi tuyến cho ổ trục từ trở thành thách thức quan trọng trong thiết kế máy móc công nghiệp chính xác.
Nghiên cứu đề xuất một mô hình ổ từ hình nón, chú trọng đến tác động xen kênh giữa các chuyển động và các thành phần bất định, đồng thời xem xét ảnh hưởng của nhiễu ngoại vi.
- Nghiên cứu, ứng dụng bộ quan sát trạng thái mở rộng có khả năng giảm thiểu ảnh hưởng của sai lệch mô hình và nhiễu ngoại vi
Nghiên cứu này đề xuất một thuật toán điều khiển trượt kết hợp với đạo hàm cấp phân số, sử dụng bộ quan sát trạng thái mở rộng để điều khiển đối tượng ổ từ chủ động có hình dạng hình nón.
Kiểm chứng chất lượng hệ thống với bộ điều khiển và bộ quan sát đề xuất nhằm giảm thiểu tác động của xen kênh và nhiễu ngoại sinh Đối tượng nghiên cứu là ổ từ chủ động dạng hình nón, tập trung vào xây dựng và thiết kế thuật toán điều khiển trượt dựa trên bộ quan sát trạng thái mở rộng kết hợp với đạo hàm cấp phân số, nhằm cải thiện chất lượng điều khiển của hệ thống.
Nghiên cứu này tập trung vào việc thiết kế bộ điều khiển trượt, sử dụng bộ quan sát trạng thái mở rộng kết hợp với đạo hàm cấp phân số, nhằm cải thiện hiệu suất trong các trường hợp có tính tới yếu.
Hệ thống ổ từ hình nón chịu ảnh hưởng của 61 yếu tố xen kênh do các trục chuyển động và nhiễu ngoại sinh Luận văn tóm tắt các nội dung chính và nhấn mạnh những đóng góp mới của tác giả, thể hiện ý nghĩa khoa học và giá trị nghiên cứu.
- Luận văn tập trung vào giải quyết vấn đề của việc thiết kế bộ điều khiển cho ổ từ chủ động dạng hình nón vốn là đối tượng phi tuyến
Bộ điều khiển trượt sử dụng bộ quan sát trạng thái mở rộng kết hợp với kỹ thuật đạo hàm phân số, giúp loại bỏ các yếu tố bất định và xen kẽ, đồng thời nâng cao độ ổn định cho hệ thống.
Bộ điều khiển trượt dựa trên bộ quan sát trạng thái mở rộng được thiết kế để kháng nhiễu tốt hơn, với thời gian xác lập ngắn hơn và sai lệch nhỏ hơn so với bộ điều khiển kháng nhiễu chủ động thông thường Kết quả mô phỏng chứng minh tính khả dụng và độ chính xác của các phân tích lý thuyết, đồng thời thể hiện hiệu quả của bộ điều khiển đề xuất.
Bố cục của luận văn:
Chương 1 của luận văn cung cấp cái nhìn tổng quan về ổ từ chủ động hình nón, bao gồm sự phát triển, phân loại, ứng dụng, cũng như ưu và nhược điểm của nó Luận văn cũng trình bày các nghiên cứu trong và ngoài nước liên quan đến ổ từ chủ động dạng hình nón và các thiết kế điều khiển tương ứng Bên cạnh đó, chương này giới thiệu phương pháp điều khiển trượt, bộ quan sát trạng thái mở rộng, và đạo hàm phân số, cùng với ứng dụng của chúng trong hệ thống ổ đỡ từ chủ động Cuối cùng, luận văn phân tích và tổng hợp để đề xuất các hướng nghiên cứu tiếp theo.
Chương 2 trình bày mô hình động học của hệ thống CAMB-Rotor, tập trung vào nguyên lý cơ bản của ổ đỡ từ chủ động Mô hình này phân tích động học và động lực học của ổ từ hình nón với 5 bậc tự do, đồng thời xem xét các thành phần xen kênh, thành phần bất định và nhiễu ngoại vi.
Chương 3 trình bày thiết kế bộ điều khiển FOSMC-ESO cho ổ từ chủ động hình nón, trong đó luận văn thảo luận về việc áp dụng bộ điều khiển trượt kết hợp với bộ quan sát trạng thái mở rộng (ESO) và điều khiển đạo hàm phân số (FO).