Chaotic Behaviour of Active Magnetic Bearing System by Time Series Analysis

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Faculty of Technology

Degree Programme in Technomathematics and Technical Physics

Alireza Fakhrizadeh Esfahani

CHAOTIC BEHAVIOUR OF ACTIVE MAGNETIC BEARING SYSTEM BY TIME SERIES ANALYSIS

Examiners: Professor Heikki Haario

Docent Ph.D. Tuomo Kauranne

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ABSTRACT

Lappeenranta University of Technology Faculty of Technology

Degree Programme in Technomathematics and Technical Physics Alireza Fakhrizadeh Esfahani

Chaotic Behaviour of Active Magnetic Bearing System by Time Series Anal- ysis

Master's thesis 2012

104 pages, 75 gures, 3 tables, 2 appendices Examiners: Professor Heikki Haario

Docent Ph.D. Tuomo Kauranne

Keywords: Chaotic Behaviour, Active Magnetic Bearings, Time Series, Signal Process- ing, Rosenstein Method, Embedding Dimension, Cao Method, 0-1 test for chaos Chaotic behaviour is one of the hardest problems that can happen in nonlinear dynamical systems with severe nonlinearities. It makes the system's responses unpredictable.

It makes the system's responses to behave similar to noise. In some applications it should be avoided. One of the approaches to detect the chaotic behaviour is nding the Lyapunov exponent through examining the dynamical equation of the system. It needs a model of the system. The goal of this study is the diagnosis of chaotic behaviour by just exploring the data (signal) without using any dynamical model of the system. In this work two methods are tested on the time series data collected from AMB (Active Magnetic Bearing) system sensors. The rst method is used to nd the largest Lyapunov exponent by Rosenstein method. The second method is a 0-1 test for identifying chaotic behaviour. These two methods are used to detect if the data is chaotic. By using Rosenstein method it is needed to nd the minimum embedding dimension. To nd the minimum embedding dimension Cao method is used. Cao method does not give just the minimum embedding dimension, it also gives the order of the nonlinear dynamical equation of the system and also it shows how the system's signals are corrupted with noise. At the end of this research a test called runs test is introduced to show that the data is not excessively noisy.

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Hereby I would like to express my sincere gratitude to my advisor Prof. Heikki Haario as he had valuable comments on my work. The nancial support the department has provided has made these two years possible for me to focus on my studies.

I want to thank Dr. Tuomo Kauranne for his nice comments. Special thanks to Prof.

Jari Hämäläinen as he showed me some other elds of mathematics.

Also I want to give special thanks to Dr. Rafal Jastrzebski as he answered my questions and he provided the valuable data.

Special thanks to Prof. Olli Pyrhonen as he makes a friendly environment and spon- sored the experimental setup.

Great thanks to Prof. Matti Heilio as he familiarized me with pure parts of mathematics.

Great thanks to Prof. Mohammad Reza Raei as his nice lecture notes and his experience make me eager to study control systems more. And his recommendation letter if I didn't have this I couldn't come to nland.

This work is dedicated to my father, Ali Fakhrizadeh, who has been our support throughout our life. To my mother, who taught us with her boundless love. To my deceased brother, who taught me aection and honesty. To my brother Omid, who taught me hope and determination. To my dear sister Shirin.

My special thanks go to my dear friend Behdad as he gave me very nice hints and comments. My special thanks go to my dear friend Ehsan as he gave me hope to continue my study.

Special thanks to all of my friends in Lappeenranta town.

Lappeenranta November 2012 Alireza Fakhrizadeh Esfahani

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Chapter 1 Introduction

Chaotic behaviour is one of the complicated problems, encountered in the highly dynamical systems. Nonlinear dynamical systems are quite unpredictable hard to control in the chaotic regime. In controlling nonlinear dynamical systems, the chaotic behaviour should be detected to be able to avoid from this region.

An active magnetic bearing (AMB) shows highly nonlinear dynamical behaviours in some regions. Even it can enter to the chaotic region in some situations. In some operating regions such as touch-down, in the rotating system with chaotic vibrations, in the magnetic circuits, and even the driving system can show chaos.

Most of the studies where have done are model based i.e. the analysis is based on the dynamical model of the system. But what should be done if the model is not accessible or hard to nd? In this study the chaotic behaviour is detected just by looking on the signal (time series) coming from the vibration sensors installed on the system. All of mechanical data is chaotic. Two methods are used, Rosenstein and a 0-1 test, to detect the chaotic behaviour.

The Rosenstein method estimates the largest Lyapunov exponent. If the largest Lya- punov exponent is positive the system will be chaotic.

The 0-1 test shows the chaos in the data if the resulted histogram is centred around one.

At the end of result chapter and the two appendices the noise problem is studied.

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CHAPTER 1. INTRODUCTION 8 The AMB system and its components are introduced in the second chapter. In the third chapter the AMB system is studied in more details and its operation is explained.

In this chapter it is assumed that the AMB system is linear. In chapter four there is an introduction to chaotic dynamics. The Rosenstein method and the 0-1 test are introduced in chapter ve. The results are in the sixth chapter. The embedding dimension and runs test are introduced in the appendices. The runs test is used to show that the data is not noisy so much.

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Chapter 2 Magnetic Bearing

2.1 Basics of magnetic bearing operation

Generation of magnetic eld forces without any contact by controlling the dynamics of an electromagnet actively is the main principle of magnetic suspension. Figures 2.1 and 2.2 show the main components of a simple magnetic bearing. Each component's application is explained as follows:

Figure 2.1: A simple structure of an active magnetic bearing [1]

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CHAPTER 2. MAGNETIC BEARING 10

Figure 2.2: The schematic of a magnetic bearing system

The sensor nds the position of the rotor from the reference, the microprocessor control system calculates the needed signal from measurements of the sensor to correct the position of the shaft, the power amplier transforms the generated control signal to a control current and this control current generates a magnetic eld in the electromagnet and this process totally keeps the shaft xed in its rotating position.

2.2 Classication of magnetic bearings

Figure 2.3 shows the classication of magnetic forces and magnetic hovering. There are two main action groups that magnetic bearings' work is based on. Reluctance force and Lorentz force are these action groups. In reluctance force group there are three dierent cases depending on their relative permeability µ_r: the case for ferromagnetic materials with µ_r 1, the case for diamagnetic materials with µ_r < 1 and the case for superconducting materials withµr = 0. In this group the force is calculated by the principle of virtual work:

f = ∂W

∂S (2.1)

Where W is the energy of the eld and S is the virtual displacement of the hovering body. In ferromagnetic material group (µ_r 1) the force is very large. In practical applications if there are control actions and active devices to keep the rotating shaft in its position then the category is the active magnetic bearing (" Classical" active magnetic bearing). In gure 2.3 all active magnetic bearing categories are designated

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Figure 2.3: Magnetic bearing and levitation classication

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CHAPTER 2. MAGNETIC BEARING 12 by " A" (grey boxes ii gure 2.3). If in bearings there are no control actions they are passive and in the gure 2.3 it is shown by " P".

Tuned LC bearing has a stable stiness characteristic in an LC circuit that is excited near its resonance point. This type of bearing comprises the electromagnetic coil bearing inductance (L) and a capacitor (C). When the position of the shaft is changed it leads to changing inductance of the electromagnet. The LC circuit works in a way that when the shaft goes o its central position, the circuit falls in its resonance. The circuit takes more current from the AC source and pulls back the shaft to its main position. Because this system works stably without any control loop it can be called passive bearing.

Permanent magnet-based bearings (type 3) have low damping, but in special applications they can be used without any control loops. So they are also passive magnetic bearings.

Type 4 devices are used when strong forces are needed. For this type of bearings superconducting materials are used.

The second main group in the classication of magnetic bearing is Lorentz force- based magnetic bearings. The Lorentz force law is:

f =Q(E+v×B) (2.2)

Here Q is an electrical charge particle E is the electrical eld v is the charge's velocity and B is the magnetic ux density.

Electro-dynamic levitation (type 5) is achieved without active control when high eddy currents are induced in a big relative speed between the stator and the moving body.

Type 6 is working according to AC and induced current interaction. This leads to a passive levitation just as type 5.

Type 7 is similar to an induction motor to some extent. However in an induction motor the forces act in the direction of circumference to produce torque. But in this type of bearings the forces act radially to support the rotor.

Type 8 acts similar to type 7 but the rotor with the internal induced current is replaced by a permanent magnet rotor.

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2.3 Some real applications of magnetic bearings

• Medical Devices: The articial heart pump is a medical application of AMB systems. This device is called left ventricular assist device (LVAD). It assists the heart to keep the pumped blood at a desired rate. (Left ventricular assist device (LVAD) Pat. No. US 2007/0265490 A1)

Figure 2.4: A Left Ventricular Assist Device (LVAD) and its location in the body [2].

• Vacuum Systems: Magnetic bearings don't have any suering from the contam- inations in the environment. Even the bearing can be located outside of the vacuum container. (Magnetic bearing device and a vacuum pump equipped with the same Pat. No. US 6,465,924 B1 & US turbomolecular vacuum pump having a magnetic bearing-supported rotor Pat. No. US 4,023,920)

• Turbo-machinery: Application of active magnetic bearings in the turbo-machinery industries gives the advantage of vibration control and damping, desired dynamic behaviour.

• Superconducting bearings: The passive stability property of this type of M.Bs makes them an alternative to the AMBs. However for the damping problem maybe it is necessary to have AMB system.

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Figure 2.5: Outside view of a turbomolecular vacuum pump [3](left) and cross section view of a turbomolecular vacuum pump [4](right)

2.4 Test rig which is used for the experimental works

Figure 2.8 shows the schematic diagram of the system under study. In the studied experiment there is an axial magnetic bearing which acts in the axial direction to counteract and act the forces in the axial direction of the shaft. The radial magnetic bearing is located on two points of the shaft to levitate the shaft in the desired position (the center point). For sensing the states of the system two radial sensors are needed near the radial bearings. Also there is an axial sensor near the axial bearing. For protecting the rotating points near the stationary point of the system from mechanical touching there are two safety bearings. There is a disc that is located in the axial bearing winding. In this study the airgap under radial bearing winding is 0,1 mm. It means it is quite necessary to have safety bearings.

The test rig which is used in our experiment is plotted in gure 2.9. The sensors sense the position of the shaft through the Sensors A and B. For sensing the position eddy current sensors are used in this project. The reading of these sensors is transmitted to the dSpace system. The dSpace is an intermediate device between computer and the AMB. According to the data coming from the sensors and also the control algorithm

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Figure 2.6: A magnetic bearing test facility which is developed by Siemens and Zit- tau/Görlitz University of Applied Sciences for developing an oil free turbine [5].

in the dSpace the necessary commands will be produced. The commands go to a power amplier to be able to excite the electrical windings of the AMB. The current of bearings is also monitored by the dSpace. Before the dSpace and AMB going in action the designed control algorithm is fed to the dSpace by a host PC. The control algorithm can be implemented in the Matlab Simulink environment simply.

2.5 Anatomy of the test rig

2.5.1 Sensors

The current sensing is done by Hall-eect sensors. For implementing a real current measurement it needs to combine a Hall-eect sensor with a current transducer (Trans + Reducer). The combination is needed because of the high value of system's currents

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Figure 2.7: A superconducting magnetic bearing which is developed in a 10 kW ywheel energy storage test system [6].

Figure 2.8: Schematic diagram of the AMB system which is used in this study. [7]

in proportion to the Hall-eect sensor's current nominal values. And also the transducer isolates the Hall-eect sensor from the main current. There are some additional circuits to compensate the nonlinearities in the whole measuring system specially the magnetic circuit.

Hall-eect sensors work according to Hall-eect phenomenon which is shown in gure 2.10. The magnetic eld passes through the Hall element. There is a current through the Hall element. If the eld is in parallel to the current there is no eect in the measurement system. But if the eld has an angle with the current the eld will exert a force on the charges carried by the current. This force causes the current to follow a curved path. The charges of the current are accumulated on two regions near the points A and B. It makes a potential dierence in these points. By measuring this voltage the current can be measured.

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Host PC dSpace

Position control Current control ADC

Power amplifiers

Bearing A Bearing B

Dis placement A Dis placement B

Current A Current B

Sensor A

Sensor B

Figure 2.9: Block diagram of the test rig that is used in this experiment [8].

The schematic diagram of a transducer is plotted in gure 2.11. The transducer con- sists of a Hall-eect sensor which is mounted in the air-gap. The secondary winding is wound on the soft magnetic core. The amplied Hall voltage triggers the transis- tors and the current ows through the secondary winding. This secondary winding produces a magnetic eld opposed to the main magnetic eld built by the primary current. Therefore the resultant magnetic eld will be zero. The secondary current goes through the resistor in the output circuit. This current is transformed to a voltage by the resistor in the output. This voltage is the measured signal that is used for measuring the ux.

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Figure 2.10: Hall-eect phenomenon [9].

Figure 2.11: A transducer's diagram [10].

2.5.2 dSpace

Figure 2.12 shows the whole system from the hardware and software point of view.

The operator can be interfaced with one of two ways. Simulink and dSPACE environment in a PC or an embedded system which is programmed under supervision of a PC. The control programme (algorithm) is programmed through one of the mentioned devices. The AMB system is controlled directly either by dSPACE system or an embedded hardware (Designed Hardware and Software block) . The control program is downloaded on these systems by the PC.

The software in the PC to programme the dSPACE is called ControlDesk which is shown in gure 2.13 . This is a user-interface software. The dSPACE board is a plat- form. The simulation is run on it. Real time control can be implemented through dSpace. dSpace environment gives opportunity to connect Matlab-Simulink with the real system to exert control and read real data from the system. Matlab-Simulink and ControlDesk software are working together and also they are compatible. As it is

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Figure 2.12: Block diagramme of the software and hardware parts of the system [7]

shown in gure 2.13 the Simulink icon can be seen in ControlDesk navigator area. Con- trolDesk environment gives the ability of downloading the applications to the dSPACE board, conguration of virtual instrumentation that can be used to control, monitor and automate experiments and develop controllers. The interaction between the Con- trolDesk and the software in the dSPACE board is done through using a number of les such as the generated executable program(.ppc), the variable description le(.sdf), and the virtual instrument panel(.lay). A ControlDesk le associates all these les with a single entity. To congure the software to interface with the real signals we need to dene analog to digital and digital to analog converters. Also to communicate between Simulink and the dSPACE board and the plant we should dene the A/D and D/A parameters.

ControlDesk also can be used to build a direct control console(Panel) to control and monitor the plant directly in the online manner. Figure 2.15 shows a very simple controller and the monitored data.

2.5.3 Actuators

Actuators are made by switching ampliers and electromagnets. The control signals are converted by the ampliers to electrical currents in the coils. An electromagnetic eld is produced by these currents.

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Figure 2.13: ControlDesk environment.

Ampliers

Here the power ampliers consist of a three phase line frequency diode rectier for producing DC link voltage and an H Bridge switching power amplier to make the current of coils.

Figure 2.16 shows the circuit diagram of the power amplier.

Electromagnets

Figure 2.17 shows the magnetic ux paths and the magnets are used in the test rig. By using the magnetic paths in gure 2.17 we can calculate the force of electromagnets.

By Ampere's law around two adjacent poles, we have:

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Figure 2.14: A very simple on/o controller which is designed in the Simulik environment

I

H.dl = Z

S

J.dS (2.3)

whereH is the magnetic eld intensity and J is the current density. By assuming that H is uniform in the magnetic circuit we have:

l_{F e}H_{F e}+ 2l_airH_air =N i (2.4)

l_{F e} B

µ₀µ_{F e} + 2l_airB

µ₀ =N i (2.5)

where B is the magnetic ux density, µ0 is magnetic permeability of the free space, µ_{F e} is the relative magnetic permeability of the iron, l_{F e} is the length of the iron path, l_air is the length of the air-gap, N is the number of winding turns and i is the current of the winding. From 2.5 it follows that:

B =µ₀ N i (_µ^l^{F e}

F e + 2l_air) ≈µ₀ N i

2lair (2.6)

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Figure 2.15: A control and monitoring console which is produced by ControlDesk To nd the force it is needed to calculate the magnetic energy. In linear magnetic circuits the magnetic energy is equal to magnetic co-energy. It can be calculated by integrating over volumetric magnetic co-energy density

W_e =W_ce = µ0

2 Z

V

H²dV = 1 2µ₀

Z

V

B²dV = 1

2µ₀B²S_air2l_air (2.7) where S_air is the area of the surface of the air-gap.

l_air =lcosχ (2.8)

where χis the force acting angle.

Figure 2.18 shows the schematic diagram of the magnetic circuit of the magnetic bearing. Force is the partial derivative of the co-energy:

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Figure 2.16: Circuit diagram of a power amplier

Figure 2.17: Flux paths of magnets (left) magnets of the test rig (right) [7]

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Figure 2.18: Geometry of a radial magnetic bearing

f = ∂W_ce

∂l = B²S_aircosχ

4l²_air (2.9)

2.5.4 Analog to digital converters

In practice signals are analog such as current, voltage, position, force and etc. But the control system works digitally so it is needed to convert analog signals to digital ones [7]. Analog to digital conversion procedure comprises three steps:

1. Sampling: It samples the continuous signal in discrete time instants. So we have:

x_a(nT)≡x(n) (2.10)

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2. Quantization: In the input of this procedure the signal is discrete time but its values can be any real number. But in the output of the quantizer the signal values are in discrete values.

3. Coding: This process converts the discrete time discrete valued signal x_q(n) , in the output of the quantizer, to binary codes.

Figure 2.19: A/D converter block diagram

Figure 2.19 shows all three needed blocks in the A/D conversion process. The analog signal in the input of A/D converter has the harmonies above half sampling frequency.

These harmonies should be ltered out by a low-pass lter which is called an anti- aliasing lter. This lter is located before the A/D converter block [11].

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Chapter 3 Principles of Active Magnetic Bearing Operation

In this chapter the operation of the magnetic bearing is explained in more detail.

3.1 Viewing magnetic bearing as a controlled suspen- sion

Basically magnetic bearings are categorized based on which magnetic laws aect their operation. The rst group works according to reluctance force and the second group operates based on the Lorentz force law. They are called reluctance force bearings and Lorentz force bearings. The reluctance force bearings are widely used in industry.

3.1.1 Active and passive magnetic bearings

After many years of magnetic bearing utilization in industries it seems active magnetic bearings are preferred to passive magnetic bearings. Active here means there is a con-

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trol loop. This controls actively by using electromagnets, sensors, and ampliers. Some advantages of active magnetic bearings over passive ones are free adjustability by the control, static and dynamic stiness, static and dynamic damping, load-independent static positioning, attenuating destabilizing forces in rotating systems and generating and monitoring the excitation force.

Here Earnshaw's theorem should be mentioned: Suspending a rigid body in all six degrees of freedom by just using permanent magnets is not possible physically because there must be at least one unstable degree of freedom.

Very low damping of the passive magnetic bearings gives them further disadvantage. So for industrial usage their applications should be limited to areas where another source of damping is available. It means for example that the levitated body is submerged in a uid. Otherwise to provide an external force with suitable damping, additional mechanical or electromagnetic elements are needed. Another solution to introduce the needed damping is the eddy current eect which is generated by the motion of the moving parts. Two examples of passive magnetic bearings(PMBs) are hybrid turbomolecular pumps (TMPs) and blood pumps or articial hearts.

3.1.2 Control loop elements

Figure 3.1 shows a simple diagram of an active magnetic bearing. A brief description of the elements of the active magnetic bearing is given here.

A rotor is levitated at a distancex₀ from the electromagnet of the bearing. Dierence between desired position x₀ and actual position of the rotor is measured steadily by a contact-less position sensor (eddy current or inductive type sensor usually). This dierence is fed to a controller (these days a digital controller) by the sensor. Keeping the rotor position in its desired value is the main goal of the controller. The controller stabilizes the magnet forcef_m and the weight of the rotor mg .

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CHAPTER 3. PRINCIPLES OF ACTIVE MAGNETIC BEARING OPERATION28 This is not the only goal of the controller because, the stabilization of the control loop has the greatest priority. The command signal of the position is sent out by the controller to the power amplier. The power amplier transforms this signal into an electric current. This current makes a magnetic eld in the coil of the bearing. The desired forcef_m is generated by the magnetic eld. The behaviour of the power amplier and electromagnet are dependent on each other. The main properties of the magnetic bearings, for example the force dynamics, depend on power amplier and electromagnetic parameters such as amplier's voltage and current, bearing's geometry and coil's inductance number of turns and inductance. So the combination of power amplier and bearing magnet is called an electromagnetic actuator.

Figure 3.1: Schematic diagram of an active magnetic bearing control loop and its elements.

For controlling rotations and transverse motions of the rotor it is needed to have a system more complex than the setup of gure 3.1. Nevertheless by this bearing setup can be used to study the basic properties of a magnetic bearing control loop. The mathematical model of this bearing setup is derived in the following section.

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3.1.3 Basic model of the magnetic bearing

For simplicity, rstly the dynamics of the sensor and power amplier are ignored. In real application, if the resulting eigen-frequencies of the closed loop system are not too high this simplication will have good results. If the stiness of the applied bearing is in a physically sensible range then the resulting eigen-frequencies of the closed loop system will not be too high. The second simplication is the bearing force characteristic. The detailed force characteristics to physical quantities such as current, rotor position and other quantities are not derived here.

Finally, the derivation of the magnetic bearing is carried out in the same way as the mechanical mass spring system is modelled. In practice the AMB design achieves system properties beyond mechanical bearing characteristics.

Figure 3.2 (a) shows a simple mechanical mass spring. The force-distance characteristic of the spring makes the mass spring system stable. In fact the characteristic of the spring has a negative slope. But as in gure 3.2 (b) and (c) the force-position and the force-current characteristic have positive slope in the operating point. The open loop magnetic bearing therefore has an inherently unstable nature.

Figure 3.2 also depicts the magnetic forcef_m with respect to air gaps and coil current i⁰ dependency. As it was shown in chapter 1 in the relation 2.9 the magnetic force is proportional to the inverse of the square of the air gap and also proportional to the square of the coil current. The following relationships therefore hold:

f = ∂Wce

∂l = B²Saircosχ

µ₀ = µ0N²i²Saircosχ

4l_air² (3.1)

For small values of the air gaps or large values of coil current the magnetic ux will be saturated. This furthermore makes the basic characteristic more nonlinear. Finally, the displacement x⁰ is limited to the air gap size geometrically.

In spite of these strong nonlinearities a linear control approach can be used quite well for a magnetic bearing system. Therefore the force/current and force/displacement relations can be linearized at the operating point. The operating point (x₀, i₀, mg) is the desired equilibrium point of the system. So as described in gure 3.3 we have

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CHAPTER 3. PRINCIPLES OF ACTIVE MAGNETIC BEARING OPERATION30 f_m(x₀, i₀) = mg .

In order to do linearization around the equilibrium point, e.g. operating point we should introduce new variables for force f, current i and displacement x as follows:

f =f_m−mg (3.2)

i=i⁰ −i₀ (3.3)

x=x⁰ −x₀ (3.4)

The resulting linearized equation should have the following form:

f(x, i) =−k_sx+k_ii (3.5)

This equation is assumed to be the fundamental description of the behaviour of an active magnetic actuator under current control. In spite of the fact that equation 2-5 is just a simple linear approximation of the real system, it works impressively well for a wide range of applications and its validation has been proved for a long time of experience. Only in special situations such as rotor-stator contact, e.g. rotor touchdown, ux saturation, very low bias currents, etc., it becomes necessary to use more specic models which are typically nonlinear models. In the equation 2-5 k_s (N/m) and k_i (N/A) are called the force/displacement and the force/current factor respectively.

3.2 Control loop

As we have seen in the section 2.1.3 the magnetic bearing system in the open loop case is unstable inherently. To stabilize this system a suitable control system and a current command signal should be used as shown in gure 3.4. The following section is devoted to describe some linear control approach to this system.

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Figure 3.2: Dierent types of forces: (a) mechanical spring (b) electromagnet (bias current is constanti⁰ =i₀) (c) electromagnet (constant air gap x⁰ =x₀) [1]

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CHAPTER 3. PRINCIPLES OF ACTIVE MAGNETIC BEARING OPERATION32

Figure 3.3: Linearization of the operating point (a) force/displacement relationship k_s <0 (b) force/ current relationship k_i >0.

3.2.1 A simple AMB control design

The main goal in the magnetic bearing control loop is the stabilization of unstable behaviours of rotor motion in the equilibrium point. Therefore, the restoring force should be provided by the control system. The control system also provides a necessary damping component to attenuate the oscillations around the operating point.

The simplest approach to this problem is to make the desired force in a way that the closed loop system's behaviour becomes similar to that of a mechanical spring-damper system. So, it is assumed the force has a desired linear relation with position x and velocity x˙:

f =−kx−dx˙ (3.6)

Assuming equations 3.5 and 3.6 to hold gives this possibility to write the control current i in terms of the rotor displacementx and its derivative x˙:

i(x) =−(k−ks)x+dx˙

k_i (3.7)

As mentioned in section 2.1.3 any other dynamics such as sensor, amplier and etc., are not included.

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Figure 3.4: Magnetic bearing control loop with appropriate control current i. The main goal in any control design problem is to achieve the desired closed-loop behaviour of the controlled system. One of the criteria for the assessment of the control loop are its eigenvalues, static and dynamic stiness and the robustness of the system. Here only the closed loop eigenvalues are analyzed. It starts by Newton's law:

m¨x=f (3.8)

To verify the basic property of the system, i.e. its open loop stability the equation (2.5) is inserted into (2.7). So the following equation is achieved:

mx¨=−k_sx+k_ii (3.9)

The contribution of the control current is zero in the open-loop case. It should be mentioned that the system is linearized at the operating point, so, the current in the coil of the electromagnet is not zero and is equal to the bias current i₀. By our experience on electromagnets with constant current we know if any ferromagnetic object which is attracted by the electromagnet, is located near enough so that it will stick to the surface of the electromagnet. This phenomenon shows the unstable behaviour of the electromagnet. It can be seen also from 3.9 by settingi= 0 and solving the equation.

We will see the characteristic equation will be:

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mλ²+k_s= 0 (3.10)

k_s is a negative number, therefore roots of equation 3.10 are λ₁ = +p

|k_s|/m and λ2 = −p

|ks|/m . Both λ1 and λ2 are real and λ1 is located in the right half of the complex plane it shows the instability of the open loop system. The locations of these routes are shown in gure 3.5.

Figure 3.5: Location of open-loop system's eigenvalues.

By inserting (2.6) into (2.8) we will nd the dierential equation of the closed loop system:

mx¨+dx˙+kx= 0 (3.11)

The characteristic polynomial of (2.11) is:

mλ²+mλ+k = 0 (3.12)

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New complex eigenvalues i.e., the solution of (2.12) are:

λ₁ =−σ+jω (3.13)

λ₂ =−σ−jω (3.14)

σ = d

2m (3.15)

ω= rk

m − d²

4m² (3.16)

Figure 3.6: Location of closed-loop system's eigenvalues (d <2√ mk).

The eigenvalues of the closed loop system are shown in gure 3.6 in the complex plane. The frequency ω is aected mainly by the stiness k i.e. the eigenvalues' λ_1,2 imaginary part. In contrast, the eigenvalues are moved to the left half of the complex plane by the dampingd. Increasing the damping value just decreases the frequencyω.

However the damping doesn't have any eect on the magnitude of the eigenvalues i.e.

(|λ_1,2|=ω₀ =p

k/m). Although from (2.16) it is clear by large damping we will have

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CHAPTER 3. PRINCIPLES OF ACTIVE MAGNETIC BEARING OPERATION36 real eigenvalues again, so it shows the oscillation removing capability of the closed loop system.

The homogeneous linear dierential equation's (2.11) solution in the undercritical, damping is:

x(t) =e^−σt(Acos(ωt) +Bsin(ωt)) (3.17) A and B should be determined by initial conditions. Another form of this solution is:

x(t) = Ce^−σtcos(ωt−φ) (3.18)

HereC is the amplitude and φ is phase angle.

Figure 3.7 shows the transient response of the system according to (2.12). In this gure zero crossing points are equidistance and this time distance isT. But the distance between peak values are not the same. SoT is called the pseudoperiod andω is called the pseudo angular frequency. In practice mostly eigenfrequency ω and eigendamping σ are used.

3.2.2 Active and passive magnetic bearings dierences

Even with simple mass-damper-spring control law for active magnetic bearings there are many benets over passive magnetic or conventional i.e. mechanical bearings:

• The magnetic bearing has low losses and more life cycle and less maintenance.

• Because no lubrication is needed, there is no contamination. This is an important advantage over conventional bearings.

• AMB systems can bear also harsh environment or even they can work in vacuum.

• The only constraint to approach high speed is the strength of the material to centrifugal force. The standard peripheral speed in AMBs is 300 m/s which is quite hard for other bearings to reach.

• Active damping of vibrations in high precision applications is very important.

And also this is very important in passing through bending critical speeds.

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• The rotor can rotate about its principal axis of inertia to cancel the resulted dynamic forces from unbalanced rotor. So even in the presence of any unbalances no vibration forces are transmitted to the machine founding. Rotors of rotating machines are equipped with AMBs do not need to balance usually.

• Important properties of AMBs such as stiness and damping can be changed adaptively and momentarily. It means without any modications in the system it can adapt itself. This on-line tuning and adaptation can be realized with the help of digital control systems.

• The rotor operating position in AMBs can be controlled in a manner independent from the stiness and external load.

• AMBs are not only used for positioning and levitation but also for some additional purposes such as monitoring, preventive maintenance and additional instrumentation.

3.2.3 Controlling by PD and PID

According to the control law (2.7) in section 2.2.1:

i(x) =−(k−k_s)x+dx˙

k_i (3.19)

Actually this control law is the same as PD controller law. It has two parts:

P = k−k_s

k_i (3.20)

D= d

k_i (3.21)

PD control: selecting the stiness and damping

In (2.19) and (2.20), for determining P and D we need to choose the stiness and damping of the closed loop system. To design a system to have a capability to handle a maximum force (load capacity) it is needed to dene the stiness parameter in the early stages of the designing process. The bearing size and the power rating of the amplier depend on the selection of the stiness.

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Figure 3.7: Time response of (2.12). ω is the pseudo frequency and decay rate σ here we have: C = 1, σ = 1, ω = 2π/s, ϕ= 2π/3.

The following sections give a suitable guide for selecting the control parametersP and D.

Very Low Stiness

For very low values of stinessk, the proportional gainP just stabilizes the system by moving the eigenvalues of the system to somewhere near to zero as shown in gure 3.8.

The proportional gain just compensates for the negative values of the bearing stiness k_s.

The k_s is a quite sensitive value to magnet gap length s₀. It can be changed by the magnet gap length,s₀, and also by the operating point current, i₀. The magnet gap is usually an uncertain value because of the manufacturing process, dierential thermal

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Figure 3.8: Eigenvalues of the closed-loop system for very low values of the stinessk (P ≈ −k_s/k_i).

growth between the rotor and stator and by the rotor centrifugal growth. Ask_sdepends ons⁻³₀ ,¹ small changes ins₀ can change k_slargely. Also, the operating point current, i₀ is related to the static load carried by the bearing: so static load variations can make great changes inks. If we put all of these problems together it will force us to have an assumption of 20% uncertainty ink_s value for design purposes.

Because of the sensitivity of the eigenvalues of the system toks the realization of very low stiness is quite hard and it needs a precise value of the system parameters, namely k_s.

Very High Stiness

1It is shown in [1] & [7]: fx=k(⁽ⁱ_(s⁰⁺ⁱ^x⁾²

0−x)² −⁽ⁱ_(s⁰⁻ⁱ^x⁾²

0+x)²) Iffx is being linearized with respect tox << s0we will have:

ki= ^∂f_∂i^x

x

ⁱx=0= ^4ki_s2⁰ 0

(cosα) ks= ^∂f_∂x^x

^x=0=−^4ki_s3²⁰ 0

(cosα)

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CHAPTER 3. PRINCIPLES OF ACTIVE MAGNETIC BEARING OPERATION40 According to (2-20) in this condition P should be very high and according to (2-16) the eigenvalues' imaginary part is very high. It means the eigenfrequencies of the rigid body are very high. In this condition the controller, the sensor should have a high bandwidth, and the power amplier shouldn't go to a dynamic saturation. Even due to the nonlinearities in the system maybe it shows an uncontrollable chattering by exerting an external disturbance. High value of P gives rise to the magnetic ux saturation and noise amplication. In this condition aP ID controller is a better choice.

Figure 3.9: Eigenvalues of the closed loop system for very large values of stiness k(P >> k_s/k_i).

Natural stiness

The easiest stiness from technical point of view has intermediate or natural values. k is at the range of the bearing stinessk_s, namely 1· · ·3× |k_s|.

In a special case when k = |k_s|(P =−2k_s/k_i) the eigenvalues' absolute value in both closed-loop and open loop are the same. Figure 3.10 shows this similarity.

Damping

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Figure 3.10: Eigenvalues of the closed loop system for a natural stiness k(P ≈

−2k_s/k_i).

The dampingdor velocity feedbackDdepend on the stiness. To achieve high damping it needs high stiness. However, high velocity feedback gain means more noise in the system since it has the derivative term x˙. The critical damping gives an upper limit for the velocity feedback. It means ω =o. So we haved= 2√

mk. PD Control: the position command in the input

By a P D control algorithm any static load changes gives a change in steady state positionx. The amount of change in thex value depends on the implemented stiness k. It can be assumed to be∆x= ∆f_e/k, the same as a mechanical spring. In technical applications such a deviation in x is undesirable and it can be modied by using the position reference. Figure 3.11 shows theP Dcontrol block diagram of the AMB system with the linearized model of the plant, sensor, current amplier and linearized actuator force.

PID controller

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Figure 3.11: The P D controller block diagram.

To overcome the problems that they are associated with P D controller, e.g. position deviation ∆x and external force disturbance it is preferable to use P ID controller.

Here we will show an example of this controller simulated by Matlab.

Figure 3.12: The P ID controller block diagram.

Figure 3.12 shows a block diagram of a P ID controller on an AMB system with a linearized current controller (k_i).

Frequency response of system's transfer function in the log scale is called the Bode plot.

Bode plots simplify the analysis of the transfer functions because the multiplication operation changes to addition in log scale. Also for the frequency axis the scale is in log. Bode plots are plotted for the magnitude and phase response of the system. These plots help to analysis and design of controllers for dynamical systems.

Bode plot of the P ID controller is depicted in the gure 3.13. The low frequencies

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of the Bode plot depend on the integrating factor K_in. Here by low frequencies it means 10 to 1k rad/s. For low frequency attenuation the K_in= 1000is the best value.

K_in = 10000doesn't have a at magnitude in low frequency range [12].

The Bode plot of the open loop system is plotted in the gure 3.14. The gain of the loop is high for low frequencies. Also for K_in = 10000 the amount of phase margin is decreased. So because of low frequency high loop gain and no decrease in phase margin the K_in = 1000is the best value [12].

Figure 3.15 shows the step response of the system for the disturbance excitation. K_in = 10000 has a poor damping. And for K_in = 100 it has a slow response. Therefore the best value isK_in = 100 [12].

The parameters of this example are in table 2.1.

Figure 3.13: Bode plot of the P ID controller.

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Figure 3.14: Bode plot of the open loop system

Figure 3.15: Step response of the system with the disturbance excitation amplitude is in (µm)

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Parameters Values

K_p 10

Kin [0,10,1000,10000]

T_d 0.01

T_f 10⁻⁴

K_sn 5×10³

K_i 158 N/m

m 3.14(π)

T_lag _2π×500¹

K_x

(Force/Displacement constant) 1.58×10⁶

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Chapter 4 Introduction to chaotic behaviour of dynamical systems

4.1 Basic Concepts in Dynamical Systems

For studying the behaviour of chaotic systems some introductory concepts need to be dened for introducing the concept of chaos.

In our case the dynamical equations are assumed in discrete time format so the dynamical equations are assumed to be in the form:

x_n+1 =f(x_n) (4.1)

Fixed Point (Equilibrium Point):

Heref is a function. A xed point of the system is dened as:

x^∗ =f(x^∗) (4.2)

In the xed point the state of the system won't change anymore.

In non-linear dynamical systems (discrete time or continuous time) the stability of the system is analyzed near the equilibrium points (xed points). because all the state

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trajectories of the system are passed through the xed point because the trajectories don't cross each other because of the uniqueness of the solution.

Contraction Mapping:

Let(X, d) , X is a set and d is its distance, be a metric space. A transformation T :X →X is called a contraction mapping if and only if:

d(T(x), T(y))≤sd(x, y), x, y ∈X (4.3) wheres is a constant satisfting 0< s <1 condition. The constants is called Contrac- tion Factor [13]. The existence condition of a xed point is that the functionf should be a contraction mapping. It should be mentioned that the continuity off is a weaker condition on the existence of the xed point and also continuity is more restrictive than (3.3) and condition (3.3) is somehow similar to continuity condition onf if (3.3) is written as:

d(T(x), T(y))

d(x, y) ≤s (4.4)

Stability Analysis of a Fixed Point:

In the single valued discrete time dynamical system 4.1 the necessary and sucient condition for local stability of xed point is:

|d(x_n+1)

d(x_n) |_x=x^∗ <1 (4.5)

Here d means derivative. In other words the derivative of the function around the equilibrium point should be located in the interval(−1,1)[14]. It analyzes the stability of the system around the equilibrium point.

Example:

x_n+1 = 0.5(x²_n−3x_n+ 6) (4.6)

|d(x_n+1)

d(x_n) |x=2 = 0.5<1 (4.7)

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CHAPTER 4. INTRO. TO CHAOTIC BEHAVIOUR 48

Figure 4.1: A typical quadratic function equation (3.6) with two intersections with the line y=x.

|d(x_n+1)

d(x_n) |_x=3 = 1.5>1 (4.8) Figure 4.1 shows the intersections of the equation 3.6 with the line y = x and it has two points. The horizontal axes is the usualx axes and the vertical axes is they axes.

For x = 2 the slope of the curve is less than unity so this point is stable. But for x= 3 because the slope at this point is more than unity the system is unstable at this equilibrium point.

This condition doesn't say anything about the stability of xed points with:

|d(x_n+1)

d(xn) |_x=x^∗ = 1 (4.9)

Note: In the multistate case we should consider these conditions for the eigenvalues of Jacobian matrix.

Web Diagrams (Cobwebs):

The sequence of consecutive state of a one dimensional discrete dynamical system can be shown on a web diagram. Figure 4.2 shows a web diagram of a quadratic equation.

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Figure 4.2: Web diagram of the above example for 4 dierent initial states.

The web diagram is plotted from an initial value on the horizontal axis. Then it is connected to the equation of the dynamical systemf(xn)then the web goes to the line y=xand then to the f(x_n) then y=xand so on.

The convergence of the sequencex1, x2,· · · , xncan be visualized on the web diagrams.

4.2 Bifurcation to chaos

Bifurcation Diagram:

Assume we have the Logistic map:

x_n+1 =ax_n(1−x_n) (4.10)

If a is increased from 1 to 4, we will see some substantial changes in the system's behaviour. Figure (3.3) shows the web diagrams of the logistic map for dierent values of aand for each one the number of iterations are 200. In gure (3.3)(a)it is just a simple stable equilibrium point. In gure (3.3)(b)the state of the system oscillates around the equilibrium point but nally it converges to it. In gure (3.3)(c)convergence happens

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CHAPTER 4. INTRO. TO CHAOTIC BEHAVIOUR 50 in an oscillating manner but with a poor convergence. In the point a= 3 the system is bifurcated as it is depicted in gure (3.3) (d). After this point (a = 3) the logistic map has two equilibrium points it is shown in gure (3.3) (e). In gure (3.3) (f) the convergence is to two points and this convergence is very high. Although the number of iterations are 200 but it has a quite rapid convergence. In gure (3.3) (g)those two equilibrium points are started to bifurcate. So two new equilibrium points are born and the number of equilibrium points are 4 in gure (3.3) (h). In gure (3.3) (i) the sequence is converged to the 4 equilibrium points so rapidly that it doesn't show any transient behaviour. In this gure there are 200 iterations. In gure (3.3)(i)there are 8 equilibrium points and in gure (3.3) (j) there are quite many. It can be said there are innite number of equilibrium points. This is the starting point of the chaotic region.

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(b)a= 2.8

(c)a= 2.9 Figure 4.3

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(d) a= 3

(e) a= 3.1

(f) a= 3.3

Figure 4.3

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(g) a= 3.4

(h) a= 3.45

(i) a= 3.52

Figure 4.3

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(j)a= 3.55

(k) a= 3.5699456718

Figure 4.3: Web diagrams of logistic map for a = 1.5, 2.8, 2.9, 3, 3.1, 3.3, 3.4, 3.45, 3.52, 3.55, 3.5699456718

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Figure 4.4: Bifurcation Diagram of logistic map for a= [3.50−3.62]

In rst 4 plots we see the system converges to a single equilibrium point from a = 3 it starts to converge to 2 equilibrium points (next 3 gures) a=3.1, 3.3, 3.4 .In the 2 next plots it is seen that there are 4 equilibrium points and in the next one there are 8 and in the last one actually there are many. This phenomenon is called bifurcation and this type of bifurcation is period doubling.

Period doubling bifurcation points have a universal property:

δ= lim

n→∞

a_n−an−1

a_n+1−a_n = 4.669201. . . (4.11) This universal constant is called the Feigenbaum constant. We can show all the equilibrium points in a single diagram with the parameter value on the horizontal axis.

This diagram is the Bifurcation Diagram . The bifurcation diagram of the Logistic map is drawn in gure 4.4.

4.3 Chaotic behaviour

As it was seen in the previous pages, as the parameter a in the logistic map is increased, the system will experience a period doubling bifurcation until the value

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Figure 4.5: The region that the zoom is done

a∞ = 3.5699456718. Beyond this point the system actually has an innite number of equilibrium points. This point is the starting point of chaotic behaviour of the system. Although the existence of the chaotic region is recognizable after increasinga, some nite equilibrium point regions can still be seen in the bifurcation diagram. In fact these regions can be seen in such a way that the system is periodic after passing the transient part. In other words we can say the system is periodic in its steady state condition and the period is the number of these equilibrium points.

Bifurcation diagrams have the fractal property. It means that if we zoom on any point, after some degree of zooming, the gure repeats itself. The fractal property of the bifurcation diagrams is shown in gures 4.5, 3.6, 3.7 and 3.8. These gures are some zooms on the bifurcation diagram which is plotted in the gure 4.5.

And other zooms are shown in gures 4.7 & 4.8.

4.4 Theoretical interpretation of chaos

Theoretically chaos is dened as [13] p.149:

Let(X, d)be a metric space, and letf :X →X be a function. The map f is chaotic

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Figure 4.6: Zooms on the bifurcation diagram of the Logistic map in the region which is shown in gure 3.5

Figure 4.7: Zooms on the on the other region

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Figure 4.8: Zooms on the on another region provided that

1. f has sensitive dependence on initial conditions 2. f is transitive.

3. Periodic points of f are dense inX .

"The map f is transitive if for any pair U, V of open sets there is an n >0 such that f⁽ⁿ⁾(U)T

V 6= " [13].

The chaotic behaviour can be shown through the exploration of the tent map:

f(x) =

3x, x≤1/2 3−3x, x > 1/2

(4.12)

If we calculate the system's equation f(x) one step beginning from some point in the open interval (¹₃,²₃), the result will be out of the interval [0,1]. Also for f²(x) the set (¹₉,²₉)∪(⁷₉,⁸₉)is mapped out of interval [0,1]by f²(x) and so forth. . .

At the end we will have exactly the Cantor set. Figure 4.11 shows the Cantor set.

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Figure 4.9: Tent mapping f(x)

Figure 4.10: Two iterations of Tent mapping (f²(x) = f(f(x)))

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Figure 4.11: The Cantor set

Each layer is an interval that made by removing the middle third part from the former interval.

It can be proved that there is a homeomorphism between every chaotic region and the Cantor set [15] p.106.

Some notes on Simulation of chaotic Systems:

There are some points that should be mentioned here:

1. For every bifurcation diagram in each a the function is iterated 2000 times and just 500 iterations are ignored because of the transient part of the response.

2. It's preferable to start every iteration process from the point where:

J acobian(f(x)) = 0 that is for one dimensional case is just f⁰(x) = 0 .

Because chaotic systems keep their states within bounds it is " the continuous stretching and folding in the state space that ensures that the state remains bounded while nearby initial conditions diverge" p. 60 [16]. Even it can be seen that the tent function has the stretching and folding eects.

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4.5 Some well-known chaotic systems

Lorenz attractor:

In early 1960s when Edward Lorenz in MIT had worked on the modeling of the atmo- spheric convection on a primitive digital computer. He simplied his 12 dimensional equations to the well known three-dimensional Lorenz system

dx

dt =σ(y−x) (4.13)

dy

dt =−xz+rx−y (4.14)

dz

dt =xy−bz (4.15)

Where σ is Prandtl number [17], r and b are parameter. r and σ are proportional to the Rayleigh number The above equation describes the convective motion of the atmosphere which is heated by the ground and cooled from above. The variable x is speed of rotation. The variable y is the temperature dierence between falling and rising uids. The variablez is the distortion from linearity of the vertical temperature prole. The parametersσ = 10, r= 28,and b= 8/3 , are usual to produce the chaotic behaviour.

The following gures show the Lorenz attractor in 3-D, X-Z projection and the time series of the y element.

This attractor is the symbol of chaos researchers. It's similarity to a buttery's wings led to the metaphor rstly used in a 1972 talk entitled " Predictability: Does the ap of buttery's wings in Brazil set o a tornado in Texas."

The phase portrait of the Lorenz system and its projection in theX−Z plane is plotted in gures 4.12 and 4.13. Figure 4.14 shows the time series of they.

Rössler attractor:

The Rössler system is a three dimensional system which is simpler than Lorenz:

dx

dt =−y−x (4.16)

dy

dt =x+ay (4.17)

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Figure 4.12: Lorenz attractor in Three dimension

Figure 4.13: Projection of Lorenz attractor in X−Z plane

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Figure 4.14: The time series of the y(t)

dz

dt =b+z(x−c) (4.18)

It has just one nonlinear term zx. According to Rössler's article, this attractor is a simplied model of the Lorenz attractor. It was introduced to explain the Lorenz equation in a simpler way [18]. Although it has such a simple equation by changing c from 2.0 to 5.7 and keepinga andb constant and equal to 0.2 (a =b = 0.2) the system experiences the chaotic behaviour.

The Rössler attractor is plotted in gure 4.15. Figure 4.16 shows Its projection in X−Y plane. The time seriesX is plotted in gure 4.17.

4.6 Lyapunov Exponent

The average exponential rate of separation of two nearby initial conditions, or the average stretching of the space are determined by Lyapunov exponent [19] p.106.

Formulation:

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Figure 4.15: The Rössler attractor in three dimension

Figure 4.16: Projection of Rössler attractor in X−Y plane

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Figure 4.17: The time series of the y(t)

∆X₁ =f(x₀+ ∆X₀)−f(x₀)∼= ∆X₀f⁰(X₀) (4.19) The Local Lyapunov Exponent is:

e^λ =|∆X₁

∆X₀| (4.20)

or

λ= ln|∆X1

∆X₀| ∼= lnf⁰(X₀) (4.21) To obtain the Global Lyapunov Exponent above equation should be averaged over many iterations:

λ= lim

N→∞

1 N

N−1

X

n=0

ln|f⁰(X₀)| (4.22)

When the Lyapunov exponent is positive it means that the exponential rate of separation of two nearby initial conditions is positive. It means the system is quite sensitive

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Figure 4.18: Lyapunov exponent vs. N for logistic map for a= 4 (N is in logarithmic scale).

to initial conditions. So the Lyapunov exponent is a measure for a chaotic system.

When the Lyapunov exponent is positive the system is in its chaotic region. To show that a system is chaotic it is enough to show that the largest Lyapunov exponent is positive [20].

For the logistic map we have:

f(x) =ax(1−x) (4.23)

f⁰(x) =a(1−2x) (4.24)

λ= lim

N→∞

1 N

N−1

X

n=0

ln|a(1−2x_n)| (4.25)

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Chapter 5 Algorithms for identifying the chaotic behaviour

In the rst section of this chapter there is a brief introduction to sources of nonlinearities in magnetic bearings. Also some special phenomena, such as bifurcation and chaos in these systems, are introduced.

In section two the Rosenstein method is introduced to measure the Largest Lyapunov exponent. The test which is used to test whether a system is chaotic or not is the cal- culation of the largest Lyapunov exponent usually. Positive largest Lyapunov exponent means the system has a chaotic behaviour: if λ > 0, nearby trajectories will diverge exponentially and if λ < 0, nearby trajectories will converge with the rate λ. This approach is used for systems with a known dynamical equation [21]. If the underlying equations are unknown, the Lyapunov exponent should be estimated [20], [22].

In the third part a simple 0-1 test is introduced to nd out if the data is chaotic or not. The method doesn't need to calculate the Lyapunov exponent.

5.1 Nonlinear dynamics of magnetic bearings

In the 2^nd chapter of this work the model used for designing the control system is a linearized model. But in special conditions the AMB system operates in a nonlinear manner. So it is necessary to predict their nonlinear behaviour.

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The main sources of nonlinearity in magnetic bearings are: [23]

1. The magnetic/force and force/coil current relations of the magnets are nonlinear.

2. There is a geometric coupling between the electromagnets, it makes a coupling between dierent orthogonal coordinate directions.

3. The saturation which is coming from the ferromagnetic core material. It makes the magnetization curve atter.

4. The hysteresis phenomenon which takes place in the magnetic core material.

5. The power amplier saturates and the control current is limited, these problems happen because of the physical limitations of the power amplier.

6. The time delays in the controller and actuators, which are inevitable. This problem happens especially in situations where the control algorithm is implemented on a digital signal processing.

7. The sensor system has nonlinearity and noise.

8. The coil inductance is nonlinear.

9. There are some nonlinear phenomena such as the fringing eect, eddy current and the leakage eect, and also theB−H magnetization curve is also nonlinear [23].

If the deections of the rotor exceed half of the gap, the net magnetic force which is produced by a pair of opposite electromagnets changes more than 44% from the linear approximation of itself [24]. Therefore, the magnetic bearings' operation changes from the equilibrium point. For small air gaps and large control currents the nonlinear characteristics become very signicant. The dynamic behaviour of the rotor can be changed from the linear model result because of the nonlinear characteristics of magnetic bearings [23].

Also there are some nonlinearities which are related to the mechanical part of the AMB systems such as touch-down phenomenon.

The spinning rotor's contact with the housing is a quite nonlinear impact/rubbing. It can lead to chaotic vibrations. This phenomenon is called touch-down. It can happen in the case of a power failure of a magnetic bearing system. During touch-down there are typically three types of phenomena These are oscillations of the rotor, chaotic

Chaotic Behaviour of Active Magnetic Bearing System by Time Series Analysis

Contents

Chapter 1 Introduction

Chapter 2

Magnetic Bearing

2.1 Basics of magnetic bearing operation

2.2 Classication of magnetic bearings

2.3 Some real applications of magnetic bearings

2.4 Test rig which is used for the experimental works

2.5 Anatomy of the test rig

2.5.1 Sensors

2.5.2 dSpace

2.5.3 Actuators

2.5.4 Analog to digital converters

Chapter 3

Principles of Active Magnetic Bearing Operation

3.1 Viewing magnetic bearing as a controlled suspen- sion

3.1.1 Active and passive magnetic bearings

3.1.2 Control loop elements

3.1.3 Basic model of the magnetic bearing

3.2 Control loop

3.2.1 A simple AMB control design

3.2.2 Active and passive magnetic bearings dierences

3.2.3 Controlling by PD and PID

Chapter 4

Introduction to chaotic behaviour of dynamical systems

4.1 Basic Concepts in Dynamical Systems

4.2 Bifurcation to chaos

4.3 Chaotic behaviour

4.4 Theoretical interpretation of chaos

4.5 Some well-known chaotic systems

4.6 Lyapunov Exponent

Chapter 5

Algorithms for identifying the chaotic behaviour

5.1 Nonlinear dynamics of magnetic bearings