vol 102 no 3 september 2011 saiee africa research journal · vol.102(3) september 2011 south...

Vol.102(3) September 2011 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 57

VOL 102 No 3 September 2011

SAIEE Africa Research Journal

SAIEE AFRICA RESEARCH JOURNAL EDITORIAL STAFF ...................... IFC

Micromachining of Optical Fibres with a Nanosecond Laser for Optical Communication and Sensor Applications by D. Schmieder, R. Samaradiwakera and J. Meyer .............................58

LTI Modelling of Active Magnetic Bearings by Means of System Identifi cation by P.A. van Vuuren, G. van Schoor and W.C. Venter ............................66

Multivariable H or Centre of Gravity PD Control for an Active Magnetic Bearing Flywheel System by S.J.M. Steyn, P.A. van Vuuren and G. van Schoor ..........................76

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Vol.102(3) September 2011SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS58

MICROMACHINING OF OPTICAL FIBRES WITH A NANOSECOND LASER FOR OPTICAL COMMUNICATION AND SENSOR APPLICATIONS D. Schmieder, R. Samaradiwakera and J. Meyer Photonics Research Group, Faculty of Engineering and the Built Environment, University of Johannesburg, P.O. Box 524, Auckland Park 2006, Johannesburg, South Africa E-mail: [email protected] phone +27 11 5589 2648; fax +27 11 559 2344 Abstract: Micromachining of single-mode telecommunication fibres (SMF28) was accomplished with a Nd:YAG laser at a wavelength of 355 nm. Micromachining is important for the manufacturing of Bragg gratings and long period gratings which are used in add-drop filters and wavelength division multiplexers. Manufacturing of miniature Fabry-Perot interferometers used for temperature sensors is also possible. A short overview of micromachining concepts is presented. The experimental setup, as well as the equipment used, is described. Alignment processes, focal point determination and centring of the laser beam onto the optical fibre are outlined. Micromachining results are presented. Keywords: Micromachining optical fibres, light feeder, beam profile displays, focal point determination, micro Fabry-Perot interferometers.

1. INTRODUCTION

To drill very small holes or slots in materials such as quartz, diamond, silicon or sapphire, short pulse lasers operating in the nanosecond, picosecond or femtosecond region are required. To produce micro features by direct machining, the laser beam is focused to spot sizes of below 30 μm. Small spot sizes are achieved using a TEMoo laser beam with a M2 value not much larger than one. The M2 factor, also called the beam quality factor or beam propagation factor, is a common measure of the beam quality of a laser beam. The M2 factor of a laser beam limits the degree to which the beam can be focused for a given beam divergence angle, which is often limited by the numerical aperture of the focusing lens. The wavelength of the laser is not so much of a concern due to the high beam intensities involved, which leads to electron plasmas and ablation of the material. It has been found that UV laser light at a wavelength of 355 nm produces significantly less thermal damage and smaller holes than longer wavelength IR lasers [1]. The research described in this paper is focused on the ablation of single-mode communication fibre (SMF) with a Nd:YAG nanosecond laser with a wavelength of 355 nm. The purpose of the ablation is to manufacture miniature holes inside a single-mode optical fibre to create in fibre Fabry-Perot interferometers. The technique can also be used for the manufacturing of Bragg gratings, long period gratings for add-drop filters and wavelength division multiplexers used in optical communication systems. The technique is also ideally suited for the manufacturing of fast response temperature sensors. The following experimental features, which are under investigation, have a significant impact on the quality and size of the holes and cavity structures.

Firstly, the selection of the machining lens, e.g. normal lenses with a focal length of 24.5, 35 or 50 mm; microscope objective lenses of 20 times magnification (NA = 0.45), 60 times magnification (NA = 0.65); or oil immersed microscope objective lenses of 100 times magnification (NA = 1.3). Secondly, the quality and kind of laser beams, such as Gaussian beams, flat top beams or Bessel beams. Thirdly, the duration and energy of the laser pulses and fourthly, the ablation speed, repetition frequency, and burst mode of the laser beam. Optical fibres are sheltered with a protective coating. For applications in optical communication and as sensors it is important to drill the holes and features right through the fibre. This is difficult because optical fibres display small hair cracks on the surface between core and cladding, which are formed during the fibre manufacturing process. A compromise has to be found between machining optical fibres with and without protective coating. The purpose of this paper is to present an approach to solve the problem of micromachining fibres with and without protective coating. The rest of the paper describes the micromachining concept, the micromachining experimental setup, laser beam profile, beam alignment process, focal point determination, laser beam positioning, and the micromachining of the optical fibres. 2. MICROMACHINING CONCEPT When intense nanosecond pulses are tightly focused, the intensity in the focal volume can become high enough to initiate absorption through nonlinear field ionization.


The nonlinear absorption of the laser radiation results in the creation of an electron-ion plasma in the focal volume of the beam [2]. In nonlinear media such as air or water a white-light super-continuum is generated in the focal volume. For optical fibres, exposed to the laser radiation, ablation takes place on the surface of the fibre. In addition to ablation with nanosecond laser pulses, thermal and shock wave effects are observed. The high intensity in the focal realm allows the machining of cavities for Fabry-Perot interferometers in the optical fibres.

3. EXPERIMENTAL MICROMACHINING SETUP WITH A HORIZONTAL LASER BEAM

A similar experimental setup as described by Rao et al. [2], Wei et al. [3], and Marshall et al. [4] is presented. For the machining process a ‘Surelite’ Nd:YAG laser from ‘Continuum’ was used. The laser operated in the frequency-doubled and frequency-tripled mode. The frequency-tripled wavelength at 355 nm was used to implement the laser micromachining of the optical fibres. The wavelength for micromachining is not so important, because ablation occurs when a highly, ionized plasma is formed in the focal region. However, the shorter the wavelength is, the smaller is the size of the holes. This is because the beam size in the focal region depends on the wavelength. Thermal distortions are suppressed when the laser pulses are faster than the development of the thermal effects. The output power of the Nd:YAG laser is selected by varying the Q-switch delay, the voltage on the flashlamps and the pulse repetition frequency. For a typical setting of Q-switch delay: 130 μs, voltage on the flashlamps: 1.2 kV and pulse repetition frequency: 10 Hz, the output power was 54 mW, and the energy/pulse 5.4 mJ/pulse. Apart from the Nd:YAG laser and the frequency doubling and tripling crystals, the system consisted of a filter and a dichroic mirror used to eliminate the frequency doubled 532 nm radiation, a 1–3 mm round aperture, a 50 mm focusing lens, a translation stage with fibre holding clamps and a setup for a camera. The camera was computer controlled and operated with the software ‘ProScope HR’. The magnification of the camera is 400 times. The schematic drawing of the micromachining setup is shown in figure 1. The three axis translation stage shown in figure 1 can be shifted back and forth between the position where the machining takes place and the position where the camera is located. The whole experimental micromachining setup is shown in picture 1. On the left side the filter (A) and the dichroic mirror (B), which reject the 532 nm laser radiation, and transmit the 355 nm laser light, are placed. In front of the dichroic mirror the head (C) of a power meter can be seen, which can be shifted into and out of the laser beam.

The power meter (D) can be seen in the background. The next items in the laser light pass are an aperture (round, 1-3 mm diameter) (E) and the machining lens (F) with a focal length of 50 mm. Next to the lens at the edge of the breadboard the light feeder (G) is visible. Behind the lens the three dimensional translation stage (H) is placed on a rail. In the background on the left side is the Nd:YAG laser (K) and on the right side the computer screen (L), displaying the image from the camera using ‘ProScope HR’ software.

Dichroic MirrorReflects 355 nm

Transmits 532 nm

50 mm Lens

Light Beam

Three Axis Translation Stage

Filter removes 532 nm

Circular Aperture1, 2 and 3 mm

ClampsFiber ClampsFiber

CameraComputer

Three Axis Translation Stage

Nd:YAG Laser

Frequency doubling andtripling crystals (2 KDPs)

Dichroic Mirror Box

Figure 1: Schematic drawing of the micromachining facility setup.

A

B

CF

D

EG

H

IK

L

Picture 1: Experimental micromachining setup An alignment screen with two clamps to mount the optical fibre is attached to the translation stage and displayed in detail in picture 2. The translation stage can be shifted towards the camera which captures the images from the machined fibre. Picture 3 displays the camera (A), pointing at the mounted optical fibre. Inside the camera is a white light LED ring, which illuminates the optical fibre. The optical fibre is inserted vertically to allow the positioning of the


camera very close to the optical fibre while preventing the image being obstructed by the fibre clamps (B). The camera can be focused onto the side surface of the fibre or onto the end surface of the fibre. During the machining process the camera is focused onto the end of the fibre. Red light from the light feeder is coupled into the fibre when the camera is focused on the fibre end surface. Picture 4 shows the camera image of the end surface of the fibre. The red light from the light feeder is clearly visible in the core of the fibre.

Picture 2: Alignment screen with fibre clamps

A

B

Picture 3: Camera setup focused onto the fibre

3.1 THE LIGHT FEEDER A prototype model of a LED/Laser Light Feeder was fabricated to illuminate the fibre with visible laser light for monitoring purposes during the micromachining process. Figure 2 shows the light feeder construction with a convex lens and a built-in rechargeable battery pack. The LED housing on the left side can be adjusted to couple

the red LED light only into the core of the fibre or into the core and the cladding of the fibre.

Picture 4: Camera display showing the core of the fibre illuminated by the light from the LED light source.

Figure 2: Light feeder schematic

4. LASER BEAM PROFILE

The laser beam was imaged with a 25.4 mm focal lens onto a white screen where the beam profile was observed.

Picture 5: Laser beam profile behind a 1 mm aperture


The laser setting was: Q-switch delay 120 μs, voltage on the flashlamps 1.2 kV and the pulse repetition frequency 10 Hz. Picture 5 shows the beam profile of the laser beam behind a 1 mm aperture. The beam profile behind the aperture displays the interference rings. With an aperture the beam profile is not as good as without aperture, but the advantage is, smaller holes can be machined, because the laser beam is curtailed.

5. THE ALIGNMENT PROCESS

To establish the optical axis the laser beam was aligned in the vertical and horizontal directions from the dichroic mirror to the three axis translation stage, where the optical fibre was placed. The vertical alignment of the laser beam was achieved by measuring the height of the laser beam near the dichroic mirror and near the translation stage. The dichroic mirror was adjusted until the height of the laser beam was parallel to the breadboard. The horizontal alignment was accomplished by following the holes on the breadboard and placing holders with apertures of 1 mm diameter in the holes. After having set up the optical axis, the optical components were inserted. The following step is to place the machining lens at a right angle to the laser beam. For this purpose an observation screen was mounted in place of the optical fibre. The lens was removed and the laser beam directed through the centre of the lens holder onto the screen. The position of the laser beam on the screen was marked. The lens was then placed back into the lens holder and aligned until the laser beam was back at the marked position on the screen.

Picture 6: Air breakdown–white light super-continuum indicated by the arrow

6. DETERMINATION OF THE FOCAL POINT USING THE AIR BREAKDOWN

The challenge is to determine the exact focal point, the correct ablation energy and the right pulse repetition

frequency. Air breakdown caused by the laser beam occurs at the position of the focal point. When air breakdown occurs a white light supercontinuum is generated, because air is a nonlinear medium. Air breakdown was used to determine the focal point of the laser beam. Air breakdown was obtained by setting the laser to: Q-switch delay 130 μs, voltage on the flashlamps 1.2 kV and pulse repetition frequency 10 Hz. There was no aperture inserted. In picture 6 the white spot where air breakdown occurs can be seen. The focal point of the machining lens is situated at the air breakdown point. The surfaces of the two clamps were brought in line with the air breakdown spot by turning the micrometer screw of the three axis translation stage, which moves the clamps towards or away from the machining lens. The fibre was installed at this position and the micrometer screw reading was taken. A further improvement of the location of the focal point can be achieved by machining holes at different positions near the focal point spot into the fibre. The focal point is situated at the location where the smallest hole is machined. 7. POSITIONING OF THE FOCUSED LASER BEAM

IN THE MIDDLE OF THE OPTICAL FIBRE As long as the optical fibre is not in the path of the laser beam a small blue spot is observed on the screen behind the optical fibre. The blue spot is the fluorescence light of the ultra violet laser beam which can’t be seen with the naked eye. When the optical fibre is in the path of the laser beam a blue line appears on the screen. The blue line is vertical to the optical fibre. The line is the diffraction pattern created by the optical fibre. A method was discovered to place the focused laser beam in the middle of the optical fibre. Positioning of the fibre is done on very low power settings of the laser beam to avoid ablation or marking of the fibre. With a laser setting of: Q-switch delay 116 μs, voltage on the flashlamps 1.14 kV and pulse repetition frequency of 10 Hz the positioning was accomplished. Method By turning the micrometer screw the optical fibre is shifted towards the focused laser beam. When the focused laser beam is reached, instead of the small blue spot a blue line appears on the screen behind the fibre, as shown in figure 3. At this position the micrometer screw reading is taken. The fibre is moved through the laser beam until the blue line disappears and the small blue spot reappears. Another reading is taken at this position. The middle of these two positions is the centre position of the fibre. To confirm, the distance between the two positions must be the width of the fibre, which is 250 μm for a fibre with


protective coating or 125 μm for a fibre without protective coating.

Focused Laser Beam

Fiber

Diffracted Laser Beam

Figure 3: Diffracted laser beam for determining the machining point

Picture 7: Blue line orthogonal to the optical fibre Picture 7 displays the image of the blue line on the screen behind the fibre. The blue line can be seen as long as the focused laser beam is on the fibre.

8. MICROMACHINING OF OPTICAL FIBRES

The micromachining of the fibre was done using a lens with a focal length of 50 mm. After machining, the fibre was shifted with the translation stage towards the camera to view the results from the micromachining.

8.1 MICROMACHINING OF OPTICAL FIBRES WITH PROTECTIVE COATING

Micromachining was started on an optical fibre with protective coating. The diameter of the fibre was 250 μm. The laser was operated at: Q-switch delay 120 μs, voltage on the flashlamps 1.15 kV, pulse repetition frequency 1 Hz, output power 1 mW, energy/pulse 1 mJ and aperture 3 mm. Red light from the light feeder was coupled into the fibre. The red light in the fibre enables visualisation of the scattered light at the machined spot. Four holes were machined. One hole is shown indicated by the arrow in picture 8. The diameter of the holes was

approximately 40 μm. The red LED light from the light feeder is scatterd from the position of the hole. With the same laser setting holes were machined with 1 shot, 30 shots, 40 shots and 50 shots. The diameters of the holes became larger with increased number of shots. The diagram in figure 4 shows the hole-diameters as a function of the number of shots.

1 shot hole size: 31 μm 30 shots hole size: 56 μm 40 shots hole size: 68 μm 50 shots hole size: 81 μm

-10 0 10 20 30 40 50 6020

30

40

50

60

70

80

90

Number of Shots

Dia

met

er o

f the

Hol

es [m

icro

met

er]

Hole Diameters versus Number of Shots

Figure 4: Hole-diameters versus number of shots

Picture 8: The camera image of the 40 μm hole machined into the fibre with illumination from the camera switched on The red light from the light feeder is coupled into the optical fibre and the output at the end face of the optical fibre is viewed with the camera. The core diameter of the single mode fibre (SMF-28) used is 8 μm, the cladding diameter is 125 μm and the total fibre diameter with protective coating is 250 μm. When the protective


coating of the fibre is removed at the coupling end in the light feeder, the light is coupled into the cladding and core as seen in picture 9. A circular aperture of 1.5 mm was inserted into the laser beam and a new machining attempt was started with a laser setting of: Q-switch delay 120 μs, voltage on the flashlamps 1.2 kV and pulse repetition frequency 1 Hz. Holes were machined into the fibre at different locations with one shot for each hole. The holes are all the same and about 25 μm in diameter, one hole is shown in picture 10.

Picture 9: Camera image of the light coupled into the core as seen from the fibre end face

Picture 10: A 25 μm hole machined into the fibre using one shot While monitoring the light from the light feeder at the end face of the fibre one hole was machined running the laser at a pulse repetition rate of 1 Hz for 15 minutes. When machining started, flickering of the light from the fibre end face could be observed, probably because of vibrations of the fibre. The hole became bigger because of thermal effects. The result is shown in picture 11.

Picture 11: Hole machined when laser operated at 1 Hz pulse repetition frequency for 15 minutes Another experiment was executed with the same laser settings. One hole was machined with 26 laser pulses, each laser pulse at an interval of 20 seconds. The hole-size achieved was about 20 μm.

8.2 MICROMACHINING OF OPTICAL FIBRES WITHOUT PROTECTIVE COATING

Optical fibres without protective coating were machined. The diameter of the fibres without protective coating is 125 μm. The setting of the laser was: Q-switch delay 115 μs, voltage on the flashlamps 1.2 kV and the pulse repetition frequency 1 Hz. An aperture of 2 mm and a 0.4 ND filter (39.41 % transmission) were inserted. Holes were machined at different positions in the optical fibre. Each hole was machined with one laser pulse. The diameter of the holes is 8 μm. Picture 12 shows a single hole.

Picture 12: Single hole machined with 1 laser pulse, the hole-diameter is 8 μm.


8.3 DRILLING HOLES RIGHT THROUGH AN OPTICAL FIBRE WITHOUT PROTECTIVE

COATING

Holes were drilled right through an optical fiber without protective coating using the following Nd:YAG laser settings: Q-switch delay 115 μs, voltage on the flashlamps 1.2 kV and a pulse repetition frequency of 10 Hz. With an aperture of 1 mm the output power behind the aperture was 1 mW.

The red LED light from the light feeder is coupled into the uncoated optical fibre and monitored at the end of the optical fibre with the camera. When drilling starts one can observe fluctuations of the red light at the fibre end. This is probably due to the laser pulse shockwaves causing the fibre to vibrate. After about 20 minutes the drilling is complete and one observes the red light at the position of the hole, in front as well as at the back of the optical fibre. The hole-size is about 20 μm. Picture 13 shows the entrance of the hole. The red light is emerging from both sides of the fibre, the entrance and the exit.

Picture 13: Entrance hole With the same laser setting another attempt was launched. The drilling took about 2 minutes, and the size of the spot was 12.5 μm as shown in picture 14. The red light emerging from the small hole can be seen.

8.4 DRILLING HOLES RIGHT THROUGH AN OPTICAL FIBRE WITH PROTECTIVE

COATING

An experiment was performed attempting to drill through an optical fibre without protective coating. The selected laser setting was: Q-switch delay 130 μs, voltage on the flashlamps 1.2 kV and the pulse repetition frequency 10 Hz. The output power was 16 mW. With this setting and with no aperture a hole

was drilled through the optical fibre with protective coating (diameter 250 μm). It took about 15 minutes. The hole was about 60 μm in diameter and is shown in picture 15. During drilling the red LED light output at the end face of the fibre was observed. When ablation occurred, flickering of the red LED light was observed.

Picture 14: A 12.5 μm hole with emerging red LED light

Picture 15: 60 μm hole drilled through an optical fibre with protective coating

9. CONCLUSION

Holes were micromachined in single-mode telecommunication fibres (SMF28) with a Nd:YAG laser at a wavelength of 355 nm. The experiments have shown, it is possible, to drill holes into optical fibres with high precision. Ways were found to determine the focal point and to centre the laser beam onto the optical fibre. The selection of the machining lens, the quality and the kind of laser beam, the duration and energy of the laser pulses, the ablation speed, repetition frequency and burst mode have a significant impact on the quality and size of


the holes and cavity structures. They are under further investigation. Figure 4: Fabry-Perot cavity in optical fibres For applications in optical communications and as sensors it is important to drill the holes and features right through the fibre.

The next step is to machine Fabry-Perot cavities into the fibres as shown in figure 4 and etch them with hydrofluoric acid to improve the surface quality. The manufactured structures have to be characterized before attempting to machine them in sapphire fibre.

10. REFERENCES [1] A. Ostendorf, K. Koerber, T. Nether, T. Temme:

“Material Processing Applications for Diode Pumped Solid State Lasers”,

In: Lambda Highlights, No. 57 (Lambda Physik, Göttingen 2000) pp. 1-2. [2] Yun-Jiang Rao, Ming deng, De-Wen Duan, Xiao- Chen Yang, Tau Zhu, Guang-Hua Cheng: “Micro Fabry-Perot Interferometers in silica fibers machined by femtosecond laser”, Optics Express 15(21), 14123-14128 (2007). [3] Tao Wei, Yukun Han, Hai-Lung Tsai, and Hai Xiao, “Miniaturized fiber inline Fabry-Perot Interferometer fabricated with a femtosecond laser,” Opt. Lett. 33(6), 536-538 (2008). [4] Graham D. Marshall, Martin Ams, and Michael J. Withford, “Point by point femtosecond laser inscription of fibre and waveguide Bragg gratings for photonic device fabrication,” Proc. PICALO, 360-362 (2006).

Fibrecore

Fabry-Perot Cavity

Top View

Micro Fabry-Perot Interferometer in Optical Fibres

Front View

Fibre (250 μm)


LTI MODELLING OF ACTIVE MAGNETIC BEARINGS BY MEANSOF SYSTEM IDENTIFICATION

P.A. van Vuuren ∗, G. van Schoor † and W.C. Venter ‡

∗ School of Electrical, Electronic and Computer Engineering, North-West University, & Private BagX6001, Potchefstroom, 2520, South-Africa. E-mail: [email protected]† E-mail: [email protected]‡ E-mail: [email protected]

Abstract: A relatively unknown phenomenon in active magnetic bearings (AMBs) is that thefrequency content of their rotor position signal can induce nonlinear behaviour in the bearings. Theexistence of such frequency-induced nonlinear behaviour is experimentally and theoretically confirmed.Frequency-induced nonlinearity is characterised by means of a novel graphical representation. Theresultant graph is quite useful in the specification of suitable excitation signals when AMBs are to bemodelled by means of system identification.

Key words: Active magnetic bearings, system identification, non-linear, LTI.

1. INTRODUCTION

Conventional bearings use lubricated contact surfaces tosupport a rotating axle. Despite centuries of development,friction remains a fact of life for conventional bearings.Friction causes loss of power and energy as well aslimited component lifetime due to wear and tear. Theseproblems can be avoided by employing magnets to supportthe rotating shaft by magnetic forces, thereby realisingcontactless and consequently frictionless rotation.

Magnetic bearings most frequently operate on the principleof an attracting magnetic force which suspends an objectagainst gravity. Such magnetic bearings exhibit negativestiffness [1] and have to be actively controlled to ensurecontactless levitation. Such bearings are called activemagnetic bearings.

As can be seen from figure 1, the basic construction ofa typical active magnetic bearing consists of a stationarystator enclosing the shaft/rotor. The stator contains severalelectromagnets that exert reluctance magnetic forces on therotor, thereby suspending it against the force of gravityand preventing contact with the stator surface. Sensorsmounted on the stator continually monitor the position ofthe rotor. These position signals are used by a controller toadjust the power amplifier that supplies each pole with thenecessary current to suspend the rotor.

Accurate and reliable position sensors are criticalcomponents of a functioning AMB system. These sensorsare however expensive. Self-sensing techniques attemptto estimate the position of the rotor from the electricalimpedance of the stator electromagnet coils [2].

For self-sensing AMBs to be of practical worth, they haveto be robust. Robustness analysis aims to quantify a controlsystem’s tolerance for uncertainty. Accurate robustnessanalysis of systems requires accurate models. Theintricacies of some systems defy conventional analytical

Figure 1: General structure of an AMB

modelling approaches and require a black-box systemidentification approach. This is the case in the robustnessanalysis of self-sensing AMBs. Self-sensing AMBs arequite sensitive to electromagnetic cross-coupling betweenthe various electromagnets in the AMB stator [3], [4]. Itis however quite difficult to obtain an accurate analyticalmodel for electromagnetic cross-coupling in an AMBstator by means of first principles deductive modellingtechniques. Fortunately, the required models can be easilyobtained by means of system identification.

More details on the robustness analysis of a self-sensingAMB by means of μ-analysis can be found in [5]. Thefocus of this contribution is on the influence of frequencyinduced nonlinear behaviour of AMBs on the choice ofthe excitation signal to be used for system identification.For this reason, this paper is only concerned with theapplication of system identification to AMBs equippedwith explicit position sensors.

System identification has been applied in various formsand for various motives to AMBs. The most basicapplication of system identification concepts is the useof parameter estimation algorithms to obtain values for


parameters that can’t be measured with conventionalsensors. Examples of such parameters in AMBs arethe damping and stiffness of the bearing that have beenestimated by both discrete-time [6] and frequency domainsystem identification techniques [7].

Frequency domain system identification [8] has also beenapplied to obtain a single multivariable model for twinradial AMBs with a flexible shaft [9]. Their identifiedmodel encapsulated the power amplifiers, both AMBs, aswell as the (non-rotating) rotor.

Discrete-time system identification [10] has been appliedto a 1-DOF AMB with the purpose of modelling thenominal plant as well as its dynamic uncertainties [11].Once again the ”plant” contained everything except thecontroller, since the objective of the whole modellingprocess was to design an H∞ controller for the 1-DOFAMB. The same research team has since extended theirwork to twin radial bearings with a rotating flexible rotor[12].

The common denominator of the previous work performedon the application of system identification to AMBs is thatthe influence of frequency-induced nonlinear behaviour inthe AMBs were ignored. This phenomenon is relativelyunknown and entails that the rotor may suddenly revealoscillatory behaviour or even delevitate if the rotor positionis perturbed with a sufficiently high frequency disturbancesignal. Evidence of this phenomenon in the literatureis sparse. Simulation results by Hegazy and Amershow that changing the amplitude of excitation signals(external disturbance forces) may induce chaotic behaviourin the rotor [13]. These simulation results are supportedby the experimental results of Jugo et al. [14]. Theyhave shown that excessive vibrations (due to mechanicalresonance at certain rotational speeds) result in loss inthe low-frequency gain of the AMB. (In other words, themagnetic force falls away if the rotor is vibrated at certainfrequencies.)

Accurate system identification however depends onpersistent excitation [15]. An excitation signal that givesrise to maximally informative input and output data isknown as a persistently exciting signal [10]. Persistentexcitation is therefore determined by the frequency contentof the excitation signal. It is consequently vital to ensurethat nonlinear behaviour isn’t induced in the AMB by thevery signal with which it is interrogated. With this in mind,the next section explores the topic of frequency inducednonlinear behaviour in more detail. Amongst other thingssection 2. also introduces a useful graphical tool thatclassifies an AMB’s behaviour into linear and nonlineardomains of operation as a function of the amplitude andfrequency content of the rotor position signal. Section 3.summarizes the application of system identification to a2-DOF AMB equipped with position sensors. Identifiedmodels are validated and evaluated in section 4., makingthe case for the inclusion of frequency induced nonlinearbehaviour in the specification of the excitation signal forLTI system identification of AMBs.

Figure 2: Simulated response of a nonlinear 1-DOF AMB to afrequency sweep

2. FREQUENCY INDUCED NONLINEARBEHAVIOUR IN AMBS

2.1 Different regions of operation

Figure 2 reveals the specific nonlinear behaviour inducedby the frequency of the shaft position. This figure showsthe simulated response of a 1-DOF sensed AMB for areference position signal that consisted of a sine-wavefrequency sweep at a constant amplitude. Also shownis the response of an identified LTI model of the 1-DOFAMB to exactly the same signal. For low frequencies,the nonlinear system behaves like a typical low-passLTI system. As the frequency of the input signal isgradually increased, the nonlinear system departs from LTIbehaviour in that its bias level starts to drift away fromits initial value. A further increase in the input frequencyhowever results in an abrupt change in the behaviour ofthe nonlinear system: it exhibits oscillatory motion witha frequency is much lower than that of the input signal.(In fact, this frequency is approximately the same as thatof the dominant pole of the closed-loop LTI model forthe system.) Not shown in figure 2 is the consequenceof a further increase in the frequency of the input signal,namely delevitation of the AMB.

From the above discussion it seems as if frequency inducednonlinear behaviour in AMBs can be classified into fourregions, namely:

• Region A: Pure LTI behaviour.• Region B: Characterized by a drifting bias level, but

otherwise similar to linear affine models.• Region C: Oscillations occuring at a much lower

frequency than that of the input signal.• Region D: Delevitation.

Up to this point, the response of a 1-DOF AMBhas been used to introduce frequency-induced nonlinearbehaviour. 2-DOF AMBs also exhibit this kind of


Figure 3: Simulated response of a nonlinear 2-DOF AMB to afrequency sweep

behaviour, as can be seen from figure 3. Only they-axis component of the rotor position is shown (sincethe desired behaviour is more pronounced in the verticaldimension due to gravity). Although the AMB responsein this figure is a function of time, it has been plottedagainst the instantaneous frequency of the input signal toaccentuate the relationship between the AMB response andfrequency. The (admittedly fuzzy) boundaries between thevarious regions of behaviour have been determined via anautomatic algorithm described in [5].

2.2 The cause of frequency induced nonlinearities

The root cause of frequency-induced nonlinearities inAMB behaviour can best be explained by making useof the force equation (1) for a 1-DOF AMB. If onlythe top coil of an AMB is allowed to carry a currentand electromagnetic cross-coupling is ignored, the forceexerted by the resulting 1-DOF can be modelled by thefollowing equation [1]:

fm = μ0

(Ni

lc/μr +2xg

)2Acos(θ), (1)

with lc the length of the magnetic path (excluding theairgap);

μ0 the permeability of free space;μr the relative permeability of the AMB stator;xg the distance of the airgap between the stator

and rotor;N the number of windings in the coil;i the current flowing in the coil;A the pole-face area; andθ the angle between the vertical axis and the

normal line to the pole face.

By collecting all constants in a single coefficient, k, theelectromagnetic force exerted by a 1-DOF AMB can be

modelled as follows (with all time-dependencies explicitlyhighlighted):

fm(t) = ki2(t)x2g(t)

. (2)

Taking the derivative of (2) with respect to time allows usto investigate the effect that changes in the position of therotor can have on the force exerted by the AMB on therotor. Application of the quotient rule of differentiationresults in the following:

d fm(t)dt

=2ki(t) di(t)dt x

2g(t)−2ki2(t)xg(t) dxg(t)dt

x4g(t). (3)

Rearranging (3) with the aim of expressing the force ofequation (2) in terms of the derivatives of position andcurrent leads to:

fm(t) =ki(t)

(di(t)

dt

)

xg(t)(

dxg(t)dt

) − xg(t)(

d fm(t)dt

)

2(

dxg(t)dt

) . (4)

From (4) it is clear that the force exerted by the AMBis inversely proportional to the derivative of the positionof the rotor∗. Sudden and large changes in the positionof the rotor will therefore result in a commensurateloss in magnetic force applied to the rotor. Anotherconclusion stemming from (4) is that both the frequencyand amplitude of the changes in the rotor position willlead to a reduction in the force. (The derivative of theposition signal is after all not only determined by howquickly the input signal changes, but also by the amplitudeof the change.)

2.3 Frequency induced nonlinear behaviour in a physicalAMB

Before proceeding with the implications of frequencyinduced nonlinearities for system identification, exper-imental proof will be given for the existence of thisphenomenon. The results presented in this section confirmthe conclusions made by Jugo et al. in [14], but in aslightly different setting and with a different methodology.(Voltage controlled AMBs were used in [14], while ourAMBs are current controlled. Furthermore, the results in[14] were obtained for a rotating shaft and required Floquetanalysis, while our results are for a non-rotating rotor.)

Experimental setup: Our experimental setup consistedof a rigid rotor suspended horizontally between twoeight-pole heteropolar radial AMBs [16]. The AMBswere designed for a peak load of 500 N and rms loadof 200 N. The characteristic parameters of the AMBs aresummarized in table 1 (some of which are elucidated infigure 4). Control of the system (and data capturing)

∗A similar result can be obtained for a 1-DOF AMB equipped withtwo actuators operating in differential driving mode [1].


Table 1: Summary of the physical AMBsParameter ValueKP (position PD controller) 20,000KD (position PD controller) 38KP (power amplifier PI controller) 1KI (power amplifier PI controller) 0.01Relative magnetic permeability 4,000Power amplifier switching frequency 100 kHzSupply voltage 300 VBias current 3 AResistance of coil wires 0.2 ΩCoil turns 50Rotor mass 12.5 kgAirgap 0.6 mmBackup bearing inner radius 250 μmAxial bearing length 49.15 mmJournal inner radius (rr) 15.88 mmJournal outer radius

(r j

)34.95 mm

Stator pole radius(rp

)35.60 mm

Stator back-iron inner radius (rc) 60.00 mmStator outer radius (rs) 75.00 mmPole width (rw) 13.89 mm

Figure 4: Physical dimensions of an AMB

is performed on a desktop PC by means of dSPACEreal time hardware and Matlab/Simulink�. Each AMBis independently controlled by means of two identicaldecoupled PD controllers (each responsible for movementalong a single dimension). All results given in this paperwere measured on the left hand AMB (and not on therotor’s centre of mass).

AMB response to a frequency sweep: The response of theleft hand AMB to a frequency sweep is shown in figure5. The reference position signal consisted of a 100 μm(peak) amplitude sinewave applied to the y-axis input (andzero to the x-axis input). This signal’s frequency waslinearly changed from 10 Hz to 5 kHz over an interval of 20seconds. (The use of a sinewave as well as a relatively longduration sweep minimizes transient effects in the AMB’sresponse.)

The presence of some of the critical resonant frequenciesof the rotor can be easily seen in the response of the AMB.Furthermore, a drifting DC level (region B behaviour)and a low frequency component which emerges at high

Figure 5: Measured AMB response to a 100 μm (peak)frequency sweep

frequency excitation (i.e. region C behaviour) are alsovisible. Clearly the response in figure 5 verifies thenonlinear behaviour predicted in the simulated responsesof figures 2 and 3.

Existence of subharmonic generation: Mere inspectionof a frequency sweep response however doesn’t compriseproof of nonlinear behaviour. Nonlinear systems arecharacterised by subharmonic generation, where the outputspectrum contains components at frequencies that arelinear combinations of the input spectral components’frequencies. (In other words, if two frequencies f1and f2 are present in the input signal, then the outputsignal contains components at the following frequenciesa f1 ± b f2, where a and b are arbitrary integers [17].)Nonlinearity can therefore be diagnosed if subharmonicgeneration occurs.

Figure 6 shows the spectrum of the steady state responseof the AMB when confronted with three sinusoidal inputs,namely two 10 μm (peak) sines at respectively 200 Hz and300 Hz as well as a 40 μm (peak) sine at 3 kHz. Thepurpose of the latter signal is to induce nonlinear behaviourin the AMB, while the former two signals were chosenin order to obtain clearly visible results of subharmonicgeneration.

Figure 6 clearly shows the existence of a significantcomponent at 100 Hz, which indicates subharmonicgeneration and consequently nonlinear behaviour. Alsovisible in this figure is a small very low-frequencycomponent (despite the fact that the data was detrendedprior to spectral analysis). Such low frequencycomponents are indicative of emergent region C behaviour.

The influence of the amplitude and frequency of thehigh-frequency signal responsible for inducing nonlinearbehaviour on the size of the low-frequency components isshown in figure 7. This figure was obtained by repeatingthe analysis performed to obtain figure 6 for various highfrequency input signals. (I.e. the 200 Hz and 300 Hz input


Figure 6: Y-axis spectrum in response to three input signals

Figure 7: Size of the 2.2 Hz component as a function of theamplitude and frequency of a high frequency inducing signal.

signals were kept unaltered, but the third input signal wasmodified in both its amplitude and frequency.) Clearlythe size of the low-frequency oscillation characteristic ofregion C behaviour is strongly influenced by both theamplitude and frequency of the inducing signal.

2.4 The frequency-amplitude graph

Whether an AMB behaves linearly or nonlinearly dependson both the amplitude and frequency of the rotor positionsignal. The behaviour of an AMB for various combinationsof frequencies and amplitudes of the input signal can besummarized in a single graph. Figure 8 presents thebasic idea of a possible graphical AMB characterizationtool. This graph is based on the response of an AMB tonumerous frequency sweeps (of various combinations ofthe amplitude and maximum frequency reached during thesweep). This graphical tool gives an immediate indicationof the expected fidelity of an LTI model for the AMB fordifferent input signals.

The conceptual graph in figure 8 can only be realized inpractice if the boundaries between regions A, B, C and D

Figure 8: Conceptual frequency-amplitude graph of an AMB

can be automatically detected. Although the progressionfrom one region to the next occurs gradually, it is possibleto detect the boundaries between them on the basis of theobserved system response. This can be done by clearlydefining the onset of region B, C and D behaviour.

Delevitation (region D) is easy to detect since it occurswhen the radial position of the point mass exceeds theinner diameter of the retainer bearing (in this case 250 μm).Similarly, region B behaviour can be defined as the firstoccurence of a trend (in any direction) in the bias levelof the AMB response. This leaves region C behaviourwhich can be defined as oscillations with a period thatis significantly longer than the period of the input signal.One possible automatic algorithm that performs the abovementioned classification is discussed in detail in [5].

As an example, figure 9 shows the frequency-amplitudegraph obtained for a representative nonlinear simulationmodel of a 2-DOF AMB. This graph shows that asurprisingly large subset of input signals will inducenonlinear behaviour in an AMB. (Most of these signalsare however outside the typical operating range of AMBapplications.)

The nonlinear simulation model with which figure 9 wasgenerated consists of a controller, four power amplifiers,a magnetic circuit model, point mass and ideal positionsensors. The accuracy of this simulation model hasbeen established in a previous study by comparison withexperimental results [18]. Similar to the physical AMBsystem, the controller consists of two identical decoupledPD controllers, each responsible for one axis of movement.Each of the four stator electromagnets is powered by itsown two-state switched mode power amplifier. The dutycycle of each of these amplifiers is constrained to remainwithin the interval of 25 % to 75 % and is controlled witha PI controller.

The remainder of the sensed-AMB simulation model is


Figure 9: Frequency-amplitude graph for a simulated AMB

concerned with the AMB plant, which is dominated bythe electromagnetic calculations required to model theforce exerted on the point mass. A reluctance networkmodel is used to model the flux distribution in the AMBmagnetic circuit [19]. The response of the reluctancenetwork model is enriched with two additional models:one responsible for predicting eddy currents and the otherfor modelling magnetic hysteresis and saturation. Thefinal electromagnetic force exerted by the AMB on thepoint mass is proportional to the square of the magneticflux density [1], [18]. Finally, the physical movement ofthe point mass is determined by means of the well-knownNewton laws.

Armed with a frequency-amplitude graph such as figure 9,it is now possible to perform system identification to obtainaccurate models for various parts of an AMB’s operatingdomain (whether linear or nonlinear).

3. SYSTEM IDENTIFICATION APPLIED TO AMBS

3.1 Injection points and measuring points

Applying system identification to AMBs is a challengingexercise due to the inherent instability of magneticbearings. Since open-loop operation of an AMB isimpossible, system identification must be performed whilethe AMB is in closed-loop operation. Various approachesexist to perform closed-loop system identification [10]. Ofthese, direct identification is the simplest to apply, yet isstill able to model unstable plants. Direct identificationentails that the control system be operated in closed-loopwhile the system is subjected to a small perturbationapplied anywhere in the loop. The desired subsystem inthe loop can then be modelled by merely extracting datafrom its specific inputs and outputs [10].

Empirical models are however only as good as the datafrom which they are derived. In system identification, thenature and quality of the model that is eventually identifiedis to a large extent determined by the excitation signalused [20]. The attribute required of excitation signals is

Figure 10: SISO closed-loop control system

persistent excitation. The main feature of a persistentlyexciting excitation signal is that its power spectrum mustbe as flat as possible (within a specified frequency band)[15].

Unfortunately the primary advantage of negative feedback,namely that feedback reduces the closed-loop system’ssensitivity for disturbances to the plant [20], complicatesmatters for closed-loop system identification. The netteffect of feedback is that the reduced sensitivity of thesystem results in less informative data for parameterestimation [10]. Even though an excitation signal mayconform to the standards of persistent excitation, thespecific point in the closed-loop control system at which itis applied may influence the extent to which the excitationsignal is distorted before it arrives at the AMB input.

As an example, consider the simple SISO (single-input,single-output) control system in figure 10, where acontroller, GC(s) is placed in series with a plant, GP(s). Ifthe excitation signal (ξ(t)) is added to the nominal systemreference signal (r̄(t)), the resultant input to the plant isa filtered version of the original excitation signal, as isevident in (5),

U(s) = GC(s)E(s)

=GC(s)R(s)

1+GC(s)GP(s)= GC(s)R̄(s)S(s)+GC(s)S(s)Ξ(s), (5)

where S(s) represents the SISO sensitivity function.

3.2 Sampling frequency and model inputs and outputs

In a typical AMB, the controller is known exactly. (In thiscase it consists of a pair of identical decentralized PIDcontrollers, each responsible for one axis of movementwithin the airgap.) The only component that has to bemodelled by means of system identification is the AMBplant which consists of the power amplifiers, AMB statorand rotor. (Separate models can be identified for thevarious components of the AMB plant if the aim is toassess the impact of the components on the total systemstability robustness [5].) In this work, the AMB plantaccepts two inputs from the controller. The first of thesetwo inputs serves as a current reference signal for thepower amplifiers of the top and bottom coils in the AMBstator, while the other input supplies a current reference tothe remaining two power amplifiers of the left and right


stator coils. The two outputs of the identified AMB plantmodel represent the x-axis and y-axis position of the rotorwithin the airgap.

At present, it is easier to model multivariable systemsby means of discrete-time system identification than withcontinuous-time techniques∗∗. This work therefore makesuse of discrete-time system identification to model a2-DOF AMB. Prior to the application of a parameterestimation algorithm, it is necessary to choose thesampling frequency of the eventual models.

System identification works best if the sampling frequencyis commensurate with the dominant time constants of theplant [10]. The lower bound on the sampling frequencyis dictated by the Nyquist frequency of the phenomenonthat has to be modelled. The natural frequency of theAMB under consideration in this work is approximately63 Hz [16]. If the sampling frequency is chosen toohigh, numerical problems may arise during parameterestimation. Furthermore, the resultant model may containtoo many high frequency dynamics and loose its focus onthe important low frequency characteristics of the system.High sampling frequencies can also lead to nonminimumphase models [10]. In this work, the sampling frequencyof the model is chosen the same as the sampling frequencyof the implemented digital control system, namely 10 kHz.

3.3 Parameterized model structures

The next step is to choose a suitable parameterized modelstructure. This choice has far reaching effects since anincorrect model structure contributes to the bias error ofthe model [10]. The two criteria that influence this choiceare firstly the fact that the AMB plant is multivariable.Secondly, the model structure must have the capabilityto model unstable plants. (The AMB plant is after allinherently unstable and should therefore also be modelledas such.)

Various model structures are available with which MIMO(multi-input, multi-output) systems can be modelled. Atpresent, the Matlab� system identification toolbox islimited to multivariable ARX (autoregressive with externalinputs) and state-space model structures. Of these two,the state-space model structure is inherently an easierand more elegant framework to extend to multivariableproblems.

The inherent instability of AMBs should however alsobe taken into consideration. ARX models are suitablefor direct closed-loop identification of unstable systems[22]. The primary drawback of ARX models is that theymay exhibit a slight high-frequency bias if the samplingfrequency is too large [10], which manifests itself in theform of very large transient responses.

It is however possible to obtain unstable state-spacemodels via system identification provided that the correct

∗∗Current Matlab� toolboxes for continuous-time system identifica-tion are limited to SISO systems [21]

weighting scheme of the residuals is used during parameterestimation. If the weighting function is calculated as theproduct of the spectrum of the input signal and the inverseof the estimated noise model, then unstable plant modelsmay ensue, provided that the correct model order is chosen.This weighting scheme is known as a ”prediction focus”,because it basically entails minimizing the one-step aheadprediction error of the model [23]. The resultant modelsadmittedly have a high-frequency bias, yet don’t exhibitthe same large transients as ARX models do.

The model structure of choice is therefore state-spacemodels, since it is possible to estimate both stable andunstable models for multivariable systems with them.

3.4 Parameter estimation algorithm

A related issue to the model structure is the specificparameter estimation algorithm with which its parameterscan be obtained. Broadly speaking, parameter estimationalgorithms for state-space models can be dichotomizedinto two slightly overlapping families of algorithms,namely prediction error methods (PEM) and subspacemethods [24].

PEM algorithms lay claim to optimal solutions of theparameter vector (in terms of the cost function that wasused). Unfortunately, this advantage comes at the cost ofincreased computation time. (PEM algorithms convergeiteratively to the ”optimal” solution.) In contrast, subspacemethods are non-iterative and therefore extremely fast,with only a slight bias error in their final estimates of theparameter vector [24]. In fact, Favoreel et al. have foundthat subspace methods performed quite satisfactorily ona range of industrial identification problems compared toPEM [24]. In practical applications, subspace methodsregularly provide the initial estimate from which PEMalgorithms proceed to the final solution [24], [23]. Thisapproach is also followed in this study.

3.5 Excitation signal

The utility of the frequency-amplitude graph becomesevident when it is used as a tool to evaluate potential exci-tation signals. Take for example the frequency-amplitudegraph of figure 9. The effect that a particular excitationsignal will have on the AMB’s behaviour can be predictedby superimposing the spectral content of the excitationsignal onto the vertical axis of the frequency-amplitudegraph. In this way it is possible to see in advance whethera particular signal will induce region B or C behaviourin the AMB. This has been done for the case of whitenoise in figure 11. The superimposed spectrum clearlyshows the characteristic flat spectrum of white noise.Furthermore, it is also evident that this excitation signalcontains significant components in region B and C (at thespecified amplitude). Consequently, it should come as nosurprise that this excitation signal does induce nonlinearbehaviour in the AMB.

What is required is an excitation signal whose spectral


Figure 11: Evaluation of potential excitation signals: white noise

Figure 12: Evaluation of potential excitation signals: rectangularpulses

content can be constrained to be within a narrow band.Examples of such signals are random phase multi-sinesignals and rectangular waves. Rectangular waves are idealexcitation signals for LTI system identification, since theirspectrum falls away rapidly for increasing frequencies.If the amplitude of the rectangular wave is specifiedprudently, the AMB will remain within its LTI operatingdomain (see figure 12).

4. RESULTS

The results presented in this section underscore theusefulness of the frequency-amplitude graph introduced insection 2.4 for characterising and modelling AMBs. Itis shown that accurate LTI models can be identified forthe AMB provided that the excitation signal is confinedto region A behaviour.

To obtain an accurate LTI model of the AMB plant, theexcitation signal should be persistently exciting, yet shortin duration and conform to the requirements of linearoperation dictated by the AMBs frequency-amplitudegraph. One such signal consists of square waves applied

Figure 13: Orbital plot of the AMB response to the excitationsignal

to the x- and y-axes independently thereby exposing thedynamics of the AMB in both directions separately whileincluding enough information of cross-coupling betweenthe horizontal and vertical dimensions. This signal’samplitude was limited to 50 μm in order to prevent theoccurence of position-induced nonlinear behaviour in theAMB.

Through a process of trial and error a tenth orderstate-space model was fitted for the AMB plant. Theclosed-loop simulation performance of this model iscompared to the response of the original nonlinearsimulation for the AMB in figures 13 and 14. Figure 13 isan orbital plot of the responses of the nonlinear simulationmodel of the AMB as well as the LTI closed-loop modelfor it. This figure clearly shows that the LTI modelcan replicate the general pattern of the original response,but can’t correctly model the DC offset in the verticaldimension. The latter shortcoming is to be expected sincethe LTI model for the plant was fitted on detrended data (asis standard practice in the system identification literature[10]). (This bias error can be easily corrected by merelyadding the mean values of the actual AMB responses to theplant state-space model, as can be seen in figure 15.) Thedynamic response of the AMB is however satisfactorilymodelled by the LTI model as can be seen in figure 14(which shows the responses of figure 13 as a function oftime).

The LTI model of figures 13 and 14 is however onlyvalid while the AMB is confined to linear behaviour.The inability of an LTI model to accurately model anAMB that exhibits frequency-induced nonlinear behaviouris visible in figure 15. This figure shows the responseof the AMB when it is exposed to a 50 μm amplitudefrequency sweep stretching from 11 Hz to 400 Hz whichis applied concurrently to both the x- and y-axes. Athigher frequencies the AMB’s behaviour clearly becomesincreasingly nonlinear.

The closed-loop response of the same LTI model of figures


Figure 14: Response of the AMB models to the excitation signal

Figure 15: Response of the AMB models to a sine sweep

13 and 14 to the frequency sweep shows that this modelis only representative of the AMB behaviour at very lowfrequencies (which is to be expected since the model wasobtained on region A data). Region C behaviour can’t bemodelled by this LTI model. The inability of LTI modelsto replicate region B and C behaviour is underscored by thefact that not a single stable closed-loop LTI model could beobtained on excitation data that contained region B and Cbehaviour. (An example of such an excitation signal is a50 μm amplitude random phase multi-sine signal appliedto both axes concurrently. This signal’s spectrum is almostuniform in a bandwidth from 11 Hz to 400 Hz and fallsaway rapidly outside this band.)

5. CONCLUSION

We have seen that nonlinear behaviour can be induced inAMBs by the frequency content and amplitude of the rotorposition signal (and therefore also by the reference positionsignal). Consequently, the characteristics of the excitationsignal used for system identification has a definite impacton the nature of the AMB (whether it can be viewedas a linear or nonlinear system). This means that theamplitude and spectral characteristics of the excitation

signal has a large influence on the quality of the resultantLTI model obtained by means of system identification.Alternatively put, LTI system identification only givesaccurate results if the excitation signal is chosen suchthat the AMB is constrained to LTI behaviour. Suitableexcitation signals can be synthesized by means of thefrequency-amplitude graph of figure 9, since this graph is aconvenient summary of the degree of nonlinearity presentin the AMB behaviour.

The frequency-amplitude graph can be of potential benefitin the design of controllers that must be able to stablysuspend high-speed rotors. Another application of thisgraph is also the robustness analysis of self-sensingAMBs by means of μ-analysis (which requires systemidentification and detailed uncertainty modelling).

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MULTIVARIABLE H OR CENTRE OF GRAVITY PD CONTROL FOR AN ACTIVE MAGNETIC BEARING FLYWHEEL SYSTEM S. J. M. Steyn*, P. A. van Vuuren* and G. van Schoor* * North-West University, School of Electrical, Electronic and Computer Engineering, Potchefstroom, South Africa, E-mail: [email protected] / [email protected] / [email protected] Abstract: A state-space model with uncertainties for an active magnetic bearing energy storage flywheel system (Fly-UPS) is developed. A multivariable robust H controller for the Fly-UPS is then synthesised. Different weighting schemes are explained and the additive uncertainties between the nominal simulation model and the physical model at varied rotational speeds are characterised. Furthermore, PD controllers are developed using the centre of gravity (COG) coordinate framework for decoupled parallel and conical modal control. Stability robustness is verified via the gain/phase margin stability robustness criterion. The performance robustness (disturbance attenuation) is analysed by assessing model sensitivity with the ISO/CD 14839-3 sensitivity standard. The results obtained show that H control and COG coordinate PD control are not fundamentally different. Keywords: H-infinity control, active magnetic bearing, flywheel, modal control, COG coordinate control.

1. INTRODUCTION

Conventional ball-bearings in rotational applications can potentially be replaced by active magnetic bearings (AMBs) [1]. AMBs levitate the rotor via contact-free, actively controlled, electromagnetic forces. The basic functioning of an AMB is shown in Figure 1 and it operates as follows [2]: sensors are used to measure the air gap between the rotor and the bearing. This measurement is sent to the controller to regulate the current in the actuator using a power amplifier (PA). The current in the actuator manipulates the magnetic force on the rotor, keeping it in suspension. In modern high-tech systems, flywheels are used as energy storage batteries [3]. In this particular case AMBs are applied to a flywheel uninterrupted power supply (Fly-UPS) system. By utilizing AMBs for the Fly-UPS, a contact- and lubrication-free ideal vacuum environment is achieved [4]. Alas, the inherently unstable nature and complexity of AMBs necessitates sophisticated feedback control [5]. Therefore, two feedback control methods will be investigated and evaluated in this article. Firstly, by introducing multivariable H control to the AMBs of the Fly-UPS, robust control can be realized [6]. The aim of H control is to synthesise a controller K such that the effects of modelling uncertainties, noise, and disturbances are minimized according to predefined performance requirements at low frequencies and robustness requirements at high frequencies [7]. H control synthesis allows frequency-dependent bounds to be specified, ensuring that the above mentioned effects remain within permissible levels [8]. H control has successfully been applied to AMBs by [1], [6] and [8] and provides promising results. However,

in [6], H control was found lacking in position deviation regulation, when compared to optimal LQR control, but showed reduced control current and provided greater stability robustness with varying rotational speed. Secondly, PD control is the most straightforward approach for rigid rotor AMBs and is frequently used [2]. The PD controllers conventionally used for AMB systems are decoupled from one another, creating a cascaded decentralised SISO (single-input, single-output) control setup. Furthermore, conventional PD control ignores the fact that the sensors and magnetic bearings are non-collocated (not on the same axial position). However, in order to compensate for non-collocation and implement a multivariable controller, an alternative PD control method is investigated. The alternative method considered is multivariable COG (centre of gravity) coordinate PD control [2]. For rigid body systems, the results obtained using either H control or COG coordinate PD control are not fundamentally different and can thus be compared [2].

Electromagnetic actuator

Power amplifier

Controller

Sensor

Rotor

Figure 1: Basic AMB functioning


In this paper multivariable robust H and COG coordinate PD controllers for an active magnetic bearing flywheel system are investigated. Because the Fly-UPS design is subject to various uncertainties as well as gyroscopic effects at varying rotational speeds, stability robustness and disturbance attenuation (performance robustness) are the primary feedback requirements.

2. FLY-UPS MODEL This section will use analytical methods to develop and describe a model for the five degrees of freedom (5-DOF) AMB Fly-UPS. The 5-DOF are the radial x and y directions for the top (‘b’) and bottom (‘a’) AMBs and the axial direction z (Figure 2). The model is able to represent the dynamic behaviour of the system within small deviations from the nominal values. The rotor displacement, gyroscopic coupling, coil currents, power amplifier bandwidth as well as sensor bandwidth are all represented within this model [4]. Some effects such as rotor touchdown during a power failure as well as rotor whirls are not represented due to their highly nonlinear nature [9].

Figure 2: Fly-UPS with bearing and sensor positions [16] 2.1 State-space model A linear time invariant (LTI) state-space model of the rigid-rotor Fly-UPS plant is developed using parameters from the existing Fly-UPS system. This model is linearised around a nominal working point, or set-point. This makes the model valid for small deviations from the set-point values. A detailed explanation to the Lagrange analysis as well as the state-space equations can be found in [2]. The state vector x represents the 5-DOF displacements and their time derivatives. The linearised state-space equation can be derived in terms of displacement and current with,

,s iF ma k x k i (1)

where F is the force on the rotor, m the rotor mass, a the acceleration, x the displacement, i the current, ks the force-displacement constant and ki the force-current constant. The input to the model is current (i) and the output is displacement (zz, xa, ya, xb, yb). The subscripts are as seen in Figure 2. The sensor measurements of the rotor are realised by representing the rotor dynamics in the bearing position coordinate system

TB z a a b bz x y x yz instead of the Euler

coordinate system T z x yz which represents displacement (z, x, y) and inclination ( , ) about the centre of mass [2]. Furthermore, the state, input, and output vectors are defined as follows:

T T TB Bx z z , and

Tz ax ay bx byi i i i iu

Tz a a b bz x y x yy

The state-space model of the two radial AMBs is

x Ax Bu (2) ,y Cx (3)

with state-space matrices

1 1 ,B s B B

0 IA

M K -M G

1 , and .

B i

0B C I 0

M K (4)

The mass and gyroscopic matrices, M and G, are transformed to bearing positions, MB and GB, with

1 and ,B B B B B B-1M T MT G T GT (5)

where

0 0 0 0 0 0 0

0 0 0 0 0 0, ,

0 0 0 0 0 0 00 0 0 0 0 0

y z

x z

mI I

mI I

GM

0 01 1 0 01and .

0 00 0 1 1

b a

Bb ab a

l l

l ll lT (6)

Variable (rad/s) represents the rotational speed of the flywheel. Matrices Ks and Ki are simply diagonal matrices of the force-displacement and force-current constants, ksA, ksB and kiA, kiB of bearings ‘a’ and ‘b’ respectively. The distance of the bearing positions from the centre of mass is represented by la and lb and the


distance of the sensor positions from the centre of mass is represented by lc and ld. The final state-space system is referenced to the sensor coordinate system by transforming the state matrices A and B to

1 and ,S S S S SA T AT B T B (7) with

1 0 01 0 00 0 10 0 1

.1 0 01 0 00 0 10 0 1

c

dB

c

dS

c

dB

c

d

ll

ll

ll

ll

T 0

T

T0

(8)

The axial (z axis) AMB state-space model is decoupled from the radial AMBs, and thus only appended to the above state-space model using:

0 1 0, = , .

0 zzz sz izk km m

A B C I

The variables used for this model were experimentally measured: m = 17.65 kg is the rotor mass, Ix = Iy = 1.16×10-1 kg·m2 are moments of inertia about the x and y axes, Iz = 1.07×10-1 kg·m2 is the z axis moment of inertia, ksA/B = 154.84×103 N/m and kiA/B = 30.5 N/A are the radial AMB force-displacement and force-current constants. ksz = 922×103 N/m and kiz = 260 N/A are the axial force-displacement and force-current constants. Finally lengths from the centre of mass are: la = -160×10-3 m, lb = 65×10-3 m, lc = -190×10-3 m and ld = 95×10-3 m. Sensor model: The sensor electronics are modelled as five cascaded second order low-pass transfer functions with bandwidths of 10 kHz, and connected to the outputs of the AMB model [9]:

2

2 2 ,2s

senss s

Ts s

(9)

with damping = 0.707 and bandwidth s = 2 10000 rad/s. Power amplifier model: The power amplifier (PA) model consists of a closed loop PI controlled system with a bandwidth of 2.5 kHz,

2

2 ( ),

(2 ) 2bus i p

PAp bus i bus

V K K sT

Ls K V R s K V (10)

where Vbus = 51 V is the bus voltage, R = 0.152 is the coil resistance, L = 6.494 mH is the coil inductance, Kp = 1 is the proportional constant and Ki = 0.1 is the integral constant [10]. The cascaded PA model is connected to the input of the AMB model.

3. H CONTROL The two basic design approaches in H control are open-loop transfer function loop shaping and closed-loop transfer function loop shaping [11]. In loop shaping, the required shape of a transfer function is defined in the frequency domain using singular values, and a controller is designed to shape the system transfer function into that required shape [1]. For open-loop shaping, the classical L=GK loop shaping is transposed to MIMO (multi-input, multi-output) systems by using the singular values of the loop transfer functions as the loop gains [12]. One such open-loop transfer function loop shaping is the Glover-McFarlane H loop shaping method [1]. Alternatively, closed-loop transfer function shaping aims to shape the sensitivity function, S = (I + GK)-1 and the complementary sensitivity function (closed-loop transfer function) T = GK(I + GK)-1 [11]. This leads to classic mixed sensitivity H control synthesis. By adhering to open-loop design objectives, it is relatively easy to estimate the closed-loop requirements over specific frequencies [11]. But, because the Fly-UPS AMBs are open-loop unstable, specifying an open-loop shape is relatively difficult. Consequently, closed-loop transfer function design using mixed sensitivity H control synthesis is the primary focus. This allows robust stability by including uncertainties in the model as well as providing robust performance by minimizing the H norm via weighting [12]. 3.1 Uncertainty The model developed in section 2 is a linear representation of the Fly-UPS system. Because the physical Fly-UPS system is non-linear, there are system dynamics and gains that are not included in this linear model. Furthermore, the Fly-UPS is a rotational system with changing dynamics over the rotational speed range from 0 – 5000 r/min. Thus it is necessary to synthesise a controller that maintains stability robustness and performance robustness for a difference between the model and the physical system, as well as the rotational speed range. The uncertainty is therefore also rotational speed dependent and thus an uncertainty set exists for each rotational speed [3].


However, because the model is an LTI model, an uncertainty boundary is considered for a difference in the plant and model over fixed rotational speeds. The speeds selected arbitrarily for uncertainty development are: 0, 1000, 2000, and 5000 r/min. Therefore, in order to ensure robustness, the additive normalised uncertainty bound ( | 1) between the linear model, physical system, and the rotational speeds above must be obtained (Figure 3).

u

Gy

W

+

Figure 3: Additive uncertainty

Denote the frequency response of the model at 0 r/min as Hm(j ) and the frequency response of the real system at 0 r/min as Hr(j ). The additive model error frequency response function (FRF) can be determined experimentally [8], and is defined as:

( ) ( ) ( ) .r mW j H j H j (11) The upper bound (maximum) of the difference between the two responses is taken as the uncertainty bound. Equation (11) is repeated for rotational speeds of 1000, 2000, and 5000 r/min. Finally, the maximum of the above four boundaries is determined and taken as the complete uncertainty bound, W . Unfortunately, by taking uncertainty over such a broad range, increases conservativeness [8]. Figure 4 shows the 4th order uncertainty bound W of a single radial AMB.

Figure 4: Uncertainty bound W 3.2 H control design The standard mixed sensitivity control scheme uses three weights or bounds in the following mixed sensitivity cost minimisation problem [13],

0

0

0

min ,e

uK

y

W SW KSW T

(12)

where K is a stabilising controller [11]. The weights are applied to the control error, e, to shape the output sensitivity, So; to the control signal, u, to shape the input sensitivity, KSo; and to the plant output, y, to shape the complimentary sensitivity, To. The weighting functions are used to shape, or penalise, the above signals into their desired shapes. However, the weighting functions must be proper to be solvable, and stable, since they are not in the loop to be stabilised by the controller [9]. Unfortunately, mixed sensitivity control has one serious drawback: pole-zero cancellation. It aims to cancel poorly damped stable poles of the system with controller zeros, making the poorly damped poles unobservable and uncontrollable [13, 14]. Furthermore, the controller designed using this method, contains the inverse of the plant, causing complications if the plant is non-invertible [9]. In order to bypass these problems another weighting scheme is applied, namely the six block problem weighting scheme [1]. This scheme includes the plant in the weighting of the sensitivity, preventing the inclusion of the inverted plant within the controller. In addition to weighting the standard So, KSo, and To signals, the input disturbance sensitivity function (GSi), and closed-loop transfer function from the plant input to the plant output (Ti) are also shaped. Thus a term involving GSi is included in the standard mixed sensitivity minimisation problem as given in (13) below [1]:

min .e o r e i d

u o r u i dK

y o r y i d

W S W W GS WW KS W W TWW T W W GS W

(13)

As can be seen in (13), the standard weights We, Wu and Wy are present with the addition of two new weights, Wr and Wd. Wr shapes the reference input to the system, and Wd shapes (attenuates) the input disturbances. Wd is important in preventing pole-zero cancelation [13]. The structure of the augmented (weighted) plant is shown in Figure 5, where the inputs are given by w = [r, d] and u = [iax, iay, ibx, iby], and the outputs are z = [z1, z2, z3] and v = [xa, ya, xb, yb]. The augmented system in Figure 5 is then reformulated into a system Tzw (Figure 6). The H control design for Tzw is now: Find a controller K that stabilizes system Tzw. The controller must minimise the H norm; .zwT

( is the ratio between the maximum and minimum singular values.)


G

K

+-

rd

y

r

VU

P

We-1

Wu-1z2

z1Wd-1Wr-1 + + Wy-1

z3

Figure 5: Augmented control formulation

K

Pz

vu

w

Tzw

Figure 6: Control configuration for system Tzw In order to solve the above minimization problem, the solutions to two Riccati equations are required. MATLAB®’s Robust Control Toolbox is used to solve these Riccati equations and synthesise the H controller [12]. 3.3 Weighting function selection There are five weighting functions to be selected in the six block problem weighting scheme, namely: We, Wu, Wr, Wy and Wd. Each weighting function is created according to predefined specifications and requirements of the AMB Fly-UPS system. As stated in section 1, the weighting functions are mostly selected on a priori knowledge of the system in a trial and error basis [3]. Once the required weighting function shape is designed, the inverse is supplied for controller synthesis [8]. This is done in order to ‘guide’ the minimisation of the H cost function. The selected weights are shown in Figure 7.

Figure 7: Selected weighting functions Weight Wr plays a part in shaping the sensitivity (So) and controller gain (KSo). In order to simplify the weighting selection, Wr is a constant with, Wr = 1.

The error weight, We, shapes the sensitivity function (So). According to the ISO/CD 14839-3 standard [15], the sensitivity for newly commissioned machines should be below 8 dB. In order to obtain an integration effect, the magnitude of the error weight at frequencies lower than 1 Hz should be small. Thus, it would be adequate to select the weighting function as a first order high-pass transfer function with a pass-band gain of 8 dB for frequencies above 1 Hz. Weight Wu represents the controller gain (KSo). However, in the case of additive uncertainty, the weight Wu is equal to the uncertainty bound W [11]. This allows the inclusion of uncertainty by replacing the weight Wu with W . Transfer function Wy, shapes the complimentary sensitivity To. In order to keep the closed loop gain close to unity, Wy is selected as a first order low-pass transfer function with a 5 kHz bandwidth and a 4 dB pass-band gain. Finally, Wd · We shapes the input disturbance signal (GSi), and Wd · Wu shapes the input closed-loop transfer function (Ti). In order to obtain good input disturbance rejection, the weight Wd is chosen as a small constant of 100×10-6. Therefore, the combined weight Wd · We bounds the change in displacement to 200 μm for an input disturbance of 1 A, and the combined weight Wd · Wu bounds Ti to a gain of 6 dB. The value achieved with the selected weighting functions during control synthesis is 3.85. This means that the synthesised controller is able to comply with the required specifications (bounds) provided by the weighting functions within 3.85 dB. Using the explained weights, a controller is synthesised. However, the 5-DOF controller has an order greater than 75. This is impractical for most applications, and thus the order of the controller is reduced using the balanced stochastic model truncation (BST) via Schur reduction method [12]. The controller order is successfully reduced to a 19th order controller with a multiplicative error bound of less than 7%.

4. COG PD CONTROL In this section, a MIMO COG (centre of gravity) coordinate PD controller will be developed. In conventional PD control for the fly-UPS, the input to the controller is the displacement error in the x or y axis and the controller output is a current reference for the electromagnets, as depicted in section 2.1. For the MIMO COG coordinate control method, instead of controlling the system in the bearing coordinate framework,

,TB z a a b bz x y x yz the system is controlled in the COG (Euler) coordinate framework,

T x yz . Unfortunately, the Fly-UPS


system functions using current as input and displacement as output. Therefore, in order to use COG control, the displacement sensor values must be transformed to COG values and the COG control output values must then be transformed to current reference values. These transformations are done using an input transformation matrix Tin and an output transformation matrix Tout [2]:

1 1 0 00 01 ,

0 0 1 10 0

d cin

d c

d c

l ll l

l l

T (14)

1 0 0

0 01 .

( ) 0 0 1

0 0

b

iA iAa

iB iBout

iA b ab

iA iAa

iB iB

l

k kl

k kk l l l

k kl

k k

T (15)

These transformation matrices are simply appended to the COG PD controller. Thus, conventional control is achieved using the COG coordinate control method as shown in Figure 8.

PIDCOG

ToutTin

x

y

ax

bxay

by

iiii

xaxbyayb

x

y

KsComp

Figure 8: Conventional control using transformed COG

control There are two advantages to using the COG control method compared to the conventional method. Firstly, by controlling in the COG coordinate framework, the parallel and conical modes can be controlled separately [2]. However, full decoupled control of the parallel and conical modes are only possible if a stiffness compensation matrix KsComp is connected in parallel with the COG control. This compensation matrix is necessary to compensate for the coupling instigated by the non-diagonal stiffness matrix introduced by the negative stiffness of each AMB as well as the non-collocated nature of the position sensors (required proof can be found in [2]). The matrix KsComp is calculated as follows:

( ) ( ) 0 0

( ) ( ) 0 01 .

0 0 ( ) ( )

0 0 ( ) ( )

sA sAd a a c

iA iA

sB sBd b b c

iB iBsComp

sA sAd cd a a c

iA iA

sB sBd b b c

iB iB

k kl l l l

k kk k

l l l lk k

Kk kl l l l l lk kk k

l l l lk k

The second advantage of MIMO COG control is that by including the transformation matrices in the feedback loop, automatic cross-coupling control between the upper and lower AMBs can be achieved. This automatic cross-coupled control occurs simply because both upper and lower AMB position values are transformed to single COG values. As a result of using the COG control scheme, new values for the proportional and derivative constants must be selected. Values f

vol 102 no 3 september 2011 saiee africa research journal · vol.102(3) september 2011 south...

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