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CONTROL OF AN ACTIVE MAGNETIC BEARING
WITH AND WITHOUT POSITION SENSING
RICHARD A. RARICK
Bachelor of Electrical Engineering
Cleveland State University
May, 1989
Submitted in partial fulfillment of requirements for the degree
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
at the
CLEVELAND STATE UNIVERSITY
August, 2007
This dissertation has been approved
for the Department of Electrical and Computer Engineering
and the College of Graduate Studies by
________________________________________________ Date__________
Thesis Committee Chairperson: Dr. Lili Dong
Department of Electrical and Computer Engineering
________________________________________________ Date__________
Committee Member: Dr. Daniel Simon
Department of Electrical and Computer Engineering
________________________________________________ Date__________
Committee Member: Dr. Charles Alexander
Department of Electrical and Computer Engineering
ACKNOWLEDGEMENTS
I would like to express my sincerest gratitude to Dr. Lili Dong for her supervision
and encouragement during the course of my studies. Her dedication to education and love
of teaching are inspiring;
To Dr. Dan Simon for his clear understanding of advanced control theory princi-
ples, and for his easy-going, yet careful and precise approach to education and scholar-
ship;
To Dr. Charles Alexander for introducing me to high-level engineering design,
the inner workings of the research community, and especially for his career guidance and
contagious optimism;
To Dr. Zhiqiang Gao for his discerning philosophical insights into engineering
and epistemology, and for introducing me to the concept of active disturbance rejection
control which has significantly influenced my approach to control theory;
To Dr F. Eugenio Villaseca for his good-natured support and stimulating conver-
sations, love of detail, dedication to education, and love of language;
To Adrienne Fox and Jan Basch for their competent, efficient, and most impor-
tant, friendly assistance in all matters administrative;
To Baixi Su-Alexander for his love of language and for the numerous deep and
fascinating conversations on all aspects of the human condition;
And to all the members of the CSU community for their dedication to the enlight-
enment of their fellow humans though the ancient and noble profession of teaching.
CONTROL OF AN ACTIVE MAGNETIC BEARING
WITH AND WITHOUT POSITION SENSING
RICHARD A. RARICK
ABSTRACT
This thesis contributes three new applications of existing control designs to the regulation
of the position of the rotor (or shaft) in an active magnetic bearing (AMB). Two of the
applications use a position sensor, while one does not. The first is an application of the
Active Disturbance Rejection Control (ADRC) concept, while the second is a variation
on the first using ADRC in conjunction with the technique of integrator backstepping
control design. Both of these use a position displacement sensor for feedback to the con-
trols. The third is an application of H-Infinity optimal control design without the use of a
position sensor. In addition, some other conventional control designs are incorporated
into the study for comparison. The performance of the controlled system is assessed in
the presence of a static force disturbance applied to the rotor and also in the presence of a
sinusoidal force disturbance. The disturbance rejection of the ADRC control design with
position-sensing was significantly better than any of the control configurations tested in
terms of the transient performance and disturbance rejection. In this preliminary study,
the benefits of the position sensing variety of AMB appear to outweigh the benefits of the iv
self-sensing type. There are many proposals in the literature for these so-called self-
sensing or sensorless controls, but because of the design challenges involved, the fact is
that only one industrial application has been proposed. The current thesis supports the
conclusion underlying this fact.
v
TABLE OF CONTENTS
Page
NOMENCLATURE........................................................................................................ IX
LIST OF TABLES ........................................................................................................... X
LIST OF FIGURES ........................................................................................................XI
I INTRODUCTION......................................................................................................... 1
1.1 History......................................................................................................... 6
1.2 Magnetic Levitation and Earnshaw's Theorem........................................... 7
1.3 Summary of the Advantages of Magnetic Bearings ................................. 10
1.4 Applications .............................................................................................. 12
1.5 Literature Review...................................................................................... 13
1.6 Contribution of Thesis .............................................................................. 16
1.7 Organization of Thesis.............................................................................. 19
II MODELING THE MAGNETIC BEARING .......................................................... 21
2.1 Principle of Operation............................................................................... 22
2.2 Mechanical Dynamics............................................................................... 24
2.3 Magnetic Circuit Analysis ........................................................................ 25
2.4 Magnetic Force on the Rotor .................................................................... 29
2.5 Electrical Circuit Analysis ........................................................................ 31
vi
2.6 The Nonlinear Model................................................................................ 32
2.7 The Linearized MIMO Model .................................................................. 34
2.8 The Linearized SISO Model ..................................................................... 39
III CONTROL WITH POSITION FEEDBACK ....................................................... 43
3.1 Control Objectives .................................................................................... 43
3.2 Open-Loop Transfer Functions................................................................. 47
3.3 Controllability and Observability ............................................................. 51
3.4 State Feedback and Stability..................................................................... 54
3.5 H-Infinity Control ..................................................................................... 66
3.6 ADRC Control .......................................................................................... 80
3.7 ADRC Integrator Backstepping Control................................................... 94
IV CONTROL WITHOUT POSITION FEEDBACK ............................................. 104
4.1 Current Feedback.................................................................................... 104
4.2 Steady-State Error................................................................................... 108
4.3 Luenberger Observer .............................................................................. 109
4.4 H-Infinity Control ................................................................................... 114
4.5 Verifying the Linearization..................................................................... 121
4.6 ADRC Integrator Backstepping Control................................................. 122
V CONCLUSION AND FUTURE WORK ............................................................... 128
vii
5.1 Summary and Conclusions ..................................................................... 128
5.2 Future Work ............................................................................................ 130
REFERENCES.............................................................................................................. 132
viii
NOMENCLATURE
ADRC: Active Disturbance Rejection Control
AMB: Active Magnetic Bearing
CFB Current Feedback
Ctrl Control
Dist Disturbance
DOF: Degree of Freedom
ESO: Extended State Observer
FB: Feedback
FI: Full Information
LP: Linear Periodic
LTI: Linear Time-Invariant
MIMO: Multiple-Input, Multiple-Output
PD: Proportional-Derivative
PID: Proportional Integral Derivative
PFB Position Feedback
RHP: Right Hand Plane
SISO: Single-Input, Single-Output
ix
LIST OF TABLES
Table Page
Table I Physical Parameters for the AMB............................................................. 48
Table II Derived Parameters for the AMB ............................................................. 49
Table III Parameter Sensitivity – State Feedback.................................................... 60
Table IV Scaling Parameters.................................................................................... 71
Table V Parameter Sensitivity – H-Infinity Full Information ................................ 76
Table VI Parameter Sensitivity – ADRC ................................................................. 90
Table VII Parameter Sensitivity – ADRC & Backstepping .................................... 100
Table VIII Parameter Sensitivity – Luenberger........................................................ 111
Table IX Parameter Sensitivity – H-Infinity Output Feedback.............................. 118
Table X Parameter Sensitivity – ADRC & Luen & Backstepping....................... 124
Table XI Performance Summary for Various Designs .......................................... 129
x
LIST OF FIGURES
Figure Page
Figure 1 Cross Section of a Simplified Radial AMB [28] ........................................ 2
Figure 2 A Five-DOF, Five-Axis AMB System [28]................................................ 2
Figure 3 Rigid Body Modes of Vibration [8]............................................................ 3
Figure 4 Bending Modes in a Rotor Assembly [46] ................................................. 4
Figure 5 A Simplified Schematic of a One-DOF AMB [44] .................................... 5
Figure 6 Rotor and Stator for a Radial Magnetic Bearing [19]................................. 5
Figure 7 Ball Bearing Model from a da Vinci Drawing [7]...................................... 7
Figure 8 Positive and Negative Stiffness [8]............................................................. 9
Figure 9 A Simplified Schematic of a One-DOF AMB [42] .................................. 10
Figure 10 Step response of rotor position [32].......................................................... 12
Figure 11 Magnetic Dipole for Spinning Electron [1] .............................................. 22
Figure 12 Schematic of a One-DOF Magnetic Bearing [29] .................................... 23
Figure 13 A Simple Magnetic Circuit [5] ................................................................. 25
Figure 14 Fringing and Leakage Flux [42] ............................................................... 26
Figure 15 Magnetic Circuit with Air Gap ................................................................. 27
Figure 16 Magnetic Force on an Object [6] .............................................................. 28
xi
Figure 17 Linearized MIMO System ........................................................................ 38
Figure 18 Third-Order SISO Subsystem................................................................... 42
Figure 19 First-Order SISO Subsystem .................................................................... 42
Figure 20 Current-Controlled Open-Loop Plant ....................................................... 45
Figure 21 Closed-Loop, Current-Controlled, Position-Sensed ................................. 45
Figure 22 Voltage-Controlled Open-Loop Plant....................................................... 46
Figure 23 Closed-Loop, Voltage-Controlled, Position-Sensed................................. 47
Figure 24 Block Diagram of Equation (3.7)............................................................. 48
Figure 25 Simulink Model of MIMO System with State Feedback.......................... 55
Figure 26 Step Disturbance, State FB, 4.6 N Load ................................................... 56
Figure 27 Zoomed View of Figure 26....................................................................... 57
Figure 28 Step Dist, State & Integral FB, 4.6 N Load .............................................. 58
Figure 29 Step Dist, State & Integral FB, 4.6 N Load .............................................. 59
Figure 30 Step Dist, State & Integral FB, 250 N Load ............................................. 59
Figure 31 Step Dist, State & Integral FB, 250 N Load ............................................. 60
Figure 32 Sine Dist, State & Integral FB (10 Hz Dist.) ............................................ 62
Figure 33 Sine Dist, State & Integral FB (10 Hz Dist.) ............................................ 63
Figure 34 Sine Dist, State & Integral FB (1,000 Hz Dist.) ....................................... 63
Figure 35 Sine Dist, State & Integral FB (1,000 Hz Dist.) ....................................... 64
xii
Figure 36 Frequency Response of AMB................................................................... 65
Figure 37 General Control Configuration ................................................................. 66
Figure 38 Interconnection Matrix Design for Full Information Feedback................ 70
Figure 39 Step Dist, H-Infinity & Integral Ctrl, 4.6 N Load..................................... 74
Figure 40 Step Dist, H-Infinity & Integral Ctrl, 4.6 N Load..................................... 74
Figure 41 Step Dist, H-Infinity & Integral Ctrl, 12 N Load...................................... 75
Figure 42 Step Dist, H-Infinity & Integral Ctrl, 12 N Load...................................... 76
Figure 43 Sine Dist, H-Infinity & Integral Ctrl (10 Hz) ........................................... 77
Figure 44 Sine Dist, H-Infinity & Integral Ctrl (10 Hz) ........................................... 78
Figure 45 Sine Dist, H-Infinity & Integral Ctrl (1000 Hz) ....................................... 78
Figure 46 Sine Dist, H-Infinity & Integral Ctrl (1000 Hz) ....................................... 79
Figure 47 Sine Dist, H-Infinity & Integral Ctrl (1000 Hz) ....................................... 79
Figure 48 Plant With Disturbance ............................................................................. 80
Figure 49 Generalized Disturbance, 0y f b u= + ..................................................... 83
Figure 50 Control Law: ( )0ˆ /u f u= − + 0b ................................................................ 83
Figure 51 Desired Dynamics: 0y u≈ ........................................................................ 83
Figure 52 ADRC Block Diagram.............................................................................. 86
Figure 53 Simulink Model for ADRC and SISO AMB ............................................ 88
Figure 54 Step Dist, ADRC, 4.6 N Load................................................................... 89
xiii
Figure 55 Step Dist, ADRC, 4.6 N Load................................................................... 89
Figure 56 Step Dist, ADRC, 15,000 N Load............................................................. 91
Figure 57 Step Dist, ADRC, 15,000 N Load............................................................. 91
Figure 58 Sine Dist, ADRC (10 Hz) ........................................................................ 92
Figure 59 Sine Dist, ADRC (10 Hz) ........................................................................ 93
Figure 60 Sine Dist, ADRC (1000 Hz) .................................................................... 93
Figure 61 Sine Dist, ADRC (1000 Hz) .................................................................... 94
Figure 62 ADRC Rotor Control Design.................................................................... 96
Figure 63 ADRC Rotor Control Subsystem.............................................................. 96
Figure 64 ADRC Coil Control Design ...................................................................... 97
Figure 65 ADRC Coil Control Subsystem................................................................ 98
Figure 66 ADRC Control for an AMB Using Integrator Backstepping.................... 98
Figure 67 Step Dist, ADRC & Backstepping, 4.6 N Load........................................ 99
Figure 68 Step Dist, ADRC & Backstepping, 6,300 N Load.................................. 101
Figure 69 Sine Dist, ADRC & Backstepping, (10 Hz) ........................................... 102
Figure 70 Sine Dist, ADRC & Backstepping, (1000 Hz) ....................................... 103
Figure 71 Current-Sensed, Voltage-Controlled AMB ............................................ 105
Figure 72 Simulink Model, Luenberger Observer and State feedback ................... 109
Figure 73 Step Dist, Luenberger Observer, 4.6 N Load.......................................... 110
xiv
Figure 74 Step Dist, Luenberger Observer, 125 N Load......................................... 112
Figure 75 Sine Dist, Luenberger (10 Hz) ................................................................ 113
Figure 76 Sine Dist, Luenberger (1000 Hz) ............................................................ 114
Figure 77 Interconnection Matrix Design, H-Infinity Output Feedback................. 115
Figure 78 Simulink Model of MIMO System with H-Infinity Output Feedback ... 116
Figure 79 Step Dist, H-Infinity Output Feedback, 4.6 N Load ............................... 117
Figure 80 Step Distt, H-Infinity Output Feedback, 4.6 N Load.............................. 117
Figure 81 Step Dist, H-Infinity Output Feedback, 10 N Load ................................ 118
Figure 82 Step Dist, H-Infinity Output Feedback, 10 N Load ................................ 119
Figure 83 Sine Dist, H-Infinity, Output Feedback, (10 Hz).................................... 119
Figure 84 Sine Dist, H-Infinity, Output Feedback, (1000 Hz)................................ 120
Figure 85 Sine Dist, H-Infinity, Output Feedback, (1000 Hz)................................ 120
Figure 86 Nonlinear MIMO Model, Current-Sensed, H-Infinity............................ 121
Figure 87 Step Dist, Linear and Nonlinear System, 4.6 N Load............................. 122
Figure 88 Simulink Model, ADRC & Luenberger & Backstepping....................... 123
Figure 89 Step Dist, ADRC & Luen & Backstepping, 4.6 N Load ........................ 124
Figure 90 Step Dist, ADRC & Luen & Backstepping, 300 N Load ....................... 125
Figure 91 Sine Dist, ADRC & Luen & Backstepping, (10 Hz) .............................. 126
Figure 92 Sine Dist, ADRC & Luen & Backstepping (1000 Hz) ........................... 127
xv
CHAPTER I
INTRODUCTION
A magnetic bearing is a device which maintains the relative position of a rotating assem-
bly (rotor and shaft) with respect to a stationary component (stator) by means of a mag-
netic field. Magnetic bearings can be broadly categorized into two types: passive and ac-
tive. In general, a passive magnetic bearing is made with permanent magnets, while an
active magnetic bearing (AMB) is made with electromagnets. The present work focuses
on AMBs, and a simplified schematic of a radial AMB is illustrated in Figure 1. The il-
lustration is shown with four pairs of coils and is a typical configuration for achieving
control of the rotor position. Note that the rotor has two degrees-of-freedom (DOF) and
can move in two dimensions in the xy-plane.
1
2
2i
4i
3i
1i2i
4i
3i
1i
Figure 1 Cross Section of a Simplified Radial AMB [28]
In practice the rotor is part of, or connected to, a rigid shaft as depicted in Figure
2. There it can be seen that the rotor assembly is free to move in the xy-plane at either end
of the shaft as well as axially along the z-direction. In this situation there are five DOFs
and five axes to control. If the rotor assembly is unbalanced, two vibration modes (or
rigid body modes) are possible: the assembly may rotate about a tilted axis (wobble) or
Figure 2 A Five-DOF, Five-Axis AMB System [28]
about a translated parallel axis as shown in Figure 3. In the latter case, the motion is an
orbit of the shaft about its center of mass. It is impossible to manufacture a perfectly bal-
anced rotor assembly, and the effect of the unbalance increases with increasing rotational
speeds. Unbalance is, therefore, one of the major factors limiting rotational speeds. It is
3
possible, however, to remove the effect of rigid body modal vibrations by using feedback
control to adjust two important parameters of AMB systems called stiffness and damping
[8].
Figure 3 Rigid Body Modes of Vibration [8]
In reality the rigid shaft assumption is not valid because all rotor assemblies are
flexible to some degree and can bend, especially at the high rotational speed to which
AMBs are typically subjected. Two such bending modes are depicted in Figure 4, where
it is clear that the motion becomes more difficult to model and analyze. The location of
the bending modes varies with the rotational speed of the rotor, so that the situation is
even more complicated than that shown. Each bending mode also has a characteristic ex-
citation frequency, and if the rotational frequency of the rotor happens to be the same,
excitation of the bending modes can cause the rotor to resonate at that frequency. This
can have catastrophic results, since the AMB actuators cannot control this oscillation.
This is a result of the fact that the AMB actuators can only control the rotor at the two
bearing ends of the shaft, so control of bending modes (which resonate along the entire
length of the rotor) is impossible with the AMB controller. Since the excited bending
modes cannot be controlled, the controller is designed to not excite them. This is done
using moving notch filters; these filters prevent any information at the bending mode fre-
4
quencies from reaching the AMB controller, thus preventing any controller output at fre-
quencies which would excite the bending modes [8].
Figure 4 Bending Modes in a Rotor Assembly [46]
The higher dimensional models are required for a more complete analysis of an
AMB system. Nevertheless, when designing a control for an AMB, many important re-
sults can be obtained by considering the simpler two-dimensional motion illustrated in
the planar model in Figure 1. In addition, the symmetry of the two-dimensional bearing
can be exploited to allow much of the analysis and control design to be performed using a
model in which the rotor is constrained to move in one dimension as shown in Figure 5.
This is, in fact, the approach that is taken in many important papers and texts on the sub-
ject, and it will also be the approach adopted in this work. An actual three-pole AMB is
shown in Figure 6.
5
Figure 5 A Simplified Schematic of a One-DOF AMB [44]
Figure 6 Rotor and Stator for a Radial Magnetic Bearing [19]
6
1.1 History
Bearings have existed since ancient times when the Egyptian pharaohs con-
structed the famous Pyramids of Giza using the ancient technology of linear bearings
moving heavy slabs overland on logs. A wooden ball bearing that was used to support a
rotating table was found in the remains of a Roman shipwreck [2]. Leonardo da Vinci
drew a design for a ball bearing set (Figure 7) for one of his machines in 1497, and Gali-
leo described caged ball bearings in the 1600s. In 1794 Philip Vaughan received the first
patent for a ball bearing, in 1898 the first patent for a Timken roller bearing was issued,
and in 1907, Sven Wingquist of SKF Bearings invented the modern self-centering ball
bearing. The importance of bearings to modern industry is illustrated by the Allied strate-
gic bombing of the ball bearing plants in Germany in WWII. The first patent for an active
magnetic bearing was issued to J. Beams and F. Holmes in 1941. This invention was in-
corporated into an ultracentrifuge used for the separation of uranium isotopes. Jesse
Beams, working at the University of Virginia’s Department of Physics in the 1940’s, is
usually given credit as the “Father of Magnetic Bearings” [23]. Numerous applications
have evolved since the 1940s, some of which are enumerated in a later section.
7
Figure 7 Ball Bearing Model from a da Vinci Drawing [7]
1.2 Magnetic Levitation and Earnshaw's Theorem
In 1842 Samuel Earnshaw proved that a collection of point charges cannot be
maintained in a stable stationary equilibrium configuration solely by the interaction of the
electrostatic forces arising from the charges [12]. The theorem can be generalized to any
force-field which satisfies an inverse-square law such as electrostatic, magnetostatic, and
gravitation force-fields [15]. These fields are derived from a potential function which sat-
isfies Laplace’s equation and as a consequence are divergenceless in free
space. Therefore, there is no point in empty space where the force from the field is
directed inward from all directions, and a stable equilibria of particles cannot exist.
( )( )0∇⋅ =F x
8
The theorem usually applies to charged particles and magnetic dipoles but also
can be extended to solid magnets. Thus, this theorem explains why it is impossible to
have a stable passive magnetic bearing using only the forces of static fields of permanent
magnets. Nevertheless, there are five known situations in which the assumptions of Earn-
shaw's Theorem are not satisfied and the theorem does not apply: time-varying fields,
diamagnetic systems, ferrofluids, superconductors, and active-feedback systems. Active
feedback control is the most commonly used method to circumvent Earnshaw's Theorem
[34], and it is this method that will be investigated to control active magnetic bearings in
the present work.
There are two major characteristics of mechanical and magnetic bearings which
must be considered when selecting a bearing for a particular application, namely, stiffness
and damping. Bearing stiffness is analogous to the spring constant or force-displacement
constant in Hooke’s Law. Unlike Hooke’s Law, however, magnetic bearing stiffness can
be negative. When a constant current is passed through the AMB’s electromagnet’s coils
so as to levitate the rotor to an equilibrium position, then, if the stiffness is positive, the
magnetic force of the electromagnets tends to return the rotor to its equilibrium position
after a disturbance or load is applied to the rotor, rendering the equilibrium point stable
(Figure 8). If the stiffness is negative, any disturbance to the rotor position is amplified
by the magnetic force, and the rotor moves away from the equilibrium point, and thus
such a system is inherently unstable. Fortunately, the stiffness can be made positive by
using feedback control in a closed-loop system. Certain applications such as high speed
(100,000 rpm) machine tool spindles require a large stiffness to withstand the large im-
pacts and loads encountered at the spindle. Active magnetic rotor bearings of up to 30
9
tons load capacity have been built and reported [11]. Since the stiffness can be controlled
electronically, AMBs are much better suited to such applications than conventional bear-
ings, where the stiffness is built into the properties of the steel and oil film.
F k x= ⋅
F k x= − ⋅
F k x= ⋅
F k x= − ⋅
Figure 8 Positive and Negative Stiffness [8]
The damping coefficient is a measure of the bearing’s ability to attenuate vibra-
tions and contributes to the stability of the system. Once again, the damping of AMBs
can be controlled electronically, allowing the bearing to adapt to different situations and
conditions. For many applications, passive magnetic bearing systems using permanent
magnets do not have sufficient stiffness and damping built in to allow them to perform
well over their entire operating range. With AMBs these parameters can be electronically
tailored to design specifications, thus AMBs have found the widest range of practical ap-
plications among the different types of magnetic bearings.
10
An active magnetic bearing uses attractive forces generated by electromagnets to
support a rotating shaft. A more detailed drawing of a one-DOF AMB is depicted in
Figure 9. Here the mechanical and electrical parameters that will be used in this work are
labeled explicitly. Not shown are the auxiliary mechanical bearings that are often co-
aligned with the magnetic bearings to assist under startup conditions or as a backup under
electrical failure or severe disturbance conditions. During the normal operation of a ro-
tor/magnetic bearing systems, contacts with auxiliary bearings or bushes are avoided.
However, auxiliary bearings are required under abnormal conditions and in malfunction
situations to prevent contact between the rotor and stator laminations.
Figure 9 A Simplified Schematic of a One-DOF AMB [42]
1.3 Summary of the Advantages of Magnetic Bearings
Magnetic bearings have numerous practical advantages over conventional bear-
ings. Some of the most important will be listed next.
11
1. The most obvious advantage is that the rotor is suspended in a magnetic field,
so there is no mechanical contact, no wear on the bearing surfaces, and no
contamination of the application environment from bearing particles.
2. No lubricating system is required. This eliminates the need for lubricants,
seals, pumps, filters, tanks, piping, and coolers.
3. Since no lubricants are used, there is no contamination of the application envi-
ronment from oil leakage at the seals.
4. Very high rotation speeds are possible because there is no friction except for
windage from the rotor. If operated in a vacuum, the windage is also elimi-
nated. The only limitation on speed in this case is the bursting speed of the ro-
tor.
5. Magnetic bearings have very small energy losses. Heat dissipated through
bearing and seal friction and the churning of the oil or other lubricant is elimi-
nated.
6. When constructed of the proper materials, magnetic bearings have an ex-
tremely wide temperature range in which they can operate. SKF claims a
range of 180 C− to 480 C .
7. For AMBs, the dynamics of the rotor can be controlled. Stiffness and damping
coefficients can be scheduled to control vibrations at critical speeds. Vibration
attenuation can be achieved by allowing the rotor to rotate about its center of
mass rather than its geometric center.
12
8. In order to control the rotor, the control system must constantly monitor the
rotor position, vibration, and load. This knowledge may also be used for im-
proved machine health monitoring, diagnosis, and maintenance.
9. The integral part of the control brings the position x of the to the same value
before and after a load step, and thus the rotor shows a behavior that cannot be
obtained with conventional mechanical bearings (Figure 10).
Figure 10 Step response of rotor position [32]
10. All of the above contribute to the long life cycle, high reliability, and eco-
nomic advantages of magnetic bearings.
1.4 Applications
Some common applications for magnetic bearings are listed below.
1. Flywheels for energy-storage space applications (high speed, low energy loss,
high reliability)
2. Watthour meters for electric utilities (low loss)
13
3. High-speed machining applications (vibration control, high stiffness)
4. Ultra high speed centrifuges
5. Medical devices such as blood pumps (contamination free)
6. Microelectronics capital equipment (contamination free)
7. Food and beverage processing equipment (contamination free)
1.5 Literature Review
The design of control systems for active magnetic bearings has been extensively
studied since their inception in the 1940s. Yet, prior to about ten years ago, more than
ninety percent of active magnetic bearing system designs were based on a decentralized
Proportional-Integral-Derivative (PID) position control. In decentralized design, the
AMB system is decoupled into single-input, single-output (SISO) loops. For example, a
bearing with two DOFs is decoupled into two SISO loops such that the current in each
loop is only determined by the shaft displacement in its corresponding axis. There are
some problems, however, with PID controllers, one of which is that they use a velocity
feedback signal, and in an AMB system the velocity signal is often difficult to measure.
System and sensor noise, as well as the small size of the air gap between the rotor and the
stator (in this work, about 0.7 mm), often makes the numerical differentiation of the posi-
tion signal very poor.
14
A variation on the conventional PID control is a cascaded PI/PD position control
as presented in [28]. The advantages of the PI/PD control are transparent design, simple
realization, and a higher closed-loop damping and stiffness in comparison with the con-
ventional PID control.
The application of H∞ optimal control design to AMBs began soon after the in-
troduction of H∞ control theory by George Zames [45] in 1981. An important require-
ment in most practical AMB applications is that the stiffness of the controlled system,
when subjected to unknown dynamic disturbance forces or loads, should not be below a
given value for some specified frequency range. The new theory was applied by Herzog
and Bleuler [18] to synthesize an H∞ control and demonstrate its effectiveness in dealing
with “worst case” disturbances within the specified frequency range. In [13], Fujita et al.
designed and experimentally tested an H∞ controller for robust stability in the presence
of various perturbations and uncertainties in plant parameters for an AMB system with
satisfactory results.
The concept of self-sensing or sensorless AMBs was introduced by Vischer [39]
in his Ph.D. dissertation in 1988. A few years later Vischer and Bleuler [40][41] pub-
lished two applications of the new concept in which the position sensor is eliminated and
the position of the levitated object is estimated by extracting position information from
the bearing coil current. The main interest of this approach consists in reducing produc-
tion costs and hardware complexity and improving reliability. Although the idea of self-
sensing itself is not new, and research has been carried out for years in this topic, it still
remains a challenge. Many self-sensing methods have been proposed in literature, how-
15
ever they are all very difficult to realize in practice. For this reason, as of 2005 there were
still no industrial applications for self-sensing magnetic bearings [31].
In 1997 Kucera [20] analyzed the sensitivity of a self-sensing AMB to variations
in the controller parameters and found that stability within the entire air gap can only be
achieved at the expense of system robustness. In the same year Noh and Maslen [26] de-
vised a technique wherein the bearing currents are presumed to be developed with a two-
state switching amplifier which produces a substantial high frequency switching ripple.
This ripple is carries information about the length of the bearing air gap. The ripple is
demodulated using a technique which extracts the length of the bearing air gap while re-
jecting the influence of control voltage. Another modulation approach was proposed by
Schammass et al. [31] in 2005. This approach is based on measuring the change of induc-
tance caused by the rotor displacement through a high frequency signal applied to the ac-
tuator coils. The current waveform is demodulated and the position is estimated from the
first harmonic component of the current.
A new approach to control design called Active Disturbance Rejection Control
(ADRC) was developed by Han [17] and Gao [14] and was simulated by Su-Alexander
[36] in a self-sensing AMB application in 2006. The controller was demonstrated to be
very robust with respect to variations in the plant parameters and disturbance rejection. It
is also easier to tune than the conventional PID controller since there are only two pa-
rameters to adjust. This is especially useful for AMBs, since the control is very difficult
to tune because of a RHP pole and zero.
In 2002 Thibeault and Smith [37] demonstrated that self-sensing magnetic bear-
ings are impractical due to fundamental limitations in the achievable closed-loop robust-
16
ness. Due to experimental data which appeared to contradict these results, Maslen et al.
[21] showed in 2003 that significantly better robustness is achievable if the magnetic
bearing is modeled as a linear periodic (LP) system rather than the linear time-invariant
(LTI) system used by Thibeault and Smith. While an intuitive explanation was given for
the cause of the improvement, a paper by Peterson et al. [27] in 2006 gives a more pre-
cise analysis to explain why modeling the self-sensing magnetic bearing as an LP system
improves the achievable robustness. This is accomplished by “lifting” the LP system to a
higher dimensional multiple-input, multiple-output (MIMO) LTI system in order to ana-
lyze the input and output directions.
This thesis has as its focus the linear control of AMBs, but it should be mentioned
that the literature is replete with references to nonlinear control designs for these devices.
A few of these are listed in the References, but a thorough review is beyond the scope of
this work. Many of the standard approaches to nonlinear control such as feedback lineari-
zation, adaptive feedback, and sliding-mode control appear to have been studied. A brief
survey does not reveal any particular type emerging as a clear leader in either perform-
ance or robustness, and an ad hoc approach seems to prevail.
1.6 Contribution of Thesis
This thesis contributes three new applications of existing control designs to the
regulation of the position of the rotor or shaft in an AMB. Although one of the major mo-
tivations for the study is the application of the controls to an energy-storage flywheel, the
17
information obtained is of a very general nature and has wide applicability. The first ap-
plication involves the use of the ADRC concept, while the second is a variation on the
first using ADRC in conjunction with the technique of integrator backstepping control
design. Both of these use a position displacement sensor for feedback to the controls.
The third is an application of H-Infinity optimal control design without the use of a posi-
tion sensor. The present evaluation of these new applications seeks to provide an initial
overview of the potential advantages or disadvantages of the three new control applica-
tions under consideration. This is of value because, although AMBs have been studied for
decades, new approaches and improvements are still needed. NASA, for example, has
been investigating magnetic bearings (active and passive, position-sensing and self-
sensing) for over ten years for use in energy-storage and combined energy-
storage/gyroscopic applications because they show promise as an alternative to batteries
and reaction wheels for space systems [9].
Because of the potential cost savings, simplicity of design, and reliability, self-
sensing AMBs are an attractive alternative to the position-sensing variety. There are
many references in the literature to H-Infinity control of AMBs with full state feedback,
but none were found in the literature search that report the results of using only current
feedback. The authors in [37] analyze the robustness of an AMB using H-Infinity and
current-sensing alone, but they do not test the results in either a simulation or an experi-
ment. Furthermore, although these authors report poor robustness for this application, the
authors in [21] and [27] refute these findings. And, as mentioned previously, no industrial
applications for self-sensing magnetic bearings currently exist, and research into this area
18
continues. It is therefore of interest to simulate a self-sensing H-Infinity controller and
record its performance and robustness.
Secondly, this thesis offers a complete derivation of the mathematical model for
the AMB. While studying the control problem for AMBs, it became clear early in the
process that a thorough understanding of the physical model of the AMB in one dimen-
sion would facilitate the analysis and design of a control system. Unfortunately, most of
the literature which addresses such analysis and design merely states the nonlinear equa-
tions and their linearized counterparts without any background or derivation. For those
works which do undertake to derive some of the equations, the treatments range from
cursory to various levels of generality and mathematical abstraction, thus adding to the
difficulty of understanding the workings of the system. Moreover, there are numerous
formulations, assumptions, and notations which differ from author to author, complicat-
ing the effort to pull the various ideas together into an understandable whole. It seemed
appropriate, therefore, to derive the AMB model from first principles and, in so doing, to
clarify some of the subtleties of the problem. This was especially true in understanding
the many assumptions underlying the electromagnetic and mechanical properties of the
AMB system.
In addition to the above, the nonlinear equations were needed in a form suitable
for simulation in Simulink so that the control designs could be tested on the nonlinear
system. By including the details of the derivation as well as the final equations, the pre-
sent work is made more accessible to readers interested duplicating the simulations and
control strategies discussed.
19
Finally, the thesis provides a comparison of conventional state-feedback, H-
Infinity, ADRC, and ADRC combined with integrator backstepping control for AMBs.
As previously stated, prior to about ten years ago, the controller of choice for most AMB
applications was the PID controller with position measurement. But this control has not
provided the performance required in some of the more modern and demanding applica-
tion in use today. The ADRC model of control design is a relatively new control strategy,
and it has produced superior disturbance rejection results in numerous second-order, mo-
tion-control applications. It has also demonstrated excellent robustness in the presence of
variations in plant parameters. However, because it is relatively new, it has not been ap-
plied and compared in as many applications as more mature and established control tech-
nologies. It is of benefit, therefore, to compare its performance as applied to the AMB
with the H-infinity and other controls studied in this document.
1.7 Organization of Thesis
The thesis is organized as follows. Chapter II derives the nonlinear mathematical
equations governing the physics of an active magnetic bearing starting from Newton’s
Laws, the Lorentz Force Law, Ampere’s Law, and Magnetic Circuit Theory. The equa-
tions are transformed into a fourth-order, MIMO system in state-space form, and the
model is then linearized by applying the Jacobian transformation. Finally, the fourth-
order MIMO system is transformed by a change of variables into a third-order SISO sys-
tem and a first-order SISO system.
20
Chapter III begins with a statement of the control problems and objectives related
specifically to AMBs. A pole in the right hand plane (RHP) renders the system inherently
unstable, and a RHP zero renders the system non-minimun phase and contributes to the
difficulty of controlling the time-domain transient performance. The control objective
consists primarily of maintaining the rotor midway between the two magnets in the pres-
ence of a static load disturbance and a sinusoidal disturbance. Next, the controllability
and observability of the system are discussed and some robustness measures are calcu-
lated. The second part of the chapter develops the four types of controls that are com-
pared in this work, namely, H-Infinity, ADRC, integrator backstepping, and PID. These
controls are then applied to the AMB in simulation and tested in the presence of a static
load disturbance and a sinusoidal disturbance. The chapter ends with a comparison of the
performance for the various control types..
In Chapter IV the same controls are applied without using position feedback, that
is, by using only the current in the magnetic coils as feedback. The proof that there can be
no linear control that will remove the steady-state error for self-sensing AMBs is dis-
cussed. The controls are applied to the self-sensing AMB and tested with disturbances as
above. The chapter ends with a comparison of performance between the position-sensing
and self-sensing AMB.
Chapter V ends the thesis ends with concluding remarks and some possible areas
of further study.
CHAPTER II
MODELING THE MAGNETIC BEARING
Much of the analysis of the magnetic bearing system and the design of a control
system for the bearing are based on a linearized model of the system. Nevertheless, it is
desirable to apply the results of the design to the nonlinear model since it more closely
approximates the real system. Moreover, a much deeper understanding of the operation of
the bearing can be obtained from a detailed derivation and examination of the nonlinear
system from fundamental principles in the theory of mechanics and electromagnetism.
Many of the primary sources cited in the References do not derive, or even state, the
nonlinear model. Of those that state the governing nonlinear equations, none transform
them into state-space form. Since the method of linearization chosen in this thesis is the
application of the Jacobian transformation, the state-space form was necessary. The few
sources that do develop a nonlinear model do so with the assumptions and notations
suited to their particular physical application. For the remaining sources, each offers a
slightly different version of the linearized model as a starting point for its respective
analysis and design. These versions differ mainly in notational conventions, but some
also differ with respect to the mechanical and electromechanical assumptions that are
made. It is important, therefore, to begin with a model for which the underlying assump-
21
22
tions and notations are clearly elucidated. The purpose of this chapter is to develop the
nonlinear physical model from fundamental principles in physics, and then to derive the
linear model by calculating the Jacobian matrix of the nonlinear system and evaluating it
at a particular operating point.
2.1 Principle of Operation
When an electric current is moving through the electromagnet’s coil, a magnetic
field is induced in the ferromagnetic core (Figure 9). The magnetic field crosses the air
gap, passes through a portion of the rotor, and back into the ferromagnetic core to form a
magnetic circuit. Within the air gap, the field is almost perpendicular to the ferromagnetic
rotor. Near the air gap, the dipole moments (Figure 11) of the spinning electrons in the
rotor will align themselves with the externally applied magnetic field, and there will be a
force on the spinning electrons in the direction of increasing magnetic field strength, that
is, in the direction toward the coil. The force is a result of the Lorentz Force Law for the
force F exerted on a charged particle q moving with velocity v in an electromagnetic field
B [30]:
( )q= + ×F E v B (2.1)
Figure 11 Magnetic Dipole for Spinning Electron [1]
23
For the development in this thesis, a simplified magnetic bearing system, namely,
the two-pole, single degree-of-freedom (DOF) magnetic bearing shown in Figure 12 will
be considered. This system is the fundamental building block for more complicated mag-
netic bearing systems and thus contains the essential design challenges of these systems
without the added complexity. For the derivation of the model, the following assumptions
about the system will be made. It is assumed that the levitated shaft or rotor moves only
in the x-direction, that no bending of the shaft occurs, and that no gravity acts on the shaft
(a satellite application).
The displacement of the shaft is measured by the distance, x , which represents
the parallel distance of the axis of the shaft from the line lying midway between the two
electromagnets. When , the shaft is at an unstable equilibrium point, and, thus, ac-
tive feedback control is required to achieve stability about this point.
0x =
Figure 12 Schematic of a One-DOF Magnetic Bearing [29]
24
2.2 Mechanical Dynamics
The parameters in Figure 12 are defined as follows:
1. 0x is the distance between the left magnet and the surface of the shaft
when the shaft is in equilibrium, that is, when the axis of the shaft is mid-
way between the two electromagnets;
2. 1x and 2x are the air gaps between the magnets and the rotor;
3. x is the displacement of the rotor from the equilibrium position, δ is the
distance between the two coils, and ρ is the radius of the rotor shaft;
4. m is the mass of the rotor;
5. 1F and 2F are electromagnetically induced forces on the rotor;
6. dF is a an external disturbance force on the rotor;
7. 1u and 2u are the control voltages applied to the magnetic coils;
8. 1i and 2i are the coil currents.
It is clear from Figure 12 that the Newtonian equation of motion governing the system is
1 2 dmx F F F= − + (2.2)
25
2.3 Magnetic Circuit Analysis
The forces on the rotor, and , are caused by the magnetic flux induced in
the ferromagnetic cores by the current flowing in the magnet coils. An expression for this
flux will now be derived from fundamental electromagnetic principles and the simple
magnetic circuit in
1F 2F
Figure 13.
Figure 13 A Simple Magnetic Circuit [5]
The basic equation governing the production of a magnetic field by a current is
Ampere’s Law which, in integral form, may be stated as
d N⋅ =∫ H l i . (2.3)
Here i is the current in the coil, N is the number of turns in the coil, H is the magnetic
field intensity produced in the ferromagnetic core of the magnet by the current, and l is
any path enclosing a surface through which the current flows. Since the permeability of
ferromagnetic materials is much larger than the permeability of air, essentially all of the
26
magnetic field produced by the current is confined to the ferromagnetic core volumes
[22]. In general, the value of H varies within the cross-sectional area A. However, it is
possible to assume a constant, average value for H across the area and to integrate in
Ampere’s Law along a closed path of mean length among all closed paths [42]. This path
is represented by the dotted line in Figure 13.
The total current passing through the surface bounded by the closed path is Ni. If
is assumed that the direction of the magnetic field is the same as the direction of the mean
path, and if leakage flux is neglected (Figure 14), then Ampere’s Law becomes
orcc
NiHl Ni Hl
= = (2.4)
Figure 14 Fringing and Leakage Flux [42]
Under the assumption that the magnetic material is linear, the magnetic flux density is
defined by
μ=B H (2.5)
so, in magnitude,
27
c
NiB Hlμμ= = (2.6)
where μ is the permeability of the magnetic medium, in this case the ferromagnetic core.
For the situation in which the core has an air gap of length g as in Figure 15, if the
fringing of the flux at the gap is ignored, and if it is assumed that the permeability of the
ferromagnetic core is much greater than the permeability of the air gap, i.e., Fe 0μ μ ,
then Ampere’s Law gives
Fe Gap
Fe FeFe 0
0
Ni d d d
B BHl Hg l g
B g
μ μ
μ
= ⋅ = ⋅ + ⋅
= + = +
≈
∫ ∫ ∫H l H l H l
(2.7)
cl
i
cl
i
Figure 15 Magnetic Circuit with Air Gap
Therefore, the magnetic flux density of the circuit is approximately the value of the flux
density in the gap, namely,
28
0NiBg
μ≈ (2.8)
Using the same reasoning, the magnetic flux density in Figure 16, where there are two air
gaps, is calculated to be
0
2NiBg
μ= (2.9)
The total flux is then
0 2 2i K iBA NAg N g
φ μ= ⋅ = = =∫ B dA (2.10)
where A is the cross-sectional area of the core (or gap), , and where B is as-
sumed to be perpendicular to the cross-sectional area vector dA.
20K Nμ= A
– u +u +
Figure 16 Magnetic Force on an Object [6]
Summarizing, the assumptions made for the magnetic circuit analysis are as fol-
lows:
1. The ferromagnetic core material has a high permeability relative to its sur-
roundings. This means that the reluctance of the core is negligible, so that all
29
of the reluctance in the circuit is contained in the air gap. It also means that
the magnetic flux density outside of the core is negligible, implying that there
is no leakage flux.
2. The magnetic core material is linear, that is, μ=B H . this assumption ignores
core nonlinearities such as hysteresis, saturation, and the effects of eddy cur-
rents.
3. The gap flux density is uniform and small, so that fringing can be neglected.
2.4 Magnetic Force on the Rotor
The induced forces on the rotor, and , can now be obtained from the above calcula-
tion of the flux in the air gap. The total magnetic force F on the object in
1F 2F
Figure 16 as a
result of the flux is determined via energy considerations. The energy stored in a mag-
netic field in a given volume V is given by
2
0
1 1 Joule2 2V V
BW dV dVφ μ= ⋅ =∫ ∫H B (2.11)
where the second equation follows from the fact that 0μ=B H and 0μ is the permeability
of free space [30]. Assuming that all of the magnetic energy is stored in the two air gaps,
the volume under consideration will be . 32 mAg
For an infinitesimal change in the air gap of length dg , the corresponding change
in magnetic energy will be
30
2 2
0 0
22 2B BdW dV Adgφ μ μ
= = ⋅ (2.12)
2
0
dW ABdg
φ
μ= (2.13)
On the other hand, from the relation between the work done in moving an object and the
force applied to that object,
dW Fdgφ = (2.14)
( )22 2
0 0
dW ABABFdg A A
φ
0
φμ μ μ
= = = = (2.15)
Substituting the flux from (2.10) into the previous equation yields
2 22 2
00
0 0
12 4 4i N A i KF NA
2i
A A g gφ μμμ μ
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠g (2.16)
So, in terms of Figure 12
2 2
11 2
1 24 4K i K iF F 2
x x⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(2.17)
and (2.2) becomes
2 2
1 21 2
1 24 4dK i K imx F F F F
x x⎛ ⎞ ⎛ ⎞
= − + = − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
d (2.18)
or
2 2
1 2
1 2
14 4 dK i K ix Fm x m x m⎛ ⎞ ⎛ ⎞
= − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(2.19)
Note that in this formulation, the total force on the rotor is a nonlinear function of the cur-
rents in the coils and the air gaps between the coils and the rotor surfaces.
31
2.5 Electrical Circuit Analysis
From Faraday’s Law, the back-EMF induced in the coil in Figure 16 because of a
change in air gap flux in is given by
dv Ndtφ
= (2.20)
Kirchoff’s Voltage Law for the coil circuits is
Sdi du iR L Ndt dt
φ= + + (2.21)
where is the voltage drop caused by coil resistance, and iR sdiLdt
is the voltage drop
caused by coil self-inductance. Using the expression for the flux again from (2.10),
2
d K d idt N dt gφ ⎛ ⎞= ⎜ ⎟
⎝ ⎠ (2.22)
the expression in (2.21) becomes
2S
di K d iu iR Ldt dt x
⎛ ⎞= + + ⎜ ⎟⎝ ⎠
(2.23)
Thus, the voltages in Figure 12 are
11 1
12sdi K d iu Ri Ldt dt x
⎛= + + ⎜
⎝ ⎠1 ⎞⎟ (2.24)
22 2
22sdi K d iu Ri Ldt dt x
⎛= + + ⎜
⎝ ⎠2 ⎞⎟ (2.25)
32
2.6 The Nonlinear Model
There are two inputs to the system, and , which control the forces on the ro-
tor. The system has four outputs,
1u 2u
1 2 1, , ,x x i and , although for a sensorless control,
only the two currents and are measurable. In addition,
2i
1i 2i 1x and 2x are not independ-
ent since
1 0
2 0
x x xx x x
= −= +
(2.26)
where 0 / 2x δ ρ≡ − (Figure 12). The states of the nonlinear system will be represented
by the column vector defined as z
1
2
3 1
4 2
z xzz iz i
x⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
z (2.27)
Then
12 and
dx dxdx dxzdt dt dt dt
= − = − = =22z (2.28)
Using (2.26) and (2.27), the acceleration equation (2.19) can be written as
2 2
3 42
0 1 0 1
14 4 dK z K zzm x z m x z m⎛ ⎞ ⎛ ⎞
= −⎜ ⎟ ⎜ ⎟− +⎝ ⎠ ⎝ ⎠F+ (2.29)
The control voltage from (2.24) can be written as
33
( ) ( )
1 11 1
1
1 1 1 11 2
1 1
1 1 11 2
1 1
3 3 20 1 0 1
2
2 2
2 2
2 2
s
s
s
s
di K d iu Ri Ldt dt xdi K di K i dxRi Ldt x dt x dt
K di K i dxRi Lx dt x dt
K K2 3Rz L z z z
x z x z
⎛ ⎞= + + ⎜ ⎟
⎝ ⎠
= + + −
⎡ ⎤= + + −⎢ ⎥
⎣ ⎦⎡ ⎤
= + + +⎢ ⎥− −⎣ ⎦
(2.30)\
So,
( )( ) ( )
0 13 3 2
0 1 0 1
22 2S
x z K2 3 1z Rz z z u
L x z K x z
⎡ ⎤−= − − +⎢ ⎥
− + −⎢ ⎥⎣ ⎦ (2.31)
Similarly,
( ) ( )
2 22 2
2
2 2 2 22 2
2 2
2 2 22 2
2 2
4 4 20 1 0 1
2
2 2
2 2
2 2
s
s
s
s
di K d iu Ri Ldt dt xdi K di K i dxRi Ldt x dt x dt
K di K i dxRi Lx dt x dt
K K2 4Rz L z z z
x z x z
⎛ ⎞= + + ⎜ ⎟
⎝ ⎠
= + + −
⎡ ⎤= + + −⎢ ⎥
⎣ ⎦⎡ ⎤
= + + −⎢ ⎥+ +⎣ ⎦
(2.32)
( )( ) ( )
0 14 4 2
0 1 0 1
22 2S
x z K2 4 2z Rz z z u
L x z K x z
⎡ ⎤+= − + +⎢ ⎥
+ + +⎢ ⎥⎣ ⎦ (2.33)
Therefore, the nonlinear system is given by
34
( )( ) ( )
( )( ) ( )
1 22 2
3 42
0 1 0 1
0 13 3 2
0 1 0 1
0 14 4 2
0 1 0 1
4 4
2
2 2
2
2 2
d
S
S
z z
z FzK Kzm x z m x z m
x z Kz RzL x z K x z
x z Kz RzL x z K x z
=
⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟− +⎝ ⎠ ⎝ ⎠
2 3 1
2 4 2
z z u
z z u
⎡ ⎤−⎢ ⎥= − −
− ++
⎢ ⎥−⎣ ⎦⎡ ⎤+⎢ ⎥= − +
+ ++
⎢ ⎥+⎣ ⎦
(2.34)
This can also be represented as
( ), dF=z f z, u (2.35)
where
1
2
uu⎡ ⎤
= ⎢ ⎥⎣ ⎦
u , (2.36)
( ) ( )( ) ( )
( )( ) ( )
22 2
3 4
0 1 0 1
0 13 2 32
0 1 0 1
0 14 2 42
0 1 0 1
4 4
2,2 2
22 2
d
d
S
S
z
K z K z Fm x z m x z m
x zF K1
2
Rz z zL x z K x z
x z K
u
Rz z zL x z K x z
⎡ ⎤⎢ ⎥
⎛ ⎞ ⎛ ⎞⎢ ⎥− +⎜ ⎟ ⎜ ⎟⎢ ⎥− +⎝ ⎠ ⎝ ⎠⎢ ⎥⎢ ⎥
u
⎡ ⎤−= ⎢ ⎥− − +⎢ ⎥− + −⎢ ⎥⎢ ⎥⎣ ⎦
⎢ ⎥⎡ ⎤⎢ ⎥+− + +⎢ ⎥⎢ ⎥+ + +⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
f z, u (2.37)
2.7 The Linearized MIMO Model
The nonlinear system in (2.37) may be transformed into a fourth-order MIMO
linear system by applying the Jacobian transformation. For the purpose of linearization,
35
the operating point is chosen so that the rotor shaft is at rest midway between the two
coils with a current of amp through each coil. Each coil has a resistance of 0 1i = 8R =
ohms, and there is no initial disturbance force, that is, ( )0 0dF t = . In matrix form
( )
( )( )( )( )
( )( )( )( )
( )( )( )( )
10 0 0
20 0 00
30 10 0
40 20 0
0011
z t x t x t
z t x t x tt
z t i t i t
z t i t i t
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦
z (2.38)
and
( )( )( )
( )( )
10 00
20 0
88
u t Ri tt
u t Ri t
⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦u (2.39)
If the nonlinear system is represented by (2.35), then the linearized system is calculated at
the operating point ( )0 0 0, , dFz u using the Jacobian transformation:
( ) ( ) ( )0 0 0 0 0 0 0 0 0, , , , , ,d dd
d
F FF
FdF
δ δ δ∂ ∂ ∂
= + +∂ ∂ ∂
f z u f z u f z uz z u
z uδ (2.40)
Replacing the differential notation with the names of the variables themselves yields the more compact notation,
1 2 dF= +z Az + B u B (2.41)
where
( ) ( ) ( )0 0 0 0 0 0 0 0 01 2
, , , , , ,, ,d d
d
F FF
∂ ∂ ∂= = =
∂ ∂f z u f z u f z u
A B Bz u
dF∂
(2.42)
Nevertheless, it must be kept in mind that the variable names are differential quantities
representing the variation of the variable from its nominal operating point.
36
The following equations are derived from (2.37) at the operating point in (2.38)
and (2.39), where the explicit dependence on the current has been retained: 0i
( )
( )
( )
20 03 20 0
0 0 00 0
0 0 0
0 0
0 0 0
0 1 0 0
02 2
, ,20 0
2 220 0
2 2
d
S S
S S
Ki Ki Kimx mx mx
FKi x R
x K x L K x LKi x R
020
x K x L K x L
⎡ ⎤⎢ ⎥⎢ ⎥−⎢ ⎥
∂ ⎢ ⎥= = ⎢ ⎥− −∂⎢ ⎥+ +⎢ ⎥⎢ ⎥−⎢ ⎥+ +⎣ ⎦
f z uA
z(2.43)
( )0 0 0 0
10
0
0
0 00 0
, , 20
22
02
d
S
S
F xK x L
xK x L
⎡ ⎤⎢ ⎥⎢ ⎥
∂ ⎢ ⎥= = ⎢ ⎥+∂ ⎢ ⎥
⎢ ⎥⎢ ⎥
+⎣ ⎦
f z uB
u (2.44)
( )0 0 0
2
0, , 1/
00
d
d
F mF
⎡ ⎤⎢ ⎥∂⎢ ⎥= =⎢ ⎥∂⎢ ⎥⎣ ⎦
f z uB (2.45)
Note that from (2.43), linearization is using the Jacobian is theoretically possible as long
as the coil bias current is nonzero.
Define (as in Kucera [20])
20 03 20 0
2, ,2 2 2
0
0
ss i
Ki Ki K x Lk k Lx x
+= = =
x (2.46)
Then
( )
0 0 02
0 0 0 0
22 2 2
i
S S
Ki Ki x kx K x L x K x L L
= ⋅+ +
= (2.47)
37
and A, , and can be written as 1B 2B
1 2
0 1 0 0 0 02 00 00
1/1 0 00 01 000 0
s i i
i
i
k k km m m m
k RLL L
k RLL L
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥− ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥= = ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥− ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
A B ⎥=B (2.48)
and (2.41) can be written as
1 1
2 2
3 3
4 4
0 1 0 0 0 02 00 00
1/1 0 00 01 000 0
s i i
di
i
k k kz zm m mz z m
Fk Rz z LL Lz zk R
LL L
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥= + ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎢ ⎥− ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
1
2
uu
+ (2.49)
or
11
1 2
22
0 1 0 0 0 02 00 00
1/1 0 00 01 000 0
s i i
di
i
xk k k xxm m m x u mdi Fk R i udt LL L
idi k Rdt LL L
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= + ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎣ ⎦ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦
+ (2.50)
It will be useful later to have (2.50) available in the following equivalent equation
forms:
1 2
11 1
22 2
2 s i i
i
i
mx k x k i k i Fdiu Ri L k xdtdiu Ri L k xdt
d= + − +
= + +
= + −
(2.51)
38
1 2
1 11
2 22
2 1s i id
i
i
k k kx x i i Fm m m m
di k R ux idt L L Ldi k R ux idt L L L
= + − +
= − − +
= − +
(2.52)
In the first equation in (2.51), the roles of sk as a force-displacement factor and
as a force-current factor are clearly seen. Both of these factors depend on the bias cur-
rent, , the air gap size at equilibrium,
ik
0i 0x , and the properties of the electromagnets as
contained in the magnetic constant, . Note also that the force-displacement
factor occurs with a positive sign, thus contributing a destabilizing mode to the system. A
block diagram representation of the linearized MIMO is shown in
20K Nμ= A
Figure 17.
1S
ik
1L
R
–
+–
1S
R
1S
1S
1m+
ik
1i
2i
xx x
1didt
2didt
+–
++–
ik
ik
ik1u
2u
F
1L 2 sk
Coil 1
Coil 2
Rotor
dF1S
1L
R
–
+–
1S
R
1S
1S
1m+
ik
1i
2i
xx x
1didt
2didt
+–
++–
ik
ik
ik1u
2u
F
1L 2 sk
Coil 1
Coil 2
Rotor
1S1S
1L1L
RR
–
+–+–
1S1S
RR
1S
1S
1m++
1i
2i
xx x
1didt
2didt
+–+–
++–+–
ik
ik
ik1u
2u
dFdF
Coil 1
F
1L1L
Rotor
2 sk2 sk
Coil 2
Figure 17 Linearized MIMO System
39
2.8 The Linearized SISO Model
The fourth-order MIMO system above may be transformed into a third-order
SISO subsystem and a first-order SISO subsystem by a change of variables. Beginning
with (2.51), the equations may be rewritten as
( )
( ) ( )
1 2
1 2 1 2 1 2
2
2
s i d
i
mx k x k i i Fdu u R i i L i i k xdt
= + − +
− = − + − + (2.53)
Using the quantities 1x , 2x , , , , and , define 1i 2i 1u 2u
1 2 1 2 1 2
1 2 1 2 1 20 0 0
2 2
2 2
2
2
x x i i ux i u u
x x i i ux i u u
− − −= = =
+ + += = =
(2.54)
Then
1 2 1 2 1
0 1 2 0 1 2 0 1
2 2 22 2 2
2
2
x x x i i i u u ux x x i i i u u u= − = − = −= + = + = +
(2.55)
so
1 0 1 0 1 0
2 0 2 0 2 0
x x x i i i u u ux x x i i i u u u= − = + = += + = − = −
(2.56)
Substituting the values in the first row of (2.54) into (2.53) yields
2 2s i d
i
mx k x k i Fdiu Ri L k xdt
= + +
= + + (2.57)
or
40
2 2 1
1
s id
i
k kx x im m m
di k R
F
x i udt L L L
= + +
= − − + (2.58)
Equations (2.57) and (2.58) exhibit the roles that sk and play in the “negative stiff-
ness” and “damping” characteristics of an AMB system. Here it is evident that that a
positive position or current displacement accelerates the rotor shaft in the positive direc-
tion, destabilizing the system. The role that plays in dampening the system is exhibited
also: an increase in the velocity of the rotor tends to diminish any increase in the current
and hence in the acceleration.
ik
ik
Define the state variables as
1
2
3
z xzz i
x===
(2.59)
Then
1 2
2
3
2 2 1
1
s id
i
z x zk k
z x x i Fm m m
kdi Rz x idt L L L
u
= =
= = + +
= = − − +
(2.60)
With these definitions, the third-order subsystem can be written as
0 1 0 0 02 2 00 1/1 00
s i
d
i
k ku mm m
k RLL L
⎡ ⎤ ⎡ ⎤⎢ ⎥
F⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= + +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎣ ⎦
z z (2.61)
or
41
0 1 0 0 02 2 00 1/1 00
s i
d
i
x xk kx x u mm mdi ik Rdt LL L
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥
F⎡ ⎤ ⎡⎢ ⎥ ⎢ ⎥⎢ ⎥
⎤⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥= +⎢ ⎥
⎥+⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥⎢ ⎥⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎦
(2.62)
This can be written compactly as
1 2 du F= +z Az + B B (2.63)
where
1 2
0 1 0 0 02 2 00 , ,1 00
s i
i
k kmm m
k RLL L
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢⎢ ⎥= =⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎣ ⎦
A B 1/ ⎥= ⎥B (2.64)
The first-order subsystem is obtained by adding the last two equations in (2.51):
( ) (1 2 1 2 1 2du u R i i L i idt
)+ = + + + . (2.65)
or, substituting the values from (2.55),
00 0
diu Ri Ldt
= + . (2.66)
Rearranging yields
00
1di R idt L L
= − + 0u (2.67)
If is a constant, the first-order subsystem is independent of rotor position and is stable
with the solution
0u
42
( ) 0 01Rt
Li t e uR
α−
= + 2.68)
where α =
(
R− . In the steady state, and close to the equilibrium position, does not vary much and can be used in the calculation of
( )0 00 /i u 0i
sk and in (2.46) [20]. Block dia-gram representations of the linearized SISO subsystems are shown in Figure 18 and Figure 19.
ik
1S
1S
1m+
xx xF
2 sk
Rotor
1S
1L
R
ididtu
Coil
ik
––+ 2 ik
dF
1S
1S
1m++
xx xF
2 sk2 sk
Rotor
1S1S
1L1L
RR
ididtu
Coil
ikik
––+–+ 2 ik2 ik
dFdF
Figure 18 Third-Order SISO Subsystem
1S
1L
R
0i0di
dt0u+–
1S1S
1L1L
RR
0i0di
dt0u+–+–
Figure 19 First-Order SISO Subsystem
CHAPTER III
CONTROL WITH POSITION FEEDBACK
3.1 Control Objectives
The primary control objective for an active magnetic bearing with one DOF is to
regulate the position of the rotor shaft so that it remains midway between the two elec-
tromagnets. A second control objective is that the system be able to reject disturbances
caused by external force loads as well as vibrations resulting from the unbalance of the
rotor. A third objective is, as is the case in all physical systems, that the control system be
robust with respect to variations of the physical parameters of the system. Finally, it is
desired that the above objectives be realized in the absence of position measurement, that
is, using only current measurement. These objectives will be discussed in the next two
chapters.
Most AMBs operate with equal bias currents in the magnet coils, and this causes
the position midway between the two magnets to be an unstable equilibrium point. This
43
44
fact is evident by recalling that the negative stiffness depends on the bias current and
nominal air gap in the coils as given in (2.46):
2030 0
,2 2s iKi Kik k 0
2x x= = (3.1)
It is also evident from the open-loop transfer function for the linearized SISO system
given in (3.10), where it is readily observed that AMBs have a real pole in the RHP, and
are therefore inherently unstable. Any perturbation from the equilibrium point in a par-
ticular direction results in an inverse-square force in the same direction, causing the bear-
ing to physically contact the auxiliary bearing. In applications with very high rotational
speeds, this situation can result in catastrophic bearing failure. Therefore, AMBs must
have feedback control in order to operate.
Adding to the difficulty of controlling AMBs is the relative size of the force-
displacement parameter to other system parameters. For example, for the system that was
used for simulation in this thesis (Table I), the sizes of stiffness parameters are
142860, 100sk ki= = (3.2)
Also, the system matrix for the third-order SISO system is
0 1 00 1 02 20 62112 0 43.4780 526.32 42.1050
s i
i
k km m
k RL L
⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ − −⎢ ⎥⎣ ⎦− −⎢ ⎥⎣ ⎦
A (3.3)
It is clear from this matrix that the coefficient of the displacement x has a very strong ef-
fect on system dynamics and on system control.
45
To begin the discussion of control for an AMB, a block diagram of the open-loop
structure is depicted in Figure 20 using a current-controlled configuration. This simple
approach assumes that there are no dynamics associated with the electromagnet and that
the force from the magnet is directly proportional to the current in the coil. This assump-
tion has the advantage that the open-loop transfer function is of second order, so the plant
can be stabilized by a PD control when position feedback is used (Figure 21).
i
Electro-magnet Rotor
Force PositionxF
Current
Magnetic Bearing
i
Electro-magnet Rotor
Force PositionxF
Current
Magnetic Bearing
Figure 20 Current-Controlled Open-Loop Plant
In practice, it is desirable to keep a given rotor position independent of the change
of load, and an integrator is added to the feedback loop yielding a PID controller. The
current-controlled, PID feedback configuration is simple and is sufficient for less de-
manding applications. It is thus widely used in applications [33].
i
Electro-magnet Rotor
Force Position
xFCurrent
Magnetic Bearing
PID+–refx i
Electro-magnet Rotor
Force Position
xFCurrent
Magnetic Bearing
PID+–+–refx
Figure 21 Closed-Loop, Current-Controlled, Position-Sensed
46
In reality, the inductance of the magnet’s coil will resist any sudden changes in
current, so for demanding applications, a more precise model of the dynamics of the elec-
tromagnets must be developed. Such a model was already stated in the second equation in
(2.57):
idiu Ri L k xdt
= + + (3.4)
It is clear from this that in order to control the current, and hence the force on the rotor,
the voltage must be controlled. Therefore, the voltage-controlled configuration in Figure
22 is the one that will be used for the remainder of this work.
uElectro-magnet Rotor
Force PositionxF
Voltage
Magnetic Bearing
uElectro-magnet Rotor
Force PositionxF
Voltage
Magnetic Bearing
Figure 22 Voltage-Controlled Open-Loop Plant
Voltage control has the advantage of yielding a more accurate model, but it has
the drawback that, by including the first-order dynamics of the coil in the open-loop plant
model, the order of the plant is raised to three. Thus the controller must be at least of or-
der two, prohibiting the use of PID control. A PD2 controller is the simplest satisfying
this requirement, but it would be very susceptible to sensor noise. For this reason, state
feedback is often used.
47
uElectro-magnet Rotor
Force Position
xFVoltage
Magnetic Bearing
Controller+–refx uElectro-magnet Rotor
Force Position
xFVoltage
Magnetic Bearing
Controller+–+–refx
Figure 23 Closed-Loop, Voltage-Controlled, Position-Sensed
3.2 Open-Loop Transfer Functions
It is useful in what follows to have the transfer function for the position-sensed
system available. Superposition will be used in order to determine the open-loop transfer
function from u to x for the SISO system. The state-space representation was derived in
(2.62) and is repeated here for convenience:
0 1 0 0 02 2 00 1/1 00
s i
d
i
x xk kx x u mm mdi ik Rdt LL L
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥
F⎡ ⎤ ⎡⎢ ⎥ ⎢ ⎥⎢ ⎥
⎤⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥= +⎢ ⎥
⎥+⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥⎢ ⎥⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎦
(3.5)
or
1 2 du F= +z Az + B B . (3.6)
Setting the disturbance force 0dF = , the open-loop transfer function from u to x can be
derived from
48
( )( ) ( ) ( )
11 3 2 2
22 2
i
i s s
X s ksU s mLs mRs k Lk s k R
−= − =+ + − −
C I A B (3.7)
A block diagram of (3.7) is shown in Figure 24.
xx x1S
u1S
1S
x
–+
++
mR
2 ik
( )22 s iLk k−
1mL
2 sk R
xx x1S1S
u1S1S
1S1S
x
–+
++–+
++
mRmR
2 ik2 ik
( )22 s iLk k−( )22 s iLk k−
1mL1
mL
2 sk R2 sk R
Figure 24 Block Diagram of Equation (3.7)
Similarly, setting , the effect the open-loop transfer function from 0u = dF to x
can be determined:
( )( ) ( ) ( )
12 3 2 22 2d i s s
X s Ls RsF s mLs mRs k Lk s k R
− += − =
+ + − −C I A B (3.8)
For reference, the following physical and derived parameters for the AMB used in
the simulations are listed next. They are the same as used in Kucera [20].
Table I PHYSICAL PARAMETERS FOR THE AMB
Parameter Name Parameter
Symbol
Value Unit
Nominal Air Gap 0x 0.0007 m
Bias Current 0i 1 A
Coil Resistance R 8 Ω
49
Rotor Mass m 4.6 kg
Coil Self-Inductance sL 0.120 H
Magnetic Constant K 59.8 10−× H m⋅
Static Disturbance Force dF 4.6 N
Table II DERIVED PARAMETERS FOR THE AMB
Parameter Name Parameter
Symbol
Formula Value Unit
Force-Displacement Constant sk 20302
Kix
142860 N/m
Force-Current Constant ik 0202
Kix
100 N/A
Total Inductance L 0
0
22
sK x Lx
+ 0.190 mH
For the above parameters, the open-loop transfer function in (3.7) is given by
( )( ) 3 2 4
2000.874 36.8 3429 10 2.286 10
X sU s s s s
=+ − × − × 6 , (3.9)
or, in zero-pole-gain form
( )( ) ( ) ( ) ( )
228.8207.6 179.5 70.19
X sU s s s s
=− + +
. (3.10)
Also, from (3.8)
( )( )
( )( ) ( ) ( )
0.21739 42.11207.6 179.5 70.19d
X s sF s s s s
+=
− + +. (3.11)
50
For the MIMO system, only the numerical open-loop transfer function using the
parameters from the tables will be given in order to save space. The state-space represen-
tation was derived in (2.50) and is repeated here for convenience:
11
1 2
22
0 1 0 0 0 02 00 00
1/1 0 00 01 000 0
s i i
di
i
xk k k xxm m m x u mdi Fk R i udt LL L
idi k Rdt LL L
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= + ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎣ ⎦ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦
+ (3.12)
or
1 2 dF= +z Az + B u B . (3.13)
In this case
(3.14)
0 1 062112 0 21.739 21.739
0 526.32 42.105 00 526.32 0 42.105
⎡ ⎤⎢ ⎥−⎢ ⎥=
− −⎢ ⎥⎢ ⎥−⎣ ⎦
A
0
⎤⎥⎥⎥⎥⎦
(3.15) 1 2
0 0 00 0 0.217391
,5.2632 0 0
0 5.2632 0
⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢= =⎢ ⎥ ⎢⎢ ⎥ ⎢⎣ ⎦ ⎣
B B
So
( ) ( )
11 12
1 21 221
31 32
41 42
( ) ( )( ) ( )1( ) ( )( )( ) ( )
G s G sG s G s
s sG s G sD sG s G s
−
⎡ ⎤⎢ ⎥⎢ ⎥= − =⎢ ⎥⎢ ⎥⎣ ⎦
G C I A B (3.16)
where
(3.17) ( ) ( 207.6) ( 179.5) ( 70.19) ( 42.11)D s s s s s= − + + +
51
(3.18)
11
21
31
41
( ) 114.4165 ( 42.11)( ) 114.4165 ( 42.11)( ) 5.2632 ( 229 ) ( 219 ) ( 52.15)( ) 60219.1979
G s sG s s sG s s s sG s s
= += += − + +=
(3.19)
12
22
32
42
( ) 114.4165 ( 42.11)( ) 114.4165 ( 42.11)( ) 60219.1979 ( ) 5.2632 ( 229 ) ( 219 ) ( 52.15)
G s sG s s sG s sG s s s s
= − += − +== − + +
3.3 Controllability and Observability
The open-loop transfer function for the SISO system in (3.7) is irreducible, there-
fore it may be deduced that, if the disturbance force is not included in the dynamics of the
system, the position-sensed AMB is controllable and observable. If the disturbance force
is constant, that is, , and if ( ) 0dF t = ( )dF t is included in the dynamics as a state, then
(2.62) becomes
0 1 0 0 02 2 1 00
10 0
00 0 0 0
s i
i
dd
x xk kx xm m m udi ik Rdt LL L FF
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ + ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦
(3.20)
Controllability
Letting
52
[
0 1 0 0 02 2 1 00
, , and 1 0 0 010 0
00 0 0 0
s i
d d di
k km m m
k RLL L
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= = =⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦
A B C ] , (3.21)
the controllability matrix is given by
2 3d d d d d d d d⎡ ⎤= ⎣ ⎦CM B A B A B A B (3.22)
( )
2
2 2 2
2 3 2
2 2 2 3
2 3 4
2 20 0
2 2 22 20
1 2 4
0 0 0 0
i i
i s ii i
d
i i
k RkLm L m
k k L k L mRk RkLm L m L m
R mR Lk LRk mRL L L m L m
⎡ ⎤−⎢ ⎥⎢ ⎥
− +⎢ ⎥−⎢ ⎥=
⎢ ⎥− −⎢ ⎥−⎢ ⎥
⎢ ⎥⎣ ⎦
CM (3.23)
Thus
( )rank 3d =CM , (3.24)
so the augmented system with a static load disturbance is not controllable.
Using the numerical values for the AMB and Matlab’s ctrbf() function, the
system can be transformed to controllability staircase form in which the system is parti-
tioned into its controllable and uncontrollable parts:
[21
,ucuc c
c c
⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
A 0 0A B = , C = C
A A B]C
)
, (3.25)
where is controllable. The function ctrbf() also returns k, the number of
controllable states. In the present case, Matlab returns
( c cA , B
3k = and
53
0 0 0 0 00 0 1 0 0
,0.21739 62112 0 43.478 0
0 0 526.32 42.105 5.2632
⎡ ⎤⎢ ⎥⎢ ⎥=− −⎢ ⎥⎢ ⎥− −⎣ ⎦
A B
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
= (3.26)
[ ]0 1 0 0C = . (3.27)
So
(3.28) 0 1 0 0
62112 0 43.478 , 00 526.32 42.105 5.2632
c
⎡ ⎤⎢ ⎥= −⎢ ⎥
− −⎢ ⎥⎣ ⎦
A c
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
B =
and
[ ]1 0 0cC = (3.29)
Since this is the same form as the non-augmented system, the conclusion is that all of the
original states are controllable, and that the augmented state is not controllable. This
is, of course, the result of the fact that the disturbance was defined to be constant.
dF
Observability
On the other hand, the observability matrix for the augmented system is given by,
2 3 T
d d d d d d d d⎡ ⎤= ⎣ ⎦OM C C A C A C A , (3.30)
( )2
1 0 00 1 0
2 20
2 20 0
sd
s i i
km m
Lk k RkLm Lm
001ikm
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥
−⎢ ⎥−⎢ ⎥⎣ ⎦
OM (3.31)
and
54
( )rank 4d =OM , (3.32)
so the augmented system is observable.
For the MIMO system in (3.16), Matlab’s ctrb() function was used to compute
the controllability matrix in the absence of disturbance dynamics.
2 31 1 1 1⎡ ⎤⎣ ⎦CM = B AB A B A B (3.33)
0 0 0 0 114.42 114.42 4817.54 4817.540 0 114.42 114.42 4817.54 4817.54 4691231 4691231
5.2631 0 221.61 0 50888.4 60219.2 4678214 5071090
0 5.2631 0 221.61 60219.2 50888.4 5071090 4678214
− −⎡ ⎤⎢ ⎥− − −⎢ ⎥=
− − −⎢ ⎥⎢ ⎥− − −⎣ ⎦
CM
( )rank 4=CM (3.34)
Therefore, the MIMO system is also controllable.
3.4 State Feedback and Stability
In order to stabilize the AMB system, one of the simplest controllers to design is
state feedback with pole-placement using Matlab’s place() function. This also serves
as a benchmark for other controller designs. The system matrix for the MIMO AMB has
the following open-loop poles:
( ) { }207.58, 179.49, 70.195, 42.105λ = − − −A (3.35)
55
The desired closed-loop poles were chosen to be larger in magnitude than the open-loop
poles but otherwise somewhat arbitrary. Let ( )0 maxλ λ= A . Then
( ) { }{ }
1 0 0 0 02 , 3 , 4 , 5
415.15, 622.73, 830.3, 1037.9
λ λ λ λ λ− = − − − −
= − − − −
A KB (3.36)
With these poles Matlab returns the following gain matrix:
(3.37) 6
6
1.7201 10 5959.7 267.16 117.451.7518 10 6057.2 119.19 268.99Place
⎡ ⎤×= ⎢ ⎥− × − −⎣ ⎦
K−
Simulation – Static Load Disturbance Rejection
The Simulink model and the results of the simulation are shown in Figure 25 and
Figure 26. A static force disturbance of 4.6 N was applied at 0.1 second. Since the mass
of the rotor is 4.6 kg, the disturbance force would yield an acceleration of the rotor of 1
m/s2 if it were not controlled.
u1
u2
Fd2x4
2x14x1
lin _state_pfb_1_p.mdl
OutputMIMOLinearMag Bearing
x' = Ax+Bu y = Cx+Du
K_place
K_place
Disturbance
Fd
Control
[0 0 0 0 ]
Figure 25 Simulink Model of MIMO System with State Feedback
56
As can be seen from the figures, the state feedback stabilizes the AMB, but there
is a large steady-state error in the position. This is because the closed-loop transfer func-
tion from to x is dF
( )( )
2 5
4 3 6 2 90.2174 631.8 3.747 10
2906 3.059 10 1 .377 10 2.228 10d CL
X s s sF s s s s s
+ + ×=
+ + × + × + × 11 (3.38)
Since the state feedback has forced the other contributions to x from the four reference
inputs to zero, the expected offset from dF in steady-state is
5 5
611 11
3.747 10 3.747 10 4.6 7.736 102.228 10 2.228 10dF −× ×
= ⋅ =× ×
× (3.39)
The zoomed view in Figure 27 shows this offset precisely.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
2
4
6
8x 10
-6 Position: x ( m )
max = 7.7362e-006min = 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
-3 Velocity: dx/dt ( m / s )
max = 0.0012957min = 0
Time( sec )
lin_state_pfb_1_para.m
Figure 26 Step Disturbance, State FB, 4.6 N Load
57
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.457.73
7.732
7.734
7.736
7.738
7.74x 10
-6 Position: x ( m )
Figure 27 Zoomed View of Figure 26
Integral Feedback Control
In order to remove the steady state error, it is necessary to add some dynamics to
the state feedback gain matrix, namely, integral feedback in the position terms. After a
small amount of tuning, the integral feedback yields the matrix in (3.40) and the results
are shown in Figure 28 and Figure 29. The integral feedback removes the steady-state
error and does so with very small currents and control voltages.
6 7
6 7
1.7201 10 7.7 10 5959.7 267.16 117.45
1.7518 10 7.7 10 6057.2 119.19 268.99Place
ssss
⎡ ⎤× + ×−⎢ ⎥
= ⎢ ⎥× + ×⎢ ⎥− − −⎢ ⎥⎣ ⎦
KI (3.40)
The reason for the removal of the steady-state error after the addition of the inte-
gral feedback becomes clear when closed-loop transfer function from dF to x is exam-
ined again:
( )( )
( ) ( )( ) ( ) ( ) ( ) ( )
0.21739 2076 830.3
111.1 181.7 830.3 856.2 926.8
d CL
X s s s sF s s s s s s
+ +=
+ + + + + (3.41)
Clearly the s in the numerator forces the contribution of dF to the position error to zero.
58
The response after integral feedback is added is shown in Figure 28 and Figure
29. Note that the peak response is about 66.5 10−× m, and the settling time is about 0.07
second.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
1
2
3
4
5
6
7x 10
-6 Position: x ( m )
max = 6.4519e-006min = 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-5
0
5
10
15x 10
-4 Velocity: dx/dt ( m / s )
max = 0.0012906min = -0.00033013
Time( sec )
lin_state_pfb_int_1_para.m
Figure 28 Step Dist, State & Integral FB, 4.6 N Load
The disturbance force was raised from 4.6 N to 250 N without changing the con-
trol design. The system was able to reject this large disturbance very well (Figure 30).
The initial rotor displacement was less than 0.0004 m, remaining well within the 0.0007
m air gap. The current remained low (Figure 31), while the voltage spiked momentarily
(for about 0.01 s) to about 140 V. This should cause no overheating in the coil. The
steady-state voltage settled at about 10 volts. These values do not seem very high, but
more knowledge of the cooling capacity of the bearing, the linear range for the magnetic
properties, etc. would be needed to accurately determine limits on these values.
59
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.04
-0.02
0
0.02
0.04i1, i2 ( A )
max = 0.037016min = -0.036348
i1i2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-3
-2
-1
0
1
2
3u1, u2 ( V )
Time( sec )
max = 2.5122min = -2.4679
u1
u2
lin_state_pfb_int_1_para.m
Figure 29 Step Dist, State & Integral FB, 4.6 N Load
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
1
2
3
4x 10
-4 Position: x ( m )
max = 0.00035064min = 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.02
0
0.02
0.04
0.06
0.08Velocity: dx/dt ( m / s )
max = 0.070142min = -0.017942
Time( sec )
lin_state_pfb_int_1_para.m
Figure 30 Step Dist, State & Integral FB, 250 N Load
60
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-2
-1
0
1
2
3i1, i2 ( A )
max = 2.0117min = -1.9755
i1i2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-150
-100
-50
0
50
100
150u1, u2 ( V )
Time( sec )
max = 136.531min = -134.1254
u1
u2
lin_state_pfb_int_1_para.m
Figure 31 Step Dist, State & Integral FB, 250 N Load
With the force disturbance returned to 4.6 N, all of the AMB’s physical parame-
ters were varied between half and twice their normal sizes. The simulated output re-
mained bounded for all of the trials except for when the nominal air gap 0x was doubled.
In that case, the simulated output became unbounded. Overall, state feedback with inte-
gral position feedback gave a very robust response with respect to these gross changes.
These simple tests were performed on the system to get an idea of the control system’s
ability to maintain a bounded output in the presence of a large variation in one of the
physical parameters. The results are summarized in the following table.
Table III PARAMETER SENSITIVITY – STATE FEEDBACK
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Unbounded Bounded
61
Bias Current 0i 1 Bounded Bounded
Coil Resistance R 8 Bounded Bounded
Rotor Mass m 4.6 Bounded Bounded
Coil Self-Inductance sL 0.120 Bounded Bounded
Magnetic Constant K 59.8 10−× Bounded Bounded
Static Disturbance Force
Max Value dF 4.6 250 N* 2.01 A**
137 V
* Load at which rotor has maximum displacement of about m 44 10−×
** Max current and voltage for max disturbance
Sinusoid Load Disturbance Rejection
Rotating machinery is never perfectly balanced. The unbalance condition in rigid
rotors can be modeled by a sinusoidal load disturbance, and this was done in the presence
of the state feedback plus integral control described above without altering the gains. At a
low frequency disturbance of 10 Hz (comparable to 600 RPM of the rotor), the sinusoi-
dal disturbance is present in the movement of the rotor at a fairly small amplitude of
about 10 percent of the rotor air gap (Figure 32). This can be attributed to the fact that the
integral factor in the state feedback did not have enough time between cycles to cancel
the sinusoid. At a higher frequency of 1000 Hz (comparable to about 60,000 RPM), there
was substantial attenuation of the sinusoid to about 0.05 percent of the air gap, probably
caused by the low-pass quality of the AMB system and integrator (Figure 34). A fre-
quency analysis of the AMB system and its control is required and listed under the Future
Work section of Chapter V in this thesis. More knowledge of the affect of these low am-
plitude vibrations on the material composition of the rotor is needed to asses the quality
of the control design. It is possible that the resonant modes of the rotor could be excited
62
and that the small amplitude oscillations could grow to an unacceptable size. The fre-
quency response from the position reference input to the position output is shown in
Figure 36. The closed-loop bandwidth is about 300 rad/s (48 Hz or 2865 RPM).
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1
-0.5
0
0.5
1x 10
-5 Position: x ( m )
max = 7.6e-006min = -7.6e-006
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5x 10
-4 Velocity: dx/dt ( m / s )
max = 0.00047752min = -0.00047752
Time( sec )
lin_state_pfb_1_para.m
Figure 32 Sine Dist, State & Integral FB (10 Hz Dist.)
63
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.04
-0.02
0
0.02
0.04i1, i2 ( A )
max = 0.034631min = -0.033858
i1i2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5
0
0.5u1, u2 ( V )
Time( sec )
max = 0.46366min = -0.45281
u1
u2
lin_state_pfb_1_para.m
Figure 33 Sine Dist, State & Integral FB (10 Hz Dist.)
0 0.005 0.01 0.015 0.02 0.025 0.03-0.5
0
0.5
1
1.5
2
2.5x 10
-7 Position: x ( m )
max = 2.3168e-007min = -2.4843e-008
0 0.005 0.01 0.015 0.02 0.025 0.03-4
-2
0
2
4x 10
-4 Velocity: dx/dt ( m / s )
max = 0.0003054min = -0.00020029
Time( sec )
lin_state_pfb_1_para.m
Figure 34 Sine Dist, State & Integral FB (1,000 Hz Dist.)
64
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-3
-2
-1
0
1
2
3x 10
-3 i1, i2 ( A )
max = 0.0028117min = -0.0027658 i1
i2
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-1.5
-1
-0.5
0
0.5
1
1.5u1, u2 ( V )
Time( sec )
max = 1.2895min = -1.2682
u1
u2
lin_state_pfb_1_para.m
Figure 35 Sine Dist, State & Integral FB (1,000 Hz Dist.)
65
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
Mag
nitu
de (
dB)
101 102 103 104-270
-225
-180
-135
-90
-45
0
Pha
se (d
eg)
Bode DiagramGm = 16 dB (at 1.16e+003 rad/sec) , Pm = 98.4 deg (at 318 rad/sec)
Frequency (rad/sec)
Figure 36 Frequency Response of AMB
Finally, it should be noted that much more tuning of the state feedback gains is
warranted. The poles were placed arbitrarily in the above case, but the poles should be
moved much further to the left in the LHP to investigate the impact on the performance
of the state feedback control, and to make the comparison with the other controls investi-
gated in this thesis more meaningful.
66
3.5 H-Infinity Control
An H-Infinity controller was designed using Matlab’s hinfsyn()function for
the MIMO AMB system with full state information. Integral feedback control on the po-
sition was also needed because H-Infinity controllers do not contain an integral term [4].
Interconnection Matrix
In order to use the hinfsyn()function, Matlab presumes that the control system
is represented in a general control configuration for a system with feedback control as
shown in Figure 37.
P
K
w
u v
qP
K
w
u v
q
Figure 37 General Control Configuration
In this representation,
- is the system interconnection structure transfer function matrix ( )s=P P
- K is the control structure to be designed using hinfsyn()
- w represents the exogenous input to the system
- u represents the control input to the system
- q represents the error to be kept small
- v represents the output measurement provided to the controller.
In matrix form
67
( )s⎡ ⎤ ⎡=
⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣
q wP
v u ⎦. (3.42)
with
=v Ku (3.43)
The overall control objective is to minimize the H-Infinity norm of the transfer
function from w to q. The controller design problem is, then, to find a controller K,
which, based on information in v, generates a control signal u which counteracts the in-
fluence of w on q, thereby minimizing the closed-loop norm from w to q: [35]
min max ∞
∞
⎛ ⎞⎜⎜⎝ ⎠K w
qw ⎟⎟ . (3.44)
The procedure to represent the AMB system with the correct interconnection ma-
trix ( )sP requires careful preparation. Recall that the MIMO system has the following
representation:
1 2 dF= +z Az + B u B (3.45)
1 2
0 1 0 0 0 02 00 00
1/1 0 00 01 000 0
s i i
i
i
k k km m m m
k RLL L
k RLL L
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤⎢ ⎥− ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥= = ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥− ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
A B ⎥=B (3.46)
Matlab requires an interconnection matrix P that satisfies the following:
68
⎡ ⎤⎡ ⎤ ⎡⎢
⎤⎥⎢ ⎥ = ⎢⎢ ⎥⎥⎢ ⎥
⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣⎣ ⎦
1 2
1 11 12
2 21 22
z A B B z
⎦
q C D D wv C D D u
P
(3.47)
Here A is the system matrix for the MIMO AMB, z is the state, and q, v, w, and u are as
described above. As is often the case in control theory, these variables play a dual role: in
(3.42) they are variables in the Laplace s-domain, while in (3.47) they represent variables
in the time domain. Note that the symbol P is used to indicate the state-space realization
of the interconnection matrix ( )sP . Matlab’s hinfsyn() function requires the state-
space realization P of . Because of certain theoretical considerations described in ( )sP
[35], it can be assumed that
11
22
==
DD 0
0 (3.48)
Also, because of the order in which Matlab expects the input u and the disturbance w, the
following hold:
1
2 1
2=
=
B BB B
(3.49)
With this assumption, (3.47) becomes
2 1
1 11 12
2 21 22
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
z A B B zq C D D wv C D D u
P
(3.50)
or in equation form
69
2 1
1 11 12
2 21 22
= + += + += + +
z Az B w B uq C z D w D uv C z D w D u
(3.51)
The vectors are given explicitly for the AMB system by:
(3.52)
1 11
2 22
1 1 3 13
2 2 4 24
dFz x z x
rz x z x u
ru u z i u
ru u z i
r
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = = = = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
w q v u 1
2
These expressions indicate that the quantity w is the input to the system including the dis-
turbance and the reference inputs; q is the output error to be minimized; v is full state (or
full information ) feedback; and u is the control input.
H-Infinity control design also requires that the inputs and outputs be scaled to
have maximum values of one. These scaled values are represented by the subscript s in
Figure 38. The quantities in Equation (3.51) and the scaling matrices and wD qD are also
represented in Figure 38. Details of these and the other matrices shown follow.
70
Scaled Model – hinfsyn() P
+
A
2B
4x1
1B
4x2
4x4
4x4
2x1
4x1
4x5
4x2
++
4x5 11D
12D
2C
zz
( )sK2x4
4x4
1C
G
++
sq
v
u
4x14x14x4
4x1
++1S
4x521D
++
4x2
22D1
2
3
4
d s
s
s
s
s
Frrrr
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1
2
uu⎡ ⎤⎢ ⎥⎣ ⎦
1
2
s
s
s
s
xxuu
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1
2
3 1
4 2
r xr xr ir i
−⎡ ⎤⎢ ⎥−⎢ ⎥
−⎢ ⎥⎢ ⎥−⎣ ⎦
4x1
4x1
4x1
ws
wDw
5x55x1
1q−D
5x1
4x4
lin_hinfsyn_fi_int_1_para.m
Figure 38 Interconnection Matrix Design for Full Information Feedback
Matrices A, B1, and B2 were given in (3.46). The P interconnection matrix is
shown inside the outer dotted line, and the original AMB system is represented by G in-
side the inner dotted line. The remaining matrices are determined by setting up the fol-
lowing equations to match the equations in the figure.
1 11 12= + +q C z D w D u (3.53)
1
12
1 13
2 24
1 11
1 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 1
dFx x
r
2
12
x x ur
u ir
u ir
⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦q C D
u
D
(3.54)
71
2 21 22= + +v C z D w D u (3.55)
(3.56)
11
2 12
3 1 13
4 2 24
2 21
1 0 0 0 0 1 0 0 0 0 00 1 0 0 0 0 1 0 0 0 00 0 1 0 0 0 0 1 0 0 00 0 0 1 0 0 0 0 1 0 0
dFr x x
rr x x u
rr i i
rr i i
r
⎡ ⎤− −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢= + +⎢ ⎥−⎢ ⎥ − ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦C Dv
2u⎡ ⎤⎢ ⎥⎣ ⎦
22
⎤⎥⎥⎥⎥⎦
D
Scaling
The scaling matrices are formed using information about the expected maximum
size of the inputs and outputs of the interconnection matrix. For the AMB under consid-
eration, the following values for these parameters are assumed:
Table IV SCALING PARAMETERS
Scaling
Parameter
Magnitude
Requirement Value Unit
_ dFσ Force Disturbance 200 N
_ xσ Displacement 0.0003 m
_ xσ Velocity 0.2 m/s
1_ iσ Coil Current 1 10 A
2_ iσ Coil Current 2 10 A
1_ uσ Coil Voltage 1 160 V
2_ uσ Coil Voltage 2 160 V
The scaling matrices are then given by
72
(3.57)
1
2
_ 0 0 0 00 _ 0 0 00 0 _ 0 00 0 0 _ 00 0 0 0 _
dFx
xi
i
σσ
σσ
σ
⎡ ⎤⎢ ⎥⎢ ⎥
= ⎢⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
wD ⎥
1
2
_ 0 0 00 _ 0 00 0 _ 00 0 0 _
xx
uu
σσ
σσ
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
qD (3.58)
Simulation – Static Load Disturbance Rejection
The Simulink model that was used for the H-Infinity full information control is
the same as that shown in Figure 25 except that the controller generated by the
hinfsyn() function is an LTI system instead of a state feedback gain matrix as in the
previous simulation. The function returns a stabilizing H-Infinity optimal LTI controller
K for the partitioned LTI plant P . The controller has the same number of states as P and
in general is not unique. The algorithm employed uses the two-Riccati formulae with
loopshifting [16][10][47]. This method uses a bisection algorithm to iterate on a cost pa-
rameter γ to determine the optimal value of γ in an effort to approach the optimal H-
Infinity controller K. Initially, the goal is to find the controller K that minimizes γ in the
relation
max γ∞
∞
<w
qw
(3.59)
The value of gamma achieved with the hinfsyn()function is 0.8852γ = , and the con-
troller returned is
73
( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )
1 2 3 3
1 2 3 3
1 k s k s k s k ss
k s k s k s k sD s−⎡ ⎤
= ⎢ ⎥− − −⎣ ⎦K (3.60)
where
(3.61)
( )( )( )
( )
12 2 14 161
9 2 11 132
5 2 8 93
3 6 2 9
1.096 10 3.027 10 1.544 10
1.033 10 4.07 10 1.637 10
4.804 10 1.367 10 6.117 10
3.879 10 3.557 10 1.23 10
k s s s
k s s s
k s s s
D s s s s
= × + × + ×
= × + × + ×
= × + × + ×
= + × + × + × 12
When integral control is added, the only change in K occurs in column 1:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 3 3
1 2 3
1k s k s k s k s
ssD s k s k s k s k s
s
α
α3
⎡ ⎤+ −⎢ ⎥⇒ ⎢ ⎥
⎢ ⎥− − − −⎢ ⎥⎣ ⎦
K (3.62)
where α is the integral gain.
The result of the simulation for a 4.6 N static load disturbance with integral feed-
back control added to the position feedback loop is shown in Figure 39. The obvious dif-
ference between the state feedback control and H-Infinity control is the much longer set-
tling time, from about 0.07 second in the former to about 0.4 second in the latter. In addi-
tion, the peak displacement increased from about 66.5 10−× to . 515 10−×
74
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-5
0
5
10
15x 10
-5 Position: x ( m )
max = 0.00014604min = -1.225e-007
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1
0
1
2
3
4
5
6x 10
-3 Velocity: dx/dt ( m / s )
max = 0.0058326min = -0.00098703
Time( sec )
lin_state_pfb_1_para.m
Figure 39 Step Dist, H-Infinity & Integral Ctrl, 4.6 N Load
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3i1, i2 ( A )
max = 0.23279min = -0.23279
i1i2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-3
-2
-1
0
1
2
3u1, u2 ( V )
Time( sec )
max = 2.0157min = -2.0157
u1
u2
lin_state_pfb_1_para.m
Figure 40 Step Dist, H-Infinity & Integral Ctrl, 4.6 N Load
75
As was done for the state feedback control, the physical parameters of the AMB
were doubled and halved while keeping the static load disturbance at 4.6 N. For the state
feedback control the system became unbounded only when the nominal air gap was dou-
bled. The H-Infinity controller, however, was more sensitive to parameter changes as
seen in the table. Most important, the maximum load disturbance which displaced the ro-
tor by 0.0003 m was only 12 N. This implies an extremely weak stiffness, especially
when compared to the 250 N disturbance rejection of the state feedback control. This
control as it is currently tuned could only be used in the least demanding of applications
with respect to stiffness. However, since the minimum value of gamma was used, further
simulations need to be performed with larger values of gamma before a fair comparison
can be made to other controllers. The current simulations only represent the most initial
tests of performance.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1
0
1
2
3
4x 10
-4 Position: x ( m )
max = 0.00038096min = -3.1957e-007
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-5
0
5
10
15
20x 10
-3 Velocity: dx/dt ( m / s )
max = 0.015215min = -0.0025749
Time( sec )
lin_state_pfb_1_para.m
Figure 41 Step Dist, H-Infinity & Integral Ctrl, 12 N Load
76
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1
-0.5
0
0.5
1i1, i2 ( A )
max = 0.60727min = -0.60727
i1i2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-6
-4
-2
0
2
4
6u1, u2 ( V )
Time( sec )
max = 5.2037min = -5.2037
u1
u2
lin_state_pfb_1_para.m
Figure 42 Step Dist, H-Infinity & Integral Ctrl, 12 N Load
Table V PARAMETER SENSITIVITY – H-INFINITY FULL INFORMATION
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Bounded Unbounded
Bias Current 0i 1 Unbounded Bounded
Coil Resistance R 8 Unbounded Unstable
Rotor Mass m 4.6 Bounded Bounded
Coil Self-Inductance sL 0.120 Unbounded Bounded
Magnetic Constant K 59.8 10−× Bounded Bounded
Static Disturbance Force
Max Value dF 4.6 12 N 0.607 A
5.20 V
Sinusoidal Disturbance Rejection
77
As was the case with state feedback, the H-Infinity control is completely unable to
reject disturbance at low frequency (Figure 43). At high frequency (Figure 45), there is a
strong attenuation of the disturbance as was the case for state feedback. The control volt-
age also oscillates at the frequency of the disturbance, but this, of course, is necessary to
reject the sinusoid. Although the amplitude is quite low, the model does not take into ac-
count any high frequency dynamics associated with the nonlinearities of the electromag-
netic system. This should be investigated in a more advanced study. Overall, the per-
formance of the H-Infinity would be judged as weak, but no conclusions should be drawn
without further tuning of the control, specifically, relaxing of the size of gamma, and
changing the weighting matrices (the Ds) in the interconnection matrix.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1.5
-1
-0.5
0
0.5
1
1.5x 10
-4 Position: x ( m )
max = 0.00011612min = -0.00012955
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.01
-0.005
0
0.005
0.01Velocity: dx/dt ( m / s )
max = 0.0071952min = -0.0074297
Time( sec )
lin_state_pfb_1_para.m
Figure 43 Sine Dist, H-Infinity & Integral Ctrl (10 Hz)
78
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3i1, i2 ( A )
max = 0.18726min = -0.18726
i1i2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-3
-2
-1
0
1
2
3u1, u2 ( V )
Time( sec )
max = 1.9918min = -1.9918
u1
u2
lin_state_pfb_1_para.m
Figure 44 Sine Dist, H-Infinity & Integral Ctrl (10 Hz)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-2
0
2
4
6
8
10x 10
-7 Position: x ( m )
max = 9.5362e-007min = -1.7557e-007
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-2
-1
0
1
2
3
4x 10
-4 Velocity: dx/dt ( m / s )
max = 0.00031802min = -0.00019499
Time( sec )
lin_state_pfb_1_para.m
Figure 45 Sine Dist, H-Infinity & Integral Ctrl (1000 Hz)
79
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
0.2
0.4
0.6
0.8
1x 10
-6 Position: x ( m )
max = 9.0204e-007min = 0
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-2
-1
0
1
2
3
4x 10
-4 Velocity: dx/dt ( m / s )
max = 0.00031802min = -0.00012249
Time( sec )
lin_state_pfb_1_para.m
Figure 46 Sine Dist, H-Infinity & Integral Ctrl (1000 Hz)
0 0.01 0.02 0.03 0.04 0.05 0.06-2
-1
0
1
2x 10
-3 i1, i2 ( A )
max = 0.0017191min = -0.0017191
i1i2
0 0.01 0.02 0.03 0.04 0.05 0.06-0.1
-0.05
0
0.05
0.1u1, u2 ( V )
Time( sec )
max = 0.096583min = -0.096583
u1
u2
lin_state_pfb_1_para.m
Figure 47 Sine Dist, H-Infinity & Integral Ctrl (1000 Hz)
80
3.6 ADRC Control
ADRC in Principle
A brief sketch will of concept of ADRC in control design is given next. For more
detail from the originators of the concept, see [17][14].
Suppose that the plant to be controlled can be represented by the block diagram in
Figure 48 where is the control signal, ( )u t ( )y t is the plant output, and is an exter-
nal disturbance. Suppose also that it is known that the plant is governed by second-order
dynamics expressed as
( )w t
( ) ( )0, , ,y f y y w b u y y= + (3.63)
Here is a parameter that is known approximately, and the function f is the mathemati-
cal model of the plant viewed as encapsulating not only the internal dynamics of the
plant, but also any internal uncertainties and disturbances, as well information about any
external disturbances contained in w. The control input is u.
0b
( )y t
( )w t
( )u tPlant
( )y t
( )w t
( )u tPlant
Figure 48 Plant With Disturbance
Suppose finally that the desired dynamics of the system are represented by the function
in the following equation: 0u
( )0 ,y u y y= . (3.64)
81
In this case, the control law which will achieve the design goal is
( ) ( ) (00
1, , ,u y y f y y w u y yb
= − + ),⎡ ⎤⎣ ⎦ , (3.65)
for then
( ) ( ) ( )0 0, , , ,y f y y w b u y y u y y= + = . (3.66)
However, an accurate model for the plant is often unavailable. One way of getting
around the modeling problem would be to estimate ( ), ,f y y w in real time. To see how
this can be accomplished, the system will be represented in state-space form.
Ordinarily, a state space representation of the system would proceed by defining
1 2[ , ] [ , ]T Tx x y y= , which gives
( )1 2
2 , ,x x
0x f y y w b uy x
=⎧⎪ =⎨⎪ =⎩
+ (3.67)
The plant model f can be estimated by augmenting the state space in (3.67) with a third
state variable, 3x , which is called an extended state and which is defined as
( )3 , ,x f y y w= (3.68)
If h is defined by , then the extended state equations for the system can be written h f=
1 2
2 3 0
3
1
x xx x b ux h
y x
=⎧⎪ = +⎨⎪ =⎩=
(3.69)
or in matrix form,
82
(3.70) e e e
e
u hy= + +⎧
⎨ =⎩
x A x B EC x
where
(3.71) [ ]0
0 1 0 0 00 0 1 , , 1 0 0 , 00 0 0 0 1
e e eb⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
A B C Ee
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
Here the e subscript indicates that the above matrices are an extended form of the
original state-space matrices.
At this point a question arises as to the whereabouts of the missing third row in A.
To date, a satisfactory explanation has not been given except to say that the third row
contains no information about the system ( )3 3x x= , so it causes no harm to drop it. In
addition, it has been demonstrated in numerous experiments and simulations that this ex-
cision does not prevent the ADRC control concept from being applied successfully in di-
vers applications. It remains for this curious maneuver to be put on a firm mathematical
basis to show why it works. Nevertheless, the present success of the application is not
unlike the use of derivatives and functions in mathematics for centuries before the
mathematics underlying their structures was well understood. The rigorous development
of calculus, for example, is credited to Augustin Louis Cauchy (1789--1857).
The goal at this point is to use this extended state-space as an observer for the
original system. The new observer is called an extended state observer (ESO). Once this
is done, the estimate of 3x f= can be used to make the correction indicated in (3.65).
At this point the ADRC approach to control can be represented visually in block
diagrams. Figure 49 represents the plant in (3.63) and shows that the combined effect of
83
the internal dynamics of the plant, together with any external disturbances, is treated as a
generalized disturbance f to a double integral plant.
+ 12s
y
f
u0b y+ 1
2sy
f
u0b y
Figure 49 Generalized Disturbance, 0y f b u= +
Next, the feedback control law from (3.65) and the ESO are incorporated into the
block diagram. This is the essence of ADRC:
++ 12s
y
f
u0b
Extended StateObserver
f̂
0u+−
y0
1b ++ 1
2s
y
f
u0b
Extended StateObserver
f̂
0u+−
+−
y0
1b
Figure 50 Control Law: ( )0 0ˆ /u f u b+ = −
Figure 50 shows the result of the generalized disturbance rejection, namely, that the de-
sired dynamics of the system have been achieved:
12s
y0u 12s
y0u
Figure 51 Desired Dynamics: 0y u≈
It remains to design the desired ADRC feedback control for the observer and
the observer gain L in the observer equations
0u
84
(3.72) ( )1
1
ˆ ˆ ˆˆ ˆ
e e
e
u y xx
⎧ = + + −⎪⎨
=⎪⎩
x A x B LC x
or
(3.73) ( )1
ˆ ˆˆ ˆ
e e e
e
ux
⎧ = − + +⎪⎨
=⎪⎩
x A LC x B LC x
y
The observability matrix for the system ( ),e eA C can be shown to have full rank,
independent of the original plant. Therefore, the eigenvalues of the observer matrix
may be placed arbitrarily. Let e −A LCe
[ ]1 2 3, , Tβ β β=L (3.74)
Then
1
2
3
1 00 10 0
e e
βββ
−⎡ ⎤⎢ ⎥− = −⎢ ⎥−⎢ ⎥⎣ ⎦
A LC (3.75)
and the characteristic polynomial is given by
( ) 3 21 2s s s s 3λ β β β= + + + . (3.76)
In the ADRC scheme, the eigenvalues are chosen to be of multiplicity three located at
oω− with 0oω > . This yields
( ) ( )3 3 2 23 3o o os s s s s 3oλ ω ω ω ω= + = + + + (3.77)
so that
21 2 33 , 3 , and o o
3oβ ω β ω β ω= = = . (3.78)
Thus the observer gains and observer matrix are given by
85
(3.79) 2
3 3
3 33 and 3 0 1
0 0
o o
o e e o
o o
ω ωωω ω
−⎡ ⎤ ⎡⎢ ⎥ ⎢= − = −⎢ ⎥ ⎢
−⎢ ⎥ ⎢⎣ ⎦ ⎣
L A LC 2
1 0ω
⎤⎥⎥⎥⎦
The final part of the development is to specify the desired control law for the
state feedback in
0u
(3.64). Since a second-order integral plant can be controlled with a sim-
ple proportional-derivative (PD) control, the feedback control is chosen to be:
( )0 1ˆpu k r x k x= − − 2ˆd (3.80)
where r is the set point, and the proportional and integral gains, and pk dk 1x̂ and 2x̂ the
estimated states. The control law chosen in (3.65) is now applied to the estimated state
3x̂ f≈ to achieve the desired design:
( ) ( ) ( )0 3 0 1 20 0 0
1 1 1ˆ ˆp du f u x u k r x k xb b b 3ˆ x̂⎡ ⎤= − + ≈ − + = − − −⎣ ⎦ (3.81)
The closed-loop transfer function for the second-order integrator in (3.64), with
this PD controller, is approximately a standard second-order transfer function given by
( )2
2 2 2P c
CLd p c
kG ss k s k s s
ω2cξω ω
= =+ + + +
(3.82)
where cω and ξ are the required closed-loop natural frequency and damping ratio. The
damping ratio ξ can be set to unity to avoid any overshoot, and cω can be adjusted to
meet system requirements. Therefore, the PD gains can be written as
2
22 2 and or2
p cd c c p c
d c
kk k
kω
ξω ω ωω
⎡⎡ ⎤= = = = ⎢⎢ ⎥
⎣ ⎦ ⎣ ⎦
⎤⎥ (3.83)
86
which means that the controller only requires one tuning parameter, cω .
The ADRC concept can be summarized in the following block diagram representation.
u
ESO
Planty
+–r
pk dk
0
1b+–+–
ˆ3x
ˆ1xˆ2x
0u u
ESO
Planty
+–+–r
pkpk dkdk
0
1b0
1b+–+–+–+–
ˆ3x
ˆ1xˆ2x
0u
Figure 52 ADRC Block Diagram
Application of ADRC to the AMB
A new application of the ADRC concept was used to control the AMB model
with position feedback. (The ADRC concept has already been applied to an AMB with-
out position feedback in [36].) It was shown in Chapter II that the fourth-order MIMO
AMB system may be decoupled into a third-order SISO system and a first-order SISO
system. The third-order system captures the electromagnetic and mechanical dynamics of
the AMB, while the first-order system captures the electrical dynamics of the bias current
in the coils. The dynamics of the first-order system are simple and easy to control. It suf-
fices, therefore, to evaluate the performance of the ADRC concept as applied to the third-
order system.
The essential component of the ADRC design is the ESO. The extended state
equations for the AMB system with h f= are given as
87
1 2
2 3
3 4 0
4
1
x xx xx x b ux h
y x
=⎧⎪ =⎪⎨ = +⎪⎪ =⎩=
(3.84)
The fourth-order ESO is then given as
(3.85) ( )1
ˆ ˆˆ ˆ
e e e
e
ux
⎧ = − + +⎪⎨
=⎪⎩
x A LC x B LC x
y
where
[ ]1
2
30
0 40 1 0 00 60 0 1 0
, , 1 0 0 0 ,0 40 0 0 1
0 0 0 0
o
oe e e
o
ob
ωωωω
⎡ ⎤ ⎡⎡ ⎤ ⎤⎢ ⎥ ⎢⎢ ⎥ ⎥⎢ ⎥ ⎢⎢ ⎥= = = ⎥=⎢ ⎥ ⎢⎢ ⎥ ⎥⎢ ⎥ ⎢⎢ ⎥
⎣ ⎦⎥
⎣ ⎦ ⎣
A B C L
⎦
(3.86)
and the PD gains are
30
21
2
33
c
PD
c
kkk
c
ωωω
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
K (3.87)
Simulation – Static Load Disturbance Rejection
The ADRC application above was simulated and tuned (Figure 53). The results
for a 4.6 N load disturbance are shown in Figure 54. One can observe that the transient
performance is much better than was the case for the state-feedback with integral control
and the H-Infinity control. The peak response was about m, whereas for the
state feedback it was
71.2 10−×
66.5 10−× m and for the H-Infinity it was m. The settling
time for ADRC was about 0.005 second, for state feedback about 0.07 second, and for H-
Infinity about 0.4 second. Finally, it should be noted that the ADRC design did not re-
41.5 10−×
88
quire the integration of the position feedback to remove steady state-error. This is be-
cause one can demonstrate that ADRC already contains an integral term in the position
feedback loop [24].
x_hat
i_hat
xdot _hat
f_hat
x
xdot
i
u
u
siso_adrc_pfb_1_para .m
ref
0
k2
k2
k1
k1
k0
k0
SISO MagBearing
x' = Ax+Bu y = Cx+Du
Output
EstimatorOutput
Disturbance
Fd
4th Order ESO
x' = Ax+Bu y = Cx+Du
1/b0
-K-
Figure 53 Simulink Model for ADRC and SISO AMB
89
0 0.005 0.01 0.015 0.02 0.025 0.03-2
-1
0
1
2
3
4
5x 10
-8 Position: x ( m )
max = 3.8839e-008min = -1.0848e-008
0 0.005 0.01 0.015 0.02 0.025 0.03-1
-0.5
0
0.5
1x 10
-4 Velocity: dx/dt ( m / s )
max = 8.2615e-005min = -5.1326e-005
Time( sec )
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 54 Step Dist, ADRC, 4.6 N Load
0 0.005 0.01 0.015 0.02 0.025 0.03-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005Coil Current: i ( A )
max = 1.3171e-031min = -0.022999
0 0.005 0.01 0.015 0.02 0.025 0.03-8
-6
-4
-2
0
2
4Control Voltage: u ( V )
max = 2.9962min = -7.5113
Time( sec )
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 55 Step Dist, ADRC, 4.6 N Load
90
The results of the parameter variation are listed in the next table. The results of
the load disturbance test were extraordinary. The peak response of the displacement re-
mained within 0.0004 m of the equilibrium position until the load exceeded 15,000 N.
This was achieved at a cost of a 76,800 V surge in the control voltage and a 144 A surge
in coil current. But the surge was still only about 0.002 second, so it is conceivable that
an AMB could be designed to withstand these brief extremes. The steady-state current
under this very large load, however, was about 75 A, so this would have to be accounted
for in the design if such large loads were expected. This test corroborates the many other
examples of superior disturbance rejection for ADRC that have been documented at
Cleveland State University.
Table VI PARAMETER SENSITIVITY – ADRC
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Unbounded Unbounded
Bias Current 0i 1 Unbounded Bounded
Coil Resistance R 8 Bounded Bounded
Rotor Mass m 4.6 Bounded Unbounded
Coil Self-Inductance sL 0.120 Bounded Bounded
Magnetic Constant K 59.8 10−× Bounded Bounded
Static Disturbance Force
Max Value dF 4.6 15,000 N
144 A
76800 V
91
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
-2
-1
0
1
2
3
4x 10
-4 Position: x ( m )
max = 0.00038103min = -0.00015218
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
-1
-0.5
0
0.5
1Velocity: dx/dt ( m / s )
max = 0.82963min = -0.58103
Time( sec )
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 56 Step Dist, ADRC, 15,000 N Load
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
-150
-100
-50
0Coil Current: i ( A )
max = 0min = -144.0792
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
-8
-6
-4
-2
0
2
4x 10
4 Control Voltage: u ( V )
max = 35752.316min = -76844.8613
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 57 Step Dist, ADRC, 15,000 N Load
Sinusoid Load Disturbance Rejection
92
The response of the ADRC controlled AMB to a sinusoid disturbance was tested,
and the results are shown in Figure 58. Evidently, ADRC control was able to attenuate
the 10 Hz signal almost completely, while maintaining a very small current and a moder-
ate voltage surge of about 14 V. The steady-state voltage was only a 0.4 V sinusoid. The
ripple in the position was only 91 10−× m. For the higher frequency, once again the sinu-
soidal was attenuated to a very small magnitude. The amplitude of the oscillation is small
enough that the rotor never comes near the bearing. However, in a more sophisticated
model, these vibrations might excite vibration modes in the rotor, so further study is nec-
essary to asses the success of the control with respect to vibration. In addition, the high
frequency oscillation of the control voltage with an amplitude of about 52 V might cause
some heat problems which also need to be studied further.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-4
-2
0
2
4
6
8
10x 10
-8 Position: x ( m )
max = 7.1048e-008min = -2.4542e-008
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
-4 Velocity: dx/dt ( m / s )
max = 0.00015332min = -0.00010184
Time( sec )
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 58 Sine Dist, ADRC (10 Hz)
93
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.03
-0.02
-0.01
0
0.01
0.02
0.03Coil Current: i ( A )
max = 0.022999min = -0.026894
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-15
-10
-5
0
5
10Control Voltage: u ( V )
max = 6.0998min = -13.9715
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 59 Sine Dist, ADRC (10 Hz)
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-1.5
-1
-0.5
0
0.5
1x 10
-7 Position: x ( m )
max = 7.2911e-008min = -1.0076e-007
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-5
0
5x 10
-4 Velocity: dx/dt ( m / s )
max = 0.00047358min = -0.00047805
Time( sec )
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 60 Sine Dist, ADRC (1000 Hz)
94
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.05
0
0.05Coil Current: i ( A )
max = 0.045891min = -0.044828
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-60
-40
-20
0
20
40
60Control Voltage: u ( V )
max = 52.7974min = -52.8755
siso_adrc_pfb_1_para.m, ωc = 2000, ωo = 6000
Figure 61 Sine Dist, ADRC (1000 Hz)
3.7 ADRC Integrator Backstepping Control
Another new application of the ADRC concept was used to control the AMB
model with a design procedure called integrator backstepping. Again, the SISO model
was used with position feedback. A brief overview of integrator backstepping design will
be given next.
In order to illustrate the idea, consider the differential equations for the SISO
AMB.
2 2 1
1
s id
i
k kx x im m m
di k R
F
x i udt L L L
= + +
= − − + (3.88)
95
The fundamental idea is to design the control in two steps. The first step is to assume that
the current variable i in the first differential equation is a “virtual control” variable or in-
put which can be used to drive the controlled variable x to a desired output response. Af-
ter a control has been designed to achieve the desired response, the second step is to
“backstep” and design the actual control so as to achieve the desired “virtual control” i.
The backstepping idea can be thought of from a signal flow point of view. In operation,
the signal u controls the current i which in turn controls the signal x. From the point of
view of design, one backsteps through this process.
This process was carried out for the SISO AMB with ADRC as the control strat-
egy. The desired output for the position is to be regulated to zero. The technique illus-
trated in the ADRC design previous to this was applied to design a third-order ADRC
controller to regulate the rotor position which is governed by second-order dynamics.
Simulink diagrams will be used to illustrate this design process. The ADRC rotor control
design is represented in Figure 62.
96
x_hatxdot _hatf_hat
i = ( -kp * x1_hat - kd * x2_hat - x3_hat ) / b0
ix
i
ADRC rotor control to develop position control with current as virtual control
x
xdot
i
siso_bkst_ADRC_pfb_1_para .m
0kp
kd
Rotor
i
Fd
x
xdot
Output
Fd
Estimator
m
3rd OrderESO
x' = Ax+Bu y = Cx+Du
1/b0
-K-
Figure 62 ADRC Rotor Control Design
The rotor is then “chopped out” of the Simulink model leaving only the control
apparatus (Figure 63). The ADRC rotor control subsystem must be supplied with the ref-
erence signal r for the position and the position feedback signal x. The subsystem must
supply the “virtual” control current which has been relabeled as for “i desired.” *i
x_hatxdot _f_hat
i*x
- x
i* = ( -kp * x - kd * x2_hat - x3_hat ) / b0
i*1
kp_rotor
-K-
kd_rotor
-K-
Rotor ESO
x' = Ax+Bu y = Cx+Du
Rotor ESOEstimator
Gotoi_star
m
1/b0_rotor
-K-
x2
r 1
Figure 63 ADRC Rotor Control Subsystem
97
Next, the ADRC control for the coil circuit is designed in the same manner except
that a second-order ADRC is required to control the first-order coil dynamics (Figure 64).
i_hatf_hat
i = ( -kp * x1_hat - x2_hat ) / b0
ui
u
u
i
ADRC coil control to develop bkst
siso_bkst_ADRC_pfb_2_para .m
kp_coil
-K-Step = 1
Output
Estimator
m
Coil ESO
x' = Ax+Bu y = Cx+Du
Coil
u i
1/b0_coil
-K-
Figure 64 ADRC Coil Control Design
Again the coil is “chopped out” leaving only the ADRC coil control (Figure 65).
The subsystem must be supplied with the desired reference current and the measured
current i. The output u is the control signal which will produce the desired current which
in turn will produce the desired position regulation.
*i
98
f_i_hat
i* - i
i_hatui
u1
kp_coil
-K-Goto
u
m
Coil ESOEstimator
Coil ESO
x' = Ax+Bu y = Cx+Du
1/b0_coil
-K-
i2
i* 1
Figure 65 ADRC Coil Control Subsystem
Lastly, the components are assembled together to produce the overall ADRC con-
trol for the AMB (Figure 66).
x
siso_bkst_ADRC_pfb_3_para.m
r = 0
0
Rotor
i
Fdx
Output
u
i
xdotDisturbance
FdCoil
u iADRC Virtual Rotor Control
r
xi*
ADRC Coil Control
i*
iu
Figure 66 ADRC Control for an AMB Using Integrator Backstepping
For an explanation of the term “integrator” in the name “integrator backstepping,”
it can be noted that higher-order systems of differential equations can be written as a sys-
tem of first-order differential equations or “integrators.” If the equations are coupled, the
above backstepping procedure can be applied in an iterative manner, working backwards
through each “integrator” equation until reaching a control variable, in which case the
iteration stops.
Simulation – Static Load Disturbance Rejection
99
The first thing to note is that it is necessary to tune four ADRC parameters (two
per ADRC control) as opposed to two for the previous “conventional” ADRC design. The
results from a 4.6 N step load are displayed in Figure 67. The peak response was about
m, and the settling time was about 0.008 second. For the “conventional”
ADRC, these values were
73.0 10−×
71.2 10−× and 0.005 second, respectively. Therefore, these
transients were only slightly degraded from what had been obtained in the previous test.
This seems plausible since two estimates of the plant f had to be made for the backstep-
ping version, causing it to be slightly slower.
0 0.01 0.02 0.03 0.04 0.05 0.06-5
0
5x 10
-7 Position: x ( m )
max = 2.9622e-007min = -3.1742e-011
0 0.01 0.02 0.03 0.04 0.05 0.06-5
0
5x 10
-4
max = 0.000394min = -0.00021059
Velocity: dx/dt ( m / s )
0 0.01 0.02 0.03 0.04 0.05 0.06
-0.04
-0.02
0
max = 0min = -0.042846
Coil Current: i ( A )
0 0.01 0.02 0.03 0.04 0.05 0.06-20
0
20
max = 7.2241min = -13.4723
Control Voltage: u ( V )
Time( sec )
siso_bkst_ADRC_pfb_3_para.m, ωcr = 2000, ωor = 6000, ωoc = 12000
Figure 67 Step Dist, ADRC & Backstepping, 4.6 N Load
100
The results of the physical parameter variation are listed next. An interesting ob-
servation is that there was a significant steady-state error with both changes in bias cur-
rent, both changes in mass, and when the magnetic constant was halved. Apparently, the
backstepping configuration for ADRC is more sensitive than ADRC alone when it comes
to compensating for steady-state error. Clearly the integrating term in the backstepping
version is affected somehow by some parameter changes but not affected in “conven-
tional” ADRC. It would be interesting to put this observation in a sound mathematical
context.
The peak response for a load disturbance exceeded m at about 6,300 N.
This is a large reduction from the 15,000 N attained without backstepping. It appears
overall that if there are no compelling reasons for choosing the backstepping design, the
“conventional” ADRC design ought to be chosen for the AMB application.
44.0 10−×
Table VII PARAMETER SENSITIVITY – ADRC & BACKSTEPPING
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Bounded Unbounded
Bias Current 0i 1 Bounded Bounded
Coil Resistance R 8 Bounded Bounded
Rotor Mass m 4.6 Bounded Bounded
Coil Self-Inductance sL 0.120 Bounded Bounded
Magnetic Constant K 59.8 10−× Unbounded Bounded
Static Disturbance Force
Max Value dF 4.6 6,300 N
58 A
9890 V
101
0 0.01 0.02 0.03 0.04 0.05 0.06-5
0
5x 10
-4 Position: x ( m )
max = 0.0004057min = -4.3472e-008
0 0.01 0.02 0.03 0.04 0.05 0.06-1
0
1max = 0.53961min = -0.28841
Velocity: dx/dt ( m / s )
0 0.01 0.02 0.03 0.04 0.05 0.06-100
-50
0 max = 0min = -58.6798
Coil Current: i ( A )
0 0.01 0.02 0.03 0.04 0.05 0.06-2
0
2x 10
4
max = 9893.8826min = -18451.1639
Control Voltage: u ( V )
Time( sec )
siso_bkst_ADRC_pfb_3_para.m, ωcr = 2000, ωor = 6000, ωoc = 12000
Figure 68 Step Dist, ADRC & Backstepping, 6,300 N Load
Sinusoid Load Disturbance Rejection
The control was tested with a sinusoidal load disturbance (Figure 69). For the 10
Hz signal, some oscillation is still visible in the response, unlike the ADRC without back-
stepping in which the ripple at 10 Hz was hardly noticeable. The amplitude has been at-
tenuated to about m, which is good, but the attenuation is slightly degraded
from the previous ADRC design. The result for the high frequency (
85.0 10−×
Figure 70) is compa-
rable to the ADRC without backstepping design, except the voltage swings on the control
102
are much smaller. This fact might be useful in applications which are sensitive to heat
buildup.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-5
0
5x 10
-7 Position: x ( m )
max = 2.8507e-007min = -4.142e-008
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-5
0
5x 10
-4
max = 0.00037739min = -0.00019718
Velocity: dx/dt ( m / s )
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.05
0
0.05
max = 0.023009min = -0.041155
Coil Current: i ( A )
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-20
0
20
max = 6.7806min = -12.8843
Control Voltage: u ( V )
Time( sec )
siso_bkst_ADRC_pfb_3_para.m, ωcr = 2000, ωor = 6000, ωoc = 12000
Figure 69 Sine Dist, ADRC & Backstepping, (10 Hz)
103
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-2
0
2x 10
-7 Position: x ( m )
max = 1.0459e-007min = -5.4195e-008
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-5
0
5x 10
-4
max = 0.00027078min = -0.00039651
Velocity: dx/dt ( m / s )
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-0.02
0
0.02max = 0.017364min = -0.01961
Coil Current: i ( A )
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-20
0
20max = 18.3978min = -13.3366
Control Voltage: u ( V )
Time( sec )
siso_bkst_ADRC_pfb_3_para.m, ωcr = 2000, ωor = 6000, ωoc = 12000
Figure 70 Sine Dist, ADRC & Backstepping, (1000 Hz)
CHAPTER IV
CONTROL WITHOUT POSITION FEEDBACK
4.1 Current Feedback
The “classical” design concept of magnetic bearing control systems, and the most
popular control configuration used since the 1940s, is based on the position-sensed, cur-
rent-controlled, and magnetic coil configuration for the SISO system in Figure 21. Since
the 1990s another actively studied configuration for the AMB is one in which the input to
the system is the coil voltage of the magnetic bearing and the output is the coil current
Figure 71.
104
105
uElectro-magnet Rotor
Force Position
xFVoltage
Magnetic Bearing
Controller+–0=refx
i
uElectro-magnet Rotor
Force Position
xFVoltage
Magnetic Bearing
Controller+–+–0=refx
i
Figure 71 Current-Sensed, Voltage-Controlled AMB
A voltage-controlled magnetic bearing which only senses the coil current is, nev-
ertheless, still controllable and observable when the static load disturbance is not in-
cluded as part of the system. This can be deduced from the irreducibility of the third-
order transfer function in (4.10). When the disturbance is included, the SISO system is
0 1 0 0 02 2 1 00
10 0
00 0 0 0
s i
i
dd
x xk kx xm m m udi ik Rdt LL L FF
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ + ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥⎣ ⎦
(4.1)
which has the same matrices as in (3.21) except for . Therefore the same controllabil-
ity result as there holds, namely, that all of the states are controllable except for the static
load disturbance.
dC
The observability matrix is
2 3 T
d d d d d d d d⎡ ⎤= ⎣ ⎦OM C C A C A C A (4.2)
106
( )
2 2
2 2
2 2 2 2 3
2 3 3
0 0 1
0 0
2 2
2 22 4
i
d i s i i i
i s ii s i i
k RL L
k k Rk mR Lk k
2
0
2
Lm L L mk k L k L mR
Lm
Rk k LRk mR RkL m L m L m
⎡ ⎤⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥= −− −⎢ ⎥⎢ ⎥⎢ ⎥− + −⎢ ⎥−⎣ ⎦
OM
L m
(4.3)
It is straightforward to verify that ( )rank 3d =OM , so the system augmented with the
static load is not observable.
Once again, it is useful to have the transfer function for the current-sensed AMB
available for analysis. Beginning with (2.58) and omitting the disturbance force, the
open-loop transfer function from u to i can be derived. Differentiating the first equation
yields
2 2s ik kx xm m
= +didt
(4.4)
Solving the second for x yields
1
i i
L di R
i
x ik dt k k
= − − + u (4.5)
Differentiating the second equation twice and substituting the previous two yields
107
3 2
3 2
2
2
2 2
2
2 2
2
2
2
1
2 2 1
2 2 1
2 1 2
2 2 2
i
i s i
i s i
i s i
i i i
s s s
d i k R d ix udt L L dt L
k k k di R d ix uL m m dt L dt LR d i k k k dix uL dt mL mL dt LR d i k k L di R k dii uL dt mL k dt k k mL dt L
R d i k di k R kiL dt m dt mL mL
= − − +
⎛ ⎞= − + − +⎜ ⎟⎝ ⎠
= − − − +
⎛ ⎞= − − − − + − +⎜ ⎟
⎝ ⎠
= − + + −
1 u
( )
2
22
2
2 1
2 2 1 2
i
s ss i
k diu umL dt L
R d i di k R kk L k i u uL dt mL dt mL L mL
− +
= − + − + + −
(4.6)
Taking the Laplace Transform yields
( ) ( ) ( )3 2 2 22 2 1 2si s
R k Rs s k Lk s I s s UL mL mL L m
⎡ ⎤+ + − − = −⎜ ⎟⎢ ⎥⎣ ⎦sk s⎛ ⎞
⎝ ⎠ (4.7)
So the open-loop transfer function is given by
( )( ) ( )
2
3 2 2
22 2
s
i s s
I s ms kU s mLs mRs k Lk s k R
−=
+ + − − (4.8)
For the parameters used in this thesis, (4.8) is given by
( )( )
2 5
3 2 45.263 3.269 10
42.11 3.923 10 2.615 10
I s sU s s s s
− ×=
+ − × − × 6 (4.9)
Or, in zero-pole-gain form
( )( )
( ) ( )( ) ( ) ( )
5.2632 249.2 249.2207.6 179.5 70.19
I s s sU s s s s
− +=
− + + (4.10)
108
4.2 Steady-State Error
One of the characteristics of a current-sensed AMB that is subjected to a static
load disturbance is that no linear controller can remove the steady-state error. This is re-
ferred to as the insolvability of the regulator problem for displacement. This theorem is
proved in [25]. Briefly, the authors use a geometric argument on the various spaces asso-
ciated with observability and linear controllers when viewed as linear transformations on
the state-space. In order to suspend a rotor stably without any steady-state position error,
Wonham [43] proved that the linear controller is subject to three necessary conditions,
the third of which is concerned with position regulation. In their paper, the authors prove
that the current-sensed AMB cannot satisfy this condition.
The magnitude of the steady-state error is easily found from the open-loop trans-
fer function from the load to the position x and is found to be dF
dss
s
Fek
= − (4.11)
If the expected loads for an application are known, and if sse is small enough to
remain within the bearing clearance tolerance, this error will not cause a problem. How-
ever, for applications which must handle large loads, the error may exceed the clearance
tolerance. It is for this reason that the 15,000 N load-carrying capacity of the position-
sensed AMB with ADRC is so attractive.
109
4.3 Luenberger Observer
One of the simplest ways to control an AMB with a current sensor is a conven-
tional Luenberger Observer and state feedback. Such an observer was designed for the
SISO AMB, and the Simulink model is shown in Figure 72.
i
u
u
siso_state_cfb_est_1_para .m
0
Output
Mag BearingSISO Linear
u
Fd
x
xdot
i
LuenbergerEstimator
x' = Ax+Bu y = Cx+Du
K_place
K_place
Estimator
Disturbance
Fd
m
Figure 72 Simulink Model, Luenberger Observer and State feedback
Simulation – Static Load Disturbance Rejection
The simulation was run with a 4.6 N static load disturbance. The results are
shown in Figure 73. Qualitatively, the position and velocity terms are almost identical to
their counterparts in the state feedback design with position-sensing. One major differ-
ence is the existence of a steady-state error. Unfortunately, for the current-sensed case
this error cannot be removed with any linear control as explained above. Nevertheless,
the steady state error can be predicted, because it can be shown that its value is /d sF k−
[41]. The negative sign occurs because the response is non-minimum phase. If the static
110
load is known not to exceed a value which would cause the rotor to get too close to the
magnet, this type of control can be used.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2x 10
-5 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5x 10
-3 Velocity: dx/dt ( m / s )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.1
0
0.1Coil Current: i ( A )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5Control Voltage: u ( V )
Time( sec )
siso_state_cfb_est_1_para.m
Figure 73 Step Dist, Luenberger Observer, 4.6 N Load
The control was then tested under variations in the system parameters, and the re-
sults are listed next. The Luenberger estimator design is very remained bounded with re-
spect to gross changes in the values of physical parameters. The maximum static load dis-
111
turbance that can be rejected, however, is lower than for the design using state feedback
with position-sensing.
Table VIII PARAMETER SENSITIVITY – LUENBERGER
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Bounded Bounded
Bias Current 0i 1 Bounded Bounded
Coil Resistance R 8 Bounded Bounded
Rotor Mass m 4.6 Bounded Bounded
Coil Self-Inductance sL 0.120 Bounded Bounded
Magnetic Constant K 59.8 10−× Bounded Bounded
Static Disturbance Force
Max Value dF 4.6 125 N
112
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5x 10
-4 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.1
0
0.1Velocity: dx/dt ( m / s )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
-1
0
1Coil Current: i ( A )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-100
-50
0
50Control Voltage: u ( V )
Time( sec )
siso_state_cfb_est_1_para.m
Figure 74 Step Dist, Luenberger Observer, 125 N Load
Sinusoid Load Disturbance Rejection
The system was subjected to a sinusoidal load disturbances with a magnitude of
4.6 N and frequencies of 10 Hz and 1000 Hz. Again, the responses (Figure 75and Figure
76) were qualitatively very similar to their counterparts in the state feedback design with
position-sensing.
113
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1
0
1x 10
-3 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.05
0
0.05Velocity: dx/dt ( m / s )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2Coil Current: i ( A )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-20
0
20Control Voltage: u ( V )
Time( sec )
siso_state_cfb_est_1_para.m
Figure 75 Sine Dist, Luenberger (10 Hz)
114
0 0.01 0.02 0.03 0.04 0.05 0.06-2
0
2x 10
-5 Position: x ( m )
0 0.01 0.02 0.03 0.04 0.05 0.06-0.01
0
0.01Velocity: dx/dt ( m / s )
0 0.01 0.02 0.03 0.04 0.05 0.06-0.1
0
0.1Coil Current: i ( A )
0 0.01 0.02 0.03 0.04 0.05 0.06-10
-5
0
5Control Voltage: u ( V )
Time( sec )
siso_state_cfb_est_1_para.m
Figure 76 Sine Dist, Luenberger (1000 Hz)
4.4 H-Infinity Control
A new application of H-Infinity control was designed for the MIMO AMB using
only current feedback from the electromagnetic coils (self-sensing control). When H-
Infinity control design is used in this situation, that is, when all of the states are not avail-
able for feedback, it is generally referred to as H-Infinity output feedback control design.
115
Once again the hinfsyn() function from Matlab was used. The design procedure is the
same as before except that there is a change in the interconnection matrix P .
Interconnection Matrix
The interconnection matrix is the critical component for the design of H-Infinity
control using Matlab. The essential change occurs in the feedback loop where it can be
observed that only the coil current is returned to the controller (Figure 77). In addition,
the various B and D matrices had to be adjusted to the proper dimensions and otherwise
modified to satisfy interconnection matrix design considerations. The detail is omitted
since it was given in Chapter III. The same is true for the scaling matrices.
+
A
2B
4x1
1B
4x2
4x4
4x4
2x1
4x1
4x3
4x2
+
4x3 11D
12D
2C
zz
( )sK2x2
4x4
1C
G
Scaled Model – Output FeedbackScaled Model – Output FeedbackPP
+
sq
v
u
2x14x12x4
4x1
+1S
2x321D
+
2x2
22D3
4
s
s
s
drr
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1
2
uu⎡ ⎤⎢ ⎥⎣ ⎦
1
2
s
s
s
s
xxuu
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
3 1
4 2
r ir i−⎡ ⎤
⎢ ⎥−⎣ ⎦
2x1
2x1
2x1
wsw wD
3x33x1
1q−D
3x1
4x4
lin_hinfsyn_cfb_1_para.m
+
A
2B
4x1
1B
4x2
4x4
4x4
2x1
4x1
4x3
4x2
++
4x3 11D
12D
2C
zz
( )sK2x2
4x4
1C
G
++
sq
v
u
2x14x12x4
4x1
++1S
2x321D
++
2x2
22D3
4
s
s
s
drr
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1
2
uu⎡ ⎤⎢ ⎥⎣ ⎦
1
2
s
s
s
s
xxuu
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
3 1
4 2
r ir i−⎡ ⎤
⎢ ⎥−⎣ ⎦
2x1
2x1
2x1
ws
wDw
3x33x1
1q−D
3x1
4x4
lin_hinfsyn_cfb_1_para.m
Figure 77 Interconnection Matrix Design, H-Infinity Output Feedback
116
Simulation – Static Load Disturbance Rejection
The simulation (Figure 78) was run using a 4.6 N load disturbance. A gamma
value of 1.0 was achieved, and the results are shown in Figure 79 and Figure 80. Like the
Luenberger observer control above, the position and velocity terms are almost identical to
their counterparts in the full information feedback design. Again the existence of a
steady-state error is evident.
u1
u2
Fd
lin _hinfsyn_cfb_1_para .m
i1, i2Reference
(0 0)
OutputMIMO LinearMagBear
u1
u2
Fd
x
x_dot
i1
i2
K_LTI
K_LTI
Disturbance
Fd
Control
Figure 78 Simulink Model of MIMO System with H-Infinity Output Feedback
117
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5
10
15x 10
-5 x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-2
0
2
4
6
8
10x 10
-3 dx/dt ( m/s )
Time( sec )
lin_hinfsyn_cfb_1_para.m
Figure 79 Step Dist, H-Infinity Output Feedback, 4.6 N Load
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3i1, i2 ( A )
i1i2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-2
0
2
4u1, u2 ( V )
Time( sec )
u1
u2
lin_hinfsyn_cfb_1_para.m
Figure 80 Step Distt, H-Infinity Output Feedback, 4.6 N Load
118
The control was then tested under variations in the system parameters, and the re-
sults are listed next.
Table IX PARAMETER SENSITIVITY – H-INFINITY OUTPUT FEEDBACK
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Bounded Unbounded
Bias Current 0i 1 Unbounded Bounded
Coil Resistance R 8 Unbounded Unstable
Rotor Mass m 4.6 Bounded Bounded
Coil Self-Inductance sL 0.120 Unbounded Unbounded
Magnetic Constant K 59.8 10−× Bounded Bounded
Static Disturbance Force
Max Value dF 4.6 10 N 0.25 A
3.0 V
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1
0
1
2
3
4
5x 10
-4 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.01
-0.005
0
0.005
0.01
0.015
0.02Velocity: dx/dt ( m / s )
Time( sec )
lin_hinfsyn_cfb_1_para.m
Figure 81 Step Dist, H-Infinity Output Feedback, 10 N Load
119
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3i1, i2 ( A )
i1i2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-2
0
2
4u1, u2 ( V )
Time( sec )
u1
u2
lin_hinfsyn_cfb_1_para.m
Figure 82 Step Dist, H-Infinity Output Feedback, 10 N Load
Sinusoid Load Disturbance Rejection
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1
-0.5
0
0.5
1x 10
-3 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.06
-0.04
-0.02
0
0.02
0.04
0.06Velocity: dx/dt ( m / s )
Time( sec )
lin_hinfsyn_cfb_1_para.m
Figure 83 Sine Dist, H-Infinity, Output Feedback, (10 Hz)
120
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1
-0.5
0
0.5
1
1.5x 10
-6 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-2
0
2
4x 10
-4 Velocity: dx/dt ( m / s )
Time( sec )
lin_hinfsyn_cfb_1_para.m
Figure 84 Sine Dist, H-Infinity, Output Feedback, (1000 Hz)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-5
0
5
10
15x 10
-7 Position: x ( m )
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-4
-2
0
2
4x 10
-4 Velocity: dx/dt ( m / s )
Time( sec )
lin_hinfsyn_cfb_1_para.m
Figure 85 Sine Dist, H-Infinity, Output Feedback, (1000 Hz)
121
4.5 Verifying the Linearization
In order to verify that the linearization of the nonlinear MIMO model developed
in Chapter II and used throughout this study was valid, the same current-sensed, H-
infinity control that was applied to the linearized MIMO model in Figure 78 was applied
to the nonlinear model. A simulation with a 4.6 N load disturbance applied to both sys-
tems was performed, and the results are shown on the same set of axes in Figure 87. The
results indicate excellent agreement between the two systems. The operating point for the
linearized system is A for the bias current and 0 1i = 0 8u = V for the bias voltage.
z Nonlinear
Terminator
Mag BearingNonlinear
u1
u2
Fd
z0
z
z_dot
K_LTI
K_LTI
InitialStates
z0
Disturbance
Fd
Control
[ u10 u20 ]
(i0 i0)
Figure 86 Nonlinear MIMO Model, Current-Sensed, H-Infinity
122
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5
10
15
20x 10
-5 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4
-2
0
2
4
6
8
10x 10
-3 Velocity: dx/dt ( m / s )
Time( sec )
nonlinearlinear
nonlin_hinfsyn_cfb_1_para.m
Figure 87 Step Dist, Linear and Nonlinear System, 4.6 N Load
4.6 ADRC Integrator Backstepping Control
The backstepping technique discussed in Chapter III was employed to combine a
Luenberger estimator with ADRC to control the SISO AMB with current feedback. This
design was developed by Su-Alexander in [36] and is included here for comparison. The
control was retuned to achieve a much better transient response than reported in [36]. The
design process will be illustrated with Simulink diagrams.
Simulink Design
As before, the rotor position control is designed first, then an ADRC control is de-
signed to control the coil. The Luenberger observer estimates the rotor position and coil
123
current using information contained in the actual coil current. The ADRC coil control
then attempts to drive the rotor to the desired position displacement, namely zero. But as
before, there is a residual steady-state error in displacement that cannot be removed with
a linear controller.
x
siso_bkst_ADRC_est_cfb_1_para.m
r = 0
0
Rotor
i
Fdx
Output
Luen EstimatorRotor Control
r
u
i
i*
u
i
xdotFd
Coil
u i
ADRC Coil Control
i*
iu
Figure 88 Simulink Model, ADRC & Luenberger & Backstepping
Simulation – Static Load Disturbance Rejection
The model was simulated with a 4.6 N load, and the results are shown in Figure
89. The response is degraded with respect to the ADRC design with position feedback,
but comparable to the H-Infinity design using current feedback alone.
124
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2x 10
-5 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2x 10
-3 Velocity: dx/dt ( m / s )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0.04
-0.02
0Coil Current: i ( A )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5Control Voltage: u ( V )
Time( sec )
siso_bkst_adrc_est_cfb_1_para.m, ωc = 9000, ωo = 27000
Figure 89 Step Dist, ADRC & Luen & Backstepping, 4.6 N Load
The control was tested with respect to parameter variation and the results are
listed next. Although this control is the only one that remained stable for all parameter
variations, the load disturbance rejection was the poorest among all ADRC control de-
signs.
Table X PARAMETER SENSITIVITY – ADRC & LUEN & BACKSTEPPING
Parameter Name
Parameter
Symbol
Nominal
Value
Value
Doubled
Value
Halved
Nominal Air Gap 0x 0.0007 Bounded Bounded
125
Bias Current 0i 1 Bounded Bounded
Coil Resistance R 8 Bounded Bounded
Rotor Mass m 4.6 Bounded Bounded
Coil Self-Inductance sL 0.120 Bounded Bounded
Magnetic Constant K 59.8 10−× Bounded Bounded
Static Disturbance Force
Max Value dF 4.6 300 N 2.6 A
225 V
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
-1
0
1x 10
-3 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.1
0
0.1Velocity: dx/dt ( m / s )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-3
-2
-1
0Coil Current: i ( A )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-400
-200
0
200Control Voltage: u ( V )
Time( sec )
siso_bkst_adrc_est_cfb_1_para.m, ωc = 9000, ωo = 27000
Figure 90 Step Dist, ADRC & Luen & Backstepping, 300 N Load
Sinusoid Load Disturbance Rejection
126
Sinusoid load disturbances were applied to this design. The results are shown in
Figure 91 and Figure 92. Again, there is a significant degradation when compared to
ADRC with position control.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2x 10
-5 Position: x ( m )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-2
0
2x 10
-3 Velocity: dx/dt ( m / s )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.05
0
0.05Coil Current: i ( A )
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5
0
5Control Voltage: u ( V )
Time( sec )
siso_bkst_adrc_est_cfb_1_para.m, ωc = 9000, ωo = 27000
Figure 91 Sine Dist, ADRC & Luen & Backstepping, (10 Hz)
127
0 0.01 0.02 0.03 0.04 0.05 0.06-5
0
5x 10
-7 Position: x ( m )
0 0.01 0.02 0.03 0.04 0.05 0.06-5
0
5x 10
-4 Velocity: dx/dt ( m / s )
0 0.01 0.02 0.03 0.04 0.05 0.06-5
0
5x 10
-3 Coil Current: i ( A )
0 0.01 0.02 0.03 0.04 0.05 0.06-2
0
2Control Voltage: u ( V )
Time( sec )
siso_bkst_adrc_est_cfb_1_para.m, ωc = 9000, ωo = 27000
Figure 92 Sine Dist, ADRC & Luen & Backstepping (1000 Hz)
CHAPTER V
CONCLUSION AND FUTURE WORK
5.1 Summary and Conclusions
Research that seeks to improve the control of active magnetic bearings remains a
worldwide undertaking by many investigators in academia and industry. This thesis gives
an initial evaluation of three new applications of existing control designs to AMBs in
comparison with each other and with some conventional design approaches. The simula-
tions performed in this thesis were designed to broadly characterize the performance of
these designs. Table XI summarizes some of the information gathered.
The results in the table represent only a preliminary assessment of some broad
performance measures. Nevertheless, it appears that the new ADRC control application
has some outstanding characteristics for AMB control, whether the position is sensed or
not. If the rejection of large static and sinusoidal load disturbances is a requirement of the
design, then ADRC appears at this stage of the study to have the best response to these
disturbances. But these results need to be investigated in more depth before ruling out H- 128
129
Infinity control for AMB applications where large load disturbances are expected. Spe-
cifically, the weighting matrices in the interconnection matrix and the value of gamma
must be tuned to optimize for disturbance rejection before drawing any conclusions.
Also, there may be other aspects of H-Infinity control that were not tested in this work
which may render it more suitable an AMB application.
FB: Feedback, P: Position, C: Current
NUP = Number of Parameters Leading to Unbounded Outputs in Simulation
Table XI PERFORMANCE SUMMARY FOR VARIOUS DESIGNS
Design
FB Max
Disp
at
4.6 N
Load
(m)
Max
Load
Reject
(N)
Max
Cur.
at
Max
Load
(A)
Max
Volt.
at
Max
Load
(V)
Steady
State
Disp
10 Hz
(m)
Steady
State
Disp
1000
Hz
(m)
NUP
State FB
& Integral P 6.5μ10-6 250 2.01 137 7.6μ10-6 5μ10-8 1
H-Infinity
& Integral P 1.5μ10-4 10 0.607 5.20 1.2μ10-4 9μ10-7 5
ADRC P 3.8μ10-8 15000 144 76800 ~ 0 7.5μ10-8 4
ADRC &
Backstep P 3.0μ10-7 6300 58 9890 4μ10-8 3μ10-8 2
Luen C 1.8μ10-5 125 1.7 75 7μ10-4 1μ10-7 0
H-Infinity C 1.5μ10-4 10 0.25 3. 1μ10-3 2μ10-8 6
ADRC &
Backstep C 5μ10-6 300 2.6 225 1.2μ10-5 6μ10-8 0
130
5.2 Future Work
The research work undertaken in this thesis has spawned numerous potential areas
for further investigation. Some will be listed now in no particular order.
1. Since most AMBs have a very high force-displacement or stiffness pa-
rameter, some of the matrices involved in the computation of various
quantities are ill-conditioned. It may prove useful in numerical simulations
to have a non-dimensional state space representation of the system to alle-
viate these effects [42].
2. The sensitivity to parameter variations in the plant was only addressed
with very coarse measurements, namely, doubling and halving the pa-
rameter size. A quantified and intense robustness analysis is warranted for
the particular controls studied, especially ADRC.
3. Many conventional approaches to nonlinear control for the AMB have
been published. Since the steady-state error problem for the current-
sensing AMB is not solvable with a linear control, some of the unexplored
approaches to nonlinear control may prove fruitful. A major problem for
all AMBs that use linear control involves the startup condition or the re-
covery condition after a major disturbance, either of which represents a
situation in which the rotor displacement is far outside the linear region of
the operating point. A nonlinear control may be able to circumvent this
problem. Also, nonlinear control schemes usually provide dramatic power
savings in the area of 90 percent, since no bias current is necessary [38].
131
4. No simulations involving sensor noise were performed. Obviously this
needs to be addressed, most likely in conjunction with robustness studies.
5. The whole area of frequency analysis was not addressed explicitly, al-
though root-locus and Bode were used in Matlab’s Linear Analysis and
Compensator Design toolboxes as an aid in tuning some of the control
simulations. An in-depth frequency analysis is required.
6. A stability analysis is warranted.
7. The tuning of various control parameters still remains somewhat of a trial-
and-error process. It is difficult to ascertain whether one control is better
than another if there is no explicit optimization of the tuning process.
8. The effect of tuning on sinusoidal rejection needs to be investigated since
vibration control is one of the major obstacles to faster rotational speeds.
9. The effect of high frequency signals on heat buildup in the laminations of
the electromagnetic cores needs to be researched.
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