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NONLINEAR CONTROL DESIGN FOR AMAGNETIC LEVITATION SYSTEM
by
Rafael Becerril Arreola
A thesis submitted in conformity with the requirements of the degree of
Master of Applied ScienceGraduate Department of Electrical and Computer Engineering
University of Toronto
c©Copyright by Rafael Becerril Arreola 2003
Nonlinear control design for a magnetic levitation systemMaster of Applied Science 2003
Rafael Becerril ArreolaGraduate Department of Electrical and Computer Engineering
University of Toronto
ABSTRACT
This thesis presents the design of two magnetically levitated high precision positioning systems based onpermanent magnet linear synchronous motors (PMLSMs). Magnetic levitation based on PMLSMs requiresthe motors to operate with a variable airgap length but previous research in the field has studied mainlymotors fastened by bearings; therefore, this work first models a motor with free normal dynamics. A 3-DOFlevitation device is then designed with basis on the resulting model of the motor. The nature of that systemconveys the need for restricting its state in order to guarantee well-definiteness of the controls. Simulationsthen evaluate the performance of three different stabilization approaches: LQR, standard nonlinear control,and invariance control. Because that 3-DOF device neither controls nor constrains rotation, a more complexmachine is introduced in order to regulate five DOF. An invariance controller completes the design of this5-DOF levitation device.
II
Acknowledgments
Almost two years of hard work produced the results presented in this thesis. However, hard work would nothave been enough without the guidance and expertise of Professor Manfredi Maggiore, who supervised it.I want to thank for his financial support and meticulous revisions but, most of all, I want to thank for hisunconditional availability and for having enjoyed this research as much as I did.
Por otro lado, dos anos de trabajo arduo no hubieran servido de nada si mis padres no me hubiesenpropocionado las bases que me permitieron llevar a cabo mis estudios de maestria. Agradezco a mis padresdos cosas mas: que me hayan mostrado el verdadero valor de la educacion; y que me hayan ensenado adisfrutar el trabajo y a verlo como una bendicion. En fin, les agradezco porque me hicieron el feliz hombrede provecho que ahora soy.
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Contents
Contents V
1 Magnetic Levitation and Linear Motors 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Review on linear motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Topologies of linear machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Principles of operation of linear motors . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Purpose and outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Literature review 7
2.1 Previous work on magnetic levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 The vehicles levitated by iron-cored PMLSMs at Kyushu University . . . . . . . . . . 72.1.2 The MIT levitated stage based on air-cored PMLSMs . . . . . . . . . . . . . . . . . . 82.1.3 Solutions not based on PMLSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 PMLSMs modelling and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 General literature on PMLSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 The algebraic models developed by Nasar et al. . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Modelling of the slots in PM motors by Zhu, Howe et al. . . . . . . . . . . . . . . . . . 102.2.4 The numerical analysis of PMLSMs at Hanyang University . . . . . . . . . . . . . . . 112.2.5 Other numerical approaches to the analysis of PMLSMs . . . . . . . . . . . . . . . . 112.2.6 Study of cogging Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.7 Control of PMLSMs with constant airgap lengths . . . . . . . . . . . . . . . . . . . . . 12
2.3 Constrained state control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Polyhedral Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Invariance control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Another constrained control approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Results of the literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 The model of a PMLSM 16
3.1 Modelling outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Magnetic field produced by the permanent magnets . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Winding’s magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Slots effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Forces calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5.1 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5.2 Normal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Single motor dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.7 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 Discussion of modelling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
IV
4 Design for three controlled degrees of freedom 35
4.1 Setup description and modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Linear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Partial feedback linearization and composed control . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Full feedback linearization with invariance control design . . . . . . . . . . . . . . . . . . . . . 48
4.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5 Controllers’ performance comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Design for five controlled degrees of freedom 54
5.1 Setup description and modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Full feedback linearization and invariance control . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2.1 Feedback linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.2 Design of the invariant set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.3 Passification and design of switched gains . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Conclusions 69
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Contribution of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4 Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Appendices 71
A Symbols description 71
B Acronyms 72
C 5-DOF setup details 73
C.1 Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.2 Motors characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
C.2.1 Stator characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.2.2 Mover characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C.3 Devices arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Bibliography 78
V
List of Tables
3.1 Parameters of the motor considered for numerical calculations . . . . . . . . . . . . . . . . . . 24
4.1 Simulation parameters for the 3-DOF system . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Estimate of the controllable region for the linearization of the 3-DOF levitation system . . . . 39
5.1 Motor parameters used for simulation of the 5-DOF device . . . . . . . . . . . . . . . . . . . 645.2 Coefficients of the invariance function for the 5-DOF system . . . . . . . . . . . . . . . . . . . 655.3 Output constants for the passification of the 5-DOF system . . . . . . . . . . . . . . . . . . . 65
C.1 Stator characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.2 Mover characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
VI
List of Figures
1.1 Configurations of flat linear motors: a) double-sided, and b) single-sided. . . . . . . . . . . . . 31.2 Configuration of the PMLSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Types of slots: a) Open slots; b) semiclosed slots . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 Frame-set considered in the analysis of a PMSLM . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Charge distribution on a permanent magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Positions of the images of the magnetic charge in an iron-cored PMSLM . . . . . . . . . . . . 203.4 Effective airgap length and its linear approximation versus actual length . . . . . . . . . . . . 213.5 Magnetic field density on the stator’s surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Magnetic field density along x axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.7 µ0Hpmy1 as function of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8 Phase currents in the armature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.9 Mover at d = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.10 Mover at d 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.11 Modeling of the slots through the function g(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 273.12 Relative permeance and its first harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.13 Thrust as a function of the air gap length and the position of the mover . . . . . . . . . . . . 303.14 Normal force as a function of the air gap length and the mover position . . . . . . . . . . . . 313.15 Normal force as a function of the mover position . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 Configuration of three LSMs for three controlled degrees of freedom . . . . . . . . . . . . . . 364.2 Forces in the 3-LSMs configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Position of the 3-DOF system’s platen under LQR control for xd = [0.055, 0, 0, 0, 0, 0]T . . . . 404.4 Speed of the 3-DOF system’s platen under LQR control for xd = [0.055, 0, 0, 0, 0, 0]T . . . . . . 404.5 Control inputs u1 and u2 in the 3-DOF system under LQR control for xd = [0.055, 0, 0, 0, 0, 0]T . 414.6 Control inputs u3 and u4 in the 3-DOF system under LQR control for xd = [0.055, 0, 0, 0, 0, 0]T . 414.7 Block decomposition of the 3-DOF system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.8 Lyapunov and invariant sets for composed control . . . . . . . . . . . . . . . . . . . . . . . . 434.9 Region for which the control input u4 of the 3-DOF composed controller is real . . . . . . . . 454.10 Estimate of the domain of attraction of the 3-DOF composed controller for first set-point . . 454.11 Estimate of the domain of attraction of the 3-DOF composed controller for second set-point . 454.12 Transient position of the 3-DOF system under composed control for a first set of parameters 464.13 Transient speed of the 3-DOF system under composed control for a first set of parameters . . 464.14 Control inputs u1, u2, and u4 of the 3-DOF system under composed control for a first set of conditions 474.15 Transient position of the 3-DOF system under composed control for a second set of conditions 474.16 Transient speed of the 3-DOF system under composed control for a second set of conditions . 474.17 Control inputs u1, u3, and u4 of the 3-DOF system under composed control for a second set of conditions 474.18 Projection of M and N into the (x1, z) plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.19 Transient position of the 3-DOF system under invariance control. . . . . . . . . . . . . . . . . 514.20 Transient speed of the 3-DOF system under invariance control. . . . . . . . . . . . . . . . . . 51
VII
4.21 Control inputs u1, u2, and u4 for the 3-DOF invariance controller. . . . . . . . . . . . . . . . 524.22 Trajectory of the 3-DOF system under invariance control. . . . . . . . . . . . . . . . . . . . . 524.23 Transient airgap length of the 3-DOF system for the three controllers. . . . . . . . . . . . . . 524.24 Transient x and z positions of the 3-DOF system for the three controllers. . . . . . . . . . . . 524.25 Control inputs u1, u2 of the 3-DOF system for the three controllers. . . . . . . . . . . . . . . 534.26 Control inputs u3, u4 of the 3-DOF system for the three controllers. . . . . . . . . . . . . . . 53
5.1 Angular response of a disturbed 4-DOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Airgap length response of a disturbed 4-DOF system. . . . . . . . . . . . . . . . . . . . . . . 555.3 Setup for five degrees of freedom with four PMLSM . . . . . . . . . . . . . . . . . . . . . . . 555.4 Disposition of the movers on the platen for the 5-DOF device . . . . . . . . . . . . . . . . . . 565.5 Fixed and rotating frames of reference for the 5-DOF device . . . . . . . . . . . . . . . . . . 585.6 Position response of the 5-DOF system under invariance control . . . . . . . . . . . . . . . . . 655.7 Speed response of the 5-DOF system under invariance control . . . . . . . . . . . . . . . . . . 655.8 Transient angles of the 5-DOF system under invariance control . . . . . . . . . . . . . . . . . 665.9 Transient angular speed of the 5-DOF system under invariance control . . . . . . . . . . . . . 665.10 Control inputs u1, u2 and u3 for the 5-DOF system under invariance control . . . . . . . . . . 665.11 Control inputs u4, u5 and u6 for the 5-DOF system under invariance control . . . . . . . . . . 665.12 Trajectory x1(t) of the 5-DOF system under invariance control . . . . . . . . . . . . . . . . . 675.13 Trajectory x3(t) of the 5-DOF system under invariance control . . . . . . . . . . . . . . . . . 675.14 Trajectory x4(t) of the 5-DOF system under invariance control . . . . . . . . . . . . . . . . . 67
C.1 5-DOF device dimensions in mm of: a) mover b) stator . . . . . . . . . . . . . . . . . . . . . 74C.2 Single layer, three pairs of poles windings distribution . . . . . . . . . . . . . . . . . . . . . . 75C.3 Upper view of the forcer of the four motors setup. All dimensions measured in mm . . . . . . 76C.4 Upper view of the platen of the four motors setup. All dimensions given in mm . . . . . . . . 77
VIII
Chapter 1
Magnetic Levitation and Linear
Motors
1.1 Motivation
Due to well-established miniaturization trends in several sectors of industry –e.g., electronics, materials, andbioengineering– the manufacturing processes of certain components rely heavily on high precision positioningsystems. The improvement of these positioning systems is particularly important for two sectors of themanufacturing industry, nanotechnology and semiconductors, both of which require very high accuracypositioning and extremely clean environments.
Nanotechnology relates to devices whose dimensions are in nanometers. This sector is at an early stageof development but the number of its applications is growing fast [2]. Its growth is manifested by theincreasing demand for workers in the area [1]. The manufacture of nanodevices employs processes likephotolithography, molding, and micromachining [3]. In many cases, these processes require positioning withnanometric accuracy.
The semiconductor industry is a more mature sector that has grown steadily over several decades [5].Despite this maturity, a large amount of waste due to manufacturing problems motivates a search for thereduction of manufacturing costs and process time. Many of these problems arise during the photolithographyof micrometric structures that form the circuitry of the chips. On industry-wide average, this process succeedsfor only one in every two chips [6]. For an economic analysis that includes the cost of waste in semiconductorfoundry, refer to [7].
Dust contamination is a problem that affects the photolithography of semiconductors and nanodevicesbecause generates waste. This particular kind of contamination occurs when free particles reach the surfacewhere the nanostructures are being created [5]. In order to alleviate this problem, ultra-clean environmentsnormally house the photolithographic procedures. These controlled environments isolate processes insidea purified space where filtration systems capture the free particles in the air and operators wear so called“bunny suits”. However, existing ultra-clean environment systems are not sufficient to eliminate every freedust particle because they do not exclude every pollution source, particularly inappropriate equipment [6].Equipment based on mechanical transmissions and/or bearings, like standard positioning stages, releasesparticles into the environment because of friction. Therefore, real ultra-clean environments require thereplacement of standard actuators by contactless ones.
In an effort to reduce dust contamination, some commercial stages use airbearings –compressed air–to maintain a constant airgap length, which is the distance between the fixed and the moving part of anactuator. However, manufacturers’ specification sheets show that these devices provide neither airgap lengthcontrol nor very high accuracy.
Magnetically levitated stages are a better alternative for positioning systems because of their more robustnature. Several researchers, e.g., D.L. Trumper and K. Yoshida, have proven that magnetically levitated
1
systems can be equipped with both airgap length control and very high accuracy. Seeking for these twodesired features, these researchers have investigated different technologies that include different types ofelectromagnets and linear motors. While electromagnets can apparently be simpler, linear motors provideboth higher efficiency, because of a shorter and more uniform airgap length, and lager travel, as Chapter 2shows. Weighing those advantages, this study focuses on the design of a magnetic levitation system thatuses linear motors.
Linear motors are already used for positioning in semiconductor photolithography [48]. However, despitethe success of experiments that employ them for levitation, industry has not adopted this technology yet.This apparent reluctance to innovate may be motivated by the practical and economic implications of imple-mentation. These implications are exemplified by the solution proposed in [14], where the authors describea positioning system that is based on non-commercially available motors. Trying to overcome the implemen-tation obstacles, the present study suggests a levitation machine based on conventional technologies. Sincemany types of conventional linear motors are available, the next section compares the most common onesand introduces their fundamentals in order to ensure the selection of the machine that best suits levitation.
1.2 Review on linear motors
The selection of the best linear motor requires understanding both their properties and their operatingprinciples. Accordingly, the next two subsections present the common motor topologies and their underlyingtheory of operation.
1.2.1 Topologies of linear machines
Even though linear motors produce straight motion, they operate very similarly to rotary ones. In fact, thereis at least one kind of linear motor for every kind of rotary one. Like rotary machines, linear ones consistof a moving and a stationary part, the mover and the stator respectively. Either the mover or the statorbecomes the armature by generating a magnetic field that travels linearly. The remaining part of the motoris called the field.
Different applications induce different motor topologies, which fall into one of the following categories:
a) flat (planar) or tubular (cylindrical)
b) single or double-sided
c) iron-cored or air-cored (ironless)
d) slotted or slotless core
e) transverse or longitudinal flux
Each topology yields different advantages in terms of modelling and control design simplicity, travellength, stability, structural strength, and efficiency. The next paragraphs explain these categories andpresent their advantages.
Whether a motor is flat or tubular mainly affects the complexity of its modelling and its freedom oftravel. Rather than lying flat, in tubular motors, the stator (or the mover) rolls around the mover (or thestator) forming a tube. Thus, the modelling of flat motors is harder than that of tubular ones because thefields at the ends of a flat motor are not uniform. On the other hand, flat machines allow lateral and normal1
displacements whereas tubular motors obstruct the travel in directions other than the longitudinal one.The length of travel also differs for single-sided and for double-sided machines. Unlike single-sided motors,
double-sided ones limit the normal travel because they confine the mover between two flat stators. On theother hand, single-sided motors might be less stable than double-sided ones because their forces are unilateral,but control of their airgap length is easier because their structure is simpler (see Figure 1.1).
1A normal displacement occurs when the airgap length varies.
2
Mover
Mover
Stator
Stator
Stator
(a)
(b)
Figure 1.1: Configurations of flat linear motors: a) double-sided, and b) single-sided.
The terms iron-cored and air-cored specify the material that fills the armature’s core. In contrast toair-cored machines, the magnetic fields in iron-cored motors attract the ferromagnetic material of the core(s)producing strong forces.
Iron-cored motors are either slotted or slotless depending on whether the windings lie inside slots oron the surface of the core. When compared to slotless motors, slotted ones lead to more complex modelsbecause the slots violate uniformity assumptions; however, slots supply stronger structures.
When either longitudinal or transversal ferromagnetic laminations form the core, they determine thedirection of the main magnetic flux. Transversal laminations imply smaller magnetizing currents but longi-tudinal laminations improve efficiency by reducing both eddy current losses and the magnitudes of higherharmonics of the flux [38].
Apart from the previous topological categorization of linear motors, their winding distribution offers adifferent classification. According to [39], the most common winding configurations are:
a) single-layer full-pitch windings with an even number of poles and one slot/pole per phase
b) triple-layer winding with an even number of poles and one slot/pole per phase
c) double-layer winding with odd number of poles and one slot/pole per phase and half filled end slots
d) economic winding for low efficiency
A large number of layers and a proper disposition of the windings aid in obtaining an almost sinusoidaldistribution of the traveling magnetic field (see [69]), which significantly simplifies modelling, as seen inChapter 3.
From the observations above, flat single-sided configurations favour magnetic levitation because theyallow larger travel in more directions. Commercial flat single-sided motors are commonly available in air-cored and longitudinally laminated iron-cored configurations.
1.2.2 Principles of operation of linear motors
The above topological classification of linear motors provides the concepts required to understand theiroperating principles, which are next explained.
3
To begin with, the forces in a linear motor are either electrodynamic or electromagnetic. Within thefirst class, also known as Lorentz type forces, are those caused by immersing current-carrying conductorsin magnetic fields as well as those caused by the interaction between a magnetic field and its reaction fieldgenerated by induced currents. On the other hand, electromagnetic forces are caused by the interactionbetween the armature’s travelling magnetic field and the field’s non-uniform magnetic properties.
Because of their planar nature, flat linear motors produce both thrust (or propulsion force) collinearto the travel, and a normal force perpendicular to the thrust. While the thrust is caused by the travel ofthe armature’s magnetic field, the normal force depends on the structure and composition of the motor. Ingeneral, this normal force has two components:
a) a repulsive force caused by the interaction of field’s and armature’s magnetic fields, and
b) an attractive force between the permanent magnets and the ferromagnetic cores.
When normal forces are present, motor designers usually fix the airgap length by employing linear bear-ings, electromagnetically controlled suspensions, or air bearings. As mentioned before, the occurrence ofthese forces depends on the motor’s structure, which determines its principle of operation. This principle ofoperation defines several categories of motors, i.e., induction, synchronous, and direct current. Since directcurrent motors cannot achieve levitation, this section studies only induction and synchronous machines.
Linear induction motors
In LIMs, as in rotary induction motors, the armature constitutes the primary and the field the secondary,i.e., the armature (the primary) is externally excited and its magnetic field induces currents on the field (thesecondary) windings . The primary windings are distributed such that, when excited by a polyphase supply,they generate a travelling field that has forward, backward, and pulsating components. The secondary iseither a metallic cage or a short-circuited three-phase winding. In less usual configurations, the secondaryis:
a) a piece of laminated or solid iron
b) a continuous variable reluctance structure
c) a conducting plate backed or not by ferromagnetic material
In any case, the currents that the primary’s magnetic field induces into the secondary generate a reactivemagnetic field. The interaction of these two fields produces a motion that can be controlled by regulatingthe phase of the primary’s currents [37].
Conventional systems based on LIMs regulate the armature currents to control thrust, speed, or positionand use bearings to restrict motion into the longitudinal single degree of freedom. Nevertheless, recent re-search showed that LIMs can simultaneously provide propulsion and suspension. As shown in [81], suspensionresults from the control of the intensity of the airgap flux, which determines the magnitude of the normalforce. However, the intensity of this flux also determines the magnitude of the thrust; thus, independentcontrol of thrust and normal force requires combined power supplies that feed the primary with two differentfrequency components simultaneously. While a low frequency component provides both thrust and normalforce, a high frequency one affects only the normal dynamics. Besides requiring complex drivers in order toachieve levitation, LIMs suffer from strong cogging forces, which are undesired forces that produce oscilla-tions and are not considered by conventional models –refer to Section 2.2.6. In addition to high complexityand oscillations, a market exploration showed that LIMs are mostly used for transportation applications;thus, most of the available LIMs are large motors designed to provide high thrust while minimizing normalforce. For these reasons, commercial LIMs are not suitable for high precision positioning stages.
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pm pm pmpm
Back Iron
MOVER
Laminated ferromagnetic material
STATOR
Slots Windings
Figure 1.2: Configuration of the PMLSM
Linear Synchronous Motors
The motion of a LSM is in synchrony with a travelling magnetic field produced by a polyphase armaturethat is excited by either AC or switched currents. The field can be a variable reluctance structure or anarrangement of permanent magnets. Three common types of LSMs result from the combination of theabove kinds of excitation with these choices of fields, namely, Linear Variable Reluctance, Linear SwitchedReluctance, and Permanent Magnet Linear Synchronous Motors.
In a Linear Variable Reluctance Motor (LVRM), the secondary is a variable reluctance structure –eithernotched or segmented– whose pitch equals the primary’s coil one. This structure’s tendency to align withthe travelling field generates movement. Since the position of the travelling field depends on the phase ofthe armature currents, the phase angle of these alternating currents determines the position of the mover.For a more detailed description of LVRMs, see [23] and [38].
Similar to LVRMs in topology and operation, Linear Switched Reluctance Motors (LSRM) differ inthat only one phase is energized at a given instant. Either linear position sensors or estimators trigger theexcitation of each phase. Since excitation is switched, the LVRM’s thrust pulsates during the overlappingof two phases and produces noise and vibration. These effects affect especially the stability of the normaldynamics because the normal force is much stronger than the thrust. Moreover, these motors provide limitedlongitudinal resolution because of their switched nature.
The third common type of LSMs is the Permanent Magnet Linear Synchronous Motor (PMLSM), inwhich an array of permanent magnets composes the field. In this case, thrust results when the field of themagnets tends to align itself with the travelling field of the armature. If, besides thrust, an attractive normalforce is desired, the armature incorporates ferromagnetic material in its core. The attraction between thecore and the magnets produces a normal force that is strong enough for levitation. If the design capitalizeson this attractive force, smaller armature currents are required. In addition to this advantage, the quasi-sinusoidal field distribution of the magnets’ arrangement facilitates a smooth travel. Therefore, both thestrong normal force and the capability of smooth travel render PMLSMs the best choice for high precisionmagnetic levitation.
1.3 Purpose and outline of the thesis
After justifying an initial goal, this chapter explored the possible paths to its completion. Now, the researchgoal can be narrowed down by choosing the best of those paths. Accordingly, this work addresses the designproblem of a magnetically levitated high precision positioning system actuated by a set of permanent magnetlinear synchronous motors. This research seeks a complete solution that includes both modelling and thedesign of controllers.
Chapter 2 introduces previous research on magnetic levitation for high accuracy positioning tasks. Thisresearch includes solutions based on both linear motors and other technologies. Next, the chapter studiesPMLSM models obtained through different methodologies while evaluating their suitability for levitationcontrol. Finally, the investigation compares different control methods in terms of their practical advantages.
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The innovative development starts in Chapter 3, which describes the derivation of a new PMLSM modelthat relies heavily on results from the literature review. The development first derives the magnetic fieldproduced by the permanent magnets and then finds the components of the travelling magnetic field in thearmature. Next, the model incorporates the effect of the slots on the armature’s core by calculating therelative permeance of the ferromagnetic body. After using the previous results to analytically find the forcesin the motor, the procedure yields a full model of the device. A brief discussion on disturbances modellingcloses the chapter.
The work in Chapter 4 uses the results in Chapter 3 to derive the model of a three degrees of freedomlevitation system actuated by three PMLSMs. After completing the modelling, the chapter presents thedesign of three alternative controllers. The first one is a simple LQR controller and the second one isobtained using methods from nonlinear control. The third controller is based on full feedback linearizationand invariance control. Chapter 4 concludes by comparing the performance of the three controllers.
Chapter 5 introduces a more realistic levitation device which controls five degrees of freedom instead ofonly three. Only one controller is considered for this more complex system. This new controller relies onfull feedback linearization and invariance control. Its simulation results close this chapter.
Finally, Chapter 6 summarizes the main contributions of this thesis and outlines future research direction.
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Chapter 2
Literature review
As seen in Chapter 1, magnetic levitation is not a new problem in engineering. Several researchers haveproposed magnetic levitation approaches to high precision positioning tasks, but their experimental solutionshave not evolved yet into popular appliances because of cost and implementation factors. A clear example isthe successful work conducted by Trumper et al. in [11]-[16], where these authors introduce a solution basedon custom technologies whose production would imply special manufacturing procedures. Nevertheless, thestudy of such antecedents serves this research in three ways: first, it shows that the objectives are feasible;secondly, it helps to foresee possible restrictions; thirdly, it exposes the advantages and limitations of thealready studied technological options.
This chapter gathers these antecedents in three main sections. The first section briefly describes andcomments on prior approaches to the high precision magnetic levitation problem. The second one presentsresearch focused on the development of models and controllers for synchronous machines. The third andlast part introduces the background required for the control design proposed in this thesis. Since this reviewembraces different areas of knowledge, the references are organized according to the field they belong to and,more specifically, by research group.
2.1 Previous work on magnetic levitation
This section introduces three independent lines of work on high precision magnetic levitation. The firstone, developed by Yoshida et al., uses PMLSMs for transportation purposes. The second one, conceived byTrumper et al., is a 6-DOF stage powered by ironless PMLSM. The third and last line of work deals with agroup of similar devices based on electromagnets.
2.1.1 The vehicles levitated by iron-cored PMLSMs at Kyushu University
Despite their travel limitations, the experiments carried out by Yoshida et al. demonstrated the feasibilityof attractive magnetic levitation with PMLSMs. Targeting transport applications, these authors designeda vehicle levitated by motors of this kind. As described in [8], they designed a system based on iron-coredPMLSMs with hybrid PMs, i.e., permanent magnets whose magnetic field is controlled by coils woundaround them. This system compensated for cogging forces (refer to Section 1.2.2) by controlling the currentsin the coils of the hybrid magnets. Since the device used a set of rubber rollers to maintain the airgapconstant, it did not prevent friction. Moreover, these rubber rollers eliminated the normal dynamics; thus,the researchers did not need to calculate the normal force and focused only on the thrust. They derived anonlinear expression for the thrust taking into account the airgap length variation. Their work, as presentedin [8], concluded by comparing 2-D FEM analysis and experimental results.
Later, Yoshida repeated the experiment without rubber rollers and reported the results in [9]. In thisnew experiment, a larger airgap compensated for the lack of bearings. Also, a 3-D FEM approach improved
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the accuracy of the analysis by accounting for the distortion of the fields at the borders of the motor. Exper-iments showed that the 3-D FEM analysis was more precise than the 2-D one because, when both methods’results were compared, 3-D FEM analysis estimated smaller forces that better matched the experimentalmeasurements.
While the research in [8] and [9] studied attractive levitation, the work in [10] focused on repulsivesuspension. Because this new problem has a different nature, air-cored PMLSMs replaced the iron-coredmotors. In order to control both propulsion and lift, the authors proposed a modified version of direct torquecontrol, a technique commonly used to control rotary motors. This modified DTC divided the dq plane1 into12 regions instead of the 6 used in conventional DTC. A look-up table stored different constant values thatdetermined the voltages applied to the motor’s windings on each region. This switched control approachshowed satisfactory performance in the face of a noisy speed output.
2.1.2 The MIT levitated stage based on air-cored PMLSMs
The work in [11] leads a series of papers on PMLSM-based levitation systems developed by Trumper et al. atthe Massachusetts Institute of Technology. This paper reviews the fundamentals of Halbach magnet arrayswhile mentioning some of their applications. The first applications are magnetic bearings for photolithog-raphy stages. As a second application, the paper introduces linear motors along with a PMLSM modelobtained through a direct solution of the magnetic fields. The last application is hybrid electromagnetsfor Maglev trains. This work remarks the higher efficiency of the Halbach arrays when compared to otherexisting magnet arrays.
The subsequent work, presented in [12], used the results in [11] to obtain a model of single sided air-coredPMLSMs. To this end, the procedure in [12] expanded the analysis of the magnet array while incorporatingthe study of the magnetic field of the armature. The authors employed the resulting motor’s model tocontrol a six-degree-of-freedom magnetically levitated xy stage driven by four custom motors. According tothe authors, implementation of the stage produced good results with accuracy of the order of hundreds ofnanometers.
As presented in [13], the authors designed a levitated stage that overcame the travel limitations of thedevice presented in [12]. The new device permitted a range of motion of ±50mm×±50mm on the horizontalplane, ±200µm on the airgap, and ±600µrad on each angle. Lead-lag compensators controlled the newlevitated stage producing the results reported in [14]. Accordingly, a linearization replaced the originalmodel decoupling the dynamics of the six degrees of freedom. The authors reported in [15] the details of themodelling and the lead-lag control design of this levitation machine. As [14] shows, the system’s responsesuffered from overshoots as large as 20µm when the device was tested with reference step input as smallas 20nm. Trying to overcome this undesired effect, Kim and Trumper designed an LQR to optimize theperformance of the linear controller. They outlined their slightly improved results in [16].
In summary, despite the significant nonlinearity of the systems, this line of research focused only on linearcontrollers thus limiting the range of operation and the robustness of the control system. The developmentin this thesis seeks to overcome these two drawbacks.
2.1.3 Solutions not based on PMLSMs
Magnetic levitation not based on PMLSMs is another prolific subject of research. The next paragraphs showhow several researchers have proposed different levitation systems that significantly differ in their theoreticalfoundations.
In [17] Menq et al. explain the design of a device that levitated a platen while controlling its six DOF.This device consisted of 10 electromagnets distributed on the walls of a cage that housed the platen. Theauthors proposed a robust feedback linearizing controller that provided the device with both a controlledrange of motion of ±4mm×±4mm on the horizontal plane, ±2mm on the vertical direction, and ±1o on eachangle, and an accuracy of the order of nanometers and microradians.
1The dq plane is used to represent the direct and quadrature components of currents and voltages through the windings ofelectric machines. For more details on dq decomposition refer to [23].
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At the National Taiwan University, research groups from three different departments worked on twosimilar levitation projects. They published their 2-DOF and 5-DOF designs in [18] and [20], respectively.Each system consisted of a levitated platen with salient PMs and a stationary platform with slots insidewhich the platen magnets were driven. The PMs fields interacted with those of hybrid magnets mountedaround the platform slots; thus, by controlling the currents on the hybrid magnets, both systems regulatedthe forces exerted on the platens. Both designs included adaptive controllers that achieved micrometricresolution (4µm×6µm×40µrad×40µrad×60µrad for the 5-DOF one) within short travelling ranges limitedby the structure of the fixed platform. The work in [19] improved the one presented in [18] by increasingthe controlled degrees of freedom up to six. In the new setup, an adaptive sliding mode controller drove theplaten within a range of motion of ±40mm in the longitudinal direction, ±1.5mm on the lateral one, and±8mm for the vertical airgap.
These results show that electromagnet-based levitation requires complex control methods and that theachievable ranges of motion are small in comparison to those of PMLSMs. Moreover, although these papersdo not evaluate either the efficiency or the energy consumption of the electromagnets, PMLSMs demonstratehigher efficiency by guiding the flux lines through shorter and well-defined paths. Therefore, PMLSMs suitmagnetic levitation better than electromagnets.
2.2 PMLSMs modelling and control
As Section 2.1 made evident, PMLSMs suit levitated positioning applications. This section concisely intro-duces several studies concerning this family of devices. After commenting on some references on PMLSMsfundamentals, the next paragraphs present the analysis of these motors as conducted by several researchgroups. The different models conceived by Nasar et al. head the survey, followed by the contribution byZhu, Howe et al. Next, an outline of the work at Hanyang University serves as a preamble to other numericalapproaches. The analysis of a few papers on cogging forces complements these numerical studies. Finally,the section discusses some recent control designs for positioning systems actuated by PMLSMs.
2.2.1 General literature on PMLSMs
In [21], McLean overviews the applications of linear motors and offers a classification of their analysismethods. While naming their advantages, the paper presents different topologies and their use in industrialapplications and transportation systems. The subsequent analysis methods overview mainly focuses on linearinduction motors including only a few paragraphs about linear synchronous motors. According to McLean,researchers have analysed LSMs through different approaches that can be grouped in three main categories:
a) one and two dimensional methods based on direct calculations of the field and magnetic circuit analysis
b) layers methods, including current sheets and Fourier analysis
c) finite element methods.
A short section on design aspects concludes the survey offered in [21].By including technical and theoretical aspects in [23], Gieras and Piech provide a more complete overview
of PMLSMs. Among other topics, the work in [23] studies PMLSMs constituting materials, electromagneticparameters, and thrust. In order to obtain an expression for the thrust, the authors derive an expressionfor the magnetomotive force of the windings. Next, they compute the force through two different methods:power balance and Lorentz equation. Although [23] does not provide a model of the normal force, it providesvaluable information on control methods, sensing, and applications. This book also presents case studiesand shorter examples along with the theory.
2.2.2 The algebraic models developed by Nasar et al.
Nasar, in collaboration with other researchers, analyzed the forces in iron-cored PMLSMs through differentanalytical methods. This section presents the results generated by two of these methods.
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By using current sheets and Maxwell equations, Nasar et al. obtained a smooth approximation to the fieldproduced by the PMs of an iron-cored PMLSM. As these authors showed in [40], an analytical expression ofthe mover’s magnetomotive force leads to the solution of Maxwell’s equations. The comparison of analytical,FEM-based, and experimental results showed that the analysis is reliable; however, its accuracy is affectedbecause the model does not completely reflect slot effects.
The work in [41] used the results from [40] to derive formulas for the normal force and the thrust. Byadapting theory from rotary motors, the authors algebraically obtained expressions for the electric parametersof the motor. These expressions and power balance identities yielded formulas for the forces. Theoreticalresults in the paper closely match experimental ones.
The book [39] gathers the methods developed in [40] and [41]. After obtaining a linear dynamic modelbased on the formulas derived in [41], the authors briefly present both rectangular and sinusoidal currentcontrols for PMLSMs. Since their method expresses the forces as infinite sums, the authors truncate themto obtain approximations that do not consider the nonlinearities of the motor’s dynamics.
Following a different approach, in [42] and [43], Nasar and Xiong developed a partly-numerical model forPMLSMs. In [43], the authors computed the magnetic field produced by PMs mounted on the surface of adisk machine. Two main ideas make this work original. The first one is the use of the concept of magneticcharge to find a system of differential equations that characterizes the PMs magnetic field. The second ideais the solution to that system through the method of images.
Extending the method in [43],[42] describes the modelling of the forces in an iron-cored PMLSM. There,an analytical procedure first yields an implicit expression for the magnetic field density of the PMs. Then,in order to find the stator’s field, the procedure uses the stator’s magnetomotive force to solve the scalarpotential equation. The combination of the fields from the PMs and the stator results in closed formexpressions for the thrust and the normal force. Finally, the authors compare the theoretical results to thoseobtained through FEM analysis and experiments. This model does not completely include the effect of thestator slots but permits its further consideration because it relies on the computation of the airgap fluxdensity, which is directly affected by the slots.
2.2.3 Modelling of the slots in PM motors by Zhu, Howe et al.
Zhu et al. wrote a series of four articles that focus on the calculation of the magnetic field in brushlesspermanent magnets DC rotary motors. Some of their results are useful to the development of the PMLSMmodel later presented in Chapter 3.
In [44], the authors calculate the PMs magnetic field in motors with either internal or external rotor. Theyoutline a two dimensional analytical method that requires the solution of the governing Laplacian/quasi-Poissonian field equations. This method generates infinite sums that describe the magnetic field densitydistribution both at the airgap and at the stator surface. According to the authors, the analytical resultsare consistent with those produced by FEM analysis simulations.
In the second article, [45], the authors compute the armature reaction field through a two dimensionalprocedure. Using the current density distribution on the coils as boundary conditions, the authors of [45]solve the governing Laplacian equation of the scalar magnetic potential and find the magnetic field densityas an infinite sum. By introducing the winding distribution factor, [45] extends these results to three-phasewinding motors. Again, theoretical results successfully predict FEM analysis ones.
The work in [46] continues this research by introducing the concept of relative permeance in order toaccount for the slot effect in the magnetic field distribution of a PM DC motor. To this end, the authorsderive a two one-dimensional and one two-dimensional model that describe the relative permeance of theslotted stator. Finally, the authors incorporate the slot effect into the motor’s model by multiplying theairgap’s field density by the relative permeance. The concepts developed in this paper apply not only topermanent magnet DC motors, but also to any kind of slotted iron-cored motors.
This line of research concludes in [47] by introducing switching controllers in the presence of unknownload. The procedure in [47] first finds the components of the open-circuit and armature reaction magneticfields. Next, it calculates the relative permeance as a function of the instantaneous position of the rotor andthe phase of the currents. A new expression of the field density distribution incorporates this permeance
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function to yield a final expression for the field in the airgap. Experimentally obtained plots indicateagreement with theoretical values.
2.2.4 The numerical analysis of PMLSMs at Hanyang University
Jung, Hyun, et al. published three articles related to this thesis. The first one, [24], evaluates the efficiencyof both short primary and short secondary PMLSMs in terms of the relationship between cogging forces andthrust. To this end, a 2-D FEM analysis relying on Maxwell’s theory is used to compute the thrust, detent,and normal forces. This analysis shows that, for cogging forces reduction, varying the width of the magnetsis better than skewing them because the first method does not reduce the thrust as much as the latter onedoes. The authors also conclude that, since force ripple is smaller in short secondary than in short primarymotors, the first type of devices suffers weaker end effects.
In [25], the authors improve the results presented in [24] by replacing the original 2-D FEM procedure witha numerical 3-D analysis based on the equivalent magnetic circuit network. The results from the numericalanalysis determine the optimal magnet length for both skewed and non-skewed secondaries. In addition, [25]provides a brief description of the new 3-D numerical method while remarking its computational advantagesover the equivalent 3-D FEM analysis.
By applying the same numerical method, [26] extends the work in [25] to skewed PMs motors. In [26],the authors explore how the motor’s parameters affect the static thrust, normal, and detent forces. Theseparameters include length of skew, overhang length, width of PMs, air-gap length, and thickness of PMs.Through repeated simulations, the authors assess the influence of the parameters on the performance of themachine. From simulation results, they also estimate the optimal length of skew.
Since the work presented in this subsection focuses on the evaluation of motor’s performance, it does notcontribute to the development of analytical models. Nevertheless, the authors’ conclusions are useful to theselection of the best motor topology and dimensions.
2.2.5 Other numerical approaches to the analysis of PMLSMs
In [27], Mizuno and Yamada explain how they used magnetic circuit analysis to determine the effects of PMsize variations on the detent force of an iron-cored PMLSM. Through a 2-D FEM procedure, the authorsobtained the forces in terms of infinite sums that are later numerically evaluated. A numerical analysisyielded an optimal relationship between the magnet width and the pole pitch. Experimental results verifiedthose obtained through numerical analysis. Like the researchers at Nanyang, Mizuno and Yamada did notderive a mathematical model of the motor.
Rather than magnetic circuit analysis, Shiying et al. used an FFT-based algorithm in order to modelthe forces in iron-cored PMLSMs. The algorithm first finds the equivalent current sheets in the motor andthen solves Maxwell’s equations for the magnetic vector potential, which is given as a Fourier series. Thesolution to this equation then yields an expression for the magnetic field density. Finally, Maxwell’s stresstensor generates expressions for the forces in the motor. In [29], the authors apply their method to bothshort primary and short secondary motors. A satisfactory experimental verification concludes their work.
According to Lim et al., FEM is too time-consuming for early stages of motor’s design. Therefore,they propose a less accurate but faster method based on equivalent magnetizing currents. In [28], theseauthors explain how this method finds the magnetic field density on the airgap of a short secondary iron-cored PMSLSM. From the field density in the airgap, they compute the forces in the motor and comparethem to those experimentally measured. According to [28], the method of equivalent magnetizing currentsalgebraically considers the slots effects; thus, the resulting model includes detent forces. Similarly to FEManalysis, this method expresses the resulting magnetic field as an infinite sum.
Because of its numerical nature, the work outlined above does not support the analytical modelling ofPMLSMs; nevertheless, it provides valuable design guidelines.
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a) b)
Figure 2.1: Types of slots: a) Open slots; b) semiclosed slots
2.2.6 Study of cogging Forces
In LSMs, the interaction between slots and permanent magnets produces residual forces known as end-effects.Cogging forces are undesired effects that include end-effects and other phenomena.
At the present time, researchers have only numerically and experimentally studied cogging forces. Theyhave not derived analytical expressions for those forces because very complex models would become necessary.Although the available results are somewhat heuristic, PMLSMs models should take them into account inorder to produce high performance designs.
In [35], Eastham et al. show how end-effects reduce LSMs’ performance. In [34], Cruise and Landyprovide both a more detailed classification and a description of such effects while analyzing the effectivenessof several techniques to reduce the cogging forces.
In [35], Akmese and Eastham provide a brief introduction to cogging forces in PMLSMs. First, theauthors explain the origin of these forces. Then, they propose a test rig that generates experimental datawithout resorting to more expensive actual linear machines. The experimental data thus generated matchesthe outcome of an FEM simulation. The work in [35] includes results from experiments and simulations ofskewed magnets motors with either NdFeB or ferrite magnets. A comparison of the results obtained withboth magnetic materials demonstrates that NdFeB provides much better efficiency.
The research in [34] gathers several useful results on cogging force reduction while extending them totwo different configurations, namely, buried and surface mounted magnets. By applying FEM analysis, theauthors estimate how much cogging forces are reduced by magnet length optimization, skewing, semiclosedslots (see Figure 2.1), and magnetic slot wedges use. Their results show that
• cogging forces are sinusoidal as predicted
• skewing helps only for surface mounted magnet configurations
• skewing and airgap increase reduce the maximum static thrust
• semiclosed slots and PI control are not efficient in reducing cogging forces
Without carrying out experimental verification, the authors of [34] concluded that skewing is the mosteffective method among those analyzed but designers should also consider semiclosed slots and “herringbone”skew of the magnets.
Despite considering only motors with constant airgap length, the concepts developed in [35] and [34] arevaluable because cogging forces occur not only in PMLSMs with fixed airgap lengths but also in those withvariable one (levitation systems).
2.2.7 Control of PMLSMs with constant airgap lengths
The control of PMLSMs with constant airgap is substantially different from the control required for levitation;however, the longitudinal dynamics present a similar structure in both cases. Therefore, constant airgapmodels provide hints that are helpful when modelling motors with variable airgap lengths. For this reason,this section presents previous research on control methods that compensate for the simplifications madeduring the modelling of PMLSMs operated under constant airgap lengths.
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In [48], Kiong et al. provide a comprehensive overview of the practical aspects of precision control ofPMLSMs. Besides listing several applications of precision positioning systems, the authors briefly describePMLSMs. The second chapter of the book studies the longitudinal dynamics with particular attention tocogging forces and friction. Considering those longitudinal dynamics, a composite controller solves the servoproblem in that chapter. This controller combines feedforward, PID, and nonlinear Radial Basis Functionbased compensators. Later, a robust adaptive controller attenuates the thrust ripple. Chapter 2 concludeswith the design of an iterative learning controller. Next, Chapter 3 models the friction in PMLSMs bymeans of relays and system identification. Simulations of an adaptive design that compensates for thisfriction complete this chapter. Chapter 4 addresses the more complicated control of xyz positioning systemsbased on PMSLMs. Although the control techniques described there target a 1-DOF problem, the bookoffers solutions to implementation aspects that have to be considered in the magnetic levitation problemtoo.
Hong, Qiang and Zehngchun address an additional problem of PMLSMs, i.e., magnetic saturation. Ac-cording to their work in [49], magnetic saturation is a detrimental phenomenon which flux-weakening controlcan compensate for. This paper explains the effects of magnetic saturation on interior magnets PMLSMs andproposes independent control strategies for the direct and quadrature currents. According to the authors,the controllers performed well in simulations but experimental results are not presented.
Following a different approach, in [50] Limei, Qingding, et al. present a robust controller that alsoconsiders friction. They propose a dynamic model of the motor that includes uncertainties in a coprimefactor characterization. Then, the authors introduce a parameter estimator and an H∞ robust control law.
In [51], a new model that also considers load variations and thrust ripple replaces the one proposedin [50]. In this case, Limei and Qingding present a disturbance observer that estimates the disturbances’amplitudes. To further improve this design, a disturbance observer based on fuzzy variable structure controlundertakes acceleration, speed and position control. Although this work focuses on motors with constantairgap, its disturbance model can be extended to higher DOF, as required by magnetic levitation.
2.3 Constrained state control
Because the control design in Chapters 4 and 5 relies on constrained stabilization, this section reviews severaldifferent approaches to the solution of this problem. First, this section briefly describes the polyhedralLyapunov functions method as introduced in several papers by Blanchini et al.. A review of invariancecontrol, as conceived by Mareczek and Buss, follows Blanchini’s work description. Finally, an approachproposed by Saberi et al. is outlined.
2.3.1 Polyhedral Lyapunov Functions
A series of papers by Blanchini et al. [53]-[56] describe numerical methods for the stabilization of linearsystems with constraints on the state. The main idea of this line of research is to replace conventionalquadratic Lyapunov functions with polyhedral ones. When compared to the ellipsoidal sets defined byquadratic Lyapunov functions, convex polyhedra provide better approximations to certain sets with non-smooth boundaries; therefore, polyhedral functions yield larger domains of attraction.
In [54], the authors propose a method that uses polyhedral Lyapunov functions to stabilize uncertainlinear systems. After describing the properties of polyhedral functions, they present a variable structurecontroller that maintains the system trajectory inside a polyhedron. According to [54], this method appliesto any plant that can be stabilized by means of a quadratic Lyapunov function.
Applying the above method, the work in [55] stabilizes a magnetically suspended metallic ball. Inthis paper, the authors constrain both the state and the input of an uncertain model that describes theexperiment. After computing a polyhedral estimate of the domain of attraction, the method generates apiece-wise continuous control that regulates the position of the ball. Experimental results show agreementwith theoretical prediction.
Including control laws for both continuous and discrete time systems, [53] offers a complete and detailedoverview of [54] and other related work. This paper outlines two algorithms that numerically compute
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polyhedral Lyapunov functions for both types of systems. Next, it provides a third algorithm that uses theresulting Lyapunov functions to design a controller. The paper finally applies this method to a two tank-system obtaining a satisfactory response. However, as mentioned in [56], this procedure is computationallyvery consuming and requires the estimate of the domain of attraction to be centred at the set point, thusrestricting its size.
2.3.2 Invariance control
Introduced by Mareczek and Buss in the following series of articles, invariance control relies mainly onpassivity and switched controls. This method first passifies the plant such that it becomes stable regardlessof the value of an additional control input. Then, it achieves invariance by switching this additional inputevery time that the system’s trajectory reaches the boundary of a predefined set. Of all the control approachespresented in this section, invariance is the simplest and the only one that also applies to certain classes ofnonlinear systems.
The work in [57] introduces the invariance method for a class of uncertain SISO non-minimum phasenonlinear systems. This class includes systems that satisfy a matching condition and whose internal dynamicscan be decoupled. Starting with a non-perturbed system, [57] presents sufficient conditions that ensureboundness and asymptotic stability inside an invariant set whose boundary is the zero set of a suitablesmooth function. Next, the paper provides similar conditions for the perturbed case by shrinking theinvariant set in order to account for the upper bound on the perturbations. This article proves the existenceof a stabilizing control law but does not provide guidelines to design it.
The work in [58] presents results akin to those in [57]. In this case, the authors propose invariance controlfor systems with a linear part and nonlinear internal dynamics. By first designing a set that should be madepositively invariant by feedback control, the authors state conditions that ensure both boundness of theinternal dynamics and asymptotic stability of the linear subsystem. They then explain the design of a timevarying control law that attains positive invariance of the set designed earlier by switching gains every timethat the trajectory of the system reaches its boundary.
References [60] and [59] present results that are more detailed and applicable. These articles outlineinvariance control methods for classes of cascade and non-cascade nonlinear systems respectively. Afterpassifying the linear part of such plants, these methods yield a switched control law based on a suitabledefinition of an invariant set. Finally, the authors provide guidelines for the design of this set.
2.3.3 Another constrained control approach
In [56], Saberi et al. outline a method that stabilizes linear plants while constraining both the state and thecontrol input inside closed convex sets. In this method, the properties of the state and input constraintsinduce a classification that determines the solvability of the problem. The authors propose a nonlinear timevarying control law for the problems that satisfy the structural conditions imposed by that classification.Although different kinds of constraints are allowed, the solvability conditions are restrictive and limit theutility of the method.
2.4 Results of the literature review
This chapter outlined the advantages and disadvantages of several technological alternatives for magneticlevitation. As a result, PMLSMs appeared to be the best option and thus Chapter 3 will focus on themodelling of these devices. The model derived in that chapter mainly relies on the developments of Nasaret al. and Zhu, Howe, et al..
As Chapter 3 will make evident, the model of a PMLSM with free normal dynamics is not globallystabilizable. Moreover, the controllers designed in Chapters 4 and 5 present singularities. Consequently,constrained state control methods will be required. An evaluation of some of these methods revealed thatinvariance control offers simplicity and lower computational cost during execution time. Therefore, thecontrol designs in Chapters 4 and 5 will rely on invariance control.
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Now that a review of literature on high precision magnetic levitation and PMLSMs has determined thedirection of this work, the next chapter will investigate the modelling of PMLSMs with free normal dynamics.
15
Chapter 3
The model of a PMLSM
In Chapter 2, a comparison of different approaches for magnetic levitation showed their suitability for highprecision positioning applications. Particularly, that study examined systems based on either electromagnetsor PMLSMs. The comparison showed that PMLSMs are better suited because they offer larger travel;consequently, the literature review thereafter focused on the modelling and control of these devices.
As Chapter 1 explained, commercial iron-cored PMLSMs better suit levitation because they provide astronger normal force. Motivated by reasons that will soon become evident, this work derives a new modelfor machines of that kind. More precisely, the new model represents a machine made of a short field (ormover in this case) moving underneath a stationary armature (or stator), as depicted by Figure 1.2. Inthis device, the permanent magnets occupy the surface of a flat structure of ferromagnetic material calledthe mover’s back iron. The stator is longitudinally laminated and transversally slotted in order to housethree-phase windings. Figure 3.1 shows the frame-set that the modelling procedure uses to abstract thismotor.
The present chapter explains the modelling procedure of the motor just described. This procedure heavilyrelies on the concepts in [42] and [46] by, to some extent, merging them. The chapter includes some of theirdetails primarily for the convenience of the reader, but also to distinguish them from the new development.
3.1 Modelling outline
Almost invariably, common applications of linear motors enforce constant airgap lengths by means of bear-ings; thus, they obstruct the motors’ normal dynamics. Researchers have developed magnetic and dynamicmodels of linear motors within this context. However, in order to achieve levitation, a design must releasethe normal degree of freedom and perform both airgap length and positioning control. Airgap length controlrequires the motor’s model to include the normal dynamics, which depend on the normal force.
As [42] explains, expressions for both the thrust and the normal force result from independent com-putation of the fields produced by the permanent magnets (Section 3.2) and the stator windings (Section3.3). The stator’s slots modulate those fields and affect the forces; thus, modelling of the slots improves thefidelity of the model (Section 3.4). In this particular case, the model incorporates the slots’ effects throughthe relative permeance of the armature, as in [46]. In order to complete the electromagnetic modelling,the procedure described in Section 3.5 computes the resultant magnetic field and the forces exerted on thesurfaces of the permanent magnets. To this end, that procedure extends the method presented in [42] byconsidering a variable airgap length. Section 3.6 shows how the normal force and thrust’s expressions yielda dynamic model. The chapter concludes by briefly introducing disturbances modelling in PMLSMs.
16
x 1i x 2i
t 1 b0
τ τ
g
hm N S
SN
NSN
S
p
y
x
z0
stator
mover
Figure 3.1: Frame-set considered in the analysis of a PMSLM
3.2 Magnetic field produced by the permanent magnets
In this section, the magnetic potential produced by the permanent magnets, Ψpm, is calculated through themethod of images and the concept of magnetic charge. Using the magnetic charge equivalency, the work inthis section first describes the magnetic potential around the magnets by means of an electrostatics’ systemof differential equations. The method of images then solves such system of equations for Ψpm leading to anexpression of the corresponding magnetic field intensity Hpm. Finally, the modelling uses Hpm to calculatethe density of the magnetic flux Bpm around the permanent magnets. In order to simplify the analysis, themodel relies upon the following assumptions:
Assumption 1. The length and thickness of the stator and mover back iron are infinite.
Assumption 2. The permeability of the stator and mover back iron is infinite.
Assumption 3. The demagnetization and recoil characteristics of the rare-earth magnets are linear andthe relative recoil permeability is 1.
Assumption 4. The permanent magnets are ideal and, consequently, exhibit a uniform magnetic potentialdistribution.
Assumption 3 requires that the permanent magnets be subject only to small variation of the externalmagnetic field, so that a straight line with slope equal to the recoil permeability replaces their hysteresischaracteristic loop.
Using the previous assumptions, Ψpm is calculated first by means of electrostatic theory and the conceptof magnetic charge. In the words of Stratton, “Within any closed region containing permanent magnetsand polarizable material, but throughout which the conduction current density J is zero, the magnetostaticproblem is mathematically equivalent to an electrostatic problem” [68]. This equivalence indicates solutionsto some magnetic problems through the application of methods developed for electric fields. Hence, theanalogy renders a solution to the field density produced by the magnets. The equivalence first defines themagnetic charge as
ρm := −∇ · M (3.1)
17
yielding the next relationshipsE → Hρ/ε → ρm
V → Ψm
p → M
where the variables are defined as
• E Electric field intensity
• H Magnetic field intensity
• ρ electric charge density
• ρm magnetic charge density
• V electric potential
• Ψm magnetic potential
• p polarization vector
• M magnetization vector
• ε permeance of the material
Accordingly, the Poisson equation for the electrostatic scalar potential describes the scalar magneticpotential produced by the permanent magnet, Ψpm, as
∇2Ψpm = ∇ · M. (3.2)
From (3.1) and (3.2), it follows that
∇ · M = −ρm (3.3)
∇2Ψpm = −ρm. (3.4)
Under the assumption of a uniform magnetization vector (Assumption 4), the next equation defines asolution to (3.4).
∇ × M = 0. (3.5)
For a body into the path of a magnetic field, the surface magnetic charge, σm, is given by
σm = M · n (3.6)
where n is the unitary vector normal to the surface (see Figure 3.2). By virtue of the magnetic charge analogyand from standard electrostatic analysis, the magnetic potential at a point P = (x, y, z) in the vicinity of aparallelepipedal permanent magnet with an uniform Ψm distribution is
Ψpm(x, y, z) =1
4π
(∫
V
ρmdV
r−
∫
S
σmdS
r
)
(3.7)
where ρm is now the volumetric charge density inside the permanent magnet, V is the volume of the magnet,S its surface perpendicular to the magnetization vector, and r the distance from P to a point inside the body.Because, for permanent magnets, M is ideally constant (M = M0), (3.3) implies ρm = 0. Moreover, onlythe surfaces S1 and S2 are non-parallel to the magnetization vector (see Figure 3.2); therefore, S = S1 + S2
and (3.7) simplifies to
Ψpm =1
4π
(∫
S1
σmdS1
r1−
∫
S2
σmdS2
r2
)
(3.8)
18
S 1
r 1
r 2dS 1
σm−
σm
S2
dS2
n
P
0M
Figure 3.2: Charge distribution on a permanent magnet
with r1 and r2 being the vectors from the centres of S1 and S2 to the point P . Since Hpm = −∇Ψpm, themagnetic field intensity becomes
Hpm =1
4π
(∫
S1
σmr1dS1
r31−
∫
S2
σmr2dS2
r32
)
. (3.9)
Because of Assumptions 3 and 2, the magnetic potential between any two points in the back iron is zero.If infinity is taken as the reference potential for which Ψpm = 0, then the potential at the back iron is alsozero. Provided that ideally the stator iron core also extends to infinity, its magnetic potential is zero as well.
Let hm be the height of the magnets and g be the length of the airgap (see Figure 3.1). Let y = 0correspond to the surface of the stator (parallel to the surface of the permanent magnet facing the stator).Then, from the previous discussion, the following boundary conditions hold for (3.4)
Ψpm
∣
∣
y=0= 0 (3.10)
Ψpm
∣
∣
y=hm+g= 0. (3.11)
The charges in the interior of the magnet and the potential on its boundary determine the potential onthe surrounding space [66]. These quantities relate to each other through the Green function that satisfies(3.4) subject to (3.10) and (3.11). Due to the symmetry of the model, this problem can be easily solvedusing the method of images.
The method of images helps finding the electric potential due to a charge in the vicinity of an equipotentialsurface, such as a conductor [67]. Its main idea is to replace the equipotential region by a charge, which isthe mirror of the actual charge with respect to such region. When it is located between two equipotentialparallel surfaces, a charge generates an infinite set of images. In iron-cored PMLSMs, due to the ferromagneticmaterial, the surfaces of the mover and the stator are equipotential surfaces.
According to [42], the polarity of the magnetic charge σm alternates from one image to another. Hence,for the i-th pole (magnet), the charge of its k-th image is
σm(i, k) = (−1)k+iσm. (3.12)
19
Lm
2
Lm
2
Lm
Lm
ge
2
Lm
2
ge
2
ge
ge
ge
Lm
2
Lm
Lm
2
ge
ge
3
ge
3
ge
ge
3
ge
Lm
2
Lm
2
Lm
ge
3
x
y
0
h
h
h
h
−1
−2
−3
−4
h0
h
h2
h3
h4
=
=
=
=
=
=
=
=1
=
4
+
+
+
−
−
−
−
...
−
−
4 −
.
.
.
.
.
.
Figure 3.3: Positions of the images of the magnetic charge in an iron-cored PMSLM
20
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
0.002
0.004
0.006
0.008
0.01
0.012
g (m)
ge
ff (
m)
geff
Linear approximation to geff
Figure 3.4: Effective airgap length and its linear approximation versus actual length
Because of the slots of the stator, different magnetic flux lines travel through different airgap spaces.Thus, the actual airgap length varies along the surface of the stator. Let hk be the position of the kthimage, ge the effective airgap length that considers the effect of the slots in the stator, and KC the Carter’scoefficient, which is a parameter specific to the topology of the motor [39]. Figure 3.3 shows the images’positions, hk, which are given by
hk =
(k + 1)hm + kge k odd
khm + (k + 1)ge k even(3.13)
where ge = gKc,
Kc =t1
t1 − gγ1, (3.14)
γ1 =4
π
b02g
arctan
(
b02g
)
− ln
√
1 +
(
b02g
)2
(3.15)
t1 is the slot pitch, and b0 the slot aperture, as depicted in Figure 3.1. Figure 3.4 shows the effective airgaplength.
Let LA be the depth of the poles along the z direction, τ the permanent magnets pole pitch (i.e. thedistance between the centres of two adjacent poles), pm the number of permanent magnets, and τp thepermanent magnets pole arc (see Figure 3.1). Respecting this nomenclature, the method of images solves(3.9) for any given point P = (x0, y0, z0) in the airgap. As a result, the y component of Hpm is given by
Hpmy(x0, y0, z0, g) =σm
4π
pm∑
i=1
∞∑
k=−∞(−1)k+i
[
arctan(x0 − x)(z0 − z)
(y0 − hk)Dk
]
∣
∣
∣
∣
∣
x=x1i
x=x2i
∣
∣
∣
∣
∣
z=LA/2
z=−LA/2
(3.16)
where
Dk =√
(x0 − x)2 + (y0 − hk)2 + (z0 − z)2, x1i =
(
i− 1 + α
2
)
τ, x2i =
(
i− 1 − α
2
)
τ, α =τpτ.
21
−0.050
0.050.1
0.150.2
0.25
−0.06−0.04
−0.020
0.020.04
0.06
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
xz
Bp
my
Figure 3.5: Magnetic field density on the stator’s surface
The previous equations produce a plot of Bpmy(x0, y0 = 0, z0, g = 0.01), shown in Figure 3.5, by firstevaluating (3.16) on a grid of values of (x0, z0), with y0 = 0 and g = 0.01m, and then multiplying theresulting array by the free space permeability, µ0.
Since a closed form for (3.16) does not exist, numerical methods are used to approximate it on the surfaceof the stator, y0 = 0. The approximation results from first substituting g = g0 and then averaging the fieldintensity along the z axis, i.e.,
Hpmyav(x0, 0) =2
LA
∫ LA/2
0
Hpmy(x0, y0 = 0, z0, g = g0)dz0. (3.17)
Numerical results revealed that equation (3.17) describes an almost sinusoidal function. Hence, a first orderFourier series approximation can replace Hpmyav without considerable loss of accuracy. The first Fouriercoefficient is given by
Hpmy1 =4
τ
∫ nτ+τ/2
nτ
Hpmyav(x0, 0) sin(π
τx0
)
dx0, (3.18)
where n = pm
2 and pm is the number of poles. Let µ0 be the permeability of free space and µr be the relativepermeability. Since Bpm = µ0(µrHpm + M0) (3.19)
and in the airgap Bpm = µ0Hpm, (3.20)
the approximation of the y component of the flux density due to the permanent magnets isBpmy ≈ Bpmy1 sin
(π
τx)
= µ0Hpmy1 sin(π
τx)
. (3.21)
Figure 3.6 shows the field intensity distribution along x and its fundamental harmonic. Note that thesinusoidal approximation is reasonably accurate in the range of interest.
Remark 1. As mentioned above, (3.21) holds for a fixed value of the airgap length g0. Since levitationinvolves a variable airgap length, Bpmy becomes a function of g. Modelling incorporates a dynamic g by firstrepeatedly calculating the coefficient µ0Hpmy1 for different values of this variable and then using the resultsto approximate the function Bpmy(g) through polynomial interpolation, as shown in Figure 3.7.
22
−0.05 0 0.05 0.1 0.15 0.2 0.25
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
x
Bp
my/B
pm
y1
Bpmy
Bpmy1
Figure 3.6: Magnetic field density along x axis
0 0.05 0.1 0.15−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
g [m]
Bp
my1(g
)
Figure 3.7: µ0Hpmy1 as function of g
23
µ0 4π × 10−7 pm 4τ 0.0571 LA 0.0500τp 0.0286 Br 1.0505Hc 836000 µr 1.1000hm 0.0020 p 2
Table 3.1: Parameters of the motor considered for numerical calculations
The figures in this section resulted from simulations that used the parameters listed in Table 3.1.
3.3 Winding’s magnetic field
The next procedure calculates the stator’s magnetic field by closely following the method described in [42],where the authors use the mmf1 of the travelling magnetic field to calculate the magnetic potential of thestator field Ψs. In turn, the magnetic potential yields an expression for the density of the stator’s magneticfield Bs. This approach requires the following assumptions.
Assumption 5. The magnetic field generated by the stator’s windings is sinusoidally distributed alongthe x axis.
Assumption 6. The magnetic field generated by the stator windings equally covers all the surface of thestator, including those areas that do not interact with the mover.
Let Ia, Ib, and Ic be the phasors of the currents flowing through each phase of the armature and let alsoIa, Ib, and Ic be their magnitudes. Denote the corresponding instantaneous currents by ia, ib, and ic, suchthat ia ≤ Ia, ib ≤ Ib, and ic ≤ Ic. Let Ia, Ib, and Ic be rotated 120o with respect to each other, as in Figure3.8. The angle θ of the phasor Ia is given by
θ = (x− d)π
τ, (3.22)
where d is the relative position in synchronous operation of the mover with respect to the stator along thex axis. Figures 3.9 and 3.10 clarify the meaning of d. Figure 3.9 shows a PMLSM when the mover is atthe origin of the x axis, i.e., the axis y of the set of coordinates of the stator and the axis y′ of the set ofcoordinates of the mover are collinear. Figure 3.10 shows a different case in wich the mover has been shiftedalong the x axis a distance d.
Let ω = θ. The authors of [23] defined a three phases armature’s magnetomotive force in terms of ω asthe Fourier series
Fs =
∞∑
n=1
Fsn sin
[(
ωt− (6n+ 1)πx
τ
)
+ (6n− 1)2π
3
]
+
∞∑
n=1
Fsn sin
[(
ωt+(6n+ 1)πx
τ
)
− (6n+ 2)2π
3
]
. (3.23)
Its coefficients are given by
Fsn =3√
2WIakwn
πKcp(6n+ 1), (3.24)
where W is the number of turns of wire on each phase, p is the number of poles, wc is the coil pitch, kwn
is the winding factor, and Kc is Carter’s coefficient, derived in Section 3.2. The winding factor results from
1The magnetomotive force can be interpreted as a magnetic potential rise (refer to [23]).
24
120 o
120 o
120 o
b
c
a
I
I
I
θ
Figure 3.8: Phase currents in the armature
y
xz
NS S
NNS
N S
z’
y’
x’
d=0
Figure 3.9: Mover at d = 0
NS S
NNS
N S
y
xz
z’
y’
x’
d
Figure 3.10: Mover at d 6= 0
25
the product of kdn, the distribution factor, and kpn the pitch factor, so that
kwn = kdnkpn (3.25)
kdn =sin
[
(6n+1)π6
]
τ3t1
sin[
3(6n+1)πt16τ
] (3.26)
kpn = sin
[
(6n+ 1)πwc
2τ
]
. (3.27)
Thus, the first harmonic of the mmf is given by
F1 = Fs1 sin[
(x− d)π
τ
]
=6√
2WIakw1
πKcpsin
[
(x − d)π
τ
](3.28)
The coefficient Fs1 serves as boundary condition for the PDE associated to the first harmonic of themagnetic potential. Hence, the approximation of the magnetic potential is given by
∇2Ψs1 = 0 (3.29)
subject to the boundary conditions
Ψs1
∣
∣
y=g= 0 (3.30)
Ψs1
∣
∣
y=0= Fs1 sin
[
(x − d)π
τ
]
. (3.31)
The solution to (3.29) is
Ψs1 = Fs1
[
cosh(π
τy)
− cosh[
πτ (hm + g)
]
sinh[
πτ (hm + g)
] sinh(π
τy)
]
sin[
(x− d)π
τ
]
(3.32)
and the components of the first harmonic of the armature’s field density are
Bs1x = −µ0Hs1x
= −µ0∂Ψs1
∂x
= −µ0π
τFs1
sinh[
πτ (hm + g − y)
]
sinh[
πτ (hm + g)
] cos[
(x− d)π
τ
]
(3.33)
andBs1y = −µ0Hs1y
= −µ0∂Ψs1
∂y
= −µ0π
τFs1
cosh[
πτ (hm + g − y)
]
sinh[
πτ (hm + g)
] sin[
(x− d)π
τ
]
(3.34)
3.4 Slots effect
The difference between the reluctance of the slots and the teeth of the stator (see Figure 3.11) affects theperformance of the system in two different ways [46]. First, it damps the magnetic field in the airgap,reducing the magnitude of the forces. This phenomenon is equivalent to having an effective airgap length
26
g
b0t1
b0
2
b0
2
b0
2
b0
2t 1
x
g(x)
− 0
g
+
Figure 3.11: Modeling of the slots through the function g(x)
larger than the actual one. Second, the slots spatially modulate the field because the flux lines travel throughlow reluctance regions (the teeth). A magnetic model accounts for the first effect by introducing Carter’scoefficient and the effective airgap (as seen in Section 3.2) and for the second effect by including the relativepermeance of the stator, which is next derived.
Remark 2. In order to obtain an expression for the relative permeance, the following analysis adapts tolinear motors the concept developed in [46] for permanent magnet DC motors.
According to [46], an expression for the magnetic field density that takes into account the slots is
Bg′ = Bgλ (3.35)
where Bg is the airgap field density of an homologous slotless machine and λ is the relative permeance, whichis given by the next expression
λ =g + hm
µrec
g(x) + hm
µrec
(3.36)
where g is the original airgap length and the function g(x) gives the shortest distance that the flux lines haveto travel through the slots before reaching the teeth when leaving the magnets from a point x (see Figure3.11). The formulation of an analytical expression for g(x) requires the following assumption.
Assumption 7. The magnetic flux lines travel vertically through the airgap and horizontally inside theslots. In other words, the width of the slot is greater than its depth.
Elementary calculations lead to the following expression for g(x)
g(x) =
g + kt1 + b02 − |x| kt1 − b0
2 ≤ x ≤ kt1 + b02
g kt1 + b02 ≤ x ≤ (k + 1)t1 − b0
2 ,(3.37)
where k is the index of each slot, b0 its width and t1 its pitch. Since (3.37) is a piecewise continuous function,its approximation by Fourier series is convenient. Figure 3.12 shows a plot of the relative permeance and its
27
−0.005 0 0.005 0.01 0.015 0.02 0.025 0.03
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Pe
rme
an
ce
Relative permeance1st harmonic
Figure 3.12: Relative permeance and its first harmonic
first harmonic approximation. Let h(x) be defined as
h(x) =1
g(x) + hm
µrec
, (3.38)
where µrec is the relative recoil permeability of the permanent magnets. Since the periodic function h(x) willbe later replaced by its Fourier series, its expression for k = 0, corresponding to one period, is henceforthused without lost of generality, i.e.,
h(x) =
1
g+b02
+ hmµrec
+ |x|(g+ hm
µrec)(g+
b02
+ hmµrec
)− b0
2 ≤ x ≤ b02
1g+ hm
µrec
b02 ≤ x ≤ t1 − b0
2 .(3.39)
In terms of h(x), the relative permeance is given by
λ =
(
g +hm
µrec
)
h(x). (3.40)
After few calculations, the Fourier series of h(x) is obtained as
h(x) = a0 +∞∑
n=1
an cos
(
2πn
t1x
)
=1
g + hm
µrec
− b204t1(g + hm
µrec)(g + b0
2 + hm
µrec)
+t1
π2(g + hm
µrec)(g + b0
2 + hm
µrec)
∞∑
n=1
[
cos
(
nπb0t1
)
− 1
]
cos
(
2πn
t1x
)
.
(3.41)
Thus, the relative permeance is given by
λ = 1 − b204t1(g + b0
2 + hm
µrec)
+t1
π2(g + b02 + hm
µrec)
∞∑
n=1
[
cos
(
nπb0t1
)
− 1
]
cos
(
2πn
t1x
)
.
(3.42)
28
3.5 Forces calculation
Now that expressions for the fields in the airgap are available, they will be used to find the forces in thePMLSM.
The superposition of the permanent magnets and the windings field densities yields the resultant fielddensity in the airgap. After incorporating the relative permeance, the resultant field density leads to expres-sions for the forces in the machine.
Remark 3. Henceforth, (3.42) is used in expressions for the thrust and the normal force. The nextprocedure partially follows the method described in [42] but it incorporates a variable airgap length.
3.5.1 Thrust
In order to find the motor’s thrust, the upper face of each magnet is henceforth viewed as a surface S uniformlycovered by magnetically charged particles. The thrust depends not only on the magnetic charge but also onthe x component of the travelling magnetic field, (3.33), which is affected by the relative permeance givenin (3.42). The thrust is given by
Fx = σm
∫
S
λBxdS. (3.43)
In terms of the fundamental harmonic of the travelling magnetic field, the thrust is given by (refer toFigure3.1
Fx ≈ pmLAσm
∫ τ2+
τp2
τ2− τp
2
λBs1xdx
≈ −pmLAσm
∫ τ2+
τp2
τ2− τp
2
λµ0π
τFs1
sinh[
πτ (hm + g − y)
]
sinh[
πτ (hm + g)
] cos[
(x− d)π
τ
]
dx
≈ pmLAσm
∫
τp2
− τp2
λµ0π
τFs1
sinh[
πτ (hm + g − y)
]
sinh[
πτ (hm + g)
] sin[
(x− d)π
τ
]
dx
, (3.44)
which yields
Fx ≈ %
Kc sinh[
πτ (g + hm)
]
[
η − ζ
g + b02 + hm
µrec
]
Ia sin(π
τd)
, (3.45)
where
% =12
√2kw1pmLAσmµ0 sinh
(
πτ hm
)
πp(3.46)
ζ =b204t1
η + Λ (3.47)
η = sin(πτp
2τ
)
(3.48)
Λ =t1
2πτ
∞∑
n=1
[
cos
(
nπb0t1
)
− 1
]
Λn (3.49)
Λn =
sin[(
2nπt1
+ πτ
)
τp
2
]
2nπt1
+ πτ
+sin
[(
πτ − 2nπ
t1
)
τp
2
]
πτ − 2nπ
t1
. (3.50)
In agreement with [42], if g → ∞, then, from (3.43),
Fx ≈ 2pmLABrFs1
sinh[
πτ hm
]
sinh[
πτ (g + hm)
]η sin(π
τd)
(3.51)
29
24
68
10
x 10−3
00.050.10.150.20.250.30.350.40.450.5
−80
−60
−40
−20
0
20
40
60
80
Airgap g [m]Displacement [m]
Fo
rce
[N
]
Figure 3.13: Thrust as a function of the air gap length and the position of the mover
3.5.2 Normal force
An expression for the normal force results from the definition of the Maxwell stress tensor [22], which gives
Fy =LA
2µ0
∫
[B2gy −B2
gx]dl, (3.52)
where l is the integration contour and Bgy , Bgx are the y and x components of the magnetic field intensityin the airgap, respectively.
Ideally, every flux line enters a magnet perpendicularly to its surface; therefore, the x component is nulland the normal force is affected only by Bgy, which is given by
B2gy = B2
s1y +B2pmy + 2Bs1yBpmy cos
(π
τd)
, (3.53)
where every field density is a function of g.In order to ease further calculations, a first order Fourier approximation replaces λ by
λ ≈ λ0 + λ1 cos
(
2π
t1x
)
. (3.54)
Substitution of (3.53) and (3.54) into (3.52) yields
Fy ≈ pmLA
2µ0
∫ τ
0
[
λ1Bg sin(π
τx)]2
dx
≈ Υ
ϕ2I2a
K2c
coth2[π
τ(g + hm)
]
+B2pmy(g) +
2ϕIaBpmy(g)
Kccoth
[π
τ(g + hm)
]
cos(π
τd)
Γ(g),
(3.55)
30
0.005
0.01
0.015
0.02
00.1
0.20.3
0.40.5
50
100
150
200
250
Airgap g [m]Displacement [m]
Fo
rce
[N
]
Figure 3.14: Normal force as a function of the air gap length and the mover position
where
Υ =pmLA
2µ0(3.56)
ϕ =6√
2kw1µ0
pτ(3.57)
Γ(g) =τ
2− ϑ
g + b02 + hm
µrec
+ς
2(g + b02 + hm
µrec)2
(3.58)
ϑ =τb204t1
+ξ − β
π2
[
cos
(
πb0t1
)
− 1
]
t1 (3.59)
ς =τb4016t21
− b20(ξ − β)
2π2
[
cos
(
πb0t1
)
− 1
]
+2$t21π4
[
cos
(
πb0t1
)
− 1
]
(3.60)
ξ =t12π
sin
(
2πτ
t1
)
(3.61)
β =sin
[
2π(τ+t1)t1
]
τt1
4π(τ + t1)+
sin[
2π(τ−t1)t1
]
τt1
4π(τ − t1)(3.62)
$ =τ
4+
t116π
sin
(
4πτ
t1
)
− 1
4
sin[
2π(2τ+t1)t1
]
τt1
2π(2τ + t1)+
sin[
2π(2τ−t1)t1
]
τt1
2π(2τ − t1)
. (3.63)
The function Γ(g) accounts for the slots. If the analysis neglects the slots, Γ(g) = Γ = τ2 and the final
expression for the normal force equals the one derived in [42]. If τ = t1, Γ(g) is given by
Γ(g) =τ
2λ2
0 −τ
2λ0λ1 +
τ
4λ2
1 (3.64)
31
0 0.1 0.2 0.3 0.4 0.50
50
100
150
200
250
300
For
ce [N
]
Displacement [m]
Figure 3.15: Normal force as a function of the mover position
3.6 Single motor dynamics
The just found expressions for the motor’s forces together with Newton laws render the dynamic model of aPMLSM as follows.
Let Mp be the mass of the mover, G the gravity constant, and x = [x1, x2, x3, x4]T = [d, d, g, g]T . Then,
the dynamics of the mover are described by
x1 = x2
x2 = %
MpKc(x3) sinh[πτ
(x3+hm)]
[
η − ζ
x3+b02
+ hmµrec
]
iq
x3 = x4
x4 =Γ(x3)Υ
Mp
ϕ2(i2d + i2q)
K2c (x3)
coth2[π
τ(x3 + hm)
]
+B2pmy(x3) +
2ϕBpmy(x3)
Kc(x3)coth
[π
τ(x3 + hm)
]
id
+G
(3.65)
where the control inputs –neglecting the current dynamics problem– are id and iq, wich represent theinstantaneous direct and quadrature currents, which, referring to Figure 3.8 can be expressed in terms ofthe instantaneous phase currents as
(
idiq
)
=2
3
(
cos(
πτ x1
)
cos(
πτ x1 − 2π
3
)
cos(
πτ x1 + 2π
3
)
− sin(
πτ x1
)
− sin(
πτ x1 − 2π
3
)
− sin(
πτ x1 + 2π
3
)
)
iaibic
(3.66)
0 = ia + ib + ic. (3.67)
The inverse transformation is given by
iaibic
=
cos(
πτ x1
)
− sin(
πτ x1
)
cos(
πτ x1 − 2π
3
)
− sin(
πτ x1 − 2π
3
)
cos(
πτ x1 + 2π
3
)
− sin(
πτ x1 + 2π
3
)
(
idiq
)
. (3.68)
32
3.7 Disturbances
In the modelling procedure described in this chapter, assumptions and simplifications yielded a PMLSMmodel that neglects magnetic saturation, end-effects, and cogging forces. The omission of these phenomenamay affect the performance of the device and can be viewed as an uncertainty. Because completely differentapproaches would become necessary, the control design in the presence of uncertainties falls outside of thescope of this thesis. Nevertheless, this section briefly describes PMLSM’s uncertainties in order to bothillustrate their complexity and provide foundations for future work. The description begins with the studyof magnetic saturation which is followed by end-effects and cogging forces.
Magnetic saturation is due to nonlinearities on the magnetization curve of ferromagnetic materials. Itdirectly affects the quadrature-axis synchronous reactance, Xsq, and the direct-axis synchronous reactance,Xsd, by making them different; consequently, the actual thrust becomes the sum of a synchronous forceand a reluctance force. While the synchronous force, Fx, has been computed previously in this chapter, thereluctance thrust is given by [23] as
F relx =
3V 2
d
(
1
Xsq− 1
Xsd
)
sin
(
2π
τd
)
, (3.69)
where V is the input voltage. In terms of an unknown parameter η, (3.69) becomes
F relx := η(id, iq, g, d) sin
(
2π
τd
)
. (3.70)
Magnetic saturation effects can be reduced either by current control, as suggested in [49] and [32], orby modelling additional forces. Nevertheless, for airgap lengths considerably larger than the slot width, thedifference between both reactances is small and the reluctance force can be neglected.
The authors of [82] and [33] found that, for sinusoidally excited motors, the end-effects uniformly reducethe thrust along the travel of the mover. In contrast, for partially excited machines, the end-effects sinu-soidally modulate the magnetic field and introduce a periodic disturbance with frequency π
τ . These resultsagree with those in [35], which were generated through experiments on a PMLSM fed with trapezoidal cur-rents. From their conclusions, modelling can characterize end-effects through a ripple of unknown magnitudeθ(id, iq, g) and known frequency as
F eex = θ(id, iq, g) sin
(π
τd)
. (3.71)
The work in [34] attributes cogging forces to the interaction between the ends of the permanent magnetsand the stator teeth. That research does not analytically derive these forces but characterizes them throughFEM analysis and experimentation. From this study, the authors conclude that cogging forces appearperiodically at every slot-pitch along the travel. Since cogging forces mostly depend on the intensity of thePM’s field in the stator, their amplitude decreases as the airgap length increases. Therefore, the coggingforces can be represented by a disturbance of the form
F cx = ξx(g) sin
(
π
t1d
)
, (3.72)
where ξx(id, iq, g) is unknown, continuous, and inversely proportional to g.From the results above, a single expression aggregates the thrust disturbances as
F dx = F ee
x + F cx
= θ(id, iq, g) sin(π
τd)
+ ξx(id, iq, g) sin
(
π
t1d
)
, ∆x(id, iq, g)ϕx(d),
(3.73)
33
where the vectors ∆x(id, iq, g) and ϕx(d) are defined by
ϕ(d)x =
[
sin(π
τd)
sin
(
π
t1d
)]
(3.74)
∆(id, iq, g)x =
[
θ(id, iq, g)ξ(id, iq, g)
]
. (3.75)
Results in [82] show that end-effects do not affect the y component of the magnetic field in the airgapand thus they do not perturb the normal force. On the other hand, cogging forces do affect the y componentof the field and must be included in the y dynamics by means of an expression of the form
F dy = F c
y
= ξy(id, iq, g) sin
(
π
t1d
)
, ∆y(id, iq, g)ϕy(d).
(3.76)
3.8 Discussion of modelling results
This chapter delineated a procedure that yields an analytical model for a PMLSM with variable airgaplength. This procedure, like its predecessors in [42] and [46], relies on numerical computations but generatesa well-defined mathematical model that describes both the longitudinal and normal dynamics of a motorwith a contactless mover. Although several approximations simplified the calculations, the modelling methodis reasonably accurate, as [42] and [46] demonstrated. The accuracy of these two references guarantees theexactness of the new model, which fully agrees with those that supported it. Specifically, when applied to amotor with a fixed airgap length and uniform armature’s permeance, the expressions for the forces derivedin Section 3.5 exactly match their counterparts in [42]. Numerical evaluation of those expressions results inthrust and normal force with expected magnitudes.
Also, in agreement with the results from [46], repeated numerical computations of the airgap field showedthat the non-uniform permeance reduced the flux per pole. Consequently, the slots weakened the forces,especially the thrust. These numerical computations also showed that, as predicted in [46], larger airgaplengths reduce the effect of the slots.
Consequently, the new model is reliable and the design of the levitated positioning system can safelybenefit from it. Accordingly, this thesis next proposes a 3-DOF levitation machine whose model is based onthe results of the present chapter. The next chapter presents this device while evaluating different controlalternatives.
34
Chapter 4
Design for three controlled degrees of
freedom
Chapter 3 presented the development of a mathematical model for PMLSMs with free normal dynamicswhich led to analytical expressions for the normal force and the thrust. The present chapter uses the sameexpressions to model and control a PMLSMs-based device that addresses the attractive levitation of a platenwith three controlled degrees of freedom.
The design of this 3-DOF levitation machine starts with a general description of the setup. Section4.1 presents both this description and the associated dynamic model. Once the modelling of the device iscompleted, the present chapter proposes three different control approaches. The first of them is an LQRcontroller based on linearization and is outlined in Section 4.2. The second one, developed in Section4.3, is a control law based on Lyapunov and LaSalle’s theorems. Section 4.4 describes the third and lastcontrol alternative, which is based on invariance control as proposed by Mareczek, Buss, and Spong. Everysection includes the results of simulations, all of which used the same motor parameters. This uniformity ofparameters permits a balanced comparison among controllers that is carried out in Section 4.5.
4.1 Setup description and modelling
The design of the levitation machine begins by first defining its general structure. The present sectiondescribes the arrangement of three motors that compose the machine and establishes standing assumptionsthat make the model valid.
Next, a standard procedure obtains the model of the machine by first performing a force analysis. Theprocedure then incorporates into the model the expressions of the forces found on Chapter 3. Finally, theresulting equations are used to write state space representation suitable for control design.
To begin with the description of the levitation device, consider the setup shown in Figure 4.1, in whichthree PMLSMs drive a floating platen. The system employs three identical motors equally spaced along astraight line and perpendicularly oriented with respect to each other. The windings of motors 1 and 3 (thoseat the ends of the structure) are connected in parallel whereas motor 2 is independently fed. In order topermit a large range of operation, the movers of the motors are significantly smaller than their stators. Inaddition to the previous specifications, the design takes into account the following constraints.
Assumption 8. The motors are far enough from each other so that their fields do not interfere with eachother.
In addition, the model developed in this chapter is valid provided that the movement of the platen neverdrives the permanent magnets outside of the areas covered by the stators’ magnetic fields. Moreover, if themovers never get too close to the borders of such regions, modelling can assume uniform effects at the endsof the movers. Making sure that such conditions are always met is a control design objective.
35
Movers
Stators
Platen
x
z
g
Figure 4.1: Configuration of three LSMs for three controlled degrees of freedom
Let x′, g′, and z′ be the principal axes of inertia of the platen, as defined in Figure 4.1.
Remark 4. Ideally, the device is symmetric with respect to the planes x′g′ and z′g′. Because Motors 1and 3 are electrically connected, their forces are identical and therefore the torque around the axis z′ is zero.Since the forces are symmetrically exerted over the entire surface of each mover, the resultant torque aroundthe axis x′ sums to zero as well. Therefore, rotation is prevented and translational dynamics yield a completemodel of the system. In Chapter 5, perfect symmetry is not assumed and five DOF are considered.
Let Mpm be the mass of each strip of permanent magnet material, Mt the mass of the platen withouttaking into account the mass of the PM strips, and Mp the mass of the whole platen. For a PM materialwith mass density ρpm, the mass of each PM is
Mpm = hmτpLAρpm, (4.1)
and the mass of each piece of back iron is
Mbi = hbτpmLAρi, (4.2)
where ρi is the density of the ferromagnetic material, hb is the thickness of the movers’ back irons and τ , τp,LA, and hm are as defined in Chapter 3 (see Figure 3.1). The mass of the platen is then given by
Mp = Mt + 3pmMpm + 3Mbi. (4.3)
Let Motor 1 and Motor 3 be the motors generating thrust in the z direction and Motor 2 the motorpowering the travel along the x axis (see Figure 4.2). Let also id1 and iq1 be the direct and quadraturecurrents fed to Motors 1 and 3. Similarly, let id2 and iq2 be the direct and quadrature currents through thewindings of Motor 2. If the forces on the motors are defined as in Figure 4.2, Newton laws yield the followingequations
Mpg = MpG− 2Fy1(g, z, id1, iq1) − Fy2(g, x, id2, iq2)
Mpx = Fx(g, x, iq2)
Mpz = 2Fz(g, z, iq1) (4.4)
where x and z are the displacements of the platen along the x and z directions respectively. Fy1 =Fy1(g, z, id1, iq1) and Fz = Fz(g, z, id1, iq1) are the forces in Motor 1 and 3, and Fy2 = Fy2(g, x, id2, iq2)and Fx = Fx(g, x, id2, iq2) are the forces in Motor 2. These forces are given by expressions (3.45) and (3.55).
36
a b
Fy1 Fy1Fy2 Fy
Fz Fz
g
Motor 1 Motor 2 Motor 3
g’
z’ x’
g
xz
G
Figure 4.2: Forces in the 3-LSMs configuration
Definition of x = [x1, x2, x3, x4, x5, x6]T = [g, g, δx, δx, δz, δz]
T leads to a state space representation for(4.4) given by
x =
x2
G− 2Fy1
Mp− Fy2
Mp
x4Fx
Mp
x6
2 Fz
Mp
. (4.5)
Substitution of the full expressions of the forces yields
x1 = x2
x2 = G− ΥΓ(x1)
Mp
ϕ2
K2c (x1)
coth2[π
τ(x1 + hm)
]
[
2(i2d1 + i2q1) + i2d2 + i2q2]
− ΥΓ(x1)
Mp
3B2pmy(x1) +
2ϕBpmy(x1)
Kc(x1)coth
[π
τ(x1 + hm)
]
[2id1 + id2]
x3 = x4
x4 = %
MpKc(x1) sinh[πτ
(x1+hm)]
[
η − ζ
x1+b02
+ hmµrec
]
iq2
x5 = x6
x6 = 2%
MpKc(x1) sinh[πτ
(x1+hm)]
[
η − ζ
x1+b02
+ hmµrec
]
iq1.
(4.6)
Introduction of the change in notation given by
i =Υ
Mpϕ2
χ =3Υ
Mp
ν =2ϕΥ
Mp
k =%
Mp
κ =b02
+hm
µrec
37
φ(x1) = iΓ(x1) coth2
[
πτ (x1 + hm)
]
K2c (x1)
(4.7)
ψ(x1) =νΓ(x1)Bpmy(x1)
Kc(x1)coth
[π
τ(x1 + hm)
]
(4.8)
γ(x1) =k
Kc(x1) sinh[
πτ (x1 + hm)
]
[
η − ζ
x1 + κ
]
(4.9)
χ(x1) = χΓ(x1)B2pmy(x1) (4.10)
and definition of u1 = iq1, u2 = iq2, u3 = id1, u4 = id2 transform those equations to the more compact form
x1 = x2
x2 = G− φ(x1)[2u21 + u2
2 + 2u23 + u2
4] − χ(x1) − ψ(x1)[2u3 + u4]x3 = x4
x4 = γ(x1)u2
x5 = x6
x6 = 2γ(x1)u1.
(4.11)
Note that the functions χ(x1) and φ(x1) share the same sign, which is determined by Γ(x1). Similarly,ψ(x1) share the same sign for x1 > −hm.
Table 4.1 lists the values of the constants used in the simulations presented in this chapter.
i 130.0188 χ 124,500ν 4645.7 k 19.5617κ 0.0079 η 0.7071ζ 0.0017 hm 0.002
Table 4.1: Simulation parameters for the 3-DOF system
4.2 Linear control
The controllability of the linearization of a nonlinear system guarantees the controllability of the originalone. Thus, in order to determine the solvability of the control problem addressed here, the system (4.11 islinearized around a set of forced equilibrium points. Finally, basic LQR performes the set-point stabilizationof the resulting linear approximation.
Consider the problem of moving the platen to a desired set-point xd = [g, 0, x, 0, z, 0]. For the problemto be well posed, there must exist one value of the control input ud = [ud
1, ud2, u
d3, u
d4]
T such that xd is anequilibrium point of (4.11). In order to find such equilibrium point, let x1 = x3 = x5 = 0. Then, (4.11)yields
0 = x2
0 = G− φ(x1)[2u21 + u2
2 + 2u23 + u2
4] − χ(x1) − ψ(x1)[2u3 + u4]0 = x4
0 = γ(x1)u2
0 = x6
0 = 2γ(x1)u1.
(4.12)
Thus, the system is in equilibrium if and only if γ(x1) = 0 or if u1 = u2 = 0. From (4.9), γ(x1) = 0 occursat values of x1 such that
0 =k
Kc(x1) sinh[
πτ (x1 + hm)
]
[
η − ζ
x1 + κ
]
, (4.13)
which is satisfied if and only if
x1 =ζ
η− κ, (4.14)
38
ifKc(x1) = ∞, (4.15)
or ifsinh
[π
τ(x1 + hm)
]
= ∞. (4.16)
For the values listed in Table 4.1, the conditions (4.14), (4.15), and (4.16) are satisfied only at x1 =−0.0055 and x1 = ∞. However, both these values of x1 fall outside of the region of interest –recall thatnegative values of the airgap length are unfeasible. Therefore, equilibrium of (4.11) requires ud
1 = ud2 = 0
but the control inputs ud3 and ud
4 can take many values. In order to obtain realistic values of the equilibriumcurrent ud
4, choose, for example, ud3 = 0. Thus, different values of xd
1 yield the values of ud4 that are listed in
Table 4.2.
xd1 ud
4 xd1 ud
4...
... 0.11 1.67490.01 -3.84491+5.30795i 0.12 1.655980.02 -2.65257+3.41723i 0.13 1.654840.03 -1.49168+1.27943i 0.14 1.672730.04 0.488849 0.15 1.686960.05 1.14496 0.16 3.15393+4.1418i0.06 1.44017 0.17 30.7896+43.5117i0.07 1.58495 0.18 169.777+240.096i0.08 1.62576 0.19 689.939+975.72i0.09 1.63242 0.2 2298.84+3251.05i
0.1 1.65698...
...
Table 4.2: Estimate of the controllable region for the linearization of the 3-DOF levitation system
Table 4.2 suggests that, since the control input ud4 is real only for certain values of xd
1, (4.11) mightnot be globally stabilizable. This limitation can be attributted to the fact that, when the airgap length iseither too small or too large, the force produced by the armature’s magnetic field cannot counteract either theattractive force between the magnets and the stator’s iron core or the gravity, respectively. For the constantsin Table 4.2, the control problem is well posed inside (xd,ud) ∈ X × R4 = xd ∈ R6|0.04 ≤ xd
1 ≤ 0.15 × R4
because (4.11) is linearizable around equilibrium points in that region; nevertheless, as the equilibrium pointapproaches the ends of X , the controllability matrix of the linearization becomes more and more singular.For example, the controllability matrix of the linearization around xd = [0.08, 0, 0, 0, 0, 0]T has a conditionnumber with respect to inversion nc = 5, 904.3 and linearization around xd = [0.15, 0, 0, 0, 0, 0]T yieldsnc = 3.2826 × 108, whereas xd = [0.16, 0, 0, 0, 0, 0]T yields nc = 3.2776× 1013. The systems that result fromthe linearization around points in X are controllable using LQR controllers with, for instance, the weightingmatrices Q = diag(100, 1, 1, 1, 1, 1) and R = I4×4.
4.2.1 Simulation results
For sake of illustration, consider the linearization of (4.11) around xd = [0.055, 0, 0, 0, 0, 0]T , given by
x =
0 1 0 0 0 0320.51 0 0 0 0 0
0 0 0 1 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 0 0
x +
0 0 0 00 0 −4.3969 −11.0630 0 0 00 1.1355 0 00 0 0 0
2.2771 0 0 0
u, (4.17)
39
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.05
0
0.05
0.1
0.15
time [s]
po
sitio
n [m
]
x1
x3
x5
Figure 4.3: Position of the 3-DOF system’s platenunder LQR control for xd = [0.055, 0, 0, 0, 0, 0]T .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
sp
ee
ds [m
/s]
time [s]
x2
x4
x6
Figure 4.4: Speed of the 3-DOF system’s platenunder LQR control for xd = [0.055, 0, 0, 0, 0, 0]T .
and the feedback gain matrix obtained by LQR design
K =
0 0 0 0 1 1.37140 0 1 1.6617 0 0
−20.55 −1.1881 0 0 0 0−51.707 −2.9893 0 0 0 0
. (4.18)
Figures 4.3 and 4.4 present results from the simulation of the respective closed-loop system when x(0) =[0.15, 0.05, 0.05, 0,−0.05, 0]T and ud = [0, 0, 0, 1.3157]. Figures 4.5 and 4.6 show the associated transientresponse of the control inputs.
Other simulations tested the response of the system linearized around different points inside the intervalgiven by Table 4.2. These simulations showed that this linear control approach suffers limitations. For exam-ple, if the system is linearized around xd = [0.045, 0, 0, 0, 0, 0]T , the LQR design cannot stabilize it for initialconditions x(0) = [0.15, 0.05, 0.05, 0,−0.05, 0]T . Similarly, with set-point at xd = [0.14, 0, 0, 0, 0, 0]T andinitial conditions x(0) = [0.04, 0.05, 0.05, 0,−0.05, 0]T the closed-loop trajectories are unstable. Moreover,the controller cannot ensure that x1(t) > 0, ∀t > 0, thus leading to non-feasible solutions.
Simulations also showed that, in certain cases, the trajectories of the system exit the set
x ∈ R6|u ∈ R4
–the set where the control inputs are physically realizable– and lead the control currents, u(t), into C4.Therefore, LQR cannot guarantee asymptotic stability.
In comparison to the linearly controlled closed-loop system that [16] presents, the present LQR designyields a much larger vertical range of operation; nevertheless, this result needs experimental verificationbecause these simulations do not consider noise and other perturbations.
4.3 Partial feedback linearization and composed control
Because in certain cases the LQR design cannot guarantee asymptotic stability of (4.11), a nonlinear con-troller becomes necessary for the 3-DOF levitation system. This section introduces a nonlinear controllerbased on Lyapunov and LaSalle’s theory.
First, this section presents a feedback controller that partially linearizes the original system while ren-dering the origin of the nonlinear dynamics attractive. Next, it provides a proof for the attractiveness of the
40
0 1 2 3 4 5 6 7 8 9 10−0.05
0
0.05
time [s]
u1, u
2 [A
]
u1
u2
Figure 4.5: Control inputs u1 and u2 in the3-DOF system under LQR control for xd =[0.055, 0, 0, 0, 0, 0]T .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
1
2
3
4
5
6
time [s]
u3, u
4 [A
]
u3
u4
Figure 4.6: Control inputs u3 and u4 in the3-DOF system under LQR control for xd =[0.055, 0, 0, 0, 0, 0]T .
closed-loop system by relying on Lyapunov and LaSalle’s theorems. Simulations of the closed-loop systemare presented at the end of this section.
Notice that system (4.11) is the composition of three subsystems, as shown in Figure 4.7. Two of thosethree subsystems are feedback linearizable.
Let xd = [xd1, . . . , x
d6]
T be the desired equilibrium point. For set-point stabilization, the form of thisvector is xd = [xd
1, 0, xd3, 0, x
d5, 0]. Define x = x − xd.
Theorem 1 The controller
u1 =v2
2γ(x1)
u2 =v1
γ(x1)
u3 = 0
u4 = −ψ(x1) ±√
ψ2(x1) + 4φ(x1)R(x1)
2φ(x1)
R(x) = ε (x1 + x2) [x21 + x2
2] + x2 − φ(x1)U(x) +G− χ(x1)
U(x) = 2
(
v2(x2)
2γ(x1 + xd1)
)2
+
(
v1(x2)
γ(x1 + xd1)
)2
[
v1v2
]
= −Kx2
(4.19)
makes xd an attractive equilibrium point of (4.11), and the set Vd × W =
x1 ∈ R2∣
∣
∣
x2
1
2 + (x1+x2)2
2 ≤ d
×
x2 ∈ R4|ω1 ≥ |xi|, i = 3, . . . , 6
is contained in its domain of attraction, where the values of d and ωi dependon the set-point xd and are specified in the proof.
41
x3 = x4
x4 = γ(x1)u2
x5 = x6
x6 = 2γ(x1)u1
x5
x6
x4
x3
x2
x1
u2
u3
u4
u1
x2 = G− φ(x1)[2u21 + u2
2 + 2u23 + u2
4] − ψ(x1)[2u3 + u4] − χ(x1)x1 = x2
Figure 4.7: Block decomposition of the 3-DOF system
Proof. The closed-loop system reads as
x1 :
˙x1 = x2
˙x2 = −ε (x1 + x2) [x21 + x2
2] − x2(4.20)
x2 :
˙x3 = x4
˙x4 = v1˙x5 = x6
˙x6 = v2
(4.21)
where x1 , [x1, x2]T and x2 , [x3, . . . , x6]
T , and the parameter ε adjusts the speed of convergence of thenonlinear dynamics.
Notice that the subsystem x2 is linear and can be represented as
˙x2 = Acx2 +Bcv (4.22)
where (Ac, Bc) are in Brunovsky normal form and v = [v1, v2]T . The origin of (4.22) is globally exponentially
stabilizable through linear state feedback of the form v = −Kx2, where the matrix K can be obtained, e.g.,by LQR design. For simulations, Q = diag(10, 1, 10, 1) and R = I2×2 were used as weighting matrices. Thecorresponding gain matrix is
K =
[ √10 2.7064 0 0
0 0√
10 2.7064
]
. (4.23)
Since the closed-loop linear subsystem is exponentially stable, the control inputs u1 and u2 are boundedprovided that γ (x1(t)) 6= 0 ∀t ≥ 0.
The stability analysis for the subsystem (4.20) is performed in z coordinates, which are defined as
z1 = x1
z2 = x1 + x2.(4.24)
In new coordinates, the system (4.20) reads as
z1 = z2 − z1z2 = −εz2[(z2 − z1)
2 + (z1)2].
(4.25)
42
z2
z1
Vd
Ω
V > 0
V < 0
Figure 4.8: Lyapunov and invariant sets for composed control
Consider the Lyapunov function candidate V (z) = 12z
21 + 1
2z22 whose derivative is given by
V (z) = −z21 + z1z2 − εz2
2 [z21 + (z2 − z1)
2] (4.26)
≤ −z21 +
z21
2+z22
2− εz2
2 [z21 + (z2 − z1)
2] (4.27)
≤ −z21
2−
ε[
z21 + (z2 − z1)
2]
− 1
2
z22 . (4.28)
V (z) is negative definite outside the set Ω =
z ∈ R2|z21 + (z2 − z1)
2 ≤ 12ε
–which, in the original coor-
dinates, is the circle
x2 ∈ R2|x21 + x2
2 ≤ 12ε
– and therefore V (z(t)) decreases in that region.Let Vd = z ∈ R2|V (z) ≤ d and notice that d = 1
4ε2(3−√
5)is the smallest value of d guaranteeing that
Ω ⊂ Vd (see Figure 4.8). Since V < 0 outside of Vd, the previous observation implies that V < 0 for allV > d and hence the set Vd is positively invariant for all d > d. The set defined by this inequality will nowbe used in the application of LaSalle’s theorem as follows.
Consider the function VE(z) =z2
2
2 , whose time derivative is
VE(z) = −εz22 [(z2 − z′1)
2 + z21 ] ≤ 0 ∀z ∈ R
2, (4.29)
and define the set E = z ∈ R2|VE(z) = 0 = z ∈ R2|z2 = 0. For any trajectory in E, the dynamics of(4.25) are given by
z1 = −z1z2 = 0
(4.30)
and hence it readily follows that E is invariant. Because of LaSalle’s theorem, every solution starting in Vd,d > d, approaches E as t→ ∞. Moreover, from (4.30), z1(t) → 0 on E. Since Vd is compact, all trajectoriesare bounded and hence, it follows that for all z(0) ∈ Vd, d > d, z(t) → 0.
The controller (4.19) is well-defined inside the set D defined as
D =
x ∈ R6|0 ≤ ψ2(x1 + xd
1) + 4φ(x1 + xd1)R(x1), φ(x1 + xd
1) 6= 0, γ(x1 + xd1) 6= 0
(4.31)
Since (4.31) includes the full state, it describes a set that is hard to visualize; however, it is possible toobtain a simple inner approximation of D by finding an upper bound to R(x1). Such inner approximation
43
can be used then to estimate the domain of attraction. Since φ(x1 + xd1) ≥ 0 in the region of interest,
bounding R(x1) implies finding upper bounds to v(x2) and x2(t).Let λ(t, x2
0), with x20 = x2(0), be the solution for the closed-loop linear system (4.21) under the action of
linear state feedback. Then, it follows that
v(x2) = −Kλ(t, x20). (4.32)
Let e1, . . . e4 denote the natural basis in R4, and let λj(t) = λ(t, ej) be the response of the systemsubject to the initial condition ej , j = 1, 2, 3, 4. Since the closed-loop linear system is stable, all solutionsλj(t) are bounded. Let xj(0) be the entries of x2
0. Because of linearity
x2(t) =
4∑
j=1
xj(0)λj(t), (4.33)
and hence
v(x2(t)) = −K4
∑
j=1
xj(0)λj(t). (4.34)
Denoting by K1 and K2 the two rows of K, i.e., K = [KT1 KT
2 ]T it follows that
‖vi(x2(t))‖2 =
∥
∥Ki[λ1(t), . . . , λ4(t)]x20
∥
∥
2i = 1, 2. (4.35)
Since vi(x2) enter into (4.31) squared, it suffices to find upper bounds to their 2-norms. These upper
bounds can be found as follows
supt
‖vi(x2)‖2 = sup
t
∥
∥Ki[λ1(t), . . . , λ4(t)]x20
∥
∥
2
≤∥
∥
∥
∥
Ki[supt
|λ1(t)|, . . . , supt
|λ4(t)|]x20
∥
∥
∥
∥
2
.(4.36)
Then, a conservative upper bound to ‖vi(x2(t))‖2 as function of x0 is given by
‖vi(x2(t))‖2 ≤
∥
∥Ki[λ1, . . . , λ4]x20
∥
∥
2, vi (4.37)
where the values λj = supt|λj(t)| are computed numerically by constraining the initial condition x20 inside
a set W =
x20 ∈ R4
∣
∣|xi| ≤ ωi, i = 3, . . . , 6
, for predefined values of ωi.Consequently, U –defined in (4.19)– satisfies
U(x) ≤ U(x1) , 2
(
v2
2γ(x1 + xd1)
)2
+
(
v1
γ(x1 + xd1)
)2
(4.38)
and it follows that
R(x) ≥ R(x1) , ε (x1 + x2) [x21 + x2
2] + x2 − φ(x1)U(x1) +G− χ(x1). (4.39)
Hence, the set
D =
x1 ∈ R2|0 ≤ ψ2(x1 + xd
1) + 4φ(x1 + xd1)R(x1), φ(x1) 6= 0, γ(x1) 6= 0
(4.40)
is an inner approximation of the projection of D onto the x1 coordinates. Moreover, since the set can nowbe easily plotted, it can be verified that on
x1 ∈ R2 | 0 ≤ ψ2(x1 + xd1) + 4φ(x1 + xd
1)R(x1)
, φ(x1) 6= 0,γ(x1) 6= 0, and hence
D =
x1 ∈ R2|0 ≤ ψ2(x1 + xd
1) + 4φ(x1 + xd1)R(x1)
. (4.41)
44
−0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20
10
20
30
40
50
60
70
80
90
100
x1
x2
Region of real controls
Figure 4.9: Region for which the control input u4 of the 3-DOF composed controller is real
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
−0.03
−0.02
−0.01
0
0.01
0.02
x1
x2
Region of real controlsLyapunov surfaceBoundary of ΩDiscontinuity
Figure 4.10: Estimate of the domain of attractionof the 3-DOF composed controller for first set-point
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
x1
x2
Region of real controlsLyapunov surfaceBoundary of ΩDiscontinuity
Figure 4.11: Estimate of the domain of attractionof the 3-DOF composed controller for second set-point
45
0 1 2 3 4 5 6 7 8 9 10−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
time [s]
Dis
pla
ce
me
nts
[m
]
x1
x3
x5
Figure 4.12: Transient position of the 3-DOF sys-tem under composed control for a first set of pa-rameters
0 1 2 3 4 5 6 7 8 9 10−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
time [s]
Sp
ee
ds [m
/s]
x2
x4
x6
Figure 4.13: Transient speed of the 3-DOF systemunder composed control for a first set of parame-ters
Choice of ω3 = 0.1, ω4 = 1, ω5 = 0.1, and ω6 = 1 yields the upper bounds v1 = v2 = 3.0226, such that D =
x1 ∈ R2∣
∣
∣ψ2(x1 + xd1) + 4φ(x1 + xd
1)[
ε (x1 + x2) (x21 + x2
2) + x2 − 13.7043φ(x1+xd
1)
γ2(x1+xd1)+G− χ(x1 + xd
1)]
≥ 0
.
Figure 4.9 shows the set D when xd1 = 0.04m and ε = 1, 650.
Next, let d be the largest value of d such that Vd, expressed in (x1, x2) coordinates, is contained in D.Given xd
1 and ε, if d is such that d > d, then Vd is an estimate of the domain of attraction of the origin of(4.20). Given x(0) ∈ Vd ×W ⊂ D, the trajectories x1(t) are bounded inside Vd. Since E is attractive andE ⊂ Vd –recall that Vd is positively invariant provided that d > d– the trajectories x(t) → E as t → ∞.Furthermore, inside E, x(t) → 0 as t → ∞. Since (4.21) is exponentially stable, the origin of the closed-loopsystem given by (4.20) and (4.21) is attractive for any initial condition lying inside Vd × W , as stated onTheorem 1.
The shaded areas in Figures 4.10 and 4.11 are parts of the set D and the solid ellipses represent theLyapunov level surfaces V = d for d = 8.5 × 10−5 and d = 3.95 × 10−4 respectively. The dashed ellipsesrepresent the boundaries of the Ω sets given by ε = 8, 500 and ε = 1, 650 in each case. The set-points arexd
1 = 0.025m and xd1 = 0.04m for Figures 4.10 and 4.11 respectively.
Remark 5. Note that the result of Theorem 1 guarantees boundedness of x(t) and attractiveness of theequilibrium xd, but not Lyapunov stability.
Remark 6. Notice, from Figures 4.10 and 4.11, that the size of the estimate of the domain of attractiondepends on the set-point. This drawback will be eliminated by the technique introduced in the next section.
4.3.1 Simulation Results
Figures 4.12 and 4.13 illustrate the state response of the closed-loop system and Figure 4.14 shows thecontrol inputs u1,u2, and u4 during transient –u3 is omitted since u3(t) = 0 for all t. For those plots, theinitial conditions are x0 = [0.047, 0.05,−0.05, 0, 0.05, 0]T and the desired set-point is xd = [0.055, 0, 0, 0, 0, 0].Figures 4.15 and 4.16 show the response of the system when x0 = [0.05, 0, 0.05, 0,−0.05, 0]T and the desiredset-point xd = [0.105, 0, 0, 0, 0, 0]. In both cases, ε = 1, 650.
46
1 2 3 4 5 6 7 8 9
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time [s]
u1,u
2,u
4 [A
]
u1
u2
u4
Figure 4.14: Control inputs u1, u2, and u4 of the3-DOF system under composed control for a firstset of conditions
0 1 2 3 4 5 6 7 8 9 10−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
time [s]
Dis
pla
ce
me
nts
[m
]
x1
x3
x5
Figure 4.15: Transient position of the 3-DOF sys-tem under composed control for a second set ofconditions
0 1 2 3 4 5 6 7 8 9 10−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
time [s]
Sp
ee
ds [m
/s]
x2
x4
x6
Figure 4.16: Transient speed of the 3-DOF sys-tem under composed control for a second set ofconditions
0 1 2 3 4 5 6 7 8 9 10−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
time [s]
u1,u
2,u
4 [A
]
u1
u2
u4
Figure 4.17: Control inputs u1, u3, and u4 of the 3-DOF system under composed control for a secondset of conditions
47
4.4 Full feedback linearization with invariance control design
Because the nonlinear controller derived in Section 4.3 yields an estimate of the domain of attraction whosesize depends on the location of the set-point xd, a third control option, based on the invariance control methodproposed in [60], is investigated here. The development begins by defining a feedback linearizing controller.Then, a linear transformation and a second feedback transformation passify the feedback linearized system.Finally, a third feedback controller yields positive invariance of a suitable set.
Consider again the controller (4.19) but replace the original control input u4 by the feedback transfor-mation
u4 = −ψ(x1) ±√
ψ(x1)2 + 4φ(x1) −u− φ(x1)U +G− χ(x1)2φ(x1)
, (4.42)
where u is the new control input. This choice of controls fully linearizes system (4.11). As a result, the x2
dynamics are still given by (4.22) and the x1 dynamics become
x1 :
˙x1 = x2
˙x2 = u.(4.43)
Control law (4.42), together with proper stabilizing feedback u = −Kx1, guarantees exponential stabilityof (4.43), provided that u4 is well-defined. In order to ensure this, the control design must guaranteeinvariance of the set M where u4 is well-defined. As seen in Section 4.3, Lyapunov control ensures positiveinvariance of level sets contained in M; however, this method produces a restrictive estimate of the domainof attraction whose size depends on the desired set-point. The procedure presented in [60] overcomes thisproblem yielding a large domain of attraction. This is achieved by switching the value of a gain α(t) everytime that the trajectories of the system reach the boundary of a set N , which in this case will be designed asa subset of M. However, the methodology in [60] is not immediately suited to the present problem becausethe control input u(t) affects the size of the set M. To clarify this, notice that, if the invariance method isapplied directly, the set in which u4 is well-defined is given by
Mu(t) = x1 ∈ R2|ψ(x1 + xd
1)2 + 4φ(x1 + xd
1)[
−u(t) − φ(x1 + xd1)U(x1) +G− χ(x1 + xd
1)]
≥ 0, (4.44)
where U(x1) is defined in (4.38). Because u(t) depends on α(t) through a linear feedback transformation,u(t) switches along with α(t); thus, the future values of u(t) are unpredictable and Mu(t) varies erratically.Therefore, it is impossible to guarantee that N ⊂ Mu(t) for every possible value of u(t).
In order to overcome this inconvenience, dynamic feedback is implemented so that the shape and size ofM depend only on the state and not on the input. In particular, consider the following dynamic extensionfor the x1 subsystem
x1 :
˙x1 = x2
˙x2 = z˙z = u,
(4.45)
which can be written as˙x1 = Ax1 + bu (4.46)
with obvious definition of A and b. From (4.42), the set where u4 is real can now be expressed as M = x1 ∈R2, z ∈ R |ψ(x1 + xd
1)2 + 4φ(x1 + xd
1)[
−z − φ(x1 + xd1)U(x1) +G− χ(x1 + xd
1)]
≥ 0. Since M does notdepend on u anymore, the invariance control presented in [60] can now be employed. Define the coordinatetransformation
ξ1ξ2y
=
x1
x2
k1x1 + k2x2 + k3z
(4.47)
and the feedback transformation
u =1
k3
(
v − (2QT12P11 −Q21)[ξ1, ξ2]
T −Q22y)
(4.48)
48
where v is an external input, k1, k2, and k3 are such that the polynomial k3λ2 + k2λ + k1 is Hurwitz, and
Q11, Q12, Q21, Q22 are such thatQ11 = TAM1 Q12 = TAM2
Q21 = kTAM1 Q22 = kTAM2
T =
[
1 0 00 1 0
]
M1 =
1 00 1
−k1
k3
−k2
k3
M2 =
001k3
,
k = [k1, k2, k3]T , and P11 is the solution to the Lyapunov equation
QT11P11 + P11Q11 = −µW11. (4.49)
The transformation (4.47) and the preliminary feedback (4.48) map (4.45) into (ξ1, ξ2, y) coordinates as
ξ =
[
0 1
−k1
k3
−k2
k3
]
ξ +
[
01k3
]
y
y = −2QT12P11[ξ1, ξ2]
T + v. (4.50)
As pointed out in [60], (4.50) is output strictly passive and hence exponentially stable for any feedbackof the form
v = −α(t)y (4.51)
provided α(t) > 0 for all t ≥ 0. As presented later in this chapter, if α(t) is a piecewise constant signalthat switches every time that the state trajectory hits the boundary of a predesigned set, then such setcan become invariant. Since the procedure presented in [60] requires the invariant set to satisfy certainproperties, an inner approximation N ⊂ M, rather than the original M, will define this invariant set. Likein [60], let Φ : R3 → R be a C1 function such that N = (ξ1, ξ2, y)|Φ(ξ1, ξ2, y) ≤ 0. A choice of Φ thatsatisfies the continuous differentiability requirement while allowing the design of N is
Φ(ξ1, ξ2, y) = a(ξ1 + xd1 − x′1)
2 + bξ22 + c(x′1 − ξ1 − xd1)ξ2 + dy2 − r2, (4.52)
which describes a three-dimensional ellipsoid centred at (x′1, 0, 0). In this case, the values of the positiveconstants a, b, c, d, and x′1 define the shape and size of the set N . Figure 4.18 compares the sets M andN for a given set of parameters and makes evident the main advantage of the invariance control method;namely, the resulting domain of attraction (the invariant set) is large and fixed for any set-point.
In order to ensure positive invariance of the set N , the trajectories of the system at ∂N (the boundaryof N ) must point inwards such set, i.e., Φ(ξ1, ξ2, y) < −ε for some small positive constant ε. Later, it willbecome evident that fulfillment of this condition requires independent study of the two cases when y = 0and y 6= 0. The time derivative of Φ(ξ1, ξ2, y) is given by
Φ =[
2a(ξ1 − x′1 + xd1) − cξ2, 2bξ2 + c(x′1 − ξ1 − xd
1), 2dy]
ξ2−k1
k3
ξ1 − k2
k3
ξ2 + 1k3
y
−2QT12P11[ξ1, ξ2]
T − αy
≤
a+c
2k3(k2 + k1)
(ξ1 − x′1 + xd1)
2 +
a− c+ck2
2k3+
b
k3(k1 − 2k2)
ξ22
+k1
k3
(
b +c
2
)
ξ21 + F (y),
(4.53)
where F (0) = 0.If y = 0, the invariance condition Φ(ξ1, ξ2, y = 0) < 0 is met if k1, k2, and k3 are chosen such that
a+c
2k3(k2 + k1)
(ξ1 − x′1 + xd1)
2 +
a− c+ck2
2k3+
b
k3(k1 − 2k2)
ξ22 +k1
k3
(
b+c
2
)
ξ21 < 0. (4.54)
49
−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08−15
−10
−5
0
5
10
x1
z
Region of real controlsInvariant set boundaryDiscontinuity
Figure 4.18: Projection of M and N into the (x1, z) plane.
Inequality (4.54) yields a set of linear inequalities on k1, k2, and k3 whose solution can be found by solvingthe linear programming problem given by
minAopk ≤ bop
k > 0
CT k (4.55)
where C = [c1, c2, c3] is a vector of design parameters, bop = −[κ, κ]T , κ ∈ R+, and
Aop =
[
c c 2a2b c− 4b 2a− 2c
]
. (4.56)
Because the weighting vector C introduces extra degrees of freedom, solving the set of inequalities as alinear programming problem conveys two advantages. First, it individually affects the speeds of convergenceof x1, x2 and z. Second, it permits free design of the geometric characteristics of the zero dynamics planey = 0. Notice that, if the LP algorithm generates values of the k constants that do not satisfy the Hurwitzcondition, different values of C and bop should be tried.
In the second case, y 6= 0, invariance of N is guaranteed by switching the value of α(t) every time thatthe system’s trajectory hits ∂N , i.e., every time Φ ≥ 0. Recall that, in order to guarantee stability of theclosed-loop system, α(t) must be strictly positive. A choice of α(t) that satisfies both stability and invarianceconstraints is
α(tk) =ε+ 2a(ξ1 − x′1) − cξ2 ξ1 + 2bξ2 + c(x′1 − ξ1) ξ2 − 4dyQT
12P11[ξ1, ξ2]T
2dy2
=ε+ 2a(x1 − x′1) − cx2 x2 + 2bx2 + c(x′1 − x1) z − 4dyQT
12P11[x1, x2]T
2dy2
(4.57)
where tk denotes the k−th switching time. The work in [60] proves that the resulting sequence α(tk) ismonotonically increasing, eventually constant (i.e., the number of switchings is finite), and positive definiteprovided that Φ ≥ 0. The authors of [60] also proved that the controller v = −α(t)y ensures exponentialstability of the origin of (4.46).
In conclusion, for suitable values of k1, k2, k3, a, b, c, d, C, and κ, N ⊂ M (a designed set where u4 is real)is positively invariant and any trajectory of the closed-loop system is exponential stable.
50
1 2 3 4 5 6 7 8 9
−0.05
0
0.05
time [s]
Dis
pla
ce
me
nts
[m
]
x1
x3
x5
Figure 4.19: Transient position of the 3-DOF sys-tem under invariance control.
0 1 2 3 4 5 6 7 8 9 10
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
time [s]
Sp
ee
ds [m
/s]
x2
x4
x6
Figure 4.20: Transient speed of the 3-DOF systemunder invariance control.
4.4.1 Simulation Results
Consider the next set of values: r = 0.12, a = 80, b = 0.5, c = −5, d = 2, x′1 = 0.048, C = [0.1, 1 , 1]T ,κ = 1, µ = 0.001, α(0) = 1, and ε = 1 × 10−6. With this values, the LP algorithm generated k =[0.1175, 0.4025, 0.01]T .
Figures 4.19 and 4.20 show the x1 and x2 response for initial conditions x0 = [0.047, 0.05,−0.05, 0, 0.05, 0]T,z(0) = 0 and desired set-point at xd
1 = 0.055. Figure 4.21 presents the evolution of the control inputs u1,u2, and u4 against time for the same initial conditions. The bold curve in Figure 4.22 represents the state’strajectory, which is constrained inside the three-dimensional ellipsoid that delineates the boundary of theset N .
4.5 Controllers’ performance comparison
Figures 4.23 to 4.26 show the response of the 3-DOF system under the action of the three controllers forinitial conditions x0 = [0.047, 0.05,−0.05, 0, 0.05, 0]T and desired set-point at xd = [0.055, 0, 0, 0, 0, 0].
For the parameters used in simulations, the linear controller stabilizes the system around equilibriumpoints ranging from 0.04m to 0.15m but not for every initial condition in the same interval. If the set-pointor the initial conditions lie outside of that range, or if the state x1(t) exits it, then simulations producecomplex values of the control inputs. In all instances the speed of convergence is high and the magnitude ofthe current consumption is the lowest. Because its domain of attraction can be found only numerically, thisdesign does not analytically ensure well-definiteness of the control inputs.
In turn, the partial feedback linearization controller reaches higher steady-state currents (see Figure4.25) under the same circumstances and its speed of response varies significantly with the initial conditions.Moreover, in the best case, the range 0.02 < x1 < 0.06 restricts the size of its domain of attraction whichdepends on the position of the equilibrium point.
The invariance controller requires currents as large as those associated to the partial feedback linearizationcontroller. The domain of attraction of the example shown here lies inside 0.035 < x1 < 0.07. The speedof convergence of this closed-loop system depends on the initial conditions but also on the occurrence ofswitchings. Although in this particular case no switchings ocurred, the speed of convergence is the slowestone.
51
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time [s]
u1,
u2,
u4 [
A]
u1
u2
u4
Figure 4.21: Control inputs u1, u2, and u4 for the3-DOF invariance controller.
−0.2
0
0.2 0.03
0.04
0.05
0.06
0.07
0.08
−15
−10
−5
0
5
10
15
x1
x2
z
Figure 4.22: Trajectory of the 3-DOF system un-der invariance control.
0 1 2 3 4 5 6 7 8 9 100.047
0.048
0.049
0.05
0.051
0.052
0.053
0.054
0.055
time [s]
x1 [m
]
LQRComposed controlInvariance control
Figure 4.23: Transient airgap length of the 3-DOFsystem for the three controllers.
0 1 2 3 4 5 6 7 8 9 10−0.06
−0.04
−0.02
0
0.02
0.04
0.06
time [s]
x3,x
5 [m
/s]
x3 LQR
x5 LQR
x3 Composed control
x5 Composed control
x3 Invariance control
x5 Invariance control
Figure 4.24: Transient x and z positions of the3-DOF system for the three controllers.
52
0 1 2 3 4 5 6 7 8 9 10−0.06
−0.04
−0.02
0
0.02
0.04
0.06
time [s]
u1,u
2 [A
]
u1 LQR
u2 LQR
u1 Composed control
u2 Composed control
u1 Invariance control
u2 Invariance control
Figure 4.25: Control inputs u1, u2 of the 3-DOFsystem for the three controllers.
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time [s]
u4 [A
]
LQR u3
LQR u4
Composed controlInvariance control
Figure 4.26: Control inputs u3, u4 of the 3-DOFsystem for the three controllers.
Simulation results, not included here, show that careful selection of the parameters τp and hm can enlargethe set where the controls are well-defined.
In conclusion, each controller presents different advantages. While the LQR design provides the fastestresponse (see Figures 4.23 and 4.24) and the lowest power consumption (see Figures 4.25 and 4.26), thenonlinear controllers produce smoother responses and guarantee stability within well-defined sets. Whencompared to the first nonlinear controller, the invariance-based one achieves a slower response but providesa larger domain of attraction. In general, the three controllers achieve the similar performances in termsof steady-state current consumption while allowing limited tuning of speed of convergence and currentconsumption.
4.6 Discussion
This chapter dealt with stabilization of a 3-DOF levitation system based on PMLSMs. After proposing amechanical setup, this research developed a model and different control strategies. The simulation of threecontrollers yields results that Section 4.5 uses to assess their performance. The performance of the threedifferent control laws varies significantly but all of them succeed on regulating the position of the levitatedplaten. Although the 3-DOF design can neither control nor obstruct the rotation of the platen when idealconditions are not met, the encouraging control results presented here will support the development of amore sophisticated device introduced in Chapter 5.
53
Chapter 5
Design for five controlled degrees of
freedom
Chapter 4 presented both the modelling and the control design of a system conceived to control three degreesof freedom of a levitating platen. Three different control methods –LQR, a combination of Lyapunov andLaSalle’s theorems, and invariance control– were developed for the resulting dynamic model. Althoughthese controllers succeeded in simulations, the remaining degrees of freedom were neither controlled norconstrained. Such omission affects the system’s stability because the modelling assumed perfect symmetryof the device, meaning that the three different motors should have identical characteristics. In reality,even when assembled by the same manufacturer, similar motors have slightly different characteristics andthe assumption of symmetry is not satisfied. In this sense, robustness of the 3-DOF levitation system isquestionable and needs verification.
The robustness of the 3-DOF mechanism with non-identical motors can be tested through further simu-lations. In order to investigate the effects of motor-to-motor parameters’ variation, one has to augment theoriginal model with at least one more degree of freedom. Let this extra degree of freedom be the rotationaround the z axis. In order to consider the variation of some of the many parameters of the motors, increaseby 0.5% the constant Υ in motor 3. This variation originates a difference between the normal forces inMotors 1 and 3 (refer to figure 4.2) and generates a torque around the z axis. Simulation results show thatthe 3-DOF invariance controller is unable to asymptotically stabilize the system under these circumstances.In fact, even a zero initial angle evolves into bounded but sustained oscillations. Figure 5.1 shows theunstable response of the system with initial conditions x0 = [0.047, 0,−0.05, 0, 0.05, 0, 0, 0, 0] and set-pointxd = [0.055, 0,−0.05, 0, 0.05, 0, 0, 0, 0], where x8 and x9 represent the new degree of freedom and its speedrespectively. Figure 5.2 shows the corresponding transient airgap length and airgap length speed response.These figures make evident the need for both a model with additional degrees of freedom and its controller.
Consider the next 4-PMLSMs design that controls two additional degrees of freedom. The four motorsof the new apparatus lie in a way such that self-alignment of the magnets prevents rotation of the platenaround the y axis. Therefore, only five degrees of freedom need to be studied.
Section 5.1 introduces the new 5-DOF device and its model. That section first describes the mechanicalstructure of the apparatus and then provides expressions for the principal moments of inertia of the platen.Section 5.1 concludes by presenting the device’s equations of motion, which yield a system of ten nonlinearODEs. Later, in Section 5.2, feedback linearization transforms the system into a set of four lower orderlinear subsystems. The invariance control method (see [60]) stabilizes these four subsystems producing thesimulation results that Section 5.3 describes. Section 5.4 concludes the present chapter by discussing theperformance of the control design.
54
0 2 4 6 8 10 12 14 16 18 20
−6
−4
−2
0
2
4
6
x 10−4
time [s]
An
gle
[m
] a
nd
sp
ee
d [
m/s
]
x8
x9
Figure 5.1: Angular response of a disturbed 4-DOF system.
0 5 10 15 20
0
0.01
0.02
0.03
0.04
0.05
time [s]
Dis
pla
ce
me
nt [m
] a
nd
sp
ee
d [m
/s]
x1
x2
Figure 5.2: Airgap length response of a disturbed4-DOF system.
Stators
Movers
Platen
z
x
y
Figure 5.3: Setup for five degrees of freedom with four PMLSM
55
LA
GM
LCL
Bτ= p
m
yG
hm
hb
ht
LC
xz
y
motor 2motor 1
τ
motor 3
Figure 5.4: Disposition of the movers on the platen for the 5-DOF device
5.1 Setup description and modelling
This section derives a model for the 5-DOF levitation device by first finding the mechanical parameters ofits floating platen. Then, assumptions and design conditions are stated in order to simplify the dynamicanalysis. Finally, an analysis of forces and torques incorporates the results from Chapter 3 yielding a tendimensional model.
Consider the device shown in Figure 5.3, which includes a tray that holds four PMLSM movers on itstop face. The movers lie in pairs along two of the main axes of symmetry of the platen, i.e., axes x and z.Let MG, CL, ht, and hb be the center of mass of the platen, the clearance between motors, the height ofthe tray housing the movers, and the height of the back iron to which the permanent magnets are attached,respectively (refer to Figure 5.4). Provided that the platen is symmetric with respect to axes x and z, itscenter of mass lies on the y axis. Let yG be the shortest distance between the center of mass and the tray’sbottom face. The value of yG is
yG =4pmMpm
(
ht + hb + hm
2
)
+Mtht
2 + 4Mb
(
ht + hb
2
)
4pmMpm +Mt + 4Mb. (5.1)
Then, the principal moments of inertia of the platen are given by
Ix = Itx +
4∑
n=1
(Ipm)n +
4∑
n=1
(Ib)n (5.2)
Iy = Ity + 4Iby + 4
pm∑
n=1
[
Ipmy +
(
LA
2+ CL + nτ − τp
2
)2
Mpm
]
(5.3)
Iz = Itz +
4∑
n=1
(Ipm)n +
4∑
n=1
(Ib)n (5.4)
where
Itx = Itz =Mt
12
[
(2LB + 2CL + LA)2
+ h2t
]
+Mt
(
ht
2− yG
2
)2
(5.5)
Ipm1 = Ipm3 = Mpm
pm∑
n=1
[
1
12
(
h2m + τ2
p
)
+
(
LA
2+ CL + nτ − τp
2
)2
+
(
ht − yG +hm
2+ hb
)2]
(5.6)
56
Ipm2 = Ipm4 = pmMpm
[
1
12
(
L2A + h2
m
)
+
(
ht − yG +hm
2+ hb
)2]
(5.7)
Ib1 = Ib3 = Mb
[
1
12
(
L2B + h2
b
)
+
(
ht − yG +hb
2
)2
+
(
LA
2+ CL +
LB
2
)2]
(5.8)
Ib2 = Ib4 = Mb
[
1
12
(
h2b + L2
A
)
+
(
ht − yG +hb
2
)2]
(5.9)
Ity =Mt
6(LA + 2CL + 2LB)
2(5.10)
Iby = Mb
[
1
12
(
L2A + L2
B
)
+
(
LA
2+ CL +
LB
2
)2]
(5.11)
Ipmy =Mpm
12
(
L2A + τ2
p
)
(5.12)
LB = pmτ (5.13)
and Mt is the mass of the tray, Mpm is the mass of each permanent magnet and Mb is the mass of the backiron on each mover.
Useful model simplifications result from design specifications and assumptions, like the next one reducingto five the number of degrees of freedom of the platen.
Assumption 9. Because the magnetic fields of the permanent magnets on the movers tend to align withthe fields produced by the stator, the resultant torque around the y axis is zero when movers and statorsare aligned.
Now, refer to Figure 5.5 and notice that tilting the platen prevents the effective airgap length from beinguniform along the motors’ surfaces. As a result, the magnetic fields do not interact as expected and thenormal force distribution is irregular. The model can not analytically include this fact because of its highcomplexity but limiting the magnitude of the tilting angle justifies the following simplifying assumption.
Assumption 10. Given small platen tilting, equivalent resultant forces exerted on the centres of themovers’ surfaces can replace the non-uniform force distributions on each motor.
Assumption 10 validates expressions for Fx and Fy in (3.45) and (3.55), respectively, for any small tiltingangles and for any position of the platen. Consequently, the origin of the inertial reference frame can beidentified with the origin of the moving one (henceforth the frame x′y′z′). Let these two points coincide atthe center of mass of the platen; thus, matching the axes of the moving reference frame and the principalaxes of inertia diagonalizes the inertia tensor.
Let θx, θy and θz be the angles of rotation of the platen around its x′, y′, and z′ axes respectively, i.e.,the angles between the x′y′z′ and the xyz reference frames . Because the inertia tensor is diagonal, the Eulerequations of motion for the platen are
∑
Mx = Ixθx − (Iy − Iz)θy θz (5.14)∑
My = Iy θy − (Iz − Ix)θz θx (5.15)∑
Mz = Iz θz − (Ix − Iy)θxθy (5.16)
where∑
Mx,∑
My and∑
Mz are the sum of the moments about the x′, y′ and z′ axes respectively.
Because of Assumption 9, θy = 0 and thus, for θy(0) = 0, it follows that θy(t) = 0 ∀t ≥ 0. Therefore,the axis x′ lies on the xy plane (see Figure 5.5) and, in a similar manner, z′ lies on the yz plane. Theequations of motion become
∑
Mx = Ixθx (5.17)∑
My = 0 (5.18)∑
Mz = Iz θz. (5.19)
57
Figure 5.5: Fixed and rotating frames of reference for the 5-DOF device
Since the normal forces are assumed perpendicular to the movers’ surfaces, they are perpendicular to theplane x′z′ and no projection is necessary. Figure 5.5 depicts these forces, from which it follows that
rFy3 − rFy1 = Iz θz (5.20)
where r is the shortest distance between the y axis and the center of mass of any of the four movers of thePMLSMs. Analogously,
rFy4 − rFy2 = Ixθx. (5.21)
Then, the full set of equations of motion of the platen is given by
Mpg = MpG− Fy1 − Fy2 − Fy3 − Fy4
Mpx = Fx1 + Fx3
Mpz = Fz2 + Fz4
Ixθx = rFy4 − rFy2
Iz θz = rFy3 − rFy1.
(5.22)
Let g be the airgap length at the point that lies at the intersection of the y′ axis and the plane tangentto the upper surfaces of the movers, so that
g1 = g − r sin(θx)
g2 = g + r sin(θx)
g3 = g − r sin(θz)
g4 = g + r sin(θz).
(5.23)
58
Also, consider the functions
φ(·) =Υϕ2Γ(·) coth2
πτ [(·) + hm]
K2c (·) (5.24)
ψ(·) =2ϕΥΓ(·)Bpmy(·)
Kc(·)coth
π
τ[(·) + hm]
(5.25)
γ(·) =%
Kc(·) sinh
πτ [(·) + hm]
[
η − ζ
(·) + κ
]
(5.26)
χ(·) = ΥΓ(·)B2pmy(·), (5.27)
where Υ, ϕ,Γ(·),Kc(·), %, Bpmy, η, ζ, and κ are as defined in Section 3.5. Then, the forces in Motor n are
Fxn = γ(gn)idn n = 1, 3 (5.28)
Fyn = φ(gn)(
i2dn + i2qn
)
+ χ(gn) + ψ(gn)iqn n = 1, . . . , 4 (5.29)
Fzn = γ(gn)idn n = 2, 4. (5.30)
Hence, substitution of equations (5.28),(5.29), and (5.30) into (5.22) yields
Mpg = MpG− φ(g1)(
i2d1 + i2q1)
− χ(g1) − ψ(g1)iq1 − φ(g2)(
i2d2 + i2q2)
− χ(g2) − ψ(g2)iq2
− φ(g3)(
i2d3 + i2q3)
− χ(g3) − ψ(g3)iq3 − φ(g4)(
i2d4 + i2q4)
− χ(g4) − ψ(g4)iq4
Mpx = γ(g1)id1 + γ(g3)id3
Mpz = γ(g2)id2 + γ(g4)id4
Ixθx = rφ(g4)(
i2d4 + i2q4)
+ rχ(g4) + rψ(g4)iq4 − rφ(g2)(
i2d2 + i2q2)
− rχ(g2) − rψ(g2)iq2
Iz θz = rφ(g3)(
i2d3 + i2q3)
+ rχ(g3) + rψ(g3)iq3 − rφ(g1)(
i2d1 + i2q1)
− rχ(g1) − rψ(g1)iq1
(5.31)
Recall that, because of symmetry, Iz = Ix , J and define u = [u1, . . . , u8]T = [iq1, . . . , iq4, id1, . . . , id4]
T
and x = [g, g, x, x, z, z, θx, θx, θz, θz]T . After the change of notation given by φ(gi) = φi, χ(gi) = χi, ψ(gi) =
ψi, and γ(gi) = γi, the equations of motion (5.31) become
x1 = x2
x2 = G− φ1
Mp
(
u25 + u2
1
)
− χ1
Mp− ψ1
Mpu1 −
φ2
Mp
(
u26 + u2
2
)
− χ2
Mp− ψ2
Mpu2
− φ3
Mp
(
u27 + u2
3
)
− χ3
Mp− ψ3
Mpu3 −
φ4
Mp
(
u28 + u2
4
)
− χ4
Mp− ψ4
Mpiq4
x3 = x4
x4 = γ1u5 + γ3u7
x5 = x6
x6 = γ2u6 + γ4u8
x7 = x8
x8 =rφ4
J
(
u28 + u2
4
)
+rχ4
J+rψ4
Ju4 −
rφ2
J
(
u26 + u2
2
)
− rχ2
J− rψ2
Ju2
x9 = x10
x10 =rφ3
J
(
u27 + u2
3
)
+rχ3
J+rψ3
Ju3 −
rφ1
J
(
u25 + u2
1
)
− rχ1
J− rψ1
Ju1.
(5.32)
59
5.2 Full feedback linearization and invariance control
Because of the high complexity of the model given by (5.32), the 3-DOF Lyapunov-based method presentedin Chapter 4 cannot be extended to this 5-DOF case. Nevertheless, the invariance control method does applyto this more complex problem, as described in the present section.
The following procedure –a straightforward generalization of the idea presented in Chapter 4– designsan invariance controller for the system given by (5.32). First, a dynamic feedback transformation linearizesthe system while decoupling it into four subsystems. Secondly, sufficient conditions are found for the well-definiteness of this feedback transformation. Finally, the same idea presented in Chapter 4 is used to define aninvariance controller that stabilizes the four subsystems while ensuring the well-definiteness of the feedbacktransformation.
5.2.1 Feedback linearization
The feedback transformation
u1 = − ψ1
2φ1+Mp
2φ1
√
ψ21
M2p
− 4φ1
Mp
(
φ1u25
Mp+χ1
Mp+x11
2+
J
2rMpx12 −
G
4
)
u2 = − ψ2
2φ2+Mp
2φ2
√
ψ22
M2p
− 4φ2
Mp
(
φ2u26
Mp+χ2
Mp+x11
2− J
2rMpx12 −
G
4
)
u3 = − ψ3
2φ3+Mp
2φ3
√
ψ23
M2p
− 4φ3
Mp
(
χ3
Mp+
J
2rMpx13 −
G
4
)
u4 = − ψ4
2φ4+Mp
2φ4
√
ψ24
M2p
− 4φ4
Mp
(
χ4
Mp− J
2rMpx13 −
G
4
)
u5 =Mpv1γ1
u6 =Mpv2γ2
u7 = 0
u8 = 0
(5.33)
and the dynamic feedback given byx11 = v3
x12 = v4
x13 = v5
(5.34)
transform system (5.32) into four lower-order linear subsystems given by
x1 :
˙x1 = x2
˙x2 = x11
˙x11 = v3
(5.35) x2 :
˙x3 = x4
˙x4 = v1˙x5 = x6
˙x6 = v2
(5.36)
x3 :
˙x7 = x8
˙x8 = x12
˙x12 = v4
(5.37) x4 :
˙x9 = x10
˙x10 = x13
˙x13 = v5,
(5.38)
60
where the equilibrium point in R13 is given by xd = [xd1 , . . . , x
d13]
T and x = x−xd. For set-point stabilization,this vector has the form xd = [xd
1 , 0, xd3, 0, x
d5, 0, x
d7, 0, x
d9, 0, 0, 0, 0].
The decoupling of the system partitions the state into x1 := [x1, x2, x11]T , x2 := [x3, . . . , x6]
T , x3 :=[x7, x8, x12]
T , and x4 := [x9, x10, x13]T . The subsystem x2 is linear and can be globally exponentially
stabilized by LQR feedback of the form[
v1v2
]
= −Kx2. (5.39)
On the other hand, LQR feedback can not stabilize the x1, x3, and x4 dynamics because the square rootsin the control law (5.33) impose constraints on these states. Specifically, these constraints require thetrajectories of the system to be contained inside a set where the arguments of the square roots are positive.Because this set is not defined in the form that the invariance control method requires, the next subsectionpresents the design of a suitable inner approximation.
5.2.2 Design of the invariant set
After addition of the three integrators (5.34), the arguments of the radicals in (5.33) do not depend onthe control inputs v1, . . . , v5 anymore. Consequently, constraining the state ensures real controls withoutimposing constraints on the control inputs v1, . . . , v5. From the definition of the controller (5.33), theconstraints on the state are
0 ≤ ψ21
M2p
− 4φ1
Mp
(
Mpφ1v21
γ21
+χ1
Mp+x11
2+
J
2rMpx12 −
G
4
)
0 ≤ ψ22
M2p
− 4φ2
Mp
(
Mpφ2v22
γ22
+χ2
Mp+x11
2− J
2rMpx12 −
G
4
)
0 ≤ ψ33
M2p
− 4φ3
Mp
(
χ3
Mp+
J
2rMpx13 −
G
4
)
0 ≤ ψ44
M2p
− 4φ4
Mp
(
χ4
Mp− J
2rMpx13 −
G
4
)
.
(5.40)
Now, the problem faced is similar to the one solved in the previous chapter; namely, the need of havingpositive arguments for the square roots in (5.33) imposes nonlinear constraints on the states. In this case,these constraints depends on different groups of states that are not related to the partitions previouslydefined. This cross-dependency obstructs the decomposition of the invariant set into lower dimensional setscorresponding to each state partition. By calculating upper bounds to x2(t), like in Chapter 4, one reducesthe number of states on which each constraint depends. This way, the trajectories of the system must beconstrained within the intersection of the sets now defined by
0 ≤ ψ21
M2p
− 4φ1
Mp
(
Mpφ1(v1)2
γ21
+χ1
Mp+x11
2+
J
2rMpx12 −
G
4
)
0 ≤ ψ22
M2p
− 4φ2
Mp
(
Mpφ2(v2)2
γ22
+χ2
Mp+x11
2− J
2rMpx12 −
G
4
)
0 ≤ ψ33
M2p
− 4φ3
Mp
(
χ3
Mp+
J
2rMpx13 −
G
4
)
0 ≤ ψ44
M2p
− 4φ4
Mp
(
χ4
Mp− J
2rMpx13 −
G
4
)
,
(5.41)
where v1 and v2 are as defined in (4.37).Since the properties of the sets defined by (5.41) are unknown, the intersection of these sets will be
61
numerically approximated by an interior ellipsoid of the form
Φ(x1, x3, x4) =
x1
x3
x4
− x′
T
H
x1
x3
x4
− x′
, (5.42)
where x′ is the center of the ellipsoid in R9.To simplify the design of the invariance controllers for x1, x3, and x4, which otherwise would be unman-
ageable, the structure of H is constrained to be block-diagonal as follows
H = block diag(H1, H3, H4), Hi =
aici
2 0ci
2 bi 00 0 di
, i = 1, 3, 4. (5.43)
where ai, bi, ci, and di are positive constants to be designed. With this choice for the structure of H , theproblem of containing x1, x3, and x4 inside the set defined by the inequalities in (5.41) has been convertedto the problem of constraining xi inside the ellipsoid (xi − x′i)THi(x
i − x′i) ≤ r2i , i = 1, 3, 5, which issignificantly simpler, as shown next.
Provided that H is in block diagonal form, the invariance function Φ(x) can be written as
Φ(x1, x3, x4) = Φ1(x1) + Φ3(x
3) + Φ4(x4). (5.44)
It follows from (5.44) that the set N = x|Φ(x) ≤ 0 can be divided into three three-dimensional subsets,i.e., N1 = x1|Φ1(x
1) ≤ r21, N3 = x3|Φ3(x3) ≤ r23, and N4 = x4|Φ4(x
4) ≤ r24. This simplificationdecouples the control problem into three parts to which the ideas in Chapter 4 can be applied.
In order to find the largest domain of attraction of (5.32), one should find the parameters ai, bi, ci, di, ri,i = 1, 3, 4, and the vector x′i, guaranteeing that the set defined by
Φ(x1, x3, x4) ≤ r21 + r23 + r24 (5.45)
be the maximum volume ellipsoid contained in the set defined by the inequalities in (5.41). Unfortunately,this problem is very difficult and goes beyond the scope of this thesis. To better clarify this point, the nextsubsection presents a brief overview of set estimation methods.
The Ellipsoid Method
Set estimation is a multidisciplinary problem that has been addressed by the communities of applied math-ematics, operations research, computer science, and electrical engineering. The reason for such popularity isthe wide diversity of its applications, which include filtering, computer graphics, and optimization. Despitethis much attention, certain problems remain open and others have been solved by means of costly proce-dures. Given the requirements of the problem targeted in this chapter, this exploration of set estimationmethods covers exclusively ellipsoidal set estimation.
The most well-know method for ellipsoidal set estimation is the so-called ellipsoid method. This schemeoffers two similar versions: the method of the inscribed ellipsoid and the method of the exterior ellipsoid.The first version finds the maximum volume ellipsoid contained in a given set while the second version findsthe minimum volume ellipsoid containing the set.
The ellipsoid method is an iterative procedure initialized with a large approximation, which is known tocontain the set to estimate. The algorithm converges along with successive bisections of the approximation,which are known as cuts. The authors of [63] explain how two versions of the basic method estimate convexpolytopes by means of two different kinds of cuts.
The study presented in [64] offers a complete survey of the ellipsoid method by discussing several non-technical aspects as well. First, the article outlines the foundations of the method and analyses solvability.An historical overview follows this analysis and precedes a brief study of different cuts. Then the authorspresent the solution of linear optimization problems through this method. A section that relates the ellipsoid
62
method and other alternatives complements the previous material. Finally, the combinatorial implicationsand appendixes including examples conclude the survey.
Similarly to [64], [62] introduces ellipsoid estimation methods that are applicable only to linear problems.This work uses path following algorithms to solve the problems of the maximum inscribed ellipsoid and theminimum volume exterior ellipsoid. The authors overview the applications –including robust control– of thecategory of convex optimization problems that comprises these two ellipsoidal estimation methods.
In [65], Sabharwal and Potter improved the previous work by adapting the ellipsoid method to sets definedby nonlinear constraints. They propose two different versions of the method which differ in the way thatthe cuts are effected. Since the optimal version of the method is hard to implement and even unfeasible, theauthors propose a suboptimal algorithm that overcomes this drawback. Both algorithms iteratively computethe matrix that defines the quadratic form of the ellipsoidal estimate –H in this case. Since the procedureupdates H using numerically computed vectors, the algorithms do not control the structure of the matrix.
Although the work in [65] considers nonlinear constraints, it proposes a method that does not apply tothe design of the invariant set because it generates matrices whose structure can not be constrained. Sinceno rigorous method is available for the design of the invariant set, one has to approximate by trial and errorthe constants ai, bi, ci, di, and ri, i = 1, 3, 4, that maximize the volume of the ellipsoid.
5.2.3 Passification and design of switched gains
Following the same procedure outlined in Chapter 4, the subsystems (5.35), (5.37), and (5.38) are passifiedby the transformations defined as
ξ1ξ2y1
=
x1
x2
k11x1 + k1
2x2 + k13x11
(5.46)
ξ7ξ8y3
=
x7
x8
k31x7 + k3
2x8 + k33x12
(5.47)
ξ9ξ10y4
=
x9
x10
k41x9 + k4
2x10 + k43x13
, (5.48)
and the feedback transformation
v3 =1
k13
(
v3 − (2(Q112)
TP 111 −Q1
21)[ξ1, ξ2]T −Q1
22y1)
(5.49)
v4 =1
k33
(
v4 − (2(Q312)
TP 311 −Q3
21)[ξ7, ξ8]T −Q3
22y3)
(5.50)
v5 =1
k43
(
v5 − (2(Q412)
TP 411 −Q4
21)[ξ9, ξ10]T −Q4
22y4)
(5.51)
where v4, v5 and v6 are external inputs, ki1, k
i2, and ki
3 for i = 1, 3, 4 are such that the polynomials ki3λ
2 +ki2λ+ ki
1 are Hurwitz, and Qi11, Q
i12, Q
i21, Q
i22 are such that
Qi11 = TAM i
1 Qi12 = TAM i
2
Qi21 = kT
i AMi1 Qi
22 = kTi AM
i2
i = 1, 3, 4
T =
[
1 0 00 1 0
]
M i1 =
1 00 1
−ki1
ki3
−ki2
ki3
M i
2 =
001ki3
,
ki = [ki1, k
i2, k
i3]
T , P i11 for i = 1, 3, 4 are the solutions to the Lyapunov equations given by
(Qi11)
TP i11 + P i
11Qi11 = −µW i
11 i = 1, 3, 4, (5.52)
63
and W i11 identity matrices. The transformations (5.46), (5.47), and (5.48) together with the preliminary feed-
back (5.49), (5.50), and (5.51) map (5.35), (5.37), and (5.38) into the (ξ1, ξ2, y1), (ξ7, ξ8, y3), and (ξ9, ξ10, y4)coordinates, respectively. As a result, the original subsystems become
ξi =
[
0 1
−ki1
ki3
−ki2
ki3
]
ξi +
[
01ki3
]
yi
yi = −2(Qi12)
TP i11ξ
i + vj , (5.53)
where ξ1 = [ξ1, ξ2]T , ξ3 = [ξ7, ξ8]
T , and ξ4 = [ξ9, ξ10]T , for i = 1, 3, 4 and j = 3, 4, 5. As pointed out in [60],
(5.53) are exponentially stable for any feedback of the form
v3 = −α1(t)y1 (5.54)
v4 = −α3(t)y3 (5.55)
v5 = −α4(t)y4, (5.56)
where the positive piecewise constant signals αi(t), i = 1, 3, 4, switch in order to redirect the state trajectoriesinto the sets Ni every time that the boundaries ∂Ni are reached.
Like in Chapter 4, an LP algorithm finds the values of the vectors k1, k3, and k4. The switched values ofαi(t) that ensure invariance are
α1(tk) =ε+ 2a1(x1 − x′1) − c1x2 x2 + 2b1x2 + c1(x
′1 − x1) x11 − 4d1y1(Q
112)
TP 111[x1, x2]
T
2d1y21
(5.57)
α3(tk) =ε+ 2a3x7 − c3x8 x8 + 2b3x8 − c3x1 x12 − 4d3y3(Q
312)
TP 311[x7, x8]
T
2d3y23
(5.58)
α4(tk) =ε+ 2a4x9 − c4x10 x10 + 2b4x10 − c4x9 x13 − 4d4y4(Q
412)
TP 411[x9, x10]
T
2d4y24
(5.59)
5.3 Simulation results
Simulation of the 5-DOF levitated stage considered a motor whose parameters are listed in Table 5.1.
µ0 4π × 10−7 Br 1.0505Hc 836000 µr 1.1000W/p 300 p 2t1 19.05× 10−3 b0 12.07 × 10−3
kw1 1 wc 57.15 × 10−3
δpm 7500 δi 7200τ 57.15× 10−3 LA 0.0500τp 28.575× 10−3 pm 4hm 2 × 10−3 hb 4.7 × 10−3
ht 0.0100 CL 0.2500Mt 0.5 J 0.1766
Table 5.1: Motor parameters used for simulation of the 5-DOF device
A trial and error procedure yielded the constants that define the invariance sets (refer to Table 5.2). Thatprocedure tried to maximize the volume of the invariant set while satisfying the conditions imposed on thefunction Φ.
Set i = 1 Set i = 3 Set i = 4
64
0 1 2 3 4 5 6−0.05
0
0.05
time [s]
Dis
pla
ce
me
nts
[m
]
x1
x3
x5
Figure 5.6: Position response of the 5-DOF systemunder invariance control
0 1 2 3 4 5 6
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
time [s]
Sp
ee
ds [m
/s]
x2
x4
x6
Figure 5.7: Speed response of the 5-DOF systemunder invariance control
ai 100 15 15bi 19.8 4.9 4.9ci -40 -10 -10di 50 10 10r2i 0.0005 0.0001 0.0001
x′i
0.02800
03×1 03×1
Table 5.2: Coefficients of the invariance function for the 5-DOF system
For the passification process, the LP algorithm generated the constants listed in Table 5.3.
kij j = 1 j = 2 j = 3
i = 1 0.0190 0.0335 0.0100i = 3 0.0136 0.0266 0.0100i = 4 0.0136 0.0266 0.0100
Table 5.3: Output constants for the passification of the 5-DOF system
Several simulations tested the controller under different initial conditions and set-points. Figures 5.6 to5.14 present the particular results from simulations run with ε = 1×10−3, initial conditions α1(0) = α3(0) =α4(0) = 1, x(0) = [0.029,−0.004, 0.05, 0,−0.05, 0, 0.0025,−0.004,−0.0025, 0.004, 0,−0.1, 0.1]T , and set-pointat xd = [0.026, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]T .
Figures 5.6 and 5.7 show the transient positions and speeds of the platen, respectively. The controlinputs are plotted against time in Figures 5.10 and 5.11. Figures 5.12 to 5.14 present the trajectories ofthe subsystems together with the boundaries of the respective invariant sets. For this particular simulation,the system reached steady state in about 7s and after 2 switchings of the αi(t) gains. Increasing the initialvalues of the gains αi(0) to 100 reduced the settling time to 5s without increasing the maximum magnitudesof the control inputs.
65
0 1 2 3 4 5 6
−2
−1
0
1
2
x 10−3
time [s]
An
gle
s [
rad
]
x7
x9
Figure 5.8: Transient angles of the 5-DOF systemunder invariance control
0 1 2 3 4 5 6
−6
−4
−2
0
2
4
6
x 10−3
time [s]
Sp
ee
ds [
rad
/s]
x8
x10
Figure 5.9: Transient angular speed of the 5-DOFsystem under invariance control
0 1 2 3 4 5 6
−1
−0.8
−0.6
time [s]
u1, u
2, u
3 [A
]
u1
u2
u3
Figure 5.10: Control inputs u1, u2 and u3 for the5-DOF system under invariance control
0 1 2 3 4 5 6−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
time [s]
u4, u
5, u
6 [A
]
u4
u5
u6
Figure 5.11: Control inputs u4, u5 and u6 for the5-DOF system under invariance control
66
−0.01
−0.005
0
0.005
0.01
0.024
0.026
0.028
0.03
0.032−0.6
−0.4
−0.2
0
0.2
0.4
x2
x1
x1
1
Figure 5.12: Trajectory x1(t) of the 5-DOF system under invariance control
−0.01
−0.005
0
0.005
0.01−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01
−0.5
0
0.5
x7
x8
x1
2
Figure 5.13: Trajectory x3(t) of the 5-DOF systemunder invariance control
−0.01
−0.005
0
0.005
0.01−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01
−0.5
0
0.5
x9
x10
x1
3
Figure 5.14: Trajectory x4(t) of the 5-DOF systemunder invariance control
67
5.4 Discussion
This chapter presented a levitation device actuated by a set of four PMLSMs. Such device controls five of thesix degrees of freedom of a floating platen while assuming that the remaining one is fixed. After introducingthe arrangement of the four PMLSMs, the present chapter showed the derivation of its dynamic model.Then, it described the design of a nonlinear controller that achieves set-point stabilization. Such nonlinearcontroller, based on invariance control, ensures the well-definiteness of the controls by confining the state ofthe system into a safe set experimentally defined. In simulations, the controller performed satisfactorily fora large set of initial conditions and set points. The steady state was reached after several seconds but thecurrent consumption never overpassed 1.5A. Thus, the controller attains reliability at the cost of mediumconsumption and medium speed response.
68
Chapter 6
Conclusions
6.1 Summary
By designing a device based on PMLSMs, this work addressed the problem of high precision positioningvia magnetic levitation. Considering commercial applications for the solution of that problem, the researchlimited the technological candidates to those that are readily available on the market. In Chapter 3, a newPMLSM model that considers variable airgap length was derived because common models are not suited formagnetic levitation. The resulting model served as the foundation for two different designs. The first designattempted to control three DOF while neglecting the remaining ones. Chapter 4 reported the test of threedifferent control approaches applied to the model of that device. Although the controllers performed wellin simulations, the design could not avoid undesired rotations whose dynamics were neglected during themodelling of the system. In order to overcome this drawback, a second apparatus was designed to controlfive DOF, as presented in Chapter 5. The new design performs position control of those five DOF whileassuming the sixth one to be constrained. To ensure that the controls were well defined, the design employedthe invariance control method in order to constrain the system’s trajectories within a suitable set. Accordingto simulations, the system achieved the main goal in a feasible manner, i.e., it produced well-defined controlinputs within physically reasonable limits.
6.2 Contribution of this work
This work introduced a full model that describes longitudinal as well as normal dynamics of slotted iron-cored PMLSMs. Inclusion of the slot effects on the distribution of the airgap’s magnetic field enhanced thatmodel. Up to that point, modelling put together the results from Nasar et al and Zhu et al, but it alsoextended them by incorporating a variable airgap.
In addition to generalizing the model by Nasar et al and combining it with the work by Zhu et al,this thesis studied disturbances in PMLSMs. Based on a literature investigation, Chapter 3 described andcharacterized those disturbances in a form that suits control theory.
The main genuine contribution of this research is the design and control of two levitation mechanismsthat considered real applications requirements. The control method proposed by Mareczek, Buss, and Spongwas applied to both mechanisms and a basic nonlinear controller was devised for the 3-DOF machine.
6.3 Future research
The achievements presented in this thesis are far from becoming ready-to-use products. The controllers heredesigned do not take into account either disturbances or parameter variations; thus, future control designsshould be able to cope with such perturbations. The invariance control method does neither attenuatedisturbances nor compensate for parameter variations, raising a problem which could be addressed by either
69
of two approaches. The first one entails extending that method by proposing either an adaptive or a robustversion. The second one is the replacement of the control approach.
The control designs developed in this thesis assumed that the full state is measured. For practical reasons,part of the state should be estimated and observers should be designed. Moreover, current control must beperformed since current power supplies are more expensive than voltage ones.
Implementation provides another direction of expansion for this work. Although simulations proved theeffectiveness of the control designs, the new electromagnetic model has not been fully verified. Even thoughthis new model very closely followed already verified models, minor errors might lead to ineffective controllaws.
6.4 Importance
This thesis presented two designs that are not ready for industrial applications mainly because they do notconsider disturbances. However, the purpose of this work was not to develop a final product, but to proveits feasibility through the design of a basic prototype. Not only the control design succeeded in simulations,but low energy consumption and the use of conventional motors also supported this feasibility proof.
In conclusion, this work does not offer a definitive solution to the problem it addressed but it providesthe foundations of a relatively inexpensive and efficient technology that eventually might solve it. Futurework can rely on the models and the control methods here presented in order to complete a design endowedwith all the characteristics that industry requires, such as robustness, improved speed, and reasonable cost.If provided with such features, this design will lead to the implementation of equipment that will reduce thegreat costs caused by waste in the semiconductors industry. Moreover, it will promote the manufacturingof nanodevices. Ultimately, these consequences will translate into benefits for both manufacturers andconsumers.
70
Appendix A
Symbols description
B Magnetic flux density
Br Remanent flux density
b0 Slot width
c1 Tooth width
CL Minimum space between motors
Fx Thrust
Fy Normal Force
F Magnetomotive force
G Gravity constant
g Length of the air gap
ge Effective length of the air gap
H Magnetic field intensity
hb Height of the back iron on the movers
hm Permanent magnets height
ht Height of the tray
Ia armature current
ia Phase a current
ib Phase b current
ic Phase c current
id Direct current component
iq Quadrature current component
Kc Carter’s coefficient
kdn Distribution factor
kpn Pitch factor.
kwn Winding factor
LA Length of the poles
M Magnetization vector
M Mass of the platen
Mb Mass of the back iron on every mover
Mpm Mass of every permanent magnet
Mt Mass of the tray
p Number of pole pairs on the stators
pm Number of permanent magnets
t1 Slot pitch
W Number of turns of wire per phase
wc Coil pitch
δi Density of iron
δpm Density of the permanent magnets
λ Relative permeance
µ0 Permeability of the free space
µr Relative permeability
µrec Relative recoil permeability of the permanentmagnets
Ψ Magnetic field potential
ρm Magnetic volume charge density.
σm Superficial magnetic charge
τ Permanent magnets pole pitch
θ Relative position angle of the mover
71
Appendix B
Acronyms
AC Alternating current
DC Direct current
DOF Degree(s) of freedom
DTC Direct torque control
FEM Finite element methods
FFT Fast Fourier transform
LIM Linear induction motor
LP Linear programming
LQR Linear quadratic regulator
LSM Linear synchronous motor
LSRM Linear switched reluctance motor
LTV Linear time varying
LVRM Linear variable reluctance motor
MIT Massachusetts Institute of Technology
mmf Magnetomotive force
ODE Ordinary differential equation
PDE Partial differential equation
PI Proportional-integral
PID Proportional-integral-derivative
PM Permanent magnet
PMLSM Permanent magnet linear synchronousmotor
SISO Single input single output
TFLM Transverse flux linear motor
72
Appendix C
5-DOF setup details
This appendix gathers the details of the hardware considered by the modelling and the control designs inthis thesis. It first specifies the requirements of the system in terms of length of travel and accuracy. Next,it provides the details of the commercial PMLSMs whose parameters were provided by the manufacturers.The appendix concludes by presenting the mechanical arrangement of the motors in both forcer and platen.
C.1 Specifications
The maximum expected travel along the x and z directions is ±50mm, and ±5mm in the airgap. Anaccuracy higher than 0.1mm is desired. Power dissipation is not considered because the design did not takeinto account current control so that the power requirements of the controllers are not known.
Because the applications targeted by this work do not involve the operation of the device under heavyloads, the design did not significantly limit the weight of the levitated platen.
C.2 Motors characteristics
The motors under consideration are iron-cored flat single-sided PMLSM with three-phase single layer wind-ings. Figure C.1 provides the dimensions of both stator and mover as viewed from above.
C.2.1 Stator characteristics
The stator is longitudinally laminated and transversally slotted to house single layer three-phase windings.Figure C.2 depicts the distribution of the single layer windings with three pole-pairs that the modellingconsidered.
Parameter Symbol units value
Slot width b0 mm 12.7Slot pitch t1 mm 19.05Turns per phase W 900Coil pitch wc mm 57.15Pole pairs p 3Number of slots z1 18
Table C.1: Stator characteristics
73
200.0
N
S
N
S
a)
b)
slottooth
toothslot
50.0
19.05
Permanen magnet
28.57
28.58
200.0
387.35
Figure C.1: 5-DOF device dimensions in mm of: a) mover b) stator
74
c a’ b’ba c’
1 2 3 4 5 6 7 8 9 1110 12 13 14 15 16 17 18Slot number:
Figure C.2: Single layer, three pairs of poles windings distribution
C.2.2 Mover characteristics
In the motors considered here, a laminated ferromagnetic mover houses non-skewed permanent magnets.The design took into account NdFeB grade N27 magnets whose properties are listed in Table C.2.
Parameter Symbol units value
PM height hm mm 2PM length LA mm 50Number of PMs pm 4Pole pitch τ mm 57.15Pole arc (PM width) τp mm 28.58PM Coercitivity Hc kA/m 836,0500Back-iron height (thickness) hb mm 4.7Back-iron width LA mm 50Back-iron length mm 200.0
Table C.2: Mover characteristics
C.3 Devices arrangement
In the device here presented, the forcer holds four stators and the platen houses four movers. Figure C.3shows the arrangement of the four stators. Figure C.4 depicts the movers mounted on the platen. Noticethat, while the stator provides three pairs of poles, the mover generates only two pairs. The missing pole inthe mover accounts for the spare space along which the motor moves.
75
1200.0
200.0 1200.0
387.35 200.0100.0
Figure C.3: Upper view of the forcer of the four motors setup. All dimensions measured in mm
76
50.0
50.0
200.0
200.0
987.34
268.67
987.34
Figure C.4: Upper view of the platen of the four motors setup. All dimensions given in mm
77
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