NONLINEAR CONTROL DESIGN FOR A MAGNETIC LEVITATION SYSTEM

Transcription

1 NONLINEAR CONTROL DESIGN FOR A MAGNETIC LEVITATION SYSTEM by Rafael Becerril Arreola A thesis submitted in conformity with the requirements of the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto c Copyright by Rafael Becerril Arreola 3

2 Nonlinear control design for a magnetic levitation system Master of Applied Science 3 Rafael Becerril Arreola Graduate Department of Electrical and Computer Engineering University of Toronto ABSTRACT This thesis presents the design of two magnetically levitated high precision positioning systems based on permanent magnet linear synchronous motors (PMLSMs). Magnetic levitation based on PMLSMs requires the motors to operate with a variable airgap length but previous research in the field has studied mainly motors fastened by bearings; therefore, this work first models a motor with free normal dynamics. A 3-DOF levitation device is then designed with basis on the resulting model of the motor. The nature of that system conveys the need for restricting its state in order to guarantee well-definiteness of the controls. Simulations then 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 complex machine is introduced in order to regulate five DOF. An invariance controller completes the design of this 5-DOF levitation device. II

3 Acknowledgments Almost two years of hard work produced the results presented in this thesis. However, hard work would not have 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 his unconditional availability and for having enjoyed this research as much as I did. Por otro lado, dos años de trabajo arduo no hubieran servido de nada si mis padres no me hubiesen propocionado las bases que me permitieron llevar a cabo mis estudios de maestria. Agradezco a mis padres dos cosas mas: que me hayan mostrado el verdadero valor de la educación; y que me hayan enseñado a disfrutar el trabajo y a verlo como una bendición. En fin, les agradezco porque me hicieron el feliz hombre de provecho que ahora soy. III

4 Contents Contents V 1 Magnetic Levitation and Linear Motors Motivation Review on linear motors Topologies of linear machines Principles of operation of linear motors Purpose and outline of the thesis Literature review 7.1 Previous work on magnetic levitation The vehicles levitated by iron-cored PMLSMs at Kyushu University The MIT levitated stage based on air-cored PMLSMs Solutions not based on PMLSMs PMLSMs modelling and control General literature on PMLSMs The algebraic models developed by Nasar et al Modelling of the slots in PM motors by Zhu, Howe et al The numerical analysis of PMLSMs at Hanyang University Other numerical approaches to the analysis of PMLSMs Study of cogging Forces Control of PMLSMs with constant airgap lengths Constrained state control Polyhedral Lyapunov Functions Invariance control Another constrained control approach Results of the literature review The model of a PMLSM Modelling outline Magnetic field produced by the permanent magnets Winding s magnetic field Slots effect Forces calculation Thrust Normal force Single motor dynamics Disturbances Discussion of modelling results IV

5 4 Design for three controlled degrees of freedom Setup description and modelling Linear control Simulation results Partial feedback linearization and composed control Simulation Results Full feedback linearization with invariance control design Simulation Results Controllers performance comparison Discussion Design for five controlled degrees of freedom Setup description and modelling Full feedback linearization and invariance control Feedback linearization Design of the invariant set Passification and design of switched gains Simulation results Discussion Conclusions Summary Contribution of this work Future research Importance Appendices 71 A Symbols description 71 B Acronyms 7 C 5-DOF setup details 73 C.1 Specifications C. Motors characteristics C..1 Stator characteristics C.. Mover characteristics C.3 Devices arrangement Bibliography 78 V

6 List of Tables 3.1 Parameters of the motor considered for numerical calculations Simulation parameters for the 3-DOF system Estimate of the controllable region for the linearization of the 3-DOF levitation system Motor parameters used for simulation of the 5-DOF device Coefficients of the invariance function for the 5-DOF system Output constants for the passification of the 5-DOF system C.1 Stator characteristics C. Mover characteristics VI

7 List of Figures 1.1 Configurations of flat linear motors: a) double-sided, and b) single-sided Configuration of the PMLSM Types of slots: a) Open slots; b) semiclosed slots Frame-set considered in the analysis of a PMSLM Charge distribution on a permanent magnet Positions of the images of the magnetic charge in an iron-cored PMSLM Effective airgap length and its linear approximation versus actual length Magnetic field density on the stator s surface Magnetic field density along x axis µ H pmy1 as function of g Phase currents in the armature Mover at d = Mover at d Modeling of the slots through the function g(x) Relative permeance and its first harmonic Thrust as a function of the air gap length and the position of the mover Normal force as a function of the air gap length and the mover position Normal force as a function of the mover position Configuration of three LSMs for three controlled degrees of freedom Forces in the 3-LSMs configuration Position of the 3-DOF system s platen under LQR control for x d = [.55,,,,, ] T Speed of the 3-DOF system s platen under LQR control for x d = [.55,,,,, ] T Control inputs u 1 and u in the 3-DOF system under LQR control for x d = [.55,,,,, ] T Control inputs u 3 and u 4 in the 3-DOF system under LQR control for x d = [.55,,,,, ] T Block decomposition of the 3-DOF system Lyapunov and invariant sets for composed control Region for which the control input u 4 of the 3-DOF composed controller is real Estimate of the domain of attraction of the 3-DOF composed controller for first set-point Estimate of the domain of attraction of the 3-DOF composed controller for second set-point Transient position of the 3-DOF system under composed control for a first set of parameters Transient speed of the 3-DOF system under composed control for a first set of parameters Control inputs u 1, u, and u 4 of the 3-DOF system under composed control for a first set of conditions Transient position of the 3-DOF system under composed control for a second set of conditions Transient speed of the 3-DOF system under composed control for a second set of conditions Control inputs u 1, u 3, and u 4 of the 3-DOF system under composed control for a second set of conditions Projection of M and N into the (x 1, z) plane Transient position of the 3-DOF system under invariance control Transient speed of the 3-DOF system under invariance control VII

8 4.1 Control inputs u 1, u, and u 4 for the 3-DOF invariance controller Trajectory of the 3-DOF system under invariance control Transient airgap length of the 3-DOF system for the three controllers Transient x and z positions of the 3-DOF system for the three controllers Control inputs u 1, u of the 3-DOF system for the three controllers Control inputs u 3, u 4 of the 3-DOF system for the three controllers Angular response of a disturbed 4-DOF system Airgap length response of a disturbed 4-DOF system Setup for five degrees of freedom with four PMLSM Disposition of the movers on the platen for the 5-DOF device Fixed and rotating frames of reference for the 5-DOF device Position response of the 5-DOF system under invariance control Speed response of the 5-DOF system under invariance control Transient angles of the 5-DOF system under invariance control Transient angular speed of the 5-DOF system under invariance control Control inputs u 1, u and u 3 for the 5-DOF system under invariance control Control inputs u 4, u 5 and u 6 for the 5-DOF system under invariance control Trajectory x 1 (t) of the 5-DOF system under invariance control Trajectory x 3 (t) of the 5-DOF system under invariance control Trajectory x 4 (t) of the 5-DOF system under invariance control C.1 5-DOF device dimensions in mm of: a) mover b) stator C. Single layer, three pairs of poles windings distribution C.3 Upper view of the forcer of the four motors setup. All dimensions measured in mm C.4 Upper view of the platen of the four motors setup. All dimensions given in mm VIII

9 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, and bioengineering the manufacturing processes of certain components rely heavily on high precision positioning systems. The improvement of these positioning systems is particularly important for two sectors of the manufacturing industry, nanotechnology and semiconductors, both of which require very high accuracy positioning and extremely clean environments. Nanotechnology relates to devices whose dimensions are in nanometers. This sector is at an early stage of development but the number of its applications is growing fast []. Its growth is manifested by the increasing demand for workers in the area [1]. The manufacture of nanodevices employs processes like photolithography, molding, and micromachining [3]. In many cases, these processes require positioning with nanometric 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 the reduction of manufacturing costs and process time. Many of these problems arise during the photolithography of micrometric structures that form the circuitry of the chips. On industry-wide average, this process succeeds for only one in every two chips [6]. For an economic analysis that includes the cost of waste in semiconductor foundry, refer to [7]. Dust contamination is a problem that affects the photolithography of semiconductors and nanodevices because generates waste. This particular kind of contamination occurs when free particles reach the surface where the nanostructures are being created [5]. In order to alleviate this problem, ultra-clean environments normally house the photolithographic procedures. These controlled environments isolate processes inside a 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 free dust 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, releases particles into the environment because of friction. Therefore, real ultra-clean environments require the replacement 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 an actuator. However, manufacturers specification sheets show that these devices provide neither airgap length control nor very high accuracy. Magnetically levitated stages are a better alternative for positioning systems because of their more robust nature. Several researchers, e.g., D.L. Trumper and K. Yoshida, have proven that magnetically levitated 1

10 systems can be equipped with both airgap length control and very high accuracy. Seeking for these two desired features, these researchers have investigated different technologies that include different types of electromagnets and linear motors. While electromagnets can apparently be simpler, linear motors provide both higher efficiency, because of a shorter and more uniform airgap length, and lager travel, as Chapter shows. Weighing those advantages, this study focuses on the design of a magnetic levitation system that uses linear motors. Linear motors are already used for positioning in semiconductor photolithography [48]. However, despite the 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 implementation. These implications are exemplified by the solution proposed in [14], where the authors describe a positioning system that is based on non-commercially available motors. Trying to overcome the implementation obstacles, the present study suggests a levitation machine based on conventional technologies. Since many types of conventional linear motors are available, the next section compares the most common ones and introduces their fundamentals in order to ensure the selection of the machine that best suits levitation. 1. Review on linear motors The selection of the best linear motor requires understanding both their properties and their operating principles. Accordingly, the next two subsections present the common motor topologies and their underlying theory of operation Topologies of linear machines Even though linear motors produce straight motion, they operate very similarly to rotary ones. In fact, there is at least one kind of linear motor for every kind of rotary one. Like rotary machines, linear ones consist of a moving and a stationary part, the mover and the stator respectively. Either the mover or the stator becomes the armature by generating a magnetic field that travels linearly. The remaining part of the motor is 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, travel length, stability, structural strength, and efficiency. The next paragraphs explain these categories and present their advantages. Whether a motor is flat or tubular mainly affects the complexity of its modelling and its freedom of travel. Rather than lying flat, in tubular motors, the stator (or the mover) rolls around the mover (or the stator) forming a tube. Thus, the modelling of flat motors is harder than that of tubular ones because the fields at the ends of a flat motor are not uniform. On the other hand, flat machines allow lateral and normal 1 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 the other 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). 1 A normal displacement occurs when the airgap length varies.

11 Stator Mover Stator (a) Mover Stator (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 to air-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 or on the surface of the core. When compared to slotless motors, slotted ones lead to more complex models because the slots violate uniformity assumptions; however, slots supply stronger structures. When either longitudinal or transversal ferromagnetic laminations form the core, they determine the direction of the main magnetic flux. Transversal laminations imply smaller magnetizing currents but longitudinal laminations improve efficiency by reducing both eddy current losses and the magnitudes of higher harmonics of the flux [38]. Apart from the previous topological categorization of linear motors, their winding distribution offers a different 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 sinusoidal distribution of the traveling magnetic field (see [69]), which significantly simplifies modelling, as seen in Chapter 3. From the observations above, flat single-sided configurations favour magnetic levitation because they allow larger travel in more directions. Commercial flat single-sided motors are commonly available in aircored and longitudinally laminated iron-cored configurations. 1.. Principles of operation of linear motors The above topological classification of linear motors provides the concepts required to understand their operating principles, which are next explained. 3

12 To begin with, the forces in a linear motor are either electrodynamic or electromagnetic. Within the first class, also known as Lorentz type forces, are those caused by immersing current-carrying conductors in magnetic fields as well as those caused by the interaction between a magnetic field and its reaction field generated by induced currents. On the other hand, electromagnetic forces are caused by the interaction between 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) collinear to the travel, and a normal force perpendicular to the thrust. While the thrust is caused by the travel of the armature s magnetic field, the normal force depends on the structure and composition of the motor. In general, 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 bearings, electromagnetically controlled suspensions, or air bearings. As mentioned before, the occurrence of these forces depends on the motor s structure, which determines its principle of operation. This principle of operation defines several categories of motors, i.e., induction, synchronous, and direct current. Since direct current 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 (the secondary) 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 is either a metallic cage or a short-circuited three-phase winding. In less usual configurations, the secondary is: 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 reactive magnetic field. The interaction of these two fields produces a motion that can be controlled by regulating the phase of the primary s currents [37]. Conventional systems based on LIMs regulate the armature currents to control thrust, speed, or position and use bearings to restrict motion into the longitudinal single degree of freedom. Nevertheless, recent research showed that LIMs can simultaneously provide propulsion and suspension. As shown in [81], suspension results from the control of the intensity of the airgap flux, which determines the magnitude of the normal force. However, the intensity of this flux also determines the magnitude of the thrust; thus, independent control of thrust and normal force requires combined power supplies that feed the primary with two different frequency components simultaneously. While a low frequency component provides both thrust and normal force, a high frequency one affects only the normal dynamics. Besides requiring complex drivers in order to achieve levitation, LIMs suffer from strong cogging forces, which are undesired forces that produce oscillations and are not considered by conventional models refer to Section..6. In addition to high complexity and 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 normal force. For these reasons, commercial LIMs are not suitable for high precision positioning stages. 4

13 STATOR Laminated ferromagnetic material Slots pm pm pm pm Back Iron Windings MOVER Figure 1.: Configuration of the PMLSM Linear Synchronous Motors The motion of a LSM is in synchrony with a travelling magnetic field produced by a polyphase armature that is excited by either AC or switched currents. The field can be a variable reluctance structure or an arrangement of permanent magnets. Three common types of LSMs result from the combination of the above kinds of excitation with these choices of fields, namely, Linear Variable Reluctance, Linear Switched Reluctance, and Permanent Magnet Linear Synchronous Motors. In a Linear Variable Reluctance Motor (LVRM), the secondary is a variable reluctance structure either notched or segmented whose pitch equals the primary s coil one. This structure s tendency to align with the travelling field generates movement. Since the position of the travelling field depends on the phase of the armature currents, the phase angle of these alternating currents determines the position of the mover. For a more detailed description of LVRMs, see [3] and [38]. Similar to LVRMs in topology and operation, Linear Switched Reluctance Motors (LSRM) differ in that only one phase is energized at a given instant. Either linear position sensors or estimators trigger the excitation of each phase. Since excitation is switched, the LVRM s thrust pulsates during the overlapping of two phases and produces noise and vibration. These effects affect especially the stability of the normal dynamics because the normal force is much stronger than the thrust. Moreover, these motors provide limited longitudinal resolution because of their switched nature. The third common type of LSMs is the Permanent Magnet Linear Synchronous Motor (PMLSM), in which an array of permanent magnets composes the field. In this case, thrust results when the field of the magnets tends to align itself with the travelling field of the armature. If, besides thrust, an attractive normal force is desired, the armature incorporates ferromagnetic material in its core. The attraction between the core and the magnets produces a normal force that is strong enough for levitation. If the design capitalizes on this attractive force, smaller armature currents are required. In addition to this advantage, the quasisinusoidal field distribution of the magnets arrangement facilitates a smooth travel. Therefore, both the strong normal force and the capability of smooth travel render PMLSMs the best choice for high precision magnetic 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 research goal can be narrowed down by choosing the best of those paths. Accordingly, this work addresses the design problem of a magnetically levitated high precision positioning system actuated by a set of permanent magnet linear synchronous motors. This research seeks a complete solution that includes both modelling and the design of controllers. Chapter introduces previous research on magnetic levitation for high accuracy positioning tasks. This research includes solutions based on both linear motors and other technologies. Next, the chapter studies PMLSM models obtained through different methodologies while evaluating their suitability for levitation control. Finally, the investigation compares different control methods in terms of their practical advantages. 5

14 The innovative development starts in Chapter 3, which describes the derivation of a new PMLSM model that relies heavily on results from the literature review. The development first derives the magnetic field produced by the permanent magnets and then finds the components of the travelling magnetic field in the armature. Next, the model incorporates the effect of the slots on the armature s core by calculating the relative permeance of the ferromagnetic body. After using the previous results to analytically find the forces in the motor, the procedure yields a full model of the device. A brief discussion on disturbances modelling closes the chapter. The work in Chapter 4 uses the results in Chapter 3 to derive the model of a three degrees of freedom levitation system actuated by three PMLSMs. After completing the modelling, the chapter presents the design of three alternative controllers. The first one is a simple LQR controller and the second one is obtained using methods from nonlinear control. The third controller is based on full feedback linearization and 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 of only three. Only one controller is considered for this more complex system. This new controller relies on full 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. 6

15 Chapter Literature review As seen in Chapter 1, magnetic levitation is not a new problem in engineering. Several researchers have proposed magnetic levitation approaches to high precision positioning tasks, but their experimental solutions have not evolved yet into popular appliances because of cost and implementation factors. A clear example is the successful work conducted by Trumper et al. in [11]-[16], where these authors introduce a solution based on custom technologies whose production would imply special manufacturing procedures. Nevertheless, the study 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 the already studied technological options. This chapter gathers these antecedents in three main sections. The first section briefly describes and comments on prior approaches to the high precision magnetic levitation problem. The second one presents research focused on the development of models and controllers for synchronous machines. The third and last part introduces the background required for the control design proposed in this thesis. Since this review embraces different areas of knowledge, the references are organized according to the field they belong to and, more specifically, by research group..1 Previous work on magnetic levitation This section introduces three independent lines of work on high precision magnetic levitation. The first one, developed by Yoshida et al., uses PMLSMs for transportation purposes. The second one, conceived by Trumper et al., is a 6-DOF stage powered by ironless PMLSM. The third and last line of work deals with a group of similar devices based on electromagnets..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 feasibility of attractive magnetic levitation with PMLSMs. Targeting transport applications, these authors designed a vehicle levitated by motors of this kind. As described in [8], they designed a system based on iron-cored PMLSMs with hybrid PMs, i.e., permanent magnets whose magnetic field is controlled by coils wound around them. This system compensated for cogging forces (refer to Section 1..) by controlling the currents in the coils of the hybrid magnets. Since the device used a set of rubber rollers to maintain the airgap constant, 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 a nonlinear expression for the thrust taking into account the airgap length variation. Their work, as presented in [8], concluded by comparing -D FEM analysis and experimental results. Later, Yoshida repeated the experiment without rubber rollers and reported the results in [9]. In this new experiment, a larger airgap compensated for the lack of bearings. Also, a 3-D FEM approach improved 7

16 the accuracy of the analysis by accounting for the distortion of the fields at the borders of the motor. Experiments showed that the 3-D FEM analysis was more precise than the -D one because, when both methods results were compared, 3-D FEM analysis estimated smaller forces that better matched the experimental measurements. While the research in [8] and [9] studied attractive levitation, the work in [1] focused on repulsive suspension. Because this new problem has a different nature, air-cored PMLSMs replaced the iron-cored motors. In order to control both propulsion and lift, the authors proposed a modified version of direct torque control, a technique commonly used to control rotary motors. This modified DTC divided the dq plane 1 into 1 regions instead of the 6 used in conventional DTC. A look-up table stored different constant values that determined the voltages applied to the motor s windings on each region. This switched control approach showed satisfactory performance in the face of a noisy speed output..1. 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. at the Massachusetts Institute of Technology. This paper reviews the fundamentals of Halbach magnet arrays while mentioning some of their applications. The first applications are magnetic bearings for photolithography stages. As a second application, the paper introduces linear motors along with a PMLSM model obtained through a direct solution of the magnetic fields. The last application is hybrid electromagnets for Maglev trains. This work remarks the higher efficiency of the Halbach arrays when compared to other existing magnet arrays. The subsequent work, presented in [1], used the results in [11] to obtain a model of single sided air-cored PMLSMs. To this end, the procedure in [1] expanded the analysis of the magnet array while incorporating the study of the magnetic field of the armature. The authors employed the resulting motor s model to control a six-degree-of-freedom magnetically levitated xy stage driven by four custom motors. According to the authors, implementation of the stage produced good results with accuracy of the order of hundreds of nanometers. As presented in [13], the authors designed a levitated stage that overcame the travel limitations of the device presented in [1]. The new device permitted a range of motion of ±5mm ±5mm on the horizontal plane, ±µm on the airgap, and ±6µrad on each angle. Lead-lag compensators controlled the new levitated stage producing the results reported in [14]. Accordingly, a linearization replaced the original model decoupling the dynamics of the six degrees of freedom. The authors reported in [15] the details of the modelling and the lead-lag control design of this levitation machine. As [14] shows, the system s response suffered from overshoots as large as µm when the device was tested with reference step input as small as nm. Trying to overcome this undesired effect, Kim and Trumper designed an LQR to optimize the performance 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 linear controllers thus limiting the range of operation and the robustness of the control system. The development in this thesis seeks to overcome these two drawbacks..1.3 Solutions not based on PMLSMs Magnetic levitation not based on PMLSMs is another prolific subject of research. The next paragraphs show how several researchers have proposed different levitation systems that significantly differ in their theoretical foundations. In [17] Menq et al. explain the design of a device that levitated a platen while controlling its six DOF. This device consisted of 1 electromagnets distributed on the walls of a cage that housed the platen. The authors proposed a robust feedback linearizing controller that provided the device with both a controlled range of motion of ±4mm ±4mm on the horizontal plane, ±mm on the vertical direction, and ±1 o on each angle, and an accuracy of the order of nanometers and microradians. 1 The dq plane is used to represent the direct and quadrature components of currents and voltages through the windings of electric machines. For more details on dq decomposition refer to [3]. 8

17 At the National Taiwan University, research groups from three different departments worked on two similar levitation projects. They published their -DOF and 5-DOF designs in [18] and [], respectively. Each system consisted of a levitated platen with salient PMs and a stationary platform with slots inside which the platen magnets were driven. The PMs fields interacted with those of hybrid magnets mounted around the platform slots; thus, by controlling the currents on the hybrid magnets, both systems regulated the forces exerted on the platens. Both designs included adaptive controllers that achieved micrometric resolution (4µm 6µm 4µrad 4µrad 6µrad for the 5-DOF one) within short travelling ranges limited by the structure of the fixed platform. The work in [19] improved the one presented in [18] by increasing the controlled degrees of freedom up to six. In the new setup, an adaptive sliding mode controller drove the platen within a range of motion of ±4mm 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 the achievable ranges of motion are small in comparison to those of PMLSMs. Moreover, although these papers do not evaluate either the efficiency or the energy consumption of the electromagnets, PMLSMs demonstrate higher efficiency by guiding the flux lines through shorter and well-defined paths. Therefore, PMLSMs suit magnetic levitation better than electromagnets.. PMLSMs modelling and control As Section.1 made evident, PMLSMs suit levitated positioning applications. This section concisely introduces several studies concerning this family of devices. After commenting on some references on PMLSMs fundamentals, the next paragraphs present the analysis of these motors as conducted by several research groups. The different models conceived by Nasar et al. head the survey, followed by the contribution by Zhu, Howe et al. Next, an outline of the work at Hanyang University serves as a preamble to other numerical approaches. 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...1 General literature on PMLSMs In [1], McLean overviews the applications of linear motors and offers a classification of their analysis methods. While naming their advantages, the paper presents different topologies and their use in industrial applications and transportation systems. The subsequent analysis methods overview mainly focuses on linear induction 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 [1]. By including technical and theoretical aspects in [3], Gieras and Piech provide a more complete overview of PMLSMs. Among other topics, the work in [3] studies PMLSMs constituting materials, electromagnetic parameters, and thrust. In order to obtain an expression for the thrust, the authors derive an expression for the magnetomotive force of the windings. Next, they compute the force through two different methods: power balance and Lorentz equation. Although [3] does not provide a model of the normal force, it provides valuable information on control methods, sensing, and applications. This book also presents case studies and shorter examples along with the theory... The algebraic models developed by Nasar et al. Nasar, in collaboration with other researchers, analyzed the forces in iron-cored PMLSMs through different analytical methods. This section presents the results generated by two of these methods. 9

18 By using current sheets and Maxwell equations, Nasar et al. obtained a smooth approximation to the field produced by the PMs of an iron-cored PMLSM. As these authors showed in [4], an analytical expression of the 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 affected because the model does not completely reflect slot effects. The work in [41] used the results from [4] to derive formulas for the normal force and the thrust. By adapting theory from rotary motors, the authors algebraically obtained expressions for the electric parameters of the motor. These expressions and power balance identities yielded formulas for the forces. Theoretical results in the paper closely match experimental ones. The book [39] gathers the methods developed in [4] and [41]. After obtaining a linear dynamic model based on the formulas derived in [41], the authors briefly present both rectangular and sinusoidal current controls for PMLSMs. Since their method expresses the forces as infinite sums, the authors truncate them to obtain approximations that do not consider the nonlinearities of the motor s dynamics. Following a different approach, in [4] and [43], Nasar and Xiong developed a partly-numerical model for PMLSMs. In [43], the authors computed the magnetic field produced by PMs mounted on the surface of a disk machine. Two main ideas make this work original. The first one is the use of the concept of magnetic charge to find a system of differential equations that characterizes the PMs magnetic field. The second idea is the solution to that system through the method of images. Extending the method in [43],[4] 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 scalar potential equation. The combination of the fields from the PMs and the stator results in closed form expressions for the thrust and the normal force. Finally, the authors compare the theoretical results to those obtained through FEM analysis and experiments. This model does not completely include the effect of the stator slots but permits its further consideration because it relies on the computation of the airgap flux density, which is directly affected by the slots...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 brushless permanent magnets DC rotary motors. Some of their results are useful to the development of the PMLSM model later presented in Chapter 3. In [44], the authors calculate the PMs magnetic field in motors with either internal or external rotor. They outline 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 density distribution both at the airgap and at the stator surface. According to the authors, the analytical results are consistent with those produced by FEM analysis simulations. In the second article, [45], the authors compute the armature reaction field through a two dimensional procedure. 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 density as an infinite sum. By introducing the winding distribution factor, [45] extends these results to three-phase winding 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 to account for the slot effect in the magnetic field distribution of a PM DC motor. To this end, the authors derive a two one-dimensional and one two-dimensional model that describe the relative permeance of the slotted stator. Finally, the authors incorporate the slot effect into the motor s model by multiplying the airgap s field density by the relative permeance. The concepts developed in this paper apply not only to permanent 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 unknown load. The procedure in [47] first finds the components of the open-circuit and armature reaction magnetic fields. Next, it calculates the relative permeance as a function of the instantaneous position of the rotor and the phase of the currents. A new expression of the field density distribution incorporates this permeance 1

19 function to yield a final expression for the field in the airgap. Experimentally obtained plots indicate agreement with theoretical values...4 The numerical analysis of PMLSMs at Hanyang University Jung, Hyun, et al. published three articles related to this thesis. The first one, [4], evaluates the efficiency of both short primary and short secondary PMLSMs in terms of the relationship between cogging forces and thrust. To this end, a -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 magnets is better than skewing them because the first method does not reduce the thrust as much as the latter one does. The authors also conclude that, since force ripple is smaller in short secondary than in short primary motors, the first type of devices suffers weaker end effects. In [5], the authors improve the results presented in [4] by replacing the original -D FEM procedure with a numerical 3-D analysis based on the equivalent magnetic circuit network. The results from the numerical analysis determine the optimal magnet length for both skewed and non-skewed secondaries. In addition, [5] provides a brief description of the new 3-D numerical method while remarking its computational advantages over the equivalent 3-D FEM analysis. By applying the same numerical method, [6] extends the work in [5] to skewed PMs motors. In [6], the authors explore how the motor s parameters affect the static thrust, normal, and detent forces. These parameters 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 the machine. 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 not contribute to the development of analytical models. Nevertheless, the authors conclusions are useful to the selection of the best motor topology and dimensions...5 Other numerical approaches to the analysis of PMLSMs In [7], Mizuno and Yamada explain how they used magnetic circuit analysis to determine the effects of PM size variations on the detent force of an iron-cored PMLSM. Through a -D FEM procedure, the authors obtained the forces in terms of infinite sums that are later numerically evaluated. A numerical analysis yielded an optimal relationship between the magnet width and the pole pitch. Experimental results verified those obtained through numerical analysis. Like the researchers at Nanyang, Mizuno and Yamada did not derive a mathematical model of the motor. Rather than magnetic circuit analysis, Shiying et al. used an FFT-based algorithm in order to model the forces in iron-cored PMLSMs. The algorithm first finds the equivalent current sheets in the motor and then solves Maxwell s equations for the magnetic vector potential, which is given as a Fourier series. The solution to this equation then yields an expression for the magnetic field density. Finally, Maxwell s stress tensor generates expressions for the forces in the motor. In [9], the authors apply their method to both short 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 [8], these authors explain how this method finds the magnetic field density on the airgap of a short secondary ironcored PMSLSM. From the field density in the airgap, they compute the forces in the motor and compare them to those experimentally measured. According to [8], the method of equivalent magnetizing currents algebraically considers the slots effects; thus, the resulting model includes detent forces. Similarly to FEM analysis, 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 of PMLSMs; nevertheless, it provides valuable design guidelines. 11