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DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/DateDuepSS-p. 15 ADAPTIV‘ I.‘ ADAPTIVE COMPENSATION OF SENSOR RUNOUT AND MASS UNBALANCE IN ACTIVE MAGNETIC BEARINGS By J oga Dharma Setz'awan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 2001 ADAPT] Active ma :1 SM high- [it-r W355 and f“, I e . . b“ ”Time ("it mQ‘le's ha"?! hm SRO; h?“ l)?” Din: r . . nit dmm} ABSTRACT ADAPTIVE COMPENSATION OF SENSOR RUNOUT AND MASS UNBALANCE IN ACTIVE MAGNETIC BEARINGS By J oga Dharma Setz'awan Active magnetic bearings (AMBs) have increasingly become the choice for high- speed, high-performance rotating machinery because they provide the scope for con- tactless and frictionless operation. Since magnetic bearings are open-100p unstable, they require careful control system design. Although general feedback control tech- niques have been proposed for precise shaft levitation, the problem of sensor runout (SRO) has been largely overlooked due to its similarities with unbalance in creating periodic disturbances. Ehrthermore, the important problem of synchronous SRO and unbalance compensation has not been adequately investigated. To improve the accuracy of magnetically levitated rotors, we pr0pose for the first time an adaptive control framework that can compensate SRO and unbalance, both individually and simultaneously, while providing shaft stabilization about the geo- metric center. In our approach, bias currents in the magnetic coils are periodically perturbed to create persistency of excitation that guarantees individual identification of the harmonic components of the synchronous disturbances. Through feed-forward cancellation of tin tides geometric C! values. While Ly; astiid IhHth‘Iii'n establish the sim; applies to both 5 mexpfl'lt’d [U l: cancellation of the disturbances and careful control system design, the algorithm pro- vides geometric center stabilization that is robust to uncertainty in plant parameters values. While Lyapunov stability theory and its derived passivity formalism provide a solid theoretical framework for the algorithm, corroborating experimental results establish the simplicity of the design and implementation procedure. The algorithm applies to both 8180 and MIMO systems involving a rigid rotor and future studies are expected to broaden its applicability to flexible rotor models. Copyright © by J oga Dharma Setiawan 2001 To my parents lthanlt all ti their cnntri': ninmnlete. In ; llnlherjt-e. it: amt pleats}: Clark J. ml (/2 law. for his nonlinear ('(IIL' magnetism at; in- data l’nln‘ersity of ' salable rm»; mum in the l llldllll ll , rlilting am; ”all Hi)” typo: lW‘mltl lllu r“ .- Mi and low ACKNOWLEDGMENTS I thank all those who contributed to the completion of this research. Without each of their contributions, this work would have been more difficult, if not impossible, to complete. In particular, the author wishes to thank my major advisor, Dr. Ranjan Mukherjee, for his support, guidance, enthusiasm, and patience. It has always been a great pleasure to work with him. I thank the members of the PhD committee: Dr. Clark J. Radcliffe, for sharing his mechatronics experimental experience; Dr. Steve Shaw, for his input on rotor vibration; Dr. Hassan M. Khalil, for making sure the nonlinear control aspects went smoothly; and Dr. Mahanti, for his input on electro- magnetism and its principles. My deepest gratitude goes to Dr. Eric H. Maslen, from the ROMAC Laboratory, University of Virginia, whose expertise in active magnetic bearings has been an in- valuable resource. His kindhearted assistance and encouragement have been prime movers in the completion of this research. I thank Roy Bailiff, a very reliable staff in the department. His prompt assistance during experimental set-up is greatly appreciated. I thank Craig Gunn, who proof- read my typo-ridden first draft, and focused my later editions. I would like to give special thanks to my wife Esti K. Wardhani for her faithfulness, care, and love, and to my parents, who have morally supported me throughout. vi List 01 Tabli List 01 Figur 1 Introduct 1.1 111.1111 1.? Liter.- 1.3 501;» 1.1 Now. 2 Magnetic 2.1 1mm ..2. Sing} 2-3 Stan.- 2-4 301151 ‘25 0111 26 [Illa 2.7 Um Adapliw 3'1 111m 3'2 C110 3.3 PM 3.4 Ru}, 3;] CW 3'6 Sim CA) Fan) .C’A’ (17' — I I I ‘ . ‘w‘ . .4 c}: f) TABLE OF CONTENTS List of Tables List of Figures 1 Introduction 1.1 Motivation ................................. 1.2 Literature Review ............................. 1.3 Scope and Content of the Thesis ..................... 1.4 Nomenclature ............................... 2 Magnetic Bearing Modeling 2.1 Introduction ................................ 2.2 Single Degree-of-Freedom Model ..................... 2.3 Standard PD Controller ......................... 2.4 Sensor Runout (SRO) Disturbance ................... 2.5 Off-line SRO Identification ........................ 2.6 Unbalance Verification Using a Trial Mass ............... 2.7 Definition of Most Commonly Used Terms ............... 3 Adaptive Sensor Runout Compensation (ASRC) 3.1 Introduction ................................ 3.2 Choice of Adaptive Controller ...................... 3.3 Proof of Stability and Parameter Convergence ............. 3.4 Robustness to Parameter Uncertainties ................. 3.5 Convergence Rate Analysis Using Averaging .............. 3.6 Simulation Results ............................ 3.6.1 Known Plant Parameters Case .................. 3.6.2 Uncertain Plant Parameters Case ................ 3.7 Experimental Procedure ......................... 3.8 Experimental Results ........................... 3.9 Summary and Remarks .......................... vii xi master-Iii COCO 12 13 13 15 15 18 18 18 20 24 28 31 31 33 37 41 44 1 Simultan Using .\1' 1.1 Intril 1.? CW1 1.3 Sim. 1.1 Praxil 1.5 Sun: 5 Simultan u'a Bias I 5.1 Intru- 52 Add; 5.3 Pm. 5.1 Rub 5.1.1 5.4.3 5.1.3 5-5 Slum 5-6 Sinm 5'7 Expo- ‘38 EXpt- 5-9 inter 5‘10 Sumx 6 EXtenSiOJ 6-1 1min, 6.2 .\11.\1 6-3 Asm 6.3.1 6.3.2 SRU 6.1.1 6.4.2 181:1 811111! 6.4 6.5 5.6 C . ~ 01mins, ( L1 Rf‘gpk Using Multiple Speeds 4.1 Introduction ................................ 4.2 Controller Design ............................. 4.3 Simulation Results ............................ 4.4 Practical Implementation Issues ..................... 4.5 Summary and Remarks .......................... 4 Simultaneous Sensor Runout and Unbalance Compensation (SRUC) 47 47 47 53 54 56 Simultaneous Sensor Runout and Unbalance Compensation (SRUC) via Bias Current Excitation (BCE) 5.1 Introduction ................................ 5.2 Adaptive Control with Bias Current Excitation ............ 5.3 Proof of Stability and Parameter Convergence ............. 5.4 Robustness to Parameter Uncertainties ................. 5.4.1 Mass Uncertainty ......................... 5.4.2 Magnetic Stiffness Uncertainty .................. 5.4.3 Actuator Gain Uncertainty .................... 5.5 Simulation Results ............................ 5.6 Simulation Studies of the Convergence Rate .............. 5.7 Experimental Procedure ......................... 5.8 Experimental Results ........................... 5.9 Interpretation of Experimental Results ................. 5.10 Summary and Remarks .......................... Extension to MIMO Systems 6.1 Introduction ................................ 6.2 MIMO Model of Magnetic Bearing with Rigid Rotor ......... 6.3 ASRC for 2—DOF systems ........................ 6.3.1 Controller Design ......................... 6.3.2 Simulation Results ........................ 6.4 SRUC-BCE for 2—DOF Systems ..................... 6.4.1 Controller Design ......................... 6.4.2 Simulation Results ........................ 6.5 ASRC-BCE for 2-DOF systems ..................... 6.6 Summary and Remarks .......................... Conclusions 7. 1 Research Summary ............................ viii 57 57 58 62 66 66 69 71 72 78 81 83 88 92 94 94 94 100 100 103 103 103 107 108 111 114 114 7.? fun: Appendicesl A. Experzt .11 .‘ 12> 1.3 1 M 1 .17) 'il .16 i A} 1 .18 . B. Anabs. Bl .9 8.2 i 8.3.5 C. Paran... Cl ( C? ( D- Passix': 7.2 Future Work ................................ 116 Appendices 118 A. Experimental Set—up ............................ 118 A.1 Magnetic Bearing Set-Up ...................... 118 A2 Structural-Dynamic Analysis of Two-Bearing Rotor ........ 118 A.3 Plant Parameters ........................... 123 AA Power Amplifier Data ........................ 126 A5 Analog PD Control Circuit ..................... 127 A.6 Radial Position Sensor ........................ 129 A.7 Digital Signal Processors ....................... 129 A8 Absolute Encoder .......................... 130 B. Analysis of Persistently Exciting Condition ................ 131 B.1 ASRC: Equation 3.14 ........................ 132 B2 SRUC Using Multiple Speeds: Equation 4.25 ........... 132 B.3 SRUC-BCE: Equation 5.41 ..................... 133 C. Parameter Convergence Using Averaging Method ............ 138 0.] Convergence Rate of ASRC ..................... 138 0.2 Convergence Rate of SRUC—BCE .................. 140 D. Passivity ................................... 144 Bibliography 152 ix 3.1 3.2 3.3 4.1 5.1 5.2 5.3 5.4 6.1 A.1 A.2 A.3 A.4 LIST OF TABLES Time constants of the averaged system ................. Parameters for ASRC simulation .................... Numerical values of variables assuming m, c, K ,, K c are over-estimated by 100% in the actual case ........................ Parameters for Simulation ........................ Experimental results with the balance disk located at the rotor midspan: part (a) ............................ Experimental results with the balance disk located at the rotor midspan: part (b) ............................. Experimental results with the balance disk located closer to bearing B: part (a) ................................. Experimental results with the balance disk located closer to bearing B: part (b) ................................. Simulation parameters for 2—DOF magnetic bearing systems ..... Two—bearing rotor data .......................... Free-free undamped natural frequencies of rotor ............ Magnetic bearing parameters for single-DOF Model .......... Servo amplifier specifications ....................... 30 32 33 53 90 91 102 119 121 123 127 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 4.1 4.2 4.3 5.1 5.2 5.3 5.4 LIST OF FIGURES Diagram of a typical active magnetic bearing system ......... Single-DOF model of magnetic bearing with rotor schematic ..... Off-line SRO identification ........................ Unbalance verification using a trial mass ................ Block diagram of magnetic bearing system with ASRC ........ Block diagram of closed-loop system in the presence of uncertain plant parameters ................................. Estimated parameters of sensor runout ................. Stabilization of rotor geometric center using ASRC .......... Transients of currents of ASRC ..................... Averaged approximation of ASRC .................... Time constants of estimated parameters in ASRC ........... Sensor runout estimation in uncertain plant .............. Stabilization of rotor geometric center using ASRC in uncertain plant Transients of currents of ASRC in uncertain plant ........... Averaged approximation of ASRC in uncertain plant ......... Time constants of estimated parameters in uncertain plant ...... Trajectory of estimated rotor geometric center and regenerated sensor signals ................................... Trajectory of estimated Fourier coefficients of sensor runout ..... Trajectory of control currents ...................... 'Ilajectory of estimated rotor geometric center and regenerated sensor signals ................................... 'Ilajectory of estimated Fourier coefficients of sensor runout ..... 'llajectory of control currents ...................... Block diagram of SRO and unbalance compensation framework . . . . Geometric center :1: and sensor signal with runout 1:, ......... Estimated Fourier coefficients ...................... Block diagram of SRUC via bias current excitation .......... Geometric center a: and sensor signal with runout :c, ......... Estimated Fourier coefficients ...................... Top and bottom bias currents ...................... xi 11 14 15 19 25 34 35 37 38 38 39 39 40 42 43 43 44 45 45 48 5.3 E11»- 513 1311:“ 5.7 E110 5.8 Eiiwl 3.9 Elli-2| 5.10 Effv 5.11 [fie] 5.13 Tin-l 73.131111“ 6.1 PM? 62?. Geo: for? 6.3 High‘ 6.1 GM BCE 5-5 Higl 5-5 Geo BC] 6.7 111-; .11 Ma :12 P“. A.3 Di: M FH- 15 R1, A.6 [3,, .17 3.[ .18 M .511) M MOB, “1.31: A12 8‘ M3 S: “-1 r. 5.5 Effect of mass uncertainty to SRUC-BCE ................ 75 5.6 Effect of magnetic stiffness uncertainty to SRUC-BCE ......... 76 5.7 Effect of actuator gain uncertainty to SRUC-BCE ........... 77 5.8 Effect of excitation amplitude; fe = 0.5 f ................ 79 5.9 Effect of excitation amplitude when fe # 0.5 f ............. 79 5.10 Effect of excitation frequency ...................... 80 5.11 Effect the number of excitation harmonics ............... 80 5.12 Time history of geometric center as and position sensor signal 1:, . . . 85 5.13 Time trace of estimated Fourier coefficients for Expt. 2 in Table 5.1 . 87 6.1 Freebody diagram of a magnetically levitated rigid-rotor ....... 95 6.2 Geometric position, sensor signal, and DC component error of ASRC for 2-DOF system model ......................... 104 6.3 Higher harmonics error of ASRC for 2-DOF system model ...... 105 6.4 Geometric position, sensor signal, and DC component error of SRUC- BCE for 2—DOF system model ...................... 109 6.5 Higher harmonics error of SRUC-BCE for 2-DOF system model . . . 110 6.6 Geometric position, sensor signal and DC component error of ASRC- BCE for 2-DOF system model ...................... 112 6.7 Higher harmonics error of ASRC-BCE for 2-DOF system model . . . 113 A.1 Magnetic bearing rig schematic ..................... 118 A2 Picture of magnetic bearing set-up ................... 119 A.3 Dimension of two-bearing rotor with balanced disk .......... 119 AA Free-free modes shapes of rotor ..................... 120 A.5 Rotor critical speed map ......................... 120 A.6 Bode plot of single-DOF magnetic bearing model ........... 124 A.7 3-D Plot of magnetic force surface .................... 124 A8 Magnetic force vs. Position ....................... 125 A9 Magnetic force vs. Current ........................ 125 A.10 Bode plot of servo amplifier ....................... 126 A.11 Analog PD control circuit ........................ 127 A.12 Bode plot of analog PD controller VW¢(S)/V,~n(s) ........... 128 A.13 Simulink block diagram for the absolute encoder ............ 130 DJ Feedback configuration for passivity analysis .............. 145 xii CHAP Introd 1.1 ML Active mam 19$ «:xperaiii 1! higher 9151101 include the F 3111121111111)‘ 11)] tiaracteristii live featurm. in rotating 8' “591%. mm The (1N: 5:, fl . MAPS InliU'. v" ,r. -1 mm} 3 CHAPTER 1 Introduction 1 .1 Motivation Active magnetic bearings (AMBs) levitate rotors and enable contactless and friction- less operation. They have a number of advantages over conventional bearing including higher efficiency. longer life, and ability to operate at higher rpm. Other advantages include the elimination of mechanical maintenance of the bearing and lubrication, suitability for clean or vacuum room operation, and adjustable stiffness and damping characteristics achieved through active control of bearing forces. Due to the attrac- tive features, magnetic bearings have been implemented in a variety of applications in rotating systems. These applications include flywheel energy storage, momentum wheels, precision machinery, turbomachinery, vacuum pumps, and medical devices. The design of magnetic bearing systems requires the knowledge of several disci- plines including mechanical/ rotor dynamics, electromagnetics, electronics and feed- back controls. This is due to the fact that a rotor supported by magnetic bearings is an open-loop unstable system, stabilized through feedback control. The integration of the feedback control strategy must carefully consider the dynamics of the rotor. Moreover, the feedback control in magnetic bearings necessitates the knowledge of Position sensors with signal conditioning, switching-power amplifier, either analog circuits or digital signal processor (DSP), and magnetic coils as the actuators. Due to 1 abundant re and 3111111101. 111 the recent areas such a; as mulri-rarz. murder to n :eusing Hirer temp su‘er r11 11115 rhes‘ sauce of rm rurrug prom 10 {938:1 the 12 Li Pemuu; dis "11110 the 1 118m}- Jdlltj‘fl mi . Q‘.. a)?“ ‘3 01 1110f $111 111. Lil ~ 1. {011%,};- {1"}er 1'. ‘iidrilff S vaiiiik. ‘ \ {Cr “Jig; ‘1 N V v . .1, ‘. 14’? - « Fil‘i. “111' i- Due to the rapid progress in electronics including DSPs technology, there are abundant research problems in AMBs. Performance improvement, cost reduction, and additional design objectives within specific applications are some of the examples. In the recent years, researchers working on magnetic bearings have been focusing in areas such as (1) rotor vibration minimization using modern control method such as multi-variable controls, robust controls, non-linear controls, and adaptive controls in order to minimize rotor vibration; (2) levitation of flexible rotors;(3) robust self- sensing schemes; (4) integration of magnetic bearings with electric motor; and (5) zero-power magnetic bearings using superconductor materials. This thesis focuses on utilizing modern control methods to improve the perfor- mance of magnetic bearings without demanding additional precision in the manufac- turing process. In particular, this thesis explores a new adaptive control framework to reject the effect of the most common periodic disturbances in AMBs. 1.2 Literature Review Periodic disturbances are common in rotating machinery. Such disturbances are crit- ical to the performance of systems using AMBs. The dominant sources of periodic disturbance in magnetic bearings are mass unbalance and sensor runout (SRO). Mass unbalance results from lack of alignment between the geometric axis and the principal axis of inertia, which results in an unbalance force synchronous with rotor angular Speed. Mass unbalance can be significantly reduced in industrial applications, if n0t completely eliminated by rotor balancing. In comparison, sensor runout is un- avoidable since it results from manufacturing imperfections in the magnetic bearing assembly. Specifically, SRO disturbance originates from a lack of concentricity of the sensing surface and non-uniform electrical or magnetic properties around the sens- ing Surface. Unlike mass unbalance, SRO also generates a disturbance at multiple harruuuits 11': ruuuut distt‘. 2131111123111111 compensatit -: Thuugh t' eemhirred ur.‘ 00111091154111“: peusation h.»C drau'hat‘k L11 which can re :15. 3. ‘21 A forward cont estimated an resemhlante mmideratie .r third-loop \ P4112 :‘1241 pm- 11131C0mpt‘1 19 and out; rater stahiii 711.11.; ~ u Zdtll-Jh 151111 ' - g- MU} ZCAI’.7~'»_ ' J...““'F.‘1r“: {5‘ 5‘11 11191715 rt 11"! ‘ a ‘41: ding”) .4; "3. I‘Pl «A (t harmonics of angular speed. Despite differences between mass unbalance and sensor runout disturbances, the control objective for their compensation is often similar. Stabilization of the rotor about the geometric center, which is the objective for SRO compensation, is often the objective for unbalance compensation. Though few researchers [11, 33] have addressed the problem of compensation of combined unbalance and SRO disturbances, there exists a large volume of research on compensation of individual disturbances. Some of the early work on unbalance com- pensation has been based on the insertion of a notch filter in the control loop [2]. The drawback of this approach stems from negative phase of the notch transfer function, which can reduce stability margin of the closed-loop system and lead to instability [15, 3, 24]. Another approach for periodic disturbance cancellation is adaptive feed- forward control [34, 9], where Fourier coefficients of the disturbance are continually estimated and used for cancellation. These adaptive controllers, operationally bear resemblance to the notch filters [24] and can result in instability if designed without a consideration for the underlying structure of the system. To preserve stability of the closed-loop system, Herzog et al. [7] developed the generalized notch filter and Na and Park [24] proposed a variation of the least mean square algorithm. Other approaches that compensate unbalance while ensuring stability include adaptive auto-centering [19] and output regulation with internal stability [21]. Both these approaches achieve rotor stabilization about the center-of-mass. Though unbalance compensation has been widely studied with the objective of stabilization about the mass center, most users and vendors push for geometric cen- tering, accepting that the real objective is to avoid seal or aero tip collisions. While geometric center stabilization has been addressed by a few researchers [8, 37] both problems were investigated in references [28, 22]. These results indicate that stabiliza- tion about mass center or geometric center can be achieved through cancellation of disturbance in the current signal or the displacement signal, respectively. In a general and experirru 1 form of rOMI of the prtLU . 1 the a1gurttht mmhsuvn‘ into two inti- teasfuhy. tht Some of 1111' mmudthsh: [hfurturg sdwstorun. hm.whhly“ duhwnahfi (1181111911511 1; Phfltorits retor augula] ”“151 apphr‘a 1'3 80 Our 911910801 .u appueat t... . ““1195. T1,. 3‘1", - metnc (.9, 999." him if. 1"‘Tiiv , ”WIMP r f . _,‘ [LP 11"»19? - drf-‘bhgr and experimental approach for disturbance attenuation by Knospe et al. [16, 17], any form of rotor vibration that can be measured can be attenuated using pseudo-inverse of the pro-computed influence coefficient matrix. The stability and performance of the algorithm in the presence of uncertainties were investigated, and experimental results were used to demonstrate effectiveness. The method decouples the problem into two independent control tasks; and while it has been demonstrated to work suc- cessfully, there is no theoretical basis for stability of the two interacting processes. Some of the other approaches employed for unbalance compensation include robust control designs [6, 30], Q—parameterization control [23], and neural networks [27]. Unfortunately, most of the approaches found in the literature do not lend them- selves to runout estimation in the presence of significant mass unbalance. This prob- lem, widely acknowledged in the literature but essentially unsolved, stems from a lack of observability of disturbances with the same frequency content. A credible way to distinguish between these disturbances is to perturb the operating conditions of the plant or its parameters. However, recent studies [11, 33] that propose variation in rotor angular speed as a means to enhance observability may not be acceptable for most applications. 1.3 Scope and Content of the Thesis Our approach to the problem is based on traditional adaptive control designs that has seen applications with a variety of electromechanical systems [4, 31] but not magnetic bearings. The objective of rotor stabilization is to precisely spin the rotor about the geometric center in the presence of SRO and unbalance. In our approach, we individ- ually identify synchronous mass unbalance and SRO at constant rotor speed through Persistence of excitation. Our adaptive control framework enable us to uniquely excite the regressor vector, if necessary, to provide the persistently exciting (PE) condition. 1115 “111119 r aimpltlllc 1 [me101lum The tilt ‘ and unhahu in 011-11119 K | I | 1 1 | Map“)? dL red tariahh 1 Mn 31181111" | flahilit): put averaging 31‘ and experirr: I theeha11eng1 I 01 adaptive t1. asnlution t: turrems. Thl mmergent‘e urination at 1119 adaptive hearing 5131' for future rm It is widely known in adaptive control field that PE conditions can contribute to an asymptotic convergence of estimated parameters to the true values and provide some level of robustness [25]. The thesis is organized as follows. In Chapter 2 we introduce the problem of SRO and unbalance using a simplified single-degree of freedom magnetic bearing model. An off-line SRO identification that is useful for verification of the results given by adaptive algorithms is also presented. In Chapter 2 we include the most commonly used variables to avoid repeating their definition in the next chapters. In Chapter 3 an adaptive sensor runout compensation (ASRC) is presented including proof of stability, parameter convergence, robustness to plant parameters uncertainty, and averaging analysis to approximate the convergence rate of adaptation. Simulation and experimental results are also provided in Chapter 3. In Chapter 4 we discuss the challenge posed by the combined SRO and unbalance problem and the limitation of adaptive control implementation via multiple angular speeds. Chapter 5 present a solution to the combined SRO and unbalance problem by excitation of the bias currents. The robustness of the algorithm is studied using a passivity analysis. The convergence rate of adaptation is investigated using the averaging method. Both simulation and experimental results are provided in Chapter 5. Chapter 6 extends the adaptive algorithms in Chapters 3 and 5 for implementation in MIMO magnetic bearing systems. Chapter 7 provides concluding remarks and provides suggestions for future research problems. 1.4 N Arabic 8} 1 .4 Nomenclature Arabic Symbols at" bi C fu fc Prq Harmonic Fourier coefficients of sensor runout Constant Sensor runout disturbance Weighted sum of position error and velocity error Magnetic force Disturbance force due to mass unbalance Force due to controlled current Gravity Controlled current Identity matrix Axial moment of inertia of rotor Transverse moment of inertia of rotor Harmonic number Top bias current Bottom bias current Electromagnetic constant Actuator gain or current stiffness Bearing stiffness or position stiffness Nominal air gap Highest harmonic number considered Mass Harmonic Fourier coefficients of mass unbalance Time Greek Symbols 7i 7p: 7q Geometric center position Position sensor signal Regressor Vector Lyapunov’s function Bias current excitation Dimensionless parameter Eccentricity Adaptation gain matrix for SRO part Adaptation gain matrix for unbalance part Adaptation gain constants for SRO part Adaptation gain constants for unbalance part Error gain Rotor angular speed Vector containing Fourier coefficients of sensor runout Vector containing Fourier coefficients of mass unbalanced Phase of mass unbalance Nominal value Estimated parameter Difference between the actual parameter and the estimated parameter Acronym 11113 ASRC | BC E DSP 111110 SPR SRO SRL'C Operate; Ret] Acronyms AMB ASRC BCE DSP MIMO PE RPM SISO SPR SRO SRUC Operators Ref ) (' )7 (° )‘1 Active magnetic bearing Adaptive sensor runout compensation Bias current excitation Digital signal processor Multi-input multi-output Persistently exciting Revolutions per minute Single-input single-output Strictly positive real Sensor runout Sensor runout and unbalance compensation Real part Matrix or vector transpose Matrix inverse cHA Magnt 2.1 11‘ In this ehu‘ nur researt“: hearing 111‘ ’ the outline M terrns 2.2 S .1118 systt 93th of the gene D111 are sintiiar In our stutl at? mam -" xiii”: l CHAPTER 2 Magnetic Bearing Modeling 2.1 Introduction In this chapter, we present the dynamics of a rigid rotor magnetic bearing used in our research. We start our study by considering a single degree of freedom magnetic bearing model. The effects of unbalance and sensor runout (SRO) are then introduced. We outline the procedure to manually identify SRO and present the most commonly used terms in the development of our adaptive algorithms. 2.2 Single Degree-of-Freedom Model AMB systems have, in general, five degrees-of-freedom (DOF): two radial DOF at each of the shaft ends and perhaps one axial DOF. However, this chapter considers a one DOF system only by assuming that the dynamics in all four radial directions are similar and can be controlled independently and that the rotor is axially fixed. In our study we assume that the effect of non-collocation between the gap sensor and the magnetic coil actuator is negligible. The typical diagram of a magnetic bearing System with a decentralized feedback control is shown in Figure 2.1. The feedback control stabilizes the rotor position in the following manner. The differential gap Sensors measure the location of the center of geometry relative to the stator. After comparing ‘ sent me the . provides rh- magnetic (F arranged u. NOW (.0! [W9 (4192me miaillrrrd } "furl- ', , .L‘t‘h at“ 1 it- , . h" I ms. Li‘rF‘ v. ‘ u ' "1 1.x K‘ ‘1191"7v,'__‘. T.“ (Jr. f comparing the position measurement to the reference position, the resulting error is sent to the compensator. The compensator, which can be an analog circuit or a DSP, provides the necessary command to the power amplifiers that drive currents in the magnetic coils. The signs of the signal sent to the top and the bottom amplifiers are arranged to be opposite to each other. To magnetic coil 1 Stator Power ~ 4?. Compensator — Powe *Ei“ amplifier V Fremgapsensor Differential sensing Figure 2.1. Diagram of a typical active magnetic bearing system Now consider the magnetically levitated rigid rotor in Figure 2.2. The rotor has two degrees-of-freedom along the :1: and y axes; the displacements along these axes are measured by non-contact gap sensors. The dynamics of the rotor along these axes, which are both inclined at 45° with the horizontal, are decoupled but similar. Along the 2: axis, one may write mit=F—m§+f.., gég/x/é (2.1) where m is the mass of the rotor, a: is the position of rotor geometric center, F is the magnetic force, fu is the unbalance force, and g is the acceleration due to gravity. 10 Figure '2 The - magnetic i” w _ Q} t iifie ’ ' .drllmg Eq . The .IP - ‘ fc 15 [h (J of. (10‘ , f ' 519i]: Note: .1: Geometric Center. :2: Mass Center Figure 2.2. Single-DOF model of magnetic bearing with rotor schematic The magnetic force can be expressed as (3°31)? _ (30:31)? (2.2) where k is the magnetic force constant, I is the nominal air gap, 2'10, 2'20 are the bias F=k currents in the top and bottom electromagnets, and I is the control current. By linearizing Eq.(2.2) about :1: = O, I = 0, Eq.(2.l) can be written as mftszx+fc+fu (2.3) f6 2 Kc] (2.4) K, 9.: 21: (if, + 2'30) /13 (2.5) K, é 2k (in, + 2'20) /l"’ (2.6) where fc is the control force, K, and K6 are the magnetic stiffness and actuator gain of the magnetic bearing respectively. The unbalance force due to mass eccentricity 11 can be mC’d' wht’fP p = ' angular SP“ 23 St Ignoring ii“ be mum i it can b? 5‘ The plant 1 To St aiii [rid 2L9 til“ where HP 3 Cirmd‘luop. ill the [)IP.‘ the ori gi r1 9.7-." J *r-Q‘J‘d a can be modeled as f“ = mw25 cos(wt + 0n) = mu)2 [psin(wt) + qcos(wt)] (2.7) where p = —5 sin(0u) , q = 5cos(l9u) , 0,, is the phase of unbalance, w is the rotor angular speed, and 5 is the eccentricity of the rotor. 2.3 Standard PD Controller Ignoring the external disturbances in Eq.(2.3), the transfer function of the plant can be written as X (s) K 6 1(3) : ms2 —— K, (28) It can be seen that without a closed-loop controller the linearized system is unstable. K The plant has eigenvalues at :t‘ / 7:. To stabilize the rotor, a proportional-derivative (PD) controller Gc(s) is commonly used as the compensator shown in Figure 2.1. K 06(3) = K, (1 + —“— s) (2.9) Kr where K, > 0 is the proportional gain and K; > 0 is the derivative gain. The stable closed-loop system can be described by _ (KP + Kd 3)Kc K3 061(3) — ms? + Kchs + (Kn Kc _ Ks) where KP > Kc (2.10) In the presence of the unbalance, the rotor geometric center will fluctuate around the origin if this standard PD controller is used. The rotor performance is further degraded as the sensor signal contains periodic disturbance due to sensor runout. 12 2.4 Sen: The {mt3 lut‘aii with sensor ru where. d. the : 2.4 Sensor Runout (SRO) Disturbance The true location of the rotor’s geometric center is not available for a magnetic bearing with sensor runout. Instead, the gap sensors provide the signal :r,. 2:, = a: + d (2.11) where, d, the sensor runout disturbance, can be expressed by the Fourier series d é a0 + i [a,- sin(z'wt) + b,- cos(iwt)] = YTqS (2.12) Y e [1 sin(wt) cos(wt) sin(nwt) cos(nwt) )T (2.13) Q5 2 [00 a1 b1 . . . an bn ]T (2.14) In the above expression, n is the number of harmonics, no is the DC component, and a,-, b,, i = 1, 2,. . . , n, are the harmonic Fourier coefficients. 2.5 Off-line SRO Identification In this section we present a method for manual off-line identification of sensor runout. This method will be used to verify the accuracy of the on-line adaptive sensor runout compensation scheme (ASRC) in Chapter 3 and the combined sensor runout and un- balance compensation scheme (SRUC) in Chapter 5. Off-line SRO identification, which has to be performed separately for each axis of the bearing, requires the rotor to be spun at low speed to avoid the effects of unbalance. We will first levitate the rotor using a PD controller, as shown in Figure 2.3. Using a DSP to generate function E0 = A0, we close the feedback loop using the signal (2:, — E0). We then adjust the magnitude of A0 such that (x, — E0) has a zero mean. Once this is accomplished, we 13 will have itlt‘ni Next. we 31, angular with phase with tlit I, - £0 - E1 order harmun Having iti second harnn Eventually. it E: This signal ( mum. to r,- we plot E. , Sfe‘fiilicant r will have identified the DC component of SRO. Next, we generate the signal E1 = A1 sin(0, + 61), where 0,, é wt is the rotor angular position obtained from the shaft encoder. We select 01 such that E1 is in phase with the first harmonic of (x, — E0). We then change the feedback signal to :r, — E0 — E1 and adjust A1 by trial and error such that (2:, — E0 — E) has no first order harmonics. Having identified the first harmonic of SRO, we then sequentially identify the second harmonic E2 = A2 sin(200 + 02) and higher order terms in the same manner. Eventually, we will have the complete SRO signal E = (E0 + E, + E2) = A0 + A, sin(0, + 01) + A, sin(29, + 02) (2.15) This signal can be subtracted from the sensor signal 22,, preferably using an analog circuit, to recover the position of the geometric center, E, = :r. In our experiments, we plot E, to verify rotor stabilization about the geometric center in the presence of significant unbalance. Unbalance Sensor Manual SRO u Geometnc d l E = A,+A,sin(e,+e,) Center 0 + PD I Magnetic Bearing *‘ 3,... ‘ It Es=0 for all 00, then E = d Figure 2.3. Off-line SRO identification 14 2.6 Unh’ in this section _ I by our adaptrt the initial unh. unbalance u>ir. was. we can to a \H‘i(if Slllll (i Eexplanned Wi 2.7 D91 1; the d “‘1‘ i... ’l 2.6 Unbalance Verification Using a Trial Mass In this section we outline the procedure for verification of rotor unbalance estimated by our adaptive algorithm. Let cu, 0U be the estimated magnitude and phase of the initial unbalance. We will add a trial mass mT to the rotor and re-estimate the unbalance using our algorithm. If 67, 07- are the magnitude and phase of the trial mass, we can verify the efficacy of our algorithm if the new unbalance vector £3, 03 is a vector sum of the initial unbalance and the unbalance due to the trial mass. This is explained with the help of Figure 2.4. = initial unbalance vector m = initial unbalance T = unbalance due to trial mass m-r = trial weight mass s9 = resultant vector Figure 2.4. Unbalance verification using a trial mass 2.7 Definition of Most Commonly Used Terms In the development of our adaptive algorithms, there are several variables that will be used repeatedly. We introduce the variables here to avoid repeating the same 15 definition in 1“ We define i where it the estimati- is the estimate 1, are estirnatr (2.16). i can : ithere. d. tht relation The pal 21mm. definition in next chapters. We define the estimated geometric position or position error as 5: e 2:, — d (2.16) where d 3 [to + Z [(1,- sin(z'wt) + f),- cos(z’wt)] = YTd (2.17) i=1 is the estimated SRO disturbance, is [do a, (3, an 5,17" (2.18) is the estimated parameter vector of the SRO, 610 is the estimated value of a0, and (1,, f),- are estimated values of 0,, b,, respectively, for z' = 1, 2,. . . , 71. Using Eqs.(2.ll) and (2.16), it can also be expressed as HI H H + 9—2 (2.19) ~ where, d, the error in the estimate of sensor runout disturbance, is given by the relation is (d — d) = YT$ (2.20) The parameter error vector 6 is defined as 00 5% 45-13: 5, (2.21) 511 Where 50 g [51 51 1T, 53 g [52 52 an En 1T, 00 _ (a0 - [10), and at _ 16 Other mos petition error where 1 is a (1... and the adapt Wilf’l’e q“ l _ ihdi0<3‘ Other most commonly used terms include the weighted sum of the estimated position error and the estimated velocity error 6 2 :1 + A515 (2.22) where A is a positive constant, the regressor vector Y... 2 K,Y — mi'r, Ym 6 32‘2"“) (2.23) and the adaptation gain for SRO components F g diag(703 712 717 ' ' ° 2 7n) ,7"), F E R(2fl+l)X(2n+1) (224) where 7,, i = 0,1, 2,. . .,n are positive constants. These constants are chosen such that 0 < A < 1 where A is a dimensionless parameter defined by A 2 WW", = Z 7,- (K, + m(iw)2) (2.25) i=0 17 CHAP“. Adapth (ASRC 3.1 Int] hmnmuwt in the single d absence of in: is designed t1 stability and the unt‘ertai; estimated pe then verified 3.2 C \ iii-Rt”); . at 11 '0 It.) 1 hifi’rJHLC-x CHAPTER 3 Adaptive Sensor Runout Compensation (ASRC) 3.1 Introduction In this section we present an adaptive algorithm to reject the effect of sensor runout in the single degree of freedom magnetic bearing model presented in Chapter 2. In the absence of mass unbalance, the adaptive sensor runout compensation (ASRC) scheme is designed to stabilize the geometric center of the rotor to the origin. The proof of stability and parameter convergence is provided. The robustness of the algorithm to the uncertainties in plant parameters is evaluated and the convergence rate of the estimated parameters is approximated through averaging analysis. The algorithm is then verified through simulations and experiments. 3.2 Choice of Adaptive Controller Assuming the unbalance force, fu is negligible, the equation of motion in Eq.(2.3) becomes m2? = K,:1:+KCI (3.1) 18 for estirnatiut metric center. 31‘;ng Willi illt Where ("a and A are defined C is a Witiv dim'aiire of Show) in Fig Ol'lgin in the file provider) I I I I I / :7 For estimation and cancellation of sensor runout, and stabilization of the rotor geo- metric center, we propose the control action 1 I: Kc (K,:r + mAri: + cé) (3.2) along with the adaptation law 5 = r Y... (E (3.3) where K, and K, are defined in Eqs.(2.5) and (2.6), 2‘: is defined in Eq.(2.16), E and )1 are defined in Eq.(2.22), I‘ is defined in Eq.(2.24), Y... is defined in Eq.(2.23), and c is a positive constant. The derivative term :i: is assumed available by taking the derivative of the measured signal 5:. The block diagram of the closed-loop system is shown in Figure 3.1. The controller can stabilize the rotor geometric center to the origin in the presence of sensor runout. The stability proofs and convergence analysis are provided in the next section. Sensor Unbalance Runout iu = O Geometric I 1 Center ' x ’------- ---q Figure 3.1. Block diagram of magnetic bearing system with ASRC 19 3.3 P" from the (ME scribed b." [sing the “‘1 above “luau“ The term (I ('3 and substitut in". ‘ Ring the l 111* , ,- ., «textures can 3.3 Proof of Stability and Parameter Convergence From the definition of Y and Y... in Eqs.(2.13) and (2.23), we can establish YTrY... = 0, YTI‘Y... = 0 (3.4) Substituting Eq.(3.2) into Eq.(3.1), the dynamics of the controlled rotor can be de- scribed by mi} = —K,J— mAi: — cé (3.5) Using the relation If 2 :‘i‘ + (I from Eq.(2.19), and e = 57} + Air from Eq.(2.22), the above equation can be rewritten as mézmg— K,c7—cé (3.6) The term (I can be derived from Eq.(2.20) if = YTJ + 2Y7}; + YT; = WE + Aé (3.7) and substituted in Eq.(3.6) to obtain m (1 — A)é’ = —Y,Tn$ — cé (3.8) knowing the relations in Eqs.(2.25), (3.3) and (3.4). Thus, the closed loop system dynamics can now be described by is = —A:r: + 6 (3.9a) l ______m (1 _ A) (Y,E 0 as t ——> 00. This implies ('3' —> 0 as t —> 00. By differentiating Eq.(3.9b), we can show that 5 = at, 25,13) is bounded. This implies that e is uniformly continuous. Since 6 —) 0 as t —) 00, we once again use Barbalat’s lemma [13] to deduce é -—) 0 as t —-> oo. Knowing E, E? —> 0 as t —> 00, we can conclude from Eq.(3.9b) v3.13" —> 0 (3.13) 21 furtherniu that where l is ll. exeiting ( PE This implies: f we can now a concludes our Lemma 3.1 an (lit/triplollt. Proof: T farm Where .31 no A! he know {run t e sub-sister Fq‘iili‘ "9r , . 13(- 3' ,l- 1511);“ Theorem 3 .nl . 403‘ 1 (”JP 51). proOf: Furthermore, we can show that there exist positive constants al, 02, and To, such that 1+7}, (121 Z Yngdr 2 ml (3.14) t where I is the identity matrix, I E 32(2"+1)(2"+1). Therefore Y... is a persistently exciting (PE) signal [13]. The proof of the PE condition is provided in Appendix 3.1. This implies from Eq.(3.13) that a —) 0, as t —> oo. Knowing 5.3 —-> 0, as t —+ 00, we can now assert that (e, o) E (0,0) is an asymptotically stable equilibrium. This concludes our proof. ~ Lemma 3.1 The origin of the closed-loop system in Eq.(3.9), (’,é,q§) E (0,0,0), is an asymptotically stable equilibrium point. Proof: The closed loop system in Eq.(3.9) is an interconnected system of the form 21 = f1(t, 21,22) (3.153.) 22 = f2(t, 22) (3.15b) where z. é 5:, and 22 3 (6 6T )T are the state variables of the two sub-systems. We know from Theorem 3.1 that 22 = 0 is an asymptotically stable equilibrium of the sub—system in Eq.(3.15b). Also, 21 = f2(t, 21,0) has an asymptotically stable equilibrium point at 21 = 0. This can be readily established from Eqs.(3.15b) and (3.9c). Using the asymptotic stability theorem for cascaded systems [13], we conclude that (5:, e, (ii) E (0,0,0), is an asymptotically stable equilibrium. Theorem 3.2 The coordinate (x, a}, 5) E (O, 0, 0) is a stable equilibrium point for the closed loop system defined by Eqs.(2.3), (3.2), and (3.3). Proof: : Using Eqs.(2.19), (2.20), (3.2), (3.3), and (2.25), we can show that at 22 (1.13.0) E (U Also. atlr. r (3.2). and (I. an Pquilihrii now be dmll 1- The e rium. 2- The t (aging) E (0, 0,0), we have J=YT$=0, i=$+d=0, 3: (WE + YTZ'J) = YTI‘Ymé = A6 = A0}? + Ar) = A5; Also, at(:r,:i:,$) 5 (0,0, 0), d: (:i: — at) = is = 0, since 0 < A < 1. From Eqs.(2.3), (3.2), and (3.3), it follows that (i,:ii,$) = (0,0,0). Therefore, (aging) E (0,0,0) is an equilibrium point. The fact that (x, it, a) E (0,0,0) is asymptotically stable can now be deduced from: 1. The equilibrium point (58,6,3) E (0, 0,0) is an asymptotically stable equilib- rium. This fact follows from Lemma 3.1, 2. The transformation matrix P that maps (5:, e, a?) to (11:, :ir, a?) a: 1‘: a: = P E (3.16) 5 J 1 0 —YT Pé —,\ (l—A) —YT (3.17) 0 0 E where E E 32(2"+1)x(2"+1l is the identity matrix, is well defined and upper bounded, and 3. The inverse transformation P‘1 exists, and [l P’1 [I is also upper bounded. Theorem 3.2 establishes that the adaptive controller proposed herein guarantees stabilization of the rotor geometric center through identification and cancellation of sensor runout disturbance. 23 "‘1 3.4 R0 ln this sectio tion in plant tion of rotor of exact lino To this end. modify our ( Where ~L , . . Smmnllllu form ‘Sllto A") the 3.4 Robustness to Parameter Uncertainties In this section we establish that the ASRC scheme is robust to uncertainties or varia- tion in plant parameters. Specifically, we shall show that ASRC guarantees stabiliza- tion of rotor geometric center and exact cancellation of sensor runout in the absence of exact knowledge of rotor mass, m; magnetic stiffness, K,; and actuator gain, Kc. To this end, we estimate the values of these parameters to be m, K” and Kc and modify our control action and adaptation law in Eqs.(3.2) and (3.3), as follows: 1 - . I = —-I={— (K59? + one + 06) (3.18) 35' = I“ Ym E (3.19) where Y... é I‘m! - mi? (3.20) Substitution of Eq.(3.18) in Eq.(3.1) indicates that the closed-loop system takes the form mi} + gfum + c)i‘: + [LE-(R, + cA) — IQ] if: = md— K33 (3.21) C Using Eqs.(2.20), (2.23) and (3.19), the right hand side of Eq.(3.21) can be sim- plified as follows: mZi— KJ: m (2% + 2Y7"; + YTJJ) - K,YTq~3 (3.22) = _Ygi + m [2YTI‘i'fme + YTI‘ (Ymé' + 37%)] (323) Using the identities YTI‘Ym = 0 and YTP?m = 0, we get md— K,d= —Y;,$ + mAé (3.24) 24 Substituting Where M, C The (“ll the adapta Wthh l8 a f can be mat g7”? 3.2 ~ .~ 1 51 *1 QUIET-Pr A é YTI‘?,,. = 271-03. + info?) (3.25) i=0 Substituting Eq.(3.24) into Eq.(3.21), we get the closed-100p system dynamics M53 + Cf + Ki: = —Y,T,,$ (3.26) where M, C and K are defined as follows: M é m(1 — A) (3.27a) A K. _ - C = R—(mh + c) — m/\ A (3.27b) K £- [gfu'g + c/\) — K3] (3.27c) The closed loop system, described by the dynamics of the rotor in Eq.(3.26) and the adaptation law in Eq.(3.19) is represented by the block diagram in Figure 3.2, which is a feedback inter-connection of two linear systems. The following observations can be made regarding these linear systems. Stricflypassivasystem / -------------- \ l - \ V=0 + “1 1 i l Y=9 z :> 5+7. r 1 > - ‘ M32+Cs+K ) \____-_-__.. _____ / Passivesystem T.. Y2=Ym¢ 1- r— A U2 Yang-Y", V Figure 3.2. Block diagram of closed-loop system in the presence of uncertain plant parameters 25 2“. ."-" err-w Lemma 3.2 transfer fum' fistnctly pi; Proof: 00: Therefore. reference 7 \ Lemma: Proof l’-; Lemma 3.2 The linear system in the forward path in Figure 3.2, defined by the transfer function 3 + A M 32 + Cs + K 0(3) = (3.28) is strictly passive if M, C, K > 0, and C — AM > 0. Proof: If M, C, K > 0, (3(3) is Hurwitz. Furthermore, if C — AM > 0, we have G(jw) + G(—jw) > 0, C(00) = 0, 3110M [C(jw) + G(—jw)] > 0 Therefore, C(s) is strictly positive real (SPR) according to lemma D.4 obtained from reference [13, 18];thus, C(s) is strictly passive. Lemma 3.3 The linear system in the feedback path of Figure 3.2 is passive. Proof: The adaptation law in Eq.(3.19) can be written as 5}, = 70K, 6? (3.293.) ii,- = 7.- [R, + mi2w2] sin(iwt) e (3.2%) 3.: 7,-[K + mm 2] cos(iwt) é, i = 1,2,. . .,n (3.29c) Using these relations, and defining K, + mill.)2 1. = _ . 3.30 p 7,-(K, + 172221.22) ( ) for i = 0, 1, 2, . . . , n, we can express the net energy flow into the system as t t ~ [31211th =/ YT¢édt o t _ —K, fat" aoé dt + 123 K + mi2 312)] [:sin(iwt)é + b,cos(iwt)é] dt 0 ‘Pofaoaodt'l'Z/h/ [aias+ +315] 26 where “'18 ' ll”! From Eq.(3-3 We nUW Theorem 1 in. Eqs.( 3.1 10.0.0) of rotor mow adaptatu‘m Proof .1 1.. i = 0. Lemmas I. iriterconn theorem Holilf‘. ” x (AV , —_—%[Eff,(t)— 00(l0) +Z:;g—’-[Eff t()+bf(t)-a(0)—gf(0)] = W[36(t)] — Wtc‘b'to )1 (331’ where W[$(t)] is a positive definite energy storage function given by the relation ... 1... ~ _ W[¢(t)l é §¢TM¢1 M 2 dlag(P0,P1,P1,P2,P2,---,~ . 9pmpn) From Eq.(3.31) we claim passivity [13, 18]. We now present our final result with the help of the following theorem. Theorem 3.3 Asymptotic stability and robustness The control and adaptation laws in Eqs.(3.18) and (3.19) guarantee asymptotic stability of the equilibrium (2:,i:, 3') E (0,0, 0) of the magnetic bearing system in Eq.(2.3) in the presence of uncertainty in rotor mass, m; magnetic stiffness, K ,; and actuator gain, Kc, provided the error and adaptation gains are chosen to satisfy M, C, K > 0, and C — AM > 0. Proof: Through proper choice of the error gains (c, A), and adaptation gains (’71, i = 0, 1,2, . . . ,n), we can easily guarantee M, C, K > 0 and C - AM > 0. Using Lemmas 3.2 and 3.3, we can then conclude that the closed-loop system is a feedback interconnection of a strictly passive system and a passive system. Using the passivity theorem from the appendix of [18], we claim (it, it, a) E (0, 0, 0) is globally uniformly stable, and 5,5: —+ 0 as t —+ 00. Now, to show (as, 2:, (if) E (0, 0, 0) is an asymptotically stable equilibrium, we first need to show that (2,2,3) 5 (0,0,0) is an equilibrium. This can be verified using Eqs.(2.3), (2.20), (3.18), and (3.19). The fact that (23,23, a) E (0,0,0) is stable follows from: ~ 1. The equilibrium point (5:, :‘r, <25) E (0, 0, 0) is globally uniformly stable. ~ 2. The transformation matrix P(t) that maps (2:, :r,¢) to (:12, :‘r, (3) is well defined 27 and t l) 3. The lli'- Fina”? we 1 Figure 3.2 \1 'm Figure 3'; mums 35‘ by simpl." 5h 3.5 0 The methut‘ qualitative l [em Oblall‘tt‘ method for small parat meMllC ()l’ of the time The ()l)‘; mated para averaging a Price of um the come Ti 1; ‘ w obtain t krilswn T1 {19 FT and II P H is upper bounded. 3. The inverse transformation P‘l exists, and II P‘1 H is also upper bounded. Finally, we prove (3,2,3) —+ (0,0,0) as t —+ 00. Since 513,52 —+ 0, E —-> 0. Also, from Figure 3.2 we claim u1 = y2 —> 0. This is true since the mass-spring—damper system in Figure 3.2 cannot have zero output for nonzero input. Since Y3; is persistently exciting, as discussed in the proof of Theorem 3.1, y2 = 0 implies 3 = 0, and d = YT; = 0. Also, (if = (YT; + YT¢) = YT; == YTI‘Ymé = 0. We conclude the proof by simply showing a: g (:7: - if) —> 0 and :i: é (:2 — if) ——> 0. 3.5 Convergence Rate Analysis Using Averaging The method of averaging is an asymptotic method, which permits the analysis of qualitative behavior of time-varying systems through a time-invariant (averaged) sys- tem obtained by time-averaging of the system. This method has become the general method for the analysis of nonlinear ordinary differential equations (ODEs) with a small parameter [38], including the determination of the existence and stability of periodic or almost periodic solutions as well as the analysis of the transient behavior of the time-varying system. The objective of this section is to approximate the convergence rate of the esti- mated parameters as represented by time constants. We first apply the two-time scale averaging analysis given in reference [32] to the proposed adaptive system in the pres- ence of uncertainties in the plant parameters. We determine the variables that affect the convergence rate of the estimated parameters. These results are finally reduced to obtain the time constants of estimated parameters when the plant parameters are known. ~ The error signal in Figure 3.2 can be seen as the result of signal (—Yf,;¢) being 28 passed thror where Gt’s) Eq.(3,323 tr T0 appl adaptation is Sufficient and (Timex Zr\‘i‘dmprim Kale 0f 1h approxj m 3 Where T \ m I II (4 311mm, where; R l, passed through an SPR transfer function G (s). é = —G.tY?.‘.$] % —G(Y3.1)-[$t (3.32) where G(s) is defined by Eq.(3.28). Note that C(Yi) is a signal vector. Substituting Eq.(3.32) to the adaptation law in Eq.(3.19) we obtain <3: 4‘37... [G(Y3,",) <25] (3.33) To apply the averaging method, we need to treat the above equation as a slow adaptation process. For this purpose we must assume that the adaptation gain I‘ is sufficiently small and the true values of parameter of belongs to a given compact and convex set for which the frozen (1‘ = 0) closed-loop system is stable. Using this assumption, we can separate the slow time scale of adaptation from the fast time scale of the other signals. By applying the averaging method given in [32], we can approximate the original system in Eq.(3.33) by using an averaged system such as A, 1 t0+T_ _ ~ e..=— [— Ymctvzwdt] T to = 4“ 11(0) 37..., (3.34) where T = 33 , $0., is the averaged estimated error vector, 30,, E 320"“), and R(0) is a symmetric positive-definite cross correlation matrix provided that A < 1 and c > 0. 11(0) —_— diag(Ro, R1, R1, R2, R2, . . . , Rn, 11,.) (3.35) where 11(0) 6 31(2"+1)(2"+1), and its components are R0 = K,K, C(32) = 0), (3.36) 29 for i = l.‘2. the derix'atir here. hower. of each com b." a SF! of s Waging tl. tion matrix Stable with Particular Expfithemh c” // = [K, + m(iw)2] [K, + m(iw)2] 2 Re{G(J'(iw))} (337) R,- for i = 1, 2,. . ., n. It can be observed that R0, R,- > 0 if G(s) is SPR. The details of the derivation of the correlation matrix is given in Appendix C. It should be noted here, however, that both R(0) and I‘ are square diagonal matrices; thus, the dynamics of each component in $0” is decoupled, exponentially stable, and can be represented by a set of simple first-order dynamics with time constants shown in Table 3.1. The averaging theorem in Sastry and Bodson [32] proved that, as long as the cross correla- tion matrix R(0) exists and l" is sufficiently small, the original system is exponentially stable within a finite ball B). if the averaged system is exponentially stable. In this particular case the equilibrium point 6 = 0 for the system in Eq.(3.33) is locally, exponentially stable. Therefore, we can conclude that the estimated parameters (5 ultimately converge to the true values (1) exponentially fast. Table 3.1. Time constants of the averaged system Known plant parameters Uncertain plant parameters 1' __ c r _ K 0 ’70 K32 0 70 [KsKsA] . __ 2[m(1— A)(iw) + c] r- _ f 2 [M(iw)2 + C(iw) + K] ‘ 7.. (K, + m(iw)2]2 ' 7.. [1%, + fir(iw)2] (K, + 722(23):!) [(iw) + A] Note: i=l,2,...,n Several observations can be made regarding the results shown in Table 3.1. First, we can easily derive the time constants for the known plant parameters case by remov- ing the bar signs and knowing that instead of going through the filter in Eq.(3.28), 30 magu( as it can be R... R. > t time must; in both ca- angular Sp! numerical ; 3.6 C 5.4 3.6.1 1. Slululzitio “‘3 of [h m‘l’df’l of real Sling) d‘plermin Parameu “morn we can “Failing: [33‘ “311‘: the signal (——Yf,",$) passes through 1 _ m(1—A)s+c 0(3) (3.38) as it can be obtained from Eq.(3.8). The above transfer function is also SPR. Thus, R0, R.- > 0. It can also be concluded that if the plant parameters are known, the time constants are independent of the gain A. Furthermore, we can observe that in both cases the time constant of the DC component is not affected by the rotor angular speed w, while the time constants of higher harmonics are subject to w. The numerical results of the time constants are presented in the next section. 3.6 Simulation Results 3.6.1 Known Plant Parameters Case Simulations are performed using Matlab/SimulinkTM to demonstrate the effective- ness of the ASRC algorithm. Though the controller was designed using a linearized model of the plant, we use the nonlinear plant model in Eq.(2.1) to simulate the real situation. The bearing parameters were assumed to be known. These values, determined in our experimental hardware, can be referenced from Table A.3. Other parameters used in simulations are shown in Table 3.2. The SRO compensation was performed up to the third harmonics. For the magnitude of SRO given in Table 3.2, we can assume that the effect of SRO to the system will not introduce significant nonlinear dynamics at steady state, even without adaptation. This can be evaluated by using the resulting magnetic force as function of I and 2: shown in Figures A.7 to A.9. Note that using Eq.(2.25), A is found to be 0.235, which is relatively far from the stability limit of 1. The simulation results are shown in Figures 3.3 through 3.5. These figures indicate 31 that estim.( geometric 1. adaptation that estimation of runout is successfully completed within 0.3 second, and the rotor geometric center is effectively stabilized to the origin. The transient-currents during adaptation are within a reasonable range as seen in Figure 3.5. Table 3.2. Parameters for ASRC simulation Angular speed: 02 = 2r x 20 rad/s Sensor runout: a0 = 10 um al = 67.615 urn; bl = 18.117 um; a2 = 7.071 um; b2 = 7.071 pm; a,=b,=0 fori23 Error gains: A = 400 s“; c = 1200 kg/s Adaptation gain matrix: I‘ = diag(1, 2,2,1,1,0.5,0.5) x 10“7 m/N Plant initial conditions: :1:(t =0) = —0.lx10‘3 m; i:(t =0) = 0 Initial conditions of estimated parameters: &,(t=0)=0; b,(t=0)=0 fori=0, 1, 2, and3 Using the averaging method explained in section 3.5, the time constants of the estimated parameters are To = 0.064 s, n = 0.065 s, 72 = 0.097 s, and r3 = 0.13 s. The dynamics of the averaged system are compared to the original dynamics as shown in Figure 3.6. In this figure without loss of generality, we omit to show results in the third harmonic components. It can be concluded that the averaged dynamics are quite well matched with the dynamics of the original systems. The effect of changing angular speed to the convergence rate of estimated param— eters can be explained using the results shown in Figure 3.7. The results suggest that we can choose adaptation gains such that all estimated parameters converge at 32 the same rat) the converfé‘“ convergence 3.6.2 U1 To demonstr ulation resu‘; esmnation c; actuator gai in Table A; gnlar \‘Ploci be verified [011C stabil 3-8 throng) the Sf‘lly’ir addition, 1 Table 3.3, lit-11f} in t' f\ , Var) the same rate for a given operating rotor speed w. We can also notice that at low w the convergence rates are relatively sensitive to the change of 0). It is clear that the convergence rate of the higher harmonics are more sensitive to the change of 0). 3.6.2 Uncertain Plant Parameters Case To demonstrate robustness of the ASRC to parameter uncertainties, we present sim- ulation results using the nonlinear model of our magnetic bearing; assuming over- estimation of the uncertain parameters, namely, rotor mass, magnetic stiffness, and actuator gain by 100%. The nominal parameter values were assumed to be the ones in Table A.3. The Fourier coefficients of sensor runout, rotor initial conditions, an- gular velocity, error gains, and adaptation gains were chosen as in Table 3.2. It can be verified from the calculation results in Table 3.3 that the conditions for asymp— totic stability in theorem 3.3 are always satisfied. The simulation results in Figures 3.8 through 3.10 show that the ASRC remains stable and able to correctly identify the sensor runout despite of the quite large over-estimation in plant parameters. In addition, the coil currents are still within a reasonably range as seen in Figure 3.10. Table 3.3. Numerical values of variables assuming m, c, K ,, K c are over-estimated by 100% in the actual case Variable Nominal value Actual value due to Unit 100 % over-estimation of parameters A 0.24 0.47 -— M 1.86 1.29 (kg) C 1.94 x 103 1.12 x 103 (kg/s) K 4.8 x 105 2.4 x 105 (N/m) (0' — AM) 1.2 x 103 0.6 x 103 (kg/s) 33 Act: 0% t 3:: ...w True values are shown by dashed lines 0.4 20 ___ Em <5 0 r . r O . A L O 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 time (s) 1 r E 3 «‘3‘ ,5 . . . . . . 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 time (s) time (s) 5 , e . 5 g owe”— <3? -5 . A . -5 - ‘ A 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 time (s) time (s) Figure 3.3. Estimated parameters of sensor runout 34 100- - d A x. g SOr- _ X § °’ ; .50 x .4 .1m 1 L 1 1 1 L 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Time (s) 0.4 Figure 3.4. Stabilization of rotor geometric center using ASRC 4 r v 4 v a 2 ‘ .' 1 2 We 1 1 . ‘ i 0 ; ; . 0 . ; ; 0 0.1 0.2 0.3 0.4 o 0.1 0.2 0.3 time (s) time (s) Figure 3.5. Transients of currents of ASRC 35 0.4 m) inlu I") a! (H 1.. In m) O b1 0”") 0.1 0.2 0.3 0.4 Solid lines = original system Dotted lines = averaged system o 0.1 0.2 0.3 0.4 time (s) -15 , 1 ‘ 0 0.1 0.2 0.3 0.4 time (s) Figure 3.6. Averaged approximation of ASRC 36 that the (‘il gen“? {an} ‘0 about f ‘ 3hr) l— 1' T— T I 1- » § ~ Slabiiity-iimit—e——e----l——— —— / 0 too 200 800 400 500 I) 600 (t) (rad/s) 0.3 I I I T T l ......... . j - ' ‘ , | 02_., ...Stabnmyumn‘..._ : l 30.15-....12_ _ . ., _ . W'— “ "“‘~\ R, » ~ I 0.1 ... ,. 4.. 1 1“ ,\ ‘3'“. ,1 I . _‘ mil \ "*"x‘ To I 0.05-., 7' t ....... A,r..._4 o 100 200 300 400 500 I 600 Figure 3.7. Time constants of estimated parameters in ASRC A comparison of Figures 3.3 through 3.5 and Figures 3.8 through 3.10 indicates that the closed-loop system with 100% over-estimated parameters has a faster conver- gence rate than the closed-loop system with known parameters. The approximation results using the averaging analysis as shown in Figure 3.11 also confirm this faster transient performance. However, the stability limit cum” of the ASRC has reduced to about 350 rad/s as seen in Figure 3.12. Previously, wmaz was about 560 rad/s as shown in Figure 3.7. 3.7 Experimental Procedure To experimentally verify the effectiveness of the ASRC, the control action and the adaptation law in Eqs.(3.18) and (3.19) were implemented in the Matlab/SimulinkTM environment and downloaded to a Digital Signal Processor (DSP) board manufactured 37 ass 5 . True values are shown by dashed lines «1° 10 ----------- - 0 A A A A 0 0.05 0.1 0.15 0.2 0.25 time(s) 100 e E 50» «11' o A A L A O A _A_ A A 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 time(s) time(s) . 20 a f . l A A A A .10 A A A A 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 time(s) time(s) 10 . . . . 5 4 i E 5 O 4%.. (0 W v (.0 .5» .5 1 L . - - . 1 . 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 time(s) time(s) Figure 3.8. Sensor runout estimation in uncertain plant 150 T V I T V~ V1 l l 1 l 0.05 0.1 0.15 02 0.25 time (s) Figure 3.9. Stabilization of rotor geometric center using ASRC in uncertain plant 38 3% $12. . . . 1» 1 o A A A A 0 0.05 0.1 0.15 0.2 time(s) 0.25 52%” .‘V : A O 0 0.05 0.1 0.15 time (s) 0.2 0.25 Figure 3.10. Transients of currents of ASRC in uncertain plant Solid lines = original system to = 0.010 s Dotted lines = averaged system 0 0.05 0.1 0.15 0.2 time (s) -20 i V f A O ’ A 10 > -20 . . m 40 1 m '50 ‘ "0 t1=0.0178 -80 A A A L .20 A A A A 0 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 time (s) time (5) . . 1o - - - E 3 N _ . . . . in t = 0.033 8 12 = 0.033 s .10 . 2 l .10 A A A A A A A A 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 time (s) time (s) Figure 3.11. Averaged approximation of ASRC in uncertain plant 39 I I I T T 1 »- Stability limit —————————— ~ - - ~ 02» _ 0 . A . o 100 200 300 \ 400 500 600 cured/s) m...“ 0.1 , , , L j I Ta Stability limitI ' 093p . . . Winn”: _ A. l 0.x" ‘ ‘ ‘ -1 E I I “004— ,,-—2--~ . i . -. T1 \ x g... . , . ._ 0 1 l r ' » 'T-F-r.’ 0 100 200 300 400 500 600 (0 (rad/s) 03mm: Figure 3.12. Time constants of estimated parameters in uncertain plant by DSpace. The DSP board, sampling approximately at 13 KHz, was used to control the rotor along both bearing axes, independently. The specifications of the DSP are presented in Appendix AA. The sensor runout estimation was performed up to the second harmonics. Under this computation load, the DSP allowed us to store ten signals in real time. For the :2: axis we stored the estimated position, :2; current, 1,; and the DC component and estimated Fourier coefficients of the first harmonic: ("103, 61,, (31:. The estimated coefficients of the second harmonic were found to be negligible but could not be stored due to DSP limitations. The sensor signal, 2,, was regenerated from stored data :2, (10;, 613, and 511, using Eq.(2.16). Since the second harmonic coefficients were used in the computation of fit, regeneration of z, from 5: in the absence of these coefficients may lack some accuracy at relatively very low order. Our choice of acquiring signals for the y axis was exactly the same as that for the x axis. 40 The schematic of the bearing-rotor rig used for the experiment is shown in Figures A.1. and A.2. The rotor is arranged such that there are node points between a sensor- actuator pair for the mode shape closest to the controller bandwidth. In this case, we avoid the non-collocation effect of sensor and actuator in the lowest flexible mode as suggested by Figure AA and Table 7 .2. The calculation of the modes shape was performed using the finite element analysis software from ROMAC, program MODAL with the input shown in Appendix A.1. The critical speed map of the rotor, obtained using program CRTSP.2 of ROMAC, is shown in Figure A.5 The electromagnets were driven by switching power amplifiers, a product of Ad- vanced Motion Control, Operating with 1.6 KHz bandwidth. The Bode plot of the power amplifiers is shown in Figure A.10. To ensure negligible effects due to unbal- ance, the rotor was well balanced and spun at the relatively low rpm of 1200. This speed is 20 times less than the first critical speed of the rotor, and guarantees negligi- ble effects due to flexibility, which was not considered in our model. We used analog PD controllers shown in Appendix A.3 to stabilize the rotor in bearing B. Thus, only bearing A is the interest of our study, controlled by the DSP. The error and adap- tations gains of the adaptive controller were chosen as A = 400 s“, c = 1200 kg/s, and F = diag(1.0, 1.7,1.7, 1.5, 1.5) x 10‘7 m/N. In controller implementation, the derivative of the estimated position signal, 5:, was numerically computed by passing the signal :2 through the transfer function 25003/ (3 + 2500). This eliminates potential problems arising from infiltration of wide- band noise into the sensor signal. 3.8 Experimental Results We first present experimental results based on our best knowledge of the values of the plant parameters m, K 3, and Kc as provided in Table A.3. Therefore, in this case 41 we can assume that the control action and adaptation law are in accordance with Eqs.(3.2) and (3.3). The trajectory of the rotor’s estimated geometric center (5:, 37) and regenerated trajectory of the geometric center provided by the position sensors (3,, 31,) are shown in Figure 3.13. These trajectories indicate that while the sensors continue to provide geometric center positions corrupt with runout disturbance, their estimated values are stabilized to the origin with ASRC. It is seen from Figure 3.14 that the estimation of Fourier coefficients of runout is completed in 0.3 seconds. In the same time, sinusoidal variation in the control currents vanish in Figure 3.15. These zero steady-state control currents imply stabilization of rotor geometric center to the origin in the absence of mass unbalance. Indeed, we can verify from Eq.(3.1) that the rotor would become unstable if this was not the case. We ensured negligible mass unbalance effects in our experiments through rotor balancing and by spinning the rotor at low rpm. Knowing that the rotor geometric center has stabilized to the origin, runout disturbance was obtained from the Fourier coefficients in Figure 3.14. The trajectories of :53, y, in Figure 3.13 also provide this information. 0.8 l . A 0.4 r i g g o i‘ 1 I); )3 l 1 -0.4 —0.8 > Scale1V=SOum Figure 3.13. Trajectory of estimated rotor geometric center and regenerated sensor Signals 42 20 AA 10AAA 10 . 5" 0 4 E, o ' «1'1- -10. - A A A -10 A A A -1 A A A -0.2 0 0.2 0.4 —0.2 0 0.2 0.4 .2 0 0.2 0.4 Time (s) Time (3) Tune (3) 20 A A A A A A 10 A A A 10 g 0 g 0 lé in: -10 -2 a A A .30 A - - _ - - 1 -0.2 0 02 0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 Time (s) Time (s) Time (s) Figure 3.14. Trajectory of estimated Fourier coefficients of sensor runout 1,, (A) -o.2 - A -0.2 o 0.2 0.4 0.6 0.2 o 0.2 0.4 0.6 Time (s) Time (s) Figure 3.15. Trajectory of control currents 43 To demonstrate the robustness of ASRC to parameter uncertainty, we used control and adaptation laws in Eqs.(3.18) and (3.19). The parameter values 771, If” and Kc in the control law were chosen to be 25% larger than the values of m, K,, and Kc provided in Table A.3. The results obtained from our experiments are shown in Figures 3.16 to 3.18. These results indicate that runout is eliminated and the rotor geometric center is successfully stabilized to the origin despite error in the model used to construct the controller. 'without ASRC 0.8 with ASRC 0.8 A 0.4 A 0.4 s s a 0 2. 0 . 1>A l >1' 0.4, K «0.4 0.8. -o.s -1 -0.5 _o 0.5 1 -1 35 0 0.5 1 xrvom x'Nolt) ScaleiV=500m Figure 3.16. Trajectory of estimated rotor geometric center and regenerated sensor signals 3.9 Summary and Remarks This chapter presents a simple, yet robust, algorithm for adaptive compensation of sensor runout in active magnetic bearings. The algorithm is based on a rigid rotor model with no mass unbalance and assumes the angular speed of the rotor to be known and constant. Using powerful tools such as Lyapunov stability, persistence of excitation, and passivity, the algorithm is shown to guarantee perfect cancellation of runout harmonics and stabilization of the rotor geometric center. Through modeling, estimation, and cancellation of the DC component of runout, the algorithm generates 44 ‘M.";'LA 4O 60 20 20. 40. i O A A A .20i g o 320» l g , "5- ,5 40 i -20 0 -60 40 A A A -20 A A A -80 A A A -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 ;. Time (s) Time (s) Time (s) VI If 40 A - A 80 - ~ ~ 80 - . - I 20 , 50. 60 . A A40, A 40 S. o S. 5. ,5? .s' 20. ,5 20 '20 o 0 4% A A A -20 A -20 A A .2 0 02 0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 Time (s) Time (s) Time (s) Figure 3.17. Trajectory of estimated Fourier coefficients of sensor runout -o.2 0 0.2 0.4 0.6 02 o 0.2 0.4 0.6 Time (3) TM (8) Figure 3.18. Trajectory of control currents 45 the H1! algoriti mini. r The the are] gence r2 during ( ASRC t method values 9) The 6 Well as (A; geometrir HIOdeled mmiwnsa film its (filler m l the equivalent action of integral feedback for elimination of steady state errors. The algorithm is robust to significant variation in plant parameters that include rotor mass, magnetic stiffness, and actuator gain. The effect of controller gains to the system performance has been evaluated using the averaging method. This approximation method successfully predicts the conver- gence rate of estimated parameters. Thus, the averaging method can be very useful during control design, in particular during the selection of controller gains of the ASRC to achieve an optimum performance. Furthermore, by using the averaging method we can claim that the estimated parameters ultimately converge to the true values exponentially fast. The effectiveness of our algorithm is validated through numerical simulations, as well as experiments. We present experimental data that confirm stabilization of the geometric center of a rotor with negligible mass unbalance effects, even when the modeled plant parameters are quite different. Our algorithm can also be used for compensation of mass unbalance, but in such applications the rotor will be stabilized about its inertial center. The problem of rotor stabilization about the geometric center in the presence of both unbalance and rounout will be addressed in Chapter 5. 46 CHJ Sim1 Unb Mull 4.1 1 In [his ch, cOmponen mode] d” can lildiri, 4.2 C The ”War 1‘. CHAPTER 4 Simultaneous Sensor Runout and Unbalance Compensation (SRUC) Using Multiple Speeds 4.1 Introduction In this chapter we present a technique to compensate the effect of the first harmonic components of sensor runout and the unbalance in the single DOF magnetic bearing model described in Chapter 2. Using the technique in this chapter, theoretically, we can individually identify the harmonics components of the two disturbances. Simu- lation results are presented and implementation issues are also discussed. 4.2 Controller Design The equation of motion, in this case, as given in Eq.(2.3) is mm? = K32: + fc + fu (4.1) 47 The i Where “i (.r Adhrlid 3‘1. “fir“ Unbalance Sensor I l" Geometric gunout i 7 § Center I! " Fam‘bauk c _1_ I 'c + 1 X ‘9 _l - Law - Kc ' Kc ms? . K. r r. a A V in Magnetic Bearing System x. Adaptation Law Controller Figure 4.1. Block diagram of SRO and unbalance compensation framework The estimate of the unbalance force can be written as f. 5 we. (42) where A Y: é m0)2 [sin(wt) cos(wt)], (15,, é [,6 (HT (4.3) The terms 15 and (j are estimates of Fourier coefficients p and q respectively. We define the errors in the estimation as 5 = p - 13, E = q — (j and ~ «5.. é [:3 (7] (4.4) With the objective of converging :2: to zero, we propose the Lyapunov function candidate V = [(1 - A)mé2 + (WP-15+ Jfrjau] (4.5) (\DIl—t I‘u é diag<7pi 7q)1 I.‘u E 322x2 (46) where 7,, and 7., are positive constants. 48 In SiflfliS‘ and (2? Further 15 ("how Substitm di'flamics ii Mg the at X 0 JV} Wm? F . mm fhp n In this Multiple Speeds approach, the top and bottom bias currents 2'10, 2'20 are con- stants. Thus, the term A in Eq.(2.25) is constant. From the definition of Eqs.(2.13) and (2.23), we can establish YTer = 0, and WW", = o (4.7) Further, the control force f6 = — [Ksi + mAi‘: + cé + mw2 [fisin(wt) + écos(wt)]] (4.8) is chosen along with the adaptation laws 45 = F Ym 6 (4.9) 25,, = —r,,v., 6 (4.10) Substituting Eq.(4.8) into Eq.(4.1) and using Eqs.(2.20) and (4.4), the closed-loop dynamics can be described by mr'i = —K,YT$—mAi-cé+Yf$u (4.11) Using the relations 57} = if + c? from Eq.(2.16), and I? = 5: + A5: from Eq.(2.22), the above equation can be rewritten as m. 2 m3— KsYTg—cé+YI$u (4.12) Cbl From the relations in Eqs.(2.20), (2.25), (4.7) and (4.9), we can write 5 = {(755 + 23?qu + Y7}; (4.13) 49 and El The de hilizin definite? l. a) 02 35bu “hue DUE m t}. fightr ha = WE + Aé‘ (4.14) and Eq.(4.l2) becomes m(1 — A)é = —Y3,",¢3' — cé + YZ‘J, (4.15) The derivative of the Lyapunov function in Eq.(4.5) is V = m(1 — A)éé + JTP"$+ (ERIE... (4.16) Utilizing Eqs.(4.9), (4.10) and (4.15), we obtain V = —c€§2 S 0. Since V is positive definite and V is negative semi-definite, by Barbalat’s lemma we can conclude that V —) 0 and E —> 0. Taking the derivative of Eq.(4.15), we can show that e is bounded, => é is uniformly continuous, => e —+ 0. Therefore, from Eq.(4.15) ~ Y3}, 35 —>0 as t—)oo (4.17) 45.. where Yfmé [Y,7,‘, —Y,'f] (4.18) Due to the orthogonality of components, we can separate Ym into Ya, which contains the first harmonic components only, and YE, which contains the DC, second and higher harmonic components. Thus, the following conditions also hold: YZ‘J. — YZJ. a 0 (4.19) YEJE —> 0 (4.20) 50 wher It can 1 Where I ”191611); and 6) fl; 51$!th 00 and 0 Th 5 CUHI Wh‘fm where Y: 3 (K, + mw2) (sin(wt) cos(wt) )T (4.21) Y; é (K, Yg) (4.22) f (K, + m(2w)2) sin(2wt) V) (K, + m(2w)2) cos(2wt) 1 Y5 é (4.23) (K, + m(nw)2) sin(nwt) J K (K, + m(nw)2)cos(nwt) } ~ ~ ~ T ¢E 2 (cm 423) (4.24) It can be shown that there exist positive constants al, 02, and To, such that 1+7}, (121 Z YEYng Z ml (4.25) t where 12”-; is a 2n-1 identity matrix. Therefore, YE is a persistently exciting signal [13] as shown in Appendix 0.2. This implies from Eq.(4.20) that $1.; —> 0 as t —> 00. Therefore, using the definitions in Eqs.(2.14) and (4.24), we can conclude that do, 61,-, and 3.- for i = 2,. . . , n converge to their true values. On the other hand, in Eq.(4.19) the signal vector Yam 2 (Ya Y“ )T is not per- sistently exciting in the subspace of B“. This implies that the estimated parameters 3,, and Eu, instead of converging to zero, converge to a plane in the parameters’ space. This condition can be alternatively described by Eil = 115 and Z, = ”a (4.26) where 2 _A mw 14w) —— _‘K. + 2 (4.27) the : tion: am when matrh )Vi M01111 By operating at two different angular speeds, two values of p can be obtained; and the four unknowns a1, b1, p, and q can be determined by solving four algebraic equa- tions given by the expressions in Eq.(4.26). The results of this algebraic calculation are then used to update the adaptation laws to 3., = FaYaé + nacho — 43..) (4.28a) :53 = FEYEé (4.28b) $19 = _FuYué + 7721(43u "' 43a) (4.28C) where Pa 6 2112“, FE E 31(2"'1)X(2"‘1), no, 6 312”, and 17., 6 912”" are positive definite matrices. With the new adaptation laws in Eq.(4.28), the Lyapunov derivative in Eq.(4.16) becomes V 1' —Cé2 — ~Zna$a _ $577114; S 0 (429) Since V is positive definite and V is negative semi-definite, by Barbalat’s lemma we can conclude that V —> 0 and 6 —-) 0. Taking the derivative of Eq.(4.15), we can show that 6, $0, and a.) converge to zero as t -—) 00. Therefore, using Eqs.(2.14) and (4.4) we can show that 611, (31, 13, and (j converge to their true values. Furthermore, the convergence of 53 to zero depends on the persistent excitation of the signal vector YE which can be claimed using the same arguments as before. From Eq.(4.15), as before it implies that éis bounded, => 8? is uniformly continuous, => 6": —-> 0. Therefore, the condition in Eq.(4.20) still holds and the signal vector YE is persistently exciting => $5; —> 0 as t —) 0, which implies that do, 61,-, and I}, for z' = 2, . . . , n converge to their true values. In summary,all estimated parameters will converge to zero as t —> O. 52 4.3 Simulation Results A numerical simulation of the Multiple Speeds approach was performed using the parameters shown in Table 4.1. Though the controller was designed using a linearized model of the plant, we used the nonlinear plant model in Eqs.(2.1) and (2.2) to simulate the real situation. In the simulation, the SRO identification was arranged to Table 4.1. Parameters for Simulation Gains: A = 400 3“, c = 1200 kg/s F = diag(1.4, 3, 3,3, 3) x 10‘7 m/N I“u = diag(3, 3) x 10'5 m/N ICs: x(t = 0) = —100pm/s :i:(t = 0) = O ¢(t=0)=0, ¢u(t=0)=0 SRO: a0 = 2.5pm a1 = 18.35pm, bl = 4.92pm a2 =1.77pm, ()2 = 1.77pm Unbalance: p = 86.6 p m, q = 50.0 p m adapt up to the second harmonic. The bearing parameters were assumed to be the ones in our experimental hardware listed in Table A.3. In this approach iio = ifo, 2'20 = 2'30, Kc = K3, and K, = K} The rotor angular speed was initially set to 1500 rpm and then increased linearly at t = 0.43 to 2100 rpm within 0.2 s. The angular speed was held constant after t = 0.6 3. After computation of the true values of 0;, b1, p, and q had been accomplished, the adaptation law was switched from using Eqs.(4.9) and (4.10) to Eq.(4.28) at t = 13. We used [‘0 = diag(3, 3) x 10‘7 m/N, P5- = diag(1.4,3, 3) x 10‘7 m/N, 17,, = diag(10, 10) s“, and 1),, = diag(10, 10) s". It can be observed in Figures 4.2 and 4.3 that (“11, I31, 15, and q“ reach different steady states values for each angular speed while the DC and second harmonics components 53 of SRO, do, 612, 52 converge to their true values within 0.23 for each speed. After t = 1.5 s, all estimated parameters have converged to the true values. At this time, as shown in Figure 4.2 the rotor is stabilized to the origin. Unmodeled effects due to angular speed acceleration can be seen t between 0.43 and 0.6 s. We avoid further discussion on this topic but advice to change the angular speed slowly. 40 40 20' 20* g o E o. l l l i " a? -20 -20' .40 ' . i 1 * -40 ' ‘ t ‘ ' 0 0.5 1 1.5 0 0.5 1 1.5 time(s) time(s) Figure 4.2. Geometric center :1: and sensor signal with runout x, 4.4 Practical Implementation Issues In implementation, the Multiple Speeds approach suffers several major drawbacks. First, the results of the algebraic calculation is very sensitive to the value of [2. Un- certainty in m, K,, and w may easily yield to a large calculation error. Secondly, the approach practically requires two far away operating speeds in order to decouple the four algebraic equations. Third, the controller was derived using the Lyapunov method by assuming a constant angular speed. Thus, the stability during the chang- ing angular speed is not guaranteed. Moreover, in many applications changing the angular speed may not be desirable. 54 Dashed lines are the true values time (s) O ' m (s) 1.5 0 0.5 1 1.5 0 0.5 . 1 1.5 time (s) time (s) Figure 4.3. Estimated Fourier coefficients 55 4.5 Summary and Remarks We have presented the technique that can be used to identify the synchronous distur- bance of sensor runout and unbalance. The technique requires the rotor to be spun at two different angular speeds because only partial convergence of the estimated pa- rameters is achieved in Eq.(4.17). In the adaptive control, this problem is attributed to the lack of “sufficient richness” [31], which means that the regressor vector does not contain enough frequencies for the parameter error to converge to zero. In the partial convergence condition, estimated parameters may drift even with small ex- ternal disturbance [31]. It is generally known [25, 31, 36] that for linear systems, r sinusoids or frequencies in the regressor vector provides for the convergent estimation of (27‘ + 1) parameters. It can be observed that the regressor vector in Eq.(4.17) has 1' = n frequencies while the number of parameters to be estimated is (2n + 3). This observation suggests that additional perturbations with frequencies other than the frequencies already contained in the regressor vector will facilitate convergence of the parameters. 56 W" ..- i‘f’f‘fi CHAPTER 5 Simultaneous Sensor Runout and Unbalance Compensation (SRUC) via Bias Current Excitation (BCE) 5.1 Introduction In this section we present a new algorithm for rotor stabilization about its geometric center in the presence of both unbalance and sensor runout. The new algorithm effec— tively compensates the effects of both unbalance and runout at constant rotor speed and is based on the adaptive control framework presented in Chapter 4. The control framework is recognized to have a unique feature that allows us to directly perturb the parameter in the regressor vector to create a persistently exciting condition. This is achieved through variation of the bias currents in opposing electromagnetic coils in a manner that does not alter the equilibrium condition of the rotor. The method of bias current excitation is discussed in section 5.2. The mathematical foundation of the adaptive algorithm is laid in section 5.3 and robustness of the algorithm is inves- tigated in section 5.4. Simulation results are presented in sections 5.5 and 5.6, and experimental results are presented and discussed in section 5.7, 5.8 and 5.9. Section 5,10 provides a summary of the main contribution and concluding remarks. 57 5.2 Cons In Eq. the [0‘ and its “(1(fo 5.2 Adaptive Control with Bias Current Excita- tion Consider the magnetic bearing model shown in Eqs.(2.1) and (2.2) m4=F—mg+f.. gag/«i (5.1) (i;o_+xl)2 _ (ii-11):] (52) Without changing the angular speed, in order to generate persistence excitation, we F=k intend to perturb the top and bottom bias currents 2'10 and 2'20 by 61 and 62 in manner that does note alter the equilibrium condition of the rotor: 310 = 2:0 + (51, 7:20 = 230 + 62 (5.3) In Eq.(5.3) 2'10 and 2'30 are constant currents in the top and bottom coils. Therefore, the total magnetic force shown Eq.(5.2) becomes _ 110+61+I 2_ 230+62—I 2 F($,I,61,62)—k [(_—_l—x ) (——_-l+$ (5.4) and its linearization about a: = 0 and I = 0 yields 0F 6F F(1‘,I,61,62)zF(0,0,61,62)+ — 127+ — I (5.5) ‘9” 7:8 61 7:8 where ... 2 ., 2 F(0,0,51,52) = k [(11%) " (22%) ] (5-6) 6F 2k .. ... 5; = K.(61.62) = 73‘ [(2.0 + 61)2 + (220 + 62V] (5-7) 2:0 I 0 58 wl. ll 0; the ( I: 1m} 8F 2k ,, ... 57 723 = K461, 62) = 1—2 [’10 + 61+ 22. + 62] (58) Additionally, the bias force in Eq.(5.6) can also be approximated by linearization about 61 = O and 62 = 0. 0 6 6 F 6 6 F(0,0,61,62)zF(0)+ an ’0’ 1’ 2) 61+ a (”’0’ 1’ 2) 52 (5.9) 061 61:0 062 61:0 62:0 62:0 where k ., .. me) = ,—. (2.3 — 2.3) (5.10) 0F(0,0,61,62) __ 21:0 861 61:0 — (2 61 (5.11) 62:0 6F(0, 0, 61, 62) 2250 662 51:0 — - l2 62 (5.12) 62:0 It can be seen that if we choose 5, = (+05, (5.13) 220 the effect of the bias current excitation to the bias force in Eq.(5.9) is negligible F (0,0,61,62 = SEQ-61) Ft: F(0) (5.14) 20 Furthermore, the open loop stiffness and the actuator gain in Eqs.(5.7) and (5.8) can be expressed as function of 61 only and approximated by 8K, K, 6 m K; + 6 5.15 ( 1) 6,1 M . ( > BKC [C(61) x K; + 61 (5.16) 861 61:0 where K: é 2k (2'13 + 2‘53) /l3 K; é 2k (2'1. + 2225/12 (5.17) 59 “in let to 1. later 11 9X69) whf’fl and I lie bl 8K8 81:2"0 = ’ 5.18 5‘51 51:0 13 ( ) 8Kc 2k ( 2’0) = —--— 1 + 4— 5.19 661 61:0 12 250 ( ) For simplicity, we may choose the excitation as 61 = Asin(wet), we a 27rfe (5.20) where A is the amplitude of excitation, fe is the frequency of the excitation. The value of A should be chosen such that the above linearization is valid; A at about 10 to 15 % of 2'10 may be used. The effect of using several different values of A and fa is later discussed in sections 5.2 and 5.3. In summary we again obtain the same linearized model as in Eq.(2.3) with the exception that K , and Kc are now time varying. mi = st + KcI + fu (5.21) where a: - A Skifo K3 2 K3 + 6381110418”, £3 = TA (5.22) K; = K; + £6 sin(wet), {c 2 2:6—2/4 (1+ ;zl—Q) (5.23) 20 and fu is defined in Eq.(2.7). The proposed framework of SRUC using bias current excitation is illustrated by the block diagram in Figure 5.1 in which we define the feedback law as C I = —K1_ [K3 + mAf + (c + émAVz + deiu] (5.24) 60 Who 54mm»! names.) BiasCurrents i K.=K;+§,5,: E II II L ————— -..—J Sensor Unbalmce Jlu Geometric :mwt - '— + COM“ :Ax I 'c+ 1 X + + '1. 7 Kc ‘ "tel-K, _> d l MagneticBearlngSystem x. Adaptation Figure 5.1. Block diagram of SRUC via bias current excitation along with adaptation laws 37: FY... E (5.25) 5, = —I‘,Y.,é (5.26) where i, é, I‘ and Y... are defined in Eqs.(2.16), (2.22), (2.24) and (2.23) respectively. A and c are positive constants I‘u 24‘: diag(yp,7q), I}, E W” (5.27) where 7,, and 7,, are positive constants, and Y3" é mw2[sin(wt) cos(wt)], 43, £- [13 (HT (5.28) The terms 15 and (j are estimates of Fourier coefficients p and q. The parameter error vector for the unbalance component can be written as ~ 41-9-15 if]. 5310-1“). (7361-4 (5-29) 61 -...A‘J‘ ‘1: ~51 OI - K £3.— ('7‘ 35' PM The term :5 is obtained by taking the derivative of the measured signal it. The term A’can be derived from Eq.(2.25) knowing that YTI‘Ym = O, YTI‘Y = 0, and YTrY<3> = o. A = YTI‘Ym = K, 2 '7,- where K, = {,w, cos(w,t) (5.30) i=0 It should be noticed that 7,- is the diagonal components of the adaptation gain I‘; and the adaptation law in Eq.(5.25) has the regressor vector Ym that now varies according to K,. 5.3 Proof of Stability and Parameter Convergence The control and adaptation laws presented in Eqs.(5.24), ({refadpsrobce), and (5.26) stabilizes the closed-loop system and converges all estimated parameters to their true values. This can be proven in the following manner. Substituting Eq.(5.24) into Eq.(5.21) and using Eq.(2.20), the dynamics of the controlled rotor can be described by mi = ——K,YT$ — m A15 — (c + émAfi + YIE, (5.31) Using the relations If: 2 it + (7 from Eq.(2.16), and e 2 Pi: + A5: from Eq.(2.22), the above equation can be rewritten as me: = mg. K,YT$ _ (c + gnaw + 34.3., (5.32) From Eqs.(2.20) and (5.25), we can write if = YTJ + 237T; + YT; (5.33) 2 WE + 2YTeré + YTI‘Ymé + YTI‘Ymé (5.34) 62 d. U W Knowing YTI‘Ym --- 0 and Eq.(2.25) the above equation becomes El: YTEB + A2 + Aé‘ (5.35) Utilizing Eqs.(2.23) and (5.35), we can express Eq.(5.32) as m(1 — A)é = 4,7,5 + émAé _ cé + 1735., (5.36) Using the Lyapunov function candidate v = g [(1 — A)mé2 + (PT-13' + $51731] (5.37) we obtain V = m(1 — ma; — émAéz + $Tr-‘25 + $31135, (5.38) From Eqs.(5.25), (5.26) and (5.36), we get V = —082 S 0. Again, positive definite in V and negative semi-definite in V imply e —+ 0 as t —> oo. Knowing V is uniformly continuous, from Barbalat’s Lemma [13], we can claim that V —+ 0 => 6 —) 0. Taking derivative of Eq.(5.36), we can show that 5 is bounded => 5 is uniformly continuous => 5? -> 0. Therefore, similar to the conditions in Eqs.(4.19) and (4.20) except now we do not need to separate the components of Ym, from Eq.(5.36) we can argue that ~ Y3", j) —+0 as t—~>oo (5.39) 45.. where may}; —Y.T] (5.40) As shown in Appendix B.3, if we ¢ on then Y3,“ is a persistently exciting signal [13], 63 which can be verified from the condition t+To C121 2 YmuYZde Z ml (5.41) t where 01, 02, and T, are positive constants and I is a 2n+3 identity matrix, we can conclude that 3 —) 0 and 3,, —+ 0. Therefore (to, 61,-, and 5.- for i = 1, . . . , n and 13 and (1‘ all converge to their true values. From Eqs.(5.25), (5.26) and (5.36), the closed-loop dynamics can be written as :7: = -).2‘: + e (5.42a) m(1 — 71).“; = —Y,"’,",$ + émAé — c'é + Y3}, (5.42b) 3: I‘ Ym é (5.42c) 5,, = I“, Y, e (5.42d) Thus, we can make the following observations. ~ Lemma 5.1 The origin of the closed-loop system in Eq.(5.42), (12,345, I,,) E (0,0,0,0), is an asymptotically stable equilibrium point. Proof: The closed loop system in Eq.(5.42) is an interconnected system of the form :21 : f1(ti 21, 22) (5.433.) i2 = f2(t, 22) (5.43b) where 21 9- :2, and 22 é (6 5T 5: )T are the state variables of the two sub—systems. We know previously that Z2 = 0 is an asymptotically stable equilibrium of the sub- system in Eq.(5.43b). Also, 2'1 = f2(t, 21, 0) has an asymptotically stable equilibrium point at 21 = 0. This can be readily established from Eqs.(5.42b) and (5.43a). Us- 64 Th far thu ing the asymptotic stability theorem for cascaded systems [13], we conclude that (it, e, 3, a") E (0, 0, 0, 0), is an asymptotically stable equilibrium. Theorem 5.1 The coordinate (x,:t,$,$u) 5 (0,0,0, 0) is a stable equilibrium point for the closed loop system defined by Eqs.(5. 21 ) through (5.26). Proof: : Using Eqs.(2.19), (2.20), (2.25), (5.420) and (5.42b) , we can show that at (23,57, 5, 5,) _=_ (0,0,0,0), we have £7 = (YT3 + YTQE) = YTI‘Ymé —_- Aé = 23(3 + Ar) = A5 Also, at(z,3,5,$,) E (0,0,0,0), if: (55 - :3) = 3 = 0, since 0 < A < 1. From Eqs.(5.31), (5.42c), and (5.42d), it follows that (22,553,) = (0,0,0, 0). There- fore, (3,13,36,50 E (0, 0, 0,0) is an equilibrium point. The fact that ($2,555,) '5 (0, 0,0, 0) is asymptotically stable can now be deduced from: 1. The equilibrium point (15,6, 5, (5,) E (0, 0,0,0) is an asymptotically stable equi- librium. This fact follows from Lemma 5.1, 2. The transformation matrix P that maps (5:,6, a, an) to (23,2, 5, an) ( x) l i) ”f, z p i ...... 1 ¢ * <25 l3.) W ( 1 0 —YT 0 ) —,\ (l—A) —YT 0 Pa (5.45) O 0 E2n+l 0 K o o 0 E2 ) lb al)‘. 0U! par 5.4 [H il assu Eda; where E2n+1 E R<2n+llx<2fill and E2 E 332””) are the identity matrices, is well defined and upper bounded, and 3. The inverse transformation P-l exists, and H P’1 H is also upper bounded. The above theorem establishes that our adaptive controller, in conjunction with si- nusoidal excitation of the bias current, guarantees stabilization of the rotor geometric center in the presence of both SRO and unbalance. This achieved through individual identification and feedforward cancellation of both SRO and unbalance disturbances. 5.4 Robustness to Parameter Uncertainties The effect of uncertainties in plant parameters namely mass m, open-loop stiffness K,, and actuator gain Kc, to the performance of the SRUC-BCE algorithm is an- alyzed in this section by considering uncertainty in each parameter independently. Our particular interest is to evaluate the behavior of the rotor geometric center and parameter estimates. 5.4.1 Mass Uncertainty In implementation, we used a mass value of m 2 pm + m instead of the nominal value m, where m is the nominal mass and pm is the mass uncertainty. It is reasonable to assume that pm is constant and leI < m. The feedback law in Eq.(5.24) and the adaptation laws in Eqs.(5.25) and (5.26) become _L 1-» 8'21“ I — Kc [K,1:+m)\:c+ (c+ 2mA)e+ (1 + m)Y“¢"] (5.46) 5: Pine (.3, = —(1 + pfifl‘u Yué (5.47) Y", é K,Y - m? (5.48) Substituting Eqs.(2.19), (2.20) and (5.46) to Eq.(5.21) we obtain m5 — ma“: —K,YT$ — 771 A5 — (c + gazing + Yfau — 95x75, (5.49) From Eqs.(2.20) and (5.47), we can write 5' = WE + 2YT$ + YT; .. .. . _ (5.50) = YT¢ + A5 + A"? where A :9— YTer = Z 7.- (K, + m(iw)2) (5.51) i=0 A=52K327i (5.52) ‘=0 Therefore, in view of Eqs.(2.22), (5.48), and (5.50) we can rewrite the dynamics in Eq.(5.49) as a linear time-varying system with an input of ul M (t)5‘é + C(t);i: + K (t):r: = u1(t) (5.53) where “1“) g —Y77n$ + ngu — BTBWLYIQ’Su (5'54) M(t) .9.- m(1 — A) C(t) é c+ $77213. + (m — mA)/\ — mA The left hand side of Eq.(5.53) can always be recast to a state space representation H1(A(t),]B(t),C(t)) shown in Eq.(D.6) such that its input u; is defined by Eq.(5.54) and the output y1 = e knowing Eq.(2.22), The closed—loop system can be viewed as 67 a feedback configuration illustrated in Figure D.1. Notice that A(t), B(t), and C(t) are bounded since M (t), C(t), and K (t) can be seen as having nominal parameters with the addition of relatively small periodic perturbations. Using the adaptation laws in Eqs.(5.47) and (5.26) we can view the system in the feedback path H2 having 112 = e as the input and y2 = ng-beu + gin-Y3; $14 as the output. For a sufficiently small pm we can prove H2 is a passive system as shown by lemma D.2. in Appendix D. Assuming that H1 remains strictly passive and satisfies the conditions in definition D.3, the following observation can be made regarding the convergence of the estimated parameters. 1. As t ——> 00, by theorem D.1 we can claim that i, i, if: and E converges to zero. 2. From Eq.(5.53) ( v5. -YT ) ¢ pm . —> o (5.55) ¢u - (1+ Elgbu 3. The persistence of excitation in the regressor vector yields ~ ¢—>0 ¢u-(l+p—m')é§u_)0 m Therefore the estimate parameters of the SRO converge to the true values while m m+pm¢u the estimate parameters of the unbalance (bu —-> 4. Since :7: —> O and (b -—> 0, from Eqs.(2.19) and (2.20) we can argue that x —) O. In conclusion, the rotor geometric center is still stabilized to the origin despite of mass uncertainty. However, this conclusion is true only as long as H1 remains strictly passive. The passivity of H1 depends on the nominal plant parameters and the choice of controller gains besides the level of the uncertainty in the mass parameter. 68 Su Flo; 5.4.2 Magnetic Stiffness Uncertainty The uncertainty in the open-loop stiffness parameter is studied by assuming that the controller uses the K, 3 p, + K, where p, is the amount of the uncertainty. Due to the excitation on the bias currents, it is logical to assume that p8 is time varying and Ipslm < K,. The feedback law in Eq.(5.24) and the adaptation law for the SRO in Eq.(5.25) become 1 - . 1 . . I = ——-K— [K,i + mAi' + (c + -2-mA)é + qufiu] (5.56) 5 = r Y", a, (5.57) Y", 9: K,.Y — mi? (5.58) Note that the adaptation law due to the unbalance part is the same as in Eq.(5.26). Since ,5, 74 0 we obtain the following. 3?". = Ym + p,Y + p.Y (5.59) A = ’17"er = A + p.YTrv = A + p, :7.- (5.60) i=0 A = A+p,§:7. (5.61) i=0 Substituting Eqs.(2.20) and (5.56) to Eq.(5.21), we can write mfv' = —K,YT5 — m )5: — (c + émfkfi + Yfis', (5.62) From Eqs.(2.20) and (5.57), we can derive 2' = w; + 2YT$ + YT; (5.63) 69 = ’i'r'TJ + A5 + A5 (5.64) Using Eqs.(2.22), (5.48) and (5.50) we rewrite the dynamics in Eq.(5.62) to M(t)5i: + C(t)§: + K(t):5 = u1(t) (5.65) where M(t) é -Y3{.$+ ngu (5-66) M(t) 2“: m(1 — A) C(t) é c+§m5+m [(1— [AM—[A] K(t)é [Alma—:1]; [0 Using similar procedures as in the previous section we can use the feedback con- figuration Figure D.1 in which H1 is the state space representation defined by the left-hand side terms in Eq.(5.65) having ul shown in Eq.(5.66) as the input and E as the output knowing Eq.(2.22). The system H2 has the input 112 = e and the output y; = Ygd - deu. As shown by lemma D3 in Appendix D2, H2 is passive for a sufficiently small p,. Assuming the system H1 in the forward path remains strictly passive and satisfies the conditions in definition D.3, by theorem D.l we can argue that i,i,§3,é—+O as t—>oo which implies ~ (3?; 4(3) :1) —+0 (5.67) 45.. Since the regressor vector is PE, 5,5,, —+ 0. Therefore, all estimated parameters 70 ”1.0% .1 6‘57" ’!‘-| converge to their true values and the rotor geometric center converges to the origin. 5.4.3 Actuator Gain Uncertainty The controller uses Kc é pc + Kc where pc is the amount of the uncertainty in the actuator gain. We assume that Ipclmaz < K,. Due to the bias current excitation, )06 may vary at the frequency of the excitation. The feedback law in Eq.(5.24) becomes I = _Ki [K315 + mAi: + (c + émA)é + deu] (5.68) C and the adaptation laws shown in Eqs.(5.25) and (5.26) are still valid. Substituting Eq.(5.68) to Eq.(5.21) and using Eqs.(2.19), (2.20), (2.22) and (5.35) we obtain M (t);i": + C(t)ft + K (t):i: = u1(t) (5.69) where m(t) g 4’55 + Yfifi'u + {-iinéu (5.70) M(t) é:- m(1 — A) C(t) é (1— 7%) [c + émfk + mA] — m(A + AA) C K(t) s A [(1— %)(c+ émA) — mA] — 7% , Again, we can recast Eq.(5.69) such that the passivity argument can be used. In this case, the system H2 has the input u; = 6 and the output y2 = Yfi—Yfi, — fiEqubu. As before, we assume that H1 remains strictly passive. Knowing H2 is passive for a sufficiently small pc, shown by lemma DA in Appendix D, by theorem D.1 we can argue that —+O as t—>oo ”HI “HP HI ”(bl 71 which implies ( an‘ _Y: ) Pc . —+ 0 (5.71) ¢u "’( - krwbu ' Since the term 1% is function of time, the estimate parameter 43., is perturbed. Since the harmonics of the unbalance are coupled with the first harmonics of the SRO, the estimate parameter 03 is also perturbed and so is d: Knowing 5i: -) 0, from Eq.(2.19) we can conclude that uncertainty in actuator gain may cause a fluctuation in the rotor geometric center. 5.5 Simulation Results Simulation results are presented in Figures 5.2, 5.3, and 5.4 to demonstrate the ef- fectiveness of the SRUC-BCE when the plant parameters are known. As before, for simulation, we used the nonlinear plant model in Eqs.(2.l) and (2.2), parameters in Tables 4.1 and A.3, and the rotor angular speed of 1500 rpm. The SRO identification was performed for up to the second harmonic. In the simulation, we excite the bias currents at time interval 5 < t < 35 s using an amplitude of 02/1 and frequency of 10 Hz. As seen in Figure 5.2, for t < 5 s, the geometric center oscillates with constant amplitude due to sensor runout and unbalance. Without bias current excitation, Fig- ure 5.3 indicates that for t < 5 s the estimated parameters 61, b1, 13, and q converge to arbitrary values while do, (12, and 62 converge to the true values. After we turned on the bias current excitation, the parameters 61, b1, 15, and q converge to the true values within 303, which results to the stabilization of the geometric center about the origin shown in Figure 5.2. During the excitation, in Figure 5.3 we can observe a diminishing slight fluctuation of the parameters do, {12, and b2 about the true values. As shown in Figures 5.2 and 5.3, after all parameters have converge to the true values i.e. t > 35, turning off the excitation has a negligible effect. The geometric center 72 remains at the origin and the estimated parameters stay at the true values. Turning off the excitation at this time has an advantage of causing less effort in the power amplifier as indicated by Figure 5.4. The effect of plant parameters uncertainties to the performance of the SRUC-BCE is illustrated in Figures 5.5 to 5.7 assuming that each parameter is over-estimated by 30 %. The strictly passivity condition discussed in section 5.4 is still satisfied knowing that the closed-loop system is still stable in the simulation results. As seen in Figure (5.5), the mass uncertainty does not affect the convergence of the geometric center to the origin. However, it should be noticed that parameters 13 and 5 do not converge to the true values. The simulation results in Figure 5.6 show that the convergence of parameters to the true values is not affected by the uncertainty in the magnetic stiffness. In Figure 5.7 we can observe that the uncertainty in the actuator gain cause a relatively small and steady fluctuation in the rotor geometric center. It can be seen that the parameters (11, b1, 13, and q do not converge to the true values while other estimate parameters converge to the true values. x(m) 20 10 20 30 40 time (s) time (s) Figure 5.2. Geometric center at and sensor signal with runout z, 73 . . ...“! rr—«n M m) A L»: ; Dashed lines are the true values 2 1 . : < A a2( ml .5 M l -1oo ' A + A ' -20 A A A O 10 20 30 40 0 1O 20 30 40 time (s) time (s) Figure 5.3. Estimated Fourier coefficients 4 . . 4 . 3 1 « 3 r . $2 . $2 "-— -'-' .N 1 1 . o ' A s A o i A A A o 10 20 30 4o 0 10 20 30 40 time (s) time (5) Figure 5.4. Top and bottom bias currents 74 A 20 20 " E E o v o x X- .20 -20 40 -40 o 10 20 30 40 0 1o 20 30 40 time (s) time (s) 4 4 3 3 s 2—_— s 2-—_— v- N 1 1 c c o 10 20 30 4o 0 1o 20 30 40 time (s) time (s) 4 E - Dashed lines are the true values <33 2 20 time (s) time (s) 4 4 a 3 E E <6“ <5“ 1 1 “0 1o 20 30 4o "0 1o 20 so 40 time(s) time(s) 100 W ------------------------ )-—-\--—------------———-—-( 50 40 E o 520 D. (U .50 O -1cc 20 0 1o 20 30 40 o 10 20 30 40 time (s) time (s) Figure 5.5. Effect of mass uncertainty to SRUC-BCE 75 A {I‘m O 10 20 30 40 time (s) r time (s) time (s) P(m) A 2“ M 0 10 20 30 4O 0 0 10 20 30 40 time (s) 4 3 0 10 20 30 40 time (s) Dashed lines are the true values 0 10 20 30 4O 0 10 20 30 40 time (s) O (NM) 0 8 8 53 A B 0 1O 20 30 40 time (s) h C O 10 20 30 40 time (s) Figure 5.6. Effect of magnetic stiffness uncertainty to SRUC-BCE 20 E o x -20 0 20 40 time (s) 4 3 (A time (s) time (s) 0 10 20 30 40 time(s) 3 E V 2 <fl~ 1 O 0 10 20 30 40 time (s) E100 _____________________ J z. 0 , 1w 0 10 20 30 40 time (s) Figure 5.7. 77 0 10 2O 30 40 0 10 20 30 40 20 E v 0 x0 -20 "”0 1 20 40 time (s) 4 f 3 . s 2 —- N 1 e 0 10 20 30 40 time (s) Dashed lines are the true values 0 1 O 20 30 40 time (s) 4 3 E v 2 < .9" 1 “0 1o 20 30 40 time (s) 0 10 20 30 40 time (5) Effect of actuator gain uncertainty to SRUC-BCE 5.6 Simulation Studies of the Convergence Rate To approximate the convergence rate of the SRUC via bias current excitation, we utilize the averaging method that has been useful in the ASRC system as seen in Chapter 3. The detail of derivation of the averaging method for the SRUC-BCE is provided in Appendix 0.2. In simulation we used the parameters in Tables 4.1 and A.3, and the rotor angular speed of 1500 rpm. As seen in Figures 5.8 and 5.9 it is clear that a higher amplitude of bias current excitation can result to faster convergence rate. The simulations shown in Figures 5.8 and 5.9 also indicate that the averaging method can accurately predict the convergence rate of the original system when the excitation frequency fe is about one-half of the rotor frequency f. It was realized by simulations during our investigations that this occurrence is specific to the choice of adaptation gains. We investigate the effect of the frequency of bias current excitation to the conver- gence rate using 0 < fe < f. It can be seen in Figure 5.10 that for the rotor frequency of f = 25 Hz, the convergence rate is highest when the frequency of the bias current excitation is about 15 Hz. Therefore, there exists an Optimum excitation frequency that can result to the fastest convergence rate. However, the analytical solution to the optimum frequency is still an open problem. The effect of having more harmonics in the excitation can be investigated using the simulation results in Figure 5.11. It can be seen that having two harmonics in the excitation and imposing the same total excitation amplitude A = A1 + A2 may decrease the convergence rate of the adaptation. Thus, excitation with one harmonic only, in this case, is preferable. 78 l = 25 Hz lo = 0.5 l f 3 Solid lines = Original system 5 Dashed lines = Averaged system 4 o no 50 $0 to 55 time (s) Figure 5.8. Effect of excitation amplitude; f, = 0.5 f 60 .20 - . A -40 5 « E :L to. -60 .. d -80 ~ . « ,.- i r=25 Hz -1m - f°=8HZ " 5 _. - Solid lines = Original system -120 ' Dashed lines =Averaged system. 0 10 20 30 4O 50 60 time (s) Figure 5.9. Effect of excitation amplitude when fe 96 0.5 f 79 A-_-o.2Amp t=25Hz 0 10 20 30 40 50 time (s) Figure 5.10. Effect of excitation frequency I I T T T ,_—— co.- . gull" ’ ' ”re 5151-123” ' Solid lines have one harmonic in BCE With A = 0.2 Amp Dashed line has two harmonics in BCE With A1: A2 = 0.1 Amp, 42° """" ' ‘ ' ‘ te1 =15 142,192 =10 Hz 0 10 20 30 40 50 time (s) Figure 5.11. Effect the number of excitation harmonics 80 5.7 Experimental Procedure We performed experiments to validate the efficacy of our algorithm on synchronous runout and unbalance compensation. The schematic of our test rig is shown in Figure A.1. We used a steel rotor, 43.2 cm in length and 2.5 cm in diameter, with a balance disk for adding trial masses for unbalance. As shown in Appendix A.2, the rotor was quite rigid with the first flexible mode frequency at approximately 438 Hz, which was six times higher than the bandwidth of the closed-loop system. At one end, the rotor was connected to an absolute encoder using a bellows-type torsionally rigid coupling. Without introducing significant radial forces on the rotor, the coupling accommodates lateral misalignments. The encoder output was used in generating the feed-forward terms in our adaptive algorithm. At the other end, our rotor was connected to a motor via a flexible rubber coupling. An optical speed sensor was used to provide feedback to an analog controller unit to maintain the speed of the rotor at a constant desired value. The rotor was levitated using two bearings, A and B. Among them, both axes of bearing B were controlled using analog PD controllers shown in Appendix A.3. Although both axes of bearing A were computer-controlled, unbalance and runout was compensated in one of the axes. A PD controller was used to control the ro- tor along the other axis. The currents in the electromagnets of both bearings were driven by switching power amplifiers, operating with a bandwidth of 1.6 KHz. The physical parameters and operating conditions of bearing A are provided in Table A.3. The rotor mass enumerated in this table pertains to that of the whole rotor. We programmed our adaptive algorithm for synchronous runout and unbalance com- pensation in Matlab/SimulinkTM environment and downloaded it to a Digital Signal Processor (DSP) board, manufactured by dSPACETM. The sampling rate of the board was set at 13 KHz for on-line identification and control. A separate DSP board 81 sampling at 5 KHz, along with suitable analog circuits, was used for manual identifi- cation of runout. The manually identified runout was used to determine the position of the rotor geometric center from the sensor signal. Before implementation of our algorithm, we levitated the rotor using a PD con- troller and manually identified runout following the procedures outlined in Section 2.5. Although the first harmonic of runout was significant, higher harmonics of runout were negligible. On the basis of these results, we set it = 1 in our algorithm for esti- mation of runout. We performed experiments with the balance disk at two different locations, shown in Figure A.l . For each location, we implemented our algorithm three times. In the first experiment, Expt. 1, we did not introduce any unbalance but estimated the initial unbalance of the rotor. In line with our discussion in section 2.6, we added a trial mass in the second experiment, Expt. 2, and re—estimated unbalance. The third experiment, Expt. 3, was performed by introducing the trial mass at a different phase angle. Although a trial mass was added to the balance disk, unbalance was compen- sated only in bearing A. Since bearing B did not have unbalance compensation, we conducted two sets of experiments with the balance disk at two different locations to gain a high level of confidence in our results. We performed our experiments at constant rotor speed of 1500 rpm (25 Hz). We used the control law in Eq.(5.24) and the adaptation laws in Eqs.(5.25) and (5.26)with the following choice of gains A = 400 s“, c =1200 kg/s, r = diag (1.4, 3,3, 3,3) x 10'7 m/N l"u = diag (3, 3) x 10"5 m/N The derivative term i: in the control law was numerically computed using the transfer function 2500 s/(s + 2500). This eliminates potential problems arising from infiltration of wideband noise into the sensor signal. During adaptation, the top bias 82 current was excited using 61 = 0.2 sin(201rt) Amperes. The excitation frequency was therefore less than half of the rotor frequency. After estimated parameters reached steady state, adaptation and bias current excitation were both discontinued. In our algorithm, bias currents are excited concurrently with estimation. This eliminates drift in the estimated Fourier coefficients of unbalance and the first har- monics of runout in the absence of persistent excitation. Compared to standard implementation, our algorithm requires an extra D/ A channel for every axis of im- plementation since both coils of each axis are excited independently. 5.8 Experimental Results We first performed experiments with the balance disk located at rotor midspan. The results are provided in Tables 5.1 and 5.2. The first column of data in Tables 5.1 and 5.2 (Expt. 0) pertains to the manually identified values of sensor runout. This data includes the DC component and the first harmonics of runout only since second and higher harmonics were found to be negligible. The phase of the first harmonic was set to zero through encoder calibration. The second column of data (Expt. 1) corresponds to our experiment performed without a trial mass. This data includes the DC component and first harmonics of runout, and the harmonics of initial unbalance of the rotor. The two columns, labeled Expt. 2 and Expts. 1, 2 in Table 5.1, of data pertain to our experiment with the addition of a trial mass of eccentricity 61 = 91.1 pm. The eccentricity value was computed from the mass of the trial weight, which was 10 gms, the radial distance of the trial weight, 4.45 cms, and the total mass of the rotor, which Was 4.87 kgs, in the following manner: 67* = {0.01 / (4.87+0.01)} x 0.04445 = 91.1um. The phase of the trial weight was 07 = —56°. The Expt. 2 column provides experimentally obtained values of runout and unbal- 83 ance. The column marked Expt. 1, 2 provides computed values of unbalance solely due to the trial mass. The computed values were obtained in line with our discussion in section 2.6, as follows CT 407‘ = (63 £03 — 6U 40(1) = (86.74—61.30 - 12.94—91.80) = 75.94—56.40 The last two columns of data in Table 5.2 pertain to experimental results obtained with the same trial mass, located at the same radial distance, but at the new phase angle OT = —146°. Among these two columns, the left column provides Fourier coefficients of runout and unbalance obtained through experiments (Expt. 3). The right column provides computed values of unbalance solely due to the trial mass. This data was obtained as follows 67‘ [07' = (63 £03 - ((1 40(1) 2 (82.1 1—14290 - 12.91—91.80) = 7474-15060 The time history of the rotor geometric center position, 2:, and sensor signal, 2,, are provided in Figure 5.12 for one of the experiments, Expt. 2. The geometric center position was evaluated from the sensor signal through cancellation of manually identified runout. The time scale in Figure 5.12 is divided into three distinct regions: (a) t S 0, where runout and unbalance were not compensated, (b) 0 S t S 300, where runout and unbalance were adaptively estimated and compensated, and (c) t 2 300, where runout and unbalance were completely compensated and bias current excitation terminated. Due to the relatively long duration of the experiment, we acquired data over the sub-intervals -—0.2 S t g 0.3, 120.0 g t 3 120.5, and 299.6 3 t 5 300.2. The time trajectories of the estimated Fourier coefficients of runout and unbalance are shown in Figure 5.13, with final values of the coefficients shown with dashed lines. The sensor runout coefficients show larger fluctuations than those of unbalance. This can be primarily attributed to the difference in scale of the plots. 84 Table 5.1. Experimental results with the balance disk located at the rotor midspan: part (a) initial unbalance estimation S plus of r5333: manual trial weight trial weight and SRO initial GT = 91.1 5T = 91.1 unbalance identification unbalance 97' = -56 97‘ = -56 Expt. 0 Expt. 1 Expt. 2 Expts. 1, 2 £10 0.0 -0.5 -0.1 — 61 40.0 38.5 39.0 - b1 0.0 -0.6 0.2 - Mag, A1 40.0 38.5 39.0 — p — 12.9 76.1 — q — -04 41.6 — Mag, 5,, — 12.9 86.7 75.9 Phase, 0., — -91.8 -61.3 -56.4 Note: units of measurement are pm and degrees up 111) X, (ii In) time (s) Figure 5.12. Time history of geometric center :1: and position sensor signal at, 85 Table 5.2. Experimental results with the balance disk located at the rotor midspan: part (b) initial unbalance estimation plus of $3333: manual trial weight trial weight and SRO initial 67 = 91.1 67 = 91.1 unbalance identification unbalance 97‘ = -146 or = -146 Expt. 0 Expt. 1 Expt. 3 Expts. 1, 3 £10 0.0 -0.5 -4.4 — 51 40.0 38.5 37.2 — b1 0.0 -O.6 -0.3 — Mag, A1 40.0 38.5 37.2 — Phase, 01 0.0 1.0 -0.4 — p — 12.9 49.5 — q — -0.4 -65.5 — Mag, 6,, — 12.9 82.1 74.7 Phase, 0,, — -91.8 -l42.9 —150.6 Note: units of measurement are pm and degrees 86 4O 60 80 100 120 time(s) Figure 5.13. Time trace of estimated Fourier coefficients for Expt. 2 in Table 5.1 87 0 20 Dashed lines = steady state values 4O 60 80 100 120 time(s) 5.9 Interpretation of Experimental Results First consider the Fourier coefficients of sensor runout presented in Tables 5.1 and 5.2. These values, identified by our algorithm, are very similar for Expts. 1, 2, and 3, performed with varying degrees of unbalance. Furthermore, the identified coefficients closely match the manually identified values of runout, Expt. 0. We can therefore claim that sensor runout has been correctly identified. Next, investigate the estimated magnitude and phase of unbalance due to the trial mass alone, for the two cases in Tables 5.1 and 5.2. The estimated magnitudes, 75.9 pm and 74.7 um, are similar and therefore consistent, and their respective phases, —56.4° and —150.6°, compare very well with the true values, —56° and —146°, re- spectively. The average value of the estimated magnitudes of unbalance, 75.3 pm, is approximately 82% of the trial mass eccentricity of 91.1 um, added at rotor midspan. Other than this percentage factor, which will be discussed later, the above data indi- cates that our algorithm determines the phase of unbalance accurately and provides consistent estimates for eccentricity over repeated trials. Now consider the plot of the rotor geometric center position, 1:, in Figure 5.12. Although this plot specifically pertains to Expt. 2, it is representative of the general behavior of the rotor geometric center with our algorithm. It can be seen from Figure 5.12 that the geometric center initially fluctuates about a nonzero mean value but this fluctuation is virtually eliminated with our algorithm. The stabilization of the rotor geometric center to the origin convinces us that both mass unbalance and sensor runout have been correctly estimated and compensated. A second set of experiments were performed with the balance disk closer to Bearing B, as shown in Figure A.1. The results of these experiments, provided in Tables 5.3 and 5.4, are very consistent with the results in Tables 5.1 and 5.2. Specifically, the estimated values of sensor runout are very similar to the values in Table 5.1 and 88 5.2, and closely match the manually identified values. The magnitude of estimated unbalance is consistent over repeated trials and the phase of unbalance closely match the phase of the trial mass for both experiments. The plots of the rotor geometric center, not shown here due to their similarity with the plot in Figure 5.12, also indicate geometric center stabilization. The ratio between the average magnitude of estimated unbalance and trial mass eccentricity is 0.51 for the experimental results in Table 3. Although this value is less than the 0.82 ratio obtained with the balance 'disk at rotor midspan, as one would expect, both values are higher than expected. An explanation of the higher values would require further analysis that takes into consideration: (a) characteristics of the support provided by Bearing B under PD control, in the absence of unbalance and runout compensation, (b) performance of our adaptive algorithm, formally de- veloped for a single degree-of-freedom rotor with collocated sensor and actuator, in our experimental test-rig, and (c) additional stiffness and unbalance introduced by the couplers at the two ends of the rotor. It will however not be worthwhile pursuing such analysis since our adaptive algorithm will have to be extended to a complete rotor model before it can be implemented in any industrial rig. Our experimental results amply demonstrate the basic feasibility of our algorithm but significant work remains to be done before it can be adopted by commercial vendors. We conclude this section with our comments on the time taken for synchronous disturbance compensation. It can be seen from Figures 5.2 and 5.12 that compensa- tion in simulation requires a shorter time than compensation in experiments. This can be attributed to the fact that only one of the bearings in our experimental setup was compensating the disturbances. The other bearing, in the absence of unbalance and runout compensation, acted as a source of additional periodic disturbances. We expect the time to reduce significantly when both bearings compensate for distur- bances, and amplitude and frequency of bias current excitation are chosen optimally. 89 The time taken for compensation in our experiments, nevertheless, should not be construed as significant. This time will be required during rotor spinup only. During steady state operation, adaptation and bias current excitation will be implemented for a few seconds periodically to account for possible drift in runout and unbalance. Depending upon the type of application, periodic implementation may occur few times every hour to once every few hours. Table 5.3. EXperimental results with the balance disk located closer to bearing B: part (a) initial unbalance estimation Sensor - plus. - 0f . run out manual trial welght trial weight and SRO initial 67 = 91.1 (T = 91.1 unbalance identification unbalance 9'1“ = -56 9T = -50 Expt. 0 Expt. 1 Expt. 2 Expts. 1, 2 £10 0.0 -0.2 -1.3 — til 40.0 40.0 40.0 — b1 0.0 -0.7 1.5 — Mag, A1 40.0 40.0 40.0 — Phase, 91 0.0 -1.0 2.1 — 13 — 10.9 45.0 ~— 5 — 4.1 36.8 - Mag, 6,, -— 11.6 58.2 47.3 Phase, 0,, - -69.4 -50.7 -46.2 Note: units of measurement are pm and degrees 90 Table 5.4. Experimental results with the balance disk located closer to bearing B: part 0)) initial unbalance estimation Sensor - plus. - Of- run out manual trial weight trial weight and SRO initial CT = 91-1 6T = 91.1 unbalance identification unbalance 9T = -146 9T = 40 Expt. 0 Expt. 1 Expt. 3 Expts. 1, 3 (“to 0.0 -0.2 -1.1 — (11 40.0 40.0 38.6 - b1 0.0 -0.7 0.9 — Mag, A1 40.0 40.0 38.7 - Phase, 61 0.0 -1.0 1.4 — p — 10.9 -21.2 — q“ — 4.1 36.5 - Mag, 6,, — 11.6 42.2 45.6 Phase, 0,, — -69.4 30.2 44.7 Note: units of measurement are pm and degrees 91 5.10 Summary and Remarks A new adaptive algorithm has been designed to identify the harmonic components of sensor runout and unbalance at constant rotor speed. The algorithm relies on persistency of excitation generated by methodical bias current excitation. The bias current excitation amplitude is small and carried out in a manner that does not alter the equilibrium condition of the rotor. The algorithm enables us to stabilize the rotor geometric center to the origin in the presence of simultaneous sensor runout and unbalance. After the geometric center has been stabilized to the origin, one may stop the bias current excitation without causing problems to the closed-loop system. If the harmonic components of the disturbances drift, the bias current excitation procedure may be invoked for a brief duration to identify the new values of the harmonics. Depending upon the application, bias current excitation may be invoked few times an hour to once every few hours. The efficacy of our algorithm was demonstrated both through simulations as well as experiments. The effect of plant parameter uncertainties on the performance of the algorithm has been investigated. In general, as shown by simulations, the algorithm can with- stand to small uncertainties in the plant parameters . It was observed that the strict passivity condition of II; should be evaluated case by case. The passivity of H1 is important for ensuring closed-loop system stability and convergence of estimated pa- rameters to their true values. Further investigation is needed to show the interaction between the nominal plant parameters and choice of gains in the algorithm to meet the strict passivity condition in H1. Furthermore, it is realized that more work should be done to reduce the effect of the uncertainty in the actuator gain. In Chapter 3, we had shown that the averaging method was a good analytical tool to predict the convergence rate of parameters in ASRC. Unfortunately, this method is not directly applicable to the SRUC algorithm and further research is required to 92 predict convergence rates of parameters. Our simulations show that the optimum adaptation speed can be achieved when the frequency of excitation is between 50 to 75 % of the rotor frequency. However, further studies are needed to understand the effect of nominal plant parameters and controller gains on the optimum frequency of excitation. Our simulation results also show that for a fixed total amplitude of bias current excitation, faster convergence is achieved with one harmonic of excitation rather than multiple harmonics. This should however be verified analytically. 93 CHAPTER 6 Extension to MIMO Systems 6.1 Introduction In this section we investigate the effectiveness of ASRC and SRUC-BCE algorithms in MIMO magnetic bearing systems. By assuming the gyroscopic effect to the system to be negligible, we decouple the 4-DOG MIMO system into two identical 2-DOF systems. We then analyze the 2—DOF dynamics of the rotor being controlled by the MIMO versions of the adaptive algorithms. The stability analysis of the 2-DOF systems is provided. The convergence of estimated parameters is studied through simulations. 6.2 MIMO Model of Magnetic Bearing with Rigid Rotor A free body diagram of the rigid rotor is shown in Figure 6.1. The forces F,- for j = 1, ..., 4 are provided by the two radial bearings, which are at distances L from the rotor center of geometric. 0 — X YZ is the inertial frame fixed in the space, 2:, y are the displacements of the center of geometric along the X and Y directions. Both X and Y axes are inclined at 45° with the horizontal. The rotor is assumed 94 constrained along the Z directions. The a: and y displacements are given with respect to the inertial frame. It is assumed in this analysis that the nominal gaps and forces in the bearings are in the same radial plane; thus, the effects of non-collocation are ignored. 0 and ill are angles of rotation about the X and the Y axis respectively. These angles are assumed to be small. We also assume that the center of mass does not coincide with the geometric center. However, the axis of rotation is still aligned with the rotor’s major principal axis; therefore, the dynamic unbalance is neglected and only static unbalance is assumed. In Figure 6.1, the static unbalance e is the Figure 6.1. Freebody diagram of a magnetically levitated rigid-rotor distance between the rotor’s geometric center 0 and its center of mass 0m on the x—y plane. The resulting dynamics equations, in terms of 2:, y, 0, and w, are given by mi} 2 F; + F; + "1.4025 cos(wt + 0,,) — mg/\/2 (6.1a) my 2 F3 + E, + maize sin(wt + 0,) — mg/\/2 (6.1b) 175' = (F1 -— F2)L — Iain]; (6.1c) 95 175 = (F, — F4)L — 1,520 (6.1d) where m is the total mass of the rotor, I, and IT are the axial and transverse mass moment of inertias, and the forces i-+i- 2 i-—i- 2 szk[(—E’l——J—) -(fl”———’)] j=1,...,4 (6.2) l- 2:,- l+ 1:,- Linearization of the external forces about 2:,- = 0 and i, = 0 gives F) = FOj "l' ch'lj + Kijj (6.3) Since the bearings are identical and symmetrical, we may write 2'10,- = 210 and izoj = izo. Also, we can obtain ch = K, and K,,- = K, that are defined by Eqs.(2.6) and (2.5). Assuming the static unbalance e is small compared to L, the following relations can be obtained from geometry: x1:a:+Ll9, xgzx—LO, x3=y+Lw, x4=y—Li/) (6.4) and we can write the equation of motion about a: = 0, y = 0, 0 = 0, and 1/1 = O as 5'1 i1 331 :31 :ii i :1: 0': m 2 = x, 2 +5, 2 +_<3_ 2 +5, (6.5) 5133 i3 $3 1'33 (1'34 54 $4 $4 where K 0 K, 1 + -'—"—L—’ —K, 1— m2 5. a —“ , and a ( ’* 2) ( “2‘ ) (6.6) 0 K. — .(1—";’,:) K.(1+fl,£-) 96 are the open loop stiffness matrices, K o K,1+'"L2 —K,l—E£2- 1.2% —" , and 5.2 ( ”3 ( '1) (6.7) o 1:, —K,(1 — '3; ) K,(1+ 1%) are the actuator gain matrices, . M 1921 . 0 0 _21T 211~ 0 0 M Jafl Q g 211‘ 217‘ (68) no no. 21T _211 0 0 19.01 I w .. —2IT 217‘ O 0 . is the gyroscopic matrix, and cos(wt + 4)..) cos wt + (1)., fu = mw2e ( ) (6.9) sin(wt + (bu) sin(wt + 45“) is the unbalance force vector. In practice, the gyroscopic effect is likely to be very small. It becomes important only when the rotor spins at extremely high speed, which is not the interest of our study. As the results, the coupled 4—DOF AMB system model can be decoupled as two 2-DOF system models. Let x = [x1 2:2]T = [23A $3]T, 1 2 [2'1 i2]T = [iA iB]T, then the equation of motion on one plane can be represented by mi L1 + _ng + 13¢” (6-10) 97 where Xfémwz sin(wt) cos(wt) (6.11) sin(wt) cos(wt) It should be noticed that the unbalance disturbance, (bu, affects both axis at equal magnitude. Furthermore, the SRO disturbance vector is now represented by d 2 (dA dB)T (6.12) d,- 9—- 00]“ + 20'? sin(z'wt) + bij cos(z'wt) j = A, B (6.13) i=1 Using YT, defined in Eqs.(2.13), we can also write d = 1%, d e 822 (6.14) where YT 0 1T é YT , X’I‘ E 322x(4n+2), yT E R2n+l (6.15) 0 A ¢A 4n+2 2n+1 = ¢ , e§R , ¢A,¢Be§R (6.16) B and (15 is defined by Eq.(2.14) for j = A and B. The following vectored equations used in the 2-DOF system model are generalized from the definitions in the single-DOF system model provided in section 2.5. Position sensor signal: x, = x + d, x, E 3? (6.17) Estimate of geometric position: i 3 x, — d, equivalently x = x + d, i E 322 (6.18) 98 Estimate of SRO disturbance: a é (ciA ciB)T, E1 6 ER" where n 62,- g £sz + 251.5 sin(iwt) + (3,3 cos(z'wt) j = A, B i=1 SRO disturbance parameter error: Regressor matrix: I; g KsXT _ "2:7", X; E 32(4n+2)x2 Adaptation gain matrix for the SRO components: |"‘J “D P E a(4n+2)x(4n+2) F E m(2n+1)x(2n+l) 1 (6.19) (6.20) (6.21) (6.22) (6.23) (6.24) where I‘ is defined in Eq.(2.24). The components of I‘ are chosen such that 0 < A < I where 99 (6.25) 2 " L AAA = A33 = Z ’7,- [K,(1 + -"—;;-) + m(z'w)2] (6.26) i=0 mL2 A... = - 27.1w — 7;) (6.27) i=0 6.3 ASRC for 2-DOF systems 6.3.1 Controller Design For AMB systems with negligible unbalance 155,, = O, Eq.(6.10) becomes mii = 1ch + K x (6.28) -—8 The following feedback control action and the adaptation law can be directly extended from the ASRC in the 3180 case shown by Eqs.(3.2) and (3.3). I = {$1 [Lac + m)“: + cé] (6.29) ~ (P 2 Elm E (6.30) The proof of stability for the 2-DOF system model is similar the proof of the ASRC for single-DOF system model in section 3.3. Briefly, we can summarize the proof below. Substituting Eq.(6.29) to Eq.(6.28) we obtain miiz—Kd—mAx—cé (6.31) ——8 Using the relation 1:: = i + d from Eq.(6.18), and 6 2 5‘2 + A)? from Eq.(6.22), the 100 above equation can be rewritten as mé=md—_Ig,d—cé (6.32) The term 6 can be derived from Eq.(6.21). YT (6.33) T: Y é + Tgxmé (6.34) In the above equation, the adaptation law in Eq.(6.30) is used to derive 5. Knowing XTI‘X", = 0, XTI‘Xm = 0 and Eq.(6.25), we can write ~ (1 = {(7713 + _Aja (6.35) Therefore using Eqs.(6.23) and(6.35) we can express Eq.(6.32) as ~ m (I —- Q)é :2 —XT — cé (6.36) m Using the Lyapunov function candidate V = éméTU — A) E + $571463, 0 < A < I (6.37) we can find its derivative along the trajectory of Eq.(6.36) as 0 ~ V=méT(1—g_)é+£f>TL-1<1> = -méT (:56 + cé) + 3137‘ E" ci> (6.38) 101 The use of the adaptation law in Eq.(6.30) results to V = —céTé _<_ 0 (639) Using Barbalat’s lemma we can conclude that é -—> 0, 5i —) 0 and 5 is bounded. Furthermore, ones can find that the regressor vector 1; e 322x(4"+2) does not meet the persistency of excitation condition. The analytical proof is a straightforward extension to higher dimensions of its counterpart in Eq.(3.14), which is not provided in this thesis since it is beyond the scope of this thesis. However, the convergence of estimated parameters and the performance of the closed-loop system are investigated by simulations in the next section. Table 6.1. Simulation parameters for 2-DOF magnetic bearing systems Angular speed: w = 21r x 25 rad/s Total rotor mass: m = 4.86 kg Distance from rotor center to sensor-actuator: L = 0.072 m Transverse moment of inertia : IT = 0.064 kg/m2 Sensor runout in 2:; axis (bearing A): a0 = 2.5 pm a1 = 18.35 pm; bl = 4.92 pm; a,=b,-=0 f01'222 Sensor runout in 1:2 axis (bearing B): do = -1 pm 01 = —12.07 pm; bl = 3.24 pm; a.-=b.-=0 fori22 Error gains: A = 400 s“; c 21200 kg/s Plant initial conditions: x(t = 0) = — 0.1.x 10’3 m; 0(t = 0) = 3 x 10“ rad :i:(t=0)=0; 0(t=0)=0 102 6.3.2 Simulation Results For simulations we use the bearing parameters shown in Table A.3 with the total mass of the rotor, m = 4.86 kg. The controller parameters and the assumed SRO disturbance are shown in Table 6.1. The adaptation is performed up to the third harmonic using the adaptation gain matrix I‘ = diag( 1, 2, 2, 1, 1) x 10‘7 m / N for both axes as shown in Eq. (6.24). We can verify that 0.4385 —0.1467 A = —0.1467 0.4385 which has eigenvalues of 0.585 and 0.292; thus the condition 0 < A < I is satisfied. All estimated parameters in both axes were initialized zero. The simulation result in Figure 6.2 shows that at the steady state the geometric centers of both axes, am and 2:3, become constant; thus the controller is able to remove the periodic disturbance due to the SRO. However, 16,; and 2:3 converge to positions other than the origin. As seen in Figure 6.2 the DC parameters do not converge to the true values while the parameters of higher harmonics seen in Figure 6.3 converge to the true values. Therefore, generalizing the ASRC scheme from SISO to the MIMO system model can not guarantee the convergence of all estimated parameters to the true values. In section 6.5, we will present the effect of bias current excitation on ASRC in 2-DOF systems to solve this problem. 6.4 SRUC-BCE for 2-DOF Systems 6.4.1 Controller Design In the presence of SRO and unbalance, the system dynamics are given by Eq.(6.10). The following feedback and adaptation laws, generalized from Eqs.(5.24) through 103 x1 axis. bearing A X2 axis, bearingB 100 - f . . 100 ... 5° A 50 S 0 V33 0 1 X x -50 -50 «:3 L «11° F r- ....................... time (s) Dashed lines = true values Figure 6.2. Geometric position, sensor signal, and DC component error of ASRC for 2—DOF system model 104 x1 axis, bearing A :42 axis, bearing B fl Y '7 T A m L - " - - - - A20 P g 510” ---- __-_ (‘0’ m as" O <13" 0 0 1 2 3 4 5 0 1 2 3 4 time (s) time (s) 20 r 20 - A 10 Em 5 _ 0 <9" _____ _ _ _ <51" 0 0 4 + ‘ ‘ -10 0 1 2 3 4 5 1 2 3 4 time (s) time (s) A o — - k A o _ k 5 -5 . 5.. .. < <66“I .10 4 -10 1 2 3 4 5 O 1 2 3 4 time (s) time(s) ._' 5 ‘ r .5) .5 a: «a:3 <13" .10 .10 . 1 - - -15 . . . - 0 1 2 3 4 5 0 1 2 3 4 time (s) time (s) Dashed lines = true values Figure 6.3. Higher harmonics error of ASRC for 2-DOF system model 105 (5.26), are considered. I = —-£;1 [Lac + mAx + é-mA + cé + 13:6,] (6.40) ~ 4 = Exmé 33,, = 4,1,5 (6.41) In each axis, we may excite the bias currents using the procedure outlined in Eqs.(5.3) through (5.20). We may also assume that the amplitude, frequency, and phase of the excitation are the same on both axes. The following is the proof of stability that closely follows its counterpart in the 1-DOF case described in section 5.3. m5: = —x,1" — m H: — (c + é—mA)é — 13:55., (6.42) Using the relations i": = 52 + d from Eq.(6.18), and 6 = STE + xi from Eq.(6.22), the above equation can be rewritten as . 2 ~ 1 - ~ m e = md — L_T — (c + §mA)é - X3451. (6.43) From Eqs.(6.21) and (6.41), we can write T313 (6.44) is + YTLjfim 6 + TLYmé (6.45) 5=v<1>+'é+2\_é (6.46) 106 Utilizing Eqs.(6.23) and (6.46), we can expressed Eq.(6.43) as m(I — A): = 4&6 + émAé — cé — 156', (6.47) Using the Lyapunov function candidate V = g [5% — Ama- + EFL-15> + Efrfau] (6.48) we can write T ° é + EVE—14> + 65mg. (6.49) 7‘5 g 0 (6.50) Again, using Barbalat’s lemma we conclude that i —> 0, 6 —> 0, and (I) and an are bounded. Moreover, ones can find that the regressor vector [1; — XI] 6 szl‘m“) satisfies the persistency of excitation condition. The analytical proof is a straight- forward extension to higher dimensions of its counterpart in Eq.(5.40), which is not provided in this thesis since it is beyond the scope of this thesis. The effectiveness of SRUC-BCE for 2-DOF systems is further investigated by simulations. 6.4.2 Simulation Results Simulations of the SRUC-BCE for 2-DOF model were performed using the parame- ters shown in Tables A.3 and 6.1. The adaptive controller considers up the second harmonic of the SRO and uses F = diag(l, 2, 2) x 10‘7 m/N for both axes. We can 107 verify that the condition 0 < A < I is satisfied knowing 0.2869 —0.11 A = -0.11 0.1869 and the eigenvalues of A are 0.397 and 0.179. For the unbalance identification, the adaptation gain I‘u = diag(3, 3) x 10‘5 m/N was used. We assume that the true values of the unbalance: p = 86.6 p m, and q = 50.0 p m. We initialized all estimated parameters to zero and excite the bias currents in both axes after t = 53. The amplitude of bias current excitation for both axes were set to 0.22 A with the frequency of 10 Hz. The simulation result in Figure 6.4 shows that the controller is able to remove the periodic disturbance due to the SRO. However, the geometric centers of both axes converge to positions other than the origin when t < 5. After the bias current excitation, as seen in Figures (6.4) and (6.5), all estimated parameters converge to the true values within 250 s. 6.5 ASRC-BCE for 2-DOF systems By removing all variables related to the unbalance, the controller proposed in section 6.4 may be reduced to obtain an ASRC with BCE. Therefore, assuming that the unbalance force is negligible we may use following controller together with bias current excitation. 1 . I = ——_I_{_;1 flax + mAx + -2-mA + cé] (6.51) 6 = 113.... 6 (6.52) The proof of stability for this system is trivial. It can be easily established from section 6.4 by removing all variables related to unbalance. Therefore, we can also conclude 108 x1 axis. bearing A x2 axis. bearing 8 100 _ - . . - 100 , x1 A ( m) T x29 ( m) 0' ~50» -5o»§ 400: - - - . 400: A - - 4 0 so 100 150 200 250 0 so 100 150 200 250 time(s) time(s) time (s) 0 ‘ ‘ ‘ . -10 - ‘ ‘ ‘ 0 50 100 150 200 250 O 50 100 150 200 250 time (s) time (s) Dashed lines = true values Figure 6.4. Geometric position, sensor signal, and DC component error of SRUC-BCE for 2-DOF system model 109 x; axis, bearing B x1 axis, bearing A ‘Y 1 1 100 150 200 250 0 50 100 150 200 250 O 50 time (s) time (s) 20- 20 E 3 ( <9" -10' ‘ * . . O 50 100 150 200 250 time (s) time (s) .50 A A A A -50 L . A A 0 50 100 150 200 250 o 50 100 150 200 250 time (s) time (S) Dashed lines = true values Figure 6.5. Higher harmonics error of SRUC-BCE for 2-DOF system model 110 that 6 -> 0, x —-> 0 and is bounded using the Barbalat’s lemma. Similarly, ones can find that the regressor vector 1; 6 R2x<4n+2l satisfies the persistency of excitation condition. The analytical proof is a straightforward extension to higher dimensions of its counterpart in Eq.(3.14), which is not provided in this thesis since it is beyond the scope of this thesis.Figures 6.6 and 6.7 show that a relatively small amplitude of bias current excitation, 0.05A with frequency of 10 Hz can effectively stabilize the rotor geometric center in both axes to the origin. The excitation is started at t = 2 second. It can be seen that the DC components converge to their true values in 2 seconds after the bias current excitation commences. 6.6 Summary and Remarks The effectiveness of the adaptive algorithms in MIMO systems has been investigated. The stability proofs of the adaptive algorithms for the MIMO model are very simi- lar to their counterparts in the SISO model. The proof of convergence of estimated parameters was however not pursued analytically. Instead we used numerical sim- ulations to observe that ASRC in MIMO systems, unlike in SISO systems, requires bias current excitation for proper estimation of all parameters and geometric center stabilization. In the absence of bias current excitation the DC components of SRO are not estimated correctly. A relatively small amplitude of bias current excitation, however, results is very fast convergence of the DC components to their true values. The extension of our SRUC-BCE algorithm to MIMO systems from SISO system was straightforward and results in geometric center stabilization with proper identification of SRO in both bearings as well as mass unbalance. 111 x1 axis, bearing A x2 axis, bearing 8 too I . . . 100 . ._ - . Dashed lines = true values Figure 6.6. Geometric position, sensor signal and DC component error of ASRC-BCE for 2—DOF system model 112 x1 axis, bearing A x; axis, bearing 8 fir fl 20 E )— - _ 3101 V 9 4 “u 0+ 2 2 4 6 1O 0 2 4 6 8 10 time (s) time (s) 20 - 20 . E ; E 10» i v10 . 3' - - A _- ( a: <3" 2. - —-- <9" 0 . ‘ o A -10 A A A 0 2 4 6 10 0 2 4 6 8 10 time (s) time (s) 0'” r 0_ ~ ’e‘ E 5 a a ’5 i l ‘3 '5b «a «UN -10 1 . ~10 A 4 . 0 2 4 6 10 0 2 4 6 8 10 time(s) time (s) 5 0A- k 0 _ S S. a '5‘ a '5 ‘ <13 <13 -10 -10 l A A .15 L A A A 0 2 4 6 10 0 2 4 6 8 10 time (s) time (s) Dashed lines = true values Figure 6.7. Higher harmonics error of ASRC-BCE for 2-DOF system model 113 CHAPTER 7 Conclusions 7.1 Research Summary The results presented in this thesis establish a new adaptive control framework for identification and compensation of periodic disturbances in active magnetic bearing systems with a rigid rotor. The advantages of the algorithms include robustness to uncertain plant parameters, simplicity of design, and ease of implementation. The algorithms are shown to apply to magnetic bearing systems modeled both as SISO and MIMO systems. Within this framework, we first design an algorithm for adaptive compensation of sensor runout in SISO model, assuming that the mass unbalance is negligible. The algorithm is developed using powerful tools such as Lyapunov stability theory and persistency of excitation concept. We prove that the algorithm guarantees stability of the rotor geometric center about the origin and correct identification of the harmonics of sensor runout disturbance. Using passivity analysis we show that the algorithm is robust to plant parameter uncertainties. The averaging method successfully predicts the convergence rate of the adaptation; thus, the averaging method can be useful in the selection of controller gains in our algorithm. Simulation and experimental results validate the effectiveness of the algorithm. We next address the problem of rotor stabilization about the geometric center in 114 the presence of combined sensor runout and mass unbalance. Our first approach to this problem is based on multiple speeds. This approach lacks robustness and has a number of drawbacks including the need for speed alteration, which will not be permissible in many applications. To overcome the limitations, we develop a new method for simultaneous on-line identification of both disturbances at constant rotor speed. This is achieved through persistency of excitation generated by methodical bias current excitation that does not alter the equilibrium condition of the rotor. After successful demonstration of the approach through simulations and experiments, we study the effects of excitation frequency, excitation amplitude, and harmonic content of excitation on the convergence of parameters. The robustness of the algorithm to parametric uncertainties and convergence rate of the parameters was also investigated. We further develop an adaptive algorithm that allows identification of both sen— sor runout and unbalance simultaneously without changing the angular speed. Using bounded external excitation such as bias current excitation we show that the regressor vector can be made persistently exciting to guarantee the convergence of estimated parameters to the true values. We verify that the excitation with one harmonic at the frequency about a half of the rotor frequency can be used. With the help of the pas- sivity analysis, the effect of uncertain plant parameters to the convergence of estimate parameters are studied. It reveals that the algorithm is robust to the uncertainties in mass and magnetic stiffness. However, the rotor center of geometric may slightly fluctuate due to the uncertainty in the actuator gain. In order to predict the effect of the control and plant parameters to the convergence rate of the adaptation, the averaging method is again utilized. However, it was found that the averaging system could not capture the important dynamics of the system; thus simulations are still preferred to investigate the convergence rate of the adaptation. Both simulation and experimental results validate the effectiveness of the algorithm. Finally, we investigate the extension of the algorithms for MIMO system model. 115 The stability proofs for using the adaptive algorithms are presented. However, due to the complexity of the MIMO system, we evaluate the convergence of the estimated parameters by simulations only to illustrate the possibility of extension. 7.2 Future Work The present work has revealed a number of areas that warrant additional investi- gation and research. In the SISO system model, one may modify the algorithm to include a robust control term such that the stability is still guaranteed in the presence of unmodeled dynamics. The robust control term may also be designed to improve the algorithm in the presence of time varying uncertainty in the actuator gain. In the MIMO system model, modification of the adaptive controller will be required if gyroscopic forces are taken into account. Within saturation limits, the optimum bias current excitation for rapid parameter estimation is a subject that needs further investigation. Since the stability proof of the proposed algorithms assume that the operation is at a constant angular speed, one may also investigate the stability of the closed-loop system when the angular speed varies. The effectiveness of the adaptive algorithms using MIMO system model still needs to be verified experimentally. Ex- tending the MIMO system model to include the effect of rotor flexibility may also be pursued. 116 APPENDICES 117 APPENDIX A Experimental Set-up A.1 Magnetic Bearing Set-Up Bearing Bearing A 8 rubber actuator ', _____ 1 sensor l _____ } / coupling - abs<())l(tlite ' ' [l EH? motor enc er :[ : , ; T I Elli.- til [I .. (fr—5‘: I i \ \ l l torscilonallyr l 0051—». ..— ngl coup mg other position 0 103’ 0'0" of balanced disk . 0.216 : balanced disk. at rotor midspan 0.432 Note: unit length is in meter Figure A.1. Magnetic bearing rig schematic A.2 Structural-Dynamic Analysis of Two-Bearing Rotor Structural-dynamic analysis was performed to determine the mode shapes and the resonant frequencies of the two-bearing rotor. The rotor dimensions are shown in Figure A.3 and Table A.1. A program ”MODAL” is used to generate a modally reduced state space model of a single free—free rotor. This program is provided by ROMAC Laboratory, University of Virginia. The detail instruction for the program is available in reference [20]. To utilize the MODAL program, we discretize the rotor as shown in Figure A.3. The input to the program is provided by file name msu4.dat. The results of the analysis are summarized in Table A.l and Figure A.4. To evaluate the effect of bearing stiffness to the rotor resonant frequency, the use of rotor critical speed map is useful as described in reference [5]. We generated the rotor critical speed map using a ROMAC’s program ”CRTSP_2” with the input file msu4a.dat. In this program gyroscopic effect is suppressed [20]. The result is shown in Figure A.5. 118 Figure A.2. Picture of magnetic bearing set-up Left Right Balanced disk Flight Figure A.3. Dimension of two-bearing rotor with balanced disk Table A.1. Two-bearing rotor data Total weight of the two-bearing rotor with balanced disk : 10.7 lb,n Rotor shaft: length = 17 in, diameter = 1 in Journal bearing: length = 3 in , diameter = 2.4 in Left-tip: length = 1 in , diameter = 0.25 in Right-tip: length = 1 in , diameter = 0.3 in Balanced disk: length :2 0.75in , diameter = 4 in 119 '8 1.5 1 T I 1 I I r 3 § 1 mode1 a: as mode 5 E 05 _ ............... -1 E1 0 it. . lllllllll * 1 .76 ‘ ....................... 3 ............. ‘~.m0d64 E .0'5 mode 2 o 4. C '1 .-l .E D 4'50 2 4 6 a 10 12 14 16 Distance along rotor (in) Figure A.4. Free-free modes shapes of rotor A 5 1.94 kHz E10 f 1.24 kHz #4,»? L g C I 3 g » 438 Hz . U. f / K3 = 4'33 x 105 mm 4 10 L A 1 L . . I 1 A A l 105 10° Bearing stillness (N/m) Figure A.5. Rotor critical speed map 120 Table A.2. Free-free undamped natural frequencies of rotor Mode Frequency 1 0 2 3 438 Hz 4 1.24 kHz 5 1.94 kHz Input for ROMAC’S software version 1.5, program MODAL. ROMAC is copyright of University of Virginia. File name: msu4.dat HSU’: 17in rotor model jds. a June 2000 derived from 23 5 -1 -3 0.0 1.00 0.0 1.00 1.033 0.50 0.0 0.50 1.033 0.50 0.0 0.50 1.033 1.00 0.0 1.00 0.0 1.00 0.0 1.00 0.0 0.50 0.9 0.50 0.0 1.00 0.0 1 00 0.0 1 0.0 1. 1.033 0.50 0.0 0.50 1.033 0.50 0.0 0.50 1.033 1.00 0.0 1.00 0.0 0.00 10. 200000. 0. .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 .9843 NOOOOOOOOOOOOOOOOOOOOOO 9843 POOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOO 29. :3 2318 :3 29. 29. 29. 2318 tn with our» corn oven even oven 01th Olin aria bitn burn 2318 29. 121 OOOOOOOOOOOOOOOOOOOOOOO .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 .28 1 1 0 0 "Loft actuator" 0 1 0 0 “Left sensor” 0 1 0 0 "Right sensor” 1 1 0 0 ”Right actuator" “Hui-circa; 714.15.“ ‘ , L . Input for ROMAC’S software version 1.5, program CTRSP-2. ROMAC is copyright of University of Virginia. File name: msu4a.dat HSU’e 171a rotor model jde. 8 June 2000 derived from 23 2 1 23 1 2 2 0 2 -10 0 5 -1 -3 2 0 1 -3 O 0 5 10000. 0. 19 10000. 0. 1 1.8+10 0 1.E+10 0. 6 1.055 O. 19 1.085 0. 1 1.1B+10 0. 2 1.1E+10 0. 0.0 1.00 0 9843 0 0. 0. 29.6 0.28 0.0 1.00 0.9843 0 0. 0. 29.5 0.28 1.033 0.50 0.9843 0 0. 0. 29.6 0.28 0.0 0.50 0.9843 0 0. 0. 29.6 0.28 1.033 0.50 0 9843 0 0. 0. 29.5 0.28 1 1 0 0 “Left ectuetor' 0.0 0.50 0.9843 0 0. 0. 29.6 0.28 1.033 1.00 0.9843 0 0. 0. 29.6 0.28 0 1 0 0 ”Left eeneer“ 0.0 1.00 0 9843 0 0. 0. 29.5 0.28 0.0 1.00 0 9843 0 0. 0. 29.6 0.28 0.0 1.00 0.9843 0 0. 0. 29.6 0.28 0.0 0.50 0 9843 0 0. 0. 29.5 0.28 0.9 0.60 0 9843 0 0. 0. 29.5 0.28 0.0 1.00 0 9843 0 0. 0. 29.6 0.28 0.0 1.00 0 9843 0 0. 0. 29.6 0.28 0.0 1.00 0.9843 0 0. 0. 29.5 0.28 0.0 1.00 0.9843 0 O. 0. 29.5 0.28 1.033 0.60 0.9843 0 0. 0. 29.6 0.28 0 1 0 0 “Right eeneor“ 0.0 0.60 0 9843 O 0. 0. 29.5 0.28 1.033 0.60 0 9843 0 O. 0. 29.5 0.28 1 1 0 0 ”Right actuator” 0.0 0.60 0.9843 0 0. 0. 29.5 0.28 1.033 1.00 0.9843 0 O. 0. 29.5 0.28 0.0 1.00 0.9843 0 O. 0. 29.5 0.28 0.0 0.00 0.9843 0 0. 0. 29.5 0.28 0.0 1.00 0.9843 0 0. 0. 29.5 0.00028 0.0 1.00 0.9843 0 0. O. 29.5 0.00028 100 20000. 50 122 A.3 Plant Parameters Table A.3. Magnetic bearing parameters for single-DOF Model Parameter Value Half-rotor mass, m 2.43 kg Electromagnetic force constant, k 2.82 x 10’6 N m2 /A2 Nominal air gap, 1 0.508 x 10‘3 m Top bias current, 2'10 2.41 A Bottom bias current, 2'50 2.06 A Actuator gain, K g 97.71 N /A Sensor gain, G, '2 x 104 V/m Open-loop stiffness, K f; 4.33 x 105 N / m Note that for constant bias currents cases as in Chapters 3 and 4, and section 6.3: . _ 0‘ e - a. _ * __ ‘ 110 — 110, 220 — 120, K,. — Kc, and K, — K,. 123 Magnitude dB 433883533 § 0 Phase (deg) -100 _200 . . ......1 . . ;.....r 101 16" 103 Frequency (Hz) Figure A.6. Bode plot of single-DOF magnetic bearing model Figure A.7. 3—D Plot of magnetic force surface 124 400-- 1 I Fo '= 16.86 N Linear region i 100 150 200 F0=_.16-.86 N. .. _ 1= - FO(N) -200 .. Linear region l l 1 1 l -1.5 -1 -o.5 ((1 IA) Figure A.9. Magnetic force vs. Current 125 A.4 Power Amplifier Data The power amplifiers are manufactured by Advanced Motion Controls Inc. Each power amplifier unit consists of a power supply that converts 120 VAC to 80 VDC with a maximum current of 15 A. This power supply hosts four 25A-Series PWM servo amplifiers with model number 12A8. The specifications of the PWM servo amplifiers are given in Table A.4. The servo amplifier is set to the current-mode. Each servo amplifier requires that the resistor labeled R30 on the board be replaced with 900 K-Ohm in order to achieve 1.6 kHz bandwidth for the corresponding magnetic coil inductance of 13 mH. The bode plot of the servo amplifier is shown in Figure A.10. Transfer function of the servo amplifier can be approximated by 10,.(3) 0.5(27r x 1600)2 G s =——= A.1 amp() Vin(s) 31’+2(0.5)(21rx1600)s+(271>—> - 1 Abs Unit Delay Constanfl @ COM (0) 0W“ CosFon CosSuutol'c' Figure A.13. Simulink block diagram for the absolute encoder 130 APPENDIX B Analysis of Persistently Exciting Condition Definition of Persistency of Excitation (PE) [25, 32]: A vector valued u 6 R" is said to be persistently exciting (PE), if there exists con- stants To, 01, (12 > 0 such that 1 t+To 021 Z T] uTudT _>_ all, ‘v’t > 0 (8.1) o t where I is the identity matrix, I E 32”“. Alternatively, the above definition of PE means that the matrix Q 9- %; ftHT" uTu (17' is bounded and positive definite. For the sake of brevity, without loss of generality, we consider up to the second har- monic components to prove the PE conditions. There are several ways to determine the positive definiteness of a matrix [26] such as: 1. Use the Sylvester’s Criterion for Positive Definiteness: A necessary and suflicient condition for a real symmetric matrix A E 32”" to be positive definite is that the determinant of A be positive and the successive principal minors of the determinant of A be positive the all principal minors are positive; that is, we must have 011 01 A1=011>0, A2; 2 20, 021 022 011 012 013 A A A3 = 021 0.22 023 > 0, . . . , An = IAI > 0 (8.2) 031 032 033 2. Find the minimum eigenvalue Am," of the matrix: If Re[Am,n(A)] > 0 then the matrix A is positive definite. The Sylvester’s criterion is useful to find the analytical conditions that guarantee positive definiteness of a matrix, in particular when the matrix has few nonzero off- diagonal terms. For a large matrix with many non-zero off-diagonal terms, finding the minimum eigenvalue numerically may become more practical. 131 B.1 ASRC: Equation 3.14 From Eq.(2.23) we can write { «E. \ J51 sin(wt) Ym= J31 cos(wt) 4 (8.3) J52 sin(2wt) K fizcos(2wt) ) where 0,- é [K , + m(z'w)2]2 > 0 for i =0, 1 and 2. In this case 0,- is constant. By taking To = 23" , the middle term of Eq.(3.14) becomes ( 200 0 0 0 0 \ ..2_« 0 01 0 o 0 (A) w T l A ‘2—1; YmYde = E O 0 0’1 0 0 = Q (8.4) ‘ 0 0 0 02 0 K 0 0 0 0 02 ) It is clear that Q is bounded. We can easily verify using the Sylvester’s criterion that Q is positive definite. This concludes the proof that Ym is PE. B.2 SRUC Using Multiple Speeds: Equation 4.25 From Eq.(4.22) J30 YE = J52 sin(2wt) (35) J52 cos(2wt) where 0,- 3 [K, + m(iw)2]2 > O for i =0 and 2. In this case K, is constant. By taking To = 3f , the middle term of Eq.(4.25) becomes H24: 200 0 0 w w T l A — YEYEdT = — o 02 o = Q (86) 27r , 2 0 0 02 Q is bounded and positive definite. This concludes the proof that YE is PE. 132 B.3 SRUC-BCE: Equation 5.41 In this case K, varies according to K, = K ; +5, sin(wet) > 0 where f, — £1395 > 0 is the resulting amplitude of variation in bearing stiffness and we 3- 27rf,B is the frequency of excitation. The Eq.(5.40) can be written as ( J33 + g, sin(wet) \ 1 [JO—f + E, sin(wet)] sin(wt) 1 [fl + 5, sin(w,t)] cos(wt) Ymu = [E + 6, sin(wet)] sin(2wt) (B.7) [J33- + E, sin(w,t)] cos(2wt) —mw2 sin(wt) K —mw2 cos(wt) ) where a: 2 [K; + m(z'w)2]2 > 0 for 2' = 1 and 2. We can also write ( «0‘5 + a. sin(w.t) \ l fifsinwt) — é-f, cos(wh1t) + g, cos(wnt) «(I—f sin(wt) + %€, sin(wh1t) + %£, sin(wut) Ymu = J5;— sin(2wt) — g, cos(wh2t) + £6, cos(w,2t) (B.8) J3; sin(2wt) + g, sin(w,,2t) + g, sin(w,2t) —mw2 sin(wt) K —mw2 cos(wt) } where wh, = (w, + iw) and w“ = (we — iw) for 2' = 1 and 2. Let the middle term of Eq.(5.39) t+To é — YmuYZmd‘r (8.9) To , The structure of Q depends on the value we relative to the value of w. We can calculate Q by letting T, as the period in which all signals of different frequencies in the regressor vector Ymu complete their full cycle. In the following cases, we de- fine 0, é a§+§€f = [K; + m(iw)2]2+%€f > 0 fori =0, 1 and 2; and C 3 mw'fl/o? > 0. 133 Casel:0 0, Q2 = 0001> 0, Q3 = 000? > 0, Q4 = 0001,02 > 0, 1 1 Q5 = ooofog > 0, Q6 = Zmzwifooolag > 0, Q7 = §m4wgffiooa§ > 0 Thus, we can conclude that Q is a positive definite matrix and Y"... is PE. In this case, ones may also recognize that in general Ymu is PE since for Q 6 9‘2““ we can easily find that Q,- > O for 2' = 1,2, ..., n. Case 2: w, = %w 2 8 0 01 0 —-1— 3 0 —( 0 1 --‘,-53 0 01 0 -%3 0 -C Q = 5 0 —§ 3 0 02 0 0 0 (3.11) 0 0 —§ 3 0 02 o 0 0 —c 0 0 0 m2w4 0 \ 0 o —( 0 0 0 m2w4 ) Using the Sylvester’s criterion, we can find that Q1=00>0, Q2=0001>0 134 1 1 Q3 = $01 (200 [(K; + mm?)2 + 55,2] -— 35:) > 0 since 00 > £5? Q4 = $- (02 [(K: + W)? + $53] — 7163) x (200 [(K; + m2)2 + $632] — if?) > 0 since 02 > £5: and 00 > $632 1 Q5 = g (02 [(K; + mw2)2 + 53’] — $5,?) x Q5, > 0 since Q5. = 20(K;)5mw2 + 66(K;)4m2w4 + 80(K;)3m3w6 + 32(K;)2m4w8 + 16637714028 + géfimzw" + l 2(K:)6 + E53 + 20(K;)3mw2£f + §(KI)2.§;4 + 50(K;)2m2w4§3 + 40 xi2m3w6K; + 3{:K;mw2 + 3(K;)4§3 > 0 and 02 > 163 8 1 Q6 = §Q60Q50 > 0 1 where Q60 = §(K:)2m2w4£f + 4€§m3w6K; + 8£fm4w8 + 126537712004 > 0 1 Q7 = §Q6aQ7a > 0 l where Q7, = 4m4wsf;1 + 16m4<418(K:)2£,2 + ‘l'gm2w452 + (K;)4m2w4{3 + 8(K;)3m3w6€f + —:-m2(.u4(K:)2£;1 + 2m3w6K:§: > 0 Therefore, we can conclude that Q is a positive definite matrix and Ymu is PE. 135 Case 3: we = %w ( 200 0 0 0 0 o 0 K 0 01 0 g 3 0 —( o 1 0 0 01 0 —§3 o —4 (i=5, 0 fig} 0 02 0 0 o (8.12) 0 0 fig: 0 02 0 0 0 —§ 0 0 0 mm 0 K 0 0 —( 0 0 0 m2w4 ) Using the Sylvester’s criterion, we can find that Q1=00>0, Q2=0001>0, Q3=0003>0 0001 Q4: 2 where Q4, = (02 [(K; + m2)2 + $63] - %E:) > 0 since 02 > "if? Q4a >0 0 Q5 = 30Q4a >0 Q6 = 00Q40Q6a > O 1 3 —(K;)2m2w4€3 + 4Efm3w6K: + 8€fm4w8 + fifimzw“ > 0 where Q5, = 2 Q7 : UOQGa > 0 Therefore, we can conclude that Q is a positive definite matrix and Y"rm is PE. 136 Pu.- - .mnm._. “any Case 4: for w, = 2w, ( 200 0 Q02 0 01 {,0} 0 1 0 {,0} 01 O Q = 5 Q02 0 05 + %EE 0 O 0 0‘; 0 —C _Qlu 0 K 0 -Q... ——< 0 Where Q02 = {sh/327+ V06): Qlu = %€smw2 Using the Sylvester’s criterion, we can also find that Q is positive definite. “Qlu —Qlu -C (13.13) The analytical proof here is omitted. The use of computer program with symbolic manipulation capability will show that Q satisfies the Sylvester’s criterion for positive definiteness. However, finding the eigenvalues numerically is more practi- cal. We can easily find that the minimum eigenvalue Am," > 0. Therefore, Yum, is PE. Case 5: for w, = w, ( 200 Q01 Q01 of + 125,2 0 O 1 Q = 5 0 0 ”if? “Q12 —Q1u —C K 0 0 where Q01 = (AK; + vaf), Qlu = firm”. and Q12 = %€8(\/3I+ v02)- 0 -< 0 _Qlu -%53 ~01. -Q12 -C 0 0 0 0 02 Qlu Qlu "12604 0 0 0 l O -C —Qlu O 0 mzw" ) (13.14) In this case, numerical evaluation shows that Q has two eigenvalues at the origin. Therefore Ymu is not PE. 137 APPENDIX C Parameter Convergence Using Averaging Method The following two-times scale averaging method is adopted from Sastry and Bodson [32] for a system having a regressor vector. C.l Convergence Rate of ASRC Consider the system in Chapter 3 of the form $=rvma (on M§+Czic+Kr = -Y,{,~ (0.2) E = :i: + A5: (03) where 3,?771, Ym E 322"“, 6, i E 32, and M, C, K, A > 0. The above three expressions are obtained from Eqs.(3.19), (3.26) and (2.22) respectively. In short, we can denote Eqs.(C.1) and (C2) as s + A —_ _ T~__ T~A_" T~ where C(.) is a signal vector and G (s) is an SPR transfer function s + A 0(8) = M32 + Cs + K (05) and Eq.(C.1) becomes . £17 = 41?... [G(Y,i¢)] (C.6) ~ When I‘ —+ 0, d>(t) varies slowly compared to e, the time scales of their variation ~ become separated. ¢(t) is called the slow state, é(t) the fast state and the system in Equations (C.1) to (Q3) a two-time scale system. In the limit as I‘ —> 0, ¢(t) may be considered frozen in Equations (C2) and (0.3), so that Calla) = GM.) J (0.7) The result of averaging theory, Eq.(C.6) therefore can be approximated by 55,, = —r AVG{Y,,, 6( Y5.» $0,, (0.8) 138 where l to+T AVG{?.,. (:(Y,’€,)} é T Yméwg) dt (0.9) where T = £1. The function AVG{Y,,. C’( Y5,» is commonly written as R(0) and called as the correlation matrix. For simplicity, we will consider 11 up to the second harmonic; thus, we use Ym defined in Eq.(B.3). Furthermore, from the definition in Eq.(3.20), we can write ( «a. \ J51 sin(wt) Ym = J51 cos(0Jt) (C.10) J52 sin(2wt) K V52 cos(20Jt) ) where 6: % [K ; + m(iw)2]2. We then obtain ( $5.6. \ _ IG(jw)|\/518in(wt + 1G(J'w)) GM) = 1000061 cos> 1 (0“) |G(j20))|\/523in(201t + [6'02 02)) . K |G(j202)|\/52 cos(20)t + 40020)) ) The product of 17'". C'( Y5) may be expanded as the sum of products of sinusoids. Further, for z' = l and 2 sin(z'02t + [C(j 101)) = sin(z'02t) cos(AG(j 2'02)) + cos(z'02t) sin(AG(j 2'01)) (C.12) cos(z'01t + AGU 201)) = cos(z'02t) cos(AG(j 2'01)) — sin(z'01t) sin(£G(j iw)) (C.13) Since the products of sinusoids at different frequencies have zero average, as do prod- ucts of sin’s with cos’s of any frequency, we obtain (R0 0 0 0 0 K 0 R1 0 0 0 0001220 0000122) K 139 '4 m: enumxfiml 5' 5 where R0 = "1%fi0fi0 ((7-15) 121 = gm «611000» «304000» = és/En/EleGUwH (0.16) R. = §fi2fi210(jzw)lcos(4002w» = Wax/6.1146020» (0.17) Using the familiar SPR condition, we know G(s) is real for real 3 and Re{G(jw)} > 0, V0: 6 (—oo, 00). Therefore R0, R1,R2 > 0, V02 6 (—oo,oo) . The final results are tabulated in Table 3.1. 0.2 Convergence Rate of SRUC-BCE Consider the system from Chapter 5 of the form 5,, = I“. Y, 6 (0.19) m(1 — me + (c — émAw = «3,6 + via, (0.20) where $,Ym E 523; ngu E 322; é,:E E R; A,c,A > O; and (c— §mA) > 0. The above expressions are obtained from Eqs.(5.25), (5.26), and (5.36). We may write these equations as —Y7’Iy‘zu$mu -_ —- s T~ é-" T~ . e—m(1-A)3+(c-%mA)_ G()[Y....¢...1 G By assuming F and Fu are relatively small, 45"," varies slowly compared to 'é. Thus on", can be considered frozen in Eq.(C.21), so that 003036) = (2.03.1); (0.27) The result of averaging theory, Eq.(C.25) therefore can be approximated by 35.. = 4.... AVG{Y...GO(Y£.,)} 5.. (0.23) where _ 1 to+T _ AVG{Y,,.., Go( Y,’,,,,)} é T Ymuaowgu) dt = 11(0) (0.29) to In the following we consider several cases of 0:, relative to 02. Casel:0 Passive system Y2 u H2 |.__?-_ Figure D.1. Feedback configuration for passivity analysis The following result is due to Kalman-Yakubovich lemma that shows the closeness of the Lyapunov stability and passivity concepts. Definition D.3 A linear time varying system Hl : :1:1:(::)zz+ IB(t)u1 (D6) is called strictly passive if it satisfies the following relations: P(t) + P(t)A(t) + AT(t)P(t) = —Q(t) < 0 (13.7) P(t)lB(t) = (‘5’ (t) (D.8) larfia fi'l‘ijflm ' . A for some uniformly bounded positive definite matrices P( t) and Q(t). For linear time-invariant systems, the following definition is commonly used to rep- resent the strictly passive condition. Definition D.4 A rational transfer function G(s) is said to be positive real if G(s) is real for all real 3, and Re{G(s)} Z 0 for all Re{s} Z O. The following lemmas are used in the study of the effect of plant parameter un- certainties to the SRUC-BCE algorithm in Chapter 5. Lemma D.2 The system H2 that has u2 = 6 as the input and y; 2 Y3}; — Yfau + flfldegu as the output, obtained from Eq.(5.53) when m is uncertain, is passivrcpfor sufficiently small lpml and relatively small amplitude of bias current excitation. Proof: To simplify the problem, we define new variables gap é ( {p ) = (bu _ (1+ %)$u (D-g) (In Therefore the output equation can be written as y,» = Y3} — Yfaup and the adap- tation law in Eq.(5.47) become (10 : 70Ks E d,- = 7,- [K, + 77123022] sin(iwt) e b,- = '7,- [K, + mizwz] cos(iwt) e, i = 1, 2,. . ., n 0 ~ pp -(1 + $3”? fizw2 sin(wt) 6 ~ Pm 9p : ”(1 + “fil'l'q mu)?! cos(wt) (2‘ It should be noted that K, is time varying as shown in Eq.(5.22) and pm is constant. For a sufficiently small Ipml, we may assume (1 + a?) > 0 and m > 0. We further define new variables K,+mi20J2 . -t = >0 f =0,1,2,..., ‘0'“ 7,-(K,+fiii2w2) or 7’ n 7pm E" "qu 7T1.— 146 For a relatively small bias current excitation we may assume that there exist a con- stant 01 such that p,-(t) > c, for all t. Therefore, recalling the definition DJ for a passive system we can obtain t t ~ ~ / ygugdt= / [Yqu—YTssup] edt 0 =1) K,aoe dt + 2 {f (K, + mi2w2) [a, 81n(iwt) + b,- cos( (iwt) )e] dt}— i=1 m0)2 / p'sin(wt)e'dt — m.022 / ZIcos(02t)e'dt 0 0 t . n t . ...: = / Wing” (......ii] ...}. 0 i=1 0 t e t e pp]; fipfipdt'l'pq/o‘ apapdt > W070). 45.0), «13.0)1 — W00). 00)). 0(0)] where W00) «5.0) 00)) = 9,—‘650 )+ :91, [630+ 330 )]+ i=1 pm I) 2 WW0 0) 42.0)) 0(0): 33(0) + 2302—1 [63(0) +1200] + 315’,— p.(0) + -—q,,(0) Since W[$(t),$u(t),qiu(t)] is a positive definite storage function, by definition DI the system H; is passive. Lemma D.3 The system H2 that has 11.2 = E as the input and y2 = Y3“; - YEW. as the output, obtained from Eq.(5.65) when K, is uncertain, is passive for sufi‘iciently small |p,| and relatively small amplitude of bias current excitation. Proof: The adaptation law in Eqs.(5.57) and (5.26) can be written as WI? 2,- 7,-,[K + mizwz] sin(iwt) e 147 b,- = 7,- [K, + mi2w2] cos(iwt) e, i = 1, 2, . . . ,n p' = —7,, m0)2 sin(wt) e if = —7, m0)2 cos(wt) (E It should be noted that K, is time varying as shown in Eq.(5.22). We define new variables K,+mi"’022 it = — . >0 f '=0,1,2,..., p() 7,-(K,+fiiz2w2) or z n 1 1 p =—>O p =—>0 p 7P q 741 For a relatively small bias current excitation we may assume that there exist a con- stant c1 such that p,(t) > CI for all t. Therefore, recalling the definition D.1 for a passive system we can obtain t t .. ~ / ygug dt = / [Ygd — Y3¢uJ édt 0 0 t ’3 t .... = / Kfioé dt + E {f (K, + mizwz) [5, sin(i02t)é + b,cos(iwt)é] dt} - 0 0 i=1 t t 172022 / p'sin(02t)edt—- m0)2 / Eicos(wt)edt 0 0 t n t ~..: 0 i=1 0 t . t _ pin/0 55dt+quo Eidt > W60), 0.0), «3.0)! — W[$(0).¢.(0),a3.(0)1 where W00).$.0),i.0)1 = 922.530) + 20,4 6’50) +3.00] + 1Pq~2 —q (t) pp~2 l: — t 2100+2 W00), 5.0)), 0(0)) = “353(0) + Z 9;- [a?(0) + 330)] + p _ )0 {239(0) + gem) 148 Since W[5(t),$u(t),4iu(t)] is a positive definite storage function, by definition DI the system H2 is passive. Lemma DA The system H; that has U2 = e as the input and y; :- ~ ~ c t " . . Yggb — Yz¢u -— p1; )Yzm as the output, obtained from Eq.(5.69) when K6 is uncertain, is passive for a sufl‘iciently small chI and relatively small amplitude of bias current excitation. Proof: We first define new variables qP gup é (1:1) ) : ¢u - (1 " 'I—I-éiklgu (13.14) Therefore the output equation can be written as y2 = Y3}; — Y: (17,, For sufficiently small pc we may assume that (1 -- 721) > 0. C From the adaptation law in Eqs.(5.57) and (5.26) we obtain 21:0 = 70K: 6 ii,- = '7,- [K, + mi2w2] sin(i02t) e b,- = 7,- [K, + mi2w2] cos(iwt) e, i = 1, 2,. . .,n 5 = —'y,, m(1 — Lie—)0)" sin(02t) é Kc if: —*y, m(1 — QM)2 cos(wt) é Kc We further define K, + mi2022 it = , >0 f .=0,1,2,..., p ( ) 7,-(K, + mz2022) or z n l 1 Pp“) = p > 0 [M(t) = p > 0 7p“ " -—c '7 (1 - Tc Kc q Kc For a relatively small bias current excitation we may assume that there exist constants c1, c2, and c3 such that p,(t) > c1, pp(t) > c2, and pq(t) > c3 for all t. Recall the 149 definition D.1 for a passive system t t ~ ~ f y2U2 dt =/ [YZlcS—qufiup] édt 0 0 t n 1 ~ = / K,Eioé dt + Z {I (K, + mi2022) [5,- sin(iwt)é’ + b,- cos(i02t)‘é] dt} - 0 i=1 0 t t 772.022] psin(wt)‘édt—mw2/ ficos(wt)édt ‘ 0 n t . 0 ...: 1: p055 dt+ {/ p, 5,5,+b,~b,- dt}+ f0 . . z 0 [ ] f“ ppfipfp dt + j; anp 6p dt > W00 )0 000)] — W00) 00)). 0(0)] where W030) 0301-35306: [’4 0+1»? t)]+ 6,2260) + —q,,0) W00) 00)) $.(0)1=—&’3 (0+2?- [01% 0)+b?(0)]+ .jp 5?,(0) + 6,3620) Since W[$(t), 5,0), ebu(t)] is a positive definite storage function, by definition D.1 the system H2 is passive. 150 BIBLIOGRAPHY 151 BIBLIOGRAPHY [1] Advanced Motion Controls PWM Servo Amplifier, Catalog and Technical Manual 1996, Camarillo, CA 93012. [2] Batty, R., 1988, Notch Filter Control of Magnetic Bearings, MS Thesis, Mas- sachusetts Institute of Technology, Cambridge, MA. 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