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vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Mechanical Engineering, Department of"@en ;
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dcterms:creator "Alizadegan, Alireza"@en ;
dcterms:issued "2019-08-31T07:00:00Z"@en, "2016"@en ;
vivo:relatedDegree "Master of Applied Science - MASc"@en ;
ns0:degreeGrantor "University of British Columbia"@en ;
dcterms:description """Cameras in mobile phones are the most popular due to their availability and portability; however, image blur caused by involuntary hand-shakes of the photographer degrades their image quality as mobile phones become lighter, smaller, and high-resolution. The optical image stabilizer (OIS) is a hardware-based alternative to conventional software-based de-bluring algorithms that offer superior de-blur; however, they are set back for mobile phone applications by cost, size, and power limitations.
The magnetically-actuated lens-tilting OIS is a novel miniaturizable and low-power conceptual design which is suitable for low-cost micro manufacturing methods; however, significant product variabilities caused by these methods, along with the strict performance requirements to outperform software-based algorithms, and the limited controller implementation capabilities of mobile phone devices pose a challenging control problem that is solved by the modeling and controller design method proposed in this thesis.
To solve the problem, practical manufacturing tolerances are simulated through computer-aided design and analyzed by finite-element methods to obtain the structure of the dynamics of OIS and uncertainties in dynamics. A dynamic uncertainty model is developed based on the analysis results and the robust H∞ control theory is applied to guarantee the closed-loop stability and optimize the closed-loop performance against uncertainties with constrained controller order.
The proposed method is demonstrated in two steps. First, it is applied to a set of large-scale OIS prototypes to demonstrate its feasibility in an experimental setting and its capability to deal with physical product variabilities. Then, it is applied to a set of small-scale OIS prototypes containing mass-produced parts to verify its applicability to real OISs. In both cases, the experimental results suggest that the robust H∞ controller outperforms the conventional nominal controllers and the μ-synthesis controller. By dealing with control challenges of the magnetically-actuated lens-tilting OIS, the application of this conceptual design to mobile phone cameras is expanded. Substitution of the conventional post-processing algorithms in mobile phone cameras with OIS has significant impact on their image quality."""@en ;
edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/58778?expand=metadata"@en ;
skos:note "Robust Control of MiniaturizedOptical Image Stabilizers for MobilePhone CamerasbyAlireza AlizadeganB.Sc., Sharif University of Technology, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Alireza Alizadegan 2016AbstractCameras in mobile phones are the most popular due to their availabilityand portability; however, image blur caused by involuntary hand-shakes ofthe photographer degrades their image quality as mobile phones becomelighter, smaller, and high-resolution. The optical image stabilizer (OIS)is a hardware-based alternative to conventional software-based de-bluringalgorithms that offer superior de-blur; however, they are set back for mobilephone applications by cost, size, and power limitations.The magnetically-actuated lens-tilting OIS is a novel miniaturizable andlow-power conceptual design which is suitable for low-cost micro manufac-turing methods; however, significant product variabilities caused by thesemethods, along with the strict performance requirements to outperformsoftware-based algorithms, and the limited controller implementation ca-pabilities of mobile phone devices pose a challenging control problem that issolved by the modeling and controller design method proposed in this thesis.To solve the problem, practical manufacturing tolerances are simulatedthrough computer-aided design and analyzed by finite-element methods toobtain the structure of the dynamics of OIS and uncertainties in dynamics.A dynamic uncertainty model is developed based on the analysis results andthe robust H∞ control theory is applied to guarantee the closed-loop sta-bility and optimize the closed-loop performance against uncertainties withconstrained controller order.The proposed method is demonstrated in two steps. First, it is applied toa set of large-scale OIS prototypes to demonstrate its feasibility in an exper-imental setting and its capability to deal with physical product variabilities.Then, it is applied to a set of small-scale OIS prototypes containing mass-produced parts to verify its applicability to real OISs. In both cases, theexperimental results suggest that the robust H∞ controller outperforms theconventional nominal controllers and the µ-synthesis controller. By dealingwith control challenges of the magnetically-actuated lens-tilting OIS, theapplication of this conceptual design to mobile phone cameras is expanded.Substitution of the conventional post-processing algorithms in mobile phonecameras with OIS has significant impact on their image quality.iiPrefaceThis thesis is original intellectual property of the author, Alireza Alizade-gan, working under supervision of Dr. Ryozo Nagamune and co-supervisionof Dr. Mu Chiao. This work has been conducted in Control EngineeringLaboratory and Micro-Electro-Mechanical Systems Laboratory at the Uni-versity of British Columbia. The results presented are going to be submittedfor publication.The simulation-based results in Chapter 3 was published in the followingconference proceedings:• A. Alizadegan, P. Zhao, R. Nagamune, and M. Chiao. “Modelingand robust control of miniaturized magnetically-actuated optical im-age stabilizers.” in Proceedings of the IEEE International Conferenceon Advanced Intelligent Mechatronics (AIM), pp. 734-739., July 12-15,2016.Chapter 4 includes experimental results. In this chapter, first the pro-posed method is applied to large-scale prototypes in Section 4.3 to demon-strate feasibility of the proposed method in an experimental setting andits capability to deal with physical product variabilities. In application tolarge-scale OIS, the µ-synthesis controller is successfully implemented withrobustness for the first time. This result was presented in the followingconference proceedings:• P. Zhao, A. Alizadegan, R. Nagamune, and M. Chiao. “Robust con-trol of large-scale prototypes for miniaturized optical image stabilizerswith product variations.” in Proceedings of the IEEE InternationalConference on Society of Instrument and Control Engineers of Japan(SICE), pp. 734-739., July 28-30, 2015.Next, the large-scale results are extended by introduction of robust H∞controller, extensive comparison with different controller design methods,and frequency-domain closed-loop validations.iiiPreface• A. Alizadegan, P. Zhao, R. Nagamune, and M. Chiao. “Robust controlof miniaturized optical image stabilizers against product variabilties.”IEEE Transactions on Mechatronics.Finally, to make experimental results more realistic, they are extendedto small-scale OIS prototypes in Section 4.4 to show applicibility of theproposed method to miniaturized OISs with mass-produced parts.• A. Alizadegan, P. Zhao, R. Nagamune, and M. Chiao. “Applicationof Robust Control Method to Miniaturized OISs for Mobile PhoneApplications.”, to be submitted for journal publicationsivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 OIS Control System Architecture . . . . . . . . . . . . . . . 31.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Sensor-shifting OIS . . . . . . . . . . . . . . . . . . . 41.3.2 Lens-tilting OIS . . . . . . . . . . . . . . . . . . . . . 61.4 Controller Design Theories . . . . . . . . . . . . . . . . . . . 91.4.1 Nominal Controller Design Methods . . . . . . . . . . 91.4.2 Robust Controller Design Methods . . . . . . . . . . 91.5 Research Objectives and Methodologies . . . . . . . . . . . . 101.6 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . 132 Optical Image Stabilizer . . . . . . . . . . . . . . . . . . . . . 152.1 Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Control System Configuration . . . . . . . . . . . . . . . . . 162.3 Performance Objectives and Specifications . . . . . . . . . . 17vTable of Contents3 Robust Controller Design Method for OISs . . . . . . . . . 193.1 Finite Element Analysis of Dynamics and Product Variability 193.2 Dynamic Uncertainty Modeling of OISs . . . . . . . . . . . . 223.3 Robust Controller Design for OISs . . . . . . . . . . . . . . . 253.3.1 State-feedback Controller Design . . . . . . . . . . . . 293.3.2 Output-feedback Controller Design . . . . . . . . . . 293.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 303.4.1 Modeling, Validation, and Robust Controller Design . 303.4.2 Closed-loop Stability Analysis of Model Samples . . . 323.4.3 Time- and Frequency-domain Performance Analysis . 344 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 394.1 Experimental Setups . . . . . . . . . . . . . . . . . . . . . . 394.1.1 Open-loop Frequency-domian System Identification . 394.1.2 Closed-loop Time-domain Controller Implementation 404.1.3 Closed-loop Frequency-domian System Identification 414.2 Experimental Methodologies . . . . . . . . . . . . . . . . . . 424.2.1 System Identification Tests . . . . . . . . . . . . . . . 424.2.2 Closed-loop Stability Tests . . . . . . . . . . . . . . . 424.2.3 Time-domain Performance Tests . . . . . . . . . . . . 434.2.4 Frequency-domain Performance Test . . . . . . . . . 444.3 Application to Large-scale OIS prototypes . . . . . . . . . . 444.3.1 Large-scale OIS Prototypes . . . . . . . . . . . . . . . 444.3.2 Robust Control of Large-scale OIS Prototypes . . . . 464.3.3 Assessments of Robustness for Closed-loop Stabilityand Performance . . . . . . . . . . . . . . . . . . . . . 474.4 Application to Miniaturized OIS Prototypes . . . . . . . . . 534.4.1 Miniaturized OIS Prototypes . . . . . . . . . . . . . . 534.4.2 Robust Control of Miniaturized OIS Prototypes . . . 554.4.3 Assessments of Robustness for Closed-loop Stabilityand Performance . . . . . . . . . . . . . . . . . . . . . 574.5 Practical Implications of the Experimental Result . . . . . . 635 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.1 Multiple Robust H∞ Control . . . . . . . . . . . . . . 685.3.2 Fixed-order Robust H∞ Control . . . . . . . . . . . . 695.3.3 Implementation on Physical Mobile Phone Cameras . 69viTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70AppendicesA Controller Design Methodologies and Parameter Values . 78A.1 Notch Filter Design . . . . . . . . . . . . . . . . . . . . . . . 78A.2 PID Controller Design . . . . . . . . . . . . . . . . . . . . . . 79A.3 Lead-lag Controller Design . . . . . . . . . . . . . . . . . . . 80A.4 LQG Controller Design . . . . . . . . . . . . . . . . . . . . . 81A.5 H∞ Controller Design . . . . . . . . . . . . . . . . . . . . . . 82A.6 Robust Controller Design . . . . . . . . . . . . . . . . . . . . 83B Hand-shake Disturbance Database . . . . . . . . . . . . . . . 85B.1 Time-domain Signals . . . . . . . . . . . . . . . . . . . . . . 85B.2 Frequency-domain Spectra . . . . . . . . . . . . . . . . . . . 86C Model Parameter Values . . . . . . . . . . . . . . . . . . . . . 88C.1 Uncertainty Model Parameter Values . . . . . . . . . . . . . 88C.2 Actuator Model Parameter Values . . . . . . . . . . . . . . . 89viiList of Tables3.1 Summary comparison of µ-synthesis with RobustH∞ on minia-turized OIS simulation model samples . . . . . . . . . . . . . 364.1 Assessment of closed-loop stability robustness in large-scaleOISs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Summary comparison of µ-synthesis with RobustH∞ on large-scale OISs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Assessment of closed-loop stability robustness in miniaturizedOISs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Summary comparison of µ-synthesis with RobustH∞ on minia-turized OISs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Percent improvements achieved by robust H∞ controller com-pared to conventional controllers for different case studies . . 63A.1 Notch-filter design parameter values . . . . . . . . . . . . . . 78A.2 Controller design parameters of the PID controller . . . . . . 80A.3 Controller design parameters of the lead-lag controller . . . . 81A.4 Controller design parameters of the LQG controller . . . . . . 82A.5 Controller design parameters of the H∞ controller . . . . . . 83A.6 Controller design parameters of the µ-synthesis controller . . 83A.7 Controller design parameters of the robust H∞ controller . . 84A.8 Summary of performance trade-offs associated with controllerdesign parameters . . . . . . . . . . . . . . . . . . . . . . . . 84C.1 Model parameter values . . . . . . . . . . . . . . . . . . . . . 88C.2 Moving-magnet actuator parameter values . . . . . . . . . . . 89viiiList of Figures1.1 Performance gap between software-based and hardware-basedimage stabilization approaches [18] . . . . . . . . . . . . . . . 21.2 Principle of operation of lens-tilting and sensor-shifting OISs 21.3 Mobile phone camera coordinate system . . . . . . . . . . . . 31.4 The general OIS control system architecture . . . . . . . . . . 41.5 Mechanical design of a sensor-shifting OIS in one axis of imagestabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Summary of limitations of the advanced control methods forthe sensor-shifting OIS . . . . . . . . . . . . . . . . . . . . . . 51.7 The principle of operation of the liquid-lens OIS [54] . . . . . 61.8 Mechanical design of a lens-tilting OIS [46] . . . . . . . . . . 71.9 The target experimental implementation layout for the pro-posed method . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.10 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Mechanical design of the miniature OIS [69] . . . . . . . . . . 162.2 Control system configuration of magnetically-actuated lens-tilting OIS in one axis of image stabilization . . . . . . . . . . 173.1 Eigen frequency analysis results . . . . . . . . . . . . . . . . . 203.2 Nominal dynamics of OIS . . . . . . . . . . . . . . . . . . . . 213.3 Effect of uncertainties on tilting dynamics . . . . . . . . . . . 233.4 Illustration of uncertainty region and definitions of uncer-tainty models . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Weighted control system block-diagram . . . . . . . . . . . . 263.6 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 313.7 Multi-model dynamic uncertainty model for FEA model ofminiaturized OIS . . . . . . . . . . . . . . . . . . . . . . . . . 323.8 Analysis of robust closed-loop stability of conventional con-trollers on OIS model samples . . . . . . . . . . . . . . . . . . 333.9 Time-domain performance assessment of different controllerson model samples of miniaturized OISs . . . . . . . . . . . . . 35ixList of Figures3.10 Comparison of hand-shake disturbance suppression of the ro-bust H∞ controller with the PID controller in time-domain . 363.11 Frequency response of transfer function from reference to track-ing error on nominal model . . . . . . . . . . . . . . . . . . . 373.12 Frequency response of transfer function from reference to con-trol input on nominal model . . . . . . . . . . . . . . . . . . . 384.1 The experimental setup . . . . . . . . . . . . . . . . . . . . . 404.2 Open-loop frequency-domian system identification setup . . . 404.3 Closed-loop time-domain controller implementation setup . . 414.4 Closed-loop time-domain controller implementation setup . . 424.5 Large-scale OIS prototype . . . . . . . . . . . . . . . . . . . . 454.6 Product variabilities in large-scale lens platforms . . . . . . . 464.7 Frequency responses of large-scale OISs . . . . . . . . . . . . 474.8 Multi-model dynamic uncertainty model of large-scale OIS . . 484.9 Time-domain performance assessment of different controllerson large-scale OISs . . . . . . . . . . . . . . . . . . . . . . . . 504.10 Frequency response of transfer function from reference to track-ing error on nominal model . . . . . . . . . . . . . . . . . . . 524.11 Frequency response of transfer function from reference to con-trol input on nominal model . . . . . . . . . . . . . . . . . . . 534.12 Miniaturized OIS prototype . . . . . . . . . . . . . . . . . . . 544.13 Product variabilities in miniaturized OISs . . . . . . . . . . . 554.14 Frequency responses of miniaturized OISs . . . . . . . . . . . 564.15 Multi-model dynamic uncertainty model of miniaturized OIS 574.16 Time-domain performance assessment of different controllerson miniaturized OISs . . . . . . . . . . . . . . . . . . . . . . . 594.17 Frequency response of transfer function from reference to track-ing error on nominal model . . . . . . . . . . . . . . . . . . . 614.18 Frequency response of transfer function from reference to con-trol input on nominal model . . . . . . . . . . . . . . . . . . . 62A.1 Weighted control system configuration H∞ controller design . 82B.1 Time-domain signals of hand-shake disturbance database . . 86B.2 Frequency-domain spectra of hand-shake disturbance database 87C.1 Tilt actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 89C.2 Force–air-gap relationship (i = 1 A) . . . . . . . . . . . . . . 90xList of AcronymsSymbol DescriptionBMI Bilinear matrix inequalityCAD Computer-aided designDOF Degree of freedomDSA Dynamic signal analyzerDSP Digital signal processorFEA Finite element analysisFEMM Finite element method magneticLTI Linear time-invariantLMI Linear matrix inequalityLQG Linear quadratic GaussianLDV Laser Doppler vibrometerICICS Institute for Computing, Information and Cognitive SystemsOIS Optical image stabilizermAh milli-ampere hourNSERC Natural Sciences and Engineering Research CouncilPID Proportional-integral-derivativeRMS Root mean squarexiAcknowledgementsThis work was supported by the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) and the Institute for Computing, Informationand Cognitive Systems (ICICS) at the University of British Columbia. Iwould like to thank my supervisor Dr. Ryozo Nagamune and co-supervisorDr. Mu Chiao who tried their best to create a productive and informativeresearch opportunity. My gratitude also goes to the members of ControlEngineering Laboratory and Micro-electro-mechanical Systems Laboratoryat the University of British Columbia for their kind assistance.xiiChapter 1Introduction1.1 MotivationsMobile phone cameras are optically mature technologies that have beenfinely tuned and optimized over the past decade to improve image qualityby increasing pixel density [12]; however, image quality of mobile phonecameras is still inferior to that of point-and-shoot cameras [69].The main source of the quality gap between mobile phone cameras andpoint-and-shoot cameras is the image blur due to involuntary hand-shakesof the photographer. As allowed space for OIS is smaller for mobile phonescompared to point-and-shoot cameras, increasing pixel density leads to verysmall pixel size [61], and therefore longer exposure time is necessary to cap-ture a certain amount of light. As exposure time increases, image qualitybecomes more affected by the blur caused by hand-shakes of the photogra-pher during exposure time. Moreover, as mobile phone devices get lighterand smaller, they become more susceptible to hand-shakes of the photogra-pher. Image stabilization is referred to techniques, either software-based orhardware-based, employed to alleviate this blur.So far, majority of mobile phones industries implement software-basedimage stabilization algorithms [47] because it is an inexpensive, compact,and low-power solution; however, image quality is degraded during the post-processing. Research shows that this performance limit can only be im-proved through a hardware-based approach [18]. Figure 1.1 shows the per-formance gap between the software-based and hardware-based approachesto image stabilization. Hardware-based approach to image stabilization andthe corresponding device are respectively called optical image stabilizationand optical image stabilizer (OIS).OIS compensates for hand-shake by restoring image to its original poitionon the image sensor. This can be achieved by either tilting the lens or shiftingthe image sensor as illustrated in Figure 1.2. In order to replace software-based image stabilizers with OIS in mobile phone devices, robust feedbackcontrol against variabilities in low-cost miniaturized OISs plays a pivotalrole.11.1. MotivationsFigure 1.1: Performance gap between software-based and hardware-basedimage stabilization approaches [18]Figure 1.2: Principle of operation of lens-tilting and sensor-shifting OISs21.2. OIS Control System Architecture1.2 OIS Control System ArchitectureThis section discusses the DOFs of OIS in a mobile phone device and theirrelationship with image blur. Then, a general control system architecturefor OIS is introduced.Involuntary hand-shakes of the photographer can occur all 6 DOFs of amobile phone device. Figure 1.3 defines a coordinate system for a mobilephone camera. In this coordinate system only rotation about x- and y-axesare important for OIS. Rotation about z-axis and translation along none ofthe axes do not introduce image blur because they do not affect the positionof the image on the image sensor.Figure 1.3: Mobile phone camera coordinate systemThe block diagram of the general control system architecture is shownin Figure 1.4, where x- and y-axes in this figure refer to the coordinatesystem defined on the mobile phone device in Figure 1.3. As shown inFigure 1.4, similar independent feedback control loops are specified for eachof x- and y-axes because typically OISs are symmetric with respect to x-and y-axis and the couplings between these two DOFs are negligible. Ineach loop, hand-shake is detected by gyro sensor of the mobile phone andpassed through a signal processor that calculates the desired adjustment forthe optical element. Then a standard feedback loop is used to drive theoptical element to the desire position.1.3 Literature ReviewThis chapter reviews existing works on OIS and its control. As introduced inSection 1.1, two main OISs applicable to mobile phone cameras are sensor-shifting OISs and lens-tilting OISs. In the following, we review mechanicaldesign of each of these OISs, discuss their performance advantages and dis-advantages, and point out their control challenges in the literature.31.3. Literature ReviewFigure 1.4: The general OIS control system architecture1.3.1 Sensor-shifting OISSo far, majority of OIS research has been focused on sensor-shifting design.The sensor-shifting OIS is size- and cost-efficient for large-scale camerasbecause it does not need to be replaced with the lens and actuate the heavylens.Figure 1.5 shows mechanical design of the sensor-shifting OIS. The imagesensor is mounted on a moving frame. The moving frame can shift by ballbearings on a stationary frame, and is actuated by a voice coil motor (VCM).Figure 1.5: Mechanical design of a sensor-shifting OIS in one axis of imagestabilizationThe sensor-shifting OIS has miniaturization and performance limitationsfor mobile phone cameras. The VCMs used in sensor-shifting OIS are diffi-cult to be miniaturized [29]. Besides, the hand-shake information gatheredat the camera body might be different from that at the camera lens, andthus, performance of the sensor-shifting OIS is often inferior to that of lens-41.3. Literature Reviewtilting OIS [8].In addition, the sensor-shifting OIS suffers from a challenging controlproblem due to friction disturbances caused by ball bearings and time-varying nonlinear dynamics due to VCM hysteresis and variations in thetilt angle as shown in Figure 1.5. Primarily, open-loop controllers have beenused for position control of moving frame [32]; however, open-loop controllerscannot deal with friction disturbances inherent in ball bearnings. Therefore,feedback controllers such as PID and lead/lag are preferred [64]. Moreover,voice coil motors (VCMs) has nonlinear dynamics which may dictate con-servative controller design. To address this issue, a modified PID controllerwith a feedforward component was proposed to quickly respond to distur-bances [31]. Another potential issue of sensor-shifting design is time-varyingdynamics caused by time-varying effects of ball bearings and gravity on themoving frames. To cope with these dynamics variations, numerous tech-niques such as fuzzy [9], gain-scheduling [63], and sliding mode [45, 60, 66],adaptive [65] and a combination of these techniques [33, 34] have been em-ployed to increase robustness of the closed-loop system against these varia-tions; however, there are disadvantages corresponding to each of these meth-ods as summarized in Table 1.6. Moreover, they yield a nonlinear/time-varying controller that might not be suitable for practical implementationon a mobile phone device with limited processing capabilities.Figure 1.6: Summary of limitations of the advanced control methods for thesensor-shifting OIS51.3. Literature Review1.3.2 Lens-tilting OISFor mobile phone cameras, a lens-tilting OIS may be a better option be-cause the lens is lighter than the sensor, and thus easier to actuate, therebyexpanding the battery life. In addition, since the lens is closer to the objectthan sensor, it requires smaller shifting for the same hand-shake. Finally,since the lens in mobile phone cameras will not be replaced, a lens-tiltingOIS is not so cost-inefficient as it would be for professional cameras.Based on this motivation, recently some conceptual designs has beenproposed that relies on manipulation of the lens to achieve optical imagestabilization. For instance, [54] proposed an OIS based on a liquid lens.The geometry of the liquid lens is adjusted by the electrowetting principlesto control lens focal length and position to achieve optical image stabiliza-tion. The principle of operation of the liquid-lens OIS is illustrated in theFigure 1.7. In Figure 1.7, A shows the schematic of the liquid-lens principlewhere (o) is a drop of oil with α contact angle on an insulation coating ofthickness d designated by the green line. The oil drop is surrounded bya conducting fluid (w). B and C are the images of the same oil drop ona parylene coating at 0 and 60V rms with 1 kHz frequency. The oil dropreacts to this excitation by changing the contact angle which in turn leadsto change of the focal length and position for optical image stabilization.Figure 1.7: The principle of operation of the liquid-lens OIS [54]The construction and miniaturization of the liquid lens assembly is quiteelegant. It also has minimal hysteresis with properly designed liquid andcoating; however, high speed (about 1 kHz) AC voltage fields (about 50 V)are required to drive the lens, which may need challenging circuit design.Furthermore, since liquid lens is involved, it might be quite difficult for themanufacturing process to obtain consistent results.Recently, conceptual design of a novel lens-tilting OIS for mobile phonecameras focusing on size and cost has recently been published [46]. Fig-ure 1.8 shows mechanical design of this lens-tilting OIS. The lens is heldby a monolithic flat structure referred to as the lens-platform. The lens-61.3. Literature Reviewplatform encompasses a plate supported by four folded beams connectedto the base. The plate has the majority of the inertia of the system andfolded beams work as linear springs providing sufficient stiffness for tiltingof the plate. Damping comes merely from air damping and material damp-ing that is fairly insignificant. The plate has 3 DOFs: two rotational DOFsabout x-axis and y-axis for image stabilization as well as a translationalDOF along z-axis for auto-focus. Actuation of 3 DOFs is achieved by fourmoving-magnet actuators installed at four corners of the device marked by1–4 located in equal distances from central axes. Each moving-magnet ac-tuator consists of a pair of coil and magnet. Magnets are attached to thelens platform whereas coils are fixed to the base. Actuation is achieved byforces generated on each magnet in response to the magnetic fields createdby the current in its corresponding coil.Figure 1.8: Mechanical design of a lens-tilting OIS [46]This mechanical design has several advantages. Since the lens-platformis designed as a monolithic flat structure, it is suitable for cheap fabrica-tion techniques such as micro-molding [21] and 3D printed molding [67].In addition, it generates tilt actuation torque by a set of moving-magnetactuators. The moving-magnet actuators are smaller than common voice-coil-motor actuators [11, 29, 57] providing the same performance because of71.3. Literature Reviewthe lighter moving part (magnets) and removal of moving wires comparedto voice coil motors with moving coils. It is also faster, cheaper, and morepower efficient than many other actuator types published in the literaturesuch as thermal [10, 38], piezo-electric [28, 44] and moving-mirror [19], andshape memory alloy actuators. The miniaturizability and low-power char-acteristics of this conceptual design has been demonstrated on a MEMSscanner [55, 56]. The actuator force is nonlinear with respect to the dis-tance between permanent magnet and the coil. However, this nonlinearityis substantially less compared to electrostatic and variable-reluctance actu-ators due to absence of ferromagnetic core in the center of the coil [4,41,50].Friction mainly includes air-damping and material-damping causing mini-mal energy loss. Moreover, since OIS is mounted on the lens, hand-shakeinformation on the lens are collected directly and used for compensationthat promises better performance. Finally, no additional actuators are nec-essary for auto-focus as this design can simultaneously accomplish imagestabilization and auto-focus by tilt and shift degrees-of-freedom (DOFs),respectively.Despite the aforementioned advantages, there are two shortcomings con-cerning control of this device which has not been addressed in [46]: (1) Acontroller was designed for a 1 DOF prototype rather than the original 3DOF conceptual design. (2) A controller was designed and tested on onlyone prototype rather than multiple prototypes. The 1 DOF prototype usedin this work cannot capture full complexity and coupling of dynamics be-tween different DOFs of the original 3 DOF conceptual design. Besides, thecontroller design approach used in this work completely ignores product vari-abilities which are inherent in cheap fabrication techniques for small-scale de-vices. Similar product variabilities are observed and considered in controllerdesign for numerous similar small-scale devices in the past decade, such assingle-stage [20] or dual-stage [13, 14, 22–25, 37, 42] hard disk drives micro-actuators, parallel-plate electrostatic micro-actuators [35, 71], micro/nano-positioning systems [30], micro-probes [59, 68], optical switches [26], gyro-scopes [15,16,36], resonators [51,52], relays [39], and capacitors [53]. Thesetwo shortcomings are specifically crucial for the magnetically-actuated lens-tilting design due to its inherent lightly-damped mechanical design. Inlightly-damped systems, robust stability typically limits the bandwidth [1].In addition, robust performance is essential for OIS application because im-proving the image blur in one product may deteriorate it in other platform.Covering these two shortcomings is the main topic of this thesis.81.4. Controller Design Theories1.4 Controller Design TheoriesThe controller design theories used in this thesis are briefly reviewed inthis section. For comparison purposes, the nominal controller design meth-ods such as proportional-integral-derivative (PID), lead-lag, linear-quadraticGaussian (LQG), and H∞ controllers are introduced in Section 1.4.1. Therobust control methods, being the emphasis of this thesis, is reviewed inSection 1.4.2.1.4.1 Nominal Controller Design MethodsNominal controllers in this thesis refer to controllers that only consider oneplant (referred to as the nominal plant) in design stage. In industry, classicalcontroller design methods such as PID and lead-lag controllers are typicallyemployed mainly because they do not need model to for design, are simpleto implement, and often generate satisfactory performance. However, thesemethods are inefficient in some applications because they are not necessar-ily an optimal controller design method. In addition, when designed with-out model, it is difficult to ensure closed-loop stability in the design stage.Therefore, they are typically used in series with notch filters to suppress theresonance modes of the plant.To cover these two limitations, model-based controller design meth-ods such as LQG and H∞ control are introduced. LQG and H∞ con-trollers are more complex compared to classical controllers, however they aredesigned optimally (in time-domain and frequency-domain, respectively).These methods also guarantee stability of the closed-loop system based onLyapunov’s quadratic stability theory through different mathematical for-mulations.Despite a number of successful industrial applications, all these nominalcontroller design methods only take one plant into consideration in the de-sign stage. More specifically, the closed-loop stability is not guaranteed andperformance is not optimized against product variabilities.1.4.2 Robust Controller Design MethodsRobust control theories explicitly deal with uncertainties in plants and dis-turbances through creating an uncertainty model [5, 6]. They typically for-mulate the robust controller design problem as an optimization problemwhere the constraint function represents closed-loop stability for the en-tire uncertainty set and the objective function represents the worst-case91.5. Research Objectives and Methodologiesclosed-loop performance considering uncertainties. Therefore, closed-loopstability can be guaranteed against product variabilities while optimizingthe worst-case performance. These methods typically lead to a dynamicoutput-feedback linear time-invariant (LTI) controller that is easily imple-mentable on mobile phones compared to adaptive or nonlinear controllers.µ-synthesis is a conventional robust controller design method that han-dles parametric uncertainties for which there are numerically-efficient off-the-shelf algorithms available [3]; however, this method suffers from somelimitations. Controller order and computational cost of design are typicallyhigh and increases drastically as the complexity of the robust control prob-lem increases. This is typically dealt with model order reduction; however,model order reduction methods may not preserve robustness; therefore, te-dious trial–and–error including order reduction and analysis should be per-formed to design a practically implementable controller. Since the order con-straint was not imposed in the designed stage, the reduced-order controlleris often far from optimal. Therefore, a small number of uncertainty parame-ters are desirable to model the uncertainties of the system which can lead toconservatism of the model. The controller designed based on a conservativemodel will have low closed-loop performance. In addition, the optimizationproblem associated with this method is non-convex. Non-convex optimiza-tion problems often yield conservative solutions. The conservativeness of thesolution depends on the choice of initial guess; however, µ-synthesis does notprovide any systematic means of solving for a decent initial guess. Further-more, as uncertainties become larger and more complex, conservativeness,complexity, and computational cost of µ-synthesis increase.1.5 Research Objectives and MethodologiesTo take advantage of the lens-tilting OIS over software-based image stabi-lization in mobile phone devices, its control challenges (discussed in Sec-tion 1.3.2) summarized below should be addressed simultaneously:C1 strict tracking performance requirements to compete with software-based image stabilization methods in image de-blur;C2 low control effort requirements to compete with software-based imagestabilization methods in battery life;C3 dynamics uncertainties due to product variabilities in low-cost micro-manufacturing of miniaturized devices;101.5. Research Objectives and MethodologiesC4 low controller order requirements for practical implementation on mo-bile phone devices.To overcome these challenges, the objective of this thesis is to developa robust controller design method for the 3 DOF miniaturized conceptualdesign that is implementable in an experimental setting as illustrated inFigure 1.9.Figure 1.9: The target experimental implementation layout for the proposedmethodSample OISs are collected from the OIS production line and identifiedin frequency-domain to generate a frequency response database that repre-sents dynamics uncertainties (C3). The frequency response database shouldbe used by proposed method to design a robust controller that can be im-plemented in the original OIS production line. To safely implement thedesigned controller on the original production line, the proposed methodshould guarantee closed-loop stability for entire uncertainty region. To min-imize conservatism of designed controller, the proposed method should op-timize closed-loop performance against uncertainties (C1 & 2). Finally, theproposed method should constrain controller order for practical implemen-tation on mobile phone device (C4).To develop the controller design method and demonstrate performanceof the designed controller, the following procedure is used:1. Dynamics variations of 3 DOF miniaturized OISs are analysed underpractical manufacturing tolerances by finite-element methods.2. A robust controller design method is developed based on finite-elementanalysis results, mechanical design, and control objectives of OIS.3. The proposed method is validated by creating uniform, comprehensive,and practical product variabilities using simulations.4. An experimental methodology is developed to compare the perfor-mance of the robust controller with nominal controllers introduced inSection 1.4.111.5. Research Objectives and Methodologies5. Feasibility of the method in an experimental setting is tested by appli-cation to large-scale OISs with by physical product variabilities usingthe experimental methodology.6. Applicability of the method in real OIS production lines is proved byimplementation on miniaturized OISs containing mass-produced parts.121.6. Organization of Thesis1.6 Organization of ThesisThis section presents organization of the thesis as well as the interconnec-tions between sections as illustrated in Figure 1.10.In Chapter 2, details of lens-tilting OIS including its mechanical de-sign (Section 2.1) is reviewed. Then, the control system configuration (Sec-tion 2.2) is described, and the control objectives and specifications (Sec-tion 2.3) are defined based on the research objectives defined in Section 1.5.Chapter 3 presents the proposed method. In this chapter, Section 3.1performs a finite element analysis (FEA) of dynamics and product variabili-ties on the OIS described in Section 2.1 based on its control system configu-ration proposed in Section 2.2. Then, Section 3.2 develops a dynamic uncer-tainty model based on the combination of finite element analysis results andmechanical design. Finally, robust controller design method is proposed inSection 3.3 to deal with the dynamic uncertainty model. There, the robustcontrol theories introduced in Section 1.4.2 are applied considering controlobjectives and control system configuration to design the controller.Chapter 4 presented the experimental results and discussions. In thischapter, Section 4.1 describes the experimental setups used for system iden-tification (Section 4.1.1) and closed-loop implementation in time-domain(Section 4.1.2) and frequency-domain (Section 4.1.3). Then, some exper-imental methodologies are explained based on the setups for performingthe tests and analysing the results, including system identification (Sec-tion 4.2.1), closed-loop stability (Section 4.2.2), and closed-loop implemen-tation in time-domain (Section 4.2.3) and frequency-domain (Section 4.2.4).Then, the setups and methodologies are used to implement the method onlarge-scale (Section 4.3) and miniaturized (Section 4.4) OISs.In Chapter 5, conclusions from experimental results are drawn (Sec-tion 5.1), contributions are summarized (Section 5.2), and outstanding re-search directions and potential future work are pointed out (Section 5.3).Details of parameter values and supporting information are deferred toappendices. Appendix A reports design methodologies and resulting pa-rameter values for baseline controllers. In Appendix B time-domain andfrequency-domain data of hand-shake signals collected from a mobile phonedevice are presented. These data are first used in Section 2.3 to define con-trol objectives and later in experimental setup to assess the performance ofthe proposed method.Finally, Appendix C reports parameter values of the dynamic uncertaintymodels of the lens-platform as well as the moving-magnet actuators that areused in the proposed method.131.6. Organization of ThesisFigure 1.10: Thesis organization14Chapter 2Optical Image StabilizerIn this chapter, details of lens-tilting OIS including its mechanical design(Section 2.1), control system configuration (Section 2.2), and control objec-tives and specifications (Section 2.3) are reviewed.2.1 Mechanical DesignFigure 2.1 shows a schematic for mechanical design of the miniature OISpresented in [46]. The lens is mounted a monolithic flat structure referredto as the lens-platform. The lens-platform encompasses a plate supportedby four folded beams connecting the plate to the base. The plate has themajority of the inertia of the system and folded beams work as linear springsproviding sufficient stiffness for tilting of the plate. Damping comes merelyfrom air damping and material damping that is fairly insignificant. Theplate has 3 DOFs: two rotational DOFs about x-axis and y-axis for imagestabilization as well as a translational DOF along z-axis for auto-focus.Actuation of 3 DOFs is achieved by four moving-magnet actuators installedin four corners of the device marked by 1–4 located in equal distances fromcentral axes. Each moving-magnet actuator consists of a pair of coil andmagnet. Magnets are attached to the lens platform whereas coils are fixedto the base. Actuation is achieved by forces generated on each magnet inresponse to the magnetic fields created by the current in its correspondingcoil. The coils’ currents are the inputs and the motions corresponding tothe 3 DOFs are the outputs of the OIS.This conceptual design has pros and cons. The monolithic flat struc-ture of the lens-platform makes it suitable for low-cost MEMS fabricationmethods; however, these methods are afflicted by significant product vari-abilities. Besides, using moving-magnet actuators to achieve tilt actuation isa quite low-cost solution; however, couplings in dynamics of different DOFscan arise due to unavoidable asymmetry in fabrication of four independentmoving-magnet actuators. Moreover, these couplings are uncertain becauseasymmetry of installation can appear in different ways for different products.152.2. Control System ConfigurationFigure 2.1: Mechanical design of the miniature OIS [69]2.2 Control System ConfigurationAccording to Section 1.2, as far as image stabilization is concerned, the con-trol goal is to tilt the lens-platform about x-/y-axis to a desired angle in anindependent manner. Therefore, only one axis is considered hereafter. Foreach axis, the control goal is to use current of individual coils appropriatelyto tilt the lens-platform to a desired angle according to hand-shake distur-bance data supplied by the mobile phone built-in gyro sensor. To achievethis goal, a control system configuration shown in Figure 2.2 was proposedin [46] specific for this OIS.The hand-shakes disturbances are detected by the gyro sensor of themobile phone and integrated once to obtain the tilt angle of the camera dueto hand-shake. The camera angle is used to calculate, in ‘Signal processor’block in Figure 2.2, the desired tilt angle of the lens that compensates forthe tilt angle of the camera. The error signal is acquired by subtractingactual tilt angle from reference tilt angle. Based on the error, the controllersends appropriate current commands to four coils in order to tilt the lens-162.3. Performance Objectives and SpecificationsFigure 2.2: Control system configuration of magnetically-actuated lens-tilting OIS in one axis of image stabilizationplatform for minimizing the error. The moving-magnet actuators describedin Section 2.1 are non-linear; however, since non-linearity is static, the non-linearity is compensated for via a static map, that is the inverse of the mapfrom the current to the torque. This is designated as ‘Look-up Table’ inFigure 2.2, and explained in detail in Appendix C. Therefore, throughoutthis thesis, controllers are designed by considering the torque as the controlinput, i.e the red box in Figure 2.2.2.3 Performance Objectives and SpecificationsThe performance considerations proposed in this thesis include three speci-fications and three objectives. Performance specifications refer to hard con-straints that have to be met, while performance objectives are performancemeasures that we wish to improve. Specifications are precedent to objec-tives and both are listed below in a descending order of priority. Performancespecifications include:(S1) maximum of 1% steady-state error to ensure acceptable reference track-ing and fair comparison of different controllers in this thesis,(S2) maximum of 6 dB peak sensitivity gain for closed-loop systems to allowsufficient stability margin for modeling errors,(S3) maximum of unity high-frequency controller gain to avoid control inputsaturation due to sensor noise amplification.After satisfaction of performance specifications, the objectives are to:(O1) minimize tracking error to improve image blur,172.3. Performance Objectives and Specifications(O2) minimize control effort to increase battery life,(O3) reduce controller order for practical implementation and computationpower consumption considerations on a mobile phone.These objectives and specifications are defined for a generic magnetically-actuated lens-tilting OIS; however, objective of the robust controller designmethod developed in the next section is to consider them for multiple OISs.18Chapter 3Robust Controller DesignMethod for OISsThis chapter presents the proposed robust controller design method. In thischapter, first a finite element analysis of dynamics and product variabilities isperformed on a computer-aided design (CAD) model of the OIS described inSection 2.1 based on its control system configuration proposed in Section 2.2(Section 3.1). Then, a general dynamic uncertainty model is developed basedon the combination of finite element analysis results and mechanical design(Section 3.2). Finally, robust control theories introduced in Section 1.4.2 areapplied considering control system configuration as well as control objectivesand specifications to deal with the dynamic uncertainty model.3.1 Finite Element Analysis of Dynamics andProduct VariabilityTo propose a physics-based dynamic uncertainty model in the next section,an FEA is performed in this section to identify dynamics of OIS as well aseffect of uncertainties in beam width and actuator installation on dynamicsof OIS. Figure 3.1 shows eigen frequency analysis results. There are fourmodes which are associated with four eigen frequencies denoted by ωni ,i = 1, 2, 3, 4. The first and second modes are respectively corresponding toshifting and tilting DOFs. The DOFs associated with the third and fourthmode are not trivial to explain based on mode shapes. Therefore, they areidentified through performing frequency response analyses on dynamics ofshifting and tilting DOFs in the next step.193.1. Finite Element Analysis of Dynamics and Product VariabilityFigure 3.1: Eigen frequency analysis resultsTo identify dynamics of shifting and tilting DOFs, frequency responseof lens-platform to translational and rotational actuation is obtained. InFigures 3.2(a) and 3.2(b), blue solid lines indicate frequency responses ofshifting and tilting dynamics of OIS, respectively. Shifting dynamics onlyincludes shifting mode, while tilting dynamics has tilting mode as well as3rd and 4th modes. Dynamics of OIS is lightly-damped because of its low-friction mechanical design discussed in Section 2.1. The bandwidth of thetilting dynamics is 260 Hz which is sufficiently larger than the hand-shakedisturbance bandwidth (20 Hz) shown in Figure B.2.203.1. Finite Element Analysis of Dynamics and Product VariabilityFrequency (Hz)101 102 103Magnitude (dB)708090100110120130140150DataModel(a) Shifting dynamicsFrequency (Hz)101 102 103Magnitude (dB)60708090100110120130140150DataModel(b) Tilting dynamicsFigure 3.2: Nominal dynamics of OIS213.2. Dynamic Uncertainty Modeling of OISsSince the focus in OIS is the tilting, and not the shifting, effects of beamwidth variation and actuator imbalance are studied on tilting dynamics (Fig-ure 3.2(b)). Figure 3.3 shows the results of this study where Figure 3.3(a)and Figure 3.3(b) show the tilting dynamics (Figure 3.2(b)) under pertur-bation of beam width and force imbalance, respectively. Figure 3.3(a) showstilting frequency responses of OIS under ±10 % beam width variations ofthe nominal width. Beam width variations lead to uncertainties in DC gainof tilting dynamics as well as all natural frequencies because the beam widthis related to spring stiffness. To be more specific, the increase of beam widthincreases stiffness, leading to the decrease in DC gain and the increase innatural frequencies. Figure 3.3(b) shows tilting frequency response of OISunder ±10 % imbalance in actuator forces. Force imbalance brings aboutcoupling of shifting dynamics with tilting dynamics. As percentage of forceimbalance increases, magnitude of coupling increases. The damping ratiosdo not change in either scenarios because they are affected by material ofthe lens-platform rather than its geometry.3.2 Dynamic Uncertainty Modeling of OISsThe objective of dynamic uncertainty modeling is to develop a mathemati-cal representation of the dynamic behaviour of the OIS as well as productvariabilities based on FEA results presented in Section 3.1. According toSection 2.2, the model considers the torque generated on the lens-platformas control input and the tilt angle of the lens-platform as measured output.Based on FEA results, a linear model of the following structure is proposed:G(s, δ) = [Gshift(s, δ) +Gtilt(s, δ)]e−sτ , (3.1)where Gshift and Gtilt denote the transfer functions corresponding to thecoupling of shift dynamics and the tilt dynamics, respectively, and δ des-ignates the uncertainty parameter vector representing product variabilities.For each of these terms, a linear model of following structure is proposedfrom Figure 3.2:Gshift(s, δ) =kshiftω2n1s2 + 2ξ1ωn1s+ ω2n1, (3.2)Gtilt(s, δ) =ktiltω2n2s2 + 2ξ2ωn2s+ ω2n2·ω2n4s2+2ξ4ωn4s+ω2n4ω2n3s2+2ξ3ωn3s+ω2n3, (3.3)223.2. Dynamic Uncertainty Modeling of OISsFrequency (Hz)101 102 103Magnitude (dB)5060708090100110120130140150160-10%-5%Nominal+5%+10%(a) Beam width uncertaintiesFrequency (Hz)101 102 103Magnitude (dB)5060708090100110120130140150160-10%-5%Nominal+5%+10%(b) Actuator installation uncertaintiesFigure 3.3: Effect of uncertainties on tilting dynamics233.2. Dynamic Uncertainty Modeling of OISswhere ωni and ξi are respectively natural frequency and damping ratio ofthe ith mode, Kshift is DC-gain of the coupling mode, and Ktilt denotesDC-gain of the tilt dynamics.Product variabilities are represented by parametric uncertainties in modelG. Based on FEA results, natural frequencies and DC-gains are considereduncertain while the damping ratios are assumed to be fixed. Kshift is con-sidered as a standalone uncertainty parameter representing force imbalance(Figure 3.3(b)), while ωn2 is the uncertainty parameter representing beamwidth variations (Figure 3.3(a)). Therefore, two independent uncertaintyparameters are defined asδ = [kshift, ωn2 ]. (3.4)The independent uncertainty parameters vary between an upper-bound andlower-bound as follows:kshift < kshift < kshift, ωn2 < ωn2 < ωn2 , (3.5)where underline and overline represent respectively the lower-bound andupper-bound of each uncertainty parameter. The upper-bound and lower-bound values should be identified experimentally.Uncertainties in ωni , i = 1, 3, 4, and ktilt are considered to be correlatedwith ωn2 because they are due to the same mechanical origin (beam widthvariations). Furthermore, this correlation leads to a control-oriented modelthat can be dealt with by the robust control theories introduced in Sec-tion 1.4.2 effectively. The correlations in all other parameters are defined asfollows:ωni = ω∗ni(ωn2ω∗n2)ci , i = 1, 3, 4, (3.6)ktilt = k∗tilt(ω∗n2ωn2)2ck , (3.7)where the superscript ∗ denotes the nominal values of dynamics parame-ters to be identified experimentally. These equations can be derived fromfundamental equations ωn =√K/J and k = 1/K for a simple second-order mass-spring-damper system assuming only stiffness variations (beamwidth). Addition of correction factors ci, i = 1, 3, 4 and ck are added toaccount for the increased complexity of the OIS compared to a simple mass-spring-damper system.243.3. Robust Controller Design for OISsBased on independent uncertainty parameters kshift and ωn2 , the un-certainty region can be represented by a rectangle as illustrated in Fig-ure 3.4. With this uncertainty region, two uncertainty models can be de-fined: real parametric uncertainty model G(s, δ) and multi-model Gi(s) =Dyu,i + Cy,i(sI − Ai)−1Bu,i, i = 1, 2, 3, 4. The real parametric uncertaintymodel considers the area inside the rectangle while the multi-model uncer-tainty model only considers the vertices. In this thesis, the emphasis is puton multi-model uncertainty model to design the robust H∞ controller de-scribed in the next section. The real parametric uncertainty model is merelyused to design the µ-synthesis controller (introduced in Subsection 1.4.2) forcomparison purposes.Figure 3.4: Illustration of uncertainty region and definitions of uncertaintymodels3.3 Robust Controller Design for OISsIn this section a robust controller design method is developed to deal withthe multi-model. The robust H∞ controller design method is based on thetheory published in [27]. This theory can consider uncertainties modeledin the previous section while providing the following additional advantagescompared to conventional off-the-shelf robust control methods such as µ-synthesis:253.3. Robust Controller Design for OISs• The controller order is theoretically constrained by order of plantmodel. Therefore, arbitrarily low-order controller can be designed asopposed to µ-synthesis control theory.• The robust controller design problem is formulated as an bilinear ma-trix inequality (BMI) optimization problem which can be solved usingefficient algorithms.• A systematic way of solving for a good initial guess for solution of thenon-convex optimization problem that is critical for performance ofthe designed controller.• Non-integer values for correction factors ci, i = 1, 3, 4 and ck (intro-duced in Equation 3.6 and Equation 3.7) are handled to reduce con-servatism of the designed controller.The robust control problem is illustrated in Figure 3.5 as a weightedcontrol system block-diagram, where K(s) and G(s, δ) denote the output-feedback robust H∞ controller and a set of OISs, respectively. e is thetracking error, u is the control input (torque), and y is the measured output(tilt angle). r is an uncertain reference signal generated by the hand-shakesignal, and δ is uncertainty parameter representing product variabilities. ewand uw are weighted tracking error and control input signals based on whichthe performance channel z is defined as follows:z :=[ew uw]T(3.8)Figure 3.5: Weighted control system block-diagramThe state-space realization of the generalized plant corresponding to this263.3. Robust Controller Design for OISsweighted control system configuration is denoted as:x˙zy = Ai Br,i Bu,iCz,i Dzr,i Dzu,iCy,i Dyr,i Dyu,ixru (3.9)The robust H∞ controller design method guarantees robust closed-loopstability and performance by considering uncertainty parameter δ (C3). Theweighting function We is used to penalize tracking error (O1), reduce steady-state error (S1), and meet stability margin specifications (S2). Wu is aweighting functions used to penalize control effort (O2) and avoid saturationdue to noise amplification (S3). To avoid undue increase of controller order(O3), we propose using constant Wu and first order We parameterized asfollows [70]:We(s) =s/MH + ωbs+ ωbML, (3.10)where ML, MH , and ωb are design parameters which, respectively, denotelow-frequency gain, high-frequency gain, and 0 dB crossing frequency ofW−1. To reduce steady state error (S1), ML should be decreased. Trackingerror (O1) can be reduced by increasing ωb. To increase stability margin(S2), peak of sensitivity MH should be reduced. Finally, Wu is taken asa constant gain, and it is increased to save control effort (O2) and avoidsaturation due to noise amplification (S3).Due to inherent trade-off between performance objectives and specifica-tions, controller design parameters ML, MH , ωb, and Wu are selected suchthat performance objectives improve while performance specifications aresatisfied. To select the controller design parameters, trial-and-error is per-formed because there is no explicit relationship between the controller designparameters and the performance objectives and specifications to formulateand optimization problem.The control problem is to find a robustly stabilizing controller parametermatrix Θ of formΘ =[AK BKCK DK](3.11)that solvesminΘmaxi||Trz,i(Θ)||∞ (3.12)273.3. Robust Controller Design for OISswhere Trz,i is the closed-loop system from r to z corresponding to Gi(s)expressed as:Trz,i(s) :=[A0,i + BiΘCi B0,i + BiΘDyr,iC0,i +Dzu,iΘCi Dyu,i +Dzu,iΘDyr,i](3.13)where the matrices are defined asA0,i =[Ai 00 0](3.14)Bi =[0 Bu,iI 0], B0,i =[Br,i0](3.15)Ci =[0 ICy,i 0],Dyr,i =[0Dyr,i](3.16)C0,i =[Cz,i 0],Dzu,i[0 Dzu,i](3.17)By applying the bounded real lemma to 3.13, this problem can be for-mulated as the following optimization problem [17]:minΘ,Piγ (3.18)s.t. Ψi(Pi, γ) +QTi ΘTPi(Pi) + Pi(Pi)TΘQi < 0 (3.19)where Pi > 0 is the closed-loop Lyapunov parameter matrix and matricesand other matrices can be written as:Ψi(Pi, γ) =AT0,iPi + PiA0,i PiB0,i CT0,i∗ −γI DTzr,i∗ ∗ −γI (3.20)Pi(Pi) =[BTPi 0 DTzu,i] (3.21)Qi =[C DTyr,i 0] (3.22)In above equations, Ψi are affine with respect to Pi and γ, Pi are affinewith respect to Pi, andQi are constant matrices merely dependent on systemparameters (see [17] for derivations). Therefore, even though the problemis non-convex due to coupling of Pi and Θ; it is convex with respect to eitherPi and Θ individually. Using this properties, a sequential design procedureis employed to solve for a local optimum as follows:283.3. Robust Controller Design for OISs1 Employ the procedure proposed in [27] to solve for an initial feasiblesolution. Set the result of this procedure as Θ1 and set k = 1.2 Set Θ := Θk and solve the convex problem 3.18 with respect to γ andP . Denote the solution for γ and Pi as γk and Pi,k, respectively.3 Set Pi := Pi,k and solve the convex problem 3.18 with respect to γ andΘ. Denote the solution for γ and Θ as γk+1 and Θk+1, respectively.4 If γk − γk+1 < , set k := k + 1 and go to step 2; otherwise, Θk+1 isthe parameter matrix of the designed controller.In this section we present the procedure used for design of initially-feasible controller as the initial guess to solve the non-convex optimiza-tion problem (3.18) based on the method proposed in [27]. The procedureinvolves two steps corresponding to design of state-feedback and initially-feasible output-feedback controller as follows.3.3.1 State-feedback Controller DesignIn this subsection, the objective is to design a state-feedback controller CKfor multi-model (3.9) such that it stabilizes the closed-loop system and op-timizes the robust H∞ performancemaxi||Trz,i(CK)||∞ (3.23)Theorem 6 in [27] states that this problem can be formulated as a convexoptimization problem based on LMIs:minL,Qγ, s.t.−Ai − ATi Br,i CTi∗ I DTzr,i∗ ∗ γI < 0 (3.24)where Ai := AiQ + Bu,iL, Ci := Cz,iQ + Dzu,iL, and Q > 0. Using thesolutions L and Q, the optimal state-feedback controller is given as CK =LQ−1.3.3.2 Output-feedback Controller DesignTo design the initially-feasible output-feedback controller, the designed state-feedback controller is set for CK and DK = 0 in (3.11). By doing this,293.4. Simulation Resultsproblem (3.13) is formulated as a convex problem based on LMIs (Theorem8 in [27]):minL,Qγ, s.t.−Mi −MTi Ni ΩTi∗ I DTzr,i∗ ∗ γI < 0 (3.25)whereMi :=[Xi(Ai +Bu,iCK) −XiBu,iCKY (Ai +Bu,iCK)− Z −G(Cy,i +Dyu,iCK)) Z +GDyu,iF − Y Bu,iCK],(3.26)Ni :=[XiBr,iY Br,i −GDyr,i](3.27)Xi, Y > 0 and Ωi :=[Cz,i +Dzu,iCK −Dzu,iCK]. Using the solutionsZ, G, and Y , controller parameters can be obtained as AK = Y−1Z andBK = Y−1G.3.4 Simulation ResultsTo preliminarily validate the proposed method, simulation studies are con-ducted. To this goal, first the proposed uncertainty model is validated bycomparing model samples with FEA samples. Then a controller is designedbased on the uncertainty model and its performance is compared to somebaseline controllers in time-domain and frequency-domain.3.4.1 Modeling, Validation, and Robust Controller DesignThe parameters of the model (3.1) are obtained based on FEA results. Thenumeric values of the model parameters are listed in the first column ofTable C.1. The model is validated in two stages: validation of nominal modelstructure and validation of uncertainty model structure. To validate modelstructure (3.1), nominal frequency responses of linear model and FEA arecompared in Figure 3.2. As shown, the proposed model structure decentlydescribes dynamics of the system.To validate uncertainty modeling, samples of frequency response ob-tained by varying Kshift and ωn2 in uncertainty model are compared withsamples of frequency response obtained by varying force imbalance and beam303.4. Simulation Resultswidth in FEA model, respectively. The results of uncertainty model valida-tion are presented in Figure 3.6.101 102 103Magnitude (dB)5060708090100110120130140150160Frequency (Hz)(a) Model with variations of ωn2Frequency (Hz)101 102 103Magnitude (dB)5060708090100110120130140150160-10%-5%Nominal+5%+10%(b) Data with force imbalance101 102 103Magnitude (dB)5060708090100110120130140150160Frequency (Hz)(c) Model with variations of KshiftFrequency (Hz)101 102 103Magnitude (dB)5060708090100110120130140150160-10%-5%Nominal+5%+10%(d) Data with beam width variationsFigure 3.6: Model validationIn this figure, Figures 3.6(a) and 3.6(c) are samples of uncertainty model313.4. Simulation Resultswhen ωn2 andKshift are perturbed, respectively, while Figures 3.6(b) and 3.6(d)are samples of FEA model when beam width and force imbalance are per-turbed, respectively. By comparing these samples, it can be seen that un-certainties in Kshift and ωn2 represent force imbalance and beam widthvariation, respectively.The multi-model uncertainty model obtained based on these parame-ter values are compared with the combination of FEA data in Figure 3.7.The multi-model uncertainty model successfully captures the variations offrequency responses. A robust H∞ controller is designed based on the multi-model uncertainty model. The controller design parameters (introduced inFigure 3.5 and Equation 3.10) for the robust H∞ controller are listed in thefirst column of Table A.7.102 103Magnitude (dB)406080100120140160180DataModelFrequency (Hz)Figure 3.7: Multi-model dynamic uncertainty model for FEA model ofminiaturized OIS3.4.2 Closed-loop Stability Analysis of Model SamplesThe designed robust H∞ controller was compared with µ-synthesis con-troller and four nominal controllers, namely PID, lead-lag, LQG, and H∞controllers. The robust H∞ controller is designed based on multi-model,323.4. Simulation Resultswhereas µ-synthesis controller is designed based on real-parametric uncer-tainty model. The nominal controllers are designed for the nominal plantmodel (Ks = 0 and ωnt = 453 Hz). The closed-loop system correspond-ing to different controllers is analyzed for different values of uncertaintyparameters in the uncertainty region 3.5. The design parameters for thesecontrollers are listed in Table A.2–A.7.Figure 3.8 shows the analysis results. The closed-loop stability of PID,lead-lag, LQG, and H∞ are illustrated in Figure 3.8(a), Figure 3.8(b), Fig-ure 3.8(c), Figure 3.8(d), respectively. In these figures, the green circle markand the red cross mark indicate that the closed-loop system is stable andunstable, respectively.Kshift-500 0 500ωn2260027002800290030003100(a) PIDKshift-500 0 500ωn2260027002800290030003100(b) lead-lagKshift-500 0 500ωn2260027002800290030003100(c) LQGKshift-500 0 500ωn2260027002800290030003100(d) H∞Figure 3.8: Analysis of robust closed-loop stability of conventional con-trollers on OIS model samples333.4. Simulation ResultsAs shown, different nominal controllers can stabilize different subregions ofthe uncertainty region including the nominal plant model; however, noneof them are capable of stabilizing the closed-loop system for the entire un-certainty region due to model-plant mismatch (C3). This is because thenominal controller design methods do not take uncertainties into account.On the other hand, both robust controllers (robust H∞ controller and µ-synthesis controller) successfully stabilized the closed-loop system for theentire uncertainty region because they guarantee robust closed-loop stabil-ity in design stage.3.4.3 Time- and Frequency-domain Performance AnalysisTime-domain performance is assessed by calculating root mean square (RMS)values of the tracking error signals. A set of tracking error signals are ob-tained for three cases corresponding to variabilities in beam width, force im-balances, and hand-shake disturbances. For each case, mean and worst-caseof RMS values of the corresponding set is calculated. Figure 3.9 comparestime-domain performance of different controllers, where the solid bars showsthe mean and the upper limit of the error bars show the worst-cases. In thefigure, nominal controllers are de-tuned versions of controllers in Figure 3.8to recover robust stability for uncertainty region (3.5).343.4. Simulation ResultsBeam width Force imbalance Hand-shake signalsTracking error (deg)00.020.040.060.080.10.120.140.160.180.2PIDLead-lagLQGH∞µ-synthesisRobust H∞Figure 3.9: Time-domain performance assessment of different controllers onmodel samples of miniaturized OISsAs shown, the robust controller reduces the mean value of tracking erroras well as its degradation (C1) compared to nominal controllers because itsystematically optimizes the performance by considering uncertainties andutilizing optimization tools in design. This improvement is shown in threecases corresponding to variabilities in beam width, force imbalances, andhand-shake disturbances. More specifically, analysis shows that hand-shakedisturbances could be suppressed about 44% better than average of nominalcontrollers.Figure 3.10 demonstrates time-domain hand-shake suppression. In thisfigure, the black solid line is the plot of a real hand-shake signal collectedby a mobile phone device. The blue dashed line and red dash-dot line aretracking error signals of the PID controller and the robust H∞ controller,respectively. As can be seen, the robust H∞ controller suppresses hand-shake disturbance more effectively than the PID controller does.353.4. Simulation ResultsTime (sec)0 1 2 3 4 5Error (deg)-1-0.500.51Not controlledPIDµ -synthesisTime (sec)0 1 2 3 4 5Current (A)-0.500.5Classical controllerRobust controllerFigure 3.10: Comparison of hand-shake disturbance suppression of the ro-bust H∞ controller with the PID controller in time-domainThe robust H∞ controller is compared with µ-synthesis controller morecarefully because both of them are robust controllers that achieve robustclosed-loop stability and optimized closed-loop performance. These con-trollers are compared in terms of controller order, tracking error, and controleffort both in frequency-domain and time-domains.Table 3.1 compares the robust H∞ controller with the µ-synthesis con-troller. The robust H∞ controller offers much lower controller order thanthe µ-synthesis controller does (C4) because it mathematically constraintsthe controller order. In Table 3.1, the values outside parenthesis for trackingerror and control effort are mean values, while the ones inside parenthesis arethe worst-case degradations. The robust H∞ controller improves the mean(C1) and degradation (C3) of the tracking error as well as the mean value(C2) and degradation (C3) of the control effort compared to the µ-synthesiscontroller.Table 3.1: Summary comparison of µ-synthesis with Robust H∞ on minia-turized OIS simulation model samplesµ-synthesis Robust H∞Controller order (–) 68 11Tracking error (deg) 0.0188 (0.0044) 0.0128 (0.0034)Control effort (mAh) 0.1711 (0.0504) 0.1537 (0.0406)Computation time (s) 43 201Figure 3.11 and Figure 3.12 compares the robust H∞ controller with theµ-synthesis controller in terms of the magnitude Bode plot of the transferfunctions from r to e and from r to u, respectively. As shown in Figure 3.11,363.4. Simulation Resultsthe robust H∞ controller has better error rejection than the µ-synthesiscontroller (C1). More specifically, 0 dB crossing frequency is 42 Hz forrobust H∞ controller, while the one for µ-synthesis controller is 38 Hz.100 101 102 103 104Magnitude (dB)-30-25-20-15-10-50510µ-synthesisRobust H∞Frequency (Hz)Figure 3.11: Frequency response of transfer function from reference to track-ing error on nominal modelIn Figure 3.12, the robust H∞ controller uses less control effort (C2)compared to the µ-synthesis controller in low-frequency band. The im-provements achieved in tracking error and control effort can be associatedwith the ability of the proposed robust H∞ controller design method tosystematically generate decent initial guess for solution of the optimizationproblem corresponding to the proposed controller design problem, and tohandle non-integer correction factors in the proposed uncertainty model.On the other hand, the computation time for design of the robust H∞ con-troller is slightly longer than that for the µ-synthesis controller because theµ-synthesis controller is designed using numerically-optimized off-the-shelfalgorithms while custom-made algorithms are developed and used for designof the robust H∞ controller.373.4. Simulation Results100 105Magnitude (dB)-150-140-130-120-110-100-90-80-70-60-50µ-synthesisRobust H∞Frequency (Hz)Figure 3.12: Frequency response of transfer function from reference to con-trol input on nominal model38Chapter 4Experimental Results4.1 Experimental SetupsThis chapter presents the experimental results. The experimental resultsinclude detailed explanation of the experimental setups and methodologiesused for conducting experiments. Then, same setups and methodologiesare used to implement the proposed method on large-scale (Section 4.3)and miniaturized (Section 4.4) OISs and assess the results. The experimen-tal setups are two folds: open-loop and closed-loop. The open-loop setupis for system identification and is presented in Section 4.1.1. The closed-loop setups are for controller implementation and assessments performedin time-domain (Section 4.1.2) and frequency-domains (Section 4.1.3). Theequipment used in these experimental setups are shown in Figure 4.1. Theexperimental setups include a dynamic signal analyser (DSA) (not shown inFigure 4.1), a digital signal processor (DSP), an amplifier, an OIS (plant),and a laser Doppler vibrometer (LDV).4.1.1 Open-loop Frequency-domian System IdentificationIn this subsection, the procedure and experimental setup used for frequency-domain system identification of OISs are described in details. The data ob-tained from this setup is used for parameter identification of the dynamicuncertainty model (3.1). More specifically, the objective is to identify nom-inal dynamics parameters in (3.1) (k∗tilt, ω∗ni , ξi, i = 1, ..., 4) and bounds onuncertainties in (3.5) (kshift, kshift, ωn2 , ωn2).Figure 4.2 shows connection of equipments for open-loop frequency-domian system identification. The dynamic signal analyzer generates si-nusoidal analog control input signals that are passed to the digital signalprocessor for signal processing. Then, the output of the digital signal pro-cessor is passed to an amplifier that generates sufficient current to be appliedto coils of the OIS. The tilt angle of the OIS is measured using laser DopplerVibrometer and fed back to the dynamic signal analyzer for analysis. Thedynamic signal analyzer calculates frequency response data based on its in-394.1. Experimental SetupsFigure 4.1: The experimental setupput and output signals. According to specifications, the LDV has 320 nmresolution, 20 kHz bandwidth which is sufficient to accurately characterizedynamics of the system in high-frequency band.The output frequency response data from this procedure characterizesk∗tilt, ω∗ni , ξi, i = 1, ..., 4. Re-iteration of this procedure for different OISsgenerates a data set that characterizes kshift, kshift, ωn2 , ωn2 .Figure 4.2: Open-loop frequency-domian system identification setup4.1.2 Closed-loop Time-domain Controller ImplementationIn this subsection, the procedure and experimental setup used for closed-loop time-domain implementation of the designed controllers are describedin detail. The data from this setup is used for assessment of closed-loopperformance of controllers in time-domain.Figure 4.3 shows connection of equipments for closed-loop time-domaincontroller implementation. The digital signal processor is used to apply404.1. Experimental Setupsnegative of the hand-shake signals (described in Appendix B) as referencesignal. The actual tilt angle of the OIS is measured by the laser Dopplervibrometer and subtracted from the reference signal. The resulting errorsignal is passed to the designed controller implemented on dSPACE digitalsignal processor. The command control input generated by the designedcontroller is passed to an amplifier that generates sufficient current to beapplied to coils of the OIS. The tilt angle of the OIS is measured using laserDoppler Vibrometer and fed back.The measurement signals of the laser Doppler vibrometer and currentsapplied to coils are logged to analyze the tracking error (e) and control effort(ik, k = 1, ..., 4) for assessment purposes.Figure 4.3: Closed-loop time-domain controller implementation setup4.1.3 Closed-loop Frequency-domian System IdentificationIn this subsection, the procedure and experimental setup used for identifi-cation of the closed-loop in frequency-domain are described in details. Thedata from this setup is used for assessment of closed-loop performance ofcontrollers in frequency-domain.Figure 4.4 shows connection of the equipments for closed-loop frequency-domian system identification. The dynamic signal analyzer generates sinu-soidal analog reference signal r. The actual tilt angle of the OIS is measuredby the laser Doppler vibrometer and subtracted from the reference signal.The resulting error signal is passed to the designed controller implementedon dSPACE digital signal processor. The command control input generatedby the designed controller is passed to an amplifier that generates sufficientcurrent to be applied to coils of the OIS. The tilt angle of the OIS is mea-sured using laser Doppler Vibrometer and fed back to the dynamic signalanalyzer for analysis. The dynamic signal analyzer calculates frequency re-sponse data based on its input and output signals.414.2. Experimental MethodologiesFigure 4.4: Closed-loop time-domain controller implementation setup4.2 Experimental MethodologiesThis subsection presents the methodologies used for performing experimentsand analysing experimental results. These methodologies are similarly usedfor application of the proposed method to large-scale and miniaturized OISspresented in the following sections. Experimental methodologies associatedwith four tests are explained: system identification tests, closed-loop sta-bility tests, time-domain tests, and frequency-domain tests. These explana-tions includes the objectives of the tests, the experimental setups used inthe tests, parameter settings of different equipments in the setups, param-eters changed during the experiments, and how the experimental data areanalysed, interpreted, and compared.4.2.1 System Identification TestsThe system identification tests are used to obtain the frequency responsedatabase in Figure 1.9. The frequency response database is obtained bymeasuring frequency response of 5 physical OIS prototypes that representsamples collected from the OIS production line in Figure 1.9. The frequencyresponses are measured using the open-loop frequency-domain system identi-fication setup introduced in Section 4.1.1. The frequency response databaseis used to identify the parameters of model (3.1).4.2.2 Closed-loop Stability TestsThe closed-loop stability tests check the closed-loop stability of a controllerfor perturbed OIS prototypes. The controllers participating in the experi-ment are the proposed robust H∞ controller, µ-synthesis controller, and fourconventional nominal controllers, namely PID, lead-lag, LQG, and H∞ con-trollers. The robust H∞ controller is designed based on multi-model uncer-tainty model, the µ-synthesis controller is designed based on real-parametricuncertainty model, and all nominal controllers are designed for OIS 3. Thedesign parameters for these controllers are listed in Table A.2–A.7. The424.2. Experimental Methodologiescontrollers are implemented in the closed-loop time-domain controller im-plementation setup introduced in Section 4.1.2.4.2.3 Time-domain Performance TestsThe time-domain performance tests are conducted to assess the trackingerror, the control effort, and their degradation with uncertainties for thecontrollers mentioned in previous subsection. To this goal, the nominal con-trollers designed for closed-loop stability tests are de-tuned such that theyrecover closed-loop stability for the uncertainty region (3.5). To de-tune, adesign parameter that best represents trade-off of closed-loop performanceand robustness in each controller design method (such as proportional gainin classical controller design methods) is adjusted in the corresponding con-troller designed for OIS 3 to increase robustness such that the controllerwould stabilize the closed-loop system for the entire uncertainty region. Thedesign parameters of the de-tuned controllers are listed in Tables A.2–A.5in parenthesis next to the controller deigns parameters corresponding to thenominal controllers.These de-tuned controllers as well as the robust controllers are imple-mented in the closed-loop time-domain controller implementation setup in-troduced in Section 4.1.2. In the setup, each controller is tested 25 timeswith 5 OIS prototypes 5 hand-shake disturbance signals and the correspond-ing tracking error and control input signals are measured. The results areanalysed for three cases. In each case, a certain combination of the OISsand the hand-shake disturbance signals are used, then the mean value andthe worst-case value of the performance measure is calculated. The combi-nation of the OISs and hand-shake disturbance signals used for each caseare explained below:Case 1 In this case, OIS 3 is used subject to the hand-shake disturbance sig-nals 1-5 to assess the robustness of the closed-loop performance touncertainties in the hand-shake disturbance signals.Case 2 In case 2, OISs 1-5 are used subject to hand-shake disturbance signal 1to assess the robustness of the closed-loop performance to uncertaintiesin the OIS dynamics.Case 3 In this case, OISs 1-5 is used subject to the hand-shake disturbancesignals 1-5 and the overall performance is analyzed to assess the overallrobustness of the closed-loop performance.434.3. Application to Large-scale OIS prototypesThe performance measure for the tracking error is chosen as the RMSvalue of the tracking error signal in degrees (deg), while the performancemeasure for the control effort is chosen as the integral of the absolute currentvalues over time in milli-ampere hour (mAh).4.2.4 Frequency-domain Performance TestThe frequency-domain performance tests are performed to compare closed-loop performance of the designed robust controllers with more details and tovalidate their time-domain performance results. To this goal, the designedrobust controllers are implemented in the closed-loop frequency-domian sys-tem identification setup (introduced in Section 4.1.3) including OIS 3. Thefrequency responses from r to e and from r to u (in Figure 3.5) are measuredto evaluate tracking error and control effort.4.3 Application to Large-scale OIS prototypesThe proposed method is experimentally validated on stand-alone prototypes(rather than OISs integrated with a mobile phone device) because there arenot available mobile phone devices designed to incorporate a lens-tilting OIS,nor there are any detailed design of this conceptual design in the literaturethat focuses on implementation on a mobile phone camera. The objectiveof implementation of the proposed method on large-scale OIS prototypes isto demonstrate feasibility of the method in an experimental setting and itscapability to deal with physical product variabilities. To achieve this goal,OIS prototypes are built in large-scale for practical fabrication by avail-able general-purpose prototyping facilities. Section 4.3.1 describes details ofthe large-scale prototypes used in these experiments. In Section 4.3.2, theapplication of the proposed method to the large-scale OIS prototypes areexplained. Finally, experimental methodologies (introduced in Section 4.2)are used to obtain experimental results in Section 4.3.3.4.3.1 Large-scale OIS PrototypesIn this subsection, details of the large-scale prototypes used in the experi-ments of this section are explained. The explanations include the mechanicaldesign details, the product variabilities in large-scale OISs, and the systemidentification results obtained by implementation of system identificationtests (introduced in Section 4.2.1) on large-scale prototypes.444.3. Application to Large-scale OIS prototypesFigure 4.5 shows a large-scale OIS prototype. The prototype is scaledabout 5 times. The lens platforms are made of 17-7 stainless steel andfabricated using a water-jet cutter. The lens platform is installed on analuminium base with screws. The coils are custom-made with copper wireswound around a core. The material of the core is copper, which has almostthe same permeability as air. So the coil can be considered as a air-core coil.The number of turns for one of the four coils are counted and the numberof turns of others are estimated based on the number of the layers and thenumber of turns in one layer. The permanent magnets are commercial rare-earth magnets attached to the lens platform by magnetic force. The numericvalues for design parameters of the moving magnet actuators are tabulatedin the second column of Table C.2 in Appendix C.Figure 4.5: Large-scale OIS prototypeTo mimic product variabilities in the miniaturized OISs, 5 lens platformsare fabricated with synthetic errors of -10%, -5%, 0%, 5%, and 10% in beamwidth as shown in Figure 4.6. The prototypes generated by assembling eachof these lens platform on the base (shown in Figure 4.5) are referred to asOIS 1, OIS 2, OIS 3, OIS 4, and OIS 5, respectively.The system identification tests (described in Section 4.2.1) are performed454.3. Application to Large-scale OIS prototypesFigure 4.6: Product variabilities in large-scale lens platformson the large-scale prototypes. Figure 4.7 shows the frequency responses ofOIS 1–OIS 5. The line colors in this figure are chosen correspondingly withsimulation results in Figure 3.3(a). In general the structure of the experi-mental frequency responses are similar to simulation ones because concep-tual design is the same. The bandwidth is reduced to about 150 Hz dueto scaling; however, it is still sufficiently larger than the hand-shake distur-bance bandwidth ( 20 Hz) shown in Figure B.2. In Figure 4.7, uncertaintiesin natural frequencies and DC gain can be observed. These uncertaintiesare due to beam width variations. As can be seen, with an increase in beamwidth from OIS 1 to OIS 5, the natural frequencies increase and DC gainsdecrease. In addition, the coupling of shifting mode due to force imbalancein simulations (Figure 3.3(b)) appears around 60 Hz in Figure 4.7 due toasymmetric actuation.4.3.2 Robust Control of Large-scale OIS PrototypesThis subsection explains the application of the proposed method in Chap-ter 3 to the large-scale OIS prototypes introduced in the previous subsec-464.3. Application to Large-scale OIS prototypes102Magnitude (dB)-50050100OIS 1OIS 2OIS 3OIS 4OIS 5Frequency (Hz)Figure 4.7: Frequency responses of large-scale OISstion. This explanation includes dynamic uncertainty modelling (introducedin Section 3.2) and robust H∞ controller design (introduced in Section 3.3)for large-scale OIS prototypes.The parameters of the model (3.1) are obtained based on the systemidentification results. The numeric values of the model parameters are listedin the second column of the Table C.1. The multi-model uncertainty modelobtained based on these parameter values are compared with the systemidentification results in Figure 4.8. The multi-model uncertainty model suc-cessfully captures the variations of frequency responses. A robust H∞ con-troller is designed based on the multi-model uncertainty model. The con-troller design parameters (introduced in Figure 3.5 and Equation 3.10) forthe robust H∞ controller are listed in the second column of Table A.7.4.3.3 Assessments of Robustness for Closed-loop Stabilityand PerformanceIn this subsection, the experimental methodologies introduced in Section 4.2are applied to the large-scale OIS prototypes described in Section 4.3.1.474.3. Application to Large-scale OIS prototypes102Magnitude (dB)-50050100DataModelFrequency (Hz)Figure 4.8: Multi-model dynamic uncertainty model of large-scale OISMore specifically, this subsection includes results obtained from implemen-tation of closed-loop stability, time-domain, and frequency-domain tests thatwere discussed in Section 4.2.2-4.2.4 of Section 4.2.Table 4.1 shows the results for the closed-loop stability tests. In thisexperiment, the designed controllers in each row are implemented on dif-ferent prototypes in each column. If the closed-loop system correspondingto the controller and the prototype was stable, the cell at intersection ofthe row associated with the controller and the column associated with theprototype is designated by a check mark (!), otherwise it is designated bya cross mark (×) is used to show that the closed-loop system was unsta-ble. According to Table 4.1, None of nominal controllers can stabilize theclosed-loop system corresponding to all OISs because the nominal controllerdesign methods do not take uncertainties into account. On the other hand,both robust controllers (robust H∞ controller and µ-synthesis controller)stabilized the closed-loop system for all prototypes (C3) because the robustcontroller design methods take uncertainties into account and guaranteesrobust closed-loop stability against uncertainties in the design stage.Figure 4.9 illustrates the results of time-domain performance tests. In484.3. Application to Large-scale OIS prototypesTable 4.1: Assessment of closed-loop stability robustness in large-scale OISsOIS 1 OIS 2 OIS 3 OIS 4 OIS 5PID × × ! ! !lead-lag × × ! ! !LQG ! ! ! × ×H∞ × × ! ! !µ-synthesis ! ! ! ! !Robust H∞ ! ! ! ! !the bar diagrams, the solid bars indicates the mean values of the RMS er-ror while the upper limit of error bars denote the worst-case RMS error.The results are consistent with simulation results (Figure 3.9). The ro-bust controllers reduce the mean values (C1) and degradation of trackingerror (C3) compared to nominal controllers because the robust controllerdesign methods take uncertainties into account and systematically optimizerobust closed-loop performance against uncertainties to minimize conser-vatism. This improvement is shown against variations in hand-shake signals,OIS uncertainties, and both. The improvements are slightly less than im-provements achieved in simulations. More specifically, analysis shows thathand-shake disturbances can be suppressed about 41% better than averageof nominal controllers.The robust H∞ controller is compared with µ-synthesis controller morecarefully because both of them are robust controllers that achieve robustclosed-loop stability and optimized closed-loop performance. These con-trollers are compared in terms of controller order, tracking error, and controleffort both in frequency-domain and time-domains.Table 4.2 summarizes comparison of the robust H∞ controller with theµ-synthesis controller. The controller order for robust H∞ controller is muchsmaller than that of the µ-synthesis controller (C4) because the robust H∞controller design method mathematically constraints the controller order.The reduction in controller order is comparable with simulation results. Thecontroller order reduction is performed by standard model reduction tech-niques. The controller order is reduced as long as closed-loop stability andclosed-loop performance specifications are not violated for the entire uncer-tainty region (3.5). Then, the tracking error and control effort measurementsare performed based on the reduced order controller. For tracking error andcontrol effort, the values outside parenthesis are mean values, while the ones494.3. Application to Large-scale OIS prototypesCase 1 Case 2 Case 3Tracking error (deg)00.050.10.150.20.25 PIDLead-lagLQGH∞µ-synthesisRobust H∞Figure 4.9: Time-domain performance assessment of different controllers onlarge-scale OISsinside parenthesis are the worst-case degradations. The robust H∞ con-troller reduces mean (C1) and degradation (C3) of the tracking error as wellas the mean (C2) and the degradation (C3) of the control effort comparedto the µ-synthesis controller. The improvements achieved in large-scale ex-periments are similar to improvements achieved in simulations; however, thetracking errors are all higher compared to simulation because more stabilitymargine was necessary to account for modeling errors in an experimentalsetting.504.3. Application to Large-scale OIS prototypesTable 4.2: Summary comparison of µ-synthesis with Robust H∞ on large-scale OISsµ-synthesis Robust H∞Designed order (–) 76 11Reduced order (–) 18 9Tracking error (deg) 0.0451 (0.0082) 0.0385 (0.0079)Control effort (mAh) 0.4556 (0.1153) 0.4555 (0.1152)Computation time (s) 75 269Figure 4.10 and Figure 4.11 show the results of the frequency-domainperformance tests. Figure 4.10 and Figure 4.11 compares the robust H∞controller with the µ-synthesis controller in terms of the magnitude Bodeplot of the transfer functions from r to e and from r to u, respectively. Asshown in Figure 4.10, the robust H∞ controller has better error rejectioncompared to the µ-synthesis controller (C1). More specifically, 0 dB crossingfrequency is 21 Hz for robust H∞ controller, while the one for µ-synthesiscontroller is 16 Hz. These values are compatible with bandwidth of thehand-shake disturbances (Figure B.2) and is much smaller than actuatorbandwidth (which is about 150 Hz as shown in Figure 4.7).514.3. Application to Large-scale OIS prototypes10-1 100 101 102 103Magnitude (dB)-60-50-40-30-20-10010µ-synthesisRobust H∞Frequency (Hz)Figure 4.10: Frequency response of transfer function from reference to track-ing error on nominal modelIn Figure 4.11, the robust H∞ controller uses less control effort com-pared to the µ-synthesis controller in low-frequency band (C2). The im-provements achieved in tracking error and control effort can be associatedwith the ability of the proposed robust H∞ controller design method tosystematically generate decent initial guess for solution of the optimizationproblem corresponding to the proposed controller design problem as well ashandling non-integer correction factors in the proposed uncertainty model.On the other hand, the iteration time for design of the robust H∞ con-troller is slightly longer than that for the µ-synthesis controller because theµ-synthesis controller is designed using numerically-optimized off-the-shelfalgorithms while custom-made algorithms are developed and used for designof the robust H∞ controller.524.4. Application to Miniaturized OIS Prototypes10-1 100 101 102 103Magnitude (dB)-80-70-60-50-40-30-20-10010µ-synthesisRobust H∞Frequency (Hz)Figure 4.11: Frequency response of transfer function from reference to con-trol input on nominal model4.4 Application to Miniaturized OIS PrototypesThe application of the proposed method to large-scale in previous section,showed feasibility of the proposed method in an experimental setting and itscapability to deal with physical product variabilities. This section is extendsthe experimental study in the previous section to demonstrate the appli-cability of the proposed method to miniaturized OIS with mass-producedparts. Section 4.4.1 describe the miniaturized prototypes used in these ex-periments. In Section 4.4.2 the application of the proposed method to theminiaturized OIS prototypes are explained. Finally, experimental method-ologies (introduced in Section 4.2) are used to obtain experimental resultsin Section 4.4.3.4.4.1 Miniaturized OIS PrototypesThe details about miniaturized prototypes, including the mechanical designdetails, the product variabilities in miniaturized OISs, and the system iden-tification results obtained by implementation of system identification testsused in the experiments of this section are explained in this subsection.534.4. Application to Miniaturized OIS PrototypesAn example of a miniaturized OIS prototype is shown Figure 4.12. Thesize of the prototype is compatible with generic mobile phone dimensions.The lens platform is made of PlasWHITE photo-polymer and fabricatedusing a 3D printer in batch. The lens platform is installed on a base of samematerial by tolerance fitting. The coils and magnets are both commercialand mass-produced. The coils are held stationary by tolerance fitting tocylinders created in the base. The magnets are attached to the lens platformusing a general-purpose glue. The numeric values for design parameters ofthe moving magnet actuators are tabulated in the third column of Table C.2in Appendix C.Figure 4.12: Miniaturized OIS prototypeFive miniaturized OIS prototypes are built as shown in Figure 4.13 torealize product variabilities. The OIS prototypes are named OIS 1–OIS 5.The sources contributing to product varibilities in miniaturized OIS pro-totypes are more complex compared to large-scale OISs. Even though nosynthetic beam width variations are imposed in miniaturized OISs, thereare numerous natural sources of variabilities in lens platform such as 3Dprinter’s fabrication tolerances, pillar residue, the curing post-processingthat leads to clogging of the beams. Unlike large-scale OISs where differentlens platforms shared a common base, separate bases are fabricated for each544.4. Application to Miniaturized OIS Prototypeslens platform in miniaturized OISs. The coils and magnets used in miniatur-ized OISs are off-the-shelf mass-produced coils and magnets with industry-quality product variabilities. The permanent magnets are attached usingglue that adds significant variabilities specially in air-gap and coil-magnetmisalignment compared to large-scale OISs where magnets are attached bymagnetic force.Figure 4.13: Product variabilities in miniaturized OISsThe dynamics of miniaurized OISs are characterized through systemidentification tests (described in Section 4.2.1) are performed on the minia-turized prototypes. Figure 4.14 shows the frequency responses of OIS 1–OIS 5. In general the structure of the frequency responses of miniaturizedOISs are similar to simulation results (Figure 3.3) and large-scale results(Figure 4.7) because all share the same conceptual design. The bandwidthis about 300 Hz which is compatible with simulation results and sufficientlylarger than the hand-shake disturbance bandwidth ( 20 Hz) shown in Fig-ure B.2; however, since PlasWHITE has a relatively higher damping, theresonance modes have higher damping compared to stainless steel. In Fig-ure 4.14, variations in natural frequencies and DC gain can be observed.In addition, the coupling of shifting mode appears in Figure 4.14 due toasymmetric actuation. The variations in frequency responses are larger andless correlated compared to simulation results (Figure 3.3) and large-scaleresults (Figure 4.7) due to added sources of product variabilities.4.4.2 Robust Control of Miniaturized OIS PrototypesThe proposed method is applied to the miniaturized OIS prototypes based onthe frequency-response database obtained in Figure 4.14. To apply the pro-posed method, first a dynamic uncertainty model is developed (introduced554.4. Application to Miniaturized OIS Prototypes101 102 103Magnitude (dB)010203040506070OIS 1OIS 2OIS 3OIS 4OIS 5Frequency (Hz)Figure 4.14: Frequency responses of miniaturized OISsin Section 3.2), and then a robust H∞ controller is designed (introduced inSection 3.3) for miniaturized OIS prototypes.The data in Figure 4.14 suggests that in addition to uncertainties innatural frequencies and DC gains, the sharpness of resonance peaks corre-sponding to resonance modes 2, 3, and 4 also vary from one prototype tothe other. To deal with this increased complexity in miniaturized OISs, acorrelation is considered between the damping coefficients and the naturalfrequency of the second mode. To represent both increase and decreaseof the damping ratio by a simple model, a linear correlation is defined asfollows:ξi = ξi +ξi − ξiωn2 − ωn2(ωn2 − ωn2), i = 1, ..., 4. (4.1)The validity of linear assumption will be verified by comparing frequencyresponses of the model with the data. In addition to correlation of dampingcoefficient, the nominal values of the dynamics parameters and the correctionfactors are adjusted heuristically to achieve an appropriate model.564.4. Application to Miniaturized OIS PrototypesThe parameters of the model (3.1) are obtained based on the systemidentification results. The numeric values of the model parameters are listedin the third column of the Table C.1. The multi-model uncertainty modelobtained based on these parameter values are compared with the systemidentification results in Figure 4.15. The multi-model uncertainty modelsuccessfully captures the variations of frequency responses. A robust H∞controller is designed based on the multi-model uncertainty model. Thecontroller design parameters (introduced in Figure 3.5 and Equation 3.10)for the robust H∞ controller are listed in the third column of Table A.7.101 102 103Magnitude (dB)010203040506070ModelDataFrequency (Hz)Figure 4.15: Multi-model dynamic uncertainty model of miniaturized OIS4.4.3 Assessments of Robustness for Closed-loop Stabilityand PerformanceHere, the experimental methodologies introduced in Section 4.2 are appliedto the miniaturized OIS prototypes described in Section 4.4.1. The resultsof the robust closed-loop stability experiments are summarized in Table 4.3.In this experiment, the designed controllers in each row are implemented ondifferent prototypes in each column. If the closed-loop system correspond-574.4. Application to Miniaturized OIS Prototypesing to the controller and the prototype was stable, the cell at intersection ofthe row associated with the controller and the column associated with theprototype is designated by a check mark (!), otherwise it is designated bya cross mark (×) is used to show that the closed-loop system was unsta-ble. According to Table 4.3, None of nominal controllers can stabilize theclosed-loop system corresponding to all OISs because the nominal controllerdesign methods do not take uncertainties into account. On the other hand,both robust controllers (robust H∞ controller and µ-synthesis controller)stabilized the closed-loop system for all prototypes (C3) because the robustcontroller design methods take uncertainties into account and guaranteesrobust closed-loop stability against uncertainties in the design stage.Table 4.3: Assessment of closed-loop stability robustness in miniaturizedOISsOIS 1 OIS 2 OIS 3 OIS 4 OIS 5PID × ! ! × ×lead-lag ! ! ! × ×LQG × × ! × ×H∞ × × ! × ×µ-synthesis ! ! ! ! !Robust H∞ ! ! ! ! !The results of time-domain performance testes are shown in Figure 4.16in form of bar diagrams. In the bar diagrams, the solid bars indicates themean values of the RMS error while the upper limit of error bars denotethe worst-case RMS error. The results are consistent with simulation re-sults (Figure 3.9) and large-scale experimental results (Figure 4.9). Therobust controllers reduce the mean values (C1) and degradation (C3) oftracking error compared to nominal controllers because the robust controllerdesign methods take uncertainties into account and systematically optimizerobust closed-loop performance against uncertainties to minimize conser-vatism. This improvement is shown against variations in hand-shake signals,OIS uncertainties, and both. The improvements achieved in miniaturizedOIS are less than simulation results (Figure 3.9) and large-scale experimentalresults (Figure 4.9) because the uncertainties that has to be considered bythe robust controller design methods are larger and more complex. There-fore, relatively modest performance is achievable. More specifically, analy-sis shows that hand-shake disturbances could be suppressed more than 31%584.4. Application to Miniaturized OIS Prototypesbetter than nominal controllers.Case 1 Case 2 Case 3Tracking error (deg)00.020.040.060.080.10.120.140.160.180.2PIDLead-lagLQGH∞µ-synthesisRobust H∞Figure 4.16: Time-domain performance assessment of different controllerson miniaturized OISsThe robust H∞ controller is compared with the µ-synthesis controller inTable 4.4. The order of robust H∞ controller is much smaller than that ofµ-synthesis controller (C4) because it mathematically constraints the con-troller order. The improvements in order is far more significant than thelarge-scale case because as uncertainties become larger and more complex(compare Figure 4.14 with Figure 4.7), order of µ-synthesis controller dras-tically increases. In Table 4.4, the values outside parenthesis for trackingerror and control effort are mean values, while the ones inside parenthesis arethe worst-case degradations. The robust H∞ controller reduces the mean(C1) and degradation (C3) of the tracking error as well as mean (C2) andthe degradation (C3) of the control effort compared to the µ-synthesis con-troller. The improvements achieved are smaller compared to ones in simula-tions. This can be associated with the increased variabilities in miniaturizedOISs compared to large-scale OISs (compare Figure 4.14 with Figure 4.7)because as uncertainties become larger and more complex, conservativenessof robust controllers increases.594.4. Application to Miniaturized OIS PrototypesTable 4.4: Summary comparison of µ-synthesis with Robust H∞ on minia-turized OISsµ-synthesis Robust H∞Designed order (–) 289 11Reduced order (–) 61 10Tracking error (deg) 0.0470 (0.0105) 0.0376 (0.0095)Control effort (mAh) 0.9157 (0.3395) 0.8940 (0.3337)Computation time (s) 2443 303Figure 4.17 and Figure 4.18 show the results of the frequency-domainperformance tests. Figure 4.17 and Figure 4.18 compares the robust H∞controller with the µ-synthesis controller in terms of the magnitude Bodeplot of the transfer functions from r to e and from r to u, respectively. Asshown in Figure 4.17, the robust H∞ controller has better error rejectioncompared to the µ-synthesis controller (C1). More specifically, 0 dB crossingfrequency is 24 Hz for robust H∞ controller, while the one for µ-synthesiscontroller is 20 Hz. These values are compatible with bandwidth of thehand-shake disturbances (Figure B.2) and is much smaller than actuatorbandwidth (which is about 150 Hz as shown in Figure 4.14).604.4. Application to Miniaturized OIS Prototypes100 101 102 103Magnitude (dB)-30-25-20-15-10-50510Robust H∞µ-synthesisFrequency (Hz)Figure 4.17: Frequency response of transfer function from reference to track-ing error on nominal modelIn Figure 4.18, the robust H∞ controller uses less control effort comparedto the µ-synthesis controller in low-frequency band (C2). The improvementsachieved in tracking error and control effort can be associated with the abil-ity of the proposed robust H∞ controller design method to systematicallygenerate a good initial guess for the solution of the optimization problemcorresponding to the proposed controller design problem as well as handlingnon-integer correction factors in the proposed uncertainty model. On theother hand, the iteration time for design of the robust H∞ controller isslightly higher than the µ-synthesis controller because the µ-synthesis con-troller is designed using numerically-optimized off-the-shelf algorithms whilecustom-made algorithms are developed and used for design of the robust H∞controller.614.4. Application to Miniaturized OIS Prototypes100 101 102 103Magnitude (dB)-100-90-80-70-60-50-40Robust H∞µ-synthesisFrequency (Hz)Figure 4.18: Frequency response of transfer function from reference to con-trol input on nominal modelTable 4.5 compares the percent improvements achieved in tracking per-formance by the robust H∞ controller with conventional controllers in threecase studies. The conventional controllers are the nominal controllers andthe µ-synthesis controller. The nominal controllers are represented by themean value of the four (PID, lead-lag, LQG, and H∞ controller) controllers.The percent improvements are calculated by normalization of the trackingerror signal RMS values with the RMS value of the mean RMS value of thehand-shake disturbance signals (Figure B.1). In each cell, the value on topcorresponds to the improvement in nominal tracking performance (solid barsin Figure 3.9, 4.9, 4.16) while the value at bottom shows the improvement indegradation of the tracking performance error bars in Figure 3.9, 4.9, 4.16.In comparison with the nominal controllers, the improvements in minia-turized is less than large-scale and improvements in large-scale is less thansimulations because the uncertainties that has to be considered by the ro-bust controller design methods are larger and more complex. Therefore,relatively modest performance is achievable. On the other hand, comparedto the µ-synthesis controller the improvements in miniaturized is more than624.5. Practical Implications of the Experimental ResultTable 4.5: Percent improvements achieved by robust H∞ controller com-pared to conventional controllers for different case studiesSimulations Large-scale MiniaturizedNominalcontrollers43.81087.832840.85586.923430.85585.4054µ-synthesiscontroller2.99090.81083.03150.12389.93241.1936large-scale and improvements in large-scale is more than simulations. Thiscan be associated with the increased variabilities in miniaturized OISs com-pared to large-scale OISs (compare Figure 4.14 with Figure 4.7) becauseas uncertainties become larger and more complex, conservativeness of theµ-synthesis controller controllers increases.4.5 Practical Implications of the ExperimentalResultThrough experimental work presented in Section 4.3 and Section 4.4, it isdemonstrated that the proposed method is implementable in an experimen-tal setting and provides closed-loop stability and performance advantageswhen used on miniaturized OISs. On top of this, the proposed method of-fers practically economic, systematic, and universal approach to controllerdesign for the OISs.Using the proposed method in practice, a single controller can be de-signed and implemented on the entire product set. This can save manu-facturers time and cost to test, model, and design controllers for individualOISs. Specifically, all controller design methods include trial-and-error thatmakes controller design for individual OISs even more overwhelming.The proposed method offers practical advantages also over other robustcontrollers. First of all, it provides a systematic method of optimizing theclosed-loop performance. This advantage saves a lot of trial-and-error ef-forts that has to go into de-tuning of nominal controllers to recover robustclosed-loop stability/performance for the uncertainty region in Section 4.2.2.Secondly, as demonstrated in Section 4.4.3 increase in uncertainties leads todrastic increase in iteration time and controller order of the µ-synthesiscontroller. This makes the µ-synthesis controller design extremely tediousbecause on top of the long iteration time, a tedious sequential procedure634.5. Practical Implications of the Experimental Resultencompassing order reduction and closed-loop stability/performance anal-ysis for the uncertainty region is needed to design the controller. Finally,the proposed method can design a robust controller for the entire produc-tion line based on only a limited number of representative samples, thusdecreasing system identification efforts significantly.Furthermore, the proposed method takes universality into account. Theproposed method does not make any assumptions about any specific param-eter value/range. Therefore, the proposed method can be directly appliedto any detailed designs (e.g. material, dimensions, component parameters,etc.) of OIS as long as it complies with the same conceptual design reviewedin Section 2.1. The detailed design of OIS could vary rapidly to adapt thisconceptual design to different mobile phones or different models of the samemobile phone according to cost, size, performance, and power requirements.Finally, low-cost manufacturing techniques can be employed for fabrica-tion of the lens-platform and assembly of moving-magnet actuators withoutconcern about their tolerances. Therefore, final price can be reduced whichis an important factor in mobile phone market.The practical advantages of the proposed method can be exploited toreduce final cost of the OISs to replace software-based algorithms. ReplacingOISs with software-based algorithms can improve the image quality of themobile phone cameras.64Chapter 5ConclusionsIn conclusion, the proposed robust controller design method was shown tobe a systematic scheme to design a controller that can deal with productvariabilities in minaturized OISs and is simple enough to be implementablein a mobile phone device. The experimental results demonstrates feasibil-ity of the proposed method in an experimental setting, and its capabilityto deal with physical product variabilities. This can expand application ofminiaturized and low-cost OISs to replace conventional post-processing al-gorithms currently implemented on many mobile phone devices, which is abig step to improve image quality of the mobile phone cameras.In this chapter, we summarize important results that supports the drawnconclusions, list contributions of this thesis, and point out some potentialfuture work.5.1 SummaryIn this thesis, a robust controller design method was developed for a novelmagnetically-actuated lens-tilting OIS designed for mobile phone devices toovercome challenges arising from product variabilities of micro-manufacturingprocesses and implementation limitations of the controller on mobile phonedevices.The details of the OIS including its mechanical design (Section 2.1),control system configuration (Section 2.2), and performance objectives andspecifications (Section 2.3) were reviewed in Chapter 2. In Chapter 3, firsta comprehensive FEA was performed on the model of the reviewed OIS toanalyze the effect of beam width variations and force imbalances on tiltingdynamics (Section 3.1). Using combination of FEA results with knowledgeof mechanical design of OIS, a model structure and an uncertainty struc-ture were proposed for tilting dynamics in Section 3.2. To deal with themodel, a robust controller design method was proposed in Section 3.1 thatconsiders uncertainties through vertices of the uncertainty region to designa low-order controller. The proposed method transformed the controller de-sign problem to an optimization problem that could be solved employing655.1. Summaryefficient and reliable algorithms. In the optimization problem, constraintfunctions represented robust closed-loop stability criterion and the objectivefunction represented worst-case closed-loop performance. Therefore, closed-loop stability could be guaranteed against uncertainties while optimizing theworst-case performance. To solve the optimization problem, the proposedmethod provided a systematic way of obtaining a decent initially-feasiblesolution to avoid conservatism.Chapter 4 presented the experimental results and discussions. Sec-tion 4.1 described the experimental setups used for system identification(Section 4.1.1) and closed-loop implementation in time-domain (Section 4.1.2)and frequency-domain (Section 4.1.3). Then, some experimental methodolo-gies were used based on the setups for performing the tests and analysingthe results, including system identification (Section 4.2.1), closed-loop sta-bility (Section 4.2.2), and closed-loop implementation in time-domain (Sec-tion 4.2.3) and frequency-domain (Section 4.2.4). Then, same setups andmethodologies were used to implement the proposed method on large-scale(Section 4.3) and miniaturized (Section 4.4) OISs.In Section 4.3, experimental results obtained from the application of theproposed method on large-scale OISs were presented. This experiment hadtwo objectives: (1) deal with physical product variabilities (2) implementthe proposed method in an experimental setting. The large-scale prototypeswere scaled 5 times the original conceptual design. The physical productvariabilities in miniaturized OISs were mimicked by intentional beam widthvariations in large-scale lens-platforms. Moreover, the moving-magnet ac-tuators and the base were custom-made and used commonly across all lens-platforms. The proposed method was directly applied to large-scale OISs byexperimental identification of dynamics parameters and uncertainty param-eters. Through large-scale experiments, it was shown that the proposedmethod is executable in an experimental setting and capable of dealingwith physical product variabilities. The experimental results suggested thatrobust controllers designed based on the proposed method could achieveclosed-loop stability for all OISs while nominal controllers cannot. Moreover,the robust controllers reduced the tracking error to almost half comparedto de-tuned versions of the nominal controllers in general. The propose ro-bust H∞ method also outperformed conventional robust controller designmethods, namely µ-synthesis. The proposed robust H∞ method reducedthe nominal tracking error by about 3% and its variabilities by about 0.1%with almost the same control effort. The controller order was decreased tohalf while the iteration time was increased slightly compared to µ-synthesis.Section 4.4 explained experimental results obtained from the application665.1. Summaryof the proposed method to miniaturized OISs to demonstrate applicabil-ity of the method to miniaturized OISs containing mass-produced parts.The dimensions of the minaturized prototypes were compatible with mobilephones, and their parts were fabricated in a batch-based manner with nointentional uncertainties. System identification results showed increase inmagnitude and complexity of the product variabilities compared to simu-lation and large-scale results; however, the proposed method was shown tobe applicable to miniaturized OISs through some mild modifications. Theexperimental results suggested that robust controllers designed based onthe proposed method could achieve closed-loop stability for all OISs whileconventional nominal controllers cannot. Moreover, the robust controllersreduced the tracking error to almost half compared to de-tuned versions ofthe conventional nominal controllers in general. The complexity of the un-certainties in miniaturized OISs made µ-synthesis controller design methodpractically challenging considering controller order and iteration time. Theproposed robust H∞ method outperformed µ-synthesis in terms of controllerorder, iteration time, and closed-loop performance due to increased uncer-tainties. The proposed robust H∞ method reduced the nominal trackingerror by about 10% and its variabilities by about 1.2% while also reducingcontrol effort by 2.4% and its variabilities by 1.7%. The controller order isreduced 6 folds, and iteration times 8 folds.In comparison with the conventional nominal controllers, the improve-ments in miniaturized OISs is less than large-scale OISs, and improvementsin large-scale is less than simulations because the uncertainties that has to beconsidered by the robust controller design methods are larger and more com-plex. Therefore, relatively modest performance is achievable. On the otherhand, compared to the µ-synthesis controller the improvements in minia-turized is more than large-scale and improvements in large-scale is morethan simulations. This can be associated with the increased variabilitiesin miniaturized OISs compared to large-scale OISs because as uncertain-ties become larger and more complex, conservativeness of the µ-synthesiscontroller controllers increases.The improvements achieved in the control of the magnetically-actuatedlens-tilting optical image stabilizer can expand its application in mobilephone cameras to replace post-processing algorithms which is a significantstep towards superior image qualities in mobile phone cameras.675.2. Contributions5.2 ContributionsThe main contributions of this thesis can be outlined as follows:• FEA was performed on CAD model of the miniaturized magnetically-actuated lens-tilting OISs to characterize their dynamics and uncer-tainties.• A dynamic uncertainty model is developed based on FEA results andmechanical design of miniaturized magnetically-actuated lens-tiltingOISs to describe dynamics and uncertainties of controller design pur-poses.• A robust controller design problem is formulated to deal with the de-veloped dynamic uncertainty model of OIS• Executability of the proposed modeling and controller design methodis demonstrated in an experimental setting with physical product vari-abilities.• Applicability of the proposed modeling and controller design methodto real miniaturized OIS production lines were demonstrated by im-plementation miniaturized OISs.• Design guidelines are developed for the conventional nominal con-trollers (i.e, PID, lead-lag, LQG, and H∞ controller) tailored for themagnetically-actuated lens-tilting OIS application (Appendix A).5.3 Future WorkThe magnetically-actuated lens-tilting OIS is a novel conceptual design in-dicating great potential for OIS in mobile phone applications as discussed inSection 1.3.2. This novel concept can benefit greatly from further researchand development on mechanical design, modeling, and controller design.In future, the research presented in this thesis can be further expanded assummarized next.5.3.1 Multiple Robust H∞ ControlThe proposed method treats the uncertainty region (Figure 3.4) in a mono-lithic way; however, this approach leads to conservatism when uncertainties685.3. Future Workare large and complex as shown in small-scale experiments. To reduce con-servatism, the multiple robust H∞ control theories [2, 62] divide the uncer-tainty region into several subregions and design a robust controller for eachsubregion.5.3.2 Fixed-order Robust H∞ ControlAnother limitation of the proposed method is that the controller order iscorrelated with order of the plant model. Therefore, the closed-loop per-formance is compromised if less controller order is desired. To improve thistrade-off, fixed-order robust H∞ control theories [48, 49] explicitly incor-porate controller order constraint in the mathematical formulation of therobust controller design problem. Therefore, an arbitrarily simple controllercan be designed based on an arbitrarily complex plant model to optimizeclosed-loop performance; however, it is not trivial whether fixed-order robustH∞ control theories can offer advantages compared to the proposed methodfor this application because in general they typically include some conser-vatism to incorporate the controller order constraint in the mathematicalformulation.5.3.3 Implementation on Physical Mobile Phone CamerasFinally, the experimental results obtained in this thesis represent the im-age blur by the RMS value of the tracking error signal that might not berealistic enough. 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Yuan, A 3D Printing and Moulding Method of the Fabrication ofa Miniature Voice Coil Motor Actuator, 2015, university of BritishColumbia.[68] X. Zhang, B. Koo, S. M. Salapaka, J. Dong, and P. M. Ferreira, “Ro-bust control of a mems probing device,” IEEE/ASME Transactions onMechatronics, vol. 19, no. 1, pp. 100–108, 2014.[69] P. Zhao, A. Alizadegan, R. Nagamune, and M. Chiao, “Robust con-trol of large-scale prototypes for miniaturized optical image stabilizerswith product variations,” in Proceedings of the 54th Annual Conferenceof the Society of Instrument and Control Engineers of Japan (SICE),2015, pp. 734–739.76[70] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control.Prentice Hall, New Jersey, 1996.[71] G. Zhu, J. Penet, and L. Saydy, “Robust control of an electrostaticallyactuated MEMS in the presence of parasitics and parametric uncer-tainties,” in Proceedings of the American Control Conference, 2006,pp. 1233–1238.77Appendix AController DesignMethodologies andParameter ValuesThis section presents methodologies used for designing baseline controllersand resulting parameter values for lens-tilting OIS.A.1 Notch Filter DesignNotch filter is used in series with classical controllers, namely PID andlead-lag, to deal with lightly-damped modes of OIS. The notch filter isparametrized as:Cnotch (s) =3∏i=1s2 + 2ξˆiωˆnis+ ωˆ2nis2 + 2ωˆnis+ ωˆ2ni, (A.1)where Cnotch (s) is the transfer function of the notch filter, and ξˆi and ωˆniare, respectively, the damping ratios and natural frequencies of the resonancemodes obtained by system identification in Section 3.4, 4.3, and 4.4. Thedesign parameters values of the notch filter for each case are summarized inTable A.1.Table A.1: Notch-filter design parameter valuesSimulations Large-scale Minaturizedωˆn1 106 64 95ξˆ1 5× 10−3 1× 10−3 2× 10−2ωˆn2 170 102 230ξˆ2 1× 10−4 1× 10−4 4.9× 10−2ωˆn3 665 252 352ξˆ3 1× 10−3 1× 10−3 4.7× 10−178A.2. PID Controller DesignA.2 PID Controller DesignThe standard PID controller is parametrized as:CPID (s) = Kp +Kis+Kds12pifcs+ 1, (A.2)where CPID (s) is the transfer function of the PID controller and Kp, Ki,and Kd, are the proportional, integral, and derivative gains, respectively.fc is the cut off frequency of the low pass filter used to deal with noiseamplification limitations of the derivative term.In PID controller, increasing proportional gain (Kp) reduces tracking er-ror (O1). The integral term eliminates stead-state error (S1) and increasingintegral gain (Ki) reduces tracking error (O1). The derivative term pro-vides damping properties for closed-loop system and increasing derivativegain (Kd) increases stability margin (S2). The low-pass filter term limitshigh high-frequency controller gain caused by the derivative term. Decteas-ing fc decreases high-frequency controller gain (S3).The procedure used for designing PID controller is listed in the following:1. set all parameters equal to zero.2. increase Kp to improve (O1) until (S2) is violated. (P-controller)3. increase Ki to improve (O1) until a acceptable bandwidth is achieved.4. set fc=10 kHz as an acceptable cut off frequency.5. set Kd =(S3)−Kp2pifcto ensure satisfaction of (S3)6. check (S2). If (S2) is satisfied, finish; else, decrease Kp slightly and goto 5.De-tuning of PID controller was done by appropriately decreasing pro-portional and integral gains until the closed-loop system associated withOISs are stable. Due to conservativeness of the de-tuned PID controller,the derivative and low-pass filter terms are unnecessary. The parametersobtained by implementation of this procedure are shown in Table A.2.79A.3. Lead-lag Controller DesignTable A.2: Controller design parameters of the PID controllerSimulations Large-scale MiniaturizedKp 7× 10−4 (6× 10−8) 1× 100 (1× 10−1) 1.2× 10−2 (3× 10−3)Ki 5× 10−1 (1× 10−2) 4× 101 (4× 101) 1× 100 (5× 10−2)Kd 1× 10−7 (−) 5× 10−3 (−) 2× 10−7 (−)fc 1× 104 (−) 3× 101 (−) 1× 104 (−)A.3 Lead-lag Controller DesignThe standard lead-lag controller is parametrized as:Cll (s) = KpK1zleads+ 11pleads+ 1+1zlags+ 11plags+ 1, (A.3)where Cll (s) is the transfer function of the lead-lag controller, Kp is theP-controller gain, K is the lead-lag controller gain, and Kd, zlead and leadare, respectively zero and pole of the lead, and zlag and lag are, respectivelyzero and pole of the lag.The procedure used for designing PID controller is listed in the following:1. set K = 102KpK∗tfor acceptable steady-state error and ωb=0.2. increase ωbw incrementally to improve (O1).3. set zlag = ωbw√K and plag = ωbw/√K.4. read ωgc from open-loop bode plot and set plead = ωgc√(S3)Kpand zlead =ωgc√Kp(S3) to ensure (S3).5. check (S2). If (S2) is satisfied, go to Step 2; else, finish.De-tuning of lead-lag controller was done by appropriately decreasingproportional gains Kp and K and bandwidth ωbw such that the closed-loop system associated with OISs are stable. The parameters obtained byimplementation of this procedure are shown in Table A.3.80A.4. LQG Controller DesignTable A.3: Controller design parameters of the lead-lag controllerSimulations Large-scale MiniaturizedKp1.0× 10−3(1.0× 10−4)1.0× 100(1.0× 10−1)1.2× 10−2(1.0× 10−2)K2.0× 101(1.0× 101)2.2× 101(2.2× 101)5.0× 101(2.0× 102)plead2.8× 103(2.8× 103)5.3× 102(5.3× 102)9.3× 102(9.3× 102)zlead2.0× 103(2.0× 103)1.8× 102(1.8× 102)3.1× 102(3.1× 102)plag2.2× 10−2(2.2× 10−2)1.7× 100(1.7× 100)2.1× 100(2.1× 100)zlag5.7× 10−1(5.7× 10−1)1.2× 102(1.2× 102)3.2× 102(3.2× 102)A.4 LQG Controller DesignThe LQG servo controller minimizes the cost functionJ = E{limτ→∞1τ∫ τ0(xTQxx+ uTQuu+ xTi Qixi)dt},subject to system dynamicsx˙ = Ax+Bu+ wy = Cx+Du+ vwhere the process noise w and measurement noise v are Gaussian whitenoises with covariances Qw and Qv, respectively. In the cost function Edesignates expected value, x is the state vector, u is the control input vector,and xi is the state variable associated with the integral of the tracking error.The design parameters Qx, Qu, and Qi are positive definite weightingmatrices used to penalize states, control input, and tracking error, respec-tively. These weighting matrices are used to achieve performance objectivesand meet performance specifications. Tracking error (O1) can be reducedby increasing Qi. To increase stability margin (S2), Qx should be increased.Finally, Qu is increased to reduce control effort (O2) and avoid saturationdue to noise amplification (S3).De-tuning of LQG controller was done by decreasing Qi until the closed-loop system associated with OISs are stable. The parameters obtained byimplementation of this procedure are shown in Table A.4.81A.5. H∞ Controller DesignTable A.4: Controller design parameters of the LQG controllerSimulations Large-scale MiniaturizedQx 1× 100 (1× 100) 1× 100 (1× 100) 1× 100 (1× 100)Qu 4× 101 (4× 101) 1× 102 (1× 102) 1× 100 (1× 101)Qw 1× 10−3 (1× 10−3) 1× 10−3 (1× 10−3) 1× 10−3 (1× 10−3)Qv 1× 10−3 (1× 10−3) 1× 10−3 (1× 10−3) 1× 10−3 (1× 10−3)Qi 1× 105 (4× 10−2) 5× 105 (1× 104) 1× 105 (1× 101)A.5 H∞ Controller DesignThe H∞ controller minimizes the H∞ norm of the system from r to z =[eTw uTw]T in A.1.Figure A.1: Weighted control system configuration H∞ controller designThe design parameters We(s) and Wu(s) are weighting functions usedto penalize tracking error and control input, respectively. These weightingfunctions are used to achieve performance objectives and meet performancespecifications. We is used to improve (O1) and meet (S1) and (S2). Wu isused to achieve (O2) and meet (S3). To avoid high order controller (O3), wepropose using constant Wu and first order We parameterized as follows [70]:We(s) =s/MH + ωbs+ ωbML, (A.4)where ML, MH , and ωb are design parameters which, respectively, denotelow-frequency gain, high-frequency gain, and bandwidth of W−1. To reducesteady state error (S1), ML should be reduced. Tracking error (O1) canbe reduced by increasing ωb which corresponds to bandwidth. To increasestability margin (S2), peak of sensitivity MH should be reduced. Finally,Wu is taken as a constant gain, and it is increased to reduce control effort82A.6. Robust Controller Design(O2) and avoid saturation due to noise amplification (S3).De-tuning of H∞ controller was done by decreasing ωb until the closed-loop system associated with OISs are stable. The parameters obtained byimplementation of this procedure are shown in Table A.5. In the following,ωb is reported in Hz and the rest of the parameters are in dB.Table A.5: Controller design parameters of the H∞ controllerSimulations Large-scale MiniaturizedML −4× 101 (−4× 101) −4× 101 (−4× 101) −4× 101 (−4× 101)ωb 1× 102 (2× 100) 7× 101 (2× 10−1) 1× 102 (1× 100)MH 2× 100 (2× 100) 2× 100 (2× 100) 2× 100 (2× 100)Wu −4× 101 (−4× 101) 1× 100 (1× 100) −3× 101 (−3× 101)A.6 Robust Controller DesignThe robust controller design theories, namely µ-synthesis and robust H∞rely on a similar design parameters and methodologies as the H∞ controllerdesign method, except for the fact that there is no de-tuning procedure in-volved because the theories themselves guarantee robust closed-loop stabil-ity and performance against uncertainties. Table A.6 and Table A.7 showsthe controller design parameters obtained for µ-synthesis and robust H∞controllers, respectively.Table A.6: Controller design parameters of the µ-synthesis controllerSimulations Large-scale MiniaturizedML −4× 101 −2× 101 −4× 101ωb 8× 101 2× 101 8× 101MH 2× 100 2× 100 2× 100Wu 2× 102 3× 101 −1× 101As the primary objective of the OIS is tracking (O1). Therefore, the gen-eral approach to design of all controllers have been to minimize the trackingerror until atleast one of the performance specifications (S2) and (S3) areviolated. In this regard, Table A.8 summarizes the general trade-off (posi-tive (+) or negative (–) effect) that each of the introduced controller designparameters have on the control objectives and specificatins.83A.6. Robust Controller DesignTable A.7: Controller design parameters of the robust H∞ controllerSimulations Large-scale MiniaturizedML −4× 101 −2× 101 −2× 101ωb 8× 101 5× 100 1× 100MH 2× 100 2× 100 8× 100Wu 2× 102 3× 101 1× 101Table A.8: Summary of performance trade-offs associated with controllerdesign parametersO1 S2 S3Kp + – –Ki + – –ωbw + – NQi + – –Qu – + +Qw + + –Qv – N +ML – + +ωb + + –MH + – –Wu + – –84Appendix BHand-shake DisturbanceDatabaseThis section explains details of hand-shake disturbance database. Thisdatabase is used in the time-domain setup (Section 4.1.2) as well as con-trol objective definition (Section 2.3). The explanation includes propertiesof hand-shake signals in time-domain and frequency-domain.B.1 Time-domain SignalsHand-shake disturbance data are collected from built-in gyro-enhanced mo-tion sensors of a mobile phone while holding the mobile phone to take aphotograph. A database is prepared by hand-shake disturbance data of 5different people. Time-domain signals of are displayed in Figure B.1, whereHSD stands for hand-shake data.85B.2. Frequency-domain SpectraTime (sec)0 1 2 3 4 5Tilt angle (deg)-1-0.8-0.6-0.4-0.200.20.40.60.81 HSD 1HSD 2HSD 3HSD 4HSD 5Figure B.1: Time-domain signals of hand-shake disturbance databaseThe DC component was removed since the goal of image stabilizationis to compensate for involuntary hand-shake and not intentional pan. Thesignals are of relatively random nature with a magnitude of typically lessthan 1 degree.B.2 Frequency-domain SpectraTo validate the database in the frequency-domain, a spectral analysis is per-formed on this database through discrete Fourier transform [7]. Figure B.2shows the spectral density for each of 5 hand-shake data in the data set. Asshown, the spectral density of different hand-shake data are consistent witheach other. In general, the dominant frequencies are below 20 Hz which iscompatible with results published in the literature [18, 58]. The samplingfrequency of the gyro-enhanced motion sensor was 100 Hz, and thus accord-ing to the Nyquist-Shannon sampling theorem [7], only frequencies below 50Hz were considered.86B.2. Frequency-domain SpectraFrequency (Hz)0 10 20 30 40 50Magnitude (dB)0102030405060HSD 1HSD 2HSD 3HSD 4HSD 5Figure B.2: Frequency-domain spectra of hand-shake disturbance database87Appendix CModel Parameter ValuesC.1 Uncertainty Model Parameter ValuesThe numeric values of the parameters used for the dynamic uncertaintymodel of OIS (G(s, δ) in 3.1) are tabulated in Table C.1. In this table, thevalues used for simulations, large-scale, prototype, and small-scale prototypeare presented in separate columns.Table C.1: Model parameter valuesSimulations Large-scale Minaturizedω∗n1 106 64 95ξ∗1/[ξ1 ξ1] 5× 10−3 1× 10−3 [2 3]× 10−2ω∗n2 170 102 230ξ∗2/[ξ2 ξ2] 1× 10−4 1× 10−4 [3 5]× 10−2ω∗n3 645 230 343ξ∗3/[ξ3 ξ3] 1× 10−3 1× 10−3 [1 3]× 10−2ω∗n4 665 252 352ξ∗4/[ξ4 ξ4] 5× 10−3 1× 10−3 [1 5]× 10−2k∗shift 0 0 0k∗tilt 4× 105 4× 100 4× 102kshift 8× 104 2× 10−1 2× 101kshift −8× 104 −2× 10−1 −2× 101ωn2 180 97 221ωn2 160 107 341ck 1.7 1.7 1.2c1 1.1 1.1 0.3c3 0.3 0.3 0.2c4 0.4 0.4 0.8τ 1.1 1.1 1.188C.2. Actuator Model Parameter ValuesC.2 Actuator Model Parameter ValuesThe numeric values of the parameters used for mechanical design and dy-namic uncertainty modeling of OIS are tabulated in Table C.2. In tables, thevalues used for simulations, large-scale, prototype, and small-scale prototypeare presented in separate columns.Table C.2: Moving-magnet actuator parameter valuesSimulations Large-scale MinaturizedInner diameter (mm) 1.0 2.06 0.5Outer diameter (mm) 4.1 14.4 3.3Height (mm) 4.0 21.5 4.0Wire gauge (AWG) 36 28 36Wire turns (-) 372 960 347Diameter (mm) 2.5 6.34 1.6Thickness (mm) 0.65 1.6 0.8Air-gap at eq. (mm) 0.25 1 1The moving-magnet actuators from current inputs to torque output dis-play static nonlinear behavior. Therefore, we will model the moving-magnetactuator as a static nonlinear map, i.e., the ’Look-up Table’ block in Figure2.2, and obtain its inverse map to compensate its nonlinearity.The model of the moving-magnet actuators is developed considering thecurrents of coils as the input and the torque generated on the lens-platformabout the x-axis as the output. Figure C.1 illustrates the tilt actuationabout one axis. A torque T is generated on the lens-platform by of forces Fgenerated on the magnets by coils with the configuration shown.Figure C.1: Tilt actuation89C.2. Actuator Model Parameter ValuesThe forces F necessary to generate a torque of T can be obtained as:F = T/4r. (C.1)The force generated on the magnet is a function of the current of the coiland the air-gap. According to Lorentzs law, force is linearly proportional tothe current, and due to Amperes law, it is inversely related to the air-gap.To quantitatively obtain this relationship, finite-element method magnetics(FEMM) software [40] is used for the parameters of moving-magnet actua-tors in Table C.2.Figure C.2 shows the inverse relationship of the force with the air-gap fora constant current of 1 A as obtained by FEMM for the large-scale prototypeas an example.Air-gap (mm)0 1 2 3 4 5Magnetic force (N)00.10.20.3Figure C.2: Force–air-gap relationship (i = 1 A)This relationship is used as a look-up table denoted by p(z) where z isthe air-gap. Hence, the force generated on each magnet can be describedas F = p(z)i. When the lens-platform is tilt about x-axis for , air-gaps ofpairs of coil/magnet on two sides of this axis can be calculated as z1,3 =z0− rsin(θ) and z2,4 = z0 + rsin(θ), where z0 is the equilibrium air-gap, r isthe distance of magnets from central axes, and the zk (k = 1, 2, 3, 4) is theair-gap of the k-th moving-magnet actuator. These air-gap values of eachactuator is used to calculate the required current to be applied to its coil(ik) by ik = F/p(zk).90"@en ;
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dcterms:title "Robust control of miniaturized optical image stabilizers for mobile phone cameras"@en ;
dcterms:type "Text"@en ;
ns0:identifierURI "http://hdl.handle.net/2429/58778"@en .