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Position Self-Sensing in the Presence of Creep, Hysteresis, and Self-Heating for Piezoelectric Actuators Islam, Mohammad Nouroz 2013

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Position Self-Sensing in the Presence ofCreep, Hysteresis, and Self-Heating forPiezoelectric ActuatorsbyMohammad Nouroz IslamB.Sc. in Industrial and Production Engineering, Bangladesh University of Engineeringand Technology, 2005M.Sc. in Industrial and Production Engineering, Bangladesh University of Engineeringand Technology, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)July 2013? Mohammad Nouroz Islam, 2013AbstractPiezoelectric ceramic actuators are widely used in micro-/nano-positioningsystems due to expedient characteristics such as fast response time, high stiff-ness, high resolution, etc. However, nonlinear effects such as hysteresis andcreep affect the position accuracy of the systems if not compensated. Often,feedback position sensors are mounted to the systems to eliminate hysteresisand creep. Nonetheless, installation of feedback sensors can be prohibitive dueto space constraints, reliability and cost. Alternatively, position self-sensing tech-niques are used to eliminate the position sensor. In this research, the objectiveis to develop a position self-sensing technique considering the nonlinear effects.To model the actuators for control or self-sensing, they are often considered ascapacitive elements. A novel real-time impedance measurement technique isdeveloped based on high frequency measurements to obtain clamped capaci-tance. Based on the real-time measurement, an improved constitutive modeland parameter identification technique is presented which includes the posi-tion dependent capacitance. As a means for position self-sensing, position islinearly related to charge. However, charge measurement is prone to drift andrequire sophisticated hardware to implement. The new relationship betweenposition and capacitance opens a new avenue for non-traditional position self-sensing; however, due to measurement noise, this new technique is only usefulfor slow operations. In this research, a novel position observer is presented thatfuses the capacitance-based self-sensing with the traditional charge-based self-sensing. This allows the position estimation over a frequency band ranges from0Hz to 125Hz where creep and rate-dependent hysteresis are observed. The es-timation error is close to 3% when compared to a position sensor. Continuousoperations at frequencies larger than 20Hz contribute to self-heat generation inthe actuators. This elevated temperature is detrimental to the performance andlife of the actuator. In this research, a self-heat generation model is presentedbased on power loss in the actuator to predict the temperature rise. The pre-dicted temperature is then used to compensate the temperature related varia-tion in the position observer. The temperature prediction error is less than 2?Cwhich creates a position estimation error close to 4% up to a temperature varia-tion of 55?C.iiPrefaceThe thesis presents the research findings on position self-sensing and con-trol of piezoelectric actuator which was conducted in the Control and Automa-tion Laboratory, at the School of Engineering, UBC, under supervision of Dr.Rudolf Seethaler. Part of the thesis has been published in peer-reviewed jour-nals and conference proceedings. The contributions highlighted in the thesisare as follows:A novel real-time piezoelectric impedance measurement technique is pre-sented in Chapter 3 which was published in World Intellectual Property Orga-nization in November 2012 for a patent application as "Apparatus and Methodfor In Situ Impedance Measurement of a Piezoelectric Actuator"1. The resultsand the measurement technique were also published in IEEE Canadian Confer-ence for Electrical and computer engineering in 2011 as "Hysteresis indepen-dent on-line capacitance measurement for piezoelectric stack actuators"2. Thetechnique was developed by Dr. Seethaler while I was responsible for experi-mental validation and writing the conference manuscript.In Chapter 4, an improved electromechanical model for piezoelectric actua-tors is presented that employs the real-time impedance measurement techniquedeveloped in Chapter 3. Part of the chapter was published in the Journal ofIntelligent Material, Systems and Structures published by Sage Journals as "AnImproved Electromechanical Model and Parameter Identification Technique forPiezoelectric Actuators"3. The article presents a novel method to identify the pa-1Rudolf J. Seethaler, and Islam, Mohammad N. "Apparatus and method for in situ impedancemeasurement of a piezoelectric actuator", WO2012149649, WIPO, (2012).2Islam, Mohammad, N., Rudolf J., Seethaler, and Mumford, David. "Hysteresis independenton-line capacitance measurement for piezoelectric stack actuators." Electrical and Computer En-gineering (CCECE), 24th Canadian Conference on. IEEE, (2011):1149?1153.3Islam, Mohammad N., and Rudolf J. Seethaler. "An improved electromechanical model andparameter identification technique for piezoelectric actuators." Journal of Intelligent MaterialiiiPrefacerameters of the electromechanical piezoelectric model. My responsibility was tosimulate the model, conduct experiments to validate the model and prepare themanuscript.A novel position observer is presented in Chapter 5 which partly is publishedin IEEE/ASME Transaction on Mechatronics as "Sensorless Position Control forPiezoelectric Actuators Using a Hybrid Position Observer"4. The observer is ableto predict the position of the piezoelectric actuator over a wide range of fre-quencies where nonlinearities such as creep and rate-dependent hysteresis takeplace. The results of creep and rate dependent hysteresis in open loop condi-tion presented in this chapter is published in Review of Scientific Instruments as"Note: Position self-sensing for piezoelectric actuators in the presence of creepand rate-dependent hysteresis"5. Experimental results of closed loop controlare presented in Chapter 6 which are partly published in IEEE/ASME Transac-tion on Mechatronics4. My roles were to conduct the experiments, analyze thedata and prepare the manuscript for the articles.Finally, a new model for self-heat generation is presented in Chapter 7 wherepower loss due to self-heating is yielded to estimate the temperature increasein piezoelectric actuators. A version of this chapter is submitted in Review ofScientific Instruments entitled as "Real-Time Temperature Estimation Due ToSelf-Heating in Piezoelectric Actuators"6. In this study, I developed the self-heatgeneration model, conduct the experimental validation of the model and pre-pare the manuscript for the article. In accordance to the copyright law, the pub-lished materials in the journals, transactions and conference proceedings areincluded with the permission of the publishers.Systems and Structures (2013):1049?1058.4Islam, Mohammad N., and Rudolf J. Seethaler. "Sensorless position control for piezoelec-tric actuators using a hybrid position observer." Mechatronics, IEEE/ASME Transactions on,(2013):1?A?S?-9.5Islam, Mohammad N., and Rudolf J. Seethaler. "Note: Position self-sensing for piezoelectricactuators in the presence of creep and rate-dependent hysteresis." Review of Scientific Instru-ments (2012): 116101?116103.6Islam, Mohammad N., and Rudolf J. Seethaler. "Real-time temperature estimation due to self-heating in piezoelectric actuators." Review of Scientific Instruments(2013): submitted.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxChapter 1: Introduction and Literature Review . . . . . . . . . . . . . . 11.1 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Piezoelectric Actuators (PEAs) . . . . . . . . . . . . . . . . . . . . . . 91.5.1 Stack Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.2 Bender Actuators . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Piezoelectric Actuator Modeling . . . . . . . . . . . . . . . . . . . . . 111.6.1 Linear IEEE Model . . . . . . . . . . . . . . . . . . . . . . . . 111.6.2 Nonlinear Physical Model . . . . . . . . . . . . . . . . . . . . 12vTABLE OF CONTENTS1.6.3 Phenomenological Model . . . . . . . . . . . . . . . . . . . . Rate-independent Model . . . . . . . . . . . . . . Rate-dependent Model . . . . . . . . . . . . . . . Other Phenomenological Hysteresis Models . . . 241.6.4 Cascaded Phenomenological Model for Hysteresis, Creep,and Vibrational Dynamics . . . . . . . . . . . . . . . . . . . 241.7 Position Control of PEA . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7.1 Feedforward (FF) Voltage Control . . . . . . . . . . . . . . . 261.7.2 Feedback (FB) Voltage Control . . . . . . . . . . . . . . . . . 271.7.3 Feedforward/Feedback (FF/FB) Control Scheme . . . . . . 301.7.4 Charge Control Scheme . . . . . . . . . . . . . . . . . . . . . 311.7.5 Integrated Voltage/Charge Control . . . . . . . . . . . . . . 321.8 Position Self-sensing (PSS) . . . . . . . . . . . . . . . . . . . . . . . . 331.8.1 PSS Using Capacitance Bridge . . . . . . . . . . . . . . . . . 331.8.2 PSS Using Charge Measurement . . . . . . . . . . . . . . . . 351.8.3 PSS Using Piezoelectricity . . . . . . . . . . . . . . . . . . . . 361.9 Errors in Piezoelectric Positioning . . . . . . . . . . . . . . . . . . . 381.10 Self-Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.11 Scope of the Work and Objectives . . . . . . . . . . . . . . . . . . . . 421.12 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Chapter 2: Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 472.1 Mechanical Components . . . . . . . . . . . . . . . . . . . . . . . . . 472.1.1 Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.2 Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.1.3 Preload Assembly . . . . . . . . . . . . . . . . . . . . . . . . . 502.1.4 Needle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.1.5 Ball Seats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2 Electrical Components . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.1 Piezoelectric Actuator Driver . . . . . . . . . . . . . . . . . . 522.2.2 Current Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.3 Force Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2.4 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 53viTABLE OF CONTENTS2.2.5 Temperature Sensor . . . . . . . . . . . . . . . . . . . . . . . 532.2.6 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 552.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Chapter 3: Impedance Measurement . . . . . . . . . . . . . . . . . . . . 573.1 Measurement Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2 Superpositioning Through Summing Circuit . . . . . . . . . . . . . 583.3 Superpositioning Through Boost Converter . . . . . . . . . . . . . . 593.4 Effect of Frequency on the Actuator Capacitance . . . . . . . . . . . 593.5 Impedance Measurement Algorithm . . . . . . . . . . . . . . . . . . 633.5.1 Selection of the Sampling Frequency . . . . . . . . . . . . . 633.5.2 Real-time Fourier Transformation . . . . . . . . . . . . . . . 673.5.3 Parameter Identification from Fourier Series Coefficients . 683.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Chapter 4: An Improved Electromechanical Model for Piezoelectric Ac-tuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.1 Proposed Piezoelectric Model . . . . . . . . . . . . . . . . . . . . . . 804.1.1 Mechanical Subsystem . . . . . . . . . . . . . . . . . . . . . 804.1.2 Electrical Subsystem . . . . . . . . . . . . . . . . . . . . . . . 804.2 Power Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.1 Identification of ?, k, b, and m for the Traditional Modelwith Constant Capacitance . . . . . . . . . . . . . . . . . . . 854.3.2 Identification of ?, k, b, and m for the Proposed Modelwith Variable Capacitance . . . . . . . . . . . . . . . . . . . 864.3.3 Identification of Free Capacitance, CT . . . . . . . . . . . . 884.3.4 Identification of Material Properties . . . . . . . . . . . . . . 894.3.4.1 Piezoelectric Strain or Charge Coefficient, d33 . . 894.3.4.2 Piezoelectric Coupling Coefficient, k33 . . . . . . 904.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.4.1 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 90viiTABLE OF CONTENTS4.4.2 Model Validation Through Charge-position Relationship . 934.4.3 Hysteresis Voltage . . . . . . . . . . . . . . . . . . . . . . . . 954.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Chapter 5: Position Self-sensing of Piezoelectric Actuators . . . . . . . 985.1 Charge Based Position Self-sensing, x?I . . . . . . . . . . . . . . . . . 1025.2 Capacitance Based Position Self-sensing, x?CP . . . . . . . . . . . . . 1045.3 Hybrid Position Observer Design . . . . . . . . . . . . . . . . . . . . 1085.4 Hybrid Position Observer Implementation . . . . . . . . . . . . . . 1115.5 Open Loop HPO Performance . . . . . . . . . . . . . . . . . . . . . . 1115.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Chapter 6: Self-sensing Position Control of Piezoelectric Actuators . . 1176.1 The Integral Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2.1 Step Profile with Variable Strokes, P1 . . . . . . . . . . . . . 1196.2.2 Sinusoidal Profile with Different Frequencies, P2 . . . . . . 1206.2.3 DC Profile with Fast Transient Section, P3 . . . . . . . . . . 1316.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Chapter 7: Self-heat Generation . . . . . . . . . . . . . . . . . . . . . . . 1337.1 Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.2 Experimental Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.3 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . 1387.4 Temperature Prediction Using a Dedicated Position Sensor . . . . 1417.5 Temperature Compensation of HPO . . . . . . . . . . . . . . . . . . 1447.6 Temperature and Position Predictions Using the HPO and the Self-Heating Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Chapter 8: Conclusions and Future Research . . . . . . . . . . . . . . . . 1538.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156viiiTABLE OF CONTENTSBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158ixList of TablesTable 1.1 Comparison of the feedback sensors . . . . . . . . . . . . . . 29Table 1.2 Error comparison between different models . . . . . . . . . 40Table 2.1 Actuator properties . . . . . . . . . . . . . . . . . . . . . . . . 49Table 3.1 Sampling frequency and folded ripple frequency at differ-ent values of ns with four times folding . . . . . . . . . . . . 67Table 3.2 Passband and bandwidth as a function of sampling peri-ods for a 48 kHz sampling frequency, and ns = 12 . . . . . . 68Table 4.1 Comparison of parameters from different piezoelectric mod-els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Table 4.2 Model parameter identification . . . . . . . . . . . . . . . . 92Table 4.3 Sensitivity of parameter identification to changes in ca-pacitance measurement . . . . . . . . . . . . . . . . . . . . . 94Table 5.1 Fitting errors in regression models of different orders . . . . 106Table 5.2 ANOVA table for capacitance based PSS regression model . 107Table 5.3 Hybrid position observer parameters . . . . . . . . . . . . . 110Table 5.4 Observer error, (?O = x? ? x?) at different frequencies . . . . . 112Table 6.1 Steady-state errors at segments ?a?? ?h? for P1 . . . . . . . . 122Table 6.2 Error comparison for different profiles . . . . . . . . . . . . 123Table 7.1 Self-heating model parameters . . . . . . . . . . . . . . . . 141Table 7.2 Comparison of the regression statistics . . . . . . . . . . . . 147xLIST OF TABLESTable 7.3 ANOVA table for capacitance based PSS regression modelwith temperature . . . . . . . . . . . . . . . . . . . . . . . . . 148xiList of FiguresFigure 1.1 (a) Direct effect of piezoelectricity (b) indirect effect ofpiezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Piezoelectric crystal structure (a) Cubic (above Curie tem-perature, TC ) (b) Tetrahedral (below Curie temperature, TC 3Figure 1.3 Effect of poling on polarization . . . . . . . . . . . . . . . . 4Figure 1.4 Nonlinear effects on piezoelectric actuator response atvarious frequencies . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.5 Polarization-electric field hysteresis . . . . . . . . . . . . . 6Figure 1.6 Strain-electric field hysteresis . . . . . . . . . . . . . . . . . 6Figure 1.7 Piezoelectric creep behaviour . . . . . . . . . . . . . . . . . 8Figure 1.8 Multilayer piezoelectric stack actuator . . . . . . . . . . . . 11Figure 1.9 Piezoelectric bender actuator . . . . . . . . . . . . . . . . . 12Figure 1.10 Electromechanical model of PEA . . . . . . . . . . . . . . . 13Figure 1.11 Elementary elasto-slide operator . . . . . . . . . . . . . . . 14Figure 1.12 Generalized Maxwell slip model for piezoelectric hysteresis 15Figure 1.13 Extended electromechanical model with a drift operator . 17Figure 1.14 Drift operator with lossy resistance . . . . . . . . . . . . . . 18Figure 1.15 Voltage-stroke hysteresis loops at different voltages . . . . 19Figure 1.16 Preisach operator and model . . . . . . . . . . . . . . . . . 20Figure 1.17 Play operators and superpositioning of the operators . . . 21Figure 1.18 Dead-zone operator . . . . . . . . . . . . . . . . . . . . . . . 22Figure 1.19 A modified structure of PI model . . . . . . . . . . . . . . . 22Figure 1.20 Cascaded model of PEA . . . . . . . . . . . . . . . . . . . . . 25Figure 1.21 Visco-elastic creep model with elastic component . . . . . 25Figure 1.22 Different position control schemes . . . . . . . . . . . . . . 26xiiLIST OF FIGURESFigure 1.23 Feedforward control scheme for the PEA . . . . . . . . . . 27Figure 1.24 Feedback control scheme . . . . . . . . . . . . . . . . . . . 30Figure 1.25 Feedforward branch with feedback control scheme . . . . 31Figure 1.26 Feedback-linearized inverse feedforward control . . . . . 31Figure 1.27 Typical charge control scheme . . . . . . . . . . . . . . . . 32Figure 1.28 Integrated voltage/charge control . . . . . . . . . . . . . . . 33Figure 1.29 Capacitance bridge based self-sensing . . . . . . . . . . . . 34Figure 1.30 Charge based position self-sensing . . . . . . . . . . . . . . 36Figure 1.31 Piezoelectric position self-sensing . . . . . . . . . . . . . . 37Figure 1.32 Maximum errors with operating frequencies . . . . . . . . 39Figure 1.33 Projection of objectives and sub-goals . . . . . . . . . . . . 44Figure 2.1 Piezoelectric actuator and ball-seats . . . . . . . . . . . . . 48Figure 2.2 Frame of the test-bed . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.3 Preload assembly . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 2.4 CAD drawing of the test-bed . . . . . . . . . . . . . . . . . . 51Figure 2.5 Schematic of the experimental setup . . . . . . . . . . . . . 54Figure 2.6 Experimental setup and the test-bed . . . . . . . . . . . . . 54Figure 3.1 Superposition of signals through summing circuit . . . . . 59Figure 3.2 Velocity measurement in clamped and free condition . . . 60Figure 3.3 Normalized voltage and current at 10 Hz . . . . . . . . . . 61Figure 3.4 Normalized ripple voltage and ripple current at 100 kHz . 62Figure 3.5 Amplifier output while supplying the driving voltage . . . 64Figure 3.6 Piezoelectric impedance measurement circuit . . . . . . . 64Figure 3.7 Ripple signal folding at different frequencies . . . . . . . . 66Figure 3.8 Resistance-capacitance model and phasor diagram . . . . 69Figure 3.9 Voltage, position, effective capacitance and resistance mea-surement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.10 Hysteresis in effective capacitance-voltage relationship . . 72Figure 3.11 Effective capacitance measurement with position . . . . . 72Figure 3.12 Hysteresis in effective position-voltage relationship . . . . 73xiiiLIST OF FIGURESFigure 3.13 Effective capacitance measurement with position at var-ious frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 3.14 Effective measurements of resistance and capacitance . . 74Figure 4.1 Schematic representation of the test setup . . . . . . . . . 81Figure 4.2 Force measurement at 50 Hz . . . . . . . . . . . . . . . . . . 81Figure 4.3 Capacitance-position relationship at various frequencies . 83Figure 4.4 Capacitance-position relationship at various frequencies . 87Figure 4.5 Driving signal, U at 10 Hz . . . . . . . . . . . . . . . . . . . 91Figure 4.6 Charge-position and current-position relationship . . . . 93Figure 4.7 Comparison of charge-position relationship . . . . . . . . 96Figure 4.8 Comparison of driving voltage and hysteresis voltage . . . 97Figure 5.1 Rate-dependent hysteresis and creep . . . . . . . . . . . . 99Figure 5.2 Charge-position relationship between 10 Hz ? 300 Hz . . . 101Figure 5.3 Magnitude and phase response . . . . . . . . . . . . . . . . 103Figure 5.4 Position estimation from charge measurement . . . . . . . 104Figure 5.5 Capacitance-position relationship at low frequencies . . . 105Figure 5.6 Capacitance-position relationship at low frequencies . . . 106Figure 5.7 Capacitance-position relationship at low frequencies . . . 107Figure 5.8 Position estimation from capacitance measurement . . . 108Figure 5.9 Hybrid position observer . . . . . . . . . . . . . . . . . . . . 109Figure 5.10 Magnitude plot of different transfer function . . . . . . . . 109Figure 5.11 Hybrid position observer implementation . . . . . . . . . . 111Figure 5.12 Measured position, HPO position and observer error be-tween 0.01 Hz and 1 Hz . . . . . . . . . . . . . . . . . . . . . 113Figure 5.13 Measured position, HPO position and observer error be-tween 10 Hz and 100 Hz . . . . . . . . . . . . . . . . . . . . 114Figure 5.14 Measured position, HPO position and observer error be-tween 50 Hz and DC signal in the presence of creep . . . . 115Figure 6.1 An integral controller schematic . . . . . . . . . . . . . . . 119Figure 6.2 Results with P1 profile . . . . . . . . . . . . . . . . . . . . . 121Figure 6.3 Results with P2a profile . . . . . . . . . . . . . . . . . . . . . 125xivLIST OF FIGURESFigure 6.4 Results with P2b profile . . . . . . . . . . . . . . . . . . . . . 126Figure 6.5 Results with P2c profile . . . . . . . . . . . . . . . . . . . . . 127Figure 6.6 Results with P2d profile . . . . . . . . . . . . . . . . . . . . . 128Figure 6.7 Results with P3 profile . . . . . . . . . . . . . . . . . . . . . 129Figure 6.8 Voltage profiles in open loop and closed loop condition . 130Figure 7.1 Effect of frequencies on self-heat generation . . . . . . . . 137Figure 7.2 Change in steady-state temperature increase with driv-ing frequency (at 100 V sinusoidal signal) . . . . . . . . . . 138Figure 7.3 Change in temperature increase with driving voltage (at150 Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Figure 7.4 Self-heat generation model with position sensor input . . 141Figure 7.5 Self-heat generation prediction and errors . . . . . . . . . 142Figure 7.6 Temperature profile using self-heat generation at variousfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Figure 7.7 Effect of temperature on (a) capacitance-position rela-tionship (b) capacitance measurement at 50Hz . . . . . . . 145Figure 7.8 Effect of frequency on capacitance-position relationship . 145Figure 7.9 (a) Self-heat generation model with HPO input (b) tem-perature compensated HPO . . . . . . . . . . . . . . . . . . 146Figure 7.10 (a) Temperature prediction by self-heat generation model(b) errors in prediction at 100 Hz . . . . . . . . . . . . . . . 149Figure 7.11 (a) Temperature prediction by self-heat generation model(b) errors in prediction at 50 Hz . . . . . . . . . . . . . . . . 149Figure 7.12 Position and temperature estimation and error compari-son . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151xvGlossary of NotationA Surface area of the actuator.CP Clamped piezoelectric capacitance.CT Free piezoelectric capacitance.Ce Effective piezoelectric capacitance.E Electric field.F External force.G(s) Real system.L(s) Observer gain.RP Piezoelectric resistance.?T Change in temperature.? Function for capacitance based position self-sensing.? Force-voltage proportionality constant.x? Measured piezoelectric position.?0 Permittivity in vaccuum.?CC Lmax Maximum controller error in closed loop.?OC Lmax Maximum observer error in closed loop.?TC Lmax Maximum total error in closed loop.?T OLmax Maximum total error in open loop.? Self-heating model parameter.G?(s) Modeled system.x? Estimated piezoelectric position using an ob-server.x?I Charge based position self-sensing.x?Cp Capacitance based position self-sensing.xviGlossary of Notation?s Switiching frequency of HPO.? density.? Time constant.?s Observer damping.a0 Charge-position relationship slope.b Actuator damping.b0 Model intercept.b j Model coefficients.bT n Self-heat geneation model coefficients.c Specific heat.d33 Piezoelectric charge/strain Ripple frequency.fs Sampling frequency.fBW Bandwidth.fPB Passband f ol ded Folded ripple frequency.g Self-heat geneation model coefficient.k Stiffness.kP Piezoelectric stiffness.kT Overall heat transfer coefficient.ks Spring stiffness.k33 Piezoelectric coupling coefficient.m Moving mass in the test-bed.n f Number of frequency Number of cycles over which Fourier transformis performed.ns Number of Samples per folded ripple frequency.q Charge.tss Timespan in steady-state.u Hysteresis loss per driving cycle per unit vol-ume.x Piezoelectric position.xviiAcknowledgmentsI would like to express my heartiest gratitude to my thesis supervisor Dr.Rudolf Seethaler who provided me the opportunity to work on an interestingtopic of piezoelectric actuators. Through his professional guidance and gener-ous support throughout the period, the goals were possible to achieve in thisstudy. His honest dedication to the research, attention to detail and quest forinnovation has set an example in front of me which I have aimed to achieve inmy professional career.I would like to thank the members of my thesis supervisory committee Drs.Abbas Milani, Solomon Tesfamariam, and Ryozo Nagamune for their valuablecomments to the research findings. I also would like to thank Drs. Lukas Bichler,Kenneth Chau and Wilson Eberle for providing access to their lab facilities.This thesis would not be smooth without the kind and timely support ofthe Laboratory manager Russell LaMountain and the machinist Alex Willer whogenerously helped me to build the test-setup for the experiments and grantedaccess to different tools and facilities.I would like to acknowledge the funding support from NSERC on a strate-gic grant project which provided the necessary financial support throughout theperiod.I also would like to thank David Mumford, from Westport Innovations Inc.for his initial support in developing test setup and setting the goals for this re-search. Also, I would like acknowledge the feedback of Dr. Nimal Rajapakse, theprincipal investigator of the project, for his valuable comments at the progressmeetings.My deepest gratitude goes to my parents who sacrificed their pleasure for thebetterment of me throughout their lives. Without their continual support froma distant place, it would be extremely difficult to continue this long journey.xviiiAcknowledgmentsFinally, I would like to take the opportunity to thank my wife, Angela Nusratfor her continual support, encouragement and appreciation since I met her.xixDedicationThis thesis is dedicatedto my ever caring parentsSelina AkhterMohd. Nazrul Islam Fakirto my beloved wifeAngela Nusratand to my loving sonNazif IslamxxChapter 1Introduction and LiteratureReviewPiezoelectric ceramic actuators are widely used in micro-/nano-positioningapplications due to their superior mechanical and electrical properties over tra-ditional actuators. The active material permits miniaturization of the actuatorswhich is a very desirable property in different micro-/nano-positioning applica-tions. However, piezoelectric ceramic actuators suffer from nonlinearities suchas hysteresis and creep which affects the positioning accuracy. Several attemptshave been made earlier to address these issues by developing different modelsfor hysteresis and creep and using these models for sensorless position control.Another branch of sensorless control is the self-sensing technique where posi-tion is reconstructed by measuring some electrical parameter such as charge,capacitance, etc. In this research, the goal is to develop a reliable position self-sensing for feedback control in the presence of hysteresis and creep nonlinear-ity. This eliminates the requirement of a dedicated position sensor in the actu-ation system. Most of the self-sensing techniques are affected by temperaturevariation. Hence, the effect of temperature on the position self-sensing is alsoinvestigated in this research.This chapter provides required background for the research which includesthe fundamentals of piezoelectricity, hysteresis behaviour and their modelingapproaches, creep nonlinearity and modeling, actuator modeling, different po-sition control schemes, self-sensing position estimation and self-heating phe-nomenon. Finally, the motivation, objective and the thesis organization is pre-sented at the end of the chapter.11.1. PiezoelectricityFigure 1.1: (a) Direct effect of piezoelectricity (b) indirect effect of piezoelectric-ity1.1 PiezoelectricityPiezoelectricity is a material property which generates charge when pres-sure is applied to the material. This property is first discovered by Jacques andPierre Curie in 1880 during a study of charge generation due to pressure in differ-ent crystalline structures such as quartz, tourmaline, etc. [1]. The term ?piezo-electricity?is first proposed by Hankel where the prefix ?piezo? is derived from aGreek word ?piezen?which means ?to press? [1]. A reverse effect of piezoelectric-ity, based on the fundamental thermodynamic principles, is proposed by Lipp-mann in 1881 which is later verified by the Curies [1].Thus, the piezoelectric effect is classified either as direct effect or the indirecteffect. Figure 1.1 (a) shows the direct effect where electric charge or voltage isgenerated due to applied mechanical stress. Piezoelectric sensing applicationssuch as displacement or force sensors are based on the direct effect of piezoelec-tricity. The indirect effect is shown in Figure 1.1 (b), where mechanical strain isgenerated due to applied electric charge or voltage. Piezoelectric actuation isbased on the indirect effect of piezoelectricity [2].The natural piezoelectric materials such as quartz, Rochelle salt etc. demon-strate little piezoelectric effect. In the 20th century, metal oxide based piezo-electric ceramic materials such as Barium titanate B aT iO3 and Lead zirconate21.1. PiezoelectricityFigure 1.2: Piezoelectric crystal structure (a) Cubic (above Curie temperature,TC ) (b) Tetrahedral (below Curie temperature, TC )[4]titanate (PZT, PbZ ix T i(1?x)O3 [0 ? x ? 1]) were developed with improved piezo-electric properties. PZT is a solid solution of PbZ iO3 and PbT iO3. Among theartificial piezoelectric materials, PZT is the most widely used for sensing andactuation due to its large sensitivity and high operating temperature in compar-ison to other ceramic materials [3]. The crystallographic structure of PZT ma-terials is similar to the perovskite structure where the oxygen ions (O2?) are facecentered in the unit cell (Figure 1.2 (a)) [5]. There are two types of metal ions: asmall tetravalent ion[4], usually Titanium (T i 4+) or Zirconium (Z r 4+), is locatedin the lattice of relatively larger divalent metal ions such as Lead (Pb2+) or Bar-ium (B a2+). Depending on the temperature, the crystal structure varies. Abovethe Curie temperature, the perovskite crystal structure demonstrates a simplecubic shape without any dipole in the structure. However, below the criticalCurie temperature, a tetragonal or a rhombohedral lattice structure is observeddepending on the composition of the material. In the tetragonal or rhombohe-dral phase, the central Titanium (T i 4+) or Zirconium (Z r 4+) ion is moved to oneside and creates an asymmetry in the structure which leads to a spontaneous po-larization, PS . Due to the polarization, a dipole moment is created in the crystal(Figure 1.2 (b)). Regions having the same dipole direction are called Weiss do-31.1. PiezoelectricityFigure 1.3: (a) No net polarization before poling (b) polarization in the directionof electric field during poling (c) polarization after poling[4]mains [6]. Although there is a dipole moment associated with the crystal, thenet effect due to the dipoles in the crystals is cancelled due to the random ori-entation of Weiss domains in the materials before the poling process (Figure 1.3(a)). Hence, no piezoelectric effect is observed in the ceramic material beforethe poling process. To pole the ceramic material, it is heated slightly below theCurie temperature and a high DC electric field is applied to the materials. Thisaligns the Weiss domains in the direction of the electric field (Figure 1.3 (b)).The electric field is then removed and a permanent polarization of the materialis achieved through poling in the direction of the applied electric field (Figure1.3 (c)). Upon cooling, this polarization remains present in the material. Whenan electric field is applied across the Weiss domains, the domains tend to alignmore to the electric field direction and this changes the dimension of the ma-terial. The relationship between applied voltage and dimensional change is notlinear for all types of PZT materials. Nonlinearities such as hysteresis are com-mon in PZT materials. Moreover, the change in dimension is also a function oftime for a constant electric field which results in creep. In addition to hysteresisand creep, vibrational dynamics also take place in applications near the reso-nant frequency. Figure 1.4 shows the change in the nonlinear behaviour of the41.2. HysteresisFigure 1.4: Nonlinear effects on piezoelectric actuator response at various fre-quenciespiezoelectric ceramics with the driving frequency. For low operating frequen-cies, the creep and hysteresis phenomena take place. At higher frequencies, thecreep phenomenon is absent but the hysteresis phenomenon is still present. Forlong time continuous operation, another phenomenon named ?self-heat gener-ation? is observed with hysteresis. These nonlinear effects are discussed in theupcoming sections.1.2 HysteresisHysteresis is a common phenomenon in piezoelectric materials due to theirdomain switching behaviour. It results from the rotation of the domains by ei-ther 180? or non-180? in the presence of an electric field or a mechanical stresslarger than a critical value. For the non-polarized piezoelectric ceramics, thepolarization-electric field (P-E) hysteresis is shown in Figure 1.5.At point ?A?, the domains are oriented randomly and hence, no net polariza-tion is present in the element. When an electric field is applied and the domainsstart to align in the direction of the electric field. At point ?B? most of the domainsare in the direction of the electric field. When the electric field is reversed, thepolarization is not zero for zero electric field. The polarization at point ?C? is51.2. HysteresisFigure 1.5: Polarization-electric field hysteresis [7]Figure 1.6: Strain-electric field hysteresis [7]61.3. Creepcalled the remnant polarization, Pr where the domains are still in the directionof the electric field as in point ?B?. To achieve zero polarization, the electric fieldis further reduced in the negative direction. The electric field at point ?D?is thecritical value beyond which domain switching takes place. The critical value ofthe electric field is known as the coercive field, Ec [7]. If the electric field is fur-ther reduced, then all the domains are reoriented again in the direction of thenegative electric field and reaches to point ?E? where most of the domains arein the opposite direction of the domains in point ?B?. In the reversal, at ?C?, theelement has negative remnant polarization, ?Pr . At point ?F?, the polarization iszero at positive coercive field, +Ec . The curve between ?A-B? is known as the vir-gin curve or initial loading curve [7]. The hysteresis loop presented by the curves?B-C-D-E-C?-F? is commonly known as the outer hysteresis loop while the looppresented by ?B-C? is known as the inner hysteresis loop. A very well know butter-fly plot (S-E) is presented in Figure 1.6 where the hysteresis between strain andelectric field is shown. The points ?A-F? have the same characteristics in termsof polarization in both P-E and S-E plots. For positioning applications, the but-terfly hysteresis (S-E) is more important since it shows the relationship betweenposition and voltage. In applications, the operation region is limited to the innerhysteresis loop where unipolar electric field is applied to achieve displacementin one direction. Hence, many researchers are interested in modeling the innerloop of the piezoelectric hysteresis [7]. This will be elaborated further in Section1.6.3 on hysteresis modeling.1.3 CreepThe creep phenomenon is defined as the drift of the piezoelectric actuator(PEA) position over time for a constant applied voltage. This is very common instatic piezoelectric operations. Figure 1.7 shows the schematic relationship ofcreep behavior. The creep nonlinearity is related to the change in remnant po-larization (increase or decrease) due to applied voltage [8]. If the voltage changeis increased the remnant polarization is also increased and continues to increaseeven though the voltage reaches at steady state [8]. A similar effect is observed inthe opposite direction of the voltage change. Several attempts have been made71.4. VibrationFigure 1.7: Piezoelectric creep behaviourto model piezoelectric creep behaviour. Since creep is logarithmic with time,a nonlinear logarithmic function can represent the creep behaviour [8?10]. Anonlinear log(t)-type creep model is shown in Equation 1.1.GCr 1 = x(t )= x0[1+? log10tt0](1.1)where, x(t ) is the model output, x0 is the nominal displacement of the actuatorat t0 seconds after applying the driving voltage, and ? is the creep parameterwhich defines the rate of logarithmic function [8]. A major limitation with thenonlinear creep model is that the creep rate is dependent on the time parameter,t0 which is used to fit the model [11]. In addition to that, model inversion is notvery convenient for feedforward control approaches [10].1.4 VibrationVibration effects in the PEAs may occur when they are operated close to thefirst natural frequency of the system. In some applications, such as scanning81.5. Piezoelectric Actuators (PEAs)tubes or cantilevers, the natural frequency is low and substantial vibrations areobserved. In some systems such as scanning tubes, the vibrations occur at sucha low frequency that the rate-dependent hysteresis effect is modeled as a vibra-tion effect [12]. Usually to incorporate the vibration dynamics, a second ordersystem model is sufficient [13?15]. However, for high levels of accuracy, higherorder models are used specially when pure inverse models are required for openloop control [12, 16]. The parameters of the vibration dynamic models are usu-ally obtained through appropriate fitting of the frequency response of the actu-ator by selecting the order in a dynamic signal analyzer [12, 16, 17]. Usually thedisplacement range is limited to 10% of the stroke to neglect the hysteresis effectin measurement [16]. In contrast to scanning tube type actuators, or bending ac-tuators, stack type actuators have a very high mechanical natural frequency. Itis recommended by the manufacturer that for positioning applications of actu-ators, resonant frequency should be avoided [18].1.5 Piezoelectric Actuators (PEAs)Piezoelectric materials are extensively used as sensors, actuators, or trans-ducers utilizing either the direct or indirect piezoelectric effect. Based on theindirect effect, an applied voltage or charge is used to create a strain in the piezo-electric system for force or position applications. Properties such as high com-pressive strength, fast rise time, high resolution, high pressure resistance, com-pactness etc. made the piezoelectric actuators (PEAs) ideal candidates for nu-merous micro-/nano-systems such as micro-positioning stages, inkjet-printing,surgical robots, fuel injection, drug delivery, etc. Due to advanced manufactur-ing technologies, piezoelectric materials can be formed almost into any shape.Different types of PEAs are developed by the manufacturers for different appli-cations. Broadly, they are classified as rod or stack type actuators (extensionalmode) and bender type or stripe actuators (flexural mode) [19] which have widevariety of applications.91.5. Piezoelectric Actuators (PEAs)1.5.1 Stack ActuatorsStack actuators are usually multilayered where several piezoelectric ceramiclayers are connected in series mechanically and in parallel electrically [14]. Inmultilayered stack type actuators, piezoelectric ceramic layers are sandwichedbetween two electrodes where the polarization of the ceramics is in the oppositedirection for two consecutive layers. The stack type actuators provide low strainand high blocking force. The stack actuators are classified into either discretetype or co-fired type. In discrete type actuators, separately prepared piezoelec-tric discs or rings are connected to the metal electrode with adhesive. The layerthickness of these actuators is usually larger than 0.1 mm (as an example in [20]).The operating voltage is usually high for these type actuators typically rangingfrom 500 V to 1000 V to obtain the required electric field. The co-fired actua-tors (monolithic type) are manufactured by high temperature sintering of theceramic material and electrode. Since they are co-fired, the layer thickness ofthe ceramics can be reduced to 0.02 mm [21]. This makes it possible to drivethese co-fired actuators with relatively low voltage typically less than 200 V. Theconstruction of a multilayered stack actuator is shown in Figure Bender ActuatorsIn comparison to the stack type actuator, the bender type actuator providelarger displacement. The generated force for bender type actuators is small incomparison to stack actuators. The natural frequency of the benders is alsosmall in comparison to stack actuators. The deflection in bender type of ac-tuator is perpendicular to the electric field direction. Usually, the bender actua-tors consist of a piezo/metal combination or a piezo-piezo combination. In thepiezo-metal combination (in Figure 1.9(a)), when the ceramic is energized theactuator deflects proportional to the voltage. This arrangement is usually usedwhen deflection is required in a single direction. In piezo/piezo combinations,both the layers can be polarized in the same direction (parallel connection (inFigure 1.9(b))) or in opposite direction (series (in Figure 1.9(c))). In the parallelconnection, the deflection is twice as large as in the series connection [22]. Thepiezo-piezo combination allows bending in both directions. Figure 1.9 shows101.6. Piezoelectric Actuator ModelingFigure 1.8: Multilayer piezoelectric stack actuatorthe construction of different bender type actuators.1.6 Piezoelectric Actuator ModelingPiezoelectric actuator models are broadly classified into linear and nonlin-ear models. Linear models do not consider hysteresis or creep related nonlinear-ities and are applicable for hard piezoelectric materials. Nonlinear models try toincorporate hysteresis and/or creep phenomena. Phenomenological models arewidely used to describe the nonlinear behaviour of the piezoelectric materialssince the nonlinear effects such as creep and hysteresis and their dependenciesare often difficult to describe.1.6.1 Linear IEEE ModelIn the classical description of piezoelectric constitutive equations [23], thepiezoelectric materials are modeled by linear equation as follows:S?= s??T ?+d?i E i , (1.2)111.6. Piezoelectric Actuator ModelingFigure 1.9: Piezoelectric bender actuator (a) Piezo-metal combination (b) Piezo-piezo combination in parallel connection (c) Piezo-piezo combination in seriesconnection [22]Di = d?i T ?+?i l E l , (1.3)where, independent variables, T denotes the applied stress and E is the electricfield which are linearly related to dependent variables, S strain and D electricaldisplacement with constants such as, mechanical compliance (s), piezoelectricconstant (d) and dielectric constant (?). The indices, ?, ? = 1,2,3. . . ,6 and i ,l = 1,2,3 are the ?tensor? expression in the material coordinate system. Basedon the operating modes of the actuators, the expression can be reduced to onedirectional case for simplicity. It is important to note that the IEEE model doesnot include the dynamics of the PEA. Moreover, the piezoelectric hysteresis isnot considered in the model.1.6.2 Nonlinear Physical ModelGoldfarb and Celanovic [13] proposed an electromechanical model for PEAsthat attempts to include both dynamic actuation and hysteresis effects. Since121.6. Piezoelectric Actuator ModelingFigure 1.10: Electromechanical model of PEA [24]their model aims to service control applications, they used readily measurablevariables of voltage, charge, force and displacement instead of electric field,electric displacement, stress and strain. A diagram of this model for stack ac-tuators is shown in Figure 1.10 and the constitutive relationships are shown inEquations 1.4-1.7:mx? +bx? +kx =?UP +F, (1.4)q =CPUP +?x, (1.5)U =UP +UH , (1.6)UH = f (q). (1.7)Equation 1.4 is known as the ?force equation? which models the mechanical sub-system of the actuator while Equation 1.5 is known as ?charge equation? whichmodels the electrical subsystem of the actuator. The electrical subsystem is cou-pled to the mechanical subsystem with the force-voltage proportionality con-stant, ?. In the mechanical subsystem, the generated piezoelectric force is lin-early related to the linear piezo voltage, UP through the force-voltage propor-tionality constant, ?. In the ?force equation?, both the generated piezoelectricforce, ?UP and the externally applied force, F drive the second order mass-spring-damper system that is comprised of a moving mass, m, viscous damp-131.6. Piezoelectric Actuator ModelingFigure 1.11: (a) An elementary elasto-slide hysteresis operator, (b) force-displacement hysteresis mapping of a single operator [13]ing with a coefficient, b and stack stiffness, k while x, x? and x? are the position,velocity and acceleration respectively. In the electrical subsystem described inthe ?charge equation?, the linear piezoelectric voltage, UP results in charge, qflowing into the actuator. The charge flow is proportional to the clamped capac-itance, CP and the elongation of the actuator. To include hysteresis, a dipole,H is introduced in the electrical subsystem which reduces the voltage availableto drive the actuator. This reduction in voltage occurs due to the polarizationvoltage associated with the dipoles within the piezoelectric ceramic [25]. Thepolarization voltage, UH opposes the applied voltage, U and reduces the piezovoltage to UP . Equation 1.6 shows that the applied voltage is a summation ofthe linear voltage and the polarization or hysteresis voltage. Equation 1.7 rep-resents the hysteresis model which describes the hysteresis voltage as a func-tion of charge. This is due to the assumption in the model that the hysteresis issolely present in the electrical subsystem. A Maxwell elasto-slide operator is thebuilding block for the generalized Maxwell slip model which is used to capture141.6. Piezoelectric Actuator ModelingFigure 1.12: Generalized Maxwell slip model for piezoelectric hysteresis [13]151.6. Piezoelectric Actuator Modelingthe rate-independent hysteresis behaviour in the proposed model [13]. Similarto hysteresis phenomenon observed in the elastic-plastic deformation of stressand strain in solid materials or electric field strength vs the flux density in mag-netic material; the model can be used for piezoelectric hysteresis modeling. Thebasic model consist of massless energy storage elements such as a mechanicalspring connected to a massless block. The massless system is considered slidingon a surface with Coulomb friction which is the rate-independent dissipative el-ement in the system. This system is called a Maxwell elasto-slide operator whichis presented in Figure 1.11(a). The constitutive equation of the system is shownin Equation 1.8:F (t )=???k{x(t )?xb(t )}, k{x(t )?xb(t )}< f =?N ,f sg n(x?), el se,(1.8)where, k is the stiffness of the spring, xb is the block displacement, x is the dis-placement of the block which is the input, f , is the break-away friction force(?N ), ? is the friction coefficient and N is the normal force. A fundamental force-displacement hysteresis behaviour, observed for a displacement input, x(t ) tothe system, is shown in Figure 1.11. To capture the complete hysteretic be-haviour, n elements are connected in parallel, where each of the elements hasa monotonically increasing break-away force, fi . The schematic of the Maxwellslip model comprising of n elasto-slide elements is shown in Figure 1.12. Thecomplete model is shown in Equation 1.9:Fi (t )=???ki {x(t )?xbi (t )}, k{x(t )?xbi (t )}< fi =?Ni ,fi sg n(x?), el se,(1.9)F (t )=n?i=1fi (t ) (1.10)where, i is the index for n number of elements connected in parallel in Figure1.12. A similar analogy can be drawn for voltage and charge in the electricaldomain which represents the piezoelectric hysteresis. In that case the stiffnessterm, ki is replaced with inverse capacitance, C?1i and the input displacementis replaced with charge, qi . The break-away force is essentially replaced with a161.6. Piezoelectric Actuator ModelingFigure 1.13: Extended electromechanical model with a drift operator [26]break-away voltage, vi . The equivalent constitutive model, known as MaxwellResistive Capacitive (MRC) model, is shown in Equation 1.11:UHi (t )=???C?1i {q(t )?qbi (t )}, k{q(t )?qbi (t )}< vi ,vi sg n(q?), el se,(1.11)UH (t )=n?i=1UHi (t ), (1.12)where, UH is the hysteresis voltage.The electromechanical model is augmented by Adriaens et al. [14] wherea mechanical operator is proposed to include the higher order dynamics for alarge frequency range. To include the higher harmonics it is necessary to modelthe mechanical system as a distributed model instead of a lumped mass system.Also, the hysteresis behaviour is modeled with a differential equation insteadof an MRC model. However, for most of the piezoelectric stack actuators, theapplication frequency range is well below the mechanical resonance frequencywhich reduces the mechanical operator into stiffness compliance [27], [28]. Anextension of the model presented in [14] is proposed in [26] which accounts forthe creep behaviour in addition to the hysteresis phenomenon with a drift op-171.6. Piezoelectric Actuator ModelingFigure 1.14: Drift operator with lossy resistance [26]erator, D . Similar to the model presented in [13], the hysteresis phenomenon ismodeled using the generalized Maxwell operator, H . The extended model witha drift operator (electrical domain) is shown in Figure 1.13.The proposed drift operator, D consists of N number of series RC elementswhich are connected in parallel shown in Figure 1.14. The creep is consideredsimilar to charge drift and hence the drift charge, qd is obtained through themodel shown in Equation 1.13:GCr1 (s)=qd (s)UP (s)=1Rs+N?i=1CiRi Ci +1, (1.13)where, UP is the linear piezoelectric voltage, R is a resistive element to accountfor the dielectric losses while the Ri and Ci elements model the creep behaviour.These linear models accurately predict the creep behaviour when the modelstarts from a known initial state. However, the past history of the hysteresis af-fects the creep behaviour which in not accounted in the linear models [29].1.6.3 Phenomenological ModelThe hysteresis behaviour between voltage and position attracts a large re-search interest since it is useful for designing feedforward controllers for posi-tion actuation applications. Figure 1.15 shows a typical hysteresis loop. It isobserved that the voltage-position hysteresis phenomenon is a function of therate of the input [27, 30?32]. Based on this, voltage-position hysteresis is clas-sified into two groups: 1) rate-independent hysteresis and 2) rate-dependent181.6. Piezoelectric Actuator ModelingFigure 1.15: Voltage-stroke hysteresis loops at different voltageshysteresis. The corresponding hysteresis models are then also classified as rate-dependent models and rate-independent models. The rate-independent hys-teresis modeling assumes that the hysteresis behavior is not influenced by therate of change of input or frequency. However, in practice rate dependence of-ten plays an important role. Rate-independent ModelInitial hysteresis models consider that the hysteresis behaviour is indepen-dent of driving frequencies or rate of operation. Several attempts were made tomodel the rate-independent hysteresis phenomenon. Among which, the classi-cal Preisach model (CPM) [12, 33], and the Prandtl-Ishlinskii (PI) operator [29]are the most studied and modified. The CPM is a phenomenological model,where a function is defined to model the piezoelectric hysteresis through nu-191.6. Piezoelectric Actuator ModelingFigure 1.16: (a) Preisach operator or hysteron (b) Preisach model with N opera-torsmerous Preisach operators, called Hysterons [34], ???[u(t )] as presented in Fig-ure 1.16 (a). For an input value, u(t ) larger than ?, the Hysteron value is set to+1and for values lower than ? the value is set to ?1. A weighting function, ?(??) ismultiplied with the hysteron. Connecting the hysterons in parallel provides theoutput shown in Figure 1.16(b). The CPM model is realized in Equation 1.14:x(t )C P M =??<??(?,?)???[u(t )]d?d?, (1.14)where, x(t ) is the output of the operator for an input of u(t ). The accuracy ofthe classical Preisach model is limited to a single operating frequency. More-over, the smoothness of the hysteresis modeling is dependent on the number ofhysterons (numerous) which is computationally expensive [28, 30].The Prandtl-Ishlinskii (PI) model is an important sub-model of the Preisachmodel that is developed to address the computational complexity and inversionproblem of CPM [29, 30]. The PI hysteresis model is based on the play or back-lash operator which is generally used in gear backlash modeling with one degreeof freedom [30]. Equation 1.15 presents an expression for the PI model with nplay operators:x(t )PI =N?i=1?iwr [u, y0](t )201.6. Piezoelectric Actuator ModelingFigure 1.17: (a), (b) Elementary play operators with different weights and thresh-old values (c) superpositioning of the play operators in (a) and (b) [35]=N?i=1w i max{x(t )? r i ,mi n{x(t )+ r i , y(t ?T )}} (1.15)where, ?iwr [u, y0](t ) is the i th play operator with w is the weight (slope) which isthe gain of the backlash operator, r is the control input threshold value, y0 ? Rand usually set to zero, T is the sampling time, u(t ) is the model input, and x(t )PIis the model output. Figure 1.17 (a) and (b) show the elementary play operatorswith different weight functions, w and threshold values, r . A simple PI model iscreated by the superposition of the two play operators shown in Figure 1.17 (a)and (b), which leads to the Figure 1.17 (c).Although the PI hysteresis model is simpler than the CPM, the classical PI islimited to the symmetric hysteresis modeling. In many practical scenarios, thepiezoelectric hysteresis is not symmetric in nature. Hence a modified PI opera-tor is proposed by [30] where a saturation operator is connected to the hysteresisoperator in series. A saturation operator, defined as a superposition of weightedlinear one-sided dead-zone operators, is shown in Equation 1.16:Sd[x](t )= [Sd0[x](t ),Sd2[x](t ), ? ? ?,Sdm[x](t )] (1.16)where,Sd [x](t )=???max{x(t )?d ,0}, f or d > 0,x(t ), f or d = 0,211.6. Piezoelectric Actuator ModelingFigure 1.18: (a) a one sided dead-zone operator (b) summation of the dead-zoneoperators for asymmetry [30]Figure 1.19: A modified structure of PI modeld= [d0,d1, ? ? ?,dm]T .The dead-zone operator is shown in Figure 1.18. The output of the modified PIoperator for asymmetric hysteresis is shown in 1.17. The complete structure ofPI operator is presented in Figure 1.19 and Equation 1.17.z(t )PI =wTs Sd [x](t ), (1.17)where,ws = [ws0, ws1, ? ? ?, wsm]T .221.6. Piezoelectric Actuator Modeling1.6.3.2 Rate-dependent ModelThe hysteresis models discussed in the previous section are rate-independentmodels which can predict the hysteresis behaviour for a limited frequency range.For large frequency ranges, the aforementioned models require modificationsto achieve acceptable levels of accuracy. Modifications have been suggested forboth the Preisach and PI hysteresis models to incorporate the rate-dependenteffects [30, 32, 33, 36, 37]. The dynamic Preisach model (DPM) includes a struc-ture where the weighting function?(?,?) is modified to address the rate-dependency.This is achieved by introducing additional dynamic operators [33, 36] or a neuralnetwork [32]. The additional dynamic operators are functions which are depen-dent on the average input voltage between two consecutive input extrema andthe rate of change of the input voltage between the input extrema [33]. More-over, an additional function, named mirror function is defined to correlate theCPM to the DPM. The output of the DPM suggests that it is a function of boththe output of the closest extremum value and the output of the CPM value atthe nearest extremum. Although, it is shown that the hysteresis can be modeledover a frequency range of 0-800 Hz within reasonable error (6.4%), it requireda priori knowledge of the input voltage waveform to attain the dynamic oper-ators [33]. Unfortunately, this is not very convenient when the future input isnot known. A slightly different method is proposed in [36] where the weight-ing function is extended to include a parameter which is a function of the rateof the input voltage. The function is simplified with the assumption that thehigher order derivatives are negligible. This limits the model accuracy to a fre-quency range of 0.01-10 Hz. A neural network to address the rate dependencyis proposed in [32]. In this case, the weighting function is modified using a neu-ral network. It is shown that the modeling accuracy is achieved within 4% ofthe maximum displacement for a frequency range of 2-32 Hz. The PI model canalso be modified to incorporate rate dependent phenomena [30]. The weights inEquation 1.16 are modified with the rate of actuation, u(t ). The percentage errorfor a frequency range of 1-19 Hz continuous operation is measured at 5.3%. Oneof the major drawbacks of the PI operators is a singularity problem when the PIweight turns to zero. A similar problem occurs if the slope is negative. Then the231.6. Piezoelectric Actuator Modelingpreliminary assumption of a monotonically increased loading curve is violatedand the model fails. In the proposed model, the singularity occurs at 40 Hz [30]. Other Phenomenological Hysteresis ModelsOther phenomenological models in the literature include the Bouc-Wen (BW)model [38], the Duhem model [39], a memory based model (MBM) [40], a poly-nomial based model [41], a first order differential equation model, [42] etc. Al-though these models can predict the hysteresis behaviour, significant improve-ment in accuracy is not realized over the previously discussed methods. More-over, some of these models require more complex computation in comparisonto their modeling performance which limits their wide acceptability.1.6.4 Cascaded Phenomenological Model for Hysteresis, Creep, andVibrational DynamicsA phenomenological model structure is proposed in [12] where the hystere-sis and creep phenomena are modeled using hysteresis and creep operators.Once the models are developed they are cascaded to complete the piezoelec-tric model shown in Figure 1.20. In the proposed model by [12], CPM is used tomodel the hysteresis.To overcome the limitations of the model described in Equation 1.1, a linearcreep model is presented in [12] which is a series connection of several springsand dampers as shown in Equation 1.18:GCr 2(s)=x(s)U (s)=1k0+N?i=11ci s+ki, (1.18)where, k0 is the elastic constant, ci and ki are the dampers and springs of the i thcreep element. The model is presented in Figure 1.20.The vibrational dynamics is modeled by a higher order transfer function. Todevelop the complete model, first the creep submodel is constructed at low fre-quency condition where the vibrational dynamics is not present. Then the vi-bration submodel is developed at high frequencies when the creep is neglected.Finally the hysteresis model is developed and cascaded to the other submod-241.7. Position Control of PEAFigure 1.20: Cascaded model of PEA [12]Figure 1.21: Visco-elastic creep model with elastic component [12]els. The input voltage, U generates a mechanical output through the hysteresismodel which is the input for the creep and vibration submodel. Finally, the posi-tion, x is obtained from the complete model. It is important to note that the rate-dependent hysteresis is embedded in the vibration submodel since the model isaimed for a system (piezo tube scanner for atomic force microscope) whose firstnatural frequency is very small (typically in the range of 500-1000 Hz). Hence, vi-bration occurs at relatively low frequencies even before observing considerablerate dependency in the hysteresis behaviour. The advantage of the model is thatit is purely phenomenological which does not require the underlying physics forthe system. Moreover, the hysteresis model is not restricted to a particular op-erator rather any model previously discussed (CPM, PI or MRC) can be used torepresent hysteresis. Similarly, any linear model can be implemented to modelthe creep behaviour. The major drawback of this type of model is that the mod-eling uncertainty limits the accuracy of the position controller. Moreover, if thehysteresis behaviour changes due to some external factors such as temperatureor aging the model cannot predict the piezoelectric behaviour accurately.1.7 Position Control of PEAPEAs are widely used in many micro-nano positioning applications. How-ever, due to nonlinearities such as creep or hysteresis, position control of these251.7. Position Control of PEAFigure 1.22: Different position control schemesactuators is challenging. Hysteresis can result in up to 15% position error [43].The positioning error due to creep varies from 1-40% [43, 44]. For high accu-racy positioning applications, the nonlinearities must be compensated. Thecontrol schemes to counter the nonlinearities are classified into five major cat-egories: 1) feedforward voltage control [12, 27, 30, 38, 43, 45, 46], 2) feedbackvoltage control [47?50], 3) feedforward with feedback control [24, 27, 47, 48, 50],4) charge control [51?53], and 5) integrated voltage/charge control [17, 54]. Theschematic shown in Figure 1.22 presents different position control principles forPEAs which are discussed in the following sub-sections.1.7.1 Feedforward (FF) Voltage ControlFeedforward voltage control scheme is a model based control scheme wherea dedicated position sensor cannot be incorporated due to space and/or costconstraints. Moreover, additional sensors can lead to reliability problems whichcan be avoided through a feedforward control scheme. The broad idea is pre-sented in [12] where an accurate phenomenological model of the PEA is devel-oped and inverted. The inverted model is then fed with the desired position,xd input and a voltage, Um is obtained from the inverted model. The invertedmodel output, Um is the input voltage for the piezoelectric plant which is re-sponsible for the required displacement, x. The feedforward control structureis presented in Figure 1.23. All three effects (hysteresis, creep and vibration) areconsidered in [12, 44] where the hysteresis is compensated by inverse CPM [12]and an inverse PI model [44], respectively. The creep is modeled by a Kelvin-viogt system in [12] and an ARMAX (Auto Regressive Moving Average with eXter-nal inputs) model in [44], respectively. Finally, the vibration is compensated by261.7. Position Control of PEAFigure 1.23: Feedforward control scheme for the PEA [12]a higher order model obtained from a signal analyzer in [12]. In [44], the oscilla-tions are damped with an input shaping technique. In most of the feedforwardschemes, the hysteresis compensation is stressed since this is the major non-linearity present in the piezoelectric positioning in all frequency ranges. Differ-ent hysteresis models and their inversions with rate dependent compensationis implemented by [38, 40, 46, 55]. The creep compensation is often added withthe hysteresis compensation in the feedforward control scheme [10, 29, 44]. Themajor limitation of the feedforward approach is that the model has to be very ac-curate to control the position. Hence, any model uncertainty or any disturbancein the plant causes error in positioning. The other limitation of the inversionmodel is that the convergence of the inversion is not guaranteed all the time.To mitigate the requirement for an accurate model and in order to allow dis-turbances, the feedforward control scheme is often augmented with a feedbacksystem. In the following section, the pure feedback and feedforward/feedbacksystem is discussed.1.7.2 Feedback (FB) Voltage ControlIn applications where the space and cost in not a limiting factor, the PEAposition control system is often equipped with a dedicated feedback sensor thatis used to compensate for hysteresis and creep. Different sensors such as LVDT[56], capacitive probes [57], Hall sensors [58], laser interferometers [55], straingauges [59], laser triangulation methods [60], or laser vibrometers [61], etc. are271.7. Position Control of PEAused as position sensors. A comparison of the sensors is presented in Table 1.1[62].281.7. Position Control of PEATable1.1:ComparisonofthefeedbacksensorsTypeCostSizePrecisionResolutionOthersTriangulationHighLargeHighHighLimitedmeasu-rementrangeLaserVeryhighLargeHighHighHighbandwidth,vibrometerSensordriftStrainguageLowSmallLowMediumFragile,temperatureeffectLVDTLowSmallMediumMediumRequirescontactCapacitiveorLowMediumHighHighRequiresinductiveprobeorMediumproximityHalleffectLowMediumMediumHighRequiressensororMediumorLargeproximity291.7. Position Control of PEAFigure 1.24: Feedback control scheme [11]A Block diagram of the feedback (FB) control scheme is presented in Figure1.24. For low frequency applications or point to point control, classical Proportiona-Intergral-Derivative (PID) controllers or multiple integrators are commonly usedfor tracking [11, 63?66]. The advantage of PID or integral controllers is that theyprovide high gain feedback which can substantially minimize the hysteresis andcreep at low frequencies. However, at large frequencies, the PID controller islimited by the phase lag and limited gain margin [11]. This results in tracking er-ror in classical PID controllers at high frequencies. To overcome this limitation,PID controllers can be augmented with feedforward control schemes [47, 48], orNeural networks [67]. Advanced controllers such as state feedback [60, 68] slid-ing model control [27], H-infinity control [69], etc. have also been developed fortracking control of PEAs.1.7.3 Feedforward/Feedback (FF/FB) Control SchemeTo improve the gain margin of the classical PID controller, a feedforwardmodel is often included with feedback control. This not only improves the lowgain margin of the feedback control, but it also improves the performance ofthe feedforward control since the feedback loop accounts only for model un-certainty and plant disturbance [11]. A common feedforward/feedback controlscheme is presented in Figure 1.25. This FF/FB structure is used by [15, 24, 46?48]. To model the feedforward branch, different inverse hysteresis models havebeen used, such as CPM [15, 48], MRC [13, 70], PI [30, 44], or Boc-Wen [46]. In[47], a high gain feedback is used to compensate for the nonlinear hysteresisand creep while the feedforward branch accounts for linear vibration dynam-301.7. Position Control of PEAFigure 1.25: Feedforward branch with feedback control scheme [11]Figure 1.26: Feedback-linearized inverse feedforward control [47]ics. Another FF/FB structure is presented in [47] and is shown in Figure 1.26.Here, an inverse model of the closed loop system is cascaded to the closed loopsystem. In the closed loop system, the feedback controller uses a high gain tocompensate the non-linear hysteresis. Once a linearized closed loop system,GC L is obtained, the feedforward branch is implemented by inverting the linearsystem, GC L . The advantage of the system lies in the simplicity of the inversionof the linear system, GC L in contrast to the inversion of the complex nonlinearhysteresis models such as CPM and higher order dynamic models. However, thelow gain margin of the feedback is still present in this structure [11].1.7.4 Charge Control SchemeUnlike the voltage-position relationship, charge is linearly related to posi-tion in PEAs [13, 17, 53, 62]. Charge based controllers are based on this lin-311.7. Position Control of PEAFigure 1.27: Typical charge control scheme [52]ear charge-position relationship. Significant hysteresis reduction is possible bycharge based position control. A typical charge control scheme is presented inFigure 1.27. A sensing capacitance, CS is added to the piezoelectric capacitiveload, CL . An amplifier with a gain, K A compensates the difference between thereference voltage, Ui n and voltage across the sensing capacitance. To includethe parasitic resistances, RL and RS are added to the model shown in Figure 1.27.The major problem with charge based controllers is the poor low frequency orDC performance due to charge drift. Improved charge control schemes requireadditional circuitry which adds complexity in the implementation [52, 53]. Adifferent architecture for charge control is shown in [71]. In this case, additionalelectrode layers are added to the PEA and the induced charge in the electrodeswere used for position estimation through an inverse function between chargeand position. However, long-term charge drift is not considered in this study.1.7.5 Integrated Voltage/Charge ControlAn interesting integration of the charge based control and voltage basedfeedfroward technique is presented in [17] for piezoelectric tube scanners. Inthis article, the charge based technique handles the hysteresis nonlinearity whilethe voltage feedforward accounts for the dynamics. In this architecture, thecharge amplifier is used for a traditional charge based control. This can suc-cessfully reduce the hysteresis non-linearity. However, in the presence of higher321.8. Position Self-sensing (PSS)Figure 1.28: Integrated voltage/charge control [17]dynamics, the charge based control cannot compensate for these errors. So, asimple inverse function of the hysteresis linearized charge control is developedwhich produces the feedforward voltage, UF F . The advantage of this system isthat it avoids complicated and sensitive models for vibration and hysteresis. Thestructure for this type of control structure is shown in Figure 1.28. However, thisalso requires a sophisticated charge amplifier.1.8 Position Self-sensing (PSS)Self-sensing is a technique where position or force is obtained in PEAs with-out having dedicated sensors. Such a technique is first proposed in [72] forvibration suppression of structures. Later, this technique is used in vibrationsuppression of beams [73?75], scanning tubes [76], etc. Self-sensing techniquesare also used for position reconstruction and control applications. Broadly, theself-sensing technique can be classified into three types: 1) capacitance bridgeself-sensing, 2) charge based self-sensing, and 3) piezoelectric self-sensing.1.8.1 PSS Using Capacitance BridgeIn capacitance based self-sensing, a capacitance of equal value of the PEA isused in a capacitance bridge [72, 73]. Since, piezoelectric materials act as bothsensor and actuator, in the presence of any deformation or strain in the actua-tor, a voltage, UPE A , proportional to the strain or deformation is induced to theactuator. A schematic of the capacitance bridge is shown in Figure 1.29. Theobjective of the self-sensing structure is to obtain a sensing signal, Us which isproportional to the induced voltage which can then be used as a feedback signal331.8. Position Self-sensing (PSS)Figure 1.29: Capacitance bridge based self-sensing [73]to control the vibration through a gain Ks . To obtain the sensing signal, a capac-itance bridge is created similar to the Figure 1.29. A voltage source, U is appliedto both the piezoelectric capacitive load, CP and the reference capacitor, Cs . Aninduced voltage, UPE A is generated due to strain in the actuator. This creates animbalance in the bridge and the voltage difference between the two branchesprovides a sensing signal, Us proportional to the induced voltage. Consideringthe leakage resistances, R1 and R2 sufficiently large and unity gain of the op-ampthe following relationships can be deduced:U1 =Cs(C1+Cs)U , (1.19)U2 =CPC2+CP(U ?UPE A), (1.20)Us =U1?U2 =CsC2+CsU ?CP(C2+CP )(U ?UPE A). (1.21)Considering the ideal condition where Cs = CP , the strain voltage is related to341.8. Position Self-sensing (PSS)the induced voltage as follows:Us =CPC2+CPUPE A . (1.22)The sensing signal is feedback to the source voltage and hence the vibra-tion control is achieved. A slight variation of the similar structure is proposedwhere the sensing signal is obtained as a rate of change of the induced voltage[77]. Though this proposed technique is simple to implement, it has several lim-itations in practice. The size of the reference capacitance, CS has to be equallylarge as the piezoelectric capacitance, CP . Moreover, the temperature changehas an effect on the capacitance values which demand a tedious continual tun-ing of the reference capacitance [73]. The other problem is to obtain an accuratepiezoelectric capacitance, CP . Hence, any mismatch in the capacitance valuecreates instability in the feedback control. An adaptive implementation is sug-gested in [78, 79] to address this problem.1.8.2 PSS Using Charge MeasurementThe other class of position self-sensing technique relies on the linear charge-position relationship [13, 17, 53, 62]. Current is integrated through a charge am-plifier and the position is estimated from the charge measurement and the con-stitutive relationships. The estimated position can then be used in a feedbackcontrol scheme employing PID or PI controllers similar to the ones discussedin sections 1.7.4. A charge based position self-sensing structure is presented inFigure 1.30. The position is estimated in two steps: 1) an applied voltage, Ui nproduces a strain, x and charge, q . The charge is measured through an integra-tor circuit and a charge amplifier. Then the output voltage, Uout , is related tothe measured charge, 2) In the second step, the position is estimated using theconstitutive relationships between output voltage and charge as well as chargeand position. Charge drift usually occurs when charge is obtained from currentintegration in the presence of any offset voltage. The position estimator in Fig-ure 1.30 aims to compensate for this charge drift. Charge feedback approachesare presented in [25, 71].351.8. Position Self-sensing (PSS)Figure 1.30: Charge based position self-sensing [80]1.8.3 PSS Using PiezoelectricityPosition self-sensing is also obtained from the direct effect of the PEA itself.This approach is used in piezoelectric tube scanners where a part of the piezo-electric material is used for actuation while the remainder is used for sensing[81]. In this approach, the driving voltage, U creates an induced voltage, US inthe sensing part of the tube which is proportional to the piezoelectric extension.The signal can be used for position estimation or vibration suppression in thescanners. An implementation diagram of this setup is shown in Figure 1.31. Thedisadvantage of this approach is that since a part of the actuator is used for sens-ing, the range of the actuator is reduced.361.8. Position Self-sensing (PSS)Figure 1.31: Piezoelectric position self-sensing371.9. Errors in Piezoelectric Positioning1.9 Errors in Piezoelectric PositioningTo verify the performance of different position controllers for PEAs, max-imum tracking errors are usually compared as an indicator for improvement.For step responses, sometimes steady-state errors are also compared. There area number of factors which may affect the performance of the controller. Amongthem driving frequency plays an important role, since the nonlinear effects ofcreep, hysteresis and vibrational dynamics appear at different frequency ranges.A number of controllers are based on feedforward phenomenological models(in Figure 1.23) only, which are designed to operate only for a limited frequencyrange where multiple effects are not visible. Moreover, modeling inaccuracy andplant uncertainty also affect the controller performance. To obtain a more reli-able controller, often these feedforward controllers are added to a feedback con-trollers (in Figure 1.25) where the latter compensate for modeling inaccuracyand plant uncertainty while the feedforward model improves the dynamics ofthe controller [11]. Figure 1.32 shows some of performance of various positioncontrol schemes in terms of maximum tracking error where sinusoidal signalsof fixed frequencies or mixed frequencies are used as a reference signal. The er-rors and the aim for different models presented in the literature as well as theperformance of the models are presented in Table 1.2.381.9. Errors in Piezoelectric PositioningFigure 1.32: Maximum errors with operating frequencies. The circular markersindicate the FF controller while the solid markers show the FF/FB controller re-sults.391.9. Errors in Piezoelectric PositioningTable1.2:ErrorcomparisonbetweendifferentmodelsModelReferenceRate,RControlTypeofFrequency%errortypeHysteresis,Hmethodprofilesrange[Hz]maximumCreep,CVibration,VPI[45]HFFSinusoidal0.12.5PI[27]R?HFFMixedSinusoidal1?502.12?2.7FF/FBMixedSinusoidal1?1000.84?2.32CPM[48]HFFSinusoidal0.018.8PDFB6.5PDFB/FF2.5PI[30]R?HOLMixedSinusoidal1?1917.3RIFF8RDFF5.3BW[46]HFFSinusoidal0.52.6?3.1FF/FB0.88?1FFSinusoidal14.8FF/FB2.5MRC[70]HFF/SS-FBTrapezoidal2msstatic22msdynamic3Self-sensing[62]H?CFF+ModelStep0.55?2.5Transferfunction[47]H-C-VFFSinusoidal1?3004.88FF/FBTriangular2002.95FBStep1.48Memorybased[82]HFFRamp60sec0.7Triangular11.4MixedSinusoidal4.3PI[31]R?HFFMultiSinusoidalRandom1?6Sinusoidal122?3401.10. Self-Heating1.10 Self-HeatingSelf-heat generation is a phenomenon which occurs in PEAs when they arecontinuously driven at high frequency under high electric field. Substantial tem-perature rise is observed in [21, 83?85] which may affect the performance anddurability of the actuator. Moreover, the hysteresis behavior may change dueto temperature variation which affects the phenomenological model and feed-forward control of the actuator. Also, due to temperature variation, the modelparameter such as piezoelectric capacitance (in Equation 1.5) is also affected. Itis reported in [21] that the self-heating phenomenon is a function of frequency,electric field and effective volume of the actuator. In [83], it is shown that a tem-perature rise increases the current in the actuator.Self-heat generation results from losses such as mechanical damping and di-electric losses. At high frequencies close to resonance, it is thought that the me-chanical losses play a major role in heat generation while at frequencies lowerthan the resonance frequency, the dielectric losses contribute most to heat gen-eration [86]. The dielectric loss is caused by the ferroelectric hysteresis losswhich primarily occurs due to domain switching [84, 86, 87]. A theoretical modelis proposed in [21] to model the self-heating phenomenon which is based on thelaw of energy conservation. The model assumes that rate of heat generation isproportional to the frequency and the hysteresis loss per driving cycle per unitvolume, u. Zheng et al. [21] also studied the geometric changes in different ac-tuators. They conclude that the steady-state temperature rise is a linear functionof ratio of effective actuator volume to surface area. They also show that hystere-sis varies with temperature. However, the temperature effect on the hysteresisloss is not considered in their model.The self-heating model presented in [21] is extended in [84] which includesa heat sink attached to the actuator in order to reduce the actuator tempera-ture due to self-heat generation. Instead of the loss term, u presented in [21],displacement hysteresis, D f (an equivalent parameter to u calculated from thestrain-electric field hysteresis) is introduced in the model. The model predictsthe self-heating phenomenon in the presence of a heat sink and concludes thatthe self-heating temperature increase is reduced by 39% by the introduction of411.11. Scope of the Work and Objectivesthe heat-sink. However, the model does not account for the mechanical lossesdue to friction between the heat sink and the actuator. A different temperatureprediction model is proposed in [83], where it is shown that temperature changedue to self-heating increases the current flow. The change in the current mea-surement is used to measure the change in temperature.1.11 Scope of the Work and ObjectivesPEAs exhibit rate-dependent hysteresis and creep in voltage driven scheme.These phenomena occur over a wide frequency range. Charge based operationreduces the hysteresis significantly. However, it requires sophisticated hard-ware to implement charge based schemes. Position feedback sensors reducethe hysteresis and creep behavior significantly. However, implementation ofdedicated position sensors is sometimes prohibitive due to cost and space con-straints. Researchers have developed several hysteresis models for use in feed-forward position controllers. However, complex mathematical modeling is re-quired to obtain accurate models which are often affected by uncertainty of theplant. Hence, these models are only useful for a limited range of frequencies.Piezoelectric position self-sensing provides a means to predict the position fromcharge information. However, charge based PSS is limited to high frequenciesand prone to drift due to any offset present in the current measurement. Otherself-sensing approaches have limitations such as temperature instability due topiezoelectric capacitance variation, loss of stroke, etc. In addition to that, tem-perature variation due to self-heat generation phenomenon substantially affectsthe self-sensing strategies which are not very well studied in earlier research.Considering these aspects, the main objectives in this research are:o To develop a PSS scheme which is independent of phenomenological mod-els of hysteresis and creep.o To obtain a self-heat generation model for actuator temperature predic-tion.o To adapt the developed PSS for temperature variation due to self-heat gen-eration.421.11. Scope of the Work and ObjectivesTo achieve the main goals, several sub-goals are set. They are as follows:o To characterize the behaviour of piezoelectric actuator, a real-time imped-ance measurement is required. The impedance measurement provides anovel quasi-static parameter identification technique based on the con-stitutive relationship.o To obtain an improved electromechanical model with the real-time imped-ance measurement. The improved model is based on a position depen-dent capacitance which is considered constant in the original model pro-posed in [13].o To obtain an improved PSS scheme that is useful over an extended fre-quency range. A new capacitance based self-sensing method is presentedwhich is combined with a traditional charge based self-sensing to improvethe position estimation through an observer over a wide frequency range.o To obtain a closed loop position control system that employs the devel-oped PSS scheme as a feedback sensor. The newly developed position ob-server is used as a replacement of a traditional position sensor to obtain aself-sensing control strategy.o To obtain a sensorless temperature measurement for an improved self-sensing scheme in the presence of self-heat generation. A self-heat gener-ation model is proposed based on the hysteresis loss relying on the consti-tutive relationship to predict the temperature of the actuator in the pres-ence of self-heating.o To obtain a temperature compensated PSS in the presence of self-heating.This is a novel contribution where a temperature compensated positionself-sensing is proposed based on the self-heat generation model devel-oped in the previous sub-goal. The arrangement provides temperatureand position estimation in the presence of self-heat generation.The objectives and sub-goals that are connected to the chapters are presentedin Figure 1.33.431.11. Scope of the Work and ObjectivesFigure 1.33: Projection of the objectives and sub-goals connected to differentchapters441.12. Thesis Organization1.12 Thesis OrganizationThe chapters in this dissertation are organized as follows. The experimentalsetup is presented in Chapter 2 where a test-bed with a high mechanical naturalfrequency is designed. This allows to perform high frequency tests in the currenttest-bed.In Chapter 3, a novel real-time impedance measurement technique is pre-sented which provides impedance measurements during the normal operationof the actuator. High frequency ripple signals are superimposed on driving sig-nals which facilitates the real-time impedance measurement. This measure-ment technique shows that effective actuator impedance varies with actuatorstroke.In Chapter 4, position dependent capacitance is added to the traditionalelectromechanical model by Goldfarb et al. [13] and the improved model isverified experimentally. The power balance is also checked and proved unaf-fected due to the inclusion of the position dependent capacitance. Finally, anovel method to identify the model parameters is also presented in this chapter.In Chapter 5, the position-capacitance relationship is exploited in order todefine a novel position self-sensing technique. A hybrid position observer (HPO)is presented which combines two separate position self-sensing signals: 1) chargebased PSS and 2) capacitance based PSS. The charge based PSS suffers at lowfrequencies due to integration error while the capacitance based PSS is lim-ited to low frequency operation due to measurement noise. Hence, the posi-tion observer yields the advantages of each PSS technique and provides an im-proved position self-sensing over a wide frequency range. The position observeris tested in open loop and compared with a traditional position sensor from 0Hz to 100 Hz where creep and rate dependent hysteresis are present.In Chapter 6, the newly developed HPO is used as a position feedback in asimple integral position controller and results are shown for single and multipledriving frequencies as well as for a scenario with actuator creep.In Chapter 7, actuator self-heating is studied. This phenomenon occurs dur-ing continuous operation at driving frequencies higher than 20 Hz. A self-heatgeneration model is proposed based on the power loss of the PEA which em-451.12. Thesis Organizationploys the constitutive model of the PEA. The model predicts the temperaturedue to self-heating with 3% accuracy in the presence of different frequenciesfrom 0 Hz to 150 Hz. Further, the presented HPO is adapted for temperaturevariation and employed to the self-heat generation model to predict the tem-perature. A circular arrangement is proposed where the predicted temperatureis used to correct the temperature related variation in the HPO. HPO positionestimation is then compared with a traditional position sensor in the presenceof self-heat generation.Finally, the contributions of the present research and future research recom-mendations are presented in Chapter 8.46Chapter 2Experimental SetupTo investigate the characteristics and validate the model of piezoelectric ac-tuators, an experimental test-bed is designed and manufactured. The purposeof the test stand is to facilitate the preloading of the actuator for high frequencyoperations (for high speed applications, the actuator manufacture recommendsa preload of 15 MPa [18]) and mounting the sensors for parameter measure-ments. In [20], an experimental test setup is presented to characterize the piezo-electric actuator. However, the actuator used in that setup is large and the setupdesign require a large mass for the preloading. The large moving mass reducesthe natural frequency significantly which limits the high frequency operations inthat setup. Moreover, a force sensor is mounted between the large preload massand the actuator which lowered the natural frequency even further. The naturalfrequency of the test-bed presented in [20] is 511 Hz. Hence, the test setup islimited to operate in the quasi-static range.To improve the operational range, a new design presented in this researchwith an aim to reduce the moving mass. Also, the force sensor is moved betweenthe PEA and the solid base in the proposed design. In addition, a soft spring isused to provide the necessary preload to the actuator. The preload assembly isredesigned which has a much smaller mass than the one in [20]. The natural fre-quency of the newly developed test-bed is 2500 Hz which allows high frequencyoperations up to 500 Hz. In the following sections, the mechanical and the elec-trical components of the test-bed and the experimental setup are discussed.2.1 Mechanical ComponentsThe mechanical components include the actuator, frame, preload assembly,needle and ball seats. The individual components are discussed below.472.1. Mechanical ComponentsFigure 2.1: Piezoelectric actuator and ball-seats2.1.1 ActuatorThe piezoelectric stack actuator under test is manufactured by Physik In-strumente, one of the leading piezoelectric actuator and positioning device man-ufacturing companies in the world. A wide range of piezoelectric actuators isavailable for different applications. In this study, the PI-885.90 model type isused which provides a nominal displacement of 32?10?m at 100 V. The PI mono-lithic actuators are used in this research due to low driving voltage requirement.The properties and geometric dimensions are presented in Table 2.1. The actu-ator is shown in Figure FrameThe frame provides the structure of the test-bed where all the mechanicalcomponents are integrated. The frame consists of 4 plates: two vertical platesand two horizontal plates. The vertical plates (side walls) are slotted to hold thehorizontal plates which are bolted from the sides. The horizontal plates (topplate and base plate) hold the whole assembly of the mechanical components,the actuator and the force sensor. The top plate has a large threaded bore to holdthe preload assembly. Figure 2.2 shows the frame of the test-bed.482.1. Mechanical ComponentsTable 2.1: Actuator properties [18]Properties ValuesActuator dimension 4.95 mm?4.67 mm ? 36 mmStiffness 25 N?m?1Capacitance (at 1 kHz-1VPP ) 3.1 ? 20%?FBlocking force 950N at 120 VNominal displacement 32 ?10%?m at 100 VOperating range -20 ? 135 VMaximum temperature -40 ? 150?CResonant frequency (at 1VPP unloaded) 40000 HzActive volume (v) 740 mm3Surface area (A) 640 mm2Layer thickness (tP ) ?55 ?mDensity (?) 7800 kg?m?3Specific heat (c) 350 J?kg?1 ?K?1Figure 2.2: Frame of the test-bed492.1. Mechanical Components2.1.3 Preload AssemblyThe preload assembly includes a cap, a preload plate, a thrust bearing and asteel ball attached to the needle. The cap is an essential part of the preload as-sembly. The top of the cap is machined to a hexagonal nut to facilitate preload-ing. The cap is threaded externally which fits into the threaded bore in the topplate. Figure 2.3 shows that the cross sectional view of the cap. A pocket iscreated inside the cap which holds the preload spring and the thrust bearing.The preload spring is a soft helical spring of stiffness 40 N ?mm?1. To achievea preload of 15 MPa, 360 N force is required for the cross sectional area of theselected actuator. For a pitch of 1.5 mm, 6 turns of the cap is sufficient to pro-vide necessary preload to the actuator. The thrust bearing prevents any torsionalforce (torque) to the actuator while preloading it. The spring is placed betweenthe thrust bearing and the preload plate which provides the preload to the actu-ator through a steel ball. The steel ball holds the needle and it is placed on a ballseat which is attached to the actuator. Any movement in the actuator results inneedle displacement.2.1.4 NeedleA steel needle is connected to the steel ball through a tolerance fitting. Anymisalignment in the needle and the preload assembly during preloading causesfriction between the cap and the needle. A Teflon sleeve (in Figure 2.3) is placedbetween the needle and the cap which reduces the frictional force from 25 N to5 N during the needle movement. A force sensor is used to measure the forcevariation during the operation. The position measurement is performed at thefree end of the needle.2.1.5 Ball SeatsWhile mounting the actuator in the test-bed and preloading, any misalign-ment may cause damage to the actuator. Hence, it is suggested to use steel ballswhile mounting the actuator. To facilitate the mounting of the actuator betweenthe steel balls, end caps are attached to the piezoelectric actuator. The end caps502.1. Mechanical ComponentsFigure 2.3: Preload assemblyFigure 2.4: CAD drawing of the test-bed512.2. Electrical Componentsare indented with the ball-end mills having the same radius as the steel balls toensure a better contact between the balls and the caps. High temperature epoxyis used to attach the end caps to the actuator. The ball seats are shown in Figure2.1. The complete test-bed is shown in Figure Electrical ComponentsThe electrical components are responsible to drive the actuator, perform themeasurements and data acquisition. This includes the piezoelectric amplifierwhich drives the actuator. Different types of sensors are used to measure cur-rent, force, position and temperature. A simple 1:100 voltage divider is used tomeasure the voltage signal. The measured quantities are recorded with a fastdata acquisition system through dSPACE/Controldesk?. The models are run inMATLAB/SIMULINK?. The electrical components are illustrated in brief in thefollowing subsections.2.2.1 Piezoelectric Actuator DriverPiezoelectric actuator driver or amplifier are required to amplify the controlsignal to drive the actuator. The power of the amplifier is one of the decidingfactors in amplifier selection. Large driving current is required to operate theactuator at high frequencies. In this research, a high power piezoelectric ampli-fier (E617.00F) is used which provides 2000 mA of peak current while the averagecurrent output is 1000 mA. The amplifier has an input range of -2 to 12 V and again of 10. A separate 24 V DC power supply is used to feed the power amplifier.2.2.2 Current SensorTo measure the current flow, a high resolution (1 mA) current sensor (Tek-tronix TCP312) is used. Once measured, charge is calculated from the currentmeasurement through integration in the MATLAB/SIMULINK?.522.2. Electrical Components2.2.3 Force SensorA force sensor (PCB-208C2) is mounted at the bottom of the test setup whichmeasures the force variation during piezoelectric actuation. The force sensor isconsidered stiff enough (1.05 kN??m?1) to resist the deflection due to piezoelec-tric actuation and hence the complete displacement is registered at the otherend of the actuator which is connected to the needle. Contrary to setup pre-sented in [20], the force sensor is not mounted between needle and actuator,but rather between actuator and frame. This has two advantages: 1) mountingthe actuator is simplified, and 2) the moving mass is reduced and the bandwidthof the test-bed is increased.2.2.4 Position SensorA pair of high speed laser vibrometers (Polytec-HSV2002) is employed tomeasure the position of the needle through a differential measurement of lasersensors. One of the sensor is used for the piezoelectric actuator movement whilethe other sensor is used to measure the test-bed vibration due to external sources.The differential measurement cancelled any unwanted noise present in the mea-surement. The resolution of the measurement system is 0.3 ?m while the band-width of the measurement is 250 kHz. A high resolution capacitive sensor (PID510.050) is also employed to measure the displacement of the actuator. How-ever, the sensor bandwidth is not attractive for high speed measurements (closeto the mechanical bandwidth of the test-bed).2.2.5 Temperature SensorInfrared (IR) temperature sensors (Micro Epsilon CT-CF15) are used to mea-sure the temperature change in the actuator due to self-heating or environmen-tal temperature variation. The infrared sensors are calibrated for the piezoelec-tric actuator surface using thermistors at room temperature. A converging lensis used in front of the IR sensors to reduce the spot size to less than 5 mm whichis the width of the actuator. Three IR sensors are used to measure the tempera-ture at the top, middle and at the bottom of the actuator. A sensor mount is used532.2. Electrical ComponentsFigure 2.5: Schematic of the experimental setupFigure 2.6: Experimental setup and the test-bed ([88] ?2013 IEEE)542.2. Electrical Componentswhich ensures the correct spacing (10 mm) between the sensors and the piezo-electric actuator surface. The complete schematic of the experimental setup isshown Figure 2.5. Figure 2.6 presents the picture of the setup and the test-bed.2.2.6 Data AcquisitionA high speed data acquisition system (dSpace CPL1103) is employed to cap-ture the experimental data. The experiments are modeled in SIMULINK? envi-ronment and implemented through MATLAB/Real Time Workshop?. The graph-ical user interface is designed in dSpace/ControlDesk? to facilitate real-timeplotting and data recording.552.3. Summary2.3 SummaryThe experimental setup that is used to characterize and test the actuator ispresented in this chapter. The setup is carefully designed to provide a high oper-ational range by reducing the moving mass in the setup. The experimental setupis divided into two major components: 1) mechanical components and 2) elec-trical components. The mechanical components are mostly related to the partswhich facilitate the mounting of the actuator. The electrical components areresponsible for measurements and data collection. The setup is used to charac-terize the actuator first and then the control experiments are performed on thesetup.56Chapter 3Impedance MeasurementAs an electromechanical component, modeling of PEAs is an essential partin implementation of these elements in real systems. In Chapter 1 (Section1.6.2), it is shown that in the electrical domain, the piezoelectric actuators aremodeled as capacitive elements. To obtain an accurate model of the actuator,the clamped capacitance of the actuator [13, 24] needs to be measured. Usuallythe clamped capacitance is measured from the free capacitance. The principleof the capacitance measurement technique is simple and relies on the basic the-ory of electric fields:C =Ir ms2pi f ?Ur ms(3.1)where, the root mean square (rms) value of a sinusoidal driving voltage is de-noted by Ur ms , Ir ms is the rms value the resulting current, f is the frequency ofthe voltage signal, and C is the measured capacitance. This type of measure-ment is usually carried out at frequencies around 1 kHz. However this leadsto two problems when applied to piezoelectric actuators. Firstly, the appliedvoltage and resultant current include charge and voltage hysteresis that distortsthe capacitance measurement. Secondly, the measured capacitance is the un-clamped capacitance since the actuator displaces due to the applied voltage[89]. Thus, constitutive relationships are required to convert it to the clampedcapacitance value used in the standard control model by Goldfarb et al. [13]. Inthis chapter, a novel real-time impedance measurement technique is presentedwhich facilitates the clamped capacitance measurement required for the model.Also, it provides an effective resistance measurement of the piezoelectric actua-tor.573.1. Measurement Principle3.1 Measurement PrincipleTo resolve the limitations in traditional measurements, an improved approachis proposed where a high frequency low voltage signal (termed ripple voltage inthe remainder of this thesis) is used to measure the capacitance instead of a tra-ditional low frequency signal. The ripple voltage frequency, fr must be muchlarger than the resonant frequency of the test bed. This will ensure that thereis no mechanical movement of the actuator due to the ripple voltage. Also, themagnitude of the ripple voltage needs to be small in order to ensure that hys-teresis is minimized [90]. To yield the additional benefit of a real-time capaci-tance measurement, the high frequency ripple signal is superimposed with thelow frequency driving signal. In this way, the capacitance measurement can takeplace in real-time without influencing the low frequency displacement of the ac-tuator or vice versa. This real-time capacitance measurement is not only novel[91], but also very useful since it provides the basis for the sensorless control al-gorithms presented in Chapters 5. There are two possible ways to superimposethe ripple onto the driving signal: 1) using a summing circuit which merges thehigh frequency ripple signal with a low frequency driving signal, and 2) usinga boost converters with fixed carrier frequency which provides a constant fre-quency ripple voltage.3.2 Superpositioning Through Summing CircuitA summing circuit similar to the one shown in Figure 3.1 may be used to su-perposition the ripple signal, Ur , on the driving voltage, U . The driving voltage,U is sourced from the piezoelectric amplifier. The ripple signal, Ur may be gen-erated through a signal generator. Transient voltage suppression (TVS) diode isused to protect the signal generator from voltage spikes. Also, a low pass filtermay be required to filter out any noise present in the driving signal. The ripplefrequency is varied through a function generator.583.3. Superpositioning Through Boost ConverterFigure 3.1: Superposition of signals through summing circuit3.3 Superpositioning Through Boost ConverterTo drive the piezoelectric actuator, a DC-DC boost converter is employed toamplify the control signal from the data acquisition board (Section 2.2.1). Thisamplifier automatically superimposes a high frequency ripple voltage (?0.1% ofthe maximum voltage range) onto the driving voltage. The magnitude and fre-quency of the amplifier output voltage ripple is close to ?100 mV and ?100 kHzrespectively. While driving the actuator, the high frequency (?100 kHz) ripplevoltage is superimposed onto the driving voltage, U , which results in a ripplecurrent, Ir . For the test bed in this thesis, the ripple voltage, Ur does not affectthe displacement of the actuator. This is due to the fact that the ripple frequencyis much larger than the mechanical bandwidth (?2500 Hz) of the setup. Figure3.2 shows the velocity of the actuator due to the ripple voltage. The free veloc-ity is measured when the actuator is not mounted in the test-bed (mechanicalbandwidth ?40 kHz) while the clamped velocity is measured when the actuatoris mounted in the test-bed (mechanical bandwidth?2500 Hz). The figure clearlyindicates that the actuator shows close to zero velocity due to the ripple voltagewhen placed in the test-bed.3.4 Effect of Frequency on the Actuator CapacitancePiezoelectric capacitance measurement is largely affected by the measure-ment frequency. Hence, it is important to specify the frequency at which the593.4. Effect of Frequency on the Actuator CapacitanceFigure 3.2: Velocity measurement in clamped and free condition [89] ?2011IEEEcapacitance measurement takes place. The piezoelectric actuator is modeled asa pure capacitor near the driving frequency since the phase angle between thevoltage, U and resultant current, I is close to 90?. The phase lag between the lowfrequency normalized voltage and normalized current is shown in Figure 3.3.The normalization of the signals are obtained by dividing the signals with theirmaximum magnitudes. So, the expressions for the capacitance measurement inEquation 3.1 are valid. However, at frequencies close to the ripple frequency of100 kHz the phase angle between the ripple voltage, Ur and ripple current, Iris close to 65?. The phase difference between the high frequency ripple voltageand current is presented in Figure 3.4.603.4. Effect of Frequency on the Actuator CapacitanceFigure 3.3: Normalized voltage and current at 10 Hz613.4. Effect of Frequency on the Actuator CapacitanceFigure 3.4: Normalized ripple voltage and ripple current at 100 kHz623.5. Impedance Measurement AlgorithmAt high frequencies, the piezoelectric actuator is modeled as a resistor-capacitorcombination connected in series instead of a pure capacitive load. To avoid am-biguity with the capacitance measured at low frequencies, the measured capac-itance at 100 kHz is called ?effective capacitance? in the remainder of the thesis.The expression for the effective capacitance measurement is as follows:Ce =Irr ms2pi fr ?Urr ms. (3.2)3.5 Impedance Measurement AlgorithmThe voltage amplifier/driver uses a boost converter, which amplifies the con-trol voltage by a factor of ten and superimposes a 100 kHz ripple voltage, Ur ,onto the driving voltage, U . The superpositioning of the low frequency drivingvoltage, U (100 Hz) and high frequency ripple voltage, Ur (100 kHz) is shown inFigure 3.5. To apply the proposed measurement technique, it is imperative toextract the high frequency ripple voltages from the low frequency driving volt-ages since the ripple quantities are employed to measure the impedance. Tofacilitate this extraction, the current and voltage signals are passed through twoidentical second order high pass active filters. The circuit diagram of the activefilter is shown in Figure 3.6. The capacitance and resistance values for the filterwere obtained from a Sullen-key topology implementation [92]. The cut-off fre-quency of the filters is set to 18000 kHz. Once the ripple quantities are extracted,the next step in the measurement is to decide on the sampling frequency of themeasurement.3.5.1 Selection of the Sampling FrequencyThe sampling frequency selection plays an important part in the proposedmeasurement technique. Shannon?s sampling theorem indicates that for a rip-ple frequency of 100 kHz, one would need to sample at more than 200 kHz inorder to ensure that the ripple signal can be reconstructed without frequencyfolding. Unfortunately, sampling at this high frequency is expensive. However,633.5. Impedance Measurement AlgorithmFigure 3.5: Amplifier output while supplying the driving voltage [89] ?2011 IEEEFigure 3.6: Piezoelectric impedance measurement circuit643.5. Impedance Measurement Algorithmsince the exact frequency of the ripple frequency is known, it is possible to pre-dict where this frequency folds to when lower sampling rates are employed. Thisfeature of the impedance measurement technique is quite novel and has beenpatented in [91]. To ensure that the sampled signals produce no leakage in thesubsequent Fourier transform, the folded ripple frequency, fr f olded , needs to bean integer multiple of the sampling frequency, fs :ns =fsfr f olded(3.3)Figure 3.7 shows the folding of the ripple frequency, fr , around the Nyquistfrequency, fN =fs2 , and the zero frequency. From the four cases with folds be-tween zero and three, one can generalize the following relationship betweenthe sampling frequency, fs , the ripple frequency, fr , the number of samples perfolded ripple period, ns , and the number of folds n f :fs =2ns frn f ns +2, f or n f = 0,2,4... (3.4)fs =2ns fr(n f +1)ns ?2, f or n f = 1,3,5... (3.5)For a ripple frequency of 100 kHz, there are many possible sampling fre-quencies for different combinations of number of samples per folded ripple fre-quency, ns , and number of frequency folds, n f . The number of samples perfolded ripple frequency, ns , needs to be chosen such that the subsequent Fouriertransform is numerically inexpensive. Both ns=4, and ns=12 lead to computa-tionally efficient Fourier transforms. Once ns has been selected, the number offrequency folds, n f determines the sampling frequency as well as the folded rip-ple frequency. One would usually aim to keep the sampling frequency at as low avalue as possible while maintaining the folded ripple frequency at a value higherthan the mechanical bandwidth of the test-bed.Table 3.1 shows possible combinations of sampling frequency, fs , and folded653.5. Impedance Measurement AlgorithmFigure 3.7: Ripple signal with (a) no fold, (b) one fold, (c) two folds, and (d) threefolds663.5. Impedance Measurement Algorithmripple frequency, fr f ol ded .Table 3.1: Sampling frequency and folded ripple frequency at different values ofns with four times foldingns n f fs [kHz] fr f olded [kHz]4 0 400 1004 2 80 204 12 16 412 0 1200 10012 4 48 43.5.2 Real-time Fourier TransformationA real-time Fourier transformation of the ripple voltage Vr and the ripplecurrent Ir is performed to obtain their magnitudes and phases. The Fourier co-efficients of the ripple voltage, AVr and BVr as well as of the ripple current, AIrand BIr can be obtained using the following discrete transfer functions:AIrIr=AVrVr=12npnp?i=1ns?j=1si n(2pinsj)z j?i ns (3.6)BIrIr=BVrVr=12npnp?i=1ns?j=1cos(2pinsj)z j?i ns (3.7)Here, np is the number of cycles the Fourier transform is performed across. Itshould be noted that the division by 2np does not actually have to be performed,since the impedances of the actuator are determined from ratios of the Fouriercoefficients which are independent of np . Selection of a larger np , has two ef-fects. First, the frequency is filtered around a narrower band. Mathematicallythis can be stated as follows:fPB =fsnsnp, (3.8)673.5. Impedance Measurement Algorithmwhere, fPB is the passband of the Fourier transform. It should be pointed outthat this passband applies to both the folded ripple frequency as well as theoriginal ripple frequency. Too wide a passband can pick up mechanical vibra-tions close to the folded ripple frequency. Too narrow a passband could miss theripple frequency, if the ripple frequency is not exactly constant. In addition tothe reduction of the passband size, the bandwidth of the capacitance measure-ment is also reduced when more cycles are employed. Assuming that the Fouriertransform provides an average capacitance measurement over the number ofcycles used, the bandwidth of the measurement can be approximated as:fBW =fsnsnppi. (3.9)Table 3.2: Passband and bandwidth as a function of sampling periods for a 48kHz sampling frequency, and ns = 12Cycles, np Passband, fPB [kHz] Bandwidth, fBW [kHz]1 4 1.272 2 0.6374 1 0.3188 0.5 0.15912 0.33 0.10616 0.25 0.08For the test-bed in this thesis, the ripple voltage frequency is not exactly con-stant at 100 kHz and so using more than four cycles could lead to a loss of fre-quency content of the ripple. Also, using more than twelve cycles would resultin a capacitance measurement with lower bandwidth than the actuator drivingfrequency [89].3.5.3 Parameter Identification from Fourier Series CoefficientsThe equivalent RC model for the piezoelectric actuator is presented in Fig-ure 3.8 (a). The phasor diagram is shown in Figure 3.8 (b). By summing up thequantities in real and imaginary directions, the following relationships can be683.6. ResultsFigure 3.8: (a) Resistance-capacitance model for the piezoelectric actuator (b)phasor diagramdeduced:AVr = AIr Re +BIr?r Ce(3.10)BVr =BIr Re ?AIr?r Ce(3.11)Once the Fourier coefficients are obtained, piezoelectric capacitance and resis-tance can be calculated using Equations 3.12 and 3.13 as follows:Ce =?1?rA2Ir +B2IrAIr BVr ? AVr BIr, (3.12)Re =AIr AVr +BVr BIrA2Ir +B2Ir. (3.13)3.6 ResultsThe measured capacitance and resistance values are observed during a 100V-10 Hz sinusoidal operation. Plots of driving voltage, actuator displacement, aswell as effective capacitance and resistance are shown in Figure 3.9. These mea-693.6. Resultssurements help to deduce relationships between different measured quantitiessuch as capacitance-voltage, capacitance-position, and capacitance-resistancerelationships. Figure 3.10 shows the relationship between the driving voltageand the effective capacitance. A large hysteresis (?10% of the voltage) is presentbetween the driving voltage and the measured capacitance. Similarly, a largehysteresis is observed between voltage and position in Figure 3.12. However,when effective capacitance is plotted against position in Figure 3.11, minimalhysteresis is observed. The hysteresis between voltage and capacitance andhysteresis between voltage and position cancel each other. In Figure 3.13, theeffect of frequency is shown on the capacitance-position relationship. It showsthat the capacitance-position relationship slightly changes at relatively large fre-quency (10 Hz). However, for quasi-static cases, the effect of frequency is small.Two important conclusions can be drawn from these figures: 1) the capacitancewhich is considered constant in the models (in Section 1.6.2) is not constant dur-ing the piezoelectric actuator movement, and 2) the capacitance-position rela-tionship provides a means for position self-sensing at least for low frequencyoperations due to low hysteresis. These findings lead to two potential contri-butions in this dissertation: 1) an improved electromechanical model is pre-sented in Chapter 4 which includes the position dependency of the capacitancein the model, and 2) an improved position self-sensing scheme is presented inChapter 5 which extends the traditional self-sensing schemes applicable in thepresence of creep and rate-dependent hysteresis. In Figure 3.14, the effectiveresistance-capacitance relationship is shown. The resistance values decreasewith increased capacitance. The hysteresis present between the resistance andcapacitance is 6.25%.Typically, capacitance is affected by temperature variation which affects theposition self-sensing strategy presented in Chapter 5. In Chapter 7, the effectof temperature on capacitance due to self-heat generation and a temperaturecompensated position self-sensing are presented.703.6. ResultsFigure 3.9: (a) Voltage, (b) position, (c) effective capacitance, and (d) effectiveresistance at 10 Hz713.6. ResultsFigure 3.10: Hysteresis in effective capacitance-voltage relationshipFigure 3.11: Effective capacitance measurement with position723.6. ResultsFigure 3.12: Hysteresis in effective position-voltage relationshipFigure 3.13: Effective capacitance measurement with position at various fre-quencies733.7. SummaryFigure 3.14: Effective resistance measurement with effective capacitance3.7 SummaryA novel real-time impedance measurement technique is presented in Chap-ter 3. To obtain a real-time measurement of the piezoelectric clamped capac-itance for an accurate model of the piezoelectric actuator, high frequency rip-ple voltage and ripple current are employed. The high frequency ripple signalsare superimposed on the driving signal and hence provide real-time measure-ment of the impedance. Since the ripple frequency is significantly higher thanthe first natural frequency of the test bed, no movement is induced by the rip-ple. Thus the ripple signals can be used to determine clamped impedance. Theimpedance of the piezoelectric actuator is modeled as an equivalent resistance-capacitance element connected in series. The impedance values are obtainedusing Fourier series transformations of the ripple voltage and ripple current.This real-time impedance measurement shows that the capacitance variation isrelated to position change in piezoelectric actuator without significant hystere-sis. Based on this finding, an extended electromechanical model is proposed inChapter 4. In addition, a position self-sensing scheme is developed in Chapter743.7. Summary5 based on the capacitance-position relationship.75Chapter 4An Improved ElectromechanicalModel for Piezoelectric ActuatorsPiezoelectric actuator models can be classified into two broad groups: 1)physical models, and 2) phenomenological models. The physical models arebased on measurable quantities such as voltage, charge, force, displacement orthe equivalent quantities such as electric field, current, stress and strain. Usu-ally, the physical models are the augmented version of the linear IEEE modeldiscussed in Section 1.6.1. The eminent Goldfarb and Celanovic model [13]includes the dynamics and hysteresis phenomenon in piezoelectric actuatorswhich are missing in the IEEE model. The dynamics are included using a sec-ond order lumped mass system. The hysteresis is modeled using generalizedMaxwell?s resistive-capacitive operator. The model is presented earlier in Sec-tion 1.6.2. Phenomenological models rely upon the input-output relationship(voltage and position) of the piezoelectric actuator. Hence,these models do notrequire knowledge of the underlying physical interaction between the variablesto implement the model which makes these models very popular. However,highly accurate phenomenological models are required to implement the feed-forward control. Due to the limitations in the phenomenological models, elec-tromechanical models are used in this study.In the physical model presented by Goldfarb and Celanovic [13], the piezo-electric actuators are modeled as capacitors in the electric domain. A hystere-sis block is added in series with the capacitor model of the actuator (Figure1.10). In this model, the piezoelectric capacitance (similar to dielectric permit-tivity in the IEEE model in Section 1.6.1) and force-voltage proportionality con-stant(similar to piezoelectric strain coefficient in the IEEE model) are assumed76Chapter 4. An Improved Electromechanical Model for Piezoelectric Actuatorsconstant. These parameters are identified from the measurements of stiffnessand DC gain between charge and displacement [13]. A frequency analyzer pro-gram is used to identify the piezoelectric capacitance. The other model param-eters such as force-voltage proportionality constant and stiffness are identifiedexperimentally from force vs voltage and force vs displacement relationships re-spectively. Badel et al. [41] employ a hyperbolic function to model the hysteresisin the Goldfarb model and identify the function coefficients for ascending anddescending hysteresis paths. Also, the force-voltage proportionality constant isidentified using one of the function parameters in relation to force and volt-age. Finally, free capacitance is measured and clamped capacitance is obtainedfrom the free capacitance using constitutive equations. Juhasz et al. [93] pro-pose a parameter identification process for an embedded piezoelectric micro-positioning system. In this study, the force-voltage proportionality constant ismeasured when the hysteresis slope between the applied voltage and the po-sition is constant. Quant et al. [94] also use a similar model as proposed byGoldfarb et al. [13] for a piezoelectric bending actuator. However, in an attemptto provide a better model fit, the force-voltage proportionality constant is con-sidered a 6th order polynomial function of voltage derived from a piezoelectricstrain coefficient measurement. The models introduced above are extensionof the traditional model proposed by Goldfarb and Celanovic [13]. In most ofthe cases, the model parameters, such as the capacitance, stiffness and force-voltage proportionality constant are assumed constant. However, all the statedmodels differ in the parameter identification technique.Table 4.1 shows how close the identification techniques of different researchgroups come to the manufacturer specifications. Every model has at least oneparameter that is significantly different from the suggested values by the man-ufacturer. Since the manufacturer measurements are based on the linear IEEEconstitutive equations, which do not include hysteresis, the variation betweenthe manufacturer stipulated data and parameters from the different models isexpected. However, these parameters should not be orders of magnitude differ-ent.This chapter presents an improved electromechanical model of the piezo-electric actuator which relies upon the real-time capacitance measurement pre-77Chapter 4. An Improved Electromechanical Model for Piezoelectric Actuatorssented in Chapter 3. A new model is developed based on the position dependentcapacitance. This leads to an improved fit between charge and position whencompared to the traditional electromechanical model. Thus, the new modelcan be used to improve the accuracy of charge based sensorless position con-trol. Another advantage of the new model is that the model parameters can beidentified from a single set of current, voltage, position and capacitance mea-surements. Finally, other material parameters such as piezoelectric strain coeffi-cient, d33, piezoelectric coupling coefficient, k33, and the permittivity constant,?33 can be deduced by employing the constitutive relationships.78Chapter 4. An Improved Electromechanical Model for Piezoelectric ActuatorsTable4.1:Comparisonofparametersfromdifferentpiezoelectricmodelswithmanufacturerdata[95]?comparisonCPcomparisonk PcomparisonModelManufacturer-?ident.?manuf.7CPident.CPmanuf.k Pident.k Pmanuf.Modelnumber[NV?1 ][NV?1 ][?F][?F][N?m?1 ][N?m?1 ]Goldfarbetal.NEC-AE0505D1610.009.311.201.406.0048.85Badeletal.NEC-AE0505D44H40F7.199.312.773.4028.7020.24Juhaszetal.PI-P885-5017.558.051.5081.50100.2950.007Obtainedfrom?=d 33As 33tPwhere,s33isthemechanicalcompliance,Aisthestackcrosssectionalarea,andt Pisthelayerthickness8Manufacturerspecifiedvalue[18]794.1. Proposed Piezoelectric Model4.1 Proposed Piezoelectric ModelIn order to better describe the behavior of the experimental test-bed and thepiezo stack used within that test-bed, both the constitutive Equations 1.4 and 1.5in the model presented by Goldfarb and Celanovic describing the mechanicaland the electrical domain require modification. The modification proposed inthe new model are outlined in the following sections.4.1.1 Mechanical SubsystemThe mechanical domain of the test setup is described by the force Equation1.4. The test setup is shown in Figure 2.3. A schematic diagram of the mechan-ical system is shown in Figure 4.1. The moving mass term includes the massof the needle assembly, mn and the mass of the piezo actuator, mP . Since thepiezoelectric actuator behaves like a heavy spring, only one third of the piezomass is also considered [96]. Traces from the force sensor indicate that the fric-tion between needle and the Teflon sleeve is negligible 4.2. Finally, the stiffnessis comprised of the preload spring stiffness, ks and the piezo stiffness, kP . Theupdated force equation describing this model is given in Equations 4.1 and 4.2:?UP =mx? +bx? +kx, (4.1)?UP =(mn +mP3)x? +bx? + (kP +ks)x. (4.2)4.1.2 Electrical SubsystemGoldfarb and Celanovic [13] model the piezoelectric actuators as capacitorsin the electrical domain. Change in piezoelectric capacitance with voltage is ob-served in [97, 98], which is not reflected in Equation 1.5. This change of capac-itance with position is confirmed by the online effective capacitance measure-ment presented in Chapter 3. Traditional capacitance measurements employ arelatively low frequency and high voltage (1 kHz, 1 VPP) in comparison to the804.1. Proposed Piezoelectric ModelFigure 4.1: Schematic representation of the test setupFigure 4.2: Force measurement at 50 Hz814.1. Proposed Piezoelectric Modelproposed effective capacitance measurement ripple voltage (Ur = 100 kHz, 0.1VPP). Since the frequency of the proposed measurement is much higher than thefirst natural mechanical frequency (2500 Hz) of the test setup, the ripple voltagedoes not induce any movement during measurement. Thus, the proposed effec-tive capacitance measurement provides a clamped capacitance measurement.In addition, the measurement has little hysteresis, since the driving voltages arevery low [90]. Figure 4.3, shows the relationship between measured effective ca-pacitance and position for driving frequencies between 10 Hz and 100 Hz.At low driving frequencies, the inertial and friction terms presented in theconstitutive Equation 1.4 can be neglected in comparison to the stiffness term[14, 28]. Hence, the mechanical constitutive equation reduces to:?UP = kx. (4.3)To include the variable capacitance as seen in Figure 4.3, the clamped ca-pacitance, CP is described as a linear function of the piezo voltage, UP , and sub-sequently is a function of position, x, since UP and x are linearly related fromEquation 4.3 for low driving frequencies:CP =CP0+?CUUP =CP0+?Cx x, (4.4)?CU =?Cx?k, (4.5)where, CP0 is the capacitance value at zero voltage and ?CU is the change incapacitance as a function of Up while ?Cx is the change in capacitance withposition. Adding the linear capacitance-position relationship (in Equation 4.4)to the charge equation presented in Equation 1.5, the following relationship isobtained:q =CPUP +?CU2U 2P +?x. (4.6)824.1. Proposed Piezoelectric ModelFigure 4.3: Capacitance-position relationship at various frequencies834.2. Power Balance4.2 Power BalanceIn the previous section, an additional term is introduced into the consti-tutive equation for the electrical domain (Equation 4.6) and no adjustment ismade to the mechanical domain (Equation 4.1). This section uses a power bal-ance to show that these equations are still consistent with each other. Ignoringthe hysteresis, the total power delivered to the stack is equal to the product ofapplied current, I , and piezo voltage, UP . The portion of the power that is notlost to friction is stored in the stack as spring, kinetic, and capacitive energy:UP I = P f r i ct i on +dd tEspr i ng +dd tEki neti c +dd tEcapaci tor . (4.7)For a nonlinear capacitor the energy stored can be calculated as follows [99]:Ecapaci tor =U qcap ??qcap dU =12CP0U 2+13?CUU 3. (4.8)Substituting Equation 4.8 into Equation 4.7 and replacing U with UP yields asimple expression for the power balance:UP I =dd t?bx?d x +dd t(12kx2)+dd t(12mx?2)+dd t(12CP0U 2P +13?CUU 3P), (4.9)UP I = bx?2+kxx? +mx?x? +CP0UPU?P +?CUU 2PU?P . (4.10)It is now demonstrated that the constitutive Equations 4.1 and 4.6 provide thesame expression for power balance. First, Equation 4.1 is divided by ? in orderto obtain an expression for the piezo voltage UP :UP =mx? +bx? +kx?. (4.11)Then, a time derivative is taken of Equation 4.6 in order obtain the current sup-plied to the stack, I :I =?x? +CP0U?P +?CUUPU?P . (4.12)844.3. Parameter IdentificationFinally, Equations 4.11 and 4.12 are multiplied to obtain the same expression asthe original power balance shown in Equation 4.10:UP I = bx?2+kxx? +mx?x? +mx? +bx? +kx?(CP0U?P +?CUUPU?P ), (4.13)UP I = bx?2+kxx? +mx?x? +CP0UPU?P +?CUU 2PU?P . (4.14)Since Equations 4.10 and 4.14 are identical, it is concluded that the mechanicalconstitutive equation is unaffected by the introduction of the nonlinear capaci-tance in the electrical constitutive equation.4.3 Parameter IdentificationTraditional models have usually assumed that CP , ?, k, and b are time in-variant. The proposed model suggests a position dependent capacitance. Inorder to show the improvements attainable with this approach, a separate iden-tification procedures for the base case with a constant capacitance and the newvariable capacitance model are defined in Sections 4.3.1 and 4.3.2. Thereafter,additional material parameters are derived in Sections 4.3.4, and Identification of ?, k, b, and m for the Traditional Model withConstant CapacitanceWhen Equations 1.4 and 1.5 are combined for quasi static conditions, anexpression linking charge linearly to displacement is obtained:q =(CPk?+?)x. (4.15)The relationship between charge and position only allows the identification ofone of the three parameters ?, k, or CP . Two of these parameters need to beidentified by other relationships. The clamped capacitance can be measuredusing the technique described in Chapter 3.The stiffness is obtained by dividing the force, Fext developed at maximum854.3. Parameter Identificationvoltage and zero stroke by the displacement achieved at maximum voltage andno externally applied force. This measurement technique is only an approxima-tion, since it assumes that the hysteresis voltage, UH , is the same for clampedand free conditions.k =Fext |Umax=0,x=0x|Umax=0,F=0(4.16)Given CP and k, one can then use Equation 4.15 to identify the force-voltageproportionality constant, ? from a quasi-static charge vs displacement exper-iment. In order to obtain values for the damping coefficient b, and mass m,one can perform dynamic experiments. In this case, a minimization of the errorbetween measured charge, q? and charge calculated from position, velocity andacceleration (using the model) is solved:er rl i n = q? ?CP?(mx? +bx? +kx)??x. (4.17)This minimization procedure can be performed with a simple quadratic errorminimization algorithm, or a nonlinear search algorithm. In this research, aMATLAB function ?fminsearch? is used which is based on the the Nelder-Meadoptimization function in [100]. Both approaches provide similar answers. Intheory, Equation 4.17 can be used to identify b, and m with the given valuesof ?, CP , and k. However, the experiments are performed well below the firstmechanical eigenfrequency of the stack assembly, and so the sensitivity for de-termining the moving mass m, is poor. Instead, the mass m, is determined byadding a third of the stack mass mP , to the needle mass mn .4.3.2 Identification of ?, k, b, and m for the Proposed Model withVariable CapacitanceThe constitutive Equations 4.1 and 4.6 are characterized by the clamped ca-pacitance, CP , the mass, m, the force-voltage proportionality constant, ?, thestiffness, k and the damping, b. Given measurements of capacitance as a func-tion of position, both stiffness k, and force-voltage proportionality constant, ?can be obtained directly from the quasi static relationship between charge and864.3. Parameter IdentificationFigure 4.4: Capacitance-position relationship at various frequenciesposition, since this is now a second order polynomial:q =CP0k?x +?Cx2k?x2+?x. (4.18)An alternative way to determine k and ? using quasi static experiments is to fitcurrent over velocity. This avoids integrating current in order to obtain charge[101]:q?x?=(CP0k?+?)+?Cxk?x. (4.19)It is shown in Equation 4.19 that the ratio of current over velocity is not a con-stant as would be the case for the original model by Goldfarb and Celanovic(compare Equation 4.15). This is confirmed by the measurements shown in Fig-ure 4.4 where driving voltages of different frequencies are applied to the actua-874.3. Parameter Identificationtor. It should be pointed out that Equation 4.19 leads to a division by zero closeto the beginning and the end of the stroke (where velocities are zero). Thus,measurements are only physically meaningful towards the center of the stroke.The mass of the actuator system is estimated by adding a third of the stackmass mP , to the needle mass mn . To obtain the damping coefficient b, dynamicexperiments need to be performed. In this case, b, is found by using the Nelder-Mead optimization algorithm (1965) to minimize the error between measuredcharge, q? and charge calculated from measured position, velocity, and accelera-tion:er rnonli near =(q? ?CP0UP ??Cx2?kU 2P ??x), (4.20)where,UP =mx? +bx? +kx?. (4.21)4.3.3 Identification of Free Capacitance, CTFree capacitance is measured under stress free condition that are presentwhen there is no or constant external load. The relationship between the clampedand free capacitance can be deduced from the constitutive charge Equation 4.6as follows:qPUP=CP +?2k=CT . (4.22)where, CT is the free capacitance. From the free capacitance, the free permittiv-ity can be obtained through Equation 4.23 where, ?T is the free permittivity, A isthe area of the electrode (? to cross sectional area of actuator), nl is the numberfor layers in the actuator and ?0 is the permittivity constant at vacuum:?T =CT A?0tP nl. (4.23)884.3. Parameter Identification4.3.4 Identification of Material PropertiesThe piezoelectric material properties such as piezoelectric strain coefficient,d33 and piezoelectric coupling coefficient, k33 can also be obtained from theconstitutive relationships once the model parameters are obtained. In this sec-tion, the material parameters are attained from the identified model parametersand compared with the manufacturer specification where available. Piezoelectric Strain or Charge Coefficient, d33The piezoelectric strain or charge coefficient, d33 is an important design pa-rameter for piezoelectric actuator systems. It is defined as the strain developedper unit of applied electric field in the piezoelectric materials. Alternatively, thisis equivalent to the polarization generated due to the applied stress. The firstsubscript 3 represents the direction of the strain or polarization while the sec-ond subscript represents direction of the applied electric field or stress. Tradi-tionally, it is measured using the linear IEEE constitutive Equation 1.2 which as-sumes the applied stress is negligible and strain is a function of applied electricfield. Measuring both strain and electric field provides the d33 measurement.However, the traditional measurement employs the applied voltage instead ofthe linearized voltage, UP , as a measure of electric field. As a result, different d33are obtained at different voltage levels due to hysteresis. To minimize this hys-teresis effect, traditional measurements are often performed at the maximumlevel of the applied voltage, U . Also, since d33 provides an estimate of the qualityof the materials, large signal values provide more practical representation. Theterm d33 is an equivalent term to the force-voltage proportionality constant, ?in the electromechanical model. The following equation [102] presents the rela-tionship between d33 and ?:d33 =?nl k=?tPLP k, (4.24)where, nl is the number of layers in the actuator and k is the stiffness of the ac-tuator. The number of layers is calculated from the thickness of the piezoelectriclayers, tP and the length of the actuator, LP .894.4. Experimental Results4.3.4.2 Piezoelectric Coupling Coefficient, k33The piezoelectric coupling coefficient, k33 is an important parameter whichrepresents the fraction of electromechanical energy conversion in the piezo-electric system. It is an indication of the effectiveness or efficiency with whichthe piezoelectric actuator converts electrical to mechanical energy or vice versa.The first subscript 3 represents the direction of the applied electric field whilethe second subscript represents the direction of the mechanical movement. Ahigher value of the coefficient is desired. It does not include the dielectric ormechanical losses. The coupling coefficient can be obtained from the followingrelationship [102] of clamped and free capacitance:k233 = 1?CPCT. (4.25)From the identified values of CP and CT , the piezoelectric coupling coefficientcan be obtained.4.4 Experimental ResultsThe model parameters are identified through experiments using the setupdiscussed in Section 2.2 and the identification algorithms outlined in Section4. Experimental ProcedureThe model identification experiments use a sinusoidal voltage profile withfrequencies between 10 Hz, 100Hz and a magnitude of 100V. The actuators areallowed sufficient time to reach a state where little hysteresis voltage is presentbefore starting the experiments. The position measurement is zeroed at the be-ginning of the measurement and data are recorded for 5 cycles shown in Fig-ure 4.5. At every frequency, 5 sets of data are recorded in order to ensure thestatistical significance. The model parameters are obtained using the methodsdescribed in Section 4.3 and listed in Table 4.2 together with the manufacturerspecifications. Two cases are presented in the table, where Case 1 presents the904.4. Experimental ResultsFigure 4.5: Driving signal, U at 10 Hztraditional model with a constant capacitance while Case 2 depicts the improvedmodel with variable capacitance.In Case 1, the value of capacitance is considered constant in the traditionalmodel. The value of the capacitance at zero V matches with the value prescribedby the manufacturer in Case 2, the new proposed model. However, in the man-ufacturer specification, it is prescribed that there is 20% variability present inthe capacitance measurement. The slope in the capacitance measurement isnot specified by the manufacturer, but similar trends have been reported by[97, 98] when a DC voltage is applied to piezoelectric components. The iden-tified stiffness values are largely different in Case 1 and Case 2. In Case 1, it ismeasured from static loading test using Equation 4.16 while in Case 2, it is ob-tained from the dynamic measurement of charge and position using Equation4.19. This identification is very close to the manufacturer specification. Also, theother parameters such as ?, d33 and k33 values are close to the values stipulated914.4. Experimental ResultsTable4.2:ModelparameteridentificationCase1Std.Case2Std.Manufacturer?sModelparametersUnit(Traditionaldev.9(Proposeddev.10SpecificationModel,Model,?Cx=0)?Cx6=0)Capacitanceat0V,CP0[?F]3.13110.033.13100.033.112Capacitanceat100V,CP100[?F]3.13100.032.16100.01-Capacitanceslope,?Cx[Fm?1 ]0?-0.036e-4-ActuatorStiffness,kP[N?m?1 ]11.2025.30.1625Actuatordamping,b[Nsm?1 ]31003772120120-Force-voltageprop.constant,?[NV?1 ],d 33[pmV?1 ]2400306952068014Piezoelectriccoupl.coeff.,k 330.9200.720.010.70149Calculatedoverthreesetsofexperiments10Calculatedoverthreesetsofexperiments11Identifiedcapacitancevaluesaremeasuredat100kHzand0.1V12Manufacturer?sspecifiedvaluesaremeasuredat1kHzand1V PP[18]13Obtainedfrom?=d 33As 33tP[102]14Manufacturerdatasheet[103]924.4. Experimental ResultsFigure 4.6: (a) Charge-position relationships modeled (proposed) and mea-sured, and (b) current-position relationships modeled and measured for variousfrequenciesfrom manufacturer?s specification with the proposed model. It is important tonote that the identified parameters are sensitive to the capacitance measure-ment which includes both the CP0 and ?Cx . The sensitivity of the identifiedparameters to changes in the capacitance measurement is shown in Table 4.3.To identify the parameters, the parameters presented in 4.2 for Case 2 are usedconsidering the capacitance measurement is unaffected.4.4.2 Model Validation Through Charge-position RelationshipFor model validation, measured values of charge-position relationships arecompared with the modeled charge-position relationship in Figure 4.6(a) usingEquation 4.18. The nearly linear relationship between charge and position islargely independent of driving frequency, since friction forces and inertial forcesare small at the tested frequencies. A more detailed comparison for the qual-ity of fit is obtained by plotting current against position as shown in in Fig-ure 4.6(b). Since current is closely related to velocity, one expects to see el-lipses that open up with increasing frequency. In Figure 4.7(a) and 4.7(b), thecharge-position relationship is compared between the traditional model (?Cx =0) and the proposed model (?Cx < 0). The modeling errors, presented in Figure934.4. Experimental ResultsTable4.3:SensitivityofparameteridentificationtochangesincapacitancemeasurementParametersNochange20%increase20%increase20%increaseinCP0in?CxinCP0and?Cxk[N?m?1 ]25.120.724.020.9?[NV?1 ]9.367.7210.779.36d 33[pmV?1 ]700697841840k 330.720.660.770.73UPmax[V]86867272944.4. Experimental Results4.7(c) and 4.7(d), are calculated using Equation 4.17 for the traditional modeland using Equation 4.20 for the proposed model. Figure 4.7 indicates that theproposed model shows slight improvements in terms of fitting the measuredcharge-position relationship. Both the 10 Hz and the 100 Hz cases show iden-tical trends where the proposed model provides lower fitting errors especiallytowards the center of the stroke. This is because of the ability of the proposedmodel to capture the curvature in the charge-displacement relationship with theadditional term in the charge constitutive equation.4.4.3 Hysteresis VoltageThe driving piezoelectric voltage, U is composed of the linear piezoelectricvoltage, UP and the hysteresis voltage, UH . Once the model parameters are iden-tified, one can obtain UP using Equation (4.14).The hysteresis voltage, UH is ob-tained through the traditional model in Equation 1.6. Figure 4.8(a) shows thedriving voltage, piezoelectric voltage and the hysteresis voltage obtained for thetraditional model with different frequency ranges. Figure 4.8(b) presents thesame voltages for the proposed model. In both cases, the hysteresis voltageand the linear piezoelectric voltage show similar trends for all tested frequen-cies. However, the hysteresis voltage obtained through the traditional model issignificantly large than for the proposed model. To be physically meaningful,hysteresis voltages should be in the order of approximately 20% of the appliedvoltage. This is the case for the parameters of the proposed model, but not forthe traditional model parameters.954.4. Experimental ResultsFigure 4.7: Comparison of charge-position relationship: measured, traditionalmodel (?Cx = 0) and proposed model (?Cx < 0) at (a) 10 Hz, (b) 100 Hz, errorsat (c) 10 Hz, and (d) 100 Hz964.5. SummaryFigure 4.8: (a) U and UH vs charge for ?Cx = 0 (traditional model), and (b) Uand UH vs charge for ?Cx < 0 (proposed model) at 10 Hz and 100 Hz4.5 SummaryIn this chapter, an improved piezoelectric model is proposed by introduc-ing a nonlinear capacitance term. This new term provides a better fit of the re-lationship between charge and position. A second advantage of this model isthat actuator stiffness and force-voltage proportionality constant can be iden-tified from quasi static measurements of position, charge, and effective capac-itance. Quantitative comparisons with manufacturer data show that the pro-posed model provides meaningful identification of the model parameters. Fur-ther, quantitative analyses through hysteresis voltage identification justify theimprovement of the proposed model over traditional model.97Chapter 5Position Self-sensing ofPiezoelectric ActuatorsPiezoelectric actuators demonstrate nonlinear phenomena such as hystere-sis and creep at different frequency ranges [26, 30, 43]. Examples of hystere-sis and creep behavior are shown in Figure 5.1, where the actuator position isplotted against applied voltage for frequencies ranging from 0 Hz (creep) to 100Hz. Usually, feedback position sensors are used to mitigate these nonlinearitiessuccessfully. However, there are many applications where dedicated positionsensors are difficult to implement due to space and/or cost constraints. Exam-ples for applications that suffer from these constraints are fuel injectors, inkjetprinting nozzles, micro-pumps etc. Moreover, an additional component may af-fect the overall reliability of the control system. Chapter 1 highlighted some ofthe model based sensorless position control techniques for piezoelectric actua-tors that have been used to avoid additional sensor elements. These techniquesare designed to compensate for hysteresis (Section 1.6.3) and creep phenom-ena (Section 1.3). However, these sensorless control techniques are typically de-signed for a very narrow frequency range. Thus, they cannot simultaneouslycompensate for both creep and rate-dependent hysteresis effects. Moreover, anaccurate model is a prerequisite for these sensorless control schemes. Thesemodels are often difficult to obtain and susceptible to modeling uncertainties[11, 12]. A number of studies include a feedforward model to a dedicated feed-back sensor (Section 1.7.3) in order to improve the dynamic properties of thefeedback system. This approach improves the bandwidth of the controller andleads to more robustness towards modeling uncertainties.Position self-sensing (PSS), which was first proposed for vibration suppres-98Chapter 5. Position Self-sensing of Piezoelectric ActuatorsFigure 5.1: Rate-dependent hysteresis and creep at different frequencies(Reprinted with permission [61]. Copyright 2012, American Institute of Physics.)99Chapter 5. Position Self-sensing of Piezoelectric Actuatorssion in different structures [72], is another technique used to improve actuatorpositioning. In this approach, a single piezoelectric element is used both as asensor and an actuator. True collocation of actuator and sensor is realized basedon the direct and indirect effect of piezoelectricity. A capacitance bridge circuitis used to extract strain or position. This approach is often used in piezoelectriccantilevers or patches. Jones et al. [104] implemented this approach with mul-tilayered actuators for micro positioning. The self-sensing approach works wellassuming ideal conditions. However, in practice, this approach poses a numberof challenges. The main problem is associated with the bridge balancing sinceit employs a similar capacitor as the piezoelectric capacitance where the accu-rate measurement of the piezoelectric capacitance is difficult [62, 73]. So, anychange in the piezoelectric actuator capacitance other than the strain may af-fect the measurement and hence the overall control system. The capacitancebridge based self-sensing is presented in Section 1.8.1. Unlike voltage, piezo-electric charge flow is found to be linearly related to the position over a widerange of frequencies [13, 52]. Figure 5.2 shows the charge-position relationshipfrom 10 Hz to 300 Hz with negligible hysteresis. A linear fitting line with a slopeof a0 is also shown in Figure 5.2. Based on this linear relationship, charge basedcontrollers are designed. Charge based PSS can also be realized relying on thelinear charge-position relationship. In this case, current is integrated and thepiezoelectric actuator position is obtained from the charge measurement. Inthis approach, the major challenge is the charge drift due to an offset currentwhich result in estimated position drift specially during slow operations (<10Hz). Section 1.8.2 highlights the charge based self-sensing scheme in more de-tail. The other self-sensing technique presented in Section 1.8.3 is based on thedirect effect of piezoelectricity where a portion of the actuator is designed tofunction as sensor. Due to the strain in the piezoelectric actuator, a voltage isinduced. The induced voltage is proportional to the actuator displacement andposition information is obtained from the induced voltage. The major short-coming in this technique is that the actuator range is reduced since some of theactuator is used as a sensor.The real-time effective capacitance measurement of the piezoelectric actua-tor presented in Chapter 3 provides yet another avenue to estimate actuator po-100Chapter 5. Position Self-sensing of Piezoelectric ActuatorsFigure 5.2: Charge-position relationship over a frequency range of 10 Hz ? 300Hz ([88] ?2013 IEEE)1015.1. Charge Based Position Self-sensing, x?Isition. Here one exploits the finding that the position of the actuator is relatedto effective capacitance with negligible (<3%) hysteresis. The problem with thecapacitance based PSS is that the signal is noisy and requires low pass filteringwhich limits its application to slow operations (<10 Hz).All of the self-sensing techniques discussed so far have advantages and short-comings. By combining charge based PSS and capacitance based PSS which ontheir own only work well for high and low driving frequencies respectively, a po-sition self-sensing for a wide frequency range can be achieved. It is important todefine the high and low frequencies at this point. By high frequency, it refers tothe range where the actuator is driven below the range to avoid the vibrationaldynamics (see Figure 1.4). To this end, a novel hybrid position observer (HPO)is developed which fuses the charge based PSS and capacitance based PSS. Itachieves good position estimates over a wide range of frequencies where bothcreep and hysteresis effects are present. The objective of this chapter is to intro-duce position estimation using the HPO. The following sections will detail thetwo PSS approaches, the operating principle of the HPO, and the implementa-tion of the HPO. Finally, the HPO results are compared with a dedicated positionsensor.5.1 Charge Based Position Self-sensing, x?ICharge based PSS is based on the linear fitting of the charge-position rela-tionship shown in Figure 5.2. The maximum position error observed in the cen-ter of the curve is less than one micrometer (<3% of the maximum stroke) for afrequency range of 10-300 Hz. This experimental linear relationship can also befound analytically. By manipulating the piezoelectric constitutive Equations 1.4and 1.5, an expression relating charge to position can be obtained,q =CPmx? +bx? +kx ?F?+?x. (5.1)1025.1. Charge Based Position Self-sensing, x?IFigure 5.3: Magnitude and phase response ([88] ?2013 IEEE)If external Force, F is neglected, then Equation 5.1 can be transformed into atransfer function in the Laplace-domain,G(s)=x(s)q(s)=x(s)sI (s)=1CP?(ms2+bs+k)+?. (5.2)Figure 5.3 shows the measured frequency response of G(s). The targeted fre-quency range (<150 Hz) in this research is well below the mechanical naturalfrequency of the test setup (2500 Hz) and the electromechanical resonance fre-quency (700 Hz). Hence, the dynamic terms in Equation 5.1 are neglected andthe following simplified expression is obtained for relating estimated position,x?I to measured charge, q? through G?(s):G?(s)=x?I (s)q?(s)=1?+CP k?=1a0. (5.3)The dynamic transfer function G(s) in Equation 5.2 reduces to a constant gainin the reduced frequency transfer function G?(s). Traditional feedforward chargecontrol schemes use Equation 5.3 by controlling the amount of charge suppliedto the piezoelectric actuator. Rather than controlling charge, charge based PSSintegrates measured current, I to obtain charge and hence position from thecharge measurement. Figure 5.2 presents the position estimation techniquefrom charge measurement based on Equation 5.3. To differentiate between the1035.2. Capacitance Based Position Self-sensing, x?CPFigure 5.4: Position estimation from charge measurement ([88] ?2013 IEEE)measured quantities and estimated quantities, a ?bar? above the variables is usedto represent the measured quantities and ?hat? is used for estimated values.The main challenge with this approach is charge leakage, which requires so-phisticated hardware architectures. Charge leakage leads to an offset current,I0 which results in a drift of identified charge and hence estimated position.To identify and compensate for the amount of offset current, a second low fre-quency position estimate based on an effective piezoelectric capacitance mea-surement is proposed. The next section details the capacitance based PSS.5.2 Capacitance Based Position Self-sensing, x?CPEffective piezoelectric capacitance measurement is discussed in Chapter 3.Figure 5.5 shows that the piezoelectric capacitance changes with stroke with lit-tle hysteresis (<3% maximum stroke). Based on a real-time measurement, effec-tive capacitance provides a means for position self-sensing. To yield the positioninformation from the effective capacitance measurement, a higher order poly-nomial is chosen. For fitting purpose, 0.1 Hz is selected which lies in between0.01 Hz to 1 Hz. Equation 5.4 presents the fitting function, ? for capacitancebased PSS,?= x?CP (C?P )= b0+5?j=1b j CjP , (5.4)where, j is the order of the polynomial and b0 and b j are the constant and thecoefficients of the polynomial. To obtain the order of the polynomial, j , the rela-tionship between capacitance and position is fitted with different orders and theresiduals and standard errors were compared to obtain a reliable fitting of the1045.2. Capacitance Based Position Self-sensing, x?CPFigure 5.5: Capacitance-position relationship at low frequenciescapacitance and position. The Microsoft Excel? Data Analysis Toolbox is usedto analyze the data. Table 5.1 presents the standard errors and maximal resid-ual errors of the fitting at different polynomial orders of capacitance-positionrelationships at 0.1 Hz.Polynomials of different orders and logarithm relationship are presented inFigure 5.6. The residual errors for different fits are presented in Figure 5.7. Theresidual plots show that the overall errors are close for different orders. However,for order 3 the error is large at the end of stroke (?1 ?m) while for order 5, theerror is close to 0.4 ?m. Although the improvement due to higher order fitting isnot significant in comparison to the maximum residual errors, the higher orderfitting is important for creep since the error due to creep is similar in magnitude(?1?m) for lower order fitting (j=3).Based on the maximum residual errors in Table 5.1 and considering the ef-fect of creep, the order of the polynomial fit is selected at 5 which provides areasonable fit for the capacitance-position relationship. In addition to the resid-ual error, the adjusted R2 values indicate that an order of 5 or 6 provides the best1055.2. Capacitance Based Position Self-sensing, x?CPFigure 5.6: Capacitance-position relationship at low frequenciesregression model. The ANOVA of the regression model ( j =5) is shown in Table5.2 where coefficients and intercept for the model are shown. The p-values in-dicate that all the coefficients are significantly important. The block diagramTable 5.1: Fitting errors in regression models of different ordersParameters j=3 j=4 j=5 j=6 lnMaximum residual error [?m] 1.34 1.24 1.17 1.15 1.49R2 value 0.99881 0.99886 0.99889 0.99889 0.997Adjusted R2 value 0.99881 0.99886 0.99889 0.99889 ?of the capacitance based PSS is shown in Figure 5.8. Due to the noise in the ca-pacitance measurement, the capacitance based PSS is only applicable for lowfrequencies.1065.2. Capacitance Based Position Self-sensing, x?CPFigure 5.7: Capacitance-position relationship at low frequenciesTable 5.2: ANOVA table for capacitance based PSS regression modelANOVA for j=5Df SS MS F Significance FRegression 5 2058890 411777.9 2581465 0Residual 14369 2292.046 0.159513Total 14374 2061182Coefficients t Stat p-valueb0 -4560.5 -19.9951 0b1 8773.922 19.99799 0b2 -6589.93 -19.6115 0b3 2445.475 19.09276 0b4 -450.855 -18.5542 0b5 33.10029 18.03204 01075.3. Hybrid Position Observer DesignFigure 5.8: Position estimation from capacitance measurement ([88] ?2013IEEE)5.3 Hybrid Position Observer DesignSince the charge based PSS provides position estimate at relatively high fre-quencies (below the high dynamic range shown in Figure 1.4) and capacitancebased PSS is suitable only at low frequencies, a hybrid position observer (HPO)is proposed which utilizes both PSS signals for position estimation. The blockdiagram of the hybrid position observer is shown in Figure 5.9 where the pre-viously discussed charge based PSS is fused with capacitance based PSS. Theshaded section of the Figure represents the real system while, the clear sectionrepresents the HPO. The structure of the HPO is very similar to a conventionalobserver. The plant model, which is represented by G?(s)s is driven by the mea-sured current, I? . The position obtained through effective capacitance, x?CP canbe thought of as a measurement unit in a traditional observer structure. In con-trary to a conventional observer however, the feedback term L(s) is not used toadjust the poles of the plant, but rather to estimate the offset current, I?0 presentin the measured current, I? . The estimated I?0 is then subtracted from the mea-sured current to eliminate the drift in charge based PSS. This is achieved throughthe use of a proportional plus integral operator:L(s)= LP +L Is. (5.5)The identified current offset is then subtracted from the current measurement.Assuming that the modeled plant, G?(s) equals the real plant, G(s), the followingtransfer functions for the estimated position signal, x? can be obtained:x?(s)xI (s)=11+ s?1G?(s)L(s)=a0s2a0s2+LP s+L I, (5.6)1085.3. Hybrid Position Observer DesignFigure 5.9: Hybrid position observer (Reprinted with permission [61]. Copyright2012, American Institute of Physics.)Figure 5.10: Magnitude plot of different transfer function (Reprinted with per-mission [61]. Copyright 2012, American Institute of Physics.)1095.3. Hybrid Position Observer DesignTable 5.3: Hybrid position observer parameters ([88] ?2013 IEEE)Parameter Units ValueCharge-position transfer fucntion, a0 C ?m?1 17Observer transition frequency, ?S r ad ? s?1 2piObserver damping, ?S 1Observer proportional gain, LP C ? r ad ? s?1 ?m?1 213.60Observer integral gain, L I C ? r ad 2 ? s?2 ?m?1 641x?(s)Io(s)=G?(s)1+ s?1G?(s)L(s)=sa0s2+LP s+L I. (5.7)Assuming further, x?CP (s)= xCP (s), the following relationship can be deduced:x?(s)xCP=s?1G?(s)L(s)1+ s?1G?(s)L(s)=LP s+L Ia0s2+LP s+L I. (5.8)Figure 5.10 presents the magnitude response of the resulting transfer functionsfor x?(s)xI (s) ,x?(s)xCP (s), x?(s)x(s) andx?(s)I0(s) . Below the switching frequency, ?S=1 Hz, the capac-itance based PSS (black line) is dominant and above ?S , charge based PSS (blueline) is favoured. The sum of the two PSS techniques has a constant unity gain(green line). Also, the offset current, I0(s) (red line) is attenuated with at least?45 dB . Constant offset currents are rejected entirely. Thus, the estimated out-put obtains the same magnitude as the actual position, x and a low frequencycurrent offset, I0 is eliminated. Moreover, the capacitance based PSS providesposition estimates at low frequencies. By choosing appropriate values for thecoefficients of L(s), a transition frequency, ?S is selected between the effectivecapacitance based PSS and the charge based PSS.LP = 2?S a0,L I =?2S a0. (5.9)The switching frequency, ?S is selected larger than the frequency of the offsetcurrent, I0 and smaller than the noise in the capacitance measurement. In theexperiments, good results are obtained with ?S = 2pi rad?s?1. A complete set ofHPO parameters is listed in Table 5.3.1105.4. Hybrid Position Observer ImplementationFigure 5.11: Hybrid position observer implementation ([88] ?2013 IEEE)5.4 Hybrid Position Observer ImplementationThe flowchart of the HPO self-sensing setup is presented in Figure 5.9. Theactuator is fed with a low frequency driving voltage, U for positioning and highfrequency ripple voltage, Ur for effective capacitance measurement. The lowfrequency voltage, U creates a driving current, I which upon integration, pro-vides a charge measurement that is employed to infer the position from Equa-tion 5.3. The high frequency voltage, Ur provides a high frequency ripple currentwhich are separated and processed to obtain the effective capacitance measure-ment through the algorithm shown in Chapter 3. A second position measure-ment is obtained from the effective capacitance measurement using Equation5.4. Based on the two position self-sensing measurements, the HPO provides areliable position estimate over a frequency range from 0 to 125 Hz.5.5 Open Loop HPO PerformanceTo evaluate the performance of the HPO, the position estimated from theHPO is compared with a position sensor (laser vibrometer) during open loopoperations. In the first experiment, the actuator is driven with sinusoidal fre-1115.5. Open Loop HPO PerformanceTable 5.4: Observer error, (?O = x? ? x?) at different frequencies (Reprinted withpermission [61]. Copyright 2012, American Institute of Physics.)Frequency Maximum error Maximum Stroke % Error[H z] [?m] [?m]0.01 0.88 33.82 2.60.1 0.42 33.56 1.31 0.59 33.11 1.810 1.00 32.56 3.150 0.79 32.41 2.4100 0.78 31.68 2.5quencies between 0.01 Hz ? 100 Hz and a magnitude of 100 V. Figures 5.12 and5.13 show the measured position, HPO position and error between the two mea-sured responses. The error percentages are also shown in Table 5.4. For a fre-quency range of 0.01 Hz to 100 Hz, the maximum error is recorded at 3.1% of themaximum stroke.In addition to the variable frequency range, the HPO is also tested in thepresence of creep. To test the creep behavior, a sinusoidal transient of 0.01 sec-onds (50 Hz) length and 100 V magnitude is followed by a constant 100 V drivingvoltage over 175 seconds. Figure 5.14 shows the results of this experiment. TheHPO shows a maximum error of 2.16% during the transient and the steady-stateportion of the experiment. In the steady-state region, the error is less than 1.5%.The amount of creep is not substantial in comparison to the hysteresis behavior.Since the observer error does not increase during the constant voltage phase,and both laser and HPO show the same trend in position, it is concluded thatthe HPO provides an estimation of creep at least qualitatively. This will be ver-ified further in Chapter 6 that describes closed loop control using the HPO as afeedback sensor.1125.5. Open Loop HPO PerformanceFigure 5.12: Measured position, HPO position and observer error between 0.01Hz and 1 Hz (Reprinted with permission [61]. Copyright 2012, American Insti-tute of Physics.)1135.5. Open Loop HPO PerformanceFigure 5.13: Measured position, HPO position and observer error between 10 Hzand 100 Hz (Reprinted with permission [61]. Copyright 2012, American Instituteof Physics.)1145.5. Open Loop HPO PerformanceFigure 5.14: Measured position, HPO position and observer error between 50 Hzand DC signal in the presence of creep (Reprinted with permission [61]. Copy-right 2012, American Institute of Physics.)1155.6. Summary5.6 SummaryPosition self-sensing technique provides an alternative to traditional posi-tion sensors where mounting a dedicated position sensor is not feasible. In thischapter, a novel HPO approach is presented which relies on two different PSSschemes: 1) PSS based on current or charge measurement and 2) PSS based oneffective capacitance measurement. The charge based PSS provides a good PSSsignal at high frequencies while the capacitance based PSS provides reliable po-sition estimate at low frequencies. The proposed HPO provides an improvedposition self-sensing over a wide frequency range by combining the advantagesof each PSS technique. The HPO performance is compared with a traditionalposition sensor in open loop condition for sinusoidal signals with frequenciesbetween 0 Hz and 100 Hz. This frequency range contains both creep and rate-dependent hysteresis. The proposed HPO provides around 3% error over thewhole range of operation which makes it applicable for real-time applicationsand a potential substitute of traditional position sensors.116Chapter 6Self-sensing Position Control ofPiezoelectric ActuatorsA novel position observer (HPO) is developed in Chapter 5 by fusing twoself-sensing signals to obtain an improved position signal over a wide frequencyrange. It is also shown how accurately the HPO can predict the position in openloop and the results are compared with a traditional sensor. The objective of thischapter is to show that the HPO can be used as a reliable substitute for a tradi-tional position feedback sensor in the control of soft piezoelectric actuators. Toshow the HPO performance in closed loop, results with HPO feedback is com-pared with the results obtained by a traditional position sensor. To minimizethe steady-state error, a simple integral controller is used in this research. Dif-ferent types of profiles including long DC signals and AC signals with variablefrequencies and lifts are part of the comparison.The advantages of the HPO feedback system are: 1) it is based on self-sensingtechnique (no position sensor is required), 2) it is voltage controlled (no sophis-ticated hardware requirement), 3) it does not require a hysteresis model, and 4)a wide frequency range from DC to 125 Hz.6.1 The Integral ControllerIntegral controllers with dedicated feedback sensors are often used in piezo-electric systems due to hysteresis and unknown force disturbances. The majoradvantage of the I?controller is it provides high gain which can overcome hys-teresis problems when operated at frequencies below the vibrational dynamics[11]. One of the pitfalls of the I?controller is a velocity dependent error at highfrequencies typical for a Type?I control system. At higher frequencies, the track-1176.1. The Integral Controllering error is quite substantial and more advanced control strategies that includea feedforward branch [15, 27, 46, 48] could be implemented for improved track-ing. However, the main objective in this chapter is to validate the performanceof the HPO in the closed loop and so no feedforward strategy is added.The simple integral control schematic is shown in Figure 6.1. The plantmodel, P (s) is obtained from the constitutive relationship in Equations 1.4 and1.6, assuming zero hysteresis voltage, UH , and neglecting small external forces,F . Since the targeted frequency range is much lower than the mechanical nat-ural frequency of the plant, the mass and damping terms can be neglected andthe following relationship is established for the plant model:P (s)=x(s)UP (s)=?ms2+bs+k??k. (6.1)The overall transfer function for this simple linear plant with an integral con-troller gain, K I is given by:x(s)xR (s)=K I ?ks+K I ?k=?Cs+?C. (6.2)The position controller gain is obtained by selecting a control bandwidth, ?C ,and using the identified model parameter, k?,K I =?C k?. (6.3)In this application, ?C is selected at 250 Hz which is at least twice the maximumtargeted frequency range. The ratio of k?can be obtained from the parameteridentification section. For simplicity, the term can also be estimated from theratio of maximum voltage to maximum stroke neglecting hysteresis at maximumdisplacement using Equation 4.3.1186.2. Experimental ResultsFigure 6.1: An integral controller schematic6.2 Experimental ResultsTo achieve sensorless position control, the HPO is used as a feedback posi-tion sensor in the piezoelectric system. To investigate the performance of theHPO in a controlled environment, different profiles have been tested: 1) stepprofile with variable lift (P1), 2) sinusoidal profiles (P2), and 3) DC profile withfast transient (P3). To gauge the accuracy of the HPO, a laser vibrometer is addedto the experiment for reference position measurements. The laser vibrometer isnot used in the control loop. It simply provides an absolute reference to com-pare the position estimates of HPO to measured position (from laser vibrome-ter). The primary objective of the experiments then is to investigate the trackingerror between the position measurement of the laser vibrometer, x?, and the po-sition estimate of the HPO, x? . This error is denoted as observer error, ?O . Theother error of interest is the total error, ?T which is calculated from the differencebetween the reference position, xR and the position measured with the refer-ence laser sensor, x?. The controller error, ?C is calculated from the differencebetween the reference position, xR , and the HPO position, x?.6.2.1 Step Profile with Variable Strokes, P1The HPO performance is evaluated for profile, P1 (in Figure 6.2(a)) which in-cludes the variable lifts (minor loops) in both ascending and descending paths.Figure 6.2(a) shows the reference signal, xR , the open loop response and the1196.2. Experimental Resultsclosed loop response. The open loop response shows a large overshoot (due tostep profile) while in the closed loop response with the HPO feedback, there isno overshoot observed. The response times are in the same order of magnitude(<5 ms) for both the open loop and the closed loop experiments. In Figure 6.2(b)the errors are plotted. The observer error in closed loop operation between thelaser vibrometer and the HPO, ?OC L is recorded at less than 0.82 ?m, both dur-ing transients and in the steady-state conditions. To compare the steady-stateerrors, steady-state segments (tss :?a???h?) are defined by Equation 6.4 where, tpis the width of the pulse and ts is the settling time of the measured signal.tss = tp ? ts . (6.4)Settling time, ts is equal to 5 time constants, 5?Cwhere, ?C is the controller band-width. The open loop total error (?TOL ) and closed loop total error (?TC L ) are alsocompared over segments ?a?? ?h? in Figure 6.2. Due to hysteresis in the openloop condition, ?TOL reaches up to 2.75 ?m. In the closed loop condition, ?TC L isless than 0.67 mum. In the segments ?a?? ?h?, maximum values of total error, ?TC Land maximum values of observer error, ?OC L are marked with circles. Due to thevariation in the experiments, 50 tests were performed and statistical inference isobtained over the segments ?a?? ?h? based on the maximum values of ?TC L , ?TOL ,and ?OC L in each test. The statistical results over the maximum errors (?TC L , ?TOL ,and ?OC L ) are shown in Table 6.1. The mean of the maximum total error duringopen loop experiments is 2.69 ?m which is 8.97% of the maximum stroke. Themean of the maximum total error in closed loop, ?TC Lmax is 0.78 ?m which is 2.6%of the maximum position and a 71% improvement over the conventional openloop control strategy. The mean of the maximum observer error, ?OC Lmax is 0.72?m which is 2.4% of the maximum lift.6.2.2 Sinusoidal Profile with Different Frequencies, P2Four types of sinusoidal profiles are used to show the tracking performanceof the control system. The first sinusoidal profile, P2a , uses a single low fre-1206.2. Experimental ResultsFigure6.2:(a)P1referenceposition,HPOposition,measuredopenloopandclosedloopposition(b)Totalerror(openloopandclosedloop)andclosedloopobservererror;maximumclosedlooptotalerrorandobservererrorineachseg-ment([88]?2013IEEE)1216.2. Experimental ResultsTable 6.1: Steady-state errors at segments ?a?? ?h? for P1 ([88] ?2013 IEEE)Test St ati st i cs Seg ment s[?m] a b c d e f g hmax(?TOLmax ) 0.89 3.08 0.89 1.75 1.73 1.62 0.76 1.85mean(?TOLmax ) 0.54 2.69 0.60 1.38 1.33 1.34 0.48 1.52?(?TOLmax ) 0.15 0.16 0.14 0.14 0.17 0.15 0.10 0.17max(?TC Lmax ) 0.83 0.99 0.83 0.82 0.89 0.66 0.95 0.63mean(?TC Lmax ) 0.45 0.78 0.58 0.34 0.40 0.36 0.63 0.31?(?TC Lmax ) 0.18 0.14 0.17 0.15 0.17 0.11 0.17 0.13max(?OC Lmax ) 0.78 0.93 0.85 0.90 0.78 0.64 0.92 0.84mean(?OC Lmax ) 0.41 0.72 0.56 0.44 0.39 0.38 0.58 0.40?(?OC Lmax ) 0.18 0.14 0.15 0.15 0.14 0.11 0.16 0.14quency at 125th of the controller bandwidth:P2a : xR = [15?15cos(20pit )]?m.The second profile, P2b , uses a single high frequency at half the controller band-width:P2b : xR = [15?15cos(250pit )]?m.The third profile, P2c , uses four separate frequencies with the highest frequencyat 15th of the controller bandwidth:P2c : xR = 17.6?4.4[cos(20pit )+ cos(30pit )+ cos(80pit )+ cos(100pit )]?m.The fourth profile, P2d , uses four separate frequencies with the highest frequencyat 12.5th of the controller bandwidth.P2d : xR = 17.6?4.4[cos(60pit )+ cos(100pit )+ cos(150pit )+ cos(200pit )])?m.Figure 6.3(a) shows the open loop and closed loop responses of P2a (at 10Hz). The errors are plotted in Figure 6.3(b). The observer error in closed loop,1226.2. Experimental ResultsTable 6.2: Error comparison for different profiles ([88] ?2013 IEEE)Profile type Error type Error [?m] % errorP2a?OC L 0.82 2.74?COL 0.69 2.3?TC L 1.46 4.86?TOL 3.51 11.72P2b?OC L 0.92 3.08?CC L 9.84 32.82?TC L 9.7 32.34?TOL 6.93 23.11P2c?OC L 0.91 3.04?COL 2.32 7.74?TC L 2.81 9.29?TOL 4.52 15.08P2d?OC L 0.89 2.95?CC L 4.42 14.74?TC L 4.74 15.81?TOL 5.86 19.541236.2. Experimental Results?OC L is less than 0.82?m which is 2.74% of the total error. There is a consid-erable (64%) improvement observed in closed loop total error, ?TC L over openloop total error, ?TC L using a simple I?controller (see Table 6.2). Since the driv-ing frequency at 10 Hz is substantially lower than the control loop frequency of250 Hz, the Type?II controller is able to substantially reduce hysteresis error andtrack the reference position closely for low frequency tracking control. When thesame controller is applied for P2b profile (125 Hz), large ?TC L is observed (Figure6.4, Table 6.2). Even though the total errors shown in Table 6.2 are in the sameorder of magnitude for both the open and the closed loop condition, the sourcesof error are different. In the open loop condition, hysteresis is the main sourceof error; in the closed loop condition the error is mainly due to the phase lag in-troduced by the Type?I control system at high frequency. This can be confirmedfrom the controller error, ?CC L in Table 6.2 for 10 Hz and 125 Hz. The controllererror, ?CC L is quite small in P2a while in case of P2b the controller error is quitehigh. However, the ?OC L is still in small ( 3%) and similar to the low frequencycase (P2a).Figures 6.5(a) and 6.6(a) show the responses in open loop and closed loopconditions for profiles P2c and P2d , respectively. The profiles are selected so thatmultiple frequency components (10 Hz to 50 Hz in P2c and 30 Hz to 100 Hz inP2d ) as well as the minor loops can be included in the tests. Similar to the fixedfrequency results, the ?TC L increases at high frequencies. However, from Table6.2, we can conclude that the HPO performance is not affected by the frequencychanges even in the presence of mixed frequency cases (rate-dependency). Theerrors for P2c and P2d are shown in Figures 6.5(b) and 6.6(b), respectively.1246.2. Experimental ResultsFigure 6.3: (a) P2a (10 Hz) reference position, HPO position, measured open loopand closed loop position (b) Total error (Open loop and closed loop) and closedloop observer error ([88] ?2013 IEEE)1256.2. Experimental ResultsFigure 6.4: (a) P2b (125 Hz) reference position, HPO position, measured openloop, and closed loop position (b) Total error (open loop and closed loop) andclosed loop observer error ([88] ?2013 IEEE)1266.2. Experimental ResultsFigure 6.5: (a) P2c ) reference position, HPO position, measured open loop, andclosed loop position (b) Total error (open loop and closed loop) and closed loopobserver error ([88] ?2013 IEEE)1276.2. Experimental ResultsFigure 6.6: (a) P2d reference position, HPO position, measured open loop, andclosed loop position (b) Total error (open loop and closed loop) and closed loopobserver error ([88] ?2013 IEEE)1286.2. Experimental ResultsFigure 6.7: (a) P3 reference position, HPO position, measured open loop, andclosed loop position (b) Total error (open loop and closed loop) and closed loopobserver error (Reprinted with permission [61]. Copyright 2012, American Insti-tute of Physics.)1296.2. Experimental ResultsFigure 6.8: Voltage profiles in open loop and closed loop condition1306.2. Experimental Results6.2.3 DC Profile with Fast Transient Section, P3This profile is selected to compare the HPO performance in the presence ofcreep with a fast transient path. The transient part of the profile is a 50 Hz sig-nal followed by a DC signal over 175 seconds. The results are shown in Figure6.7. Although the amount of creep is not large in open loop condition (<1 ?m),the improvement is visible in the error when the HPO feedback is introducedin closed loop control. In addition to that, the voltage signal is shown for bothopen loop and closed loop conditions in Figure 6.8. The voltage signal is increas-ing over time in the open loop while the controlled voltage in the closed loop isdecreasing which tries to minimize the creep voltage and hence the position.The HPO error for profile P3 is 0.53 ?m.1316.3. Summary6.3 SummaryIn this chapter, the HPO is used as a feedback sensor to achieve sensorlessfeedback control. It is shown that the HPO feedback results in a maximum ob-server error of 3% of maximum scale when compared to a laser reference sensorover a very wide frequency range (DC to 125 Hz). There is a 58% improvementin low frequency tracking operation observed compared to open loop operationwith the HPO feedback. The observer error does not increase with the frequency.However, the overall tracking error is increased due to the controller error (Ta-ble 6.2). The HPO performance compares well with other sensorless positioncontrol algorithms shown in the literature (in Section 1.9). Different profiles areselected to include the creep behavior, rate dependency and minor loops. How-ever, since a Type?I control system was used, tracking error at frequencies ap-proaching the bandwidth of the control loop are high. A feedforward controllercould be added to the control system shown here in order improve tracking per-formance at higher frequencies. However, this would distract from the HPO per-formance and so it was opted not to include a feedforward term in the controllerimplementation shown here.132Chapter 7Self-heat GenerationA large number of piezoelectric actuators are made of soft PZT materialswhich provide high piezoelectric coefficient, d . However, soft PZT material pos-sesses a high dielectric loss factor, tan? [20]. For non-resonant applications suchas in high speed positioning stages or high speed inkjet printing, it is reportedthat the total loss encountered is primarily due to dielectric loss [86]. In contrary,for resonant applications such as in ultrasonic cleaning, the total loss is relatedto the mechanical or elastic losses in the actuator [86]. Irrespective of the type ofthe application (resonant or non-resonant), during continuous operation, thetotal loss, due to dielectric loss or elastic loss, contributes to the temperaturerise of the actuator [86]. This phenomenon is known as self-heat generation.The self-heating phenomenon for PEAs was first addressed in [21] where it isshown that under continuous high frequency and high voltage loading, signifi-cant heat generation is observed. In Chapter 5, it has been shown that the HPOis able to predict position over a frequency range of 0 Hz to 100 Hz. However,self-heat generation occurs already at 50 Hz operation if the actuator is contin-uously driven over a long period of time [85]. The temperature increase due toself-heat generation affects the capacitance of the piezoelectric actuator. Thisindicates that during continuous operation even at 50 Hz, the performance ofthe HPO may deteriorate since it uses capacitance based PSS. Hence, it is neces-sary to predict the temperature rise of the piezoelectric actuator due to self-heatgeneration. Also, the estimated temperature may be used to ensure that the ac-tuator is not operated beyond the Curie temperature (150?), where it may losepiezoelectric properties.In [21, 85], it is shown that the temperature rise due to self-heating in non-resonant applications is a function of driving frequency, f, electric field, E andeffective volume to surface area ratio, veA . A model to predict the self-heating133Chapter 7. Self-heat Generationtemperature is shown in Equation (7.1):T ?T0 =u f vekT A(1?e?kT Av?c t), (7.1)where, u is the loss of the sample per driving cycle per unit volume, kT is theoverall heat transfer coefficient, ? is the density of the PZT material, v is the ac-tuator volume and t is the time [21]. The loss term u is obtained from the P-Ehysteresis loop using a Sawyer-Tower circuit, whereas P is the polarization andE is the electric field. It is reported that the parameter, u is dependent on tem-perature variation. The term kT accounts for the convection and radiation heattransfer. It is reported that kT is constant for low electric fields and varies whenexposed to high electric fields. In addition, the term kT , slightly varies with tem-perature. The prescribed range of kT is 20?40W m?2 ?K?1. To summarize, theprediction of actuator temperature by using [21, 85] requires the knowledge ofdriving frequency, as well as parameters u and kT that themselves are dependenton temperature.The literatures found to date describe how to model the temperature in-crease due to self-heat generation where a hysteresis loss component and fre-quency is known. The limitations are: 1) the hysteresis loss component itself isa temperature dependent parameter, 2) the models require the exact knowledgeof frequency which limits the model to predict the temperature for mixed fre-quency operation, and 3) the term kT is a function of electric field and drivingfrequency. In this chapter, self-heat generation is modeled in piezoelectric ac-tuators using the constitutive relationship of first law of thermodynamics andpower loss due to hysteresis voltage in the actuator. The model does not requireany known frequency and hence should be independent of the type of the pro-file and knowledge of frequency. However, since the constitutive equations areused in the model, position signal is required as an input to the model. It shouldbe noted that self-heating of non-resonant operations of the PEAs is consideredonly, since the actuators usually operate at frequencies lower than the first me-chanical resonant frequency. This leads to the assumption that the dielectriclosses are the main contributor to the total losses which result in self-heat gen-eration.1347.1. Proposed ModelThis chapter is organized as follows. In Section 7.1, a self-heat generationmodel that requires voltage, current, as well as position is presented to esti-mate the temperature. Experiments to characterize the temperature behaviorare shown in Section 7.2. The parameter identification of the model is presentedin Section 7.3. Experimental results, showing temperature estimates using adedicated position sensor input are shown in Section 7.4. To obtain a sensorless(eliminating position sensor) temperature prediction, a temperature compen-sated HPO and temperature model is shown in Section 7.5. Finally, the combi-nation of temperature compensated HPO and the new temperature model arevalidated in Section Proposed ModelThe self-heat generation model proposed in this chapter is based on the lawof energy conservation of a closed system presented in [21]:q?G ? q?D = ?cvdTd t(7.2)where, q?G is the rate of heat generation, q?D is the rate of heat dissipation, ?, c andv are the density, specific heat and total volume of the actuator, respectively. It isassumed that the system is in perfect insulation. For non-resonant applications,it is reported that the dielectric loss is equivalent to the hysteresis loss and hence,is the major contributor of heat generation [20, 21, 85, 86]. Therefore, Equation(7.2) can be rewritten as follows:PLoss ?kT A4T =UH I ?kT A4T = ?cvdTd t, (7.3)where, PLoss is the power loss due to hysteresis loss, kT is the overall heat trans-fer coefficient, ?T is the temperature difference between the actuator and theenvironment due to self-heating and A is the surface area of the actuator. Theoverall heat transfer coefficient considers the heat conduction, convection andradiation [21]. The term UH is the hysteresis voltage in [13] and I is the current.1357.2. Experimental ApproachThe transfer function between PLoss and ?T is shown in Equation 7.4:?T =PLoss ??s ??+1(7.4)where,?=?vckT A, (7.5)?=1kT A, (7.6)PLoss =UH I . (7.7)7.2 Experimental ApproachThe experimental setup discussed in Chapter 2 is used for the self-heatingexperiments. The actuator surface temperature is measured using a non-contactinfrared sensor (Micro-epsilon thermometer CT-CF15) to avoid friction betweenthe sensor and the actuator. For the current measurement, a high resolutioncurrent probe (TEKTRONIX-TCP312) is used. A high frequency laser vibrometer(POLYTEC-HSV2002) is employed to measure the position of the actuator. Thetest-bed is shown in Figure 2.3.Figure 7.1 shows the time responses of the actuator temperature when stepchanges in frequency are applied. Operating at less than 20 Hz results in neg-ligible self-heating while beyond 200 Hz, resonant vibrations start to becomenoticeable, since the first natural frequency of the test-bed is 2500 Hz. To ob-tain maximum temperature rise, a sinusoidal unipolar voltage of 100 V, whichis equivalent to 1.8 kV mm?1, is applied to the actuator. The actuators are op-erated for more than 400 seconds to reach the steady-state temperature. Figure7.1 shows that the time constant is independent of the applied frequency, butthe steady-state value of the final temperature increases with frequency. Figure7.2 shows that there is a linear relationship between the steady-state tempera-ture and the driving frequency. This is consistent with [20, 21, 83]. In agreementwith [21, 83], it is also observed in Figure 7.3 that the change in temperature1367.2. Experimental ApproachFigure 7.1: Effect of frequencies on self-heat generation. It is important to notethat the frequencies are high enough to result in self-heat generation. However,the frequencies are still well below the resonant frequency (see Figure 1.4) of thetest-bed and hence a quasi-static approximation is still valid.1377.3. Parameter IdentificationFigure 7.2: Change in steady-state temperature increase with driving frequency(at 100 V sinusoidal signal)is related to the square of the applied voltage. The temperature increase is notsubstantial when the driving voltage is less than 50 V at 150 Hz.7.3 Parameter IdentificationThe model presented in Section 7.1 requires two parameters to be identified:? and ?. The time constant, ? is 100 seconds for the frequency ranges from 20 Hzto 150 Hz obtained from Figure 7.1. The value of overall heat transfer coefficient,kT can be attained from Equation 7.5. Using the values of total surface area, Aand kT , the second model parameter, ?, is obtained from Equation 7.6. Theterm UH , in Equation 7.7 can be obtained from the piezoelectric constitutiveequations proposed by Goldfarb et al. in [13]. The constitutive force equation1387.3. Parameter IdentificationFigure 7.3: Change in temperature increase with driving voltage (at 150 Hz)1397.3. Parameter Identificationat quasi-static region and the expression for hysteresis voltage is recalled fromEquations 4.3 and 4.8 as follows:kx =?UP (7.8)UH =U ?UP (7.9)where, k is the stack stiffness, ? is the force-voltage proportionality constant,and UP is the linear piezoelectric voltage. Using Equations (7.7), (7.8) and (7.9)the following function can be obtained for power loss due to hysteresis:PLoss = (U ?k?x)I (7.10)where, x is the measured position of the actuator. At maximum supply volt-age, Umax , and maximum displacement, xmax , the term k?can be attained fromEquation (7.11):k?=Umaxxmax. (7.11)It is important to note that although self-heat generation occurs at high fre-quency operations, the range of frequency is still below than the dynamic rangementioned in Figure 1.2. Hence, the quasi-static approximation is still valid forthe frequency range where self-heating phenomenon occurs. The final modelfor self-heating is shown in Equation 7.12 where TRe f is the reference tempera-ture. The flowchart is shown in Figure 7.4.TSH = TRe f +(U ? k?x)I ??? ? s+1(7.12)The properties of the PZT material are summarized in Table 2.1 and the self-heating model parameters are shown in Table 7.1. The parameters are obtainedby fitting the experimental results shown in Figure 7.1 and Equations 7.5 and 7.6.The ratio of k?is considered unaffected since the maximum stroke is unaffecteddue to temperature variation in the setup. The value of the overall heat transfercoefficient, kT compares well with the prescribed values in [21].1407.4. Temperature Prediction Using a Dedicated Position SensorFigure 7.4: Self-heat generation model with position sensor inputTable 7.1: Self-heating model parametersParameters ValuesTime constant, ? 100 sec.Overall heat transfer coefficient, kT 28 W ?m?2 ?K?1? 57 W ?K?1Stiffness to coupling coeff. ratio, k?2.85 e6 V ?m?1Ambient temperature, TRe f 23?C7.4 Temperature Prediction Using a Dedicated PositionSensorOnce the model parameters are identified, real-time measurements of cur-rent, voltage and position provide an accurate measurement of temperature riseof the actuator due to self-heating. The experimental results and the predictedtemperatures (from the model) for various driving frequencies are plotted in Fig-ure 7.5. Both the temperature rise and fall can be predicted using the proposedself-heating model. For recording the temperature fall, the actuator is drivenat 150 Hz-50 V, 10 Hz-100 V and 0 Hz-0 V. In another experiment, the actuator isdriven by a combination of voltages and frequencies to observe the performanceof the model under complex operating conditions. The results are shown in Fig-ure 7.6. The prediction error is within?2? which is less than 3% of the maximumtemperature rise due to self-heating at 150 Hz-100 V.1417.4. Temperature Prediction Using a Dedicated Position SensorFigure 7.5: (a) Self-heat generation at different frequencies: measured and pre-dicted temperature (b) prediction error1427.4. Temperature Prediction Using a Dedicated Position SensorFigure 7.6: (a) Temperature profile using self-heating by driving at different fre-quencies and voltages: measured and predicted temperature (b) Prediction er-ror1437.5. Temperature Compensation of HPO7.5 Temperature Compensation of HPOIn Section 7.3, a self-heat generation model is proposed which predicts thetemperature increase due to self-heating based on driving current, driving volt-age and a position measurement. This limits the model to a system where a ded-icated position measurement is available. In this section, the HPO presented inChapter 5 will be employed as a position input to the self-heat generation modelto obtain a pure self-sensing temperature and position estimation.Before employing the HPO signal as an input to the self-heating model, theeffect of temperature on the capacitance based PSS needs to be evaluated. Fig-ure 7.7 (a), shows the effect of temperature variation on the capacitance-positionrelationship. The capacitance increases with temperature increase while the po-sition is unaffected at least for the investigated temperature range (upto 60?C).Figure 7.7(b) shows the corresponding capacitance-time relationships. It is ob-served that temperature creates an offset in the capacitance measurements. Thesize of the offset is linearly related to temperature. This offset in the capacitance-position relationship is expected to affect the HPO estimation. Since the self-heating phenomenon is dependent on frequency of operation, the effect of fre-quency on the capacitance-position relationship is also shown in Figure 7.8.Only small variation in the charge-position relationship is observed due to fre-quency variation.From Figure 7.7, it is evident that a new fitting function is required to de-fine the capacitance-position relationship with temperature variation to obtaina reasonable estimate of position from the capacitance based PSS. The new fit-ting function, ?T is presented in Equation 7.13 which requires a temperatureinput to predict the position from capacitance measurement in the presence oftemperature variation:?T = x?CP (CP ,T )= bT 0+N?n=1bT nC? nP + g T? , (7.13)where, bT 0 is the intercept, bT n is the expression for coefficients for capacitanceand g is the temperature coefficient. The coefficients are obtained through a re-1447.5. Temperature Compensation of HPOFigure 7.7: Effect of temperature on (a) capacitance-position relationship (b)capacitance measurement at 50HzFigure 7.8: Effect of frequency on capacitance-position relationship1457.5. Temperature Compensation of HPOFigure 7.9: (a) Self-heat generation model with HPO input (b) temperature com-pensated HPO1467.6. Temperature and Position Predictions Using the HPO and the Self-Heating Modelgression analysis. The orders are selected by comparing the adjusted R2 valuewhich indicates that an order more than 5 would not improve the adjusted R2value (Table 7.2). Also, an interaction between capacitance and temperaturedoes not have any effect on adjusted R2 value. In addition to that, the p-valuefor the interaction is quite high (0.33) which indicates that the coefficient forinteraction is redundant. The model is obtained based on 50 Hz data. The com-parison is shown in Table 7.2.Table 7.2: Comparison of the regression statisticsn=4 n=5 n=6 n=5 with CPand T interactionRegression StatisticsR Square 0.998189 0.998194 0.998194 0.998193Adjusted R Square 0.998188 0.998193 0.998193 0.998193Standard Error 0.474422 0.473808 0.47377 0.473825An updated HPO structure is shown in Figure 7.8 which includes tempera-ture compensation using the self-heat generation model (from Equation 7.13).The compensated HPO output is fed back to the self-heat generation model.This feedback mechanism between the HPO and the temperature model worksquite well, because the temperature variation is relatively slow in comparison tothe piezoelectric position changes.7.6 Temperature and Position Predictions Using the HPOand the Self-Heating ModelA sensorless temperature prediction due to self-heating was presented inSection 7.5 where the temperature compensated HPO is employed as an input tothe self-heating model to estimate the position of the actuator. In this section,experiments are carried out to validate temperature and position predictionsusing the HPO as the position input to the temperature model. To distinguishbetween the two position inputs to the temperature model, TSH (PS) is used todefine the self-heating model using the laser position input while TSH (HPO) is1477.6. Temperature and Position Predictions Using the HPO and the Self-Heating ModelTable 7.3: ANOVA table for capacitance based PSS regression model with tem-peratureANOVA for n=5df SS MS F Significance FRegression 6 992110.4 165351.7 1096054 0Residual 7993 1205.832 0.150861Total 7999 993316.2Coefficients t Stat p-valuebT 0 -870.776 -7.50918 0bT 1 1639.54 7.873573 0bT 2 -1132.54 -7.61792 0bT 3 376.6851 7.140752 0bT 4 -61.8409 -6.64742 0bT 5 4.036799 6.187723 0g 0.263115 860.7364 0used to denote the self-heating model that uses temperature compensated HPOposition input.In Figures 7.10 and 7.11, 100 Hz and 50 Hz self-heating temperature pre-dictions are shown for both TSH (PS) and TSH (HPO). The measurements arecompared with the infrared sensor and found to agree within less than 2?. InFigure 7.12, a mixed frequency case is tested where temperature cycling is ob-tained with a combination of frequencies to heat up and cool down the actuator.Heating up is achieved through a series of sections (?a???c?) with increasing fre-quencies (20 Hz?50 Hz?100 Hz). This sequence of driving frequencies leads toa final temperature larger than 50?C. To cool the actuators back down, drivingfrequencies of 50 Hz and 0 Hz (?d? and ?e?) are used. The HPO position com-pares well with the laser sensor position in Figure 7.13 (a). The laser vibrometerhas some drift in long term measurements which is visible in sections (?a? and?d?) in the profile. Although the contact between the needle and the PEA is verysmall (hence most of the heat is assumed to be transferred through convectionand conduction is neglected), combined effect of expansion of the needle andcontraction of the piezoelectric actuator due to temperature increase may con-1487.6. Temperature and Position Predictions Using the HPO and the Self-Heating ModelFigure 7.10: (a) Temperature prediction by self-heat generation model (b) errorsin prediction at 100 HzFigure 7.11: (a) Temperature prediction by self-heat generation model (b) errorsin prediction at 50 Hz1497.6. Temperature and Position Predictions Using the HPO and the Self-Heating Modeltribute to this drift. However, it is interesting to see that at the maximum temper-ature region (section ?c?), the laser measurement does not show any drift whilein section ?d? (temperature decreasing), a maximum drift of 0.6 ?m is observed.Further, careful investigation including the temperature distribution in the nee-dle assembly and actuator is required to conclude on thermal expansion of thePEA and the needle assembly; however, this is not included in the scope of thisstudy.The HPO position also shows some changes at the beginning of frequencyswitches (beginning of different sections). This is attributed to the change in theunmodeled capacitance frequency dependency shown in Figure 7.12. The HPOoutput is used in the self-heating model to predict the temperature and whichin turn is feed back to the HPO to compensate the temperature. This circularfeedback is possible due to the slow temperature variation in comparison to theposition change of the actuator. Considering the presence of laser drift and HPOerror due to frequency changes, the maximum observer error is recorded at 1.38?m (4.18% of the maximum stroke) in Figure 7.12(c) over the frequency range of0 Hz to 100 Hz and a temperature change of up to 55?C. To the best knowledge ofthe author, this is the first time in scientific literature that a successful positionestimation algorithm has been presented for a wide range of frequencies andoperating temperatures. The HPO output is used in the self-heating model topredict the temperature and which in turn is feed back to the HPO to compen-sate the temperature. This circular feedback is possible due to the slow temper-ature variation in comparison to the position change of the actuator. The errorsin temperature prediction from the two different position sources (position sen-sor and HPO) are compared in Figure 7.12 (d). In some portions, TSH (PS), whichuses the laser position measurement, provides slightly better temperature pre-dictions than the HPO based values of TSH (HPO). However, the maximum erroris recorded 1.45?C in both the cases.1507.6. Temperature and Position Predictions Using the HPO and the Self-Heating ModelFigure 7.12: (a) Position measurements (HPO and sensor), (b) Self-heating atmixed frequency cases, (c) error in HPO, and (d) errors in temperature predic-tion1517.7. Summary7.7 SummaryA novel sensorless temperature measurement is presented in this chapterwhich is based on power loss due to self-heating and the piezoelectric consti-tutive relationship. The advantage of the proposed model is that it does not re-quire the knowledge of the driving frequency in contrast to the earlier developedmodels. Moreover, the new method is much simpler than the models in [21, 85],which require the term energy loss per cycle per volume, u and the overall heattransfer coefficient, kT which are functions of the electric field and temperature.The proposed model is tested in fixed and mixed frequency cases and comparedwith a traditional temperature sensor. The temperature prediction is accuratewith in ?2?C which is close to 3% of the investigated range. When the HPOis updated to include temperature dependencies and combined with the newmodel, sensorless temperature and position estimation is obtained in the pres-ence of self-heating. This provides a maximum position error of 4.18% over thefrequency range of 0 Hz to 100 Hz and up to 55?C of temperature increase dueto self-heat generation.152Chapter 8Conclusions and Future ResearchThe foremost objective of this dissertation is to develop a PSS technique forposition feedback control in PEA over a wide frequency range where creep andrate-dependent hysteresis lead to large errors in traditional feedforward posi-tioning systems. In addition to that, a self-heat generation model is developedand temperature compensation of the position self-sensing scheme is proposedin this research. The research outcome can be extended to different applicationssome of which are outlined in the future research section.8.1 ConclusionsThe research findings are based on a novel real-time piezoelectric impedancemeasurement algorithm which facilitates the parameter identification of the elec-tromechanical models. The measurement technique superimposes a high fre-quency voltage ripple onto the normal driving voltage and extracts impedanceof PEA from the relationship between voltage and current ripples. Since themeasurement frequency is very high in comparison to the natural frequency ofthe test bed, the measurement is obtained in a clamped condition. This leads toa relationship between capacitance and position where the hysteresis is negli-gible (<3%). The capacitance value decreases with the actuator displacement.This finding leads to two major conclusions: 1) the capacitance in the elec-tromechanical model is not constant, and 2) the capacitance-position relation-ship provides a means for position self-sensing through a regression model. Thefirst conclusion indicates that an improved electromechanical model is requiredto include the position dependent capacitance. The proposed modeled is veri-fied experimentally and compared with the traditional model. Based on the up-dated electromechanical model, a novel parameter identification technique is1538.1. Conclusionsproposed where a single set of experiments provide all the required parametersin the model. In addition to that, some important material parameters such aspiezoelectric strain coefficient, coupling coefficient, and permittivity constantare also obtained through constitutive relationships. These values compare wellwith the manufacturers specified values.The second conclusion of the capacitance-position relationship (throughregression model) promotes a self-sensing technique applicable to slow oper-ations where creep and hysteresis are present. By combining this capacitancebased PSS technique with conventional charge based PSS, using a novel hybridposition observer, position is estimated over an extended frequency range wherecreep and rate dependent hysteresis are prominent. This is achieved by exploit-ing the advantages of each technique where the hysteresis free charge basedPSS provides the position estimate at high frequencies (lower than the dynamicrange) while the capacitance based PSS (?) is employed at slow operations. Themajor problem with the charge based PSS is the drift in position estimation dueto offset currents in the charge measurement. The capacitance based measure-ment then serves two purposes: 1) to provide a PSS signal at low frequency op-eration where charge is difficult to measure, 2) to eliminate offset currents in thecharge based PSS. The performance of the position observer is compared with atraditional position sensor. The comparison shows that position observer is ableto predict the position from static operations up to 100 Hz with an error close to3% of full scale.The developed HPO is then tested in a controlled environment with differentwaveforms from variable lift step profiles to mixed frequency sinusoidal pro-files. The creep profile is also tested in conjunction with a fast transient sec-tion. A simple Integral-controller is implemented to observe the performance.Similar to the open loop conditions, the HPO is compared to a reference sen-sor and the error is recorded at around 3% of full scale. Even though the HPOerror is fairly constant at all frequencies, the overall controller error increasesat higher frequencies, since a simple Type-I controller was used in the experi-ments. In the future, other control schemes like Type-II controllers or feedfor-ward/feedback controllers presented in the literature may be implemented toreduce the controller error at higher frequencies. Self-heating phenomena are1548.1. Conclusionsobserved in piezoelectric actuators when operated at relatively high frequenciesover an extended period of time. The frequency range, over which the HPO isproposed, is high enough to generate heat due to self-heating when continu-ously operated. The temperature of the actuator may increase up to 60?C at 100Hz driving frequency. The temperature rise may have several effects: 1) the lifeof the actuator is reduced if continued over an extended period of time, 2) thestroke of the actuator may be affected, and 3) the position self-sensing schememay be affected. A self-heat generation model is proposed based on the powerloss of the piezoelectric actuator that relies on the electromechanical model ofthe actuator. The advantages of the model are: 1) unlike the other self-heatingmodels, it does not require a known frequency of operation which makes themodel applicable for different profiles, and 2) it is based on the power loss of theactuator over time due to self-heating and not dependent on other temperaturedependent parameters such as overall heat transfer coefficient and hysteresisloss per unit volume. Since, self-heating is a function of voltage, frequency andlength of actuation, the experiments to validate this model contain multiple fre-quencies and voltage magnitudes. Since the temperature model requires drivingvoltage, driving current, and measured position as its inputs, the model is testedboth with a laser position measurement, and a temperature compensated HPOas the position input. The temperature prediction error is less than 2?C for bothposition sensor input and HPO input to the self-heat generation model whichis close to 3% of the maximum investigated scale. The errors position estimatesobtained with the combined HPO and temperature model is less than 5%.The major goals in this research are met with experimental validations. Thefollowing contributions are significant in this research:o A novel impedance measurement algorithm is proposed which providesreal-time clamped capacitance and resistance measurement for piezoelec-tric actuators.o An improved electromechanical model and parameter identification is pro-posed based on the real-time capacitance measurement.o A novel hybrid position observer is presented which relies upon two self-sensing techniques to provide improved position information over an ex-1558.2. Future Researchtended frequency range where creep and rate-dependent hysteresis arepresent. The HPO is helpful for sensorless position control in piezoelec-tric actuators.o A self-heat generation model is presented which predicts the temperaturerise in the piezoelectric actuator due to self-heat generation.o A novel sensorless temperature and position estimation technique is pre-sented based on temperature compensated HPO to predict self-heatingtemperature variation.8.2 Future ResearchSome of the future research possibilities are outlined as follows:o The real-time impedance measurement presented in this thesis providescapacitance and resistance measurements for piezoelectric actuators. Theimpedance measurement is obtained at constant loading condition whereforce changes are negligible. In future research, the effect of preloading orexternal load variation should be investigated.o During the work on this thesis it is observed, that capacitance and resis-tance values of new actuators stabilize over the first several thousand cy-cles of use. Even though no measurable changes are observed after thisinitial break in period during the work on this thesis, it is expected thatover the life of the actuator changes will take place. This may result dueto failing of the piezoelectric layers, or cracks developing in the actuator.Hence, by comparing the capacitance/resistance measurement, a healthmonitoring strategy can be developed for the piezoelectric actuator.o The HPO is tested in closed loop condition with a Type-I controller wherethe reference tracking performance is poor for high frequency profiles. Ithas been shown in the literature that a feedforward branch is often usedto improve the dynamic performance of the feedback controller. How-ever, most of the control systems are based on a dedicated position sen-sor. The proposed HPO can be used as a replacement of the traditional1568.2. Future Researchsensor where a simple hysteresis model can be included in the feedfor-ward branch of the controller to improve the tracking performance.o When the HPO is updated for the self-heat generation cases, a good esti-mate for piezoelectric actuator positioning is obtained. In further investi-gations, the HPO should be tested over a wide temperature range speciallyfrom subzero condition to high temperature range up to 100?C. This studywill be helpful to implement the HPO in applications such as piezoelectricfuel injectors.o Piezoelectric paint sensors (PPS) [105, 106] are a relatively new technol-ogy to obtain structural health monitoring for large infrastructures suchas bridges, buildings, etc. The main advantage of the paint sensor is that itcan be used on different surfaces where traditional strain sensors cannotbe mounted easily. 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