Vibration Control of Piezoelectric Actuator by Implementation of Optical Positioning Sensor

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1 International Journal of Optomechatronics ISSN: (Print) (Online) Journal homepage: Vibration Control of Piezoelectric Actuator by Implementation of Optical Positioning Sensor Zhaoli Hu & Gary Pierre Maul To cite this article: Zhaoli Hu & Gary Pierre Maul (2007) Vibration Control of Piezoelectric Actuator by Implementation of Optical Positioning Sensor, International Journal of Optomechatronics, 1:4, , DOI: / To link to this article: Published online: 05 Dec Submit your article to this journal Article views: 93 View related articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 03 January 2018, At: 06:16

2 International Journal of Optomechatronics, 1: , 2007 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR BY IMPLEMENTATION OF OPTICAL POSITIONING SENSOR Zhaoli Hu and Gary Pierre Maul Department of Industrial & Systems Engineering, Ohio State University, Columbus, Ohio, USA 1. INTRODUCTION PZT bimorph actuator is used to design mini-sized vibratory feeder applied in the micromedical industry. Since the vibration trajectory of PZT bimorph actuator will be controlled to obtain desired parts feeding, optical positioning sensor is used to provide non-contact measurement with precise displacement. Because the torsional vibration causes the rotary motion of the actuator, the angle between the optical sensor and the PZT plate is subjected to change during the vibration procedure. Optical loss due to a varying reflection angle is compensated by multiple-segment micrometer calibration. Satisfactory hysteresis suppression and good waveform tracking is obtained with implementation of optical sensor in piezo-actuating system. Application of smart material in the sensing and actuating field results in significant progress in micro-sized motion sensing and control. Among a wide variety of smart material devices, piezoelectric actuators are widely used for their high precision and good controllability. A consistently discussed problem with piezoelectric material is the reduced precision and repeatability due to hysteresis effect (Duparre et al. 2000; Mayergoyz 1996, 2003; Ge and Jouaneh 1995). To reduce piezoelectric actuator tracking errors and apply it in a high-precision required field such as micromotion control, previous researchers have considered using a highspeed servo system to obtain the desired positioning capability (Choi et al. 1997; Croft et al. 2000; Kim and Kim 2001; Thornhill et al. 2003; Lin et al. 2006; Sun et al. 2004; Kim 2003). However, these approaches were based on the fact that the motion frequency was relatively low and positioning error was the primary consideration. Therefore, the control algorithm is focused on stepped point tracking. In our research we worked on how to apply a piezo actuator in high frequency operation while maintaining required precision. It is well known that the hysteresis effect based on different models is well predicted in low frequency range; therefore, the piezoelectric actuator can provide precise and repeatable positioning if an effective model is established. However, for high frequency or variable frequency operation the system dynamics will make it harder to find an accurate model. Under this condition, using a piezo actuator to obtain desired operation becomes an Address correspondence to Dr. Zhaoli Hu, R & D Engineer, Manufacturing Automation Lab, Ohio State University, N. Columbine Drive, Dunlap, IL 61525, USA

3 370 Z. HU AND G. P. MAUL NOMENCLATURE d e F G I k i u actuator displacement output error external force dynamic transfer function inertial of rotation adaptive model parameters waveform generator input y dynamic system output a, b tangential lines of the actuator bending surface c tuning factor of adaptive controller h bimorph actuator bending angle g damping factor of actuator unresolved problem. If we can find a methodology to obtain reliable high frequency motion, then a mini-sized vibration actuator can be used as a key part to replace traditional actuators to achieve controllable and precise production. For instance, in micro-medical industries tiny parts need to be moved between different assembly sites automatically while the orientation and shape selection need to be done in the middle of the movement. A traditional industrial tool, a vibratory feeder, which delivers parts by providing a hopping style motion, is normally used to obtain such kinds of operation. An electromagnetic actuator is used in the current industry application for open loop actuation; therefore the feeding speed is uncontrollable and parts jam in front of different working sites. If the piezoelectric actuator can be used to decouple the vibration, we can obtain a more controllable moving speed while solving other problems in traditional feeding, such as hopping and backsliding (Hu and Maul 2003). To study the possibility of controlling a large displacement and high frequency piezo actuated vibration, a PZT-5H bimorph actuator was used to drive a mini-sized vibratory track operating at about 40 Hz while the amplitude of vibration is more than a couple of hundred microns. The bimorph design of the actuator provides required critical displacement for parts feeding. But on the other hand, it suffers from hysteresis, narrow bandwidth, and under-damped frequency response. Therefore, to apply a PZT actuator in vibratory feeding we must overcome undesired higher harmonics and dynamic hysteresis. A distinguished part of this application in comparison to other piezo actuation is that we are seeking controlled parts moving, e.g., the force provided by the piezo actuator instead of position and trajectory. A novel square-pulse-shaped driving force sequence has been designed to provide a digital controlled feeding speed (Hu et al. 2007). Using reverse engineering, we can find out that the desired driving waveform for effective parts feeding is composed of two quadratic portions with different curvature (Figure 1). As long as the actuator follows this waveform the resulting mechanical force will be (in the ideal case) a desired square-pulsed force sequence and the feeding speed of the parts can be analytically controlled (Hu and Maul 2003; Hu et al. 2007). Because the second order derivative may arouse a significant noisy signal, there is a great need to maintain the smoothness of the responding waveform. The reason for this is that a smooth but deformed waveform may cause a skewed force sequence, while a bumpy but accurate waveform may cause a scattered force sequence. The skewed force sequence can feed the part with some sacrificed speed, but a scattered force sequence willnotbeabletofeedthepartatall(huetal.2006). To solve the problem, we proposed a control methodology called indirect waveform generation and tracking (IWGT), which compares the responding waveform

4 VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR 371 Figure 1. Schematic trajectory of the quadratic driving waveform. When actuator is forced to follow the waveform, a square pulsed mechanical force will be generated to feed the part on the track. with the desired waveform shape by their frequency component error; it then uses adaptive control to change the generation function of the driving waveform for the next cycle. Because we change the waveform as a whole, the responding waveform is smooth and the second order derivative is maintained in as a pulsed sequence. This control design puts a high requirement on the dynamic precision of the vibration displacement measurement because thousands of data points need to be sampled per second, and the error between the responding waveform and the reference waveform will be used in an adaptive compensator which will drive the final vibration trajectory to the desired scheme. This dynamic non-contact waveform sampling gives us a chance to control high frequency motion of piezoelectric actuator, which were mainly applied for static positioning. Because the operating frequency is close to the resonant frequency of the actuator, higher harmonics of the driving waveform may introduce significant distortion in the responding wave if it is not well compensated (Hu and Maul 2003). The optical sensor signal is used in designing a waveform-based compensator and filter to overcome the limitation on the bandwidth and influence of hysteresis effect; this high speed and precise sensing allows us to obtain good results in consequent control. Section 2 focuses on the sensing process and control algorithm design and considerations. Section 3 shows the process to implement the sensor and control into a vibratory feeding system to obtain results of part-feeding speed. The result shows that the whole controlling process provides projected results. 2. DESIGN OF SENSING AND CONTROL 2.1. Analysis of Position Sensing Process The optical positioning sensor used in this research has a sub-micron static resolution on the far side of its measurement range. The output of the sensor is then connected to the DAQ Board with 12-bit resolution. So by properly adjusting

5 372 Z. HU AND G. P. MAUL Figure 2. The far side of the optical sensor has about 500 um linear measurement range with sensitivity of 2.6 mv=um. If a 12-bit DAQ board is used the resolution of displacement is about 0.9 um. the input voltage range, we can obtain reasonable resolution on position for control purpose. But the dynamic precision is influenced by two factors: the nonlinear sensitivity of sensor due to measurement range variation and nonlinearity due to varying angle of the reflecting surface. The first factor is intrinsic due to the design of the optical positioner (Culshaw 1989; Udd 1991), where there is a near side range and far side range with different sensitivity (Figure 2). Therefore, a feasible solution is to set the neutral position of the actuator in the center of the near side when the resolution requirement is high and displacement is small; or set the neutral position on the far side when displacement is large. In this application, the dynamic vibration amplitude is much larger than the range of the near side (< 150um) while the requirement on displacement precision is not strictly required. This leads us to set the neutral point at about 450 um from sensor tip which gives us enough linear range on the far side for vibration measurement. The second factor is caused by the rotational motion of the actuator. As shown in Figure 3, a platinum film is stuck to the free end of the actuator as the reflecting Figure 3. The PZT actuator is clamped on one end and the free end will vibrate according to the applied driving voltage. A platinum film is on the free end of the actuator to improve the reflectance. This helps to improve the S=N ratio. At same time, we noticed that the reflecting angle of the platinum film with respect to the optical positioner is varying at different periods of the vibration.

6 VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR 373 Figure 4. a) Comparison of the rigid plate vibration (solid line) and cantilever plate vibration (dotted line); and b) model of the cantilever plate into a rigid plate through the tangential lines a and b. surface to improve reflection. It is obvious that the reflection angle is varying at different periods of the vibration. So there is a need to establish an empirical function to compensate for the optical loss due to the reflection angle variation, which will then return a direct relation between actuator displacement and sensor voltage. Because the piezo actuator plate behaves like a cantilever plate under the external forces, it is not appropriate to use a torsional vibration centered at clamp end O to describe the dynamic of the actuating system. However, due to the fact that the amplitude of the vibration is small we can reduce the non-linear vibration of the cantilever plate to a linear torsional vibration centered at O through the tangential line of the non-linear vibration (see Figure 4). The dynamic equation describing the system is approximated by the small angle rigid plate rotational equation. I h þ g _ h þ kh ¼ F Although the torsional vibration can be viewed as composed of horizontal and vertical motion (with respect to the clamping surface), the vibratory feeder is designed in such a way that only the motion along the direction of the optical sensor (vertical) will contribute for feeding. Therefore, the goal of compensation is to find out the direct relationship between the sensor voltage and vertical distance from optical sensor probe to the reflecting surface. For better reference, the sensor voltage corresponding to the turn point between the near side and the far side (Figure 2) is used as the reference of origin. The reason to use this reference for calibration is because it is stable and easy to be determined each time before the experiment. The compensation process can then be divided into two steps. The first step studies the response of the optical sensor to flat surface motion. The probe of the optical positioner will be mounted on a micrometer based screw rod, which will move the probe toward or away from the platinum film (Figure 5a). To obtain the optimal resolution under restricting noise, the whole distance range of 1500 um is divided into multiple segments. The result shows that with consideration of the S=N ratio at these measured positions, the finest voltage increment we can obtain is around 3 mv. Therefore, the best resolution of displacement measurement is about 1.15 um. The linear relationship between sensed voltage and mechanical ð1þ

7 374 Z. HU AND G. P. MAUL Figure 5. The experiment s design for calibration of the distance measurement with reflection angle variation. a) a platinum film stuck to the rigid plate from reflection and the optical positioner mounted on the micrometer-based screw rod that will move the positioner closer or away from the reflection surface (platinum film). During the movement, the reflection angle remains the same (90 ). b) the positioner is fixed and the rigid plate is moved by a screw rod, therefore the reflection angle is varying when the plate rotate with respect to the pivot. measured displacement can be interpolated and described as V ¼ 0:002d þ 5:997 Here v is the sensor voltage and d is the displacement of optical probe. The second step of compensation is shown in Figure 5b. A rigid plate with its length as same as the effective length of the cantilever piezo plate described in Eq. (1) is used to model the vibration. The rigid plate will then rotate slightly with respect to the pivot when the same micrometer based screw rod is used to push the free end. Therefore, the sensor voltage collected under this condition contains influence from both distance and angle. The subtraction between the results of experiment 5a and 5b describe an empirical relationship of sensor voltage and angle. This is shown in Figure 6 where the red line represents the interpolation of distance versus sensor voltage when constant flat face reflection is considered, i.e., experiment 5a; the blue dots indicate the relation between distance and sensor voltage when reflection angle variation is considered, i.e., experiment 5b. As we can observe in Figure 6, because of the angle variation exerted optical loss the actual displacement is smaller than it shows on the linear red line in the nearby range on both sides of the neutral point. When it gets far away from the neutral point, we observe some abnormal trends that the blue dots cross over the red line. This is due to the nonlinear effect when the measurement distance is out of linear range. When we implement the calibration into the control, we map the empirical data points into a quadratic function and use it to automatically convert the measured distance to true distance. ð2þ 2.2. Design of Dynamics Compensation A synthetic compensator designed using the inversion-based control technique can use zero-pole cancellation to assign another set of zeros and poles to the system, expanding the pass-band and improving the settling time. Since the load on the vibratory feeder dynamics varies during operation, the model transfer function can t

8 VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR 375 Figure 6. Calibration of the distance measurement with reflection angle variation. The red line is an interpolation based on the flat surface reflection, e.g., the distance varies while the reflecting surface maintains perpendicular to the incident light (see Figure 5a). The blue dots are measured when the reflecting surface is rotating with respect to the pivot, so the reflection angle varies with the distance change (see Figure 5b). The fitting function of error between the blue dots and the red line is the empirical function used for calibration of the angle variation. The red point marked as the neutral point of vibration is the position where the reflection surface is perpendicular to the incident light. This point is chosen based on the flat surface measurement and the distance from the far-near side turning point. be considered as time invariant for compensator design. To design an adaptive linear compensator in the vibratory system, a recursive least square (Evans et al. 2001) system identification method was applied to analyze the system dynamically. The determined system parameters were then used to construct an inverse compensator. The dynamically identified transfer function can be described as GðsÞ ¼ AðsÞ BðsÞ The direct inverse compensator can be applied to system input u in ½nŠ to produce a modified driving signal u½nš. UðsÞ ¼ BðsÞ AðsÞ U inðsþ ð3þ ð4þ However, under most circumstances the direct inverse transfer function is a differentiator and will result in significant spikes in waveform third derivative at the point where two quadratic curves connected because the second derivative of the two quadratic curves changes dramatically at these points. Also, the compensator may be unstable when the system has zeros on the right half plane. In the practical application, these factors will either depolarize the actuator or excite resonant vibration. To avoid these problems, we propose a modified inverse compensator with excess number of parameters. G c ðsþ ¼ k 0 BðsÞ AðsÞðs 2 þ k 1 s þ k 2 Þ ð5þ

9 376 Z. HU AND G. P. MAUL Figure 7. Schematic drawing of compensator. The desired waveform shape u in is fed into the inverse compensator G c. The output of compensator u is the input to the actuator and y is the output of actuator. The bold arrow between the recursive least squares (RLS) identifier and the compensator indicates parameters being passed. Here k 0;1;2 are constants that are determined by required system bandwidth and impulse response. A block diagram of a portion of the controller implementation is shown in Figure 7. The waveform generator outputs a desired input signal u in that is passed through the inverse compensator and the modified driving signal u is then sent to the PZT bimorph actuator. The RLS Identifier analyzes u and y to determine system parameters. The output values of the RLS identifier are the coefficients of the characteristic equation, which are used to adapt the compensator design for the next cycle. The RLS identified system parameters are used to construct a linear plant model with no hysteresis influence. For an ideal linear system, the model output y mdl versus input u should be a straight line. In the operational system, however, due to parameter variation, it could be a curved line, indicating residual higher harmonic effects. Therefore, the modeled input versus output data is interpolated as a straight line by linear regression. The hysteresis disturbance v is calculated by comparing the hysteresis loop with the desired linear response. The hysteresis loop is determined by measuring input voltage versus output displacement (Figure 8). Figure 8. Schematic drawing of nonlinear disturbance analysis.

10 VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR 377 For a point B on the hysteresis loop, the applied input voltage is V B and output displacement is d B. The point on the interpolated straight line with the same input voltage is labeled as A and the corresponding output displacement is d A. The displacement difference, Dd, between A on the desired linear shape and B on the hysteresis loop is recorded for the entire cycle as nonlinear disturbance and then decomposed into Fourier series and the base frequency for the decomposed disturbance is chosen to be as same as base frequency of the driving waveform Stability Analysis A high requirement on industrial application is to maintain a stable operation under unpredictable environment. In vibratory feeding, parts are adding into or picked up from the system stochastically and therefore the system dynamics is varying consistently. One advantage of this control algorithm is that it used indirect waveform generating and tracking (IWGT) control and increased the stability and robustness of the system, e.g., the feedback error is used to change the waveform generator parameters instead of acting directly on the input signal. To avoid the mechanical jerk caused by driving waveform change between two cycles, moving average of parameters are used to maintain a smooth operation. This is reasonable since the cycle time of dynamic variation is much slower than the cycle time of the control loop; therefore, we have more than enough time to change waveform generation parameters and make sure it follows the right direction. In a simple way, we can define it i K l is a series of parameters decided the waveform generation, c is the parameter determine tuning speed, and e is the sum up error between the measured output and model predicted output and e ¼ y y mdl If we make the c small enough, the K i will adapt much slower than the other system variations. The right side of the Eq. (6) can be viewed as a derivative process computing the sensitivity of output error to the parameter K i. The left side, on the other hand, can be viewed as inputs to an integrator that gradually changes the value of K i based on the historical output error. To make the square of the error small, it is designed to allow the parameters in the direction of the negative gradient of e 2. ð6þ ð7þ 3. EXPERIMENTS AND RESULTS The vibratory parts feeding system was composed of a low damping PZT-5H bimorph actuator and a plastic rail. One end of the actuator was clamped and the plastic rail was bonded perpendicularly on the top of the other end with epoxy. When AC voltage was applied to the actuator the end of actuator deflects and drives the rail to vibrate in the horizontal direction. The probe of optical

11 378 Z. HU AND G. P. MAUL Figure 9. The hardware of vibratory feeding and feedback control. positioner is pointed to the side of rail along the direction perpendicular to the neutral position plate surface and a finely polished platinum film was attached to the side to act as a reflective target and enhance the signal-to-noise (S=N) ratio. A computer data acquisition system (NI6040 DAQ) and real-time control card (National Instruments PCI7030) were used for sensing and actuation control (shown in Figure 9). There are many different models of piezo-actuator the have been built by previous researchers in this field (Tzen et al. 2003; Song and Li 1999; Cattafeta et al. 2000; Fleming et al. 2003; Yang 1997) and each of them has its distinction based on different applications. Our derivation process for Eq. (1) shows that the system can be viewed as a torsional vibration, i.e., a second order system. The frequency response of the vibratory system was measured and is shown in Figure 10. The system was lightly damp as expected with a damping ratio of f ¼ 0:05. A simple Figure 10. The frequency response of the vibratory system. The circles are experimental measured data while the solid line is the simulated response of Eq. (6).

12 VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR 379 analytical solution of the load free system, determined from the frequency response was 1651 GðsÞ ¼ s 2 ð8þ þ 18:175s þ The Matlab simulated frequency response of Eq. (8) is shown as a solid line in Figure 10. It shows a good fit to the experimental data in the frequency range of Hz. The natural frequency of the system obtained from Eq. (6) is f 0 ¼ 72:3 Hz. The compensator according to the design of Eq. (5) was then added into the system. As mentioned before, the parameters k 0;1;2 can be determined by the system from a specification of the desired frequency response of the compensated system. In this case, the corresponding compensator parameters were tuned as k 0 ¼ 21; k 1 ¼ 280 and k 2 ¼ After implementation of the compensator, the system reduces the differentiation effect and filters out undesired spikes while maintaining most characteristics of the inverted driving waveform u. The simulated frequency response of the compensated synthetic system had a bandwidth of 250 Hz and damping factor f ¼ 0:12 as shown in Figure 11. The new system provides enough bandwidth for the vibratory operation. Besides linear harmonics, there is nonlinear hysteresis effect needs to be addressed to obtain desired vibration trajectory. Strictly speaking, the nonlinear hysteresis effect can t be described explicitly. However, for simplicity reason we view the hysteresis disturbance as a constant periodical disturbance applied to each vibration cycle. Therefore, the nonlinear compensation can be pre-determined and add in as a feed forward compensation. Figure 12 shows a hysteresis loop when driving voltage is varying from 10 V to þ 10 V. To verify the functionality of the noise suppression method, the control designed was implemented and tested. The experimental results are compared Figure 11. The frequency response modified by implementing the appropriate compensator. The solid line is the original response and the dotted line is the response after implementation of the compensator.

13 380 Z. HU AND G. P. MAUL Figure 12. The open-loop static hysteresis was measured by applying a saw-tooth voltage waveform in the range 10 V. between dynamic hysteresis loops (Figure 13a) and the static hysteresis loop (Figure 12) within same voltage extremes. The displacement range and the width of the hysteresis loop (roughly speaking, the loop area) for the 40 Hz waveform were both greater than those of static ones, indicating that the hysteresis effect was Figure 13. Hysteresis loop at 40 Hz: a) quadratic drive without compensation; b) quadratic drive with linear compensation; c) sinusoidal drive without compensation; and d) quadratic drive with hysteresis suppression.

14 VIBRATION CONTROL OF PIEZOELECTRIC ACTUATOR 381 magnified by faster vibration. It can also be observed that there was an abnormal curvature of hysteresis loop for the uncompensated vibration. This is attributed to the coupling of linear and nonlinear effects. When the linear frequency compensator was implemented, the higher harmonic effect was reduced and the abnormal curvature of the uncompensated hysteresis loop was diminished as shown in Figure 13b. However, compared with the response of the uncompensated system to a single driving frequency (Figure 13c), the discrepancy can still be observed. This indicates that linear compensator can t absolutely remove the harmonic-induced noise, because of the coupling between linear and nonlinear effects adds complexity into RLS system identification process, which will take a longer time to converge. Figure 13d shows the result of iteration of alternating process between linear identification and nonlinear filtering. It indicated that reducing noise gradually through multiple cycles will allow linear and nonlinear noise being offset by each other and total noise could be reduced to a much lower level with this methodology. Although there is still observable hysteresis loop in Figure 13d, but it is already in a low level where enough harmful disturbance has been suppressed. Experimental results have revealed that the efficiency of vibratory feeding system was significantly improved after implementation of this controller. 3. CONCLUSION In this research, a PZT-5H bimorph actuator with narrow bandwidth is used for vibratory parts feeding. It is shown that with the help of a non-contact and high precision optical sensor, we can successfully implement a synthetic feed forward compensator to overcome the resonant distortion in the tracking waveform and frequency dependent dynamic hysteresis effect. Both linear frequency dispersion and nonlinear hysteresis noise were fully compensated by inversion based control and their coupling effect was reduced by an adaptive FIR filter. At 40 Hz operation test, we observed a significant reduction of the harmonic influence and dynamic hysteresis effect. In summary, IWGT has been demonstrated to be an effective control technique that when implemented with the non-contact optical sensor would allow compact and none magnetic PZT actuators to be applied in micro and precision industry. REFERENCES Cattafeta, L., J. Mathew, and A. Kurdila Modeling and design of piezoelectric actuators for fluid flow control. SAE. 2000:1 8. Choi, G. S., H. S. Kim, and G. H. Choi A study on position control of piezoelectric actuators. IEEE Catalog Number : Croft, D., D. Mcallister, and S. Devasia High-speed scanning of piezo-actuators for nanofabrication. ASME Journal of Manufacturing Science and Engineering. 120(3): Culshaw, Brian and John Dakin Optical fiber sensors: System and applications. Boston: Artech House. Duparre, Jacques, Peter Buecker, Bernt Goetz, and Thomas Martin Theoretical and experimental investigation of significant characteristic parameters of piezoelectric actuator. Smart Structures and Materials. 3(3985):6 9.

15 382 Z. HU AND G. P. MAUL Evans, R. J., I. M. Y. Mareels, L. J. Sciacca, D. N. Cooper, R. H. Middleton, R. E. Betz, and R. A. Kennedy Adaptive servo control of large antenna structures. In Model identification and adaptive control. London: Springer-Verlag Inc. Fleming, A. J., S. Behrens, and S. O. R. Moheimani Synthetic impedance for implementation of piezoelectric shunt-damping circuits. Electronics Letters. 36(18): Ge, P. and M. Jouaneh Modeling hysteresis in piezoceramic actuators. Precision Engineering. 17(3): Hu, Z., D. F. Farson, and G. P. Maul Economic application of piezoelectric actuator in linear vibratory feeding. Proc. IMechE Part B, JEM. 220(12): Hu, Z. and G. P. Maul Design and analysis for decoupled vibratory system. Flexible Automation and Intelligent Manufacturing. 2: Hu, Z., G. P. Maul, and D. F. Farson Piezo actuated vibratory feeding with precise vibration control. Intl. Journal of Production Research. 45(5): Kim, Do-Hyung, Jae-Hung Han, Seung-Man Yang, Dae-Hyun Kim, In Lee1, Chun-Gon Kim, and Chang-Sun Hong Optimal vibration control of a plate using optical fiber sensor and PZT actuator. J. Smart Materials and Structures and Structures. 12: Kim, S. and S. H. Kim A precision linear actuator using piezoelectric driven friction force. J. Mechatronics. 11: Lin, F. J., H. J. Shieh, P. K. Huang, and L. T. Teng Adaptive control with hysteresis estimation and compensation using RFNN for piezo-actuator. IEEE Trans. Ultrasonic, Ferro and Freq Control. 53(9): Mayergoyz, Isaak D Mathematical models of hysteresis. New York: Springer- Verlag Inc. Mayergoyz, Isaak D Mathematical models of hysteresis and their applications. Amsterdam: Elsevier. Song, D. and C. J. Li Modeling of piezo actuator s nonlinear and frequency dependent dynamics. J. Mechatronics. 9: Sun, L., C. Ru, and W. Rong Hysteresis compensation for piezoelectric actuator based on adaptive inverse control. Fifth World Congress on Intelligent Control and Automation. 6(15 19): Thornhill, N. F., S. L. Shah, and B. Huang Controller performance assessment in set point tracking and regulatory control. International Journal of Adaptive Control and Signal Processing. 7(7 9): Tzen, J., S. Jeng, and W. Chieng Modeling of piezoelectric actuator for compensation and controller design. J. Precision Engineering. 27(1): Udd, Eric Fiber optic sensors: An introduction for engineers and scientists. New York: John Wiley and Sons. Yang, J. S Equations for the flexural motion of elastic plates with partially electroded piezoelectric actuators. J. Smart Materials and Structures. 6:

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