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robust model reference adaptive control for a two dimensional piezo driven micro displacement scanning platform based on the asymmetrical bouc wen model

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Robust model reference adaptive control for a two-dimensional piezo-driven microdisplacement scanning platform based on the asymmetrical Bouc-Wen model Haigen Yang, Wei Zhu, and Xiao Fu Citation: AIP Advances 6, 115308 (2016); doi: 10.1063/1.4967428 View online: http://dx.doi.org/10.1063/1.4967428 View Table of Contents: http://aip.scitation.org/toc/adv/6/11 Published by the American Institute of Physics AIP ADVANCES 6, 115308 (2016) Robust model reference adaptive control for a two-dimensional piezo-driven micro-displacement scanning platform based on the asymmetrical Bouc-Wen model Haigen Yang,1,a Wei Zhu,2 and Xiao Fu1 Engineering Research Center of Wider and Wireless Communication, Technology, Ministry of Education, Nanjing University of Posts and Telecommunications, Nanjing 210094, China Institute of Launch Dynamics, Nanjing University of Science and Technology, Nanjing 210094, China (Received 28 June 2016; accepted 24 October 2016; published online November 2016) The hysteresis characteristics resulted from piezoelectric actuators (PAs) and the residual vibration in the rapid positioning of a two-dimensional piezo-driven microdisplacement scanning platform (2D-PDMDSP) will greatly affect the positioning accuracy and speed In this paper, in order to improve the accuracy and speed of the positioning and restrain the residual vibration of 2D-PDMDSP, firstly, Utilizing an online hysteresis observer based on the asymmetrical Bouc-Wen model, the PA with the hysteresis characteristics is feedforward linearized and can be used as a linear actuator; secondly, zero vibration and derivative shaping (ZVDS) technique is used to eliminate the residual vibration of the 2D-PDMDSP; lastly, the robust model reference adaptive (RMRA) control for the 2D-PDMDSP is proposed and explored The rapid control prototype of the RMRA controller combining the proposed feedforward linearization and ZVDS control for the 2D-PDMDSP with rapid control prototyping technique based on the real-time simulation system is established and experimentally tested, and the corresponding controlled results are compared with those by the PID control method The experimental results show that the proposed RMRA control method can significantly improve the accuracy and speed of the positioning and restrain the residual vibration of 2D-PDMDSP © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4967428] I INTRODUCTION In recent years, piezoelectric actuators (PAs) based on the inverse piezoelectric effect have been widely used in precision positioning,1,2 due to their small size, high energy density, high resolution, and quick frequency response However, on the one hand, owing to the hysteresis characteristics of PAs, piezo-driven positioning systems also has a strong nonlinearity, so how to effectively control piezo-driven positioning systems is difficult and emphasized in positioning systems;3 on the other hand, in the rapid positioning, the existing residual vibration of piezo-driven positioning systems slow down the positioning speed Therefore, how to eliminate the residual vibration and improve the positioning speed of positioning systems is also one of the problems which must be urgently solved In order to reduce the influence of the hysteresis characteristics of PAs on the positioning precision of PAs and piezo-driven positioning systems, the best way is the feedforward compensation for the hysteresis behavior of PAs and piezo-driven positioning systems as realized by using the established inverse hysteresis model4–8 to track the desired displacement.9–12 Because of exist of the inverse a Electronic mail: yhg@njupt.edu.cn 2158-3226/2016/6(11)/115308/14 6, 115308-1 © Author(s) 2016 115308-2 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) modelling error of the hysteresis behavior, the closed-loop control to compensate for the model error and restrain the outside disturbances was introduced.13–17 Although the closed-loop control method could restrain the hysteresis of PAs and piezo-driven positioning systems, the inverse of the hysteresis model enlarged the model error to some extent In the existing literature, it is few reported that how to eliminate the residual vibration The general methods include increasing the damping, increase stiffness, and using complex control algorithms However, increasing damping and increase stiffness affect the response speed of positioning systems and increase energy consumption, and using complex control algorithms is difficult to realize, further exacerbate the complexity of the precision positioning control, and reduce the real-time of the systems the piezo-driven positioning systems.18,19 Therefore, in this paper, the robust model reference adaptive (RMRA) control including feedforward linearization, zero vibration and derivative shaping (ZVDS) for a two-dimensional piezo-driven micro-displacement scanning platform (2D-PDMDSP) is proposed and experimentally verified in Section II, the structure of 2D-PDMDSP is analyzed and the dynamic model is put forward; in Section III, utilizing an online hysteresis observer based on an asymmetrical Bouc-Wen model, the PA with the hysteresis characteristics is feedforward linearize to a linear actuator; in Section IV, zero vibration and derivative shaping (ZVDS) is used to eliminate the residual vibration of the 2D-PDMDSP in the rapid positioning; in Section V, a robust mode reference adaptive (RMRA) control is put forward to control the 2D-PDMDSP, and the stability and robustness of the RMRA control is theoretically proved; in Section VI, the rapid control prototype of the RMRA controller combining the proposed feedforward linearization, ZVDS, and RMRA control for the 2D-PDMDSP with rapid control prototyping technique based on the real-time simulation system are established and experimentally tested, and the corresponding controlled results are compared with those by the PID control method; in Section VII, some conclusions are drawn II 2D-PDMDSP Figure shows the photograph of a 2D-PDMDSP From figure 1, the 2D-PDMDSP is composed of: two PAs I and II, which are used to generate output displacements and forces, and promote the X-axis and Y-axis directions of the 2D-PDMDSP respectively; two oval amplification mechanisms, which can enlarge the output displacements of PAs; a flexure hinge transmits the displacements and forces, and promote the movement of a two-dimensional position stage and a load The relationship between the X axis and Y axis directions is interrelated and independent of each other, therefore the FIG Photograph of the 2D-PDMDSP 115308-3 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) X axis and Y axis directions can be separately controlled In this paper, only the X axis direction examples for controlling are presented Because the long axis of the oval amplification mechanisms is much larger than the maximum output displacement of the PAs, can be approximated as a linear amplification mechanism The dynamics model for the X axis direction of the 2D-PDMDSP can be expressed as m0 Xă + c0 X + k0 X = F (1) Fp (2) M where F is the force transmitted by the flexure hinge; M is the magnification of the oval amplification mechanism; F p is the output force of the PA-I; X is the output displacement of the X axis direction of the two-dimensional position stage and load; m0 , c0 , and k are the mass, damping, and stiffness of the X axis direction Appling a 10 Hz sinusoidal voltage, as showed in figure 2, to the PA-I, the measured inputoutput relationship curve of the 2D-PDMDSP is shown in figure Observing from figure 3, due to the hysteresis characteristics of the PA-I, the input-output relationship of the 2D-PDMDSP also presents the hysteresis behavior Appling a Hz square voltage, as showed in figure 4, to the PA-I, the measured output displacement of the 2D-PDMDSP is shown in figure Observing from figure 5, the large amplitude and the long duration of the residual vibration of the 2D-PDMDSP limit its positioning speed, and even affect its life Therefore, the hysteresis characteristics and the residual vibration in the rapid positioning of the 2D-PDMDSP greatly affect the positioning accuracy and speed In order to improve the positioning F= FIG Time histories of a sinusoidal applied voltage to the 2D-PDMDSP FIG Measured hysteresis curve of the 2D-PDMDSP with the sinusoidal applied voltage 115308-4 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) FIG Time histories of a square applied voltage to the 2D-PDMDSP FIG Time histories of the output displacement of the 2D-PDMDSP with the square applied voltage FIG Measured hysteresis curve of the PA-I with the 10 Hz sinusoidal voltage accuracy and reduce the complexity of the positioning control algorithm of the 2D-PDMDSP, the PA is feedforward linearized, which make the input-output relationship of the 2D-PDMDSP be linear At the same time, in order to increase its positioning speed, the ZVDS technique is used to eliminate residual vibration III LINEARIZATION CONTROL FOR THE PA-I BASED ON THE ASYMMETRICAL BOUC-WEN MODEL Figure shows the measured hysteresis curve of the PA-I under the no-loading condition with the 10 Hz sinusoidal voltage as shown in figure The output displacement in the stable period is different from that in the initial period due to the memory of piezoelectric ceramic materials Considering the fitted line in least-squares sense as the linear component, the hysteresis curve shown in figure can be decomposed into the linear component (x L (t)) and the hysteresis component 115308-5 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) (x H (t)) The experimentally measured hysteresis curve can be considered as the superposition of a linear component and a hysteresis component When modelling the hysteresis characteristics of the PA-I, the linear and hysteresis components can be predicted separately The output displacement from the PA-I can be given by x (t) = xL (t) + xH (t) (3) where x(t) is the output displacement of the PA-I The output force of the PA-I is Fp (t) = kx (t) (4) Considering that the hysteresis of PSAs is asymmetrical,16 utilizing a linear function and the asymmetrical Bouc-Wen hysteresis operator proposed by Zhu and Wang16 to respectively simulate the linear component and the hysteresis component The asymmetrical Bouc-Wen model can be expressed as Fp (t) = kv u (t) + F0 + FH (t) (5) F˙ H (t) = A˙u (t) − β | u˙ (t)| |FH (t)| n−1 FH (t) − γ u˙ (t) |FH (t)| n + δu (t) sgn (FH (t)) (6) where k v is a constant representing the ratio between the output force and applied voltage; F is the initial force without applied voltage; F H (t) is the hysteresis force; A, β, γ, and n are the undetermined parameters; sgn (x) = x > ; δ is the asymmetrical factor, and δ < when modeling PSAs −1 x < According to equation (5), if the hysteresis force can be compensated, the relationship between the output displacement and the applied voltage will be linear However, because the hysteresis force of PSAs cannot be directly measured by sensors, the hysteresis observer is established to estimate the hysteresis force According to equation (6), the hysteresis observer can be expressed as n−1 n Fˆ˙ H (t) = A˙u (t) − β | u˙ (t)| Fˆ H (t) Fˆ H (t) − γ u˙ (t) Fˆ H (t) + δu (t) sgn Fˆ H (t) (7) where Fˆ H (t) is the estimated value of F H (t) According to equations (5) and (7), the linearization control for the PA-I can be expressed as ˆ   uFF (t) = u (t) − FHkv(t)   (8)   Fˆ˙ H (t) = A˙u (t) − β | u˙ (t)| Fˆ H (t) n−1 Fˆ H (t) − γ u˙ (t) Fˆ H (t) n + δu (t) sgn Fˆ H (t)  According to equation (8), the block diagram of the linearization control for the PA-I based on the asymmetrical Bouc-Wen model is shown in figure By using the parameter identification method proposed by reference 17, the parameters of the linearization control given by equation (8) are identified as follows kv = 1.2∗ 10−7 , A = 4.5∗ 10−8 , β = 0.059, γ = −0.023, n = 1.055, and δ = 2.1∗ 10−8 (9) Utilizing the values of the parameters given by equation (9), the hysteresis curves between the output displacements and the applied voltages from and to the PA-I without and with the feedforward linearization control are shown in figure In figure 8, the fitted line according to the relationship between the output displacement and applied voltage of the PA-I with the feedforward linearization control in least-squares sense is also presented From figure 8, with the feedforward linearization control, the input-output curve of the PA-I is almost linear, and the PA-I can be used as a linear actuator FIG Block diagram of the linearization control for the PA-I 115308-6 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) FIG Relationships between the output displacement and the applied voltage of the PA-I with and without the linearization control IV ZERO VIBRATION AND DERIVATIVE SHAPING In order to eliminate the residual vibration of the 2D-PDMDSP in the rapid positioning, the input shaping technique is used The idea of input shaping technique is a step signal, which be sent to a system at time t1, will stimulate a vibration response of the system, and at time t2, another step signal sent to the system will stimulate another vibration response If these two vibration responses have the same amplitude with the opposite phase, they can cancel each other out, and thus the vibration is eliminated at time t2 The 2D-PDMDSP can be considered as a linear system with the linearization control for the PA-I According to equation (1), the transfer function of the 2D-PDMDSP can be written as ωn2 (10) s2 + 2ξωn s + ωn2 where k is the magnification; ξ is the damping ratio; and ωn is the natural frequency According to equation (10), assume that at time t = 0, after the linearization control, the amplitude of the step voltage to the ZVDS is U FF The output voltage up of the ZVDS is G (s) = k    UFF    + 2K + K2      + 2K UFF up =    + 2K + K      + 2K + K    UFF  + 2K + K 0≤t< T T ≤t 0, we have (36) − β1 f − |f | M sgn (ˆe) = − β1 f − β1 |f | M ≤ (37) − β1 f − |f | M sgn (ˆe) = − β1 f + β1 |f | M ≥ (38) When eˆ < 0, we have According to equations (37) and (38), we have eˆ − β1 f − β1 |f | M sgn (ˆe) ≤ (39) According to equations (36) and (39), we have V˙ ≤ − XT QX (40) In other word, V˙ (t) is a negative definite matrix Therefore, according to the Lyapunov’s second method, when t ≥ 0, for arbitrary α0 , α1 , f, and arbitrary initial conditions, when the disturbance is less than |f | M , the error e is bounded and asymptotic convergence to zero The block diagram of the RMRA control method including the linearization control based on the asymmetrical Bouc-Wen model proposed in Section III and the ZVDS proposed in Section IV is shown in figure 11 115308-10 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) FIG 11 Block diagram of the RMRA control method including the linearization control and the ZVDS for the 2D-PDMDSP VI RAPID CONTROL PROTOTYPES AND EXPERIMENTAL RESULTS A Rapid control prototypes In order to validate the effectiveness of the RMRA control method for the 2D-PDMDSP and compare the corresponding controlled results with those by the PID control method, the rapid control prototypes for the RMRA and PID controllers for the 2D-PDMDSP with rapid control prototype technique based on the real-time simulation system (the dSPACE DS1103 with Matlab/Simulink) are established The schematic diagrams of the rapid control prototypes for the RMRA and PID controllers are shown in figures 12(a) and 12(b), respectively, and the photograph is shown in figure 13 According to figure 12, the rapid control prototypes for the RMRA and PID controllers establish the control algorithm of the RMRA and PID controllers for the 2D-PDMDSP with MATLAB/Simulink of the real-time simulation system according to the input signals and the feedback displacements by the Polytech laser Doppler vibrometer (type: OFV-505/5000) and outputs the RMRA and PID controlled voltages to the power amplifier for the 2D-PDMDSP with the dSPACE DS1103 of the real-time simulation system FIG 12 Schematic diagrams of the rapid control prototypes for the controllers for the 2D-PDMDSP 115308-11 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) FIG 13 Photograph of the rapid control prototypes for the controllers for the 2D-PDMDSP B Controller parameters Parameters of the RMRA controller Utilizing the parameter identification method proposed by reference 10, the linear model of the 2D-PDMDSP given by equation (18) can be rewritten as Xă + 7.212∗ 102 X˙ + 6.423∗ 107 X = 15.234u + 1.796f (41) According to equation (41), the damping ratio ζ is 0.025, so the 2D-PDMDSP is an underdamping system The reference model given by equation (19) is set to Xă m + 12000X˙ m + × 107 Xm = 15r (42) According to equation (42), the damping ratio ζ of the reference model is 0.77, and is in the best range ∗ Let Q = 16 × 10 According to equation (24), P = 2.67 10 13.33 Therefore, eˆ = 16 13.33 0.0018 13.33e + 0.0018˙e; let the initial values of φ0 (ˆe, r),φ1 (ˆe, X), and φ2 eˆ , X˙ be 0, λ = 5000,λ = 5000, λ = 5000, |f | M = Parameters of the PID controller Utilizing the linear quadratic regulate method, the parameters of the PID controller are KP = 35, KI = 1750, and KD = 0.0008 (43) C Experimental results When the desired output displacement is the triangular signal in the first half and the sinusoidal signal in the second half as shown in figure 14(a), the output displacement of the 2D-PDMDSP with the RMRA control is shown in figure 14(a), and the output displacement error is shown in figure 14(b) The controlled results by the PID control for the 2D-PDMDSP is also shown in figure 14 Observing from figure 14, the RMRA control has an adaptive process, and then can accurately control the output displacement of the 2D-PDMDSP Because the initial values of the gain coefficients ˙ of the control law are 0, the adaptive process is used to adjust the (φ0 (ˆe, r), φ1 (ˆe, X), and φ2 (ˆe, X)) gain coefficients Without considering the adaptive process, the RMRA control can achieve more control accuracy than the PID control After the adaptive process, the stable values of the gain coefficients of the RMRA control are φ0S = 1.067, φ1S = 2.208 × 10−7 , and φ2S = 8.909 × 10−5 (44) 115308-12 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) FIG 14 Desired and controlled triangular and sinusoidal output displacements of the 2D-PDMDSP with the RMRA control and PID control The initial values of φ0 (ˆe, r), φ1 (ˆe, X), and φ2 eˆ , X˙ can be given by equation (44) When the desired output displacement is the Hz square signal as shown in figure 15(a), the output displacements of the 2D-PDMDSP with the RMRA control and the PID control are shown in figure 15(a), and the corresponding output displacement errors are shown in figure 15(b) From figure 15, when the initial values of the gain coefficients are set appropriate, the RMRA control does not require the adaptive process as shown in figure 14, which can quickly and accurately control the output displacement of the 2D-PDMDSP According to the analysis of Section II and figure 15, due to including the ZVDS, the RMRA control and the PID control delay a vibration period In vibration period, the control errors are larger Without considering the delay period, the RMRA control can also achieve more control accuracy than the PID control When the desired output displacement is the non-periodical signal as shown in figure 16(a), the output displacements of the 2D-PDMDSP with the RMRA control and the PID control are shown in figure 16(a), and the corresponding output displacement errors are shown in figure 16(b) From figure 16, the RMRA control can more effectively control the 2D-PDMDSP than the PID control In order to quantify the control error, the maximum absolute control error (MACE) and the maximum relative control error (MRCE) are defined as ∆MAX = MAX |x (t) − xm (t)| and γ = ∆MAX × 100% xFS (45) where x(t) and x m (t) the actual output displacement and the desired output displacement of the 2D-PDMDSP, respectively; x FS is the maximum of the desired output displacement According to equation (45) and figures 14–16, the MACEs and the MRCEs of the 2D-PDMDSP with the RMRA control and the PID control are listed in Table I According to Table I, when the FIG 15 Desired and controlled square output displacements of the 2D-PDMDSP with the RMRA control and PID control 115308-13 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) FIG 16 Desired and controlled non-periodical output displacements of the 2D-PDMDSP with the RMRA control and PID control TABLE I Quantitative control errors of the RMRAC and PID control Control method ∆MAX (µm) γ (%) Triangular and Sinusoidal displacement RMRA PID 0.062 0.21 0.517 1.75 Square displacement RMRA PID 0.071 0.28 0.568 2.24 Non-periodical displacement RMRA PID 0.091 0.32 0.414 1.45 Desired displacement desired output displacements are shown in figures 14(a), 15(a), and 16(a), relative to the PID control, the MACEs and the MRCEs of the RMRA control are reduced by 70.5%, 74.6%, and 71.5%, respectively Therefore, the RMRA control can more effectively control the 2D-PDMDSP than the PID control, and the control accuracy is significantly improved VII CONCLUSIONS In this paper, in order to improve the accuracy and speed of the positioning and restrain the residual vibration of 2D-PDMDSP, firstly, utilizing an online hysteresis observer based on the an asymmetrical Bouc-Wen model, the PA with the hysteresis characteristics was feedforward linearized and could be used as a linear actuator; secondly, the ZVDS technique was used to eliminate the residual vibration of the 2D-PDMDSP; lastly, the RMRA control for the 2D-PDMDSP was proposed and explored The rapid control prototype of the RMRA controller combining the proposed feedforward linearization, ZVDS, and RMRA control for 2D-PDMDSP with rapid control prototyping technique based on the real-time simulation system was established and experimentally tested, and the corresponding controlled results were compared with those by the PID control method The experimental results showed that the proposed RMRA control method could significantly improve the accuracy and speed of the positioning and restrain the residual vibration of 2D-PDMDSP ACKNOWLEDGMENTS The authors wish to acknowledge the financial support by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (Grant No 708048) and the Program for New Century Excellent Talents in University (Grant No NCET-05-0765),the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (Grant No NY213111), 115308-14 Yang, Zhu, and Fu AIP Advances 6, 115308 (2016) the Key University Science Research Project of Jiangsu Province (Grant No 15KJB130005), and the Basic Scientific Research of National Defense (JCKY201606C001) S Devasia, E Eleftheriou, and S O R Moheimani, “A survey of control issues in nanopositioning,” IEEE Transactions on Control Systems Technology 15(5), 802–823 (2007) Dong, J Tang, and Y ElDeeb, “Design of a linear-motion dual-stage actuation system for precision control,” Smart Materials and Structures 18(9), 095035 (2009) S Moriyama, F Uchida, and E Seya, “Development of a precision diamond-turning machine for fabrication of asymmetric aspheric mirrors,” Optical Engineering 27(11), 1008–1012 (1988) G Song, J Q Zhao, X Q Zhou, and J A de Abreu-Garcia, “Tracking control of a piezoceramic actuator with hysteresis compensation using inverse Preisach model,” IEEE/ASME Transactions on Mechatronics 10(2), 198–209 (2005) J P Lien, A York, T G Fang, and G D Buckner, “Modeling piezoelectric actuators with hysteretic recurrent neural networks,” Sensors and Actuators A: Physical 163(2), 516–525 (2010) M Goldfarb and N Celanovic, “Modeling piezoelectric stack actuator for control of micromanipulation,” IEEE Control Systems Magazine 17(3), 69–79 (1997) S H Lee and T J Royston, “Modeling piezo transducer hysteresis in the structural vibration control problem,” Journal of Acoustic Society of America 108(6), 2843–2855 (2000) J W Li, X B Chen, Q An, S D Tu, and W J Zhang, “Friction models incorporating thermal effects in highly precision actuators,” Review of Scientific Instruments 80(4), 045104 (2009) C H Ru and L N Sun, “Hysteresis and creep compensation for piezoelectric actuator in open-loop operation,” Sensors and Actuators A: Physical 122(1), 124–130 (2005) 10 G Schitter and A Stemmer, “Identification and open-loop tracking control of a piezoelectric tube, scanner for high-speed scanning-probe microscopy,” IEEE Transactions on Control Systems Technology 12(3), 449–454 (2004) 11 H Janocha and K Kuhnen, “Real-time compensation of hysteresis and creep in piezoelectric actuators,” Sensors and Actuators A: Physical 79(2), 83–89 (2000) 12 K K Leang, Q Zou, and S Devasia, “Feedforward control of piezoactuators in atomic force microscope systems,” IEEE Control Systems Magazine 29(1), 70–82 (2009) 13 P Ge and M Jouaneh, “Tracking control of a piezoceramic actuator,” IEEE Transactions on Control Systems Technology 4(3), 209–216 (1996) 14 J L Ha, Y S Kung, R F Fung, and S C Hsien, “A comparison of fitness functions for the identification of a piezoelectric hysteretic actuator based on the real-coded genetic algorithm,” Sensors and Actuators A: Physical 132(2), 643–650 (2006) 15 M S Tsai and J S Chen, “Robust tracking control of a piezoactuator using a new approximate hysteresis model,” Journal of Dynamic Systems Measurement and Control-transactions of the ASME 125(1), 96–102 (2003) 16 S Bashash and N Jalili, “Robust adaptive control of coupled parallel piezo-flexural nanopositioning stages,” IEEE/ASME Transactions of Mechatronics 14(1), 11–20 (2009) 17 R B Owen, M Maggiore, and J Apkarian, “A high-precision, magnetically levitated positioning stage,” IEEE Control Systems Magazine 26(3), 82–95 (2006) 18 W Zhu and D H Wang, “Non-symmetrical Bouc-Wen model for a piezoelectric ceramic actuator,” Sensors and Actuators A: Physical 181, 51–60 (2012) 19 D H Wang and W Zhu, “Phenomenological model for pre-stressed piezoelectric stack actuators,” Smart Materials and Structures 20(3), 035018 (2011) W

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