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Self-sensing control of piezoelectric positioning stage by detecting permittivity
Sensors and Actuators A 226 (2015) 76–80
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Self-sensing control of piezoelectric positioning stage by detecting permittivity Katsuhiro Saigusa ∗ , Takeshi Morita Graduate School of Frontier Science, The University of Tokyo, Kashiwa, Chiba 277-8563, Japan
a r t i c l e
i n f o
Article history: Received 15 August 2014 Received in revised form 26 January 2015 Accepted 16 February 2015 Available online 23 February 2015 Keywords: Piezoelectric actuator Hysteresis Permittivity detection Self-sensing control
a b s t r a c t Piezoelectric displacement contains hysteresis and creep properties. Therefore, a displacement sensor is indispensable in precise positioning devices; however, the additional space and cost are problems. On the other hand, self-sensing methods that utilize the piezoelectric actuator itself as the displacement sensor have been proposed. With these self-sensing methods, precise positioning becomes possible without an additional displacement sensor. We developed a self-sensing method utilizing the non-hysteresis relationship between the permittivity change and the piezoelectric displacement. Furthermore, a differential current measurement method using two piezoelectric elements with a bimorph actuator could improve the positioning accuracy. In this study, we examine the control of a positioning stage using two multilayered piezoelectric actuators by applying the differential current measurement method for selfsensing control. The results indicate that the differential current measurement method is effective for precise positioning control. The positioning errors due to hysteresis decreased from 0.8 m to 0.1 m for a 10 m displacement range. In addition, permittivity feedback control could compensate for the creep property. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Piezoelectric materials have the advantages of accuracy and rapid response. Therefore, they are used in high-precision devices as actuators. For example, a piezoscanner of AFM or a deformable mirror for a telescope adopts piezoelectric actuators [1,2]. However, piezoelectric displacement contains hysteresis and creep properties, and these properties have been studied from various viewpoints [3–5]. To eliminate the hysteresis property of a piezoelectric actuator for precise positioning control, a displacement sensor, such as a capacitive sensor or a laser interferometer, is indispensable. However, for a low-cost and simple system, self-sensing methods that utilize the piezoelectric actuator itself as a displacement sensor are required [6–9]. For example, a self-sensing method based on the non-hysteresis relationship between the piezoelectric displacement and the electric charge was proposed [6,7]. However, this method is not stable in the quasi-static state, because the electric charge is difficult to hold and the offset charge signal accumulates, due to the integration of the current signal.
∗ Corresponding author. Tel.: +81 471364615. E-mail address: [email protected] (K. Saigusa). http://dx.doi.org/10.1016/j.sna.2015.02.022 0924-4247/© 2015 Elsevier B.V. All rights reserved.
On the other hand, we have reported the non-hysteresis relationship between the piezoelectric displacement and the permittivity of piezoelectric actuators. This relationship was utilized for the self-sensing method for detecting the piezoelectric displacement from the permittivity change. By applying this method, the piezoelectric displacement of the bimorph actuator was controlled within a 0.4 m error for an 80 m displacement range [10–12]. In this study, we examine our self-sensing method for the positioning stage using multilayered actuators, which are mainly utilized in precise positioning devices. In addition, the differential current measurement method, which was quite useful in our previous study using the bimorph mechanism [12], is applied to the positioning stage for the improvement of the permittivity detection accuracy.
2. Experimental setup 2.1. Permittivity detection Self-sensing control by detecting permittivity is based on the non-hysteresis relationship between the piezoelectric displacement and the permittivity change. This is related to the fact that the permittivity follows a similar butterfly curve to that of the piezoelectric displacement versus input voltage, as shown in Fig. 1. It
K. Saigusa, T. Morita / Sensors and Actuators A 226 (2015) 76–80
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(Tektronix, TCPA300) and a lock-in amplifier (NF, LI5640). By using the permittivity detection voltage as the lock-in amplifier reference, the current amplitude signal with the permittivity detection voltage can be measured. By calibrating in advance, the leakage current phase signal can be eliminated. The permittivity is proportional to the current amplitude because the dimensions of the piezoelectric actuator, such as the thickness and electrode size, can be considered to be constant. This method requires no positioning sensor. The actual displacement was measured by a laser interferometer (Canon, DS-80) which had a bandwidth of 5 kHz within a 10 m displacement range. The actual peak-to-peak noise with our setup was 0.1 m in this displacement measurement as shown later. 2.2. Differential current measurement method
Fig. 1. Relationship between applied voltage, displacement, and permittivity.
is considered that domain structure modification strongly affects both the piezoelectric displacement and the permittivity [13]. Therefore, the domain structure information can be obtained from the permittivity change detection, and this information includes the piezoelectric displacement [12]. Fig. 2 shows an overview of the permittivity detection method. The driving voltage and the permittivity detection voltage were supplied from a function generator (NF, WF1974). These two signals were added and amplified by a high-speed amplifier (NF, 4025). The driving voltage brought about the piezoelectric displacement for positioning. The permittivity detection voltage was applied to the piezoelectric actuator for permittivity detection. It was a sinusoidal wave that should have a low amplitude and a high frequency in order not to influence the displacement. The high frequency causes the large current amplitude that improves the signal-to-noise (SN) ratio; however, the frequency must be far from the resonant frequency to avoid a current signal from the resonant effect. The current signal was picked up with a current probe
The differential measurement method can improve the selfsensing positioning accuracy, as demonstrated in our previous study using a bimorph actuator [12]. The bimorph actuator is composed of two piezoelectric plates, which have a push-pull relationship. In other words, a decreasing or increasing permittivity change is opposite between them. Basically, the initial permittivity of the piezoelectric material is large compared with the permittivity change induced by the driving voltage. As a result, we have to use a wide range for the lock-in amplifier to detect a small current amplitude change. On the other hand, if the piezoelectric actuator contains a push-pull mechanism like the bimorph actuator, only the permittivity change can be obtained by the differential current method. With this method, the base current with the essential permittivity can be canceled, and a narrow lock-in amplifier range for the current amplitude change becomes possible for a better SN ratio. 2.3. Design of positioning stage To utilize the differential measurement method, we designed a positioning stage using two multilayered actuators (NEC
Fig. 2. Overview of permittivity detection.
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Fig. 3. 3D model of the positioning stage.
TOKIN, AL3.5X3.5X9D-6F, 3.5 mm × 3.5 mm × 9 mm) to compose the push–pull mechanism, as shown in Fig. 3. When one actuator is extended to drive the movable part in one direction, the other actuator is contracted for driving the same direction. The driving voltage was from 0 V to 150 V, which induced an electrical field parallel to the polarization direction. This voltage range was determined so as not to reverse the polarization. Therefore, in the initial condition, +75 V was applied to both multilayered actuators, meaning an offset stress was applied. During operation, the current amplitude change between the two driving wires for each actuator is opposite because of the push–pull mechanism. To detect the differential current amplitude, two wires were inserted in the opposite direction into the current probe. For comparison, the stage was driven without self-sensing control. Both actuators were driven with a 0–150 V sinusoidal voltage at a frequency of 0.1 Hz. As shown in Fig. 4, there was a hysteresis relationship between the driving voltage and the piezoelectric displacement. With this open-loop control, the maximum hysteresis error was 0.8 m for a 10 m displacement range. 3. Experiments and results By using two multilayered actuators involved in the positioning stage, the relationship between the piezoelectric displacement and the current amplitude was measured. As mentioned in Section 2, the current amplitude is proportional to the permittivity. Therefore, in this study, the current amplitude signal was utilized as the permittivity information because the internal structure of the multilayered actuator is not clear, and it is difficult to calculate the permittivity from the current amplitude and the actuator dimensions. When the two actuators were driven and only one current signal was measured, small hysteresis properties were confirmed, as shown in Fig. 5. On the other hand, with the differential current measurement using two current signals, the maximum hysteresis
Fig. 4. Open-loop hysteresis of the stage.
Fig. 5. Relationship between piezoelectric displacement and current amplitude for permittivity detection (upper: without differential measurement, lower: with differential measurement).
error was improved to 0.1 m from 0.3 m. The permittivity detection voltage was a sinusoidal wave whose amplitude and frequency were 1 VP-P and 65 kHz, respectively. This frequency was as high as possible and could avoid a resonant state. The driving voltage was a sinusoidal wave whose amplitude and frequency were 150 VP-P and 0.1 Hz, respectively. To apply this result to feedback positioning control, a third-order polynomial approximation was performed using the data plotted in Fig. 5. The obtained equation was D = 0.3718 − 9.4413×10−2 I + 3.7337×10−5 I 2 + 1.3682 × 10−6 I 3 , where D is the displacement, and I is the detected current amplitude. An outline of the control block diagram is shown in Fig. 6. We adopted a PI control setting with a proportional gain of 20 V/m and an integration gain of 400 V/(m s) through trial and error. An overview of the permittivity feedback control experiment is shown in Fig. 7. A digital signal processor (DSP, dSPACE, DS1104) was used with MATLAB 7.6 and Simulink 7.1 for the control programming. The target displacement signal, stepping up and down every 1 m in an 8 m range, was input into the DSP from the function generator. As shown in Fig. 8, permittivity feedback control could reduce the positioning error to ±0.10 m at most, whereas, as shown in Fig. 9, open-loop control could only reduce it to ±0.45 m.
Fig. 6. Outline of the control block diagram.
K. Saigusa, T. Morita / Sensors and Actuators A 226 (2015) 76–80
79
Fig. 7. Overview of permittivity feedback control.
movement affects both the piezoelectric displacement and the permittivity during the creep phenomenon. Therefore, it was expected that the creep property could be compensated by the permittivity feedback control. With a constant target displacement of +2 m or −2 m for the permittivity feedback control, the actual displacement was measured as shown in Fig. 11. In the case of a −2 m target displacement, the creep property was compensated as we expected, while it remained at 0.03 m with a target displacement of +2 m.
Fig. 8. Result of permittivity feedback control stepping every 1 m.
A piezoelectric actuator has a creep property in which the piezoelectric displacement increases gradually while a constant voltage is applied. We tried to demonstrate the effectiveness of the permittivity feedback control for eliminating this creep property. To confirm the creep property, a +30 V or −30 V DC voltage added to a +75 V offset voltage was applied without feedback control. As shown in Fig. 10, creep displacements of 0.07 m (+30 V) and 0.05 m (−30 V) were measured after 50 s. The current amplitude change, which is proportional to the permittivity change, had a similar tendency to the creep displacement. This is because the domain
Fig. 9. Result of open-loop control stepping every 1 m.
Fig. 10. Stage displacement and detection current change time when a constant voltage of +30 V (upper) or −30 V (lower) was applied.
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Based on these results, positioning control was performed with a target displacement from −4 m to +4 m with a 1 m step. The positioning error was reduced to 0.10 m from 0.45 m with openloop control. Permittivity feedback control was effective at reducing the creep property. The creep displacement remained at 0.03 m when +2 m was the target displacement. However, in the case of a −2 m target displacement, the creep was successfully compensated. Such a directional dependency is not reasonable because the positioning stage was designed symmetrically including the polarization directions of the two actuators. Therefore, this result implies that the two actuators had different creep properties. In this study, we adopted 0.1 Hz for the driving frequency. However, the stage should be driven with high speed in practical applications. Therefore, a suitable control system for high-speed driving should be realized. Moreover, we should study the creep property difference in each actuator and verify how the difference influenced the experimental results for the creep property, and in future works, we should compensate the temperature change effect. References
Fig. 11. Result of compensation for creep by permittivity feedback control.
However, even in the case of a +2 m target displacement, the amount of creep was reduced to 0.03 m from 0.07 m. 4. Conclusions The permittivity feedback control was effective for the positioning stage using two multilayered actuators. From the relationship between the piezoelectric displacement and the permittivity change and the differential measurement method, it was confirmed that the hysteresis displacement could be reduced to 0.1 m from 0.8 m, which was measured with open-loop control. In particular, the differential measurement method was useful for the multilayered actuators, similar to bimorph actuators.
[1] S. Devasia, E. Eleftheriou, S.O. Reza Moheimani, A survey of control issues in nanopositioning, IEEE Trans. Control Syst. Technol. 15 (5) (2007) 802–823. [2] M.A. Ealey, J.F. Washeba, Continuous facesheet low voltage deformable mirrors, Opt. Eng. 29 (1990) 1191–1198. [3] T.S. Low, W. Guo, Modeling of a three-layer piezoelectric bimorph beam with hysteresis, J. Microelectromech. Syst. 4 (4) (1995) 230–237. [4] M. Jang, C. Chen, J. Lee, Modeling and control of a piezoelectric actuator driven system with asymmetric hysteresis, J. Franklin Inst. 346 (2009) 17–32. [5] M. Rakotondrabe, Y. Haddab, P. Lutz, Nonlinear modeling and estimation of force in a piezoelectric cantilever, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2007, pp. 1–6. [6] K. Furutani, M. Urushibata, N. Mohri, Displacement control of piezoelectric element by feedback of induced charge, Nanotechnology 9 (2) (1998) 93–98. [7] I.A. Ivan, M. Rakotondrabe, P. Lutz, N. Chaillet, Quasistatic displacement selfsensing method for cantilevered piezoelectric actuators, Rev. Sci. Instrum. 80 (2009) 065102. [8] E. Uzgur, S. Kim, E. Bryce, D. Hutson, M. Strachan, K.J. Kirk, Extension sensing of piezo actuators using time-domain ultrasonic measurement and frequencydomain impedance measurement, J. Electroceram. 27 (2011) 38–44. [9] C.V. Newcomb, I. Flinn, Improving the linearity of piezoelectric ceramic actuators, Electron. Lett. 18 (11) (1982) 442–444. [10] A. Kawamata, Y. Kadota, H. Hosaka, T. Morita, Self-sensing piezoelectric actuator using permittivity detection, Ferroelectrics 368 (1) (2008) 194–201. [11] Y. Ishikiriyama, T. Morita, Improvement of self-sensing piezoelectric actuator control using permittivity change detection, J. Adv. Mech. Des. Syst. Manuf. vol. 4 (1) (2010) 143–149. [12] H. Ikeda, T. Morita, High-precision positioning using a self-sensing piezoelectric actuator control with a differential detection method, Sens. Actuators A: Phys. 170 (2011) 147–155. [13] F. Xu, S. Trolier-McKinstry, W. Ren, B. Xu, Z.L. Xie, K.J. Hemker, Domain wall motion and its contribution to the dielectric and piezoelectric properties of lead zirconate titanate films, J. Appl. Phys. 89 (2) (2001) 1336–1348.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Self-sensing control of piezoelectric positioning stage by detecting permittivity Katsuhiro Saigusa ∗ , Takeshi Morita Graduate School of Frontier Science, The University of Tokyo, Kashiwa, Chiba 277-8563, Japan
a r t i c l e
i n f o
Article history: Received 15 August 2014 Received in revised form 26 January 2015 Accepted 16 February 2015 Available online 23 February 2015 Keywords: Piezoelectric actuator Hysteresis Permittivity detection Self-sensing control
a b s t r a c t Piezoelectric displacement contains hysteresis and creep properties. Therefore, a displacement sensor is indispensable in precise positioning devices; however, the additional space and cost are problems. On the other hand, self-sensing methods that utilize the piezoelectric actuator itself as the displacement sensor have been proposed. With these self-sensing methods, precise positioning becomes possible without an additional displacement sensor. We developed a self-sensing method utilizing the non-hysteresis relationship between the permittivity change and the piezoelectric displacement. Furthermore, a differential current measurement method using two piezoelectric elements with a bimorph actuator could improve the positioning accuracy. In this study, we examine the control of a positioning stage using two multilayered piezoelectric actuators by applying the differential current measurement method for selfsensing control. The results indicate that the differential current measurement method is effective for precise positioning control. The positioning errors due to hysteresis decreased from 0.8 m to 0.1 m for a 10 m displacement range. In addition, permittivity feedback control could compensate for the creep property. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Piezoelectric materials have the advantages of accuracy and rapid response. Therefore, they are used in high-precision devices as actuators. For example, a piezoscanner of AFM or a deformable mirror for a telescope adopts piezoelectric actuators [1,2]. However, piezoelectric displacement contains hysteresis and creep properties, and these properties have been studied from various viewpoints [3–5]. To eliminate the hysteresis property of a piezoelectric actuator for precise positioning control, a displacement sensor, such as a capacitive sensor or a laser interferometer, is indispensable. However, for a low-cost and simple system, self-sensing methods that utilize the piezoelectric actuator itself as a displacement sensor are required [6–9]. For example, a self-sensing method based on the non-hysteresis relationship between the piezoelectric displacement and the electric charge was proposed [6,7]. However, this method is not stable in the quasi-static state, because the electric charge is difficult to hold and the offset charge signal accumulates, due to the integration of the current signal.
∗ Corresponding author. Tel.: +81 471364615. E-mail address: [email protected] (K. Saigusa). http://dx.doi.org/10.1016/j.sna.2015.02.022 0924-4247/© 2015 Elsevier B.V. All rights reserved.
On the other hand, we have reported the non-hysteresis relationship between the piezoelectric displacement and the permittivity of piezoelectric actuators. This relationship was utilized for the self-sensing method for detecting the piezoelectric displacement from the permittivity change. By applying this method, the piezoelectric displacement of the bimorph actuator was controlled within a 0.4 m error for an 80 m displacement range [10–12]. In this study, we examine our self-sensing method for the positioning stage using multilayered actuators, which are mainly utilized in precise positioning devices. In addition, the differential current measurement method, which was quite useful in our previous study using the bimorph mechanism [12], is applied to the positioning stage for the improvement of the permittivity detection accuracy.
2. Experimental setup 2.1. Permittivity detection Self-sensing control by detecting permittivity is based on the non-hysteresis relationship between the piezoelectric displacement and the permittivity change. This is related to the fact that the permittivity follows a similar butterfly curve to that of the piezoelectric displacement versus input voltage, as shown in Fig. 1. It
K. Saigusa, T. Morita / Sensors and Actuators A 226 (2015) 76–80
77
(Tektronix, TCPA300) and a lock-in amplifier (NF, LI5640). By using the permittivity detection voltage as the lock-in amplifier reference, the current amplitude signal with the permittivity detection voltage can be measured. By calibrating in advance, the leakage current phase signal can be eliminated. The permittivity is proportional to the current amplitude because the dimensions of the piezoelectric actuator, such as the thickness and electrode size, can be considered to be constant. This method requires no positioning sensor. The actual displacement was measured by a laser interferometer (Canon, DS-80) which had a bandwidth of 5 kHz within a 10 m displacement range. The actual peak-to-peak noise with our setup was 0.1 m in this displacement measurement as shown later. 2.2. Differential current measurement method
Fig. 1. Relationship between applied voltage, displacement, and permittivity.
is considered that domain structure modification strongly affects both the piezoelectric displacement and the permittivity [13]. Therefore, the domain structure information can be obtained from the permittivity change detection, and this information includes the piezoelectric displacement [12]. Fig. 2 shows an overview of the permittivity detection method. The driving voltage and the permittivity detection voltage were supplied from a function generator (NF, WF1974). These two signals were added and amplified by a high-speed amplifier (NF, 4025). The driving voltage brought about the piezoelectric displacement for positioning. The permittivity detection voltage was applied to the piezoelectric actuator for permittivity detection. It was a sinusoidal wave that should have a low amplitude and a high frequency in order not to influence the displacement. The high frequency causes the large current amplitude that improves the signal-to-noise (SN) ratio; however, the frequency must be far from the resonant frequency to avoid a current signal from the resonant effect. The current signal was picked up with a current probe
The differential measurement method can improve the selfsensing positioning accuracy, as demonstrated in our previous study using a bimorph actuator [12]. The bimorph actuator is composed of two piezoelectric plates, which have a push-pull relationship. In other words, a decreasing or increasing permittivity change is opposite between them. Basically, the initial permittivity of the piezoelectric material is large compared with the permittivity change induced by the driving voltage. As a result, we have to use a wide range for the lock-in amplifier to detect a small current amplitude change. On the other hand, if the piezoelectric actuator contains a push-pull mechanism like the bimorph actuator, only the permittivity change can be obtained by the differential current method. With this method, the base current with the essential permittivity can be canceled, and a narrow lock-in amplifier range for the current amplitude change becomes possible for a better SN ratio. 2.3. Design of positioning stage To utilize the differential measurement method, we designed a positioning stage using two multilayered actuators (NEC
Fig. 2. Overview of permittivity detection.
78
K. Saigusa, T. Morita / Sensors and Actuators A 226 (2015) 76–80
Fig. 3. 3D model of the positioning stage.
TOKIN, AL3.5X3.5X9D-6F, 3.5 mm × 3.5 mm × 9 mm) to compose the push–pull mechanism, as shown in Fig. 3. When one actuator is extended to drive the movable part in one direction, the other actuator is contracted for driving the same direction. The driving voltage was from 0 V to 150 V, which induced an electrical field parallel to the polarization direction. This voltage range was determined so as not to reverse the polarization. Therefore, in the initial condition, +75 V was applied to both multilayered actuators, meaning an offset stress was applied. During operation, the current amplitude change between the two driving wires for each actuator is opposite because of the push–pull mechanism. To detect the differential current amplitude, two wires were inserted in the opposite direction into the current probe. For comparison, the stage was driven without self-sensing control. Both actuators were driven with a 0–150 V sinusoidal voltage at a frequency of 0.1 Hz. As shown in Fig. 4, there was a hysteresis relationship between the driving voltage and the piezoelectric displacement. With this open-loop control, the maximum hysteresis error was 0.8 m for a 10 m displacement range. 3. Experiments and results By using two multilayered actuators involved in the positioning stage, the relationship between the piezoelectric displacement and the current amplitude was measured. As mentioned in Section 2, the current amplitude is proportional to the permittivity. Therefore, in this study, the current amplitude signal was utilized as the permittivity information because the internal structure of the multilayered actuator is not clear, and it is difficult to calculate the permittivity from the current amplitude and the actuator dimensions. When the two actuators were driven and only one current signal was measured, small hysteresis properties were confirmed, as shown in Fig. 5. On the other hand, with the differential current measurement using two current signals, the maximum hysteresis
Fig. 4. Open-loop hysteresis of the stage.
Fig. 5. Relationship between piezoelectric displacement and current amplitude for permittivity detection (upper: without differential measurement, lower: with differential measurement).
error was improved to 0.1 m from 0.3 m. The permittivity detection voltage was a sinusoidal wave whose amplitude and frequency were 1 VP-P and 65 kHz, respectively. This frequency was as high as possible and could avoid a resonant state. The driving voltage was a sinusoidal wave whose amplitude and frequency were 150 VP-P and 0.1 Hz, respectively. To apply this result to feedback positioning control, a third-order polynomial approximation was performed using the data plotted in Fig. 5. The obtained equation was D = 0.3718 − 9.4413×10−2 I + 3.7337×10−5 I 2 + 1.3682 × 10−6 I 3 , where D is the displacement, and I is the detected current amplitude. An outline of the control block diagram is shown in Fig. 6. We adopted a PI control setting with a proportional gain of 20 V/m and an integration gain of 400 V/(m s) through trial and error. An overview of the permittivity feedback control experiment is shown in Fig. 7. A digital signal processor (DSP, dSPACE, DS1104) was used with MATLAB 7.6 and Simulink 7.1 for the control programming. The target displacement signal, stepping up and down every 1 m in an 8 m range, was input into the DSP from the function generator. As shown in Fig. 8, permittivity feedback control could reduce the positioning error to ±0.10 m at most, whereas, as shown in Fig. 9, open-loop control could only reduce it to ±0.45 m.
Fig. 6. Outline of the control block diagram.
K. Saigusa, T. Morita / Sensors and Actuators A 226 (2015) 76–80
79
Fig. 7. Overview of permittivity feedback control.
movement affects both the piezoelectric displacement and the permittivity during the creep phenomenon. Therefore, it was expected that the creep property could be compensated by the permittivity feedback control. With a constant target displacement of +2 m or −2 m for the permittivity feedback control, the actual displacement was measured as shown in Fig. 11. In the case of a −2 m target displacement, the creep property was compensated as we expected, while it remained at 0.03 m with a target displacement of +2 m.
Fig. 8. Result of permittivity feedback control stepping every 1 m.
A piezoelectric actuator has a creep property in which the piezoelectric displacement increases gradually while a constant voltage is applied. We tried to demonstrate the effectiveness of the permittivity feedback control for eliminating this creep property. To confirm the creep property, a +30 V or −30 V DC voltage added to a +75 V offset voltage was applied without feedback control. As shown in Fig. 10, creep displacements of 0.07 m (+30 V) and 0.05 m (−30 V) were measured after 50 s. The current amplitude change, which is proportional to the permittivity change, had a similar tendency to the creep displacement. This is because the domain
Fig. 9. Result of open-loop control stepping every 1 m.
Fig. 10. Stage displacement and detection current change time when a constant voltage of +30 V (upper) or −30 V (lower) was applied.
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Based on these results, positioning control was performed with a target displacement from −4 m to +4 m with a 1 m step. The positioning error was reduced to 0.10 m from 0.45 m with openloop control. Permittivity feedback control was effective at reducing the creep property. The creep displacement remained at 0.03 m when +2 m was the target displacement. However, in the case of a −2 m target displacement, the creep was successfully compensated. Such a directional dependency is not reasonable because the positioning stage was designed symmetrically including the polarization directions of the two actuators. Therefore, this result implies that the two actuators had different creep properties. In this study, we adopted 0.1 Hz for the driving frequency. However, the stage should be driven with high speed in practical applications. Therefore, a suitable control system for high-speed driving should be realized. Moreover, we should study the creep property difference in each actuator and verify how the difference influenced the experimental results for the creep property, and in future works, we should compensate the temperature change effect. References
Fig. 11. Result of compensation for creep by permittivity feedback control.
However, even in the case of a +2 m target displacement, the amount of creep was reduced to 0.03 m from 0.07 m. 4. Conclusions The permittivity feedback control was effective for the positioning stage using two multilayered actuators. From the relationship between the piezoelectric displacement and the permittivity change and the differential measurement method, it was confirmed that the hysteresis displacement could be reduced to 0.1 m from 0.8 m, which was measured with open-loop control. In particular, the differential measurement method was useful for the multilayered actuators, similar to bimorph actuators.
[1] S. Devasia, E. Eleftheriou, S.O. Reza Moheimani, A survey of control issues in nanopositioning, IEEE Trans. Control Syst. Technol. 15 (5) (2007) 802–823. [2] M.A. Ealey, J.F. Washeba, Continuous facesheet low voltage deformable mirrors, Opt. Eng. 29 (1990) 1191–1198. [3] T.S. Low, W. Guo, Modeling of a three-layer piezoelectric bimorph beam with hysteresis, J. Microelectromech. Syst. 4 (4) (1995) 230–237. [4] M. Jang, C. Chen, J. Lee, Modeling and control of a piezoelectric actuator driven system with asymmetric hysteresis, J. Franklin Inst. 346 (2009) 17–32. [5] M. Rakotondrabe, Y. Haddab, P. Lutz, Nonlinear modeling and estimation of force in a piezoelectric cantilever, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2007, pp. 1–6. [6] K. Furutani, M. Urushibata, N. Mohri, Displacement control of piezoelectric element by feedback of induced charge, Nanotechnology 9 (2) (1998) 93–98. [7] I.A. Ivan, M. Rakotondrabe, P. Lutz, N. Chaillet, Quasistatic displacement selfsensing method for cantilevered piezoelectric actuators, Rev. Sci. Instrum. 80 (2009) 065102. [8] E. Uzgur, S. Kim, E. Bryce, D. Hutson, M. Strachan, K.J. Kirk, Extension sensing of piezo actuators using time-domain ultrasonic measurement and frequencydomain impedance measurement, J. Electroceram. 27 (2011) 38–44. [9] C.V. Newcomb, I. Flinn, Improving the linearity of piezoelectric ceramic actuators, Electron. Lett. 18 (11) (1982) 442–444. [10] A. Kawamata, Y. Kadota, H. Hosaka, T. Morita, Self-sensing piezoelectric actuator using permittivity detection, Ferroelectrics 368 (1) (2008) 194–201. [11] Y. Ishikiriyama, T. Morita, Improvement of self-sensing piezoelectric actuator control using permittivity change detection, J. Adv. Mech. Des. Syst. Manuf. vol. 4 (1) (2010) 143–149. [12] H. Ikeda, T. Morita, High-precision positioning using a self-sensing piezoelectric actuator control with a differential detection method, Sens. Actuators A: Phys. 170 (2011) 147–155. [13] F. Xu, S. Trolier-McKinstry, W. Ren, B. Xu, Z.L. Xie, K.J. Hemker, Domain wall motion and its contribution to the dielectric and piezoelectric properties of lead zirconate titanate films, J. Appl. Phys. 89 (2) (2001) 1336–1348.