Resonant Controller for Fast Atomic Force - Semantic Scholar
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Resonant Controller for Fast Atomic Force Microscopy Sajal. K. Das, Hemanshu. R. Pota and Ian. R. Petersen
Abstract— The imaging performance of the atomic force microscope (AFM) in higher scanning speed is limited to the one percent of the first resonant frequency of it’s scanning unit i.e., piezoelectric tube scanner (PTS). In order to speed up the functioning of the AFM for high speed imaging, a resonant controller with an integral action has been applied in the both x and y axis of the PTS for damping the resonant mode of the scanner and improve the tracking performance. The overall closed-loop system with this scheme has higher bandwidth with improved gain and phase margin than the existing PI controller. It can reduce the cross coupling of the scanner and allows faster scanning. To measure the performance improvement of the proposed scheme a comparison has been made between the proposed controller scanned image and the existing AFM PI controller scanned image.
I. I NTRODUCTION Scanning probe microscopes (SPM) are a group of instruments used for imaging and measuring the properties of materials and surfaces. There are basically two forms of SPM, scanning tunneling microscope (STM) and atomic force microscope (AFM). The STM can generate quality pictures only for conductors [1]. The AFM allows for a variety of surfaces both conducting or insulating to be imaged and characterized at an atomic level. It is extensively useing in many areas such as nano-lithiography [2], DNA nano technology and nano-fabrication [3]. The current AFM setting uses voltage source to move the PTS in the lateral and vertical directions. PTS is a thin walled tube with four electrodes in the lateral axis and one electrode in the vertical axis. Today the majority of the commercially available AFMs use PTS for x, y and z positioning because of its simplicity, large achievable scan range (>100 μm) and smaller capacitance. In spite of having such good qualities, there are some adverse effects of the PTS such as mechanical vibration, cross coupling, hysteresis, creep effects and thermal drift which inversely affects the scanning speed and limits the overall performance of the AFM [4]. Research work continues to model the dynamics of piezoelectric tube scanners and increase the scan speed of AFMs [5]. The scanning speed of an AFM is limited to about one percent of the first resonant frequency of the scanner [6]. In most of the AFMs, the first resonant frequency of the PTS is approximately 1 kHz. Therefore, scanning frequency higher than 10 Hz results distortion in image. Piezoelectric Sajal. K. Das, Hemanshu. R. Pota and Ian. R. Petersen are with the School of Engineering and Information Technology (SEIT), The University of New South Wales at Australian Defence Force Academy (UNSW@ADFA), Campbell, Canberra, Australian Capital Territory - 2600, Australia. e-mail: [email protected], h.pota @adfa.edu.au and [email protected]
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tube scanners also suffer from some nonlinearities such as hysteresis [4], creep effects [7] and there is a significant amount of cross coupling [8] between the axes of the PTS. Many feedback controllers have been already applied to damp the resonant mode of the PTS. Open-loop control techniques such as model based filtering [9] and feedforward control [10] have been utilized to damp the resonance of the scanning motion. However, in open-loop control it requires the accurate modelling of nonlinearities. Passive and active damping of a tube scanner has also been investigated in [11] and [12] respectively. A survey of detailed overview about damping controller can be found in [13]. An iterative control approach has been reported in [14] for compensating hysteresis and vibration as well as resonance of the piezo scanner. Though [14] achieves good tracking, implementation complexity and insensitivity to external disturbance make this approach unsuitable in many cases [7]. Integral resonant control (IRC) [6] is another technique to suppress the resonant mode of the PTS. In spite of providing good damping, it cannot eliminate drift and nonlinearities at low frequencies [7]. Positive position feedback control (PPF) [8] is a popular damping technique which is designed using a root locus technique where three unknown parameters are need to be determined for a second order controller [6]. A sector bound H ∞ control technique has also been applied in [4] to increase the scanning rate. However, their design methodology is computationally complex. In this work, we have proposed a high bandwidth resonant controller that is not only able to damp the resonant mode of the scanner but also can reduce the cross coupling between the axes of a PTS with improved gain and phase margin. The advantage of this controller over other damping controllers is that, the methodology used to design the controller is very simple and easy to implement. Two op-amps, three resistors, a capacitor and an inductor is sufficient to build such a controller. The rest of the paper is organized as follows. The experimental setup used for the present work is described in Section II. The identification and modeling of piezoelectric tube scanner using a black box system identification method is presented in Section III. The design and selection of controller are shown in Section IV, while Section V presents the overall performance of the proposed controller. A comparison of scanned images between proposed controller and the existing PI controller is given in Section VI. Finally, the paper is concluded with brief remarks in Section VII.
Dy (S) = Gdy vy S) = Vy (S)
III. S YSTEM I DENTIFICATION The dynamics of the piezoelectric tube actuator can be modeled either using conventional mathematical theory [5],[15] or using an experimental approach [8]. The current work uses an experimental approach to model the dynamics of the piezoelectric tube. In this approach, we used a dual channel spectrum analyzer HP35670A from where swept sine waves are generated and fed into the AFM scanner piezoelectric tube through the signal access module (SAM) and a high voltage amplifier. The generated signal from the frequency spectrum analyzer is a band limited noise signal of 10 to 2000 Hz. The displacement is measured using the capacitive sensor in this experiment. The piezoelectric tube has three capacitive sensors in the x, y and z axes. In this paper we consider only the x and y axes displacements. The following two transfer functions were fitted to the measured data. Dx (S) Gdx vx (S) = = Vx (S)
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The experimental setup used to implement the proposed controller at the University of New South Wales at Australian Defence Force Academy in Canberra is shown in Fig. 1. It consists of a frequency spectrum analyzer, high voltage amplifier (HVA), signal access module (SAM) and a dSPACE board. A frequency spectrum analyzer is used to generate band limited noise signal and that signal is applied to the piezoelectric tube scanner through the HVA and SAM. When the AFM is operated with its existing internal PI controller, it uses its internal high voltage amplifier and generates own signal for scanning. The output of internal PI controller of AFM is marked as LV-X and LV-Y for the X and Y piezo respectively in SAM. These signals can be used as a reference signal while implementing the external controller for scanning. The scanning signal also can be generated using dSPACE.
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Fig. 2. Measured and identified model frequency response of the x sensor output. 20 0 −20 −40 1 10
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II. E XPERIMENTAL S ETUP
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SPM system and experimental setup used in this work.
(2) Here Dx (S) is the output voltage from the x sensor and Vx (S) is the input to the high voltage amplifier. Similarly Dy (S) is the output from the y sensor and Vy (S) is the input to the high voltage amplifier (HV). To obtain this model, we use the prediction error method [16] from the Matlab system identification toolbox which estimates the model parameters by minimizing the optimally determined output prediction error. We consider only the first resonant mode of the piezoelectric tube scanner in our identification approach. Fig. 2 and Fig. 3 show the matching of open-loop frequency response between the measured data and the identified model for the x and y sensor outputs. It is seen that there is a good match between the measured and the identified model frequency response. It should be noted here that, there is about a 180◦ phase shift between the input and output in the frequency response, both for the x and y sensor outputs.
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Fig. 3. Measured and identified model frequency response of the y sensor output.
IV. C ONTROLLER D ESIGN AND S ELECTION The block diagram of the overall closed-loop system is shown in Fig. 4 where Gw (S) is the plant model and H1 (S), H2 (S) ... Hk (S) are the resonant controllers for different resonant modes. H1 (S) is to damp the first resonant mode, H2 (S) is to damp the second resonant mode and similarly Hk (S) is to damp the kth resonant mode. This approach was first introduced in [17], [18] to suppress the mechanical vibrations in a cantilever beam. Since we consider only to damp the first resonant mode of the PTS, we use H1 (S) as the resonant controller. 5HIHUHQFH ,QSXW
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If the damping resistor RR is not chosen carefully there may be undesirable phase shift in the closed-loop response of the system. Choosing a smaller value of RR introduces a notch in the system near the resonance frequency which decreases the system performance and stability. Again system may not have good damping by choosing a large value of the damping resistor. Therefore, we have a trade-off for choosing a good value of damping resistor so that we get a reasonable damping which can reduce vibration of the piezoelectric scanner as well as increase system stability, performance and makes it robust. The value of the damping resistor RR is chosen 250 Ω in the present work. As the cross coupling has a major effect on image rotation, we have introduced a high gain integrator in the feedforward path to reduce the effect of the cross coupling of the PTS. The use of high gain integrator in the feedforward path is possible because the resonant controller decays the resonant mode. We select the integrator gain (KI ) -1200 by using the root locus technique to achieve maximum damping with high closed-loop system bandwidth and reasonable gain and phase margin.
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Each resonant controller can be realized with the circuit shown in Fig. 5. The general form of transfer function of resonant controller circuit is as follows
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−gw CR s(RR + LR s) LR C R s 2 + R R CR s + 1 where gw is the gain of the controller. The trick with this controller is to tune it to the system resonant frequency and the key feature of this controller is its simplicity of design. By selecting the proper value of LR , RR , CR we are able to control the resonant of the piezoelectric tube scanner as well as vibration. Since ωr = √L 1 C , the value of LR and CR is chosen such R R that ωr is equal to or nearly equal to the resonant frequency of the system. The resonant frequencies of the PTS in both x and y axes are around 900 Hz which is equivalent to 5600 rad approximately. Therefore we select CR = 1μF, LR = 0.0319 H and gw = −1 (since negative feedback). The proper designing of the controller depends on selecting the proper value of the parameter RR . Selecting the value of the damping resistor is a key design step for getting a good closed-loop response.
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(b) Fig. 6. Comparison of the open-loop and closed-loop frequency response of the x and y sensor output (a) input to the x piezo, output from the x sensor, (b) input to the y piezo, output from the y sensor.
V. P ERFORMANCE OF THE OVERALL RESULTING C ONTROLLER
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A. Frequency domain performance To measure the performance of the proposed control scheme, we measured the open and closed-loop frequency response of both x and y sensors output of the scanner by using the spectrum analyzer. Fig. 6 shows the comparison of the open and closed-loop frequency response of sensors output. It can be seen that there is about more than 20 dB damping in the closed-loop system which in turns reduces the oscillation or vibration significantly. Moreover, the closedloop system bandwidth is more than 300 Hz in the x axis and 400 Hz in the y axis that ensures fast scanning.
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(b) Fig. 7. Comparison of the open-loop and closed-loop cross coupling for x and y sensors output (a) input to the y piezo, output from the x sensor, (b) input to the x piezo, output from the y sensor.
The comparison between the open-loop and closed-loop frequency response of the cross coupling between the axes is shown in Fig. 7. It is seen that there is a reduction in the cross coupling for x and y sensors. The nominal value of the closed-loop cross coupling for both axes is around −50 dB which signifies that there is almost no coupling effect between the axes. Therefore, each axis can be treated as an independent axis, i.e., single-input-single-output (SISO)
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system. The resonance in both cases of the cross coupling has been reduced significantly. B. Performance comparison between the resonant controller with integral action and the integral controller The comparison of y sensor output frequency response using the integral controller only and the integral controller with the resonant controller is shown in Fig. 8. The gain of the integral controller in both cases was chosen -1200. It can be seen that, integral controller with resonant controller is able to achieve more damping and has higher bandwidth than the integral controller only.
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(c) (d) Fig. 11. 10.42 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
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The main objective of this paper is to achieve nondistorted, nontilting good quality scan images at a higher scanning frequency compared with the existing AFM PI controller and in order to do so we first implement the resonant controller only for sample scanning. The scanned images obtained by using only the resonant controller is good in quality but seems to be tilted (see Fig. 10(a)).
A. Imaging performance of the resonant controller
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VI. I MAGE S CANNING R ESULTS
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The experimental tracking performance of the proposed controller is examined by tracking a 5.21 Hz reference triangular signal generated from the AFMs software for 8μm×8μm scan. As our system has about 180◦ phase shift between input and output signal in open-loop, hence the x sensor output signal is an inverted input signal. But to show the tracking performance of the open-loop system we purposely multiply the sensor output signal by -1 as shown in Fig. 9(a). The experimental closed-loop tracking performance is shown in Fig. 9(c) from where we can see that the proposed control scheme has achieved good control over tracking of reference signal. Fig. 9(b) and Fig. 9(d) show the comparison of the open and closed-loop cross coupling for 5.21 Hz and 31.25 Hz reference input and the comparison shows that there is almost no cross coupling between the axes in closed-loop. To measure the cross coupling, a reference triangular signal was applied to the x axis of the piezo and the output was taken from the y sensor.
controller in imaging capability. The experimental images presented in Fig. 11 to Fig. 14 show a comparison between the existing PI controller scanned image and the proposed controller scanned image at different scanning frequencies. The comparison shows the effectiveness of the proposed controller in image scanning also. The sample used for each imaging is a TGQ1 grating reference sample and the controller was implemented in x and y direction simultaneously in a dSPACE DSP control system. The reference signals applied to both x and y directions of the piezoelectric tube scanner are generated from AFM software.
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C. Experimental tracking performance
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Fig. 10. Comparison of 10.42 Hz scanned images using (a) resonant controller only and (b) resonant controller with an integral action.
B. Imaging performance of the proposed controller After improving the lateral positioning of the PTS, we moved to investigate the overall performance of the proposed
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The reason of the image rotation is the cross coupling between the axes of the scanner. The resonant controller has zero dc gain, it is used only to damp the resonant mode of the PTS. To reduce the effect of cross coupling between the axes of the PTS we have added an integrator with the resonant controller in such a place that the overall closedloop system has a higher bandwidth with improved gain and phase margin.
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(c) (d) Fig. 12. 31.25 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
performance achieved in this experiment is just by damping the first resonant mode of the x and y axes of the scanner, our future work will deal with the resonant mode damping of the z axis of the scanner and minimization of the cross coupling effect between x, y and z axes of the PTS.
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ACKNOWLEDGMENT
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The authors would like to thank Mr. Shane Brandon of the School of Engineering and Information Technology, UNSW@ADFA for his technical assistance in the experiment. R EFERENCES
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Fig. 13. 62.5 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
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Fig. 14. 125 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
VII. C ONCLUSION In this paper, the resonant controller was applied to damp the first resonant mode of the piezoelectric tube scanner. An integral action was added with the resonant controller in order to provide gain at low frequencies. Compared with the standard integral controller, the proposed controller can achieve larger bandwidth because the resonant controller can take care of the resonant mode of the scanner. The stability margin is also improved with the proposed controller compared with the standard integral controller. The controller has only a second order transfer function and can be easily implemented with simple analog circuit. Since the
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[1] G. Binnig and H. Rohrer, “Scanning tunneling microscopy from birth to adolescence,” Rev. Mod. Phys., vol. 59, pp. 615–625, Jul 1987. [2] D. L. White and O. R. Wood, “Novel alignment system for imprint lithography,” J. of Vacuum Sci. & Tech. B: Microelectronics and Nanometer Struct., vol. 18, no. 6, pp. 3552–3556, Nov 2000. [3] J. An, K. Xue, and J. B. Xu, “Fabrication hfox nanopatterns by atomic force microscopy lithography,” in Proc. IEEE Int. Conf. Electron Devices and Solid-State Circuits EDSSC, pp. 1–4, 2008. [4] N. Chuang, I. R. Petersen, and H. R. Pota, “Robust H ∞ control in fast atomic force microscopy,” in Proc. American Control Conf. (ACC), pp. 2258–2265, 2011. [5] M. E. Taylor, “Dynamics of piezoelectric tube scanners for scanning probe microscopy,” Rev. of Sci. Instrum., vol. 64, no. 1, pp. 154–158, 1993. [6] B. Bhikkaji and S. O. Moheimani, “Integral resonant control of a piezoelectric tube actuator for fast nanoscale positioning,” IEEE/ASME Trans. on Mech., vol. 13, no. 5, pp. 530–537, 2008. [7] A. J. Fleming, S. S. Aphale, and S. O. R. Moheimani, “A new method for robust damping and tracking control of scanning probe microscope positioning stages,” IEEE Trans. on Nanotechnology, vol. 9, no. 4, pp. 438–448, 2010. [8] I. A. Mahmood and S. O. R. Moheimani, “Making a commercial atomic force microscope more accurate and faster using positive position feedback control,” Rev. of Sci. Instrum., vol. 80, no. 6, pp. 063 705(1)–063 705(8), 2009. [9] D. Croft, G. Shedd, and S. Devasia, “Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application,” in Proc. American Control Conf, pp. 2123–2128, 2000. [10] K. Leang, Q. Zou, and S. Devasia, “Feedforward control of piezoactuators in atomic force microscope systems,” IEEE Control Sys. Mag., vol. 29, no. 1, pp. 70–82, 2009. [11] A. J. Fleming and S. O. R. Moheimani, “Sensorless vibration suppression and scan compensation for piezoelectric tube nanopositioners,” IEEE Trans. on Control Sys. Tech., vol. 14, no. 1, pp. 33–44, 2006. [12] N. Kodera, H. Yamashita, and T. Ando, “Active damping of the scanner for high-speed atomic force microscopy,” Rev. of Sci. Instrum., vol. 76, no. 5, pp. 053 708(1)–053 708(5), 2005. [13] S. Devasia, E. Eleftheriou, and S. Moheimani, “A survey of control issues in nanopositioning,” IEEE Trans. on Control Sys. Tech., vol. 15, no. 5, pp. 802–823, Sep. 2007. [14] S. Tien, Q. Zou, and S. Devasia, “Iterative control of dynamicscoupling-caused errors in piezoscanners during high-speed afm operation,” IEEE Trans. on Control Sys. Tech., vol. 13, no. 6, pp. 921–931, 2005. [15] G. Schitter, “Advanced mechanical design and control methods for atomic force microscopy in real-time,” in Proc. American Control Conf., pp. 3503–3508, Jul, 2007. [16] D. S. Bayard, F. Y. Hadaegh, Y. Yam, R. E. Scheid, E. Mettler, and M. H. Milman, “Frequency domain identification experiment on a large flexible structure,” in Proc. American Control Conf, pp. 2532– 2544, 1989. [17] H. Pota, S. Reza Moheimani, and M. Smith, “Resonant controller for smart structures,” Smart Materials and Structures, vol. 11, pp. 1–8, 2002. [18] H. Pota, S. R. Moheimani, and M. Smith, “Resonant controllers for flexible structures,” in Proc. Conf. of Decision and Control, vol. 1, pp. 631–636, 1999.
Resonant Controller for Fast Atomic Force Microscopy Sajal. K. Das, Hemanshu. R. Pota and Ian. R. Petersen
Abstract— The imaging performance of the atomic force microscope (AFM) in higher scanning speed is limited to the one percent of the first resonant frequency of it’s scanning unit i.e., piezoelectric tube scanner (PTS). In order to speed up the functioning of the AFM for high speed imaging, a resonant controller with an integral action has been applied in the both x and y axis of the PTS for damping the resonant mode of the scanner and improve the tracking performance. The overall closed-loop system with this scheme has higher bandwidth with improved gain and phase margin than the existing PI controller. It can reduce the cross coupling of the scanner and allows faster scanning. To measure the performance improvement of the proposed scheme a comparison has been made between the proposed controller scanned image and the existing AFM PI controller scanned image.
I. I NTRODUCTION Scanning probe microscopes (SPM) are a group of instruments used for imaging and measuring the properties of materials and surfaces. There are basically two forms of SPM, scanning tunneling microscope (STM) and atomic force microscope (AFM). The STM can generate quality pictures only for conductors [1]. The AFM allows for a variety of surfaces both conducting or insulating to be imaged and characterized at an atomic level. It is extensively useing in many areas such as nano-lithiography [2], DNA nano technology and nano-fabrication [3]. The current AFM setting uses voltage source to move the PTS in the lateral and vertical directions. PTS is a thin walled tube with four electrodes in the lateral axis and one electrode in the vertical axis. Today the majority of the commercially available AFMs use PTS for x, y and z positioning because of its simplicity, large achievable scan range (>100 μm) and smaller capacitance. In spite of having such good qualities, there are some adverse effects of the PTS such as mechanical vibration, cross coupling, hysteresis, creep effects and thermal drift which inversely affects the scanning speed and limits the overall performance of the AFM [4]. Research work continues to model the dynamics of piezoelectric tube scanners and increase the scan speed of AFMs [5]. The scanning speed of an AFM is limited to about one percent of the first resonant frequency of the scanner [6]. In most of the AFMs, the first resonant frequency of the PTS is approximately 1 kHz. Therefore, scanning frequency higher than 10 Hz results distortion in image. Piezoelectric Sajal. K. Das, Hemanshu. R. Pota and Ian. R. Petersen are with the School of Engineering and Information Technology (SEIT), The University of New South Wales at Australian Defence Force Academy (UNSW@ADFA), Campbell, Canberra, Australian Capital Territory - 2600, Australia. e-mail: [email protected], h.pota @adfa.edu.au and [email protected]
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
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tube scanners also suffer from some nonlinearities such as hysteresis [4], creep effects [7] and there is a significant amount of cross coupling [8] between the axes of the PTS. Many feedback controllers have been already applied to damp the resonant mode of the PTS. Open-loop control techniques such as model based filtering [9] and feedforward control [10] have been utilized to damp the resonance of the scanning motion. However, in open-loop control it requires the accurate modelling of nonlinearities. Passive and active damping of a tube scanner has also been investigated in [11] and [12] respectively. A survey of detailed overview about damping controller can be found in [13]. An iterative control approach has been reported in [14] for compensating hysteresis and vibration as well as resonance of the piezo scanner. Though [14] achieves good tracking, implementation complexity and insensitivity to external disturbance make this approach unsuitable in many cases [7]. Integral resonant control (IRC) [6] is another technique to suppress the resonant mode of the PTS. In spite of providing good damping, it cannot eliminate drift and nonlinearities at low frequencies [7]. Positive position feedback control (PPF) [8] is a popular damping technique which is designed using a root locus technique where three unknown parameters are need to be determined for a second order controller [6]. A sector bound H ∞ control technique has also been applied in [4] to increase the scanning rate. However, their design methodology is computationally complex. In this work, we have proposed a high bandwidth resonant controller that is not only able to damp the resonant mode of the scanner but also can reduce the cross coupling between the axes of a PTS with improved gain and phase margin. The advantage of this controller over other damping controllers is that, the methodology used to design the controller is very simple and easy to implement. Two op-amps, three resistors, a capacitor and an inductor is sufficient to build such a controller. The rest of the paper is organized as follows. The experimental setup used for the present work is described in Section II. The identification and modeling of piezoelectric tube scanner using a black box system identification method is presented in Section III. The design and selection of controller are shown in Section IV, while Section V presents the overall performance of the proposed controller. A comparison of scanned images between proposed controller and the existing PI controller is given in Section VI. Finally, the paper is concluded with brief remarks in Section VII.
Dy (S) = Gdy vy S) = Vy (S)
III. S YSTEM I DENTIFICATION The dynamics of the piezoelectric tube actuator can be modeled either using conventional mathematical theory [5],[15] or using an experimental approach [8]. The current work uses an experimental approach to model the dynamics of the piezoelectric tube. In this approach, we used a dual channel spectrum analyzer HP35670A from where swept sine waves are generated and fed into the AFM scanner piezoelectric tube through the signal access module (SAM) and a high voltage amplifier. The generated signal from the frequency spectrum analyzer is a band limited noise signal of 10 to 2000 Hz. The displacement is measured using the capacitive sensor in this experiment. The piezoelectric tube has three capacitive sensors in the x, y and z axes. In this paper we consider only the x and y axes displacements. The following two transfer functions were fitted to the measured data. Dx (S) Gdx vx (S) = = Vx (S)
2
6
−186.6s +1.348×10 s−2.412×10
Magnitude (dB)
The experimental setup used to implement the proposed controller at the University of New South Wales at Australian Defence Force Academy in Canberra is shown in Fig. 1. It consists of a frequency spectrum analyzer, high voltage amplifier (HVA), signal access module (SAM) and a dSPACE board. A frequency spectrum analyzer is used to generate band limited noise signal and that signal is applied to the piezoelectric tube scanner through the HVA and SAM. When the AFM is operated with its existing internal PI controller, it uses its internal high voltage amplifier and generates own signal for scanning. The output of internal PI controller of AFM is marked as LV-X and LV-Y for the X and Y piezo respectively in SAM. These signals can be used as a reference signal while implementing the external controller for scanning. The scanning signal also can be generated using dSPACE.
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Fig. 2. Measured and identified model frequency response of the x sensor output. 20 0 −20 −40 1 10
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II. E XPERIMENTAL S ETUP
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SPM system and experimental setup used in this work.
(2) Here Dx (S) is the output voltage from the x sensor and Vx (S) is the input to the high voltage amplifier. Similarly Dy (S) is the output from the y sensor and Vy (S) is the input to the high voltage amplifier (HV). To obtain this model, we use the prediction error method [16] from the Matlab system identification toolbox which estimates the model parameters by minimizing the optimally determined output prediction error. We consider only the first resonant mode of the piezoelectric tube scanner in our identification approach. Fig. 2 and Fig. 3 show the matching of open-loop frequency response between the measured data and the identified model for the x and y sensor outputs. It is seen that there is a good match between the measured and the identified model frequency response. It should be noted here that, there is about a 180◦ phase shift between the input and output in the frequency response, both for the x and y sensor outputs.
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Fig. 3. Measured and identified model frequency response of the y sensor output.
IV. C ONTROLLER D ESIGN AND S ELECTION The block diagram of the overall closed-loop system is shown in Fig. 4 where Gw (S) is the plant model and H1 (S), H2 (S) ... Hk (S) are the resonant controllers for different resonant modes. H1 (S) is to damp the first resonant mode, H2 (S) is to damp the second resonant mode and similarly Hk (S) is to damp the kth resonant mode. This approach was first introduced in [17], [18] to suppress the mechanical vibrations in a cantilever beam. Since we consider only to damp the first resonant mode of the PTS, we use H1 (S) as the resonant controller. 5HIHUHQFH ,QSXW
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If the damping resistor RR is not chosen carefully there may be undesirable phase shift in the closed-loop response of the system. Choosing a smaller value of RR introduces a notch in the system near the resonance frequency which decreases the system performance and stability. Again system may not have good damping by choosing a large value of the damping resistor. Therefore, we have a trade-off for choosing a good value of damping resistor so that we get a reasonable damping which can reduce vibration of the piezoelectric scanner as well as increase system stability, performance and makes it robust. The value of the damping resistor RR is chosen 250 Ω in the present work. As the cross coupling has a major effect on image rotation, we have introduced a high gain integrator in the feedforward path to reduce the effect of the cross coupling of the PTS. The use of high gain integrator in the feedforward path is possible because the resonant controller decays the resonant mode. We select the integrator gain (KI ) -1200 by using the root locus technique to achieve maximum damping with high closed-loop system bandwidth and reasonable gain and phase margin.
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Measured Open Loop Measured Close Loop 10 Frequency (Hz)
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−gw CR s(RR + LR s) LR C R s 2 + R R CR s + 1 where gw is the gain of the controller. The trick with this controller is to tune it to the system resonant frequency and the key feature of this controller is its simplicity of design. By selecting the proper value of LR , RR , CR we are able to control the resonant of the piezoelectric tube scanner as well as vibration. Since ωr = √L 1 C , the value of LR and CR is chosen such R R that ωr is equal to or nearly equal to the resonant frequency of the system. The resonant frequencies of the PTS in both x and y axes are around 900 Hz which is equivalent to 5600 rad approximately. Therefore we select CR = 1μF, LR = 0.0319 H and gw = −1 (since negative feedback). The proper designing of the controller depends on selecting the proper value of the parameter RR . Selecting the value of the damping resistor is a key design step for getting a good closed-loop response.
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(b) Fig. 6. Comparison of the open-loop and closed-loop frequency response of the x and y sensor output (a) input to the x piezo, output from the x sensor, (b) input to the y piezo, output from the y sensor.
V. P ERFORMANCE OF THE OVERALL RESULTING C ONTROLLER
20 Magnitude (dB)
A. Frequency domain performance To measure the performance of the proposed control scheme, we measured the open and closed-loop frequency response of both x and y sensors output of the scanner by using the spectrum analyzer. Fig. 6 shows the comparison of the open and closed-loop frequency response of sensors output. It can be seen that there is about more than 20 dB damping in the closed-loop system which in turns reduces the oscillation or vibration significantly. Moreover, the closedloop system bandwidth is more than 300 Hz in the x axis and 400 Hz in the y axis that ensures fast scanning.
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Fig. 8. Comparison of the measured y sensor output frequency response between integral controller and resonant integral controller.
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(b) Fig. 7. Comparison of the open-loop and closed-loop cross coupling for x and y sensors output (a) input to the y piezo, output from the x sensor, (b) input to the x piezo, output from the y sensor.
The comparison between the open-loop and closed-loop frequency response of the cross coupling between the axes is shown in Fig. 7. It is seen that there is a reduction in the cross coupling for x and y sensors. The nominal value of the closed-loop cross coupling for both axes is around −50 dB which signifies that there is almost no coupling effect between the axes. Therefore, each axis can be treated as an independent axis, i.e., single-input-single-output (SISO)
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system. The resonance in both cases of the cross coupling has been reduced significantly. B. Performance comparison between the resonant controller with integral action and the integral controller The comparison of y sensor output frequency response using the integral controller only and the integral controller with the resonant controller is shown in Fig. 8. The gain of the integral controller in both cases was chosen -1200. It can be seen that, integral controller with resonant controller is able to achieve more damping and has higher bandwidth than the integral controller only.
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(c) (d) Fig. 11. 10.42 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
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The main objective of this paper is to achieve nondistorted, nontilting good quality scan images at a higher scanning frequency compared with the existing AFM PI controller and in order to do so we first implement the resonant controller only for sample scanning. The scanned images obtained by using only the resonant controller is good in quality but seems to be tilted (see Fig. 10(a)).
A. Imaging performance of the resonant controller
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VI. I MAGE S CANNING R ESULTS
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The experimental tracking performance of the proposed controller is examined by tracking a 5.21 Hz reference triangular signal generated from the AFMs software for 8μm×8μm scan. As our system has about 180◦ phase shift between input and output signal in open-loop, hence the x sensor output signal is an inverted input signal. But to show the tracking performance of the open-loop system we purposely multiply the sensor output signal by -1 as shown in Fig. 9(a). The experimental closed-loop tracking performance is shown in Fig. 9(c) from where we can see that the proposed control scheme has achieved good control over tracking of reference signal. Fig. 9(b) and Fig. 9(d) show the comparison of the open and closed-loop cross coupling for 5.21 Hz and 31.25 Hz reference input and the comparison shows that there is almost no cross coupling between the axes in closed-loop. To measure the cross coupling, a reference triangular signal was applied to the x axis of the piezo and the output was taken from the y sensor.
controller in imaging capability. The experimental images presented in Fig. 11 to Fig. 14 show a comparison between the existing PI controller scanned image and the proposed controller scanned image at different scanning frequencies. The comparison shows the effectiveness of the proposed controller in image scanning also. The sample used for each imaging is a TGQ1 grating reference sample and the controller was implemented in x and y direction simultaneously in a dSPACE DSP control system. The reference signals applied to both x and y directions of the piezoelectric tube scanner are generated from AFM software.
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C. Experimental tracking performance
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Fig. 10. Comparison of 10.42 Hz scanned images using (a) resonant controller only and (b) resonant controller with an integral action.
B. Imaging performance of the proposed controller After improving the lateral positioning of the PTS, we moved to investigate the overall performance of the proposed
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The reason of the image rotation is the cross coupling between the axes of the scanner. The resonant controller has zero dc gain, it is used only to damp the resonant mode of the PTS. To reduce the effect of cross coupling between the axes of the PTS we have added an integrator with the resonant controller in such a place that the overall closedloop system has a higher bandwidth with improved gain and phase margin.
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(c) (d) Fig. 12. 31.25 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
performance achieved in this experiment is just by damping the first resonant mode of the x and y axes of the scanner, our future work will deal with the resonant mode damping of the z axis of the scanner and minimization of the cross coupling effect between x, y and z axes of the PTS.
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ACKNOWLEDGMENT
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The authors would like to thank Mr. Shane Brandon of the School of Engineering and Information Technology, UNSW@ADFA for his technical assistance in the experiment. R EFERENCES
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Fig. 14. 125 Hz scanned images using the existing AFM PI controller (a) 2D-image, (c) 3D-image and using the proposed controller (b) 2D-image (d) 3D-image.
VII. C ONCLUSION In this paper, the resonant controller was applied to damp the first resonant mode of the piezoelectric tube scanner. An integral action was added with the resonant controller in order to provide gain at low frequencies. Compared with the standard integral controller, the proposed controller can achieve larger bandwidth because the resonant controller can take care of the resonant mode of the scanner. The stability margin is also improved with the proposed controller compared with the standard integral controller. The controller has only a second order transfer function and can be easily implemented with simple analog circuit. Since the
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