Identification and Open-Loop Tracking Control of a ... - IEEE Xplore

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004

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Identification and Open-Loop Tracking Control of a Piezoelectric Tube Scanner for High-Speed Scanning-Probe Microscopy Georg Schitter and Andreas Stemmer

Abstract—Fast and precise positioning is a basic requirement for nanotechnology applications. Many scanning-probe microscopes (SPM) use a piezoelectric tube scanner for actuation with nanometer resolution in all three spatial directions. Due to the dynamics of the actuator, the imaging speed of the SPM is limited. By applying model-based open-loop control, the dynamic behavior of the scanner can be compensated, reducing the displacement error, topographical artifacts, modulation of the interaction force, and modulation of the relative tip-sample velocity. The open-loop controlled system enables imaging of up to 125- m-sized samples at a line scan rate of 122 Hz, which is about 15 times faster than the commercial system. Index Terms—Atomic force microscopy, fast scanning, friction force, piezoelectric transducers, scanning probe, tracking control.

I. INTRODUCTION

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CANNING-PROBE microscopy (SPM), such as atomic-force microscopy (AFM) and friction-force microscopy (FFM), is a technique to measure spatially resolved sample properties with nanometer resolution by scanning the sample laterally in close vicinity of a probing tip [1]. Usually the interaction parameter between the probing tip and the specimen is held constant by closed-loop operation. In AFM, the topography of a specimen can be measured by recording the lateral displacement of the scanning unit in combination with the deflection signal of a micromechanical cantilever supporting the scanning-probe tip [2]. AFM experiments can be done in tapping [3], noncontact [4], constant-height [5], or constant-force modes [6]. Friction experiments and topography measurements are made in the constant-force mode [7], [8], also referred to as contact mode, where the AFM tip is permanently in contact with the sample. The feedback controller compensates for variations in the cantilever deflection due to changes in the interaction force by moving the tip or sample up and down. Thus, the tiploading force is held constant, allowing for the measurement of the local friction coefficient by recording the torque of the cantilever at constant force and constant scanning velocity [8]. Nanotribology is an important field of research, but friction experiments at the nanometer scale using the AFM/FFM are limited to speeds lower than 300 m/s due to the dynamic behavior of the scanning unit (e.g., [9]). Speeding up the AFM/FFM enables friction experiments at technically relevant speeds in the mm/s range.

Manuscript received July 30, 2002; revised March 4, 2003. Manuscript received in final form May 5, 2003. Recommended by Associate Editor K. Kozlowski. The authors are with the Nanotechnology Group, Swiss Federal Institute of Technology Zurich, Zurich CH-8092, Switzerland (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCST.2004.824290

Most SPMs use a piezoelectric tube scanner [10] for actuation in all three spatial directions to perform the scanning motion and to enable closed-loop control of the tip–sample interaction. The lateral scan is usually done by applying a triangular voltage signal to the piezo tube, inducing a linear motion at a constant velocity, which is required for friction measurements. A limitation of the imaging speed is given by the lateral dynamics of the piezoelectric scanner. First efforts to improve scanning speed have been done by compensating the lateral dynamics of a scanning piezo by an optimal inverse approach [11], [12].This article presents a method to suppress the lateral oscillations of the scanning piezo by means of low-order open-loop control. Using suboptimal control is insensitive against model techniques such as uncertainties and enables order reduction of the controller model to save computation time. Due to a higher order of the controller model, the optimal inverse approach may take more computational power, which is required for other applications running simultaneously on the same signal processor, such as a model-based closed-loop controller to operate the AFM in the vertical direction [13]. A mathematical model of the piezo scanner’s dynamics is shown in Section II, which was obtained by a black-box identification [14] using subspace methods [15], [16]. In the third section, the design and implementation of a model-based loworder open-loop controller is presented, which was calculated methods. A comparison of the simulated and meautilizing sured tracking accuracy of the new open-loop controlled piezo and the uncompensated one is shown in Section IV and experimentally confirmed in Section V. Our results demonstrate that images can be recorded at scan rates up to 122 Hz, a factor that is about 15 times higher than usual. Using the open-loop controller, the artifacts in the topography and friction signal vanish, thereby enabling scanning speed in the mm/s range. II. IDENTIFICATION OF THE PIEZO SCANNER DYNAMICS System identification is a control-engineering tool to calculate a mathematical model of a dynamical system from a set of input and output data [14]. In the case of the piezoelectric tube scanner, a system identification is performed in both scanning directions (X and Y), which form the plane perpendicular to the scanner’s axis of symmetry (Z). A. Equipment Tests are performed on an AFM system (Nanoscope IIIa, Veeco) to identify a “J”-class piezoelectric tube scanner (125 m nominal scan range) in the X and Y directions. In this commercial system, the sample is scanned and the measurement

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Fig. 1. Block diagram of the measurement circuit used for the identification of the piezoscanner dynamics.

head supporting the cantilever is fixed. During the identification experiment, the measurement head of the AFM has to be removed. The input of the system is the voltage amplifier of the scanning direction to be identified. The output of the system is the movement of the piezoelectric tube scanner along the scanning direction to be identified and is measured with a capacitive displacement sensor of type PI D-015.00 (Physik Instrumente,Karlsruhe Germany). This sensor is mounted on top of the piezoelectric tube instead of the AFM measurement head. A band-limited white-noise signal is chosen to excite the piezoelectric tube scanner. White noise has a constant power spectrum, which implies that the piezoelectric tube scanner is excited at the frequencies within the bandwidth of interest. The input data are generated by a digital signal processor (DSP) system [17]. The sampling rate of the DSP is set to 80 ksample/s. The monitor signal of the sensor electronics representing the system’s response to the applied input signal is recorded by the DSP system simultaneously with the input data. Fig. 1 shows the measurement setup consisting of the piezoelectric tube, the capacitive sensor, and the DSP system used for the system identification. The resolution of the DSP’s A/D and D/A converter is 16 bit [17], which is the same resolution as that of the commercial AFM system. The lateral resolution of the scanning system can be calculated by mapping the scanner’s nominal lateral range of 125 m to the 16 bit, while the least significant bit has to be rejected. resolution

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B. System Identification From the set of measured input and output data, we calculated a linear model using a numerical subspace-based state space system-identification algorithm [15], [16]. The model order was chosen to be 5, because a higher model order did not provide substantial reduction of the modeling

Fig. 2. Comparison of the measured system output (solid line) and the simulated output of the identified system (dashed line) in response to a band-limited white-noise signal using a set of validation data.

error, but increased the calculation time of the model. The quality of the obtained model was tested by a comparison of the simulated model output to the measured output using a set of validation data (Fig. 2), which were not used for the model identification and were measured separately from the identification data. As shown by Croft et al. [12] and confirmed by our identification experiments, the dynamic behavior of the piezoelectric tube scanner along the scanning directions can be modeled linearly (compare Fig. 2). The nonlinearities of the piezo, such as hysteresis and creep, do not affect the dynamic behavior and could be compensated separately [12]. In the case of our piezoelectric tube scanner, hysteresis and creep are compensated electronically by the commercial system at the generation of the driving signal. The identified model of the piezoelectric tube scanner’s dynamics in the X direction consists of a low pass at 527 Hz, a at 4.7 kHz, and a very moderately damped resonance at 710 Hz, as shown in weakly damped resonance (2) at the bottom of the page. The nonminimum phase-transmission zeroes are artifacts due to the identification and sampling of the system. The model obtained from the identification in the Y direction looks quite similar and shows a low pass at 905 Hz, a moderately damped res, and a very weakly damped resoonance at 4.2 kHz , as shown in (3) at the bottom of nance at 700 Hz the next page. This resonant behavior impairs the tracking accuracy of the scanning piezo and can be compensated, which is the focus of the next section. Furthermore, these lateral resonances cause imaging artifacts due to crosstalk between the scanning direction and the Z direction. The crosstalk between these axes is composed of the coupling within the piezoelectric tube and the

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lateral bending of the cantilever [18]. Compensating the dynamic behavior of the piezoelectric tube scanner along the fast scanning direction reduces the crosstalk-induced errors caused by the piezo’s lateral oscillations (see Section V). III. CONTROLLER DESIGN The speed of the scanning motion of the piezoelectric tube scanner can be determined from the scan range, given in micrometers, and the scan rate, which is the frequency of the triangular scanning signal along the fast scanning direction, given in hertz. In the following, the design and results are presented for the fast scan direction in AFM/FFM, normally employed perpendicular to the cantilever. The procedure for the second scanning direction is the same and is implemented on another channel of the DSP system. At scan rates higher than 8 Hz, the piezoelectric tube scanner starts to oscillate adversely along the fast scanning direction, because the higher frequency components of the triangular scanning signal already excite the resonance of the scanning piezo (see Fig. 4). These oscillations modulate the relative tip–sample velocity and cause a substantial tracking error, impeding locally resolved measurements. Tracking accuracy can be improved using model-based control methods. Closed-loop control cannot be used due to the lack of a sensor to measure the lateral position of the employed piezoelectric tube scanner. An optimal feedforward trajectory [11] cannot be applied to the commercial system we use because we do not have access to the DSP system generating the scan trajectory and combining the scan signals and the closed-loop signal to record and illustrate the measured topography, cantilever deflection, and friction. We chose to implement a low-order realization of the new controller to save computing power and to consider a cheap solution for future implementation, such as a switched-capacitor filter. The design of a linear open-loop controller is chosen, norm [19], owing which is based on the minimization of the to the ability to specify requirements to the tracking accuracy -based controllers against model unand the insensitivity of certainties. These requirements have to be defined in the form of mathematical equations (weights) by which the model of the is extended. The structure of the piezoelectric tube scanner extended mathematical model is shown in Fig. 3. Model uncertainties can occur due to varying sample mass, which shifts the resonance frequency of the system. For measurements in air, the minimal resonance frequency of 700 Hz is that of the identified system, where the sample mass of 4.8 g is given by the sensor and mounting. The maximal resonance frequency of 950 Hz occurs at the minimal mass of 0.8 g, given by a very small sample and the sample holder, and is determined from Fig. 6(f) by calculating the time elapsed between two stripes that represent the period of the eigenfrequency. of the system is calculated from the The natural frequency

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Fig. 3. Scheme of the extended mathematical model used for the design of the is the dynamic weight of the manipulated variable feedforward controller. and is the weight of the frequency-dependent tracking accuracy.

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Fig. 4. Comparison between the simulated (solid lines) and the measured (dotted lines) tracking accuracy of the (a) uncontrolled piezo and (b) open-loop controlled system, respectively, in response to the desired triangular scanning signal (dashed lines) at a frequency of 100 Hz.

image size , the scanning rate between two stripes as

, and the lateral distance

(4) For measurements in air, the mass of the sample and, thus, the resonance frequency of the system can vary between the given values. The sample mass will never be less than 0.8 g, but can increase to several grams for heavier samples or for doing experiments in water. Thus, the model of the identified system with a sample mass of 4.8 g is chosen to be the nominal model of the piezoelectric scanner. The gain of the system can change due to the temperature dependency and a depolarization of the piezoelectric material [20]. The former effect has to be taken into account at temperature variations of tens of Centigrade, but can be neglected by working at a constant temperature in the laboratory. The latter effect has a logarithmic dependency on the time after polarization of the piezoelectric material and is in the range of about 0.01% per annum for lead zirconate titanate (PZT or PXE) materials older than three years [20]. Thus, gain variations of the system do not have to be taken into account at the design of our new controller.

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For the design of the controller, the identified discrete model is transformed using a prewarped bilinear transformation [21]. For a first implementation on the used DSP, the designed controller is transformed back to the plane using the inverse of the same bilinear transformation. The controller design is performed in the continuous time domain in view of a future implementation of a continuous analog filter. In the design step, the feedforward controller (FF) is calcunorm of the extended model lated such that the (5) gets minimized. The steady-state gain of the controller has to be scaled to 1 because this controller is just inserted into the system before the piezoelectric tube scanner’s voltage amplifier; the rest of the system is unchanged. The weight of the manipulated variable is chosen so that the system is not excited at frequencies around the first resonance frequency of the piezoelectric tube of the tracking error is chosen so that scanner. The weight the tracking error is small at low frequencies. Beyond the cutoff frequency, the system does not try to track the guidance signal. The designed controller is an eighth-order model that can be balanced [22] and reduced to a third-order model without noticeable loss of control performance. The reduced third-order model is implemented in its state-space representation on the DSP system [17] at a sampling rate of 80 ksample/s. The open-loop controlled scanning motion is delayed with respect to the guidance signal due to the calculation time of the controller implemented on the DSP and the A/D- and D/A conversion times of the DSP system. In order to use the data-acquisition system and the image-processing unit of the commercial AFM, the open-loop controlled lateral deflection of the piezoelectric tube scanner has to be synchronized with the given scan signal. To this end, the frequency of the triangular guidance signal is measured and the guidance signal is delayed by one period minus the delay time of the open-loop control system.

Fig. 5. Comparison between the simulated tracking accuracy of the uncontrolled system [solid line, (a)] and the open-loop controlled system [solid line, (b)] in response to a 30-Hz triangular scanning signal, which is disturbed by an added band-limited white noise (dotted lines).

IV. SIMULATED AND MEASURED TRACKING ACCURACY Fig. 4 shows a comparison of the simulated piezoelectric tube scanner [solid line, panel(a)] in response to a triangular guidance signal and the simulated open-loop controlled system [solid line, panel (b)] at a scanning rate of 100 Hz. In Fig. 4(a), the guidance signal (dashed line) excites the piezoelectric tube scanner at its resonant frequency when the scan direction is reversed. The natural oscillations of the piezo modulate the scan velocity and impair the tracking accuracy. The open-loop controlled system [Fig. 4(b)] tracks the guidance signal (dashed line) very well and differs from the desired position just at the edges of the scan signal. The scan speed is held constant at the predetermined value during a relatively long range of the scan. The dotted lines in Fig. 4(a) and (b) show the measured deflection of the piezoelectric tube scanner in response to the 100-Hz scanning signal for the uncompensated and the open-loop controlled system, respectively, which is in good agreement with the simulations. For this verification experiment, the triangular guidance signal was

Fig. 6. Silica-bead projection pattern imaged with the (a)–(f) uncompensated AFM and (g)–(i) open-loop controlled AFM. All images are recorded from right to left and are 13.5 13.5 m . The line-scan rate is 2 Hz for (a)–(c) and 61 Hz for (d)–(i). Sample topography: (a), (d), and (g); cantilever deflection signal: (b), (e), and (h); friction signal: (c), (f), and (i).

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generated by a function generator (Tabor, Yucaipa, CA, USA); thus, the hysteresis of the piezo was uncompensated. Fig. 5 shows a simulated scan at a rate of 30 Hz of the uncontrolled piezo and the open-loop controlled piezo, respectively, in the presence of noise. We added a 10-kHz band-limited white-noise signal to the guidance signal to show the sensitivity of the piezoelectric tube scanner in response to such excitations, which can occur in the investigated commercial system at scan ranges smaller than 200 nm. The system that is excited by the distorted guidance signal directly [solid line, Fig. 5(a)] intensifies its oscillations due to

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Fig. 7. Comparison of the friction signal imaged with the (a)–(e) uncompensated AFM and (f)–(k) open-loop controlled AFM at different scan rates. (a) and (f): 20 Hz; (b) and (g): 30.5 Hz; (c) and (h): 40.7 Hz; (d) and (i): 61 Hz; and (e) and (k): 122 Hz. All images have the same scaling, are recorded from right to left, and are 13.5 13.5 m .

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the added noise signal, modulates the scan speed, and shows poor tracking, whereas the open-loop controlled system [solid line, Fig. 5(b)] shows good tracking of the desired scan motion at constant velocity.

V. EXPERIMENTAL RESULTS To compare the performance of the open-loop controlled system to the standard commercial system, we imaged a hydrophobic/hydrophilic friction contrast on a silica bead projection pattern [23]. The sample was prepared by air drying a solution of monodisperse silica beads on a hydrophilic cover slip to produce a mask for physical vapor deposition. After evaporating a 10-nm-thick aluminum layer onto the cover slip, generating a silica bead projection pattern, the silica beads were removed from the cover slip by sonication in ultrahigh-quality water. To increase the hydrophobic/hydrophilic contrast, the sample was exposed to the vapor of a 5% solution of octadecyltrichlorosilane in ethanol at room temperature for 60 s. Fig. 6 shows a comparison between the uncompensated and compensated scan at a scan rate of 61 Hz. Fig. 6(a)–(c) shows a reference scan at a scan rate of 2 Hz. Fig. 6(d)–(f) shows the uncompensated scan at a scanning rate of 61 Hz and Fig. 6(g), (h), and (i) shows the open-loop controlled scan at 61 Hz. The loading force of the AFM tip on the sample was held constant by the PI controller of the commercial AFM system. Fig. 6(a), (d), and (g) shows the measured topography, represented by the PI-controlled voltage applied to the piezoelectric tube scanner in the Z direction. Fig. 6(b), (e), and (h) shows the deflection signal of the cantilever, representing the control error of the closed-loop system. Due to the limited bandwidth of the PI-controlled AFM, the control error increases at higher scan rates, which can be seen in Fig. 6(e) and (h), in comparison to Fig. 6(b). The lateral deflection of the cantilever is shown in the Fig. 6(c), (f), and (i), representing the torque of the cantilever due to the friction force between tip and sample. In Fig. 6(d)–(f) the displacement error due to the lateral oscillations of the piezoelectric tube scanner along the fast scanning direction is evident by the deformation of the circle of the projection pattern. In the topographical image of the uncompensated measurement [Fig. 6(d)] the coupling between the fast scanning direction and the Z direction of the piezoelectric tube can be seen in

the apparent corrugation of the sample surface, which is a topographical artifact as a result of the scanner’s lateral oscillations. But the oscillations in the Z direction due to the crosstalk between the scanning motion and the Z position of the piezoelectric tube cannot be compensated by the PI controller completely and cause an additional deflection of the cantilever [Fig. 6(e)]. This additional deflection may increase the maximum value of the cantilever deflection, resulting in a higher maximum value of the tip-sample interaction force, which may damage the AFM tip and/or the sample. However, this additional deflection always causes a modulation of the tip-loading force, which influences the friction measurement [see Fig. 6(f)]. The friction measurement is additionally influenced by the lateral oscillations of the scanner directly. After reversing the scanning direction, the oscillations superimposed on the scanning motion are strong enough to temporarily reverse the relative tip-sample movement, which turns the sign of the measured friction. This can be seen in Fig. 6(f) by the five bright stripes on the right side of the image, perpendicular to the fast scanning motion. The new open-loop controller described in Section III compensates the perturbing lateral oscillations of the piezoelectric tube scanner. Thus, the topographical artifacts due to the coupling between the axis of motion of the scanner disappear [Fig. 6(g)] and the modulation of the tip loading force due to the oscillations in the Z direction vanishes [Fig. 6(h) and (i)]. The direction of the relative tip-sample movement is not reversed anymore [Fig. 6(i)] and the relative tip-sample velocity is held constant for most of the scan (compare Fig. 4). Thus, the lateral displacement error due to the oscillations of the piezoelectric tube scanner gets minimized as well. These improvements can be seen clearly by comparing Fig. 6(g) and (d) to Fig. 6(a), showing the reference measurement imaged at 2 Hz. To demonstrate the improvement of the AFM system using the open-loop controller and to show the imaging artifacts due to the scanner’s oscillations, some friction measurements that were acquired at different scan rates with the uncompensated AFM and the open-loop controlled system, respectively, are compared in Fig. 7. The oscillations of the piezoelectric tube scanner get obvious by the vertical bright stripes in the images taken by the uncompensated AFM system [Fig. 7(a)–(e)], which result from multiple reversals of the direction of the relative tip-sample movement due to the oscillations of the piezo superimposing the scanning movement. The distance between two stripes is a function of the scan size, the scan rate, and the natural frequency of

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the piezoelectric tube scanner along the fast scan axis [see (4)]. The suppression of the lateral oscillations can clearly be seen in the images acquired with the open-loop controlled AFM system [Fig. 7(f)–(k)]. Modulation of the tip-sample loading force and of the relative tip-sample velocity vanish, enabling high-speed (mm/s) friction measurements at the nanometer scale.

REFERENCES [1] D. Sarid, Scanning Force Microscopy. New York: Oxford Univ. Press, 1994. [2] G. Binnig, C. F. Quate, and C. Gerber, “Atomic force microscope,” Phys. Rev. Lett., vol. 56, pp. 930–933, 1986. [3] Q. Zhong, D. Inniss, K. Kjoller, and V. B. Elings, “Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy,” Surface Sci., vol. 290, pp. L688–L692, 1993. [4] F. J. Giessibl, “Atomic resolution of the silicon (111) (7 7) surface by atomic force microscopy,” Sci., vol. 267, pp. 68–71, 1995. [5] R. C. Barrett and C. F. Quate, “High-speed, large-scale imaging with the atomic force microscope,” J. Vacuum Sci. Technol. B (Microelectron. Process. Phenomena), vol. 9, pp. 302–306, 1991. [6] D. Rugar and P. Hansma, “Atomic force microscopy,” Phys. Today, vol. 43, pp. 23–30, 1990. [7] E. Meyer, H. Heinzelmann, P. Grütter, T. Jung, H. R. Hidber, H. Rudin, and H. J. Güntherodt, “Atomic force microscopy for the study of tribology and adhesion,” Thin Solid Films, vol. 181, pp. 527–544, 1989. [8] U. D. Schwarz and H. Hölscher, Modern Tribology Handbook , B. Bhushan, Ed. Boca Raton, FL: CRC, 2001, vol. 1. [9] V. V. Tsukruk, V. N. Bliznyuk, J. Hazel, D. Visser, and M. P. Everson, “Organic molecular films under shear forces: Fluid and solid languir monolayers,” Langmuir, vol. 12, pp. 4840–4849, 1996. [10] G. Binnig and D. P. E. Smith, “Single-tube three-dimensional scanner for scanning tunneling microscopy,” Rev. Sci. Instrum., vol. 57, pp. 1688–1689, 1986. [11] D. Croft and S. Devasia, “Vibration compensation for high speed scanning tunneling microscopy,” Rev. Sci. Instrum., vol. 70, pp. 4600–4605, 1999. [12] D. Croft, G. Shed, and S. Devasia, “Creep, hysteresis, and vibration compensation for piezoactuators: Atomic force microscopy application,” Trans. ASME J. Dynam. Syst., Meas., Control, vol. 123, pp. 35–43, 2001. [13] G. Schitter, P. Menold, H. F. Knapp, F. Allgöwer, and A. Stemmer, “High performance feedback for fast scanning atomic force microscopes,” Rev. Sci. Instrum., vol. 72, pp. 3320–3327, 2001. [14] L. Ljung, System Identification, Theory for the User, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1999. [15] P. Van Overschee and B. De Moor, “N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems,” Automatica, vol. 30, pp. 75–93, 1994. [16] M. Viberg, “Subspace-based methods for the identification of linear time-invariant systems,” Automatica, vol. 31, pp. 1835–1851, 1995. [17] DSP-System: Processor Board DS1005, 2001. 16-bit A/D-board DS2001, 16-bit D/A-board DS2102. dSpace, Germany . [18] O. M. El-Rifai and K. Youcef-Toumi, “Coupling in piezoelectric tube scanners used in scanning probe microscopes,” in Proc. 2001 Amer. Control Conf., vol. 4, 2001, pp. 3251–3255. [19] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control. Chichester, U.K.: Wiley, 1996. [20] J. W. Waanders, Piezoelectric Ceramics, Eindhoven, The Netherlands: Philips Components, 1991. [21] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. Reading, MA: Addison-Wesley, 1990. [22] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds,” Int. J. Control, vol. 39, pp. 1115–1193, 1984. [23] U. C. Fischer, J. Heimel, H. J. Maas, M. Hartig, S. Hoeppener, and H. Fuchs, “Latex bead projection nanopatterns,” Surf. Interface Anal., vol. 33, pp. 75–80, 2002.

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VI. CONCLUSION In this article, a low-order feedforward controller is presented to compensate for the lateral oscillations of a piezoelectric tube scanner stemming from its mechanical resonances. The main interest was focused on improving the tracking accuracy of the piezo along the fast scanning movement to achieve high-speed atomic force and friction-force microscopy. Imaging with the compensated AFM/FFM minimizes the displacement error of the scanning motion and provides for a constant tip-sample velocity for most of the scanned area. Up to the maximum scan rate of our commercial AFM system, which is at 122 Hz, the new open-loop-controlled AFM deviates from the desired scanning motion just at the fringe of the scanned area when the scanning direction is reversed. The topographical artifacts due to the crosstalk between the scanning motion and the Z axis of the piezo disappear. As a result, the additional control error of the PI-controlled cantilever deflection in the Z direction due to the coupling-induced oscillations disappears and the modulation of the tip-sample loading force vanishes and no longer perturbs friction measurements. The direction of the relative tip-sample movement is no longer reversed by resonances enabling high-speed friction measurements. The presented open-loop controller is implemented on a DSP system and increases the speed of the commercial scanningprobe microscope by a factor of 15, enabling large-scale atomicand friction-force microscopy at an imaging speed in the mm/s range.

ACKNOWLEDGMENT The authors would like to thank D. Haefliger and Dr. R. W. Stark for their help with the sample preparation. They would also like to thank Prof. L. Guzzella from the Measurement and Control Laboratory, Swiss Federal Institute of Technology, Zurich, for fruitful discussions.