Improved Control of Atomic Force Microscope for High-Speed Image ...

2012 Australian Control Conference

15-16 November 2012, Sydney, Australia

Improved Control of Atomic Force Microscope for High-Speed Image Scanning M.S. Rana, H.R. Pota, and I.R. Petersen

M. S. Rana, H. R. Pota, and I. R. Petersen are with the School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia. E-mail: [email protected], [email protected], and [email protected]

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scanning (i.e., when actuating a PZT with voltage signals of low amplitude) hysteresis can be ignored [7]. Feedback control schemes have been used to compensate for hysteresis in [5], [8], [9]. Creep (see Fig. 2(b)) is another nonlinearity which is noticeable at low frequency signals. This effect

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The atomic force microscope (AFM), invented by Binnig, Quate, and Gerber [1], has the ability for generating three-dimensional (3D) images of material surfaces at an extremely high resolution down to the atomic level. The AFMs make a momentous contribution in the areas such as nanotechnology, nano-particle characterization, material science, and nano-machining [2]. Its performance is not constrained to imaging in a vacuum environment as are the scanning electron microscope (SEM) and the transmission electron microscope (TEM). A basic schematic diagram of the AFM is shown in Fig. 1. For control researchers this instrument is particularly challenging, since its ability to image the surface of a sample is entirely dependent upon the use of a feedback loop. There appears to be some scope to improve its performance. The performance of an AFM can be measured by the image quality and the higher scanning speed. For this reason a variety of different control methods have been developed to control the AFM positioning to generate controlled motion for fast and accurate positioning. For 3D positioning majority of the commercially available AFMs use a PZT scanner because of its large output force, high bandwidth, and fast response time. However, the piezoelectric material exhibits inherent nonlinearities such as creep and hysteresis, which drastically degrade the positioning performance of the PZT actuators [3]. Use of feedback control techniques other than PI controllers to improve tracking accuracy and scanning speed of AFMs has been undertaken by a number of researchers [3], [4], [5], [6]. Nonlinearity in the form of hysteresis (see Fig. 2(a)) becomes noticeable when PZTs are actuated using voltage signals of high amplitudes. On the other hand at low-range

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Abstract— In this paper the design and experimental implementation of an observer based model predictive control (OMPC) scheme for accurate tracking and fast scanning of an atomic force microscope (AFM) is presented. The design of this controller is based on an identified model of the AFM piezoelectric tube (PZT) scanner. A Kalman filter is used to obtain full-state information in the presence of position sensor noise. To evaluate the performance improvement using the proposed control scheme an experimental comparison has been made with scanned images between the proposed controller and the existing AFM PI controller. The experimental results verify the efficacy of the proposed controller.

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Fig. 2. Piezoelectric effects of PZT scanner: (a) Hysteresis and (b) Creep.

can be severe during slow operation of the AFM. A few methods have been proposed to deal with this effect [10], [11]. Mechanical vibration is also responsible for the poor image quality of the AFM and this vibration is created because of the resonant modes in the PZT actuator. To avoid this problem, the scanning speed of AFMs is often kept limited to 1% [3] of the scanner’s first resonance frequency. This vibration can be compensated by proper damping of the resonant modes. Research has been done by several authors on this issue [5], [12], [13], [14]. Capacitive sensors are commonly used in nanopositioning system because of their high-resolution measurement capability. But these sensors have their self noise which degrades the imaging performance of the AFM. In our proposed

© 2012 Engineers Australia

II. E XPERIMENTAL S ETUP Fig. 3 shows the experimental setup at the AFM laboratory, UNSW, Canberra, Australia, which is used to implement the proposed control scheme. Setup consists of an NT-MDT Ntegra scanning probe microscope (SPM), which was configured to operate as an AFM. There are some important accessories of the AFM such as signal access module (SAM), control electronics, vibration isolator, and a computer. Other parts which were used for the control implementation are signal analyzer, DSP dSPACE RT-1103 board, and a high voltage amplifier (HVA). The most important part of an AFM positioning system is

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III. M ODELING

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where Vx and Vy are the voltage signals applied in the corresponding X and Y piezo; Dx and Dy are the measured displacement from the corresponding capacitive position sensor. It is noteworthy that, there is about π radian phase shift between the input and output at low frequencies, in the both x and y sensor outputs.

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The key component of an AFM positioning system is the PZT scanner. In this work for the modeling of PZT scanner dynamics an experimental approach is considered. For simplifying the control design, the AFM scanner is treated as a two single-input single-output (SISO) system. In this process the plant is identified by using a bandlimited random noise signal within the frequency range of 10 Hz to 1100 Hz, using a dual channel HP35670A signal analyzer. This signal is supplied to the HVA (with a gain of 15) as an input and the corresponding amplified voltage is supplied to the SAM of the AFM. From SAM there is a direct connection with the PZT scanner. The displacement from the capacitive sensor is an output in this experiment. The sensor output is fed back to the signal analyzer to construct the frequency response functions. The following transfer functions are found to be a good fit by using the prediction error method (PEM) [15] as seen from Figs. 5 and 6.

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position sensors, x-sensor, y-sensor, and z-sensor which are incorporated into the scanner apparatus to allow for the direct measurements of the tube displacement in the x, y, and z directions. The block diagram of experimental connection is shown in Fig. 4.

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control scheme, we have introduced the Kalman Filter to take care this noise. In this work, we have proposed an OMPC control scheme which has got high bandwidth. This control scheme has reduced the tracking error and has significantly damped the resonant modes of the AFM PZT scanner. The special feature of MPC is that its augmented integral action reduces the nonlinear behavior of the PZT scanner which, in turn, solves the tracking error problem. MPC is also effective for nonminimum phase systems, like PZT scanner of the AFM. An useful characteristic of MPC is that it can incorporate constraints. In this control scheme a Kalman filter is used for obtaining full state information of the system. The rest of this paper is formatted as follows: Section II presents the descriptions of the AFM and the experimental setup used in this work. Section III provides the modeling and identification of the system transfer functions. In Section IV, the control scheme for the AFM scanner is presented. Section V reflects the experimental results which illustrates the significant improvement in accuracy and imaging speed which can be achieved with the proposed control scheme. Finally in Section VI, the conclusion and discussion of future work are drawn for further improving the control of the AFM.

AFM system and experimental setup.

the PZT scanner. In this experiment a scanner NT-MDT z50313cl is used to perform 3D positioning in the AFM. It is a ‘scan by sample’ type scanner. It has a scanning range of 100μ m × 100μ m × 12μ m and resonant frequency in both x and y direction is approximately 900 Hz, and in the z direction it is about 5 kHz. There are three capacitive

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IV. C ONTROLLER AND

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A. Controller Design This section presents the design of the proposed MPC control scheme undertaken in this work. An MPC control technique is designed to reduce the tracking error problem of PZT scanners to improve the positioning of fast axis (x)

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Fig. 6. Measured and identified model frequency response of the y position sensor output.

and slow axis (y). The construction of the closed-loop system is shown in Fig. 7. The MPC scheme is employed to design 5HIHUHQFH 6LJQDO



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u(k) − u(k − 1) xm (k + 1) − xm(k) xm (k) − xm (k − 1)

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where N p is the prediction horizon and Nc is the control horizon. From the predicted state variables, the predicted output variables are, y(k + 1|k) = CAx(k) + CBΔu(k) y(k + 2|k) = CA2 x(k) + CABΔu(k) +CBΔu(k + 1) .. .

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a tracking controller in which the output tracks a reference triangular signal. The basic structure of MPC is shown in Fig. 8. In MPC, the optimal control actions are found by optimizing the predicted plant behavior, under practical constraints [16]. The plant is expressed as the following statespace model

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with dimension n. In order to incorporate the integral action for disturbance rejection and tracking of a reference signal in the MPC algorithm the plant can be augmented in the following way [17]:     Δxm (k + 1) Δxm (k) =A + BΔu(k) (5) y(k + 1) y(k)   Δxm (k) y(k) = C (6) y(k)

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B. Observer Design

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It is assumed that there is no control action after time k + Nc − 1, i.e., Δu(k + i) = 0 for i > Nc − 1. The F matrix with dimension (NP , n) and Φ matrix with dimension (NP , Nc ) are as follows: ⎤ ⎡ CA ⎢ CA2 ⎥ ⎥ ⎢ 3 ⎥ ⎢ F = ⎢ CA ⎥ ⎢ .. ⎥ ⎣ . ⎦ CANp ⎡ ⎤ CB 0 ··· ··· 0 ⎢ CAB ⎥ CB ··· ··· 0 ⎢ ⎥ ⎢ CA2 B ⎥ CAB · · · · · · 0 Φ=⎢ ⎥ ⎢ ⎥ .. .. . . . . . . ⎣ ⎦ . . . . . N p −1 N p −2 N p −Nc B CA B · · · · · · CA B CA The control law is derived based on the minimization of the cost function as follows: Np

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Since the plant output is taken from the capacitive position sensor, which is noisy. For this reason we have to introduce an observer to filter the sensor noise. In a noisy environment, a state observer can act like a noise filter to reduce the effect of measurement noise. So a Kalman state observer is designed to estimate the states from the output. The Kalman observer dynamics is as follows: x(k ˆ + 1) =

(A − LC)x(k) ˆ + Bu(k) + Ly(k) ˆ y(k) ˆ = Cx(k) ˆ

The performance of the OMPC control scheme is evaluated first by measuring closed-loop frequency responses. In Figs. 9 and 10, the measured closed-loop frequency 20 0 −20

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where Q is the state weighting matrix, R is the control weighting, and Rs is the reference signal. By considering the above equations the constrained MPC problem can be expressed as a compact quadratic programming (QP) problem [17]: 1 min( ΔU T EΔU + ΔU T f ) (10) 2 s.t. MΔU ≤ γ where: E f

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M ∈ Rmc ×Nc , and γ ∈ RNc ×1 are computed by using equation (9) and where mc is the number of constraints. In this paper for constraint calculation the Hildreth’s Quadratic Programming algorithm [17] has been considered. The constrained minimization over ΔU is given by [17], [18] ΔU = −E −1 ( f + M T λ )

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where λ is the Lagrange multiplier, which is calculated by using M, γ , E, and f . The kth control input u(k) can be calculated as follows: u(k) = u(k − 1) + Δu(1)

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V. E XPERIMENTAL R ESULTS

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where x(k) ˆ is the estimated states, y(k) ˆ is the estimated state output, Cˆ is the identity matrix of dimension n × n, and L is the observer gain which depends on the Gaussian white noise, process noise covariance, and measurement noise covariance.

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responses are plotted along with the open-loop frequency responses of the X and Y piezo displacement respectively. From these figures, it can be observed that the closedloop system has got high bandwidth for both x axis and y axis. For both cases also have around 20 dB damping in the closed-loop system which in turns reduces the vibration significantly. It should be noted here that with the current experimental setup, it was not possible to measure the closed-loop frequency responses of the AFM scanner with well-tuned PI controller that is built into the AFM. The experimental tracking performance of the proposed controller is shown in Fig. 11. In Fig. 11(a) & (b) the open-loop tracking is presented at 5.21 Hz and 31.25 Hz respectively. Since our system has π radian phase shift in the open-loop output, to show the tracking performance the sensor output is multiplied by -1. The closed-loop tracking performance is shown in Fig. 11(c) & (d) at 5.21 Hz and

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10 Hz can only be set to frequencies of 10.42 Hz, 31.25 Hz, 62.50 Hz, and 125 Hz. During the imaging the AFM is operated in constant force mode by using a micro-cantilever. To show the imaging performance of the proposed control scheme over open-loop is shown in Fig. 12. In the open-loop condition, the scanner displacement is quite poor because of the uncontrolled tube resonance and results in a sluggish image, as shown in Fig. 12(a). A comparison of performance is shown in Fig. 13 between the proposed controller and the existing PI controller on the basis of the scanned images. Each of these images was obtained by scanning a NT-MDT TGQ1 grating reference sample by using NT-MDT CSG01 micro-cantilever.

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After improving the lateral positioning of the PZT scanner, we then focus on the analysis of the overall improvement in imaging capability of the AFM. By using the existing NTMDT AFM scanning NOVA software, scanning speeds above

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Fig. 13. Scanned images at 10.42 Hz, 31.25 Hz, 62.50 Hz, and 125 Hz using AFM PI controller (left) and using the proposed controller (right).

In order to analyze the AFM images, we plot their crosssectional curves taken in parallel to the square profile of the

that up to 125 Hz scanning speed the images with the proposed controller are far better than the images obtained using the existing PI controller. But it has been noted that at a scanning speed above 60 Hz there are some oscillations. So future control strategy will be to introduce a vibration compensator with the proposed control scheme for active damping of the oscillations at higher frequency. ACKNOWLEDGMENT The authors wish to thank Mr. Shane Brandon, SEIT, UNSW, Canberra, Australia for his technical support during the experimental tests.

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VI. C ONCLUSION

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In this article, we consider the use of an OMPC tracking control scheme for faster scanning of the AFM. The experimental results show that the high-precision and high-speed 3D image can be obtained by implementing the proposed controller. From comparison of scanning images it is seen

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