Chapter 19
Advanced Control of Atomic Force Microscope for Faster Image Scanning M. S. Rana, H. R. Pota and I. R. Petersen
Abstract In atomic force microscopy (AFM), the dynamics and nonlinearities of its nanopositioning stage are major sources of image distortion, especially when imaging at high scanning speed. This chapter discusses the design and experimental implementation of an observer-based model predictive control (OMPC) scheme which aims to compensate for the effects of creep, hysteresis, cross-coupling, and vibration in piezoactuators in order to improve the nanopositioning of an AFM. The controller design is based on an identified model of the piezoelectric tube scanner (PTS) for which the control scheme achieves significant compensation of its creep, hysteresis, cross-coupling, and vibration effects and ensures better tracking of the reference signal. A Kalman filter is used to obtain full-state information about the plant. The experimental results illustrate the use of this proposed control scheme.
19.1 Introduction Nanotechnology is an area of modern science which deals with the control of matter at dimensions of 100 nm or less. In recent years, of all the available microscopy techniques, atomic force microscopy (AFM) has proved itself extremely versatile as an investigative tool in this field. It is becoming a driving technology in nanomanipulation and nanoassembly [1, 2]. M. S. Rana (B) · H. R. Pota · I. R. Petersen School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT, 2600, Australia e-mail:
[email protected] H. R. Pota e-mail:
[email protected] I. R. Petersen e-mail:
[email protected] L. Liu et al. (eds.), Applied Methods and Techniques for Mechatronic Systems, Lecture Notes in Control and Information Sciences 452, DOI: 10.1007/978-3-642-36385-6_19, © Springer-Verlag Berlin Heidelberg 2014
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Fig. 19.1 Schematic view of an AFM
The AFM’s attractive features are its fast and easy sample preparation, relatively low cost, and ability to operate in various environments. It provides a 3D profile of a surface on a nanoscale by measuring forces, such as the van der Waals, capillary, electrostatic, and magnetic, between a sharp probe ( Nc − 1. The F matrix with dimensions of (N P , n) and the Φ matrix with dimensions of (N P , Nc ) are: ⎤ ⎡ CA ⎢ C A2 ⎥ ⎥ ⎢ 3 ⎥ ⎢ F = ⎢ C A ⎥; ⎢ .. ⎥ ⎣ . ⎦ C ANp
⎡ ⎢ ⎢ ⎢ Φ=⎢ ⎢ ⎣
CB C AB C A2 B .. .
0 CB C AB .. .
··· ··· ··· .. .
··· ··· ··· .. .
0 0 0 .. .
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C A N p −1 B C A N p −2 B · · · · · · C A N p −Nc B The control law is derived based on the minimization of the cost function J=
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(19.12)
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subject to the linear inequality constraints on the system inputs u min ≤ u(k + i − 1) ≤ u max , i = 1, . . . , Nc ; u min ≤ u(k + i − 1) ≤ u max , i = 1, . . . , Nc ;
(19.13a) (19.13b)
where Np is the prediction horizon, Nc is the control horizon, Q is the state weighting matrix, R is the control weighting, Rs is the reference signal, u min and u max are the low and high levels of the control action, respectively, and u min and u max are the low and high levels of the control increments, respectively. By considering the above equations, the constrained MPC problem can be expressed as a quadratic programming (QP) problem:
19 Advanced Control of Atomic Force Microscope for Faster Image Scanning
1 min( U T EU + U T f ); 2
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(19.14)
s.t. MU ≤ γ ; where E = Φ T QΦ + R; f = Φ T Q F x(k + 1|k) − Φ T Q Rs ; M ∈ Rm c ×Nc and γ ∈ R Nc ×1 are computed using Eq. (19.13), m c is the number of constraints and Rs ∈ R N p ×1 is the reference signal.
19.4.2 Observer Design Kalman observer can be used as a state observer and noise filter [37]. The displacements of the PTSs are taken from the capacitive position sensors. Capacitive sensors are commonly used in nanopositioning systems because of their high-resolution measurement capability. However, it adds unwanted noise and disturbances to the output displacement which degrades an AFM’s scanning performance. To remove of which we design a Kalman state observer as a noise filter. The Kalman state observer estimates the states from the measured output. The Kalman observer dynamics is: x(k ˆ + 1) = (A − LC)x(k) ˆ + Bu(k) + L y(k); yˆ (k) = Cˆ x(k); ˆ
(19.15) (19.16)
where x(k) ˆ is the estimated states, yˆ (k) 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.
19.5 Experimental Results 19.5.1 Creep Effect Compensation During the slow operation of the AFM in open-loop, the scanned image shows a creep effect as shown in Fig. 19.9a. Due to this effect the right side bottom edge of the vertical axis of the scanned image is rolled-off. In Fig. 19.9b, the creep effect is reduced significantly using the proposed controller.
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Fig. 19.9 Creep effect in the scanned image at open-loop (a) and compensation of creep effect using the proposed controller (b)
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Fig. 19.10 Compensation of hysteresis effect at 31.25 Hz a using the AFM PI controller and b using the proposed controller
19.5.2 Hysteresis Effect Compensation The reduction of hysteresis effect at 31.25 Hz using the AFM PI controller and the proposed controller is shown in Fig. 19.10a and b, respectively. The image of the grating has a distinct curvature for the image taken by using the AFM PI controller. In the case of proposed controller, there have no hysteresis effect in the scanned image as shown in Fig. 19.10b.
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Fig. 19.11 Comparison of measured open-loop and closed-loop frequency responses of the X piezo (a) and Y piezo (b)
19.5.3 Vibration Effect Compensation The performance of the OMPC control scheme is evaluated by measuring its closedloop frequency responses. In Fig. 19.11a and b, the measured closed-loop frequency responses are plotted along with the open-loop frequency responses of the X and Y piezo displacements, respectively. By implementing this OMPC control scheme, we have achieved a reasonable damping in the closed-loop system for both axes, which in turn reduces the vibration significantly. Therefore, the closed-loop system has higher bandwidths for both the X and Y axes. The reduction of vibration effect by using the proposed controller is shown in Fig. 19.12. Figure 19.12 illustrates the results obtained by implementing the OMPC scheme which show significantly different images from those taken in the openloop condition. 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. 19.12a, c, and e at 31.25, 62.50, and 125 Hz scanning speed and this problem is compensated by using the proposed controller as shown in Fig. 19.12b, d, and f. Fig. 19.12f demonstrates that the proposed controller can compensate the vibration effect at the higher scanning speed.
19.5.4 Cross-coupling Effect Compensation Figure 19.13a and b show comparisons of the open-loop and closed-loop crosscouplings for the 5.21 and 31.25 Hz input reference signals, respectively. They illustrate that there is significant improvement in cross-coupling in the closed-loop. To
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Fig. 19.12 Compensation of vibration effect at 31.25, 62.50, and 125 Hz a,c, and e open-loop and b, d, and f using the proposed controller
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measure this cross-coupling, a reference triangular signal is applied to the X axis of the piezo and output taken from the Y position sensor.
19.5.5 Experimental Tracking Performance The experimental tracking performance of the proposed controller is shown in Fig. 19.14. In Fig. 19.14a and b the closed-loop tracking is presented at 5.21 and 31.25 Hz, respectively, which reflect the tracking performance of the proposed controller. The closed-loop tracking error are summarized in Table 19.1.
386 Table 19.1 RMS values of tracking error in closed-loop
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5.21 31.25
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19.6 Conclusion and Future Work The main contribution of this chapter was to compensate the nonlinear effect of the PTS by using the proposed controller for proper tracking and improved imaging at higher scanning speed of an AFM. The proposed OMPC controller was implemented to compensate the effect of creep, hysteresis, cross-coupling, and vibration of the PTS. The experimental results demonstrate that the tracking control performance is greatly improved in the high-speed application using the proposed controller. In this work, the plant considered was a single-input single-output (SISO) system. However, the author is interested in working with a multi-input multi-output (MIMO) system in future. Acknowledgments The authors would like to thank Mr. Shane Brandon, SEIT, UNSW, Canberra, Australia for his technical support during the experimental tests.
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