PVPF control of piezoelectric tube scanners

doi:10.1016/j.sna.2006.07.032

Sensors and Actuators A: Physical

Volume 135, Issue 2, 15 April 2007, Pages 700-712

https://doi.org/10.1016/j.sna.2006.07.032 Get rights and content

Abstract

As in most applications of nanotechnology, speed and precision are important requirements for getting good topographical maps of material surfaces using Scanning Tunneling Microscopes (STM) and Atomic Force Microscopes (AFM). Many STMs and AFMs use piezoelectric tubes for scanning and positioning with nanometer resolution. In this work a piezoelectric tube of the type typically used in STMs and AFMs is considered. Scanning using this piezoelectric tube is hampered by the presence of a low-frequency resonance mode that is easily excited to produce unwanted vibrations. The presence of this low-frequency resonance mode restricts the scanning speed of the piezoelectric tube. Concept of a Positive Velocity and Position Feedback (PVPF) controller is introduced and a controller is designed to damp this undesired resonance mode. To achieve good precision, inputs are then shaped for the closed loop system to track a raster pattern. Experimental results reveal a significant damping of the resonance mode of interest, and consequently, a good tracking performance.

Introduction

Scanning Tunneling Microscopes (STM) and Atomic Force Microscopes (AFM) were developed in the 1980's by Binning, Roher and Co-authors [4], [5]. STMs and AFMs are used at extreme magnifications for imaging micro to atomic dimensions with high resolution. These microscopes adapt to most experimental surroundings such as general ambient air, liquids, gases, high temperatures, low temperatures, etc. This flexibility has contributed towards their extensive use in many diverse fields [3].

In both STMs and AFMs a probe is placed in close proximity, typically a few angstroms, to a material surface for which a topographic map is desired. The given surface is scanned by either moving the probe or the sample in a raster pattern, so that the probe interacts with the entire region of interest, see [3], [4], [5]. The physical unit in the STM, or the AFM, that regulates the motion of the probe or the surface is referred to as the scanning unit. Scanning, in both SPMs and AFMs, is done either by attaching the probe to a cylindrical piezoelectric tube (known as PZT tube scanner in commercial circles) or placing the sample on top of the piezoelectric tube and actuating the piezoelectric tube in a raster pattern, see [1], [3], [14], [18]. As the probe interacts with the material surface, while scanning, a “measure” of the surface topography is outputted by the scanning system. Based on this “measure” an image of the material surface is generated. The “measure” of the surface topography outputted by the scanning unit is normally the tunneling current that flows between the material surface and the probe in the case of an STM, while it may be the force experienced by the probe in the case of an AFM.

One of the advantages of using piezoelectric tubes for scanning is that under certain experimental conditions their dynamics can be well approximated by linear models, see [1], [6], [14], [15], [18], [22], [23]. However, the linear models normally possess lightly damped resonance modes, which make the piezoelectric tubes susceptible to mechanical vibrations. Furthermore non-linearities such as creep and hysteresis have to be taken into account when actuating the tube with low-frequency inputs (near DC signals) and high amplitude inputs, respectively. The presence of mechanical vibrations and the non-linearities hinder the actuation of the tube [11].

In the recent past Caughey, Fanson and Goh have introduced a control technique known as the Positive Position Feedback (PPF) control [10], [21] to suppress the mechanical vibrations in a structure. Several authors have successfully used the PPF controller in different contexts [2], [7], [8], [9], [19]. In [13] the authors have designed a PPF controller to damp the vibrations in a piezoelectric tube scanner. Therein it was noted that using a PPF controller poles of the closed loop system cannot be placed at any given set of points in the left half plane. The model structure of the PPF is such that it prevents arbitrary pole placement. In order to allow this flexibility the PPF controller has been modified in to a PVPF controller. Thus, Positive Velocity and Position Feedback (PVPF) is a control technique introduced in this paper to damp the first resonance of the piezoelectric tube. As the name suggests, the inputs to this feedback controller are position and velocity of the system output (i.e. the inputs are y and $\dot{y}$ , if y is the system output), and the controller output is feedback positively into the system.

This paper is formatted as follows: In Section 2, a description of the piezoelectric tube considered in this work is presented. In the same section details of the experimental setup and the interpretation of the piezoelectric tube setup as a linear system are also presented. A linear model is constructed for the piezoelectric tube using standard techniques of system identification [12], [16], [20] in Section 3. In order to motivate the need for a feedback controller, two direct attempts are made to actuate the piezoelectric tube in a raster pattern in Section 4. Shortcomings of both these attempts are discussed therein. In Section 5 the concept of PVPF control is introduced and a methodology is presented for designing it to robustly damp the resonance mode observed in the model. In Section 6 experimental and simulation results obtained by using the PVPF are presented. Finally, this paper is concluded in Section 7.

This paper introduces a number of innovative ideas for efficient use of the piezoelectric tube for scanning. Firstly, inbuilt electrodes in the piezoelectric tube are used for both sensing and actuation. Here, a pair of electrodes in the tube are used as actuators and their collocated counterparts are used as sensors. In relatively high bandwidth applications, collocated sensor electrodes are superior to external sensors as their resolution and bandwidth are greater. In addition, high frequency noise, which are typical of external sensors, are not as significant when using inbuilt sensors. Nevertheless, it has been a common practice to use external sensors in piezoelectric tube scanners [1], [14]. Secondly, as mentioned above, a PVPF controller is used to remove the structural vibrations resulting from the first mechanical resonance mode, and with good results. The PVPF controller due to its model structure allows for more accurate and higher frequency actuation.

Section snippets

System description

A piezoelectric tube scanner is a thin-walled cylindrical tube made of piezoelectric material. The inner and the outer walls of the piezoelectric tube are finely coated with a layer of copper. The copper coating on the inner and outer walls of the tube act as electrodes of the scanner. The outer electrode is axially quartered into four equal sections. Conventionally a pair of the opposite sections of the quartered electrode is referred as the x–x electrodes and the other pair is referred as the

System identification

In this section the modeling of the linear systems V and C are discussed. The subsystem V is of the form $Y_{v} (s) ≜ G_{v} (s) U (s),$ where $Y_{v} (s)$ is the Laplace transform of the voltages $[V_{x^{-}}, V_{y^{-}}]$ , U(s) is the Laplace transform of the input voltages ${[V_{x^{+}}, V_{y^{+}}]}^{T}$ and $G_{v} (s) = [\begin{array}{c} G_{x x} (s) & G_{x y} (s) \\ G_{y x} (s) & G_{y y} (s) \end{array}]$ is a 2 × 2 matrix of transfer functions. And the subsystem C is of the form $Y_{c} (s) ≜ G_{c} (s) U (s),$ where Y_c(s) is the Laplace transform of the capacitive sensor outputs [C_x, C_y]^T, $G_{c} (s) = [\begin{array}{c} G_{x c_{x}} (s) & G_{x c_{y}} (s) \\ G_{y c_{x}} (s) & G_{y c_{y}} (s) \end{array}]$ is a 2 × 2 matrix

Feed-forward control

As mentioned earlier the goal is to actuate the piezoelectric tube in a raster pattern. Therefore a desired trajectory for the piezoelectric tube would be to repeatedly trace straight lines back and forth in x direction, while slowly increasing its position in the y direction, see Fig. 3. A common practice to achieve such a path is to input a triangular waveform in x⁺ electrode and a “very slowly” increasing ramp in the y⁺ electrode as reference signals for the system. In fact, to have a good

Positive Velocity and Position Feedback controller

In this section, the concept of Positive velocity and Positive Position (PVPF) control is introduced and a controller of this type is designed to damp the resonant peak in the transfer function G_xx(s).

For technical ease, we rewrite G_xx(s) in standard second order form in the time domain, $\ddot{x} + 2 σ ω \dot{x} + ω^{2} x = Ψ_{1} u$ $y = Ψ_{2} x + d u$ where Ψ₁ = 1.6711 × 10⁷, Ψ₂ = 1 and d = −2.723 × 10⁻¹, see (3.5).

Positive Velocity and Position Feedback controller is defined by $\ddot{z} + 2 ξ w \dot{z} + w^{2} z = Γ_{1} \dot{v} + Γ_{2} v,$ where $v$ is the input to the controller, and ξ,

Numerical illustrations and experiments using PVPF

In this section, we construct a PVPF controller connecting the $V_{x^{-}}$ output to the input $V_{x^{+}}$ to damp the resonance in the transfer function G_xx(s), see (3.5).

Poles of G_xx(s) can be computed from (3.5), $p_{\pm} = - 30.1 \pm i 5337.3 .$

Here we set the desired closed loop poles to $P_{1 +} = P_{2 +} = - 2030.1 \pm i 5337.3, P_{1 -} = P_{2 -} = - 2030.1 \pm i 5337.3 .$

In other words we wish to push the closed loop poles of G_xx(s) further into left half plane by 2000 units.

It can be checked that the polynomial coefficients K₁ = 8.1204 × l0³, K₂ = 8.1701 × 10⁷, K₃ =

Conclusion

In this paper a piezoelectric tube of the type typically used in STMs and AFMs was considered. This piezoelectric tube was interpreted as a linear system, and a linear model was constructed for it using standard system identification techniques. A lightly damped resonant mode was observed in the linear model in the frequency region of interest. Attempts were made to actuate the piezoelectric tube in a raster pattern in open loop without a feedback controller damping the resonance mode. First by

Acknowledgements

This research was funded by the Australian Research Council (ARC), which is duly acknowledged. The authors wish to thank Dr. Andrew Fleming for setting up the Piezoelectric tube and making it simple to perform the experiments, and also for the many enlightening discussions on Piezoelectric materials.

Bharath Bhikkaji received the PhD degree in signal processing from the Uppasla University, Uppsala, Sweden, in the year 2004. He is currently a Research Academic at the School of Electrical Engineering and Computer Science, University of Newcastle, Newcastle, Australia. His research interests include System Identification, Robust Control and Active noise and Vibration control of Flexible structures.

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Marcel Ratnam received the BEng degree in computer engineering and BMath in 2003, and is currently completing the MEng degree in electrical engineering at the University of Newcastle, Australia. His research interests include control of piezoelectric tube scanners with a focus on increasing tracking bandwidth and resolution.

Reza Moheimani received the PhD degree in electrical and electronic engineering from the University of New South Wales in 1996. Following a research position at the same institution, he joined the University of Newcastle in 1997, where he is currently an Associate Professor in the School of Electrical Engineering and Computer Science, the director of Laboratory for Dynamics and Control of Smart Structures and a programme leader for the ARC Centre for Complex Dynamic Systems and Control, an Australian Government Centre of Excellence. Dr. Moheimani is an Associate Editor of several international journals including the IEEE Transactions on Control Systems Technology, and has chaired a number of international workshops and conferences. He has published two books, several edited volumes and over 150 articles in areas of robust control and estimation, smart structures, active noise and vibration control, mechatronic systems and nanotechnology.

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PVPF control of piezoelectric tube scanners

Abstract

Introduction

Section snippets

System description

System identification

Feed-forward control

Positive Velocity and Position Feedback controller

Numerical illustrations and experiments using PVPF

Conclusion

Acknowledgements

Sens. Actuators

Acta Astronautica

Piezoelectric tubes for atomic force microscopes: design of lateral sensors, identification and control

Atomic force microscopes

Phys. Rev. Lett.

Scanning tunneling microscopes

Phys. Rev. Lett.

Identification and open-loop tracking control of a piezoelectric tube scanner for high-speed scanning probe microscopy

IEEE Trans. Control Syst. Technol.

Experimental study of vibration suppression of flexible spacecraft using modular control patch

Comparison of vibration control schemes for a smart antenna

Positive position feedback control for large space structures

AIAA J.

Hysteresis, creep and vibration compensation for piezoactuators: feedback and feedforward control