Elsevier

Mechatronics

Volume 20, Issue 1, February 2010, Pages 74-84
Mechatronics

Physical-model-based control of a piezoelectric tube for nano-scale positioning applications

https://doi.org/10.1016/j.mechatronics.2009.09.006Get rights and content

Abstract

Piezoelectric tubes exhibit a highly resonant mode of vibration which, if uncontrolled, limits the maximum scan rate in nano-scale positioning applications. Highly resonant systems with collocated sensor/actuator are often controlled using resonant shunt dampers. Unfortunately, in the configuration used here, this approach is not possible due the non-minimum phase property arising from the presence of a right-half plane zero.

This problem is solved by: (i) interpreting the resonant shunt damper in the context of physical-model-based control (PMBC) and (ii) extending the PMBC approach to handle non-minimum phase systems.

The resultant controller combines the physical insight of the resonant shunt damper with the ability to control the non-minimum phase piezoelectric tube.

A digital implementation of the controller was experimentally evaluated and found to successfully eliminate the resonant mode of vibration during an accurate and fast scan using a piezoelectric tube actuator.

Introduction

Scanning tunneling microscopes (STMs) and atomic force microscopes (AFMs) are used extensively in diverse areas of science such as crystallography, cell biology, etc. [1]. When used at extreme magnifications they are capable of generating topographical maps of solid surfaces at micro to atomic resolution. In both STMs and AFMs a probe is placed in close proximity, typically a few nanometers, to the material surface for which a topographical map is desired. The given material 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. In many commercially available STMs and AFMs scanning is performed using a piezoelectric tube actuator [1]. Depending on the make, either the probe is attached to the piezoelectric tube or the sample is placed on the free end of the piezoelectric tube and then actuated in a raster pattern.

One of the advantages of using piezoelectric tubes is that under certain experimental conditions their dynamics can be well approximated by linear models, see [2], [3], [4], [5], [6]. The linear models normally reveal the presence of lightly damped resonance modes, which make the piezoelectric tubes susceptible to mechanical vibrations. Non-linear phenomenon such as creep and hysteresis also become visible when actuating the tube using low frequency inputs and high amplitude inputs, respectively [7], [8]. In such scenarios the linear approximations become inadequate. The control objective of this paper is to perform fast scans using a piezoelectric tube actuator of the type typically used in scanning probe microscopes. The main impediments to fast scanning are the presence of mechanical vibrations and hysteresis. This paper focusses on the first impediment. The second impediment is avoided using special hardware; in particular, a specially designed charge amplifier [9], [10] is used for applying signals to the piezoelectric tube in order to avoid hysteresis.

One approach to avoiding mechanical vibrations is to use band-limited input signals; [11] give a recent discussion of this approach. This feedforward approach is combined with the feedback approach of this paper. An alternative is to make use of the periodic nature of the scan and use repetitive control; see [12] for a survey. The feedback approach to active vibration control has a long history dating back to at least den Hartog [13] and has many recent developments as summarised by Preumont [14]. This paper presents a novel way of designing feedback part of the control based on physical insight.

The notion of “controller design in the physical domain” was introduced by [15] based in turn on earlier work on “impedance control” [16], [17], [18]. As discussed by those papers and [19] such controllers are naturally described in bond graph [20], [21], [22], [23] terms.1 Such “physical-model-based control” (PMBC) has been applied to the control of mechanical systems [25], [26], [27] as well as to the emerging field of hybrid numerical experimental substructuring [27]. For this reason, a model of the piezoelectric tube is developed in bond graph terms. Although such a linear model could also be developed by other means, this paper shows that the insight gained by the bond graph approach also allows a natural PMBC-based feedback solution to the problem of vibration associated with fast scanning.

In the setup used here, one side of the piezoelectric tube is used for actuation, and the other for sensing. It turns out that this asymmetric actuation leads to significant non-minimum phase behaviour which makes feedback control a more challenging problem than it would otherwise be. This problem is solved in this paper. Hitherto, the PMBC approach of [25] has been restricted to minimum-phase systems; the first purpose of this paper is to provide a novel way of extending PMBC to non-minimum phase systems. The second purpose of the paper is to apply and experimentally evaluate the PMBC approach within the context of the control of the piezoelectric tube. Thus, although the method is applied to a particular experimental system, it is applicable to mechatronic systems in general.

Section 2 summarises PMBC and Section 3 gives a bond graph based physical model of the tube. Section 4 describes the experimental equipment, presents open-loop frequency response results and uses system identification to estimate the physical model parameters of the tube; the non-minimum phase nature of the system is described. Based on this physical model, Section 5 gives a PMBC design of a feedback controller and presents a novel approach to overcoming the problems arising from the non-minimum phase behaviour. Section 6 presents experimental results and verifies that the feedback controller eliminates vibrations due to the tube resonance. Section 7 concludes the paper. The Appendix gives details of the open-loop approach, and how it can be combined with the feedback approach.

Section snippets

Physical model based control

There are three subsystems which represent:

  • Num

    the numerical subsystem implemented as software within a digital computer,

  • Phy

    the physical subsystem implemented as hardware in the physical world and

  • Tra

    the transfer system comprising sensors and actuators connecting the numerical and physical domains together with the associated control systems and signal conditioning.

Fig. 1a shows the ideal case where sensors and actuators are collocated; design is relatively straightforward in this case.

Unfortunately,

Modelling

Fig. 2a gives a schematic diagram of the piezoelectric tube where the left-hand patch acts as an actuator and the right-hand patch as a sensor. This asymmetric actuation leads to two types of motion: a bending mode where a contraction of the actuation patch leads to an extension of the sensing patch and a vertical piston mode where a contraction of the actuation patch leads to a contraction of the sensing patch. This contraction leads to the non-minimum phase behaviour discussed in the rest of

Identification

This section discusses how the physical parameters of the theoretical model of Section 3 were fitted to experimental data. Section 4.1 outlines the experimental setup and Section 4.2 shows how the parameters are extracted from the experimental data. Section 4.3 examines the non-minimum phase aspects of the identified transfer function.

Physical model based control design

With reference to Fig. 1, the physical subsystem Phy is identified with the bending mode portion of Fig. 3 and the transfer system Tra with the piston mode and associated 1 junction. In particular, the transfer function of Phy is P(s) (from Eq. (1)) and the transfer function of Tra is T(s) where:P(s)T(s)=-Gviwhere Gvi is given by (3).

As outlined in Section 2, there are two parts to the PMBC design: designing Num and eliminating the effects of Tra. These two topics are treated in the following

Experimental results

The controller designed in Section 5 was experimentally validated by two sets of experiments using the experimental equipment described in Section 4.1. The first set validated the predicted closed-loop frequency response presented in Fig. 11. The second set examined the scanning performance as measured at the tip using the capacitive sensor described in Section 4.1; this tip signal was not used in the controller but rather only for monitoring the controller performance.

Firstly, the open and

Conclusion

The general approach of physical-model-based control (PMBC) has been extended to be applicable to non-minimum phase systems in general and the non-minimum phase system arising from the feedback control of a piezoelectric tube in particular.

The controller design involves the choice of only two parameters each of which has physical significance and is thus easily chosen. The resultant third-order feedback controller then follows from these two parameters as well as four physical parameters

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