PVPF control of piezoelectric tube scanners
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1 Sensors Actuators A 135 (2007) PVPF control of piezoelectric tube scanners B. Bhikkaji, M. Ratnam, S.O.R. Moheimani School of Electrical Engineering Computer Science, University of Newcastle, NSW, Australia Received 11 December 2005; received in revised form 12 May 2006; accepted 25 July 2006 Available online 10 October 2006 Abstract As in most applications of nanotechnology, speed precision are important requirements for getting good topographical maps of material surfaces using Scanning Tunneling Microscopes (STM) Atomic Force Microscopes (AFM). Many STMs AFMs use piezoelectric tubes for scanning positioning with nanometer resolution. In this work a piezoelectric tube of the type typically used in STMs 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 Position Feedback (PVPF) controller is introduced 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, consequently, a good tracking performance Elsevier B.V. All rights reserved. Keywords: Piezoelectric tube; System identification; Resonance; Damping; Feedback control; Pole placement 1. Introduction Scanning Tunneling Microscopes (STM) Atomic Force Microscopes (AFM) were developed in the 1980 s by Binning, Roher Co-authors [4,5]. STMs 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 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 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 AFMs, is done either by Corresponding author. Tel.: ; fax: addresses: Bharath.Bhikkaji@newcastle.edu.au (B. Bhikkaji), mratnam@ee.newcastle.edu.au (M. Ratnam), Reza.Moheimani@newcastle.edu.au (S.O.R. Moheimani). 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 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 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 hysteresis have to be taken into account when actuating the tube with low-frequency inputs (near DC signals) high amplitude inputs, respectively. The presence of mechanical vibrations the non-linearities hinder the actuation of the tube [11]. In the recent past Caughey, Fanson Goh have introduced a control technique known as the Positive Position Feedback /$ see front matter 2006 Elsevier B.V. All rights reserved. doi: /j.sna
2 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) (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 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 velocity of the system output (i.e. the inputs are y ẏ,ify is the system output), 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 the interpretation of the piezoelectric tube setup as a linear system are also presented. A linear model is constructed for the piezoelectric tube using stard 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 a methodology is presented for designing it to robustly damp the resonance mode observed in the model. In Section 6 experimental 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 actuation. Here, a pair of electrodes in the tube are used as actuators their collocated counterparts are used as sensors. In relatively high bwidth applications, collocated sensor electrodes are superior to external sensors as their resolution bwidth 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, with good results. The PVPF controller due to its model structure allows for more accurate higher frequency actuation. Fig. 1. Illustration of a quartered piezoelectric tube with its dimensions exaggerated. electrodes the other pair is referred as the y y electrodes, see Fig. 1 for an illustration of the quartered tube. In order to model actuate the piezoelectric tube, an experimental set up is devised as follows: A device is constructed to hold the piezoelectric tube along the z-axis. A small aluminum cube is bonded to the upper end of the tube. This cube represents the seat where the materials that need to be scanned are placed. The heads of two ADE Technologies 4810 capacitive sensors are placed in close proximity to the adjacent faces of the aluminum cube in the x y directions, respectively, see Fig. 2. The inner electrode of the piezoelectric tube is grounded. One electrode each of the x x y y pairs, referred as x + y +, respectively, are chosen as the input ends of the piezoelectric tube, the corresponding opposite ends, referred as x y, respectively, are chosen as the output ends of the piezoelectric tube. In other words the piezo-patches corresponding to the x + y + are used as actuators the piezo-patches corresponding to the x y electrodes act as the sensors. The whole setup, i.e. the piezoelectric tube with the bonded aluminum cube the heads of the capacitive sensors, is placed in a specially constructed circular enclosure, see Fig. 4. The circular enclosure protects the experimental setup from external disturbances. 2. System description A piezoelectric tube scanner is a thin-walled cylindrical tube made of piezoelectric material. The inner the outer walls of the piezoelectric tube are finely coated with a layer of copper. The copper coating on the inner 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 Fig. 2. Schematic diagram of the experimental setup.
3 702 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Fig. 5. Input output diagram. C x C y of the subsystem C will act as indicators of the lateral displacements of the tube in the x y directions, respectively. 3. System identification Fig. 3. Illustration of the raster pattern. When input voltage signals V x + V y + are applied to the x + y + electrodes, respectively, the piezoelectric tube deforms, inducing voltages V x V y at the output electrodes x y, respectively. Furthermore, due to the deformation of the tube, the capacitance between the aluminum cube the heads of the capacitive sensors change. The change in the capacitance is measured outputted by the capacitive sensors in terms of the distance between its head the aluminum cube. This distance, denoted by C =[C x, C y ] T, is also recorded as an output. In this paper, the piezoelectric tube setup is interpreted as a linear system incorporating two linear subsystems V (V denoting voltage) C (C denoting the capacitance). With both V C having the same inputs, [V x +,V y +] T, but different outputs [V x,v y ] T [C x, C y ] T, respectively, see Fig. 5. The outputs Fig. 4. The piezoelectric tube mounted inside an aluminum shield. The x-axis capacitive sensor is shown secured at right angles to a cube mounted onto the tube tip. And the y-axis capacitive sensor is secured at right angles to the perpendicular face of the aluminum cube. In this section the modeling of the linear systems V C are discussed. The subsystem V is of the form Y v (s) G v (s)u(s), (3.1) 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 [ ] Gxx (s) G xy (s) G v (s) = (3.2) G yx (s) G yy (s) isa2 2 matrix of transfer functions. And the subsystem C is of the form Y c (s) G c (s)u(s), (3.3) where Y c (s) is the Laplace transform of the capacitive sensor outputs [C x, C y ] T, [ ] Gxcx (s) G xcy (s) G c (s) = (3.4) G ycx (s) G ycy (s) isa2 2 matrix of transfer functions. Since the tube, along with the bonded aluminum cube, is symmetric with respect to any plane containing the vertical axis of the tube, see Fig. 2, in principle, it is expected that G xx (s)=g yy (s) G xy (s)=g yx (s). Similarly, due to symmetry in the alignment of the capacitive sensors with the faces of the aluminum cube in x y directions, see Fig. 2, it is also expected that G xcx (s) = G ycy (s) G xcy (s) = G ycx (s). To determine the transfer functions G v (s) G c (s), an experiment is performed on the piezoelectric tube by inputting swept sine waves into the electrodes x + y +, recording the corresponding voltage outputs at x y the capacitive sensor outputs C x C y using a HP 35670A dual channel Spectrum Analyser. The Spectrum Analyser is also used to process the recorded input output data to obtain frequency response functions (FRF) G v (iω) G c (iω) corresponding to the transfer functions G v (s) G c (s), respectively, in the non-parametric form. In Figs. 6 7 the magnitude the phase of the FRFs G xx (iω), G yy (iω), G xy (iω) G yx (iω), obtained from the Spectrum Analyser, are plotted. It can be noted from the plots that G xx (iω) G yy (iω). However, the same cannot be said for the
4 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Fig. 6. Bode magnitude plots of the non-parametric models (solid) along with their corresponding parametric models (dashed dots). cross coupling terms G xy (iω) G yx (iω). The magnitude plots of the cross terms in Fig. 6, suggest small but noticeable differences in frequency regions near the resonance frequency. It is also worth noting that the cross coupling FRFs G xy (iω) G yx (iω), appear to have zeros close to their resonant peak. In Figs. 8 9 the magnitude the phase of the FRFs G xcx (iω), G xcy (iω), G ycx (iω) G ycy (iω) are plotted. As in the case of FRFs corresponding to the voltage subsystem V it can be noted that G xcx (iω) G ycy (iω), but the cross coupling terms G xcy (iω) G ycx (iω) differ from each other. In fact, here, the cross coupling terms differ more than their corresponding counterparts in the subsystem V. However, unlike the voltage subsystem V, the cross coupling terms G xcy (iω) G ycx (iω) in the subsystem C are negligible in magnitude, Fig. 7. Bode phase plots of the non-parametric models (solid) along with their corresponding parametric models (dashed dots).
5 704 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Fig. 8. Bode plots of the non-parametric models (solid) along with their corresponding parametric models (dashed dots). except at the frequencies around the resonance, when compared with the direct terms G xcx (iω) G ycy (iω), see Fig. 8. In other words, input V x + will have little effect on the capacitive sensor output C y in the y direction, unless the frequencies close to the resonance are excited. Similarly, input V y + will have little effect on the capacitive sensor output C x in the x direction, unless the regions close to the resonance are excited. In summary, the direct terms G xx (iω), G yy (iω), G xcx (iω) G ycy (iω) are in agreement with the expectations arising due to symmetry. However, the cross coupling terms do not concur with those expectations. The discrepancies observed in the cross coupling terms are possibly due to manufacturing defects, e.g. the tube not being a homogeneous or the walls of the tube are of non uniform in thickness, or engineering defects like extra dynamics added by the wires attached to the electrodes of the Fig. 9. Bode plots of the non-parametric models (solid) along with their corresponding parametric models (dashed dots).
6 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Table 1 Parameter values of the FRFs G xx (s), G yy (s), G cxx (s) G cyx (s) k σω ω d c c c tube or extra dynamics added by the glue that was used to bond the tube to its base to bond the aluminum cube on its top. Given the fact that cross coupling terms G xcy (iω) G ycx (iω)in subsystem C have negligible magnitude, except near resonances, these defects could have a large influence on them. Since there is only one resonance frequency in the direct term FRFs G xx (iω), G yy (iω), G xcx (iω) G ycy (iω), in the frequency regions presented in the plots, second order models are fit to their corresponding non-parametric data using stard techniques. The following models were found to fit the non-parametric data: G xx (s) = G yy (s) = k 1 s 2 + 2σωs + ω 2 + d 1, (3.5) G xcx (s) = G ycy (s) = c 1s 2 + c 2 s + c 3 s 2 + 2σωs + ω 2, (3.6) where the model parameters are as tabulated in Table 1. In Figs. 6 7 magnitude phase of the parametric fit (3.5) is plotted along with the non-parametric data. In Figs. 8 9 magnitude phase of the parametric fit (3.6) are plotted along with the corresponding non-parametric data. Parametric fits for cross coupling terms G xy (s), G yx (s), G xcy (s) G ycx (s) are not presented here as they are not used for actuating the tube. It can be inferred from Figs. 6 9 that the parametric models fit the non-parametric data reasonably well in the frequency regions plotted. Details on how the models are estimated are not presented here as the models are of low order their estimation is not difficult. It is worth noting that the static gains of the FRFs in Figs. 6 8 are not 0 db. Infact, the static gains of the direct terms are G xx (0) = G yy (0) 10.1 db, G xcx (0) = G ycy (0) 28.5 db, (3.7) while the static gains of the cross coupling term are G xy (0) = G yx (0) 15.5dB. The static gains G xcy (0) G ycx (0) were too low to be properly determined. The above mentioned values are experimentally checked the model presented for the direct terms in paper match them. In the current context, a 0 db gain with respect to the FRF G xx (iω) implies that an input V x + = A sin (ωt)atthe electrode x + would result in an output V x = A sin (ωt + φ) at the electrode x, where φ is the phase lag introduced by the system. Since the output voltage V x is caused by the deformation of the tube in the x direction it is directly related to the bending or deformation. However, a precise characterization of the relationship between the bending V x is not known. Some attempts have been made to characterize this relationship, see [23] [15], though they have not been comprehensive. In the case of the FRF G xcx (iω), a 0 db gain refers to the case where a sine wave input of V x + = A sin (ωt) atthex + electrode makes the tube oscillate as a sine wave with same frequency amplitude in the x direction. However, unlike the unit less G xx (iω), the FRF G xcx (iω) has a unit m/v. The models (3.5) (3.6) have the same set of poles but have a different set of zeros. This is similar to the case of a clamped beam where the transfer function corresponding to the collocated sensor-actuator patch the transfer function corresponding to the actuator the tip displacement have the same set of poles but different set of zeros, see [17] for details. Furthermore, the d term in (3.5) is also similar to the feed-through term in a clamped beam meant for alleviating the undesirable effects caused due to neglecting the higher order modes of the beam, see [17]. 4. 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 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 a very slowly increasing ramp in the y + electrode as reference signals for the system. In fact, to have a good scan of the surface the changes in y direction must be quasi-static with respect to the changes in the x direction. Normally for illustration purposes the slowly varying ramp in the y + electrode is either replaced by a dc signal or assumed to be earthed or open circuited, see [1,6,14]. In Figs we have plotted the capacitive sensor responses C x C y, respectively, to a triangle wave input of 30 V 40 Hz at the x + electrode with the y + electrode being Fig. 10. Response recorded by the capacitive sensor C x for a triangular waveform input with amplitude 30 V fundamental frequency 40 Hz.
7 706 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Fig. 11. Response recorded by the capacitive sensor C y for a triangular waveform input with amplitude 30 V fundamental frequency 40 Hz. earthed. Note that the capacitive sensor output C x is not exactly triangular waveform, but appears to be equal to a triangular waveform plus certain periodic corrugations. This implies that the lateral displacement of the tube along the x-axis is not really a straight line but rather a highly corrugated straight line. The distortion in the capacitive sensor output C x is due to the amplification of the 21st the 23rd harmonics of the triangular waveform which are close to the resonance frequency of the piezoelectric tube. In the case of the capacitive sensor output C y, it is apparent from Fig. 11 that C y is also a periodic triangular waveform but with an amplitude that is negligible when compared with the amplitude of C x. This is not surprising due to the weak cross coupling that exists between the input V x + the output C y. In general to eliminate the periodic corrugations in the output at C x, two different approaches are taken. In the first approach, instead of using a triangular waveform inputs to x + are shaped such that the output at C x is a triangular waveform. More specifically the input at x + is set to u(t) where a k H(iω k ) sin (ω k t), (4.1) k=1 1 H(iω) (4.2) G xcx (iω) a k ω k are such that f d (t) a k sin (ω k t) (4.3) k=1 is the desired triangular waveform output at C x in the Fourier series form. It is easy to see that inputting u(t) (4.1) at x + would give f d (t) at the output C x. The second approach is to use a feedback controller that would damp the resonances in G xcx (s), then input a triangular waveform to the closed loop system. Note that damping the resonance peaks in G xcx (s) would Fig. 12. Response recorded by the capacitive sensor C x for an input u(t) of the form (4.1) with f d (t) being a triangular waveform with amplitude 30 V fundamental frequency 40 Hz. automatically suppress the amplification of the harmonics of the triangular waveform that are close to the resonance. A first look at the two approaches would suggest that the first method is more prudent simpler than the second. In Fig. 12, we have plotted the capacitive sensor response C x to an input of the form (4.1) with f d (t) being the Fourier decomposition of a 30 V amplitude 40 Hz triangular waveform G xcx (s) asin(3.6). The capacitive sensor response plotted in Fig. 12 does appear to be a smooth triangular waveform. However, this approach, is heavily dependent on the correctness of the model G xcx (s), consequently suffers from lack of robustness towards model uncertainties. In general, due to wear tear other external influences, piezoelectric tube characteristics such as gain resonance frequency are prone to minor changes or perturbations. In particular, when the system s resonance frequency gets perturbed inputting u(t) would not result in a triangular waveform at the output. To illustrate this, in Fig. 13 the response C x of the capacitive senor to the input u(t) is plotted with the resonance of the tube perturbed artificially by placing a nonconducting mass on the bonded aluminum cube. Note that periodic corrugations, similar to the one seen in Fig. 10, appear in Fig. 12 also. Therefore to minimise the effects of these perturbations, it is imperative to damp the resonant peak. In this paper both approaches are adopted. First a feedback controller is constructed, linking the voltage output V x to the input V x +, to damp the vibrations, then an input of the type u(t) (4.1) with H being the inverse of the closed loop system is designed to get the desired triangular waveform at the output at C x, see Fig Positive Velocity Position Feedback controller In this section, the concept of Positive velocity Positive Position (PVPF) control is introduced a controller of this type is designed to damp the resonant peak in the transfer function G xx (s).
8 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) In (5.3) (5.4) z is the controller state (5.2) y(t) is system output (5.1). Using (5.3) (5.4) in (5.1) (5.2), respectively, gives ẍ + 2σωẋ + ω 2 x = Ψ 1 (z + r) y = Ψ 2 x + d(z + r) (5.5) z + 2ξwż + w 2 z = Γ 1 ẏ + Γ 2 y, (5.6) respectively. Note that (5.6) boils down to Fig. 13. Response recorded by the capacitive sensor C x, for an input u(t)ofthe form (4.1) with f d (t) being a triangular waveform with amplitude 30 V fundamental frequency 40 Hz, when the system resonance frequency is perturbed. For technical ease, we rewrite G xx (s) in stard second order form in the time domain, ẍ + 2σωẋ + ω 2 x = Ψ 1 u y = Ψ 2 x + du (5.1) where Ψ 1 = , Ψ 2 = 1 d = , see (3.5). Positive Velocity Position Feedback controller is defined by z + 2ξwż + w 2 z = Γ 1 v + Γ 2 v, (5.2) where v is the input to the controller, ξ, w, Γ 1 Γ 2 are the design parameters. If r(t) is the desired system output signal, generally referred to as the reference signal, then the corresponding input to the system (5.1) is set to u(t) z + r (5.3) the controller input is set to v(t) y(t). (5.4) Fig. 14. Closed loop system along the feed-forward input. Here r(t) denotes the reference signal, which in the current context is the triangular waveform f d (t) (4.3), u(t) denotes the designed input of the form (4.1) H, denotes the inverse of the FRF of the closed loop capacitive sensor response along the x-axis. { z+2ξwż+w 2 Γ1 ẏ + Γ 2 y z = which implies that Γ 1 (Ψ 2 ẋ+dż + dṙ) + Γ 2 (Ψ 2 x+dz + dr), (5.7) z + (2ξw Γ 1 d)ż + (w 2 Γ 2 d)z = Γ 1 (Ψ 2 ẋ + dṙ) + Γ 2 (Ψ 2 x + dr). (5.8) Setting 2ξ 1 w 1 2ξw Γ 1 d (5.9) w 2 1 w2 Γ 2 d, (5.10) Eq. (5.8) can be rewritten as z + 2ξ 1 w 1 ż + w 2 1 z = Γ 1(Ψ 2 ẋ + dṙ) + Γ 2 (Ψ 2 x + dr). (5.11) Clubbing (5.5) (5.11) rewriting it in matrix form gives [ ] [ ][ẋ ] [ ][ ] ẍ 2σω 0 ω 2 0 x + + z 0 2ξ 1 w 1 ż 0 w 2 1 z [ ][ẋ ] [ ][ ] Ψ1 x = + Γ 1 Ψ 2 0 ż Γ 2 Ψ 2 0 z [ ] [ ] 0 Ψ1 + ṙ(t) + r(t), (5.12) Γ 2 d Γ 2 d which can be rewritten as [ ] [ ][ẋ ] [ ][ ] ẍ 2σω 0 ω 2 Ψ 1 x + + z Γ 1 Ψ 2 2ξ 1 w 1 ż Γ 2 Ψ 2 w 2 1 z [ ] [ ] 0 Ψ1 = ṙ(t) + r(t). (5.13) Γ 2 d Γ 1 d Note that the system output is equal to [ ] x y(t) = [Ψ 2 d] + dr(t). (5.14) z
9 708 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Laplace transform of (5.13) (5.14) gives [ ][ ] s 2 + 2σωs + ω 2 Ψ 1 x(s) w 2 1 = w2 Γ 2 d, (5.29) (Γ 1 Ψ 2 s + Γ 2 Ψ 2 ) s 2 + 2ξ 1 w 1 + w 2 1 z(s) see also (5.9) (5.10), the choice of K 1, K 2, K 3 K 4,or [ ] alternatively the desired pole positions {p i } 4 Ψ i=1, should be such 1 = r(s) that the controller damping 2ξw the square of the controller Γ 2 ds + Γ 1 d frequency w 2 do not become negative. In other words the choices should be such that [ ] x(s) 2ξw = 2ξ y(s) = [Ψ 2 d] + dr(s) (5.15) 1 w 1 + Γ 1 d>0 (5.30) z(s) It is evident from (5.15) that the poles of the closed loop system w 2 = w 2 1 are the roots of the determinant of the polynomial matrix + Γ 2d>0. (5.31) [ ] s 2 + 2σωs + ω 2 Using constraints (5.30) (5.31) controller stability can be Ψ 1 A (Γ 1 Ψ 2 s + Γ 2 Ψ 2 ) s 2 + 2ξ 1 w 1 + w 2. (5.16) characterised in terms of linear inequalities 1 a 1 K 1 + a 2 K 2 + a 3 K 3 >a 4 (5.32) Furthermore, is easy to check that det(a) P(s) (s 2 + 2σωs + ω 2 )(s 2 + 2ξ 1 w 1 + w 2 1 ) Ψ 1Ψ 2 (Γ 1 s + Γ 2 ) (5.17) s 4 + (2σω + 2ξ 1 w 1 )s 3 + (ω 2 + 2σω2ξ 1 w 1 + w 1 )s 2 + (2σωw ξ 1w 1 ω 2 ΨΓ 1 )s + ω 2 w 2 1 ΨΓ 2, where Ψ Ψ 1 Ψ 2. (5.18) Let {p i } 4 i=1 be the desired pole positions Q(s) s 4 + K 1 s 3 + K 2 s 2 + K 3 s + K 4. (5.19) b 1 K 1 + b 2 K 2 + b 3 K 4 >b 4 (5.33) where ( a 1 = 1 d ) Ψ ((2σω)2 ω 2 ) (5.34) Matching the coefficients of (5.17) (5.19) gives 2σω + 2ξ 1 w 1 = K 1 (5.20) w 2 + 2σω2ξ 1 w 1 + w 2 1 = K 2 (5.21) a 2 = 2σω d Ψ a 3 = d Ψ (5.35) (5.36) 2σωw ξ 1w 1 ω 2 ΨΓ 1 = K 3 (5.22) ω 2 w 2 1 ΨΓ 2 = K 4. (5.23) Eqs. (5.20) (5.23) imply 2ξ 1 w 1 = K 1 2σω, (5.24) w 2 1 = { K2 ω 2 2σω2ξ 1 w 1 K 2 ω 2 2σω(K 1 2σω), (5.25) Γ 1 = 1 Ψ [w2 1 2σω + ω2 2ξ 1 w 1 K 3 ] (5.26) Γ 2 = 1 Ψ [w2 1 ω2 K 4 ] (5.27) In other words using the PVPF controller (5.2) one could place the poles of the closed loop system (5.13) at any desired location. However as 2ξ 1 w 1 = 2ξw Γ 1 d (5.28) a 4 = 2σω d Ψ ((2σω)3 4σω 3 ) (5.37) b 1 = (2σω + dψ ) 2σω3 b 2 = (1 + dψ ) ω2 b 3 = d Ψ b 4 = (5.38) (5.39) (5.40) [ ω 2 (2σω) 2 + d ] Ψ (ω4 (2σω) 2 ω 2 ). (5.41) 6. Numerical illustrations 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 ± = 30.1 ± i (6.1)
10 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Fig. 15. Frequency of the PVPF controller K PVPF. Here we set the desired closed loop poles to P 1+ = P 2+ = ± i5337.3, P 1 = P 2 = ± i (6.2) 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 K 1 = l0 3, K 2 = , K 3 = K 4 = corresponding to the desired closed loop poles P 1+, P 2+, P 1 P 2 (6.2) satisfy the inequalities (5.32) (5.33). Solving for the controller parameters Γ 1, Γ 2, ξ Fig. 16. Frequency response of the closed loop model (5.15) along with the open loop model (3.5). ω from (5.24) (5.29), we obtain the PVPF controller 1915s K PVPF (s) s s , (6.3) that would render a closed loop system having poles at P 1+, P 2+, P 1 P 2.InFig. 15, the FRF, K PVPF (iω), of the controller is plotted. The low gain the quick roll off of the FRF K PVPF (iω) suggest that the control effort needed to push the real part of the poles p ± (6.1) to P 1+, P 2+,P 1 P 2 is not very high. Hence, the controller K PVPF (s) is very easily implementable. In what follows, we will first evaluate the effectiveness of the PVPF controller in damping the resonance in G xx (s) both numerically experimentally. By numerically, we mean using Fig. 17. Non-parametric FRFs of the closed the open loop systems.
11 710 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) Fig. 18. Response recorded by the capacitive sensor C x for a triangular waveform input of 30 V 40 Hz. the expressions obtained for Γ 1, Γ 2, ξ ω in the closed loop model (5.15) determine the damping introduced by comparing it with the open loop model G xx (s) (3.5). By experimentally, we mean the case where we input swept sine waves in the x + electrode of the tube, while keeping x electrode earthed, determine the non-parametric FRFs of the closed loop transfer functions compare them with their open loop non-parametric counterparts. Furthermore, we also examine the effect of the PVPF controller on the transfer functions G xcx G xcy of the subsystem C. In Fig. 16, we have plotted the frequency response of the closed loop model (5.15), along with the open loop model (3.5). A damping of around 30 db at the resonance is evident from the plot. In Fig. 17 we have plotted the non-parametric frequency response of the closed loop system FRFs obtained experimentally along with the corresponding non-parametric open loop FRFs in G v (iω). It is evident from Figs that in the case of G xx (s) the damping predicted numerically in Fig. 16 is matched by the experimental results presented in Fig. 17. Furthermore experimental results suggest a significant damping of the resonance in the FRF of the cross coupling term G xy (s). As in the case of G xx (s) the effect of the feedback on G xy (s) can be mathematically characterised. However, that characterization is not presented here as it is neither important nor difficult. As mentioned earlier, the objective is to actuate the piezoelectric tube in a raster pattern. In Fig. 18, the capacitive sensor output C x of the closed loop system to a triangular waveform as input are plotted. It is apparent that the periodic corrugations in open loop capacitive sensor response C x are not present in the corresponding closed loop response. Nevertheless the closed loop response at C x is not really a smooth triangle waveform, in particular near the peaks of the response. In Fig. 19 the closed loop frequency responses (FRFs) of the capacitive sensors are plotted. Note that, as in the case of the closed loop frequency responses of the voltage subsystem, the closed loop frequency responses of the capacitive subsystem also have their resonances damped. As a consequence harmonics of the triangular waveform close to the resonance are not amplified. However, to get a smooth triangular waveform signal at the output, as in (4.1) we shape the input, in other words we set the input to u(t) a k H(iω k ) sin (ω k t), (6.4) k=1 Fig. 19. Closed loop magnitude response of the subsystem C (dashed dots) along with its open loop counterpart (solid).
12 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) respectively. In Fig. 21 the response C x of the perturbed closed loop system to the filtered input u(t) is plotted. It can be noted that the output is still a triangular waveform similar to the one in Fig. 20. Thus, confirming the robustness introduced into the system by the PVPF controller. 7. Conclusion Fig. 20. Response recorded by the capacitive sensor C x for an input u(t) (6.4) generated from a 30 V fundamental frequency 40 Hz. where a k ω k are the Fourier components of the desired triangle wave, 1 H(iω) G (cl) (6.5) xc x (iω) G (cl) xc x (iω) is the FRF of the transfer function G (cl) xc x (s) fitted for the non-parametric data plotted in Fig. 19. The response C x to such an input is shown in Fig. 20, which is a smooth triangular waveform. Finally, to illustrate the robustness introduced into the system by the PVPF controller, as before, the system resonance frequency is perturbed by placing a nonconducting mass on the bonded aluminum cube. The same PVPF controller K PVPF (6.3) the filtered input u(t) (6.4) used in the case of the unperturbed system, are used as the feedback controller the input, In this paper a piezoelectric tube of the type typically used in STMs AFMs was considered. This piezoelectric tube was interpreted as a linear system, a linear model was constructed for it using stard 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 using a triangular waveform as the control signal next by a custom designed control signal, which was obtained by inverting the frequency response function of the linear model. The custom designed control signal was expected to actuate the tube accurately, if the linear model for the piezoelectric tube was reasonably accurate. In the former case the piezoelectric tube appeared to track a waveform which was roughly a raster pattern plus certain periodic vibrations. It was observed that these periodic vibrations were due to the amplification of the harmonics of the control signal (triangular waveform) that were close to the resonance frequency. In the case of the custom designed control signal, even though the piezoelectric tube would track a raster pattern, it was noted the custom designed inputs were not robust to model uncertainties. Hence, it was concluded that damping the resonance mode was necessary to damp the unwanted vibrations of the open loop system also to make the system more robust to model uncertainties. Concept of a PVPF controller was introduced a methodology for designing them to robustly damp the resonance mode was presented. The PVPF controller due to its simplicity in structure is easily implementable both in hardware software. Experimental results suggested that more than a 30 db damping of the resonance mode can be achieved using a PVPF controller. The experimental results were also in agreement with theoretical predictions simulations. Custom designed control signals for the closed loop system were designed to actuated the piezoelectric tube. These control signals not only actuate the piezoelectric tube in a raster pattern but were also robust to model uncertainties. In this work, non-linearities like creep hysteresis have been avoided by not using low-frequency signals high amplitude signals as inputs. However, the design of the experimental setup is open for extensions, including extending the models to compensate for creep, as well as implementing a charge rather than voltage control to compensate for hysteresis. Acknowledgements Fig. 21. Response recorded by the capacitive sensor C x for an input u(t) (6.4) generated from a 30 V fundamental frequency 40 Hz, with the system resonance frequency perturbed. 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 making it simple to perform the experiments,
13 712 B. Bhikkaji et al. / Sensors Actuators A 135 (2007) also for the many enlightening discussions on Piezoelectric materials. References [1] A. Daniele, S. Salapaka, M.V. Salapaka, M. Daleh, Piezoelectric tubes for atomic force microscopes: design of lateral sensors, identification control, in: Proceedings of the American Control Conference, San Diego, California, 1999, pp [2] C. Choi, K. Park, Self-sensing magnetic levitation using LC resonant circuit, Sens. Actuators 72 (1999) [3] B. Bhushan (Ed.), Springer Hbook of Nanotechnology, Springer- Verlag, Heidelberg, Germany, [4] G. Binning, C.F. Quate, C. Gerber, Atomic force microscopes, Phys. Rev. Lett. 56 (9) (1986) [5] G. Binning, H. Roher, C. Gerber, E. Weibel, Scanning tunneling microscopes, Phys. Rev. Lett. 49 (1982) 57. [6] G. Schitter, A. Stemmer, Identification open-loop tracking control of a piezoelectric tube scanner for high-speed scanning probe microscopy, IEEE Trans. Control Syst. Technol. 12 (3) (2004) [7] G. Song, B.N. Agrawal, Vibration suppression of flexible spacecraft during attitude control, Acta Astronautica 49 (2) (2001) [8] G. Song, S. Schmidt, B.N. Agrawal, Experimental study of vibration suppression of flexible spacecraft using modular control patch, in: Proceedings of IEEE Aerospace Conference, Snowmass, Colarado, [9] J.J. Dosch, D.J. Leo, D.J. Inman, Comparison of vibration control schemes for a smart antenna, in: Proceedings of the 31st Conference on Decision Control, Tucson, Arizona, 1992, pp [10] J.L. Fanson, T.K. Caughey, Positive position feedback control for large space structures, AIAA J. 28 (4) (1990) [11] K.K. Leang, S. Devasia, Hysteresis, creep vibration compensation for piezoactuators: feedback feedforward control, in: Proceedings of 2nd IFAC Conference on Mechatronic Systems Berkeley, California, USA, 2002, pp [12] L. Ljung, System Identification: Theory for the User, Prenctice-Hall, Upper Saddle River, NJ, [13] M. Ratnam, B. Bhikkaji, A. Fleming, S.O.R. Moheimani, PPF control of a piezoelectric tube scanner, in: Proceedings of 44th IEEE Conference on Decision Control European Control Conference ECC 2005, Seville, Spain, December [14] N. Tamer, M. Daleh, Feedback control of piezoelectric tube scanners, in: Proceedings of the 33rd Conference on Decision Control, Lake Buena Vista, Florida, 1994, pp [15] O.M. El Rifai, K. Youcef-Toumi, Coupling of piezoelectric tube scanning in scanning probe microscopes, in: Proceedings of the American Control Conference, Arlington, Virginia, 2001, pp [16] R. Pintelon, J. Schoukens, System Identification: A Frequency Domain Approach, IEEE press, New York, [17] S.O.R. Moheimani, D. Halim, A.J. Fleming, Spatial Control of Vibration: Theory Experiments, World Scientific, Singapore, [18] S. Salapaka, A. Sebastian, Control of nanopositioning device, in: Proceedings of the 42nd IEEE Conference on Decision Control, Maui, Hawaii, 2003, pp [19] S.O.R. Moheimani, B.J.G. Vautier, B. Bhikkaji, PPF control of an active structure, in: Proceedings of 44th IEEE Conference on Decision Control European Control Conference ECC 2005, Seville, Spain, December [20] T. Soderstrom, P. Stoica, System Identification, Prentice Hall International, Hemel Hempstead, UK, [21] T.K. Caughey, C.J. Goh, Analysis control of quasi distributed parameter systems, Californial Inst. of Technology, Pasadena, CA, Dynamics Lab. Rept, DYNL-82-3, [22] T. Ohara, K. Youcef-Toumi, Dynamics control of piezo tube actuators for subnanometer precision applications, in: Proceedings of the American Control Conference, Seattle, Washington, 1995, pp [23] M.E. Taylor, Dynamics of piezoelectric tube scanners for scanning probe microscopy, Rev. Sci. Instrum. 64 (1) (1993) Biographies Bharath Bhikkaji received the PhD degree in signal processing from the Uppasla University, Uppsala, Sweden, in the year He is currently a Research Academic at the School of Electrical Engineering Computer Science, University of Newcastle, Newcastle, Australia. His research interests include System Identification, Robust Control Active noise Vibration control of Flexible structures. Marcel Ratnam received the BEng degree in computer engineering BMath in 2003, 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 bwidth resolution. Reza Moheimani received the PhD degree in electrical electronic engineering from the University of New South Wales in 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 Computer Science, the director of Laboratory for Dynamics Control of Smart Structures a programme leader for the ARC Centre for Complex Dynamic Systems 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, has chaired a number of international workshops conferences. He has published two books, several edited volumes over 150 articles in areas of robust control estimation, smart structures, active noise vibration control, mechatronic systems nanotechnology.
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