Modeling nonlinear behavior in a piezoelectric actuator

Modeling nonlinear behavior in a piezoelectric actuator

H. Richter, E.A. Misawa*, D.A. Lucca, H. Lu1

School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078-5016, USA

Received 19 March 1999; received in revised form 22 June 2000; accepted 4 October 2000

Abstract

A piezoelectric tube actuator is employed as a sample positioning device inNanocut, a cutting instrument conceived to study themechanics of nanometric cutting. Extension of functionality of the instrument as a nanometric machine tool motivates the search for anaccurate model of the actuator for implementing feedback control. A simple nonlinear model describing longitudinal expansion of thepiezoelectric tube actuator is presented in this paper. The model derivation is based on a non-formal analogy with nonlinear viscoelasticmaterials under uniaxial extension, for which the responses to a step input are similar to the piezoelectric tube. Suitability of the modelstructure for arbitrary inputs is tested by cross-verification between time and frequency domains. Two parameter estimation procedures areexamined and the results of the experimental work for characterization, estimation and validation are presented. © 2001 Elsevier ScienceInc. All rights reserved.

Keywords:Piezoelectric tube; Nonlinear dynamic models; Nanometric cutting; Rheological models; Nonlinear viscoelasticity

1. Introduction

Piezoelectric actuators have become a standard option inpositioning applications where the displacements must besmall and highly accurate. In particular, ultra-precisionmanufacturing requires exceptionally fine and repeatablemotions, making piezoelectric actuators a common choice.These actuators, however, are sensitive to environmentalchanges such as temperature; and moreover, several non-linearities are present in the behavior. The well-known phe-nomena of hysteresis and creep, along with nonlinear volt-age dependence affect the dynamic response, sometimesprecluding closed-loop operation [1]. Consequently, a pre-cise mathematical model is necessary to design feedbackcontrollers properly. This work is motivated by a particularproblem related to ultra-precision manufacturing. A nano-metric cutting instrument, denoted asNanocut, was de-signed and built as a team effort by Oklahoma State Uni-versity and the University of North Carolina at Charlotte, asa means to investigate the mechanics of ultra-precisionmachining [2]. The instrument employs a tubular piezoelec-tric actuator which achieves sub-nanometer resolution for3D positioning, but not necessarily highlyaccuratemotions

when operated in open loop. This is primarily due to theassociated nonlinear effects, which can cause errors on theorder of microns, while environmental perturbations, suchas temperature changes, cause smaller errors when experi-ments are carried out in a controlled environment. Thepresent work is focused on the axial extension of the piezo-electric actuator, hereafter referred to as “PZT tube”. Anon-formal analogy between the observed behavior and thatof nonlinear viscoelastic materials is established. The mo-tivation for such an analogy is a remarkable similarityobserved in the input/output behavior of the piezo tube andnonlinear viscoelastic materials under uniaxial extension.This analogy is used as a basis to construct the proposeddifferential model. An extensive set of experiments is car-ried out to estimate the parameters of the model. The ex-periments involve both frequency and time domain mea-surements, which are used to cross-examine the model forself-consistency between domains. Finally, an independentset of experiments is performed to validate the numericalmodel.

Piezoelectric materials and actuators have received con-siderable attention in the literature, especially in recentyears, with the advent of new applications based on scan-ning-probe microscopy. Instruments such as the scanningtunneling and atomic force microscope almost universallyuse piezoelectric tube actuators to generate the fine motionsrequired [3]. In recent years PZT actuators have also been

* Corresponding author. Tel.:11-405-744-5904; fax:11-405-744-7873.

E-mail address:[email protected] (E.A. Misawa).

Precision EngineeringJournal of the International Societies for Precision Engineering and Nanotechnology

25 (2001) 128–137

0141-6359/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved.PII: S0141-6359(00)00067-2

introduced to high-density hard disk applications as a sub-actuator for fine head displacement [4]. Much of the pub-lished research, however, is concerned only with modelsbased on the linearized constitutive laws of piezoelectricbehavior. The most widely accepted linear description ofpiezoelectric materials is given in the IEEE Standard onPiezoelectricity published in 1987 as an American NationalStandard [5]. In many instances, actuator geometry andsimple loading conditions -the term “loading” understood asimposed stress and electric field- result in simple secondorder linear transfer functions that map applied voltage toactuator displacement ([8], [7], [9], [10] among many oth-ers.) Considerable research works have been reported in theliterature that describe the linear portion of the behavior ofpiezoelectric actuators, especially for the stacked-type con-struction. The fundamentals of nonlinear piezoelectric phe-nomena are analyzed mostly in the framework of the phys-ics of crystals and thermodynamics, rendering descriptionsthat are extremely difficult to reduce to a set of nonlinearordinary differential equations [11]. As stated in [12], bas-ing model derivation on the IEEE linear constitutive rela-tions requires several assumptions, resulting in an oversim-plified description that fails to capture the essentialnonlinear behavior that is present in all piezoelectric ceram-ics. In the case of PZT tube actuators, perhaps the highestdegree of simplification is given in [13], where the axialmotion is reduced to the description

uzz5 d33eff V

L(1)

whereuzz is the axial strain,d33eff is an effective piezoelectric

constant that considers tube geometry and an average elec-tric field value,V is the applied radial voltage, andL is tubelength. From an engineering modeling perspective, severalworks have been reported that specifically address the non-linearities, for example, Goldfarb and Celanovic [12] uti-lized a Maxwell slip model to describe hysteresis. Anotherwork [21] that focuses on the modeling of hysteresis adaptsthe Preisach mathematical model to the case of piezoelectricactuators, with good agreement between model and exper-iment. In several works ([7], [19]) hysteresis is accountedfor by augmenting the linear dynamics with an empiricaldifferential equation of hysteretic behavior. The main dis-advantage of such an approach is the cumbersome mathe-matical expressions which make it difficult to find a set ofparameters that can match experimental data. An interestingapproach was taken by Tamer [3]. In that work, the lineardynamic description is augmented in the frequency domainby a describing functionof a hysteresis nonlinearity, ren-dering a model that is still linear, but includes the phase-lageffects due to the actual nonlinear phenomenon. Thus, bettercontrollers can be designed, even though the model may notbe accurate in the time domain. This is presented as anad-hoc solution to a particular problem rather than as ageneral treatment. The creep phenomenon that is observed

in piezoelectric materials has been treated in fewer publi-cations. Moreover, these works mostly deal with descrip-tions of creep that are not intended to serve as models forcontrol purposes [14]. Similarly, the nonlinear dependenceof axial displacement on input voltage has not receivedmuch attention. Arguably, a reason for these voids is thefact that both effects may be easily compensated by afeedback controller, since the creep can be regarded as aslow output disturbance and the nonlinear dependence oninput voltage as a gain uncertainty. A simple approach to thecompensation of creep and hysteresis by inserting a capac-itor in series with the piezoelectric actuator was proposed byKaizuka and Siu [6]. In fact, controllers are now commer-cially available that achieve subnanometer resolution andaccuracy for unidirectional actuators. A nonlinear three-dimensional model that encompasses creep, nonlinear volt-age dependence, and hysteresis would constitute an impor-tant contribution to the development of high precisionpiezoelectric tube actuators.

2. Observed behavior

In this section, the experimental setup used throughoutthe research is described first. Then, three nonlinear effectsare identified from experimental results.

2.1. Experimental setup

The piezo actuator is used as a sample-positioning devicein the Nanocut instrument. The instrument is capable ofthree-dimensional positioning of a sample with respect to astationary cutting tool or indenter, and is equipped withforce and displacement transducers which transmit data intoa computer for later analysis. Fig. 1 illustrates the mechan-ical hardware as viewed from the top. The piezoelectric tube

Fig. 1. TheNanocutinstrument.

129H. Richter et al. / Precision Engineering 25 (2001) 128–137

(3) is mounted on the horizontal slide (1) as a cantileverbeam. The application of voltage to the appropriate elec-trodes results in either horizontal/vertical tip displacementby bending, or longitudinal expansion of the piezoelectrictube. A capacitance gage is positioned inside the PZT tubeto measure changes in longitudinal motion. The piezoelec-tric material employed in the actuator is PZT-5 (Lead-Zirconate-Titanate).2 Fig. 2 shows a functional diagram ofthe open-loopz positioning system. The commanded posi-tion in microns is sent to a 16 bit D/A board on a PC-compatible computer. The resulting analog voltage is am-plified and applied to thez electrode of the PZT tube and theexpansion measured by the capacitance gage is amplified tobe read by the 16 bit A/D board. The capacitance gage andits signal conditioner have a combined sensitivity of 3.96 Vmm21, resulting in a A/D resolution of 0.077 nm/bit. ThePZT tube has a nominal3 sensitivity of 8.33 nmV21, and amaximum displacement of 1.5mm, imposed by the con-struction ofNanocut. This implies a maximum of 180 Vbetween electrodes. The tube and power amplifier have anaverage combined sensitivity of 0.15mmV21, resulting in aD/A resolution of 0.046 nm/bit. These resolutions are farfrom being fully effective in practice due to environmentalperturbations and the nonlinear effects under study. A moredetailed description ofNanocutis reported elsewhere [2].

2.2. Three nonlinear effects

2.2.1. Nonlinear gainWhen several constant voltage inputs are applied to the

z-electrodes, the corresponding responses are not propor-tional to the applied voltage. A set of experiments wasperformed to evidence the effect. Power amplifier transienteffects were eliminated from the experiment by using aswitch, to better simulate a step input. For each input volt-age, ranging from 15 to 120 Volts, data were recorded attwo different resolutions. Fig. 3 (left) represents short-termbehavior, sampled every 5ms, i.e., at 200 kHz. The plot onthe right shows the same experiment, with data sampledevery 500ms, (2 kHz), to evidence the slow creep phenom-enon. In both cases the response has been normalized bydividing by the input value. Physically, the source of non-linearity in this case is the voltage-dependency of the pi-ezoelectric constants. A damped oscillation near 12 kHz isobserved in the response. This is possibly a structural modeof the PZT tube, not to be modeled in this work.

2.2.2. Nonlinear creepThe steady creep feature is evident in Fig. 3 (right). The

rate of creep is sensibly constant for each input voltage, up

to the time-frame of the experiment (about 10 seconds).Physically, creep is caused by a gradual alignment of di-poles, and therefore expansion -and notrate of expansion-should converge to a steady value. Convergence isachieved, however, over a period of time that is much largerthan the time constant of any practical control system. Creepis deemed to be nonlinear based on the fact that expansionrates depend on voltage in a nonlinear fashion, as will beshown later.

2.2.3. Frequency behavior and hysteresisA set of responses corresponding to sinusoidal inputs of

several amplitudes and frequencies was recorded. The inputfrequency is observed to dominate the response up to thefirst few kilohertz. Significant distortion and appearance ofharmonics is observed beyond 5 kHz. Fig. 4 shows themagnitude and phase of the fundamental frequency compo-nent. The data represented correspond to a experimentalsinusoidal input describing function, or SIDF. A nonlineargain characteristic is clear in this representation. Fig. 5

Fig. 2. Functional Diagram.

Fig. 3. Nonlinear Response to a series of step voltage inputs.

Fig. 4. Sinusoidal Input Response.

130 H. Richter et al. / Precision Engineering 25 (2001) 128–137

shows a set of experimental hysteresis loops. A 60 V trian-gular voltage at several frequencies was applied to theactuator electrodes. As discussed next, phase lag due tolinear dynamics is embedded in the loops, but the presenceof rate-independent hysteresis makes the effect nonlinear asa whole.

2.2.4. Hysteresis in context: some important statementsThere is no complete agreement in the use of the term

“hysteresis” among the various research fields where mem-ory effects are present. In the case of piezoelectric actuators,only a few works explicitly point out the nature of thememory effect that is under consideration [12]. The term“hysteresis” is usually attributed to rate-independent mem-ory, although some material has been published in which theterm is employed to describe rate-dependent phenomena[15]. Some other publications employ the termsstatic hys-teresisand dynamic hysteresis[20] to distinguish the twoeffects. An important observation is the fact that rate-inde-pendent effects are not found isolated from rate-dependentmemory, except in idealized situations. The relative impor-tance of the two effects should be assessed before includingeither in a mathematical model. In piezoelectric actuatorsand in many other engineering systems, it is found that therate-dependent memory has a greater importance at highfrequencies, because of the common phase-lag characteris-tics of most physical devices. Rate-independent phenomenaprevails at low frequencies for the same reasons. In fact, itis a common practice to reveal the effect in experiments byapplying slow cyclic inputs, so that the phase lag due torate-dependent memory is negligible. This practice maylead to erroneous results, however, when the system is onlylinear, but contains poles at or near the origin. In this case,for sufficiently slow inputs, a loop is obtained which may beerroneously regarded as hysteretic. For the case of thepresent PZT tube, the phase-lag due to the linear portion of

the dynamics is seen to prevail over most of the frequencyrange of interest. It is because of this observation that theterm “hysteresis loop” is used throughout this work to referto thecoupledmemory effect. The proposed model, beingof a viscoelastic type, doesnotaccount for rate-independentmemory. It does provide sufficiently accurate predictions ofthe coupled hysteresis loops, however.

2.3. Models for nonlinear viscoelastic behavior

A large number of models have been proposed and testedover the years that explain the observed phenomena to acertain degree of accuracy, but there is no all-embracingtheory which is likely to accommodate all eventualities[17]. As in other fields, models can be constructed byempirical observation or by physical considerations. Phys-ical models for nonlinear viscoelastic behavior are still indevelopment, and their reduction to a set of manageableordinary differential equations constitutes a difficult task.For this reason, a phenomenological model for nonlinearviscoelasticity is employed as a starting point. The modelpresented here can be traced back to the work of Marin andPao, [16]. Even though the theory is based on empiricalobservation of the stages of creep response when a constantstress input is applied, the resulting model can still be linkedto a nonlinear Burgers rheological model [16, 18]. TheBurgers model in the linear case is of wide application inlinear viscoelasticity, and its parameters are physically well-defined quantities, such as viscosity and elastic modulus.The equations for variable input represent the behavior of anonlinear spring-dashpot system, as shown in Fig. 6. In the

Fig. 5. Experimental Hysteresis Loops.

Fig. 6. Simple Nonlinear Burgers Model.


linear case, this is known as the four-element Burgersmodel. In Fig. 6 the linear spring corresponds to an instan-taneous elastic response. The single nonlinear dashpot re-lates to the steady creep component for constant stress, andthe parallel arrangement of a nonlinear spring and dashpotrepresents a transient component.4 This parallel arrange-ment, in the linear case, is known as a Voigt element.

2.4. A non-formal analogy

The input/output nonlinear features observed in the pi-ezoelectric actuator match -at least in a qualitative sense-those observed in nonlinear viscoelastic materials underuniaxial extension. In order to construct a model, stress andstrain in the viscoelastic case can be interpreted as voltageand longitudinal expansion, respectively. A differentialmodel that describes the basic Burgers model represented inFig. 6 is given as

«̇s 5 S s

hcDm

(2)

«̇t 5 2l«t 1 lS s

mcDp

(3)

« 5 «s 1 «t (4)

where«s and «t represent the steady and transient com-ponents of the total creep,« that is the response to anapplied stresss. Note that whenp 5 1, the spring anddashpot in the linear Voigt element represent an elasticmodulusmc, and a viscosity coefficientb 5 mc/l. Simi-larly, in the linear case, when m5 1, hc represents aviscosity coefficient. Whenp . 1, as usually found ex-perimentally, the parallel arrangement must be treated asa “black-box”, and it is not possible nor relevant to findisolated parameters that correspond to viscosity and elas-ticity, in a strict sense. However, ifp is close to one, asit is usually the case, the physical significance ofm andm/l is maintained for practical purposes. The same argu-ments apply to the nonlinear dashpot representing steadycreep.

Upon application of the analogy, nonlinear dynamic be-havior of the PZT tube can be treated as nonlinear viscoelas-tic response, which has received more attention than thepiezoelectric case. Of course, the analogy is of limited usein gaining physical insight about the piezo actuator, butonce more, the goal is to produce a good control model.This approach is conceptually similar to the one proposedby Goldfarb and Celanovic [12], replacing the elasto-slideelements by Voigt units. The present approach has theadvantages of being able to capture the observed creepeffect, which is ignored in [12], as well as the nonlineardependence on input voltage.

2.5. The proposed model

The model consists of a series arrangement ofn Voigtunits and a nonlinear dashpot representing nonlinear creep.The equations presented here result from extending andmodifying the equations for nonlinear viscoelasticity pre-sented in [18]. The derivations are rather simple, but toolengthy to be included in this paper. Using then transientcomponents and the steady creep component as states, themodel equations can be written in state-space form. Notethat the states are completely decoupled.

5«̇v1

5 2l1«v11 l1r1ft~V~t!!

«̇v25 2l2«v2

1 l2r2ft~V~t!!···

···«̇vn

5 2ln«vn1 lnrnft~V~t!!

«̇s 5 fs~V~t!!

(5)

with output equation

« 5 «v1 1 «v2 1 · · ·1 «vn1 «s (6)

Also, the following restriction is imposed

Oi51

n

r i 5 1 (7)

In the above equations,l i andr i are characteristic frequen-cies and weights, respectively. The “transient” response to astep input then becomes a weighted sum of exponentialswhich converges to unity, with a common nonlinear gainrepresented by the functionft(V(t)). The overall response iscomposed of the transient response and a permanent creeprepresented by the last state. Power functions of voltage areto be tested for experimental validity. Note that when theexponents are non-integer values, which are likely to arisefrom the parameter estimation procedures, the power func-tions are complex-valued. To avoid this inconvenience, thefollowing modification is introduced:

ft~V! 5 sign~V!SuVumDp

(8)

fs~V! 5 sign~V!SuVuhDm

(9)

where sign(V) 5 uVu/V denotes the signum function. This,however, implies that the step response must be an oddfunction of V, becausef(2V) 5 2f(V) for the chosenpower functions. This was experimentally confirmed byapplying negative and positive steps of the same magnitudeand comparing the outputs.

2.5.1. Computation of the sinusoidal input describingfunction (SIDF)

The SIDF of the model is found to be particularly usefulfor cross-verification between time and frequency domains,as described later. The SIDF is provided below


N~ A, w! 5 Ns~ A, w! 1 Oi51

n

Ni~ A, w!, where: (10)

Ni~ A, w! 5Ap21

mp

2G~ p/ 2 1 1!

Îp G~ p/ 2 1 3/ 2!

l ir i

l i 1 jw(11)

Ns~ A, w! 5Am21

hm

2G~m/ 2 1 1!

Îp G~m/ 2 1 3/ 2!

1

jw(12)

The sinusoidal input amplitude is denoted byA andw is thefrequency. Also,G( x) is the Gamma (generalized factorial)function.

2.6. Parameter estimation from time domain data

Here we assume that model parameters are to be identifiedfrom a series of step-voltage responses. This is the easiest wayto estimate parameters using an oscilloscope with data storagecapability. In this case, it is assumed that the input functions inthe model given by Eq. (6) are power laws of the form

ft~V! 5 SV

mDp

fs~V! 5 SV

hDm

It is also considered that there is no “instantaneous” elasticcomponent, as such an approximation is granted only fortrue viscoelastic materials, in which the subsequent expan-sion is slower than the initial fast transient by orders ofmagnitude. The set of parameters to be estimated consists ofmodel ordern 1 1, then characteristic frequenciesli andtheir weightsr i, and the parameters from the above powerlaws, namely,p, m, m, h. These parameters are to berecovered from the step responses and creep data describedearlier for seven input voltages. Note that parameter recov-ery from a step response, which is not a frequency-richsignal, is justified by the fact that a resonance-free modelhas been chosena priori, and the estimation is based on thesolutionsof the differential equation rather than in the formof the equations themselves. Having completed the param-eter estimation, the model given by equations (6) can bereadily constructed and tested for inputs other than a step.

2.7. Experimental results 1: time-domain estimation

2.7.1. Steady creepThe steady rates of creep are extracted using a regression

line fitted to the last portion of the long-term data. Theslopes of the regression lines are the constant rates to befitted to a voltage power law. To fit the obtained slopes to apower law of voltage, another linear regression is applied.The resulting parameters are summarized in Table 1.

2.7.2. Transient componentOnce the steady creep is characterized, it is subtracted

from the overall creep response, leaving the transient com-ponent for parameter identification. To verify the constantcreep rate assumption, a regression line is fitted to the lastportion of the transient components, yielding an estimatedslope of zero and the final steady values summarized in theTable 2, with final values expressed in Volts. Fig. 7 showsa poor fit for the slopes of steady creep and a very close fitfor the final values. Moreover, it is seen that the slope datapresented may not be suitable for approximation by anyreasonable, smooth function of voltage. In simulation stud-ies for control system design, this limitation may be cir-cumvented by implementing look-up tables.

Fig. 7. Fitness of power law parameters.

Table 1Measured vs. Predicted Slopes of Steady Creep

h 5 2850.37, m5 1.5313

Voltage Measured Slope Predicted Slope

15.90 2.42903 1024 3.54203 1024

29.87 1.66883 1023 0.93023 1023

50.40 2.01943 1023 2.07243 1023

63.30 2.79633 1023 2.93773 1023

83.00 4.97993 1023 4.44853 1023

98.80 4.77103 1023 5.80883 1023

119.9 7.74453 1023 7.81313 1023

Table 2Measured vs. Predicted Final Values of Transient Component

m 5 31.3701, p5 1.1279

Voltage Final Value Final Slope Predicted F.V

15.90 0.4766 Zero, 16 digits 0.464629.87 0.9321 Zero, 16 digits 0.946250.40 1.6725 Zero, 16 digits 1.707163.30 2.1714 Zero, 16 digits 2.207483.00 2.9660 Zero, 16 digits 2.996498.80 3.6774 Zero, 16 digits 3.6472119.9 4.6681 Zero, 16 digits 4.5371


2.7.3. Estimation of characteristic frequencies, weights,and model order

The last step is to obtain the characteristic frequenciesand their corresponding weighting factors from the normal-ized transient. For this purpose, the transient response ob-tained from the 83 V input is chosen, for it has a goodsignal-to-noise ratio. An iterative process that estimates realcharacteristic frequencies from a step response known as“Procedure X” [15] is employed to obtain a first approxi-mation. Six frequencies and weights are found, resulting ina model of order seven. These estimates are employed asinitial guesses in a standard nonlinear least-squares routine(NLSQ), providing a set of refined parameters. Notice thatthe 83 V normalized data are re-sampled to contain only1000 logarithmically-spaced points, so that the NSLQ rou-tine is possible to implement. Re-sampling also has a low-pass filtering effect on the data. Table 3 summarizes theobtained parameters. Note that the sum of the weightingfactors is close to one5, as required, and that some of themare very small, so model reduction may be easy to accom-plish by approximation. A comparison between predictedand measured responses to a constant voltage input is shownin Fig. 8. The predicted responses are plots of the analyticsolutions to Eq. (6) using the estimated parameters. The

normalized rms prediction errors range from 0.64 to 2.75percent. It can be stated that the model, together with itsestimated parameters, provides an adequate fit of the ob-served time-domain data. The 95% confidence intervals forthe parameters corresponding to the power laws are pro-vided in Table 4. All parameters excepth have acceptabledeviations. The poor fit observed graphically for the creeprate is quantified by the confidence intervals.

2.8. Experimental results 2: time-frequencycross-verification

It is evident thatit is always possible, in principle, to findappropriate voltage and time functions that match the stepresponses to a good degree of accuracy. Given the modelstructure, parameter recovery is adequate from a set of stepresponses, however it is questionable if the structure itself iscompatible with the response in the frequency domain. Inorder to verify this important aspect, the analytical SIDF ofthe model is computed using parameters estimated fromtime domain. Then, analytical and experimental SIDF’s arecompared. The experimental SIDF is obtained by applyingsinewaves of various amplitudes and frequencies, and re-cording the output amplitudes and phase lags for each fre-quency. Results are shown in Fig. 9, showing a good agree-ment.

Fig. 8. Prediction of the Step Response. Fig. 9. SIDF Prediction of Sinusoidal Response.

Table 3Estimated Characteristic Frequencies and Weights

Char. Freq.lProc. X

Weight rProc. X

Char. Freq.lNLSQ

WeightNLSQ

7549.700 0.4449 7506.799 0.66141343.200 0.0785 1365.767 0.2322399.9356 0.2790 390.7856 0.028428.26340 0.0824 73.03461 0.05844.742100 0.0554 4.875001 0.03220.648000 0.0591 0.632376 0.0313

(ri 5 1.044(NLSQ)

Table 4Confidence Intervals for Parameter Estimates

Parameter Mean 95% Confidence Interval

p 1.1279 [1.0948, 1.1610]m 1.5313 [1.0480, 2.0145]n 2850.37 [1597.12, 7255.03]m 31.3701 [25.3219, 39.3701]


2.8.1. An alternative estimation procedureThe measured magnitude and phase are converted into

complex form for each of the amplitudes and frequenciesunder study. Then, in a manner analogous to removing thesteady creep from time-domain data, the creep portion of theSIDF is removed from the complex data. This is possiblebecause parametersp, m, m, h are available from estima-tion in time. Then, the remaining data are normalized by theSIDF coefficient that corresponds to the transient compo-nent. The result should be a set of data that fully matches infrequency, phase, and magnitude. Also, the magnitudeshould asymptotically approach unity (or 0 dB) as fre-quency approaches zero. This is demonstrated in Fig. 10.Note that such complex data constitutes the Bode plot of alinear transfer function in partial fraction expansion form:

G~s! 5 Oi51

n l ir i

s 1 l i

therefore any suitable method can be employed to estimatethe poles and residues. This is offered as an alternative tousing Procedure X and the NLSQ refinement.

2.8.2. Prediction of hysteresis loopsUnbiased triangular voltages with a fixed amplitude of

63 V and three representative frequencies are applied to thePZT actuator. The input/output behavior is represented inFig. 11, where it can be confirmed that, for this system,linear phase lag dominates the loop shapes, since the sim-ulated data corresponds to a model with essentially linearphase characteristics. The difference between simulated andpredicted loops should provide any rate-independent hys-teresis present in the system.

2.9. Experimental results 3: model validation

In this section, the numerical state-space model is con-fronted with a set of arbitrary test input signals. The simu-

lations are performed using the Matlab/Simulink package.The differential model is readily constructed from Eq. (6)using the identified parameters. The state-space model canbe compactly expressed as

H q̇ 5 Aq 1 F~V!« 5 Cq (13)

whereA is a 7-by-7 matrix given by

A 5 diag~2l1, 2l2, . . . ,2l6, 0!

C is a row vector of ones

C 5 @1, 1, 1, 1, 1, 1, 1#

andF is the vector function given by

F~s! 5 Fl1r1SV

mDp

, l2r2SV

mDp

, . . . ,

l6r6SV

mDp

, SV

hDmG

Fig. 10. Normalized Sinusoidal Response. Fig. 11. Prediction of Hysteresis Loops.

Fig. 12. Prediction of response to sequential voltage steps, 100 Hz.


Three kinds of signals are employed for validation: Saw-tooth, staircase and a band-limited random input. Figs. 12and 13 show the simulated and measured responses whenthe input is a staircase with nearly equal ascending steps andunequal descending steps, for low, medium and high fre-quencies. Figs. 14 and 15 show the simulated and measuredresponses to sawtooth excitation at 100 Hz. and 1 kHz.respectively. It is seen that the model adequately follows theresponse, up to its design bandwidth. Additionally, the du-ration of some experiments was extended to the secondrange, to check the assumption of a time-invariant model,with positive results. The model is tested in its ability topredict the response to biased inputs. This is shown in Fig.16. The bias is adequately predicted by the model, despitethe fact that the neglected resonance contributes to offseterror. Finally, a zero-biased, uniform random input bandlimited at roughly 1 kHz is applied. The results are depictedin Fig. 17. The general trend of the response is adequatelytracked, however there exists a moderate error in the am-

plitudes. The prediction error relative to actual output, takenas a ratio of standard deviations is roughly 32 percent.

Conclusions

The model fits experimental data and predicts arbitraryresponses with a good degree of accuracy. The results showthat power laws are reasonable assumptions for the inputfunctions, although the creep law introduces moderate er-rors. The model structure along with estimated parameters isshown to be consistent in the frequency and time domains.The model is also useful to reveal that dynamic hysteresispredominates in this particular PZT actuator, at least withinthe amplitude ranges under study. A shortcoming of themodel is, perhaps, its high order. By observing the normal-ized frequency response in Fig. 10, it is obvious that a lowerorder model can be constructed, but at the expense oftime-domain accuracy. For control applications, a lower

Fig. 13. Prediction of response to sequential voltage steps, 1 kHz.

Fig. 14. Prediction of response to sawtooth voltage, 100 Hz.

Fig. 15. Prediction of response to sawtooth voltage, 1 kHz.

Fig. 16. Prediction of response to biased sawtooth voltage, 500 Hz.


order model may be preferable, as long as it retains theessential characteristics in the frequency domain. If modelreduction produces significant error, it is the robust controlsystem’s task to account for those errors and still producedesired closed-loop performance, which is the ultimate goalof this line of research. Among the qualitative shortcomingsof the model is the fact that it predicts zero steady creep ratefor zero stress input, regardless of past input history. This isa direct consequence of the strain-hardening theory em-ployed for inducing dynamic behavior from the constantinput response. Step-down experimental data shows, how-ever, that the piezo actuator will creep even when the inputis stepped down to zero from some constant voltage. Thiseffect may be hard to capture in a differential model, for itinvolves not only past output history, but past input or inputrate history. A more general nonlinear hereditary integralrepresentation may be able to capture this effect, but anintegral representation is inconvenient for control systemdesign purposes, and it may be difficult to obtain an equiv-alent differential model. A more practically oriented draw-back is that a full identification procedure for each actuatorunder study would be required, as it happens with all em-pirical models that are tailored to a specific component. Themodel is a dynamic fit that allows the design of a controlsystem. The main conclusion is that the proposed analogydoes provide an adequate means to describe the nonlineari-ties present in this PZT tube actuator system. The PZTactuator has a quasi-linear behavior, at least in the range ofamplitudes considered in this work. Closed-loop control,therefore, is straightforward to design and implement. Theonly difficulties expected in the design process are the sameones found in linear plants, namely, unmodeled resonance,sensor noise, and actuator saturations.

Notes

1. The authors would like to thank M.J. Klopfstein forhis assistance with the use ofNanocut. The kind sup-

port by the National Science Foundation, Division ofDesign, Manufacture and Industrial Innovation isgratefully acknowledged.

2. Manufactured by Staveley Sensors Inc., E. Hartford,Connecticut.

3. The PZT tube has a nonlinear static sensitivity.4. In this context,transientrepresents a function of time

that quickly converges to some steady value.5. The NLSQ problem did not include this constraint.

References

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Fig. 17. Prediction of response to random input band limited at 1 kHz.


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Modeling nonlinear behavior in a piezoelectric actuator