Hysteresis Modeling and Tracking Control of a Piezostage for

Biological Cell Manipulation

Qingsong Xu, Member, IEEE, and Yangmin Li∗, Senior Member, IEEE

Abstract— This paper presents a new control scheme to com-pensate for the amplitude- and rate-dependent hysteresis behaviorof a piezo-driven parallel stage developed for the application ofbiological cell manipulation. A variable phase-delay model withvariable gain is established to describe the nonlinear hysteresisof the system. The proposed controller integrates an inversemodel-based preview feedforward control and a PID feedbackcontrol, which has a simple structure and is ease of real-timeimplementation. The effectiveness of the preview-based controlis demonstrated through experimental studies. Results show thatthe combined control scheme suppresses the tracking error bymore than 11 times compared to the stand-alone PID control. Itprovides a baseline of practical control of the piezostage systemfor biological cell manipulation.

I. INTRODUCTION

Biological cell manipulation plays a crucial role in biotech-nology. For instance, cell positioning [1] is important forsuch tasks of formatting tissues with different types of cells,isolating small populations of cells for testing, and in-vitrodrug testing using single cell patch clamping. Cell manip-ulation commonly requires skilled operators. The operatormanipulates the cell by resorting to the visual informationfrom the optical microscope. However, the biological cells areeasily deformable. They may be severely damaged during themanipulation owing to the exerted excessive forces or handtremors. Thus, devices capable of automatic cell manipulationare desirable to provide more feasible manipulation of biolog-ical cells.

A common cell manipulation system consists of three mod-ules including an executive module, a control module, and asensory module [2]–[4]. In order to precisely manipulate thecells with the size of several tens of microns, each of the threemodules should be carefully designed. To construct an exec-utive module, cell manipulators with ultrahigh precision areneeded. As is known, a compliant parallel micromanipulator(CPM) employing flexure hinge-based joints provides a motionof small range with ultrahigh accuracy [5]. Additionally, stack-type piezoelectric actuator (PZT) is capable of positioning with(sub)nanometer resolution, high stiffness, and rapid responsecharacteristics. Thus, a piezo-driven CPM has great potentials

This work was supported by Macao Science and Technology DevelopmentFund under Grant 016/2008/A1 and the research committee of the Universityof Macau under Grant UL016/08-Y2/EME/LYM01/FST.

The authors are with the Department of Electromechanical Engineering,Faculty of Science and Technology, University of Macau, Av. Padre TomasPereira, Taipa, Macao SAR, China.

∗Corresponding author: Y. Li is with the Department of ElectromechanicalEngineering, Faculty of Science and Technology, University of Macau, Av.Padre Tomas Pereira, Taipa, Macao SAR, China (phone: +853 83974462; fax:+853 28838314; e-mail: ymli@umac.mo).

in the domain of biological manipulation. Nevertheless, PZTintroduces nonlinearity into the system mainly owing to itshysteresis occurring at the voltage-driven strategy [6], [7]. Theadverse hysteretic behavior calls for a careful compensationto guarantee the manipulation accuracy. The objective of thisresearch is to develop a new tracking control scheme forcompensating the hysteresis nonlinearity of a piezoactuatedstage in order to accomplish a high accuracy manipulation.

The hysteretic phenomena are often rate-dependent, i.e., thehysteresis output relies on the frequency of the input. Severalrate-dependent hysteresis models are presented to capturethe rate dependency property [8]–[11] to describe the rate-dependent hysteretic characteristics. Nevertheless, most of theabove models have a lot of parameters to be determined, e.g.,the modified Prandtl-Ishlinskii model uses 28 parameters in[9], which complicate the hysteresis modeling and controlprocesses. It has been shown that the the hysteresis inducesa phase delay or a time delay to the whole system. In thisresearch, a variable phase delay with variable gain modelis presented to describe the amplitude- and rate-dependenthysteresis behavior of a micropositioning system, which hasless parameters than most of existing rate-dependent models.

By taking the system as a phase delay system, a leadcompensator is designed in [7] to compensate the hysteresisbehavior of a piezoceramic actuator system, where a look-uptable is suggested to overcome the rate dependency effect ofthe hysteresis. Considering the system as a time delay one, aSmith predictor-based controller is presented to suppress thehysteresis of a piezoactuator in [6], where a robust H∞ isdesigned by solving the variation ranges of the gain and time-delay values. For the trajectory control problem where thefuture information is available, a preview-based feedforwardcontrol is proposed alternatively in this research to compensatefor the amplitude- and rate-dependent hysteresis effects. Dueto the existence of the modeling error, a feedforward controlalone cannot totally cancel the rate-dependent hysteresis. Thus,a PID feedback control is adopted to suppress the model erroras well as other nonlinearity such as creep effects. It will beshown that the proposed controller has a simple structure thanexisting works and is easy to implement in real time control.

The rest of the paper is constructed as follows. SectionII presents the investigated system and the concerned prob-lem. Then, the rate-dependent hysteresis effect is modeledin Section III, and the identified model is verified throughexperiments. Afterwards, Section IV outlines the controllerdesign procedure, which is tested via experimental studiesaccomplished in Section V. Finally, some concluding remarks

978-1-61284-154-0/10/$26.00 © 2010 IEEE 44

Proceedings of the 2010 IEEEInternational Conference on Nano/Molecular Medicine and Engineering

December 5-9, 2010, Hong Kong

Output platformPZT

Supportingcolumn

Base

(a)

Output platformCapacitive

sensor

PZT

(b)

Fig. 1. (a) CAD model of the assembled XYZ micropositioning stagedriven by three stack PZTs; (b) a prototype flexure-based decoupled XYZmicropositioning stage whose output motions are measured by three noncon-tact capacitive sensors.

are summarized in Section VI.

II. SYSTEM AND PROBLEM DESCRIPTIONS

The CAD model of the test bed employed in this research,i.e., an XYZ parallel micropositioning stage is shown inFig. 1(a). The stage is constructed by three PPP (P representsprismatic joint) limbs and driven by three PZTs throughdisplacement amplifiers. It is known that PZT can not bearlarge transverse loads due to the risk of damage, whichusually arises from the influences of other PZT. The adopteddisplacement amplifier acts as an ideal P joint and possesses alarge ratio of stiffness in transverse direction to that in workingdirection. Hence, the amplifier also acts as a decoupler withthe roles of transmitting axial force of actuator and preventingthe actuator from suffering undesired transverse motion andloads as well. By this way, the three actuators are well isolatedand protected. Moreover, the ideal translation provided bycompound parallelogram flexures allows the generation ofdecoupled output motion of the stage. More details about thestage structure and working principle is presented in anotherpaper of the authors [12].

The experimental setup of the XYZ micropositioning systemis shown in Fig. 1(b). The stage is fabricated from Al7075 alloyby the wire-EDM process. This alloy has a higher elasticity andlighter mass than steel materials. The stage is driven by threePZTs with the stroke of 30 μm (model P-840.20 producedby Physik Instrumente Co., Ltd.). The positions of the outputplatform are measured by three noncontact capacitive sensors(model: D-510.050 from the Physik Instrumente). Concerningthe controller apparatus, a dSPACE DS1005 (from dSPACEGmbH) rapid prototyping system equipped with DS2001 A/Dand DS2102 D/A modular boards are employed. The D/Aboard produces an analogy voltage output which is then ampli-fied by a three-axis piezo voltage amplifiers (model E-503.00from the Physik Instrumente) to provide a voltage of -20–100V for the drives of the PZTs. Besides, the sensor outputvoltage signals are passed through a signal conditioner (modelE-509.E03 from the Physik Instrumente) and then acquired

0 2 4 6 8 10

0

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Input voltage (V)

Dis

plac

emen

t x (

μm

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xyz

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250

Input voltage (V)D

ispl

acem

ent x

(μ

m)

1 Hz4 Hz12 Hz

(b)

Fig. 2. (a) Hysteresis loops of the three axes motions with PZT #1 driven; (b)x-axis hysteresis loops obtained by applying sinusoidal voltage inputs withdifferent frequencies.

simultaneously by the A/D board. Control algorithms aredeveloped with MATLAB/Simulink software and downloadedto DS1005 PPC board to realize the real-time control. Asampling frequency of 5 kHz is adopted in this research toavoid signal aliasing.

A preliminary open-loop test shows that the mi-cropositioning platform has a spatial workspace around250 μm×250μm×250 μm with a maximal cross-talk of 4%between the three axes [see Fig. 2(a)], which confirms thewell-decoupling property of the XYZ stage. The hysteresisloops in Fig. 2(a) means that the shape of the hysteresis loopis dependent on the amplitude of the input. From Fig. 2(b),one can observe that the hysteresis shape also relies on theinput rate. The modeling and compensation of the complexhysteresis effects are presented in the following sections.

III. RATE-DEPENDENT HYSTERESIS MODELING

A. Hysteresis Modeling

Considering that the magnitude and phase of output areamplified by a gain and shifted by a phase angle with respectto the input, respectively, a variable gain and variable time-delay model can be established for the hysteretic system as

45

(a)

(b)

(c)

Fig. 3. Variable gain (a), phase delay (b), and residual position (c) versusamplitude and frequency of the input signals. Meshes represent the outputs ofidentified linear regression models and dots denote experimental results.

follows:X(s)

U(s)= H(s), (1)

where X(s) and U(s) denote the Laplace transforms of theoutput and input signals. H(s) represents the variable gainand variable time-delay hysteresis model, which can be furtherexpressed by:

H(s) = K(um, fr) e−τ(um,fr)s, (2)

where the gain K and time-delay τ are assumed to be functionsof the amplitude um and frequency fr of the input signal.

TABLE I

IDENTIFIED HYSTERESIS MODEL PARAMETERS OF THE

MICROPOSITIONING STAGE FOR x-AXIS MOTION

Parameter Value Parameter Value Parameter Valuea0 15.2536 b0 1.9238 c0 −1.7893a1 1.9619 b1 0.7477 c1 1.1365a2 −0.1013 b2 0.0456 c2 0.3635

b3 0.0047 c3 −0.0463b4 −0.0466 c4 −0.0140b5 0.0003 c5 −0.0150

From (1) and (2), it is seen that once the input signal isgiven, the desired output position can be obtained as follows:

X(s) = K(um, fr) e−τ(um,fr)sU(s). (3)

Assume that the input is expressed by:

u(t) = A sin(2πfrt −π

2) + A, (4)

where A and fr denote the amplitude and frequency. Then,the position output can be derived as:

x(t) = KA sin(2πfrt −π

2− θ) + KA + x0, (5)

where the variable phase delay θ relates to the time delayτ by θ = 2πfrτ . x0 denotes a residual position due to thehysteresis effect, which is not zero although the driving voltageis decreased to zero.

In order to identify a variable phase delay with variablegain model, the three parameters including the gain K , shiftedphase θ, and residual position x0 need to be determined asfunctions of the input amplitude as well as input rate.

B. Model Identification and Verification

Given the voltage input with varying amplitudes (0.5–10V) and frequencies (1–20 Hz), the output gain, phase delay,and residual position are generated as shown in Figs. 3(a)–(c). It is observed that the gain varies as the variation of theinput amplitude, whereas it remains almost constant as theincreasing of the input rate. On the other hand, the phasedelay and residual position rely on both input amplitude andinput rate. Therefore, the variable gain and phase delay can beapproximated by the linear regression functions:

K(um) = a0 + a1um + a2u2m, (6)

θ(um, fr) = b0 + b1um + b2fr + b3umfr + b4u2m + b5f

2r ,

(7)

x0(um, fr) = c0 + c1um + c2fr + c3umfr + c4u2m + c5f

2r ,

(8)

where a0—a2, b0—b5, and c0—c5 are the estimated coef-ficients for the three linear regression functions, which aredescribed in Table I.

The established variable gain and variable phase delaymodel is verified by experimental studies. To make the modelverification more persuasive, input signals that are not used inthe model identification process are employed for tests.

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1 1.2 1.4 1.6 1.8 20

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Inpu

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) an

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utpu

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(a)

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−2

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Mod

elin

g er

ror

(μm

)

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Fig. 4. For a a 5-V, 5-Hz voltage input, the comparison of model output andexperimental result.

Concerning a 5-V, 5-Hz voltage input, the model outputand experimental result are compared in Fig. 4. It is seenthat maximum and mean modeling error is 5.31 and 1.01 μm,which accounts for 4.7% and 0.9% of the output position rangeof the stage, respectively. Besides, with a 3-V, 15-Hz voltageinput, the model output and experimental result are comparedin Fig. 5, which shows a 4.9% maximum and 0.5% mean errorswith respect to the motion range. Thus, the mean modelingerror is below 1% of the motion range, which indicates themodeling accuracy of the identified hysteresis model is betterthan 99%.

IV. HYSTERESIS CONTROL

By taking the system as a phase delay system, a leadcompensator is designed in [7] to compensate the hysteresisbehavior of a piezoceramic actuator system, where a look-uptable is established to overcome the rate dependency effect ofthe hysteresis. Considering the system as a time delay one, aSmith predictor-based controller is presented to suppress thehysteresis of a piezoactuator in [6], where a robust H∞ isconstructed by solving the variation ranges of the gain andtime-delay values. An alternative preview-based feedforwardcontrol is proposed in this research to compensate for the

0.2 0.25 0.3 0.35 0.40

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Inpu

t (V

) an

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Input voltage (V)Model output (μm)Experimental (μm)

(a)

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elin

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Fig. 5. For a a 3-V, 15-Hz voltage input, the comparison of model outputand experimental result.

1 seK

τ

+

−

dx ePlantPID

controllerx

ffu

fbu u++

Fig. 6. Block diagram of the proposed preview-based feeforward inconjunction with PID feedback controller.

amplitude- and rate-dependent hysteresis effects. Owning tothe modeling errors, a feedforward (FF) control alone cannottotally cancel the rate-dependent hysteresis. Hence, a PIDfeedback (FF) control is employed to suppress the model erroras well as other nonlinearity such as creep effects. The blockdiagram of the proposed control scheme is depicted in Fig. 6.

47

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Pos

ition

x (μ

m)

(a)

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ition

err

or (μ

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Act

ual p

ositi

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(c)

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Time (s)

Inpu

t vol

tage

(V

)

FFFBFF+FB

(d)

Fig. 7. (a) A 60-μm, 5-Hz position reference input, (b) experimental results obtained by the three control approaches, (c) actual position versus the referenceposition; (d) the components of control effort of the combined controller.

A. Preview Controller Design

From (1) and (2), it is seen that once the desired output isgiven, the input signal can be obtained as follows:

U(s) =1

K(um, fr)eτ(um,fr)sX(s), (9)

which is represented by the feedforward term in Fig. 6.Assume that the desired output is given by:

xd(t) = A sin(2πfrt −π

2) + A + x0, (10)

where A and fr denote the amplitude and frequency of thereference signal.

In view of (4) and (5), the voltage input can be derived as:

uff (t) =A

Ksin(2πfrt −

π

2+ θ) +

A

K, (11)

where the variable phase lead θ is related to the preview time τ

by θ = 2πfrτ . Thus, the above equation represents a preview-type feedforward control input [13]. Instead of constructinga complex optimal preview control, the controller is imple-mented based on the phase shift θ estimated by the analyticalhysteresis model identified in the proceeding discussions.

B. Combined Controller Design

Due to the existence of modeling error as revealed in SectionIII, the hysteresis effects cannot be totally compensated bythe FF control (11). Therefore, an additional FB control isadopted to compensate for the model imperfection and otherdisturbances of the system. Specifically, a PID feedback con-troller is employed due to its robustness and easy-to-implementproperties. The FB control input can be expressed as:

uFB(t) = Kp ex(t) + Ki

∫ t

0

ex(τ)dτ + Kd

dex(t)

dt(12)

where the tracking error is defined as ex(t) = xd(t)−x(t) withx denoting the measured position. The three control parametersKp, Ki, and Kd are proportional, integral, and derivative gains,respectively.

For the convenience of real-time digital control, the overall

48

control signal is derived in a discrete-time form:

u(tk) = uFF (tk) + uFB(tk)

= uFF (tk) + Kp ex(tk) + Ki

k∑i=1

ex(ti)Ts

+Kd

ex(tk) − ex(tk−1)

Ts

(13)

where k is the index of time series and Ts represents thesampling time interval (Ts = 0.0002 s).

V. EXPERIMENTAL STUDIES

In this section, experimental studies are performed to verifythe feasibility of the proposed control scheme for the motiontracking control of the micropositioning stage.

The preview-based feedforward controller is implementedbased on (11), and the PID feedback controller is designed byexperimental tuning. To obtain better performance, the PIDparameters are tuned by trial-and-error approach, i.e., Kp =0.017, Ki = 8.62, and Kd = 2.33 × 10−5.

For example, concerning a reference input as shown inFig. 7(a), the control errors of FF, FB, and FF+FB are plottedin Fig. 7(b). The maximum errors of the three controllers are3.145 μm, 10.294 μm, and 0.894 μm, which equal to 5.2%,17.2%, and 1.5% of the motion range, respectively. Comparedto the stand-alone PID control, the FF scheme reduces thetracking error by 3.3 times while the FF+FB combined controlsignificantly alleviates the error by 11.5 times. The obtainedactual positions versus the position reference are shown inFig. 7(c), which indicates that the proposed controller suppressthe hysteresis effect to a negligible level. Concerning theFF+FB combined controller, the control effort components areplotted in Fig. 7(d). It is observed that the control voltage ismainly contributed by the FF control. Whereas the minor con-trol effort of PID controller is used to alleviate the modelingerror and other uncertainty which cannot be compensated bythe FF control alone.

VI. CONCLUSION

The results presented in this paper show that a preview-based feedforward combined with feedback control is effi-cient in compensating the complex hysteresis behavior ofa piezostage. To describe the amplitude- and rate-dependenthysteresis, a variable gain and variable phase delay model isestablished by resorting to open-loop experimental data. Themodel is verified by experimental test which shows that the

mean modeling error is less than 1% of the motion range.The identified hysteresis model is employed to construct thefeedforward controller. The strategy of feedforward combinedwith PID feedback control has suppressed the maximumcontrol error by more than 3 and 11 times with comparisonto the stand-alone FF and PID control, respectively, whichexhibits the effectiveness of the established rate-dependentmodel. The presented preview-based controller is ease of real-time implementation and can be applied for the modeling andcontrol of other types of hysteretic systems. In the future,

biological manipulation experiments will be conducted toverify the efficiency of the proposed control strategy.

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