Modeling and Compensation of Asymmetric Hysteresis Nonlinearity ...

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Modeling and Compensation of AsymmetricHysteresis Nonlinearity for Piezoceramic Actuators

with a Modified Prandtl-Ishlinskii ModelGuo-Ying Gu, Member, Li-Min Zhu, Member, and Chun-Yi Su, Senior Member

Abstract—This paper presents a modified Prandtl-Ishlinskiimodel (MPI) for the asymmetric hysteresis description andcompensation of piezoelectric actuators. Considering the fact thatthe classical Prandtl-Ishlinskii (CPI) model is only efficient for thesymmetric hysteresis description, the MPI model is proposed todescribe the asymmetric hysteresis nonlinearity of piezoceramicactuators. Different from the commonly used approach forthe development of asymmetric Prandtl-Ishlinskii models byreplacing the classical play operator with complex nonlinearoperators, the proposed MPI model still utilizes the classical playoperator as the elementary operator, while a generalized inputfunction is introduced to replace the linear input function in theCPI model. By this way, the developed MPI model has a relativesimple mathematic format with fewer parameters to characterizethe asymmetric hysteresis behavior of piezoceramic actuators.The benefit for the developed MPI model also lies in the factthat an analytic inverse model of the CPI model can be directlyapplied for the inverse compensation of the asymmetric hysteresisnonlinearity represented by the developed MPI model in real-time applications. To validate the developed MPI model andthe inverse hysteresis compensator, simulation and experimentalresults on a piezoceramic actuated platform are presented.

Index Terms—Piezoceramic actuator, asymmetric hysteresisnonlinearity, inverse hysteresis compensation, Prandtl-Ishlinskiimodel

I. INTRODUCTION

Piezoelectricity is the electromechanical phenomenon ofstrong coupling between electrical properties and mechanicalproperties of some piezoelectric materials. Mechanical stressesin the piezoelectric materials produce measurable electriccharge, which is referred as the direct piezoelectric effect.Conversely, mechanical strains are generated in response toan applied electric field and this is called the conversepiezoelectric effect. In general, the direct effect is the basisfor sensing applications and the converse effect is used for

Manuscript received November 21, 2012; revised January 31, 2013; ac-cepted March 16, 2013. This work was supported in part by the NationalNatural Science Foundation of China under Grant 91023047, the Science andTechnology Commission of Shanghai Municipality under Grant 11520701500,and the Shu Guang Project supported by Shanghai Municipal EducationCommission under Grant 10SG17.

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

G.-Y. Gu and L.-M. Zhu are with State Key Laboratory of MechanicalSystem and Vibration, Shanghai Jiao Tong University, Shanghai 200240,China (e-mail: [email protected]; [email protected]).

C.-Y. Su is with College of Automation Science and Engineering SouthChina University of Technology, Guangzhou 510641, China. He is on leavefrom Concordia University, Montreal, Quebec H3G 1M8, Canada (e-mail:[email protected]).

precision actuation and manipulation in control applications.Due to the excellent advantages of the large output force, highbandwidth, and fast response time, piezoceramic actuatorsmade of piezoelectric materials have been widely used fora long time in nanopositioning applications [1]–[4] such asscanning probe microscopes, nano-imprint equipments andmicro/nano-manipulations. However, piezoceramic materialssuffer from the strong hysteresis effect. It can significantlydegrade the performance of the piezoceramic actuators whichleads in the best case to reduce the motion accuracy and inworst case, to destabilize the control system [5]–[9].

To address such a challenge, many efforts have been madeto compensate for the hysteresis nonlinearity in piezoceramicactuators. Feedforward control is the most common usedapproach [10], [11]. The main idea of the feedforward controlapproach is to develop a mathematical model that describesthe complex hysteresis nonlinearity, and then to implement afeedforward controller based on the inverse hysteresis modelto linearize the actuator response, which can be schematicallyshown in Fig. 1. It refers, in particular, to the utilizationof a hysteresis model accurately describing the hysteresisnonlinearity. As a matter of fact, there are two kinds ofhysteresis modeling approaches, which are physics-based hys-teresis models and phenomenological hysteresis models. Thephysics-based models [12], [13] are derived on the basis ofa physical measure, such as energy, displacement, or stress-strain relationship, while phenomenological hysteresis models[14]–[16] are constructed from the experimental data withoutconsidering the physical property of the actuator. Reader mayrefer to [17]–[21] and reference therein for reviewing recentstudies on hysteresis modeling. A popular phenomenologicalhysteresis model adopted for hysteresis modeling and com-pensation of piezoceramic actuators is the Preisach model [6],[22], [23]. The main limitation is that the Preisach modelis not analytically invertible. Thus, numerical methods aregenerally adopted to obtain approximate inversions of thePreisach model [24]–[27]. As a subclass of the Preisachmodel, the Prandtl-Ishlinskii (P-I) model [28] is an alternativeeffective method to describe the hysteresis nonlinearity bya single threshold variable together with a density function.Compared with the Preisach model, the main advantages ofthe P-I model are the reduced modeling complexity and theanalytical inverse for the P-I model, thus making it moreefficient for real-time applications [29]–[33]. However, theclassical P-I (CPI) model is limited to the symmetric hysteresisdescription. It cannot be utilized when an asymmetry exists



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HysteresisInverse hysteresis

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yd(t)

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Fig. 1. Schematic illustration of the feedforward control technique forhysteresis compensation.

in the hysteresis loops, such as those observed in the input-output relations of the piezoceramic actuators depicted in Fig.2. In an attempt to overcome this drawback of the CPI model,researchers have proposed several variations of P-I modelsfor characterizing the asymmetric hysteresis loops. Kuhnen[34] in the first time proposed deadzone operators cascadedwith the classical play operator to describe the asymmetrichysteresis nonlinearity. Instead of using the symmetric playoperator, Dong and Tan [35] utilized an asymmetric backlashoperator as the elementary operator for asymmetric hysteresismodeling. As another choice in [36], two asymmetric operatorswere utilized for the ascending branch and descending branchof hysteresis loops respectively. Using two nonlinear envelopefunctions based play operators, a generalized P-I model [18]was proposed to describe a more general class of hysteresisloops which are observed in magnetostrictive actuators orshape memory alloys actuators. These reported works [18],[34]–[36] for P-I models generally replace the classical playoperator by modified or nonlinear play operators with morecomplex formats under different conditions, and experimentalresults are applied to verify the effectiveness of the asymmetrichysteresis representation and compensation.

In this paper, a novel modified P-I (MPI) model thatoriginates from the classical P-I (CPI) model is developed todescribe the asymmetric hysteresis nonlinearity of piezoce-ramic actuators. In contrast to change the formulation of theplay operators as reported in [18], [34]–[36], the classical playoperator is still utilized to serve as the elementary operator inthe developed MPI model, while a generalized input functionis introduced to replace the linear input function in the CPImodel. Thus, the hysteresis shapes can be determined by notonly the weighted play operators but also the generalizedinput function. The main advantages of the developed MPImodel lie in the facts that: i) rather than using the nonlineardeadzone operators, modified or nonlinear play operators, theclassical play operator is still utilized in the MPI model toserve as the elementary operator; ii) the MPI model has a

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ispl

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Fig. 2. Asymmetric hysteresis loops of a piezoceramic actuator.

relative simple mathematic format with fewer parameters todescribe the asymmetric hysteresis behavior of piezoceramicactuators; iii) an analytic inverse of the MPI model can bedirectly derived from the inverse of the CPI model for thepurpose of the feedforward hysteresis compensation in real-time applications. To validate the developed MPI model andthe inverse hysteresis compensator, simulation and experimen-tal results on a piezoceramic actuated platform are presented.

The remainder of this paper is organized as follows. SectionII introduces the MPI model and its analytical inverse. SectionIII presents the simulation verification of properties of thedeveloped MPI model. Finally, experimental validation of thedeveloped MPI model and its inverse hysteresis compensatoron a piezoceramic actuated platform is presented in SectionIV and Section V concludes this paper.

II. MODIFIED PRANDTL-ISHLINSKII MODEL

Before introducing the proposed MPI model, the CPI modelis firstly reviewed in brief.

A. CPI model

The CPI model [28] is a famous operator-based phenomeno-logical model, which utilizes the weighted play operators anda linear input function to describe the hysteresis nonlinearity.The play operator is the basic hysteresis operator with sym-metric and rate-independent properties. The one-dimensionalplay operator can be considered as a piston with a plungerof length 2r. The output w is the position of the center ofthe piston, and the input is the plunger position v. Accordingto the definition in [28], the play operator w(t) = Fr[v](t)with a threshold r for any piecewise monotone input functionv(t) ∈ Cm[0, tE ] is expressed as

w(0) = Fr[v](0) = fr(v(0), 0)

w(t) = Fr[v](t) = fr(v(t), w(ti))(1)

for ti < t ≤ ti+1, 0 ≤ i ≤ N − 1 with

fr(v, w) = max(v − r,min(v + r, w)) (2)

where 0 = t0 < t1 < . . . < tN = tE is a partitionof [0, tE ], such that the function v(t) is monotone on each



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of the subintervals [ti, ti+1]. The argument of the operator(1) is written in square brackets to indicate the functionaldependence, since it maps a function to another function. Theplay operator has the rate-independent property, that is, theoutput of the play operator is only influenced by the currentinput value and the past extrema of input function v(t) whilethe velocity of input variation between each extreme shall notaffect the shape of output. As an illustration, Fig. 3 shows thetransfer characteristics of the play operator (1).

0 r ( )v t

( )w t

r-

Fig. 3. Input-output relationships of the play operator (1).

Considering the fact that the piezoceramic actuators have thepositive excitation nature. Rather than using the play operatorin (2), an one-side play operator (OSP) is adopted in this work.The OSP operator For[v](t) with a threshold r ≥ 0 for anypiecewise monotone input function v(t) ∈ Cm[0, tE ] is definedas

w(0) = For[v](0) = for(v(0), 0)

w(t) = For[v](t) = for(v(t), w(ti))(3)

for ti < t ≤ ti+1, 0 ≤ i ≤ N − 1 with

for(v, w) = max(v − r,min(v, w)) (4)

for 0 = t0 < t1 < . . . < tN = tE . For comparison, Fig. 4shows the transfer characteristics of the OSP operator (3).

Remark: It should be noted that the advantage for choosingthe OSP operator lies in that when describing the hysteresisnonlinearity, we avoid to calculate the expression of v + r ateach threshold r. The following simulation and experimentalresults will further demonstrate that with the OSP operatorthe proposed MPI model is good enough for the asymmetrichysteresis description of piezoceramic actuators.

0 r ( )v t

( )w t

1 r

1

Fig. 4. Input-output relationships of the one-side play operator (3).

The CPI model utilizes the above OSP operator For[v](t) todescribe the relationship between the output yc and the input

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Fig. 5. Hysteresis loops generated by the CPI model (5).

v as [28], [29]

yc(t) = P [v](t) = p0v(t) +

∫ R

0

p(r)For[v](t)dr (5)

where p(r) is a density function that is generally calculatedfrom the experimental data, and p0 is a positive constant. Thedensity function p(r) generally vanishes for large values of r,while the choice of R = ∞ as the upper limit of integration iswidely used in the literature for the sake of convenience [18],[30].

Remark: It should be noted that there is no general criterionfor the selection of the density function p(r). Generally, it iscompletely selected by the designer. Once the structure of thedensity function p(r) is fixed, the parameters involved in thedensity function shall be determined by identification fromexperimental data in order to best reflect the shapes of thehysteresis exhibited in piezoceramic actuators.

From (5), the CPI model is defined in terms of an integralof the OSP operator For[v](t) with a density function p(r) andan linear input function p0v(t). According to the symmetricproperties of the play operator and the linear function, theCPI model can only be used to characterize the symmet-ric hysteresis nonlinearity. As an illustration, Fig. 5 showsthe hysteresis loops generated by the CPI model (5) withp0 = 2.42, p(r) = 5e−0.505(r−1)2 , r ∈ [0, 1] and the inputv(t) = 0.5 + 0.5sin(3t)/(1 + t). It can be observed that thegenerated hysteresis loops in Fig. 5 are symmetric. However,the real hysteresis nonlinearity of the piezoceramic actuatorexhibits the asymmetric shape as shown in Fig. 2. Therefore,the CPI model yields considerable errors for description of theasymmetric hysteresis nonlinearity. One of the motivations ofthis work is to find an effective model that can accurately rep-resent the asymmetric hysteresis nonlinearity of piezoceramicactuators.

B. MPI model

Considering the fact that the CPI model is developed forthe symmetric hysteresis description, a novel MPI model isproposed in this work to characterize the asymmetric hys-teresis behavior of piezoceramic actuators. It is worthy of



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yg(t

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Fig. 6. Hysteresis loops generated by the MPI model (6).

mentioning that although several models [18], [34]–[36] havebeen reported to overcome the drawback of the CPI modelthat can only describe the symmetric hysteresis, the developedMPI model in this work is different from them. Analyzing thestructure of the CPI model (5), it consists of two parts. Thefirst part is a linear input function p0v(t), and the secondone is the integral of the OSP operator For[v](t) with adensity function p(r). According to the two-part structure ofthe CPI model, the previous reported models in [18], [34]–[36] focus on the modification of play operators in the secondpart with more complex formats under different conditions.In contrast to change the play operator, we develop a modelto modify the first part of the CPI model with a generalizedinput function, keeping the second part the same with theCPI model. One reason for this alternative choice is motivatedby the fact that the classical play operator is still utilized toserve as the elementary operator. In such a way, the developedMPI model has a relative simple mathematic format withfewer parameters to characterize the asymmetric hysteresisbehavior of piezoceramic actuators. Another advantage isthat an analytic inverse of the MPI model can be directlyderived from the inverse of the CPI model for the real-timefeedforward hysteresis compensation, which will be shown inthe following development.

In this paper, the proposed MPI model is formulated as

yg(t) = Pg[v](t) = g(v(t)) +

∫ R

0

p(r)For[v](t)dr (6)

where g(v(t)) is a generalized odd input function with thememoryless and locally Lipschitz continuous properties, p(r)and For[v](t) are defined the same as the ones in theCPI model (5). Since the polynomial function [37], [38] isgenerally recognized as an effective choice to describe thehysteresis loops, a third degree polynomial input functiong(v(t)) = a1v

3(t) + a2v(t) with two coefficients a1 and a2is utilized in the developed MPI on the asymmetric hysteresischaracterization of piezoceramic actuators.

Comparing (6) with (5), the difference between the MPImodel and the CPI model lies on the selection of the inputfunction g(v(t)). If g(v(t)) is selected as g(v(t)) = p0v(t),the MPI model can be directly reduced to the classical one.

+

-

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cP

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1

gP

w

Fig. 7. Signal flow chart of the inverse compensator P−1g .

Therefore, the CPI model can be considered as a special caseof the MPI model. The benefit for choosing the generalizedinput function g(v(t)) (a third degree input function in thiswork) is that the proposed MPI model can characterize thereal hysteresis nonlinearity of the piezoceramic actuator withthe asymmetric behavior. To elucidate this advantage of theMPI model, Fig. 6 shows the hysteresis loops generated bythe MPI model (6) with g(v(t)) = −v3(t) + 2.42v(t). It isworthy of mentioning that the density function p(r) and inputv(t) are chosen the same as the ones used in the CPI modelthat generates the hysteresis loops in Fig. 5. As a contrast, itis the polynomial input function g(v(t)) that makes the MPImodel accommodate a more general class of hysteresis shapeswith the asymmetric behavior.

Remarks: i) Due to the rate-independent characteristic ofthe OSP operator and the generalized input function, theproposed MPI model is a rate-independent hysteresis modellike the previous reported models in [18], [32], [34]–[36],which can describe hysteresis loops that are invariant withrespect to the frequency of the input.

ii) It should be noted that the developed MPI model isformulated by (6), where the asymmetric part is generatedby selection of g(v(t)). A third-order polynomial is chosendue to the fact it can present reasonable complicated shapesof the hysteresis taking into consideration of the inversecalculations and number of parameters. Furthermore, the third-order polynomial [37], [38] is generally recognized as aneffective choice to describe the hysteresis loops. Certainly, theselection of g(v(t)) is not unique and other forms can also bechosen. For the development of the paper, a form has to befixed and we select a third-order polynomial due to the abovereasons.

C. Inverse MPI model

The main idea of the inverse hysteresis compensation isto cancel the hysteresis nonlinearity by cascading the inversehysteresis model with the real hysteresis as shown in Fig. 1.In the above section, the MPI model is developed based on theCPI model to describe the asymmetric hysteresis nonlinearity.In this section, we will show that the inverse of the CPI modelcan also be directly applied to the MPI case.

In order to conveniently implement the real-time inversecontroller, the definition of MPI model given in (6) can beapproximated in the form of a finite number of the OSP



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operators as follows

yg(t) = Pg[v](t) = a1v3(t)+a2v(t)+

n∑i=1

b(ri)Fri [v](t) (7)

where n is the number of the adopted play operators formodeling, and b(ri) = p(ri)(ri − ri−1) is the weightedcoefficient for the threshold ri.

On the basis of the CPI model (5) and inspired by the workin [6], the MPI model (7) can be re-expressed as the followingsuperposition

yg(t) = Pg[v](t) = Pc[v](t) + P [v](t) (8)

where Pc[v](t) = a1v3(t) is a new model component.

The synthesis of efficient real-time control algorithms forthe compensation of the asymmetric hysteresis nonlinearityrepresented by the MPI model is based on the implicit operatorequation

v(t) = P−1[w](t) (9)

for the inverse compensator

v(t) = P−1g [yg](t) (10)

where w = yd − Pc[v], yd(t) is the desired displacement,and the inverse CPI hysteresis model [30] P−1 is used as asubsystem expressed as

P−1[w](t) = a2w(t) +n∑

j=1

bjFrj [w](t) (11)

with

rj = a2rj +j−1∑i=1

bi(rj − ri)

a2 = 1a2

bj = − bj

(a2+j−1∑i=1

bi)(a2+j∑

i=1bi)

(12)

Fig. 7 shows the signal flow chart of the inverse compensatorP−1g for the developed MPI model. By this way, the analytical

inverse hysteresis model of CPI [30] can be directly appliedfor the asymmetric hysteresis compensation of piezoceramicactuators described by the developed MPI model.

III. VERIFICATION OF WIPING-OUT AND CONGRUENCYPROPERTIES BY SIMULATION

The wiping-out property and congruency property are essen-tial properties for the validity of the operator-based hysteresismodels [15], [28]. In the following development, we will showthat the proposed MPI model also fulfills these two propertiesby simulation.

A. Wiping-out property

The wiping-out property refers to the nonlocal memorylessbehavior that the hysteresis output depends not only upon thecurrent input but also the previous dominant input extrema. Inother words, only the alternating series of the dominant inputextrema are stored by the hysteresis model and all other inputextrema are wiping-out. It can be shown that the developedMPI model possesses this property.

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rr

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Fig. 8. Geometrical interpretation of the wiping-out property with thememory curve structure.

Inspired by the validation approach of the wiping-out prop-erty for the operator-based hysteresis models [15], [27], [28],[39], the wiping-out property of the proposed MPI modelis asserted in a geometrical interpretation with the memorycurve structure. To this end, we first demonstrate that the OSPoperator possesses the wiping-out property. Suppose that aninput function v satisfies v0 = 0 at time t0. At this time, wehave For[v](0) ≡ 0 for r ≥ 0. Next, suppose that the inputfunction v is increased monotonically to a value v1 at time t1.By definition of For[v] in (3), For[v](t1) = max(v1 − r, 0)is the line segment ABC as shown in Fig. 8(a). If the inputis then monotonically decreased from v1 to v2 at time t2,For[v](t2) = min(v2, For[v](t1)) is the line segment DEBCas shown in Fig. 8(b). Then, the input is monotonicallyincreased from v2 to v3 at time t3 satisfying v3 > v1,For[v](t3) = max(v3 − r, For[v](t2)) is the line segmentA′B′C as shown in Fig. 8(c). Thus, the memory behavior attime t is completely described by the memory curve For[v](t).It is evident that the dominant input extrema lower than v3have been wiping-out. It can be concluded that the OSPoperator possesses the wiping-out property. In addition, thewiping-out property follows the linear superposition [28],[39] and the introduced input function g(v(t)) is odd andmemoryless. Therefore, the developed MPI model still fulfillsthe wiping-out property, which can be further illustrated bythe following simulation example.

In Fig. 9(a), a special input function, having a given inputstring sequence of extrema (0, v1, v2, v3, v4, v5) is chosenin the simulation so as to verify the wiping-out propertydiscussed above. When the input function and density functionare, respectively, chosen as g(v)(t) = −v3(t) + 2.42v(t) andp(r) = 5e−0.505(r−1)2 , Fig. 9(b) gives the generated hysteresisloops corresponding to the input history shown in Fig. 9(a). In



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Fig. 9. Simulation of the wiping-out property.

the output-input plane of Fig. 9(b), Pi = (v(ti), yg(ti)), i =0, 1, ..., 5 represent the associated memories with the inputstring sequence (0, v1, v2, v3, v4, v5). In the beginning, theinput v(t) starts from its initial value 0, and quickly reaches itsfirst maximum value v1. The process takes the trajectory froman initial point (0, 0) to memory point P1(v1, yg1) followingthe

−−→0P1 curve of the hysteresis loops as shown in Fig. 9(b).

After that, the input v(t) decreases to its first minimum valuev2, and the corresponding output-input relationship switchesinto the

−−−→P1P2 curve. Subsequently, the input v(t) increases to

its second maximum value v3. Correspondingly, the output-input trajectory of the hysteresis loops moves from pointP2(v2, yg2) to point P3(v3, yg3). Since the second maximumvalue v3 is lower than the first maximum one v1, i.e. v3 < v1,the wiping-out phenomenon no occurs so far. Next, the inputv(t) decreases to the second minimum value v4 and thisprocess corresponds to the

−−−−−→P3P2P4 curve. At this moment,

the second minimum value v4 is lower than the first minimumone v2, i.e. v4 < v2. The memory P2(v2, yg2) is wiped out bythe memory P4(v4, yg4). In the end, the input v(t) continuesto increase from v4 to its third maximum value v5. Sincev5 > v1 > v3, from Fig. 9(b) it can be observed that thememories P1(v1, yg1) and P3(v3, yg3) are both wiped out bythe memory P5(v5, yg5). During this period, the hysteresiscurve switches to the

−−−−−→P4P1P5 branch and this process will

not be affected by any past maximum values because all thememories about the past input maximum values have been

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Fig. 10. Simulation of the congruency property.

completely wiped out. Using this example, we can see thatthe MPI model successfully fulfills the wiping-out property.

B. Congruency property

The congruency property means that two minor hysteresisloops corresponding to the the same input range are congruent.By adopting the memory curve [15], [27], [28], [39], it can beshown that the MPI model possesses the congruency propertyas well.

For example, the input v(t) for validation is given in Fig.10(a). In this case, the input string sequence of input maximaand minima is defined as (0, v1, v2, v3, v4, v5, 0), where v1 =v5 and v2 = v4. Thus, the input ranges [v2, v1] and [v4, v5]are the same. For the input voltage shown in Fig. 10(a), theassociated hysteresis loops generated by the MPI model aregiven in Fig. 10(b). In Fig. 10(b), the lower minor hysteresisloop 1 is caused by the input wavering between the values v2



7

SC PEAADC

Interface

dSPACE

InterfaceComputer

(dSPACE board)

FHGS

with PCA

DAC

Interface

PCA

Fig. 11. The experimental platform.

End-

effector

flexure

Base

PCA

F

Fig. 12. Schematic structure of the PCA-actuated one-dimensional flexuremechanism.

and v1 in the ascending process, and the upper minor hysteresisloop 2 is led by the input wavering between the values v4and v5 with the same range in the descending process. It canbe observed that one of the two minor hysteresis loops canexactly overlap the other one if shifted the loop 1 upside alongthe vertical direction by 0.5593 unit without rotation, whichdemonstrates that the two minor loops are congruent.

IV. EXPERIMENTAL VERIFICATION

In this section, an experimental platform with a piezece-ramic actuator will be established and experimental tests shallbe conducted to verify the developed MPI model and the corre-sponding inverse hysteresis compensator for the piezoceramicactuator with the asymmetric hysteresis nonlinearity.

A. Experimental setup

The experimental platform is shown in Fig. 11. It consistsof a host computer, a dSPACE-DS1103 controller board, apiezoceramic actuator, a piezoelectric amplifier, a strain gaugeposition sensor and a sensor signal conditioner. The host com-puter provides a user interface for the dSPACE-DS1103 con-troller board. The dSPACE-DS1103 controller board equipped

Signal

conditioner

Piezoceramic

Actuator with

SGS

dSPACE

board

ADC

DAC

Piezo

amplifier

Host

computer

Fig. 13. Block diagram of the experimental platform.

0 2 4 6 8 10 12 14 16−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Ide

ntif

ied

erro

r ( µ

m)

n = 8n = 10n = 15

Fig. 14. Identification errors with different selections of n.

with 16-bit DACs and 16-bit ADCs is adopted to generatecontrol codes and obtain the displacement information. Thepiezoceramic actuator (PCA) is a preloaded piezoceramicstack actuator (PSt 150/7/100 VS12 from Piezomechanikin Germany), which is used to drive the one-dimensionalflexure hinge guiding stage (FHGS) with the nominal 75µm displacement. Fig. 12 shows the schematic structure andwork principle of the PCA-actuated one-dimensional flexuremechanism, where the double parallelogram flexure is adoptedto remove the parasitic motions. The piezoelectric amplifier(PEA) with a fixed gain of 15 is used to provide excitationvoltage for the piezoceramic actuator in the 0-150 V range.To measure the real-time position of the the piezoceramicactuator on the nanometer scale, the strain gauge sensor (SGS)bonded to the piezoceramic stack actuator is used in this work.Compared with the capacitive sensor and LVDT sensor, theSGS is a contact type sensor, which offers high resolutionand bandwidth [1], [40]. The output signals of the SGS passthrough the signal conditioner (SC), which are simultane-ously sampled by the 16-bit ADC for the dSPACE-DS1103controller. Fig. 13 shows block diagram of the experimentalplatform.

B. Asymmetric hysteresis description results

In practice, the threshold values ri in the MPI model (7)are given by

ri =i− 1

n||v(t)||∞, i = 1, 2, ..., n (13)

with ||v(t)||∞ = 1 in the normalized case.To experimentally validate the proposed model, the first

step is the identification of the model parameters. Due to thehighly nonlinear, high dimensional, and multiple constraintcharacteristics, identification of the hysteresis models is a



8

0 1 2 3 4 5 60

20

40

60

80

100

120

140

Time (s)

Inp

ut v

olta

ge (

V)

(a) Input voltage

0 1 2 3 4 5 60

10

20

30

40

50

60

70

Time (s)

Dis

plac

emen

t (µm

)

Experimental dataModel prediction

(b) Displacement

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

Time (s)

Err

or (

µm)

(c) Error

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

70

Input voltage (V)

Dis

plac

emen

t (µm

)

Experimental dataModel prediction

(d) Hysteresis loops

Fig. 15. Experimental verification of the MPI model with a simple input signal.

TABLE IIDENTIFIED PARAMETERS OF THE MPI MODEL.

Number ri bi ai1 0 0.2313 -0.15692 0.1 0.3059 0.46033 0.2 0.01554 0.3 0.07525 0.4 0.06836 0.5 0.02527 0.6 0.00358 0.7 0.00949 0.8 0.0264

10 0.9 0.0032

challenging task and many identification algorithms have beendeveloped to solve the problem [6], [41]–[43] such as the leastsquare method, the genetic algorithms, and particle swarmoptimization (PSO). PSO [44] has been shown to be superiorto its competitors for identification of the P-I model. Withoutlosing generality, the PSO algorithm [45] is borrowed in thiswork to simultaneously obtain the weighted parameters b(ri),and coefficients a1 and a2 of polynomial input function withthe fixed threshold values ri in (13). Table I lists the identifiedparameters of the MPI model (7) with ten OSP operators (i.e.n = 10). It should be noted that the larger n is selected, itis more precision to describe the hysteresis loops in theory.In real applications, it is verified that further increase ofthe numbers of the OSP operators leads to no significantimprovement in the modeling accuracy. As an illustration,

Fig. 14 shows comparisons of identification errors with threedifferent values of n. From this figure, it can be observedthat the identification errors are at the same level with thethree n. In the following development, we select n = 10for a case study. To intuitively demonstrate the advantage offewer parameters in the proposed MPI model, Table II listsa quantified comparison with existing asymmetric P-I models[18], [34]–[36] when the number of threshold n is chosen as10.

TABLE IIA QUANTIFIED COMPARISON OF PARAMETERS IN DIFFERENT MODELS.

Model Parameters’ numberKuhnen [34] 29n = 10, l = 4

Dong [35] 50n = 10

Jiang [36] 51n = 10

Janaideh [18] 32n = 10

Proposed 22n = 10

With the identified parameters of the proposed MPI model,Fig. 15 shows the comparison of experimental data of thepiezoceramic actuator and associated prediction results of theMPI model. The waveforms of the input voltage, displacement,prediction error and the hysteresis loops are shown in Fig. 15for the case in which the input voltage is a simple triangle



9

0 2 4 6 8 10 12 14 160

50

100

150

Time (s)

Vol

tage

(V

)

(a) Input voltage

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

80

Time (s)

Dis

plac

emen

t ( µ

m)

Experimental dataModel simulation data

(b) Displacement

0 2 4 6 8 10 12 14 16−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Err

or (

µm)

(c) Error

0 50 100 1500

10

20

30

40

50

60

70

80

Voltage (V)

Dis

plac

emen

t ( µ

m)

Experimental dataModel simulation data

(d) Hysteresis loops

Fig. 16. Experimental verification of the MPI model with a complex input signal.

waveform with three dominant maximum values. From Fig.15(b), it is observed that the prediction results of the MPImodel agree well with the experimental measurements. Fig.15(c) shows that the maximal prediction error is about 1.4%as a percentage of the full displacement range. In addition,the comparison of the hysteresis loops is given in Fig. 15(d).It can be seen that the proposed MPI model well describesthe asymmetric hysteresis nonlinearity of the piezoceramicactuator with both major and minor loops.

To further verify the effectiveness of the MPI model, Fig. 16shows the validation of the MPI model with a complex inputsignal, where the waveforms of the input voltage, displace-ment, prediction error and the hysteresis loops are given. FromFig. 16, it is observed that the proposed MPI model presentsexcellent performance in simulating the complex hysteresisloops. In this case, the maximal error between the predictionand experimental results is about 2.1%, which is slightly largerthan the one for the simple input signal. According to theresults as shown in Figs. 15-16, the mean modeling errorswith the MPI model are all less than 1%.

To demonstrate the advantage of the developed MPI model,the CPI model is also adopted to describe the hysteresisnonlinearity of the piezoceramic actuator under the sameexcitation of the complex input signal shown in Fig. 16(a).The modeling results with the CPI model are shown in Fig.17. It can be observed that the CPI model can only characterizethe symmetric hysteresis and the prediction output of the CPI

model is far from the experimental output of the piezoceramicactuator. Therefore, using the CPI model to describe theasymmetric hysteresis in the piezoceramic actuator leads tolarge errors, which motivates the development of the MPImodel in this work to characterize the asymmetric hysteresisnonlinearity. As an illustration, Fig. 18 shows the predictionerrors of the CPI and MPI model. It can be seen that theprediction error of the CPI model is three times larger thanthe one of the MPI model. In summary, we can conclude thatthe proposed MPI model well characterizes the asymmetrichysteresis loops of the piezoceramic actuators with higheraccuracy.

C. Asymmetric hysteresis compensation results

By using the inverse compensator (9)-(12), Fig. 19 demon-strates the inverse hysteresis compensation performance forthe piezoceramic actuator with the identified model parametersin Table I. Fig. 19(a) shows a comparison of the desired andactual trajectories, where the actual trajectory well follows thedesired trajectory. Fig. 19(b) shows the tracking errors definedas the difference between the desired position and the actualposition. It is worthy of mentioning that due to the existence ofmodeling uncertainty the tracking errors exhibits asymmetryabout the zero error level. To quantify the performance ofthe inverse compensator, the root-mean-square error erms with



10

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

80

Time (s)

Dis

plac

emen

t (µm

)

Expeimental dataModel simulation data

(a) Displacement

0 50 100 1500

10

20

30

40

50

60

70

80

Desired position (µm)

Act

ual p

ositi

on (

µm

)

Expeimental dataModel simulation data


Fig. 17. Experimental verification of the CPI model with a complex inputsignal.

0 2 4 6 8 10 12 14 16−2

−1

0

1

2

3

4

Time (s)

Err

or (

µm)

Prediction error of classical P−I modelPrediction error of modified P−I model

Fig. 18. Comparison of prediction errors with the CPI model and MPI model.

respect to the full displacement range is calculated by

erms =

√1N

N∑i=1

(ydi − yi)2

maxi

(ydi)−mini(ydi)

× 100% = 1.3%, (14)

and the maximum error em is

em =max

i(ydi − yi)

maxi

(ydi)−mini(ydi)

× 100% = 3.7%, (15)

where N is the number of experimental data points, yi andydi are the actual position and desired position at the ith

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

Time (s)

Dis

plac

emen

t (µm

)

Desired trajectoryActual trajectory

(a) Trajectory following

0 1 2 3 4 5 6 7 8−1

−0.5

0

0.5

1

1.5

2

2.5

Time (s) E

rror

(µm

)

(b) Error

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

Desired position vs control actionControl action vs actual positionDesired position vs actual position

(c) Three kinds of input-output relationships

Fig. 19. Demonstration of the inverse hysteresis compensation performance.

data point. Fig. 19(c) illustrates how the inverse hysteresiscompensator linearizes the asymmetric hysteresis nonlinearity,where both the inverse major-loop and minor-loop hysteresisloops with asymmetric characteristics are produced by theinverse hysteresis compensator to cancel the correspondingreal asymmetric hysteresis effect. Furthermore, analyzing theexperimental data, the maximum hysteresis caused error emwith the inverse compensator is reduced by up to 72.7%compared with the maximum hysteresis caused error 13.6%in the open-loop response.

To further verify the inverse compensator with the developedMPI model, experiments with periodic sinusoidal referencesxd(t) = Asin(2πt)+37.5 (µm) are conducted. Table III sum-maries the tracking performances em and erms at various am-



11

Frequency (Hz)

10-2

10-1

100

101

102

103

104

-80

-70

-60

-50

-40

-30

-20

-10

0

Magnitu

de (

dB

) Without the inverse compensation

With the inverse compensation

Frequency (Hz)

101

102

103

-3.2

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

Magnitu

de (

dB

)

185 Hz844 Hz

Fig. 20. Amplitude response of the system with and without the inversehysteresis compensation.

TABLE IIITRACKING PERFORMANCE FOR DESIRED SIGNALS

xd(t) = Asin(2πt) + 37.5 (µm).

Performance em erms

A = 7.5 2.93% 1.03%A = 15 2.17% 1.06%A = 22.5 2.59% 1.08%A = 30 2.61% 1.11%

plitudes A. It, therefore, clearly demonstrates the effectivenessof inverse compensator with the MPI model for asymmetrichysteresis compensation of the piezoceramic actuator. As a lastexample, the frequency response of the system with the inversecompensator are studied. Fig. 20 shows the compensationresponse of the system with and without the inverse hysteresiscompensation. It can be seen that the -3 dB bandwidth isincreased from 185 Hz (without the inverse compensation) to884 Hz (with the inverse compensation).

V. CONCLUSION

Due to the construction of the analytical inverse, the CPImodel is an effective method to describe and compensate forthe symmetric hysteresis nonlinearity. Since the CPI modelis developed only for describing the symmetric hysteresis, ityields considerable errors for the asymmetric hysteresis. Toremedy this drawback, a MPI model is developed in thiswork to characterize the asymmetric hysteresis nonlinearityof piezoceramic actuators. The experimental results show thatwith the use of MPI the modeling error is reduced by up to66.7% compared with the CPI model. Using the MPI model,an analytic inverse hysteresis model can be directly derivedfrom the inverse of the CPI model. Experimental results ona piezoceramic actuated platform are also presented to verifythe effectiveness of the developed MPI model and the inversehysteresis compensator.

In the future, the robust feedback controller together withthe MPI model based feedforward compensator will be de-signed to further improve the tracking performance of thepiezoceramic actuators.

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Guo-Ying Gu (S’10-M’13) received the B.E. degreein electronic engineering and the Ph.D. degree inmechanical engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 2006 and 2012,respectively.

Dr. Gu is currently working as a postdoctoral re-search fellow with School of Mechanical and PowerEngineering in Shanghai Jiao Tong University. Hewas a Visiting Researcher with Concordia Univer-sity, Montreal, QC, Canada (from October 2010 toMarch 2011 and from November 2011 to March

2012). He was awarded ”Scholarship Award for Excellent Doctoral Studentgranted by Ministry of Education of China” in 2011 and Best ConferencePaper Award in The 2011 IEEE International Conference on Information andAutomation. His research interests include motion control of nanopositioningstages for scanning probe microscopy applications, robust control, modelingand control of smart material based actuators with hysteresis.

Li-Min Zhu (M’12)received the B.E. (with honors)and Ph.D. degrees in mechanical engineering fromSoutheast University, Nanjing, China, in 1994 and1999, respectively.

From November 1999 to January 2002, Dr. Zhuworked as a postdoctoral fellow in Huazhong Uni-versity of Science and Technology. In March 2002,he was appointed associate professor in the RoboticsInstitute, Shanghai Jiao Tong University. Since Au-gust 2005, he has been a professor. He was a VisitingScholar with Monash University, Clayton, Australia

(from September 1997 to May 1998), and the City University of Hong Kong,Kowloon, Hong Kong (from December 2000 to March 2001). His researchinterests include: (1) CNC machining and optical inspection of complexshaped parts, and (2) control, sensing and instrumentation for micro/nanomanufacturing.

Chun-Yi Su (SM’98) received the Ph.D. de-gree from South China University of Technology,Guangzhou, China, in 1990.

Dr. Su is currently with the College of AutomationScience and Engineering at the South China Uni-versity of Technology, Guangzhou, P. R. China, onleave from Concordia University, Montreal, Canada.Prior to joining Concordia University in 1998, hewas with the University of Victoria from 1991 to1998. Dr. Su conducts research in the application ofautomatic control theory to mechanical systems. He

is particularly interested in control of systems involving hysteresis nonlinear-ities. Dr. Su is the author or co-author of over 300 publications, which haveappeared in journals, as book chapters and in conference proceedings.

Dr. Su has served as Associate Editor of IEEE Transactions on AutomaticControl and IEEE Transactions on Control Systems Technology, and Journalof Control Theory & Applications. He is on the Editorial Board of 14 journals,including IFAC journals of Control Engineering Practice and Mechatronics.Dr. Su has also severed for many conferences as an organizing committeemember, including General Co-Chair of the 2012 IEEE International Confer-ence on Mechatronics and Automation, and Program Chair of the 2007 IEEEConference on Control Applications.

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