1792 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 21, NO. 3, JUNE 2016

Modeling and Identification of Piezoelectric-Actuated Stages CascadingHysteresis Nonlinearity With Linear Dynamics

Guo-Ying Gu, Chun-Xia Li, Li-Min Zhu, and Chun-Yi Su

Abstract—In this paper, we propose a new modeling and identi-fication approach for piezoelectric-actuated stages cascading hys-teresis nonlinearity with linear dynamics, which is described asa Hammerstein-like structure. In the proposed approach, the hys-teresis and linear dynamics together with the delay time and higherorder dynamic behaviors are obtained with three data-driven iden-tification steps under designed input signals. In the first step,the step input signal is applied to estimate the delay time of thepiezoelectric-actuated stages. In the second step, the autoregres-sion with exogenous signal identification algorithm is adopted toidentify the linear dynamics using a small-amplitude band-limitedwhite noise input signal. In the third step, with the identified lineardynamics model, the parameters of the rate-independent Prandtl–Ishlinskii hysteresis model are identified by the particle swarm opti-mization algorithm using a simple low-frequency triangle input sig-nal with different amplitudes. Finally, the experimental results ona piezoelectric-actuated stage show that both the hysteresis anddynamic behaviors of the piezoelectric-actuated stage are well pre-dicted by the proposed modeling method. In addition, we providethe analysis of quantitative prediction errors of the identified modelwith comparison to experimental data, which clearly demonstratethe effectiveness of the proposed approach.

Index Terms—Dynamics, Hammerstein model, hysteresis,piezoelectric-actuated stages.

I. INTRODUCTION

W ITH the rapid development of micro/nanopositioningtechnology, traditional stepper, DC/servo motors are not

up to the accuracy requirement. Micromotion actuation technol-ogy, for instance, Lorenz motors (such as voice coil motors andultrasonic motors) [1]–[4] and smart material-based actuators(such as piezoelectric actuators and magnetostrictive actuators)[5]–[7] become promising in the precision applications. Amongthem, the piezoelectric actuators are more popular in nanopo-sitioning applications due to their advantages of subnanometerresolution, high bandwidth, and large mechanical force. Nowa-days, piezoelectric-actuated stages have been applied in a widerange of nanopositioning equipment such as nanomanipulators[5], [8], [9] and scanning probe microscopy [6], [10]–[12].However, the inherent hysteresis nonlinearity and lightlydamped dynamics in the piezoelectric-actuated stages usually

Manuscript received February 26, 2015; revised June 15, 2015; ac-cepted July 31, 2015. Date of publication August 7, 2015; date of currentversion April 28, 2016. Recommended by Technical Editor K. K. Leang.This work was supported by the National Natural Science Foundation ofChina under Grant 51405293 and Grant 51421092.

G.-Y. Gu, C.-X. Li, and L.-M. Zhu are with the State Key Laboratoryof Mechanical System and Vibration, School of Mechanical Engineer-ing, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:guguoying@sjtu.edu.cn; lichunxia@sjtu.edu.cn; zhulm@sjtu.edu.cn).

C.-Y. Su is with the Department of Mechanical and Industrial Engi-neering, Concordia University, Montreal, QC H3G 1M8, Canada (e-mail:cysu@alcor.concordia.ca).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2015.2465868

deteriorate the system performance and make a major challengefor achieving the high-performance nanopositioning control [5],[11], [13].

Various control techniques handling the hysteresis and lightlydamping caused vibration have been proposed to improve thetracking performance of piezoelectric-actuated stages, whichcan be roughly classified into feedforward inverse controltechniques and control techniques involving with feedback.Feedforward inverse control [14]–[16] generally results intracking errors, which depends on the accuracy of the identifiedsystem models. This explains the reason why the controltechniques involving with feedback controllers are developed.The reader may refer to the recent surveys [5], [6], [11] ofpiezoelectric-actuated stages for the detailed discussions andcomparisons of different control techniques. Although, the sys-tem dynamic model may not be necessary for the PID control,precise modeling and identification of the hysteresis, and dy-namics of piezoelectric-actuated stages are generally importantfor the model-based control design, achieving high controlaccuracy.

For this purpose, several interesting studies have been re-cently reported in the literature, for instance, in [10], [17]–[22]. Analyzing these reported studies, we found that: 1) Thepiezoelectric-actuated stages were described as the linear dy-namics models, thus, either the time-domain-based axiomaticdesign theory [17], the commercial dynamic signal analyzer[10], the weighted iterative least square fitting technique [18],or the subspace-based state space system-identification tech-nique [19] was developed to identify the model parameters.However, the hysteresis nonlinearity was generally neglected in[10], [17]–[19], which made the identified models not accurate.2) As a matter of fact, the reasonable hysteresis model shouldbe the case that the hysteresis and the dynamics are coupled[20], [21]. However, it really imposes a challenge on how toidentify the parameters of the coupled dynamics, which is along standing problem even in the control society. Generally,hysteresis and linear dynamics were identified individually withtwo decoupled steps, where the high-order dynamics and delaytime behaviors were not considered. 3) Liu et al. [22] presentedan alternative approach for piezoelectric actuators described bythe Preisach hysteresis model and cascaded nonhysteretic dy-namics by constructing an inverse hysteresis as a feedforwardcompensator to cancel the hysteresis effect so that the lineardynamic part can be identified. However, the inverse construc-tion still needs the knowledge of the hysteresis part, making theidentification complicated.

It is obvious from the above analysis that the availableapproaches for the identification of piezoelectric-actuatedstages containing the hysteresis and linear dynamics are still an

1083-4435 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 21, NO. 3, JUNE 2016 1793

Fig. 1. Hammerstein-like model structure of the piezoelectric-actuatedstages.

open problem. This paper is, therefore, motivated to proposean effective modeling and identification approach to tacklethe challenge. As an initial attempt, the stage is described asa Hammerstein-like structure [23]–[25], that is, a cascade ofa nonlinear static block followed by linear dynamics. Ratherthan neglecting the hysteresis nor using the feedforwardcompensator that is generally reported in the literature [10],[17]–[22], the coupled effects together with the delay time andhigher order dynamic behaviors are taken into account in thispaper. In this paper, the proposed approach is implementedwith three data-driven identification steps to identify themodel by only using the input–output experimental data underthe designed input signals. The experimental results on acustom-built piezoelectric-actuated stage demonstrate that theprediction of the identified model agrees well with the exper-imental data. In addition, the analysis of quantitative predictionerrors of the identified model is provided with comparison toexperimental data, which is generally missed in the reportedworks.

II. IDENTIFICATION PROBLEM STATEMENT

For the purpose of identification, the nonlinear dynamic be-havior of the piezoelectric-actuated stages is described by theHammerstein-like model structure as shown in Fig. 1 [13], [21],where H[v](t) denotes the input hysteresis nonlinearity, G(z)represents the linear discrete-time dynamics, v(t) is the controlinput, w(t) is the output of the hysteresis, usually unknown, andy(t) is the output of the system.

In this study, H[v](t) is described by a rate-independent mod-ified Prandtl–Ishlinskii (MPI) hysteresis model. For any piece-wise monotone input function v(t) ∈ Cm [0, tE ], the MPI hys-teresis model can be expressed as [26],

w(t) = H[v](t) = g(v(t)) +∫ R

0p(r)Fr [v](t)dr (1)

where g(v(t)) is a generalized odd input function with the mem-oryless and locally Lipschitz continuous properties, r ≥ 0 arethresholds, p(r) is a density function that is generally calculatedfrom the experimental data, and Fr [v](t) is the one-side playoperator, defined as

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

w(t) = Fr [v](t) = fr (v(t), w(ti)) (2)

for ti < t ≤ ti+1 , 0 ≤ i ≤ N − 1 with

fr (v, w) = max(v − r,min(v, w)) (3)

for 0 = t0 < t1 < · · · < tN = tE . We should mention that theproposed modeling and identification approach in this paper isnot limited to a certain kind of hysteresis models. For verifica-tion, the MPI model is selected as an illustration because the MPIis very popular in recent studies for describing the hysteresis inpiezoelectric actuators. Certainly, other hysteresis models [5],for instance, the physics-based models and phenomenologicalmodels can also be selected.

As shown in Fig. 1, the linear dynamics G(z) is expressed as

G(z) =B(z)A(z)

z−d (4)

with

A(z) = 1 + a1z−1 + a2z

−2 + · · · + anaz−na

B(z) = b1 + b2z−1 + · · · + bnbz

−nb+1 (5)

where z−1 is unit backward shift operator, z−d is the pure delaytime of the practical system with d > 0, A(z) and B(z) arepolynomials with degrees of na and nb − 1, respectively, a1 ,a2 , . . ., ana are the coefficients of the polynomial A(z), b1 , b2 ,. . ., bnb are the coefficients of the polynomial B(z).

The following objective of this paper is to develop an effec-tive approach to identify the coefficients of the input functiong(v(t)) and density function p(r) of the MPI hysteresis modelin (1) and the coefficients ai , bi , and d of the linear dynamicsG(z) in (4) and (5) only using the experimental input–outputdata.

To verify the proposed identification approach, simulationand experimental results on a piezoelectric-actuated stage arealso presented in the following development.

III. PROPOSED IDENTIFICATION APPROACH

From the discussions in the previous section, there are threeterms in the model of the piezoelectric-actuated stages, in-cluding the hysteresis nonlinearity, conventional linear time-invariant dynamics, and pure delay time. It is obvious that thepure delay time of the system z−d in (4) can be obtained throughthe timescale of the input voltage and captured sensor response.In addition, it is well known in the literature [6], [15], [21] thatthe effect of hysteresis can be ignored in the displacement re-sponse if small-amplitude input excitation signals are applied.In this sense, the small-amplitude band-limited white noise sig-nal is able to identify coefficients of the polynomials A(z) andB(z) in (5) together with the obtained delay time using theARX identification method. Finally, with the identified parame-ters of the delay term and linear dynamics model, the parametersof the MPI hysteresis model (1) can be obtained by the parti-cle swarm optimization (PSO) algorithm. In the following, theproposed identification approach with three identification stepswill be detailed to identify the model parameters in (1)–(5) fora piezoelectric-actuate stage.

1794 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 21, NO. 3, JUNE 2016

Fig. 2. Experimental platform and identification results of the piezoelectric-actuated stage. (a) Experimental setup. (b) Step response. (c) Frequencyresponse. (d) Hysteresis nonlinearity.

A. Experimental Platform

The experimental platform consists of a custom-builtpiezoelectric-actuated stage composed of the piezoelectric ac-tuator and a one-dimensional monolithic flexure mechanism. AdSPACE-DS1103 control board equipped with the 16-bit analogto digital converters (ADCs) and 16-bit digital to analog con-verters (DACs) is used to implement the identification approach.The DACs produce the analog control input and the voltage am-plifier with a fixed gain of 15 which is adopted to provide ex-citation voltage 0–150 V for driving the piezoelectric actuator.An integrated strain gauge sensor (SGS) with the measurementrange 75 μm and resolution <1 nm is utilized to real-time mea-sure the position of the piezoelectric actuator with the sensitivityof 7.5 μm/V. The sensor output signals are then captured by asignal conditioner and simultaneously acquired by the ADCsof the dSPACE-DS1103 control board. In this study, the sam-pling frequency of the dSPACE-DS1103 control board is set as20 kHz. Fig. 2(a) shows the block diagram of the experimentalsetup.

B. Identification of the Delay Time

The delay time of the experimental platform is first estimatedby applying the step input signal to the piezoelectric-actuatedstage.

With the measured position signals of the sensors, as shownin Fig. 2(b), it can be directly obtained that the delay time of the

experimental system is 0.00015 s. Considering the fact that thesampling interval is 0.00005 s, d = 3 in (4) is obtained.

C. Identification of the Linear Dynamics

Subsequently, a small-amplitude band-limited white noisesignal is used to excite the piezoelectric-actuated stage byneglecting the hysteresis nonlinearity. In this case, H[v](t) is ap-proximated as 1. Then, the ARX algorithm is adopted to identifythe coefficients ai and bi in (5) with the obtained d = 3. Thereare two methods, i.e., least squares and instrumental variablesmethod, to estimate the coefficients ai and bi in the ARX modelstructure. It should be noted that the identification accuracy ofthe ARX algorithm would increase along with the increase ofthe polynomial degrees na and nb. However, the computationalcomplexity will increase as well. There is a tradeoff betweenthe complexity and accuracy [27], [28]. In this study, the iden-tification tool of the MATLAB is used to implement the ARXalgorithm to get the optimum model orders and parameters.

Using the sampling input and output data of the piezoelectric-actuated stage, the identified parameters are na = 10,nb = 10, a1 = −1.5157, a2 = 0.3685, a3 = 0.5132, a4 =0.02363, a5 = −0.09831, a6 = −0.2931, a7 = 0.0842, a8 =0.2921, a9 = −0.2582, a10 = 0.05803, b1 = 0.02146, b2 =0.02553, b3 = 0.01495, b4 = −0.001324, b5 = 0.001639, b6 =0.01780, b7 = 0.02112, b8 = 0.01059, b9 = 0.001185, b10 =−0.001171.

The identification results using ARX from the data show thatthe loss function is 2.92354e-008 and final prediction error is

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 21, NO. 3, JUNE 2016 1795

Fig. 3. Comparisons of the experimental responses of the piezoelectric-actuated stage and model simulation responses. (a) f = 1 Hz. (b) f =10 Hz. (c) f = 50 Hz. (d) f = 120 Hz.

2.92477e-008. Fig. 2(c) also shows the dynamic frequency ofthe identified linear dynamics model with comparison to theexperimental data, which clearly verifies the effectiveness ofthe identified model.

D. Identification of the Hysteresis

With the identified linear dynamics model, it is ready to iden-tify the parameters of the MPI hysteresis model. In this step,the frequency of the input signal is restricted as 1 Hz. Thus, theidentified linear dynamics model can be directly recognized as aconstant gain, which is obtained as G(j2π) = 0.6415 as shownin Fig. 2(c).

In order to identify the parameters of the MPI model, thedefinition of MPI model given in (1) is described in the discreteform of a finite number of the play operators as follows [26]:

w(t) = H[v](t) = p1v3(t) + p2v(t) +

n∑i=1

q(ri)Fri[v](t)

(6)where the third degree polynomial input function g(v(t)) =p1v

3(t) + p2v(t) with two coefficients p1 and p2 is utilizedfor the asymmetric hysteresis description, n is the number ofthe adopted play operators, and q(ri) = p(ri)(ri − ri−1) is theweighted coefficient for the threshold ri .

It is worth mentioning that the identification of the hysteresismodels is a challenging task and many algorithms, such as theleast square method, the genetic algorithms, and PSO have beendeveloped to solve this problem. According to the previousstudies [26], [29], [30], the PSO is adopted as an illustration

in this study. In this step, a simple triangle input signal v(t)is adopted as an illustration for the identification, which isexpressed as follows:

v(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

6t, 0 ≤ t < 0.5 s

6(1 − t), 0.5 ≤ t < 1 s

6(t − 1), 1 ≤ t < 2 s

6(3 − t), 2 ≤ t < 3 s

6(t − 3), 3 ≤ t < 4.5 s

6(6 − t), 4.5 ≤ t < 6s.

(7)

The identified parameters of the MPI model (6) withn = 10 are ri = (i − 1)/10, i = 1, 2, . . . , 10, q(r1) = 0.4777,q(r2) = 0.4473, q(r3) = 0.1113, q(r4) = 0.0671, q(r5) =0.1411, q(r6) = 0.0104, q(r7) = 0.0001, q(r8) = 0.0354,q(r9) = 0.0961, q(r10) = 0.0084, p1 = −0.2774, p2 =0.6232.Fig. 2(d) shows the hysteresis behaviors of the identified modeland the experimental results of piezoelectric-actuated stage.It can be seen that the predicted response of the identifiedmodel agrees well with the experimental results of the restedpiezoelectric-actuated stage, where the root-mean-square(RMS) predicted error is about 0.57% with respect to the strokeof the stage. It is worth mentioning that in theory the larger nof the play operators in the MPI hysteresis models is used, themore precise it may be to describe the hysteresis nonlinearity[26]. In this study, the PSO algorithm is borrowed as anillustration to simultaneously obtain the weighted parametersq(ri), and coefficients p1 and p2 with the fixed threshold values

1796 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 21, NO. 3, JUNE 2016

Fig. 4. Comparisons of the predicted errors of the dynamic model and MPI model under different frequencies. (a) f = 1 Hz. (b) f = 10 Hz. (c) f =50 Hz. (d) f = 120 Hz.

TABLE IRMS PREDICTED ERRORS OF THE PROPOSED DYNAMIC MODEL AND ONLY

MPI MODEL WITH RESPECT TO THE STROKE OF THE STAGE

Frequency 1 Hz 10 Hz 50 Hz 80 Hz 100 Hz 120 Hz

Dynamic model 1.03% 1.75% 2.73% 3.15% 3.23% 3.59%MPI model 1.03% 1.98% 4.85% 7.36% 8.37% 10.06%

ri . The convergence of the PSO algorithm is influenced by thepopulation size and weighting size of the PSO algorithms. Forthe detailed analysis and discussions of the convergence of thePSO algorithm, the reader may refer to [29], [30].

E. Experimental Verification

In this section, experiments are conducted to verify the iden-tified hysteresis and dynamics models with proposed identifi-cation approach. In the experiments, the input signals v(t) =5 + 5 sin(2πft) (V) with different frequency f were used toexcite the piezoelectric-actuated stage.

Fig. 3 shows the experimental responses of the piezoelectric-actuated stage compared with responses of the identified dy-namic model and the rate-independent MPI model under dif-ferent input frequencies, respectively. Fig. 4 also shows thepredicted errors of the dynamic model and MPI model underdifferent frequencies. For the quantitative comparison, Table Iand Fig. 5 show the RMS predicted errors of the identifieddynamic model and only MPI model with respect to the strokeof the stage.

Fig. 5. Comparisons of RMS predicted errors of the proposed dynamicmodel and only MPI model with respect to the stroke of the stage.

It can be seen that the rate-independent MPI model itself canonly describe the experimental response of the piezoelectric-actuated stage when the input frequencies are below than 10Hz. This can explain why many interesting studies have beenreported to use the rate-independent hysteresis model for char-acterizing the piezoelectric-actuated stages and then apply it forcontrol design, for instance, in [14], [31], and [26]. However,large errors exist in this case with the increase of the input fre-quencies. On the other hand, for the larger input frequencies (forinstance, >10 Hz), the combined model structure with the rate-independent hysteresis model and linear dynamics model accu-rately describes the complex hysteresis nonlinearity in terms ofboth amplitude-dependent and frequency-dependent behaviors

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 21, NO. 3, JUNE 2016 1797

and the dynamic response of the piezoelectric-actuated stage.Therefore, the dynamic model with the identified parameterscan well account for the dynamic behavior of the piezoelectric-actuated stage in a large frequency band, which clearly demon-strates the effectiveness and feasibility of proposed modelingand identification approach.

IV. CONCLUSION

This paper proposes a new modeling and identification ap-proach for the piezoelectric-actuated stages by comprehensivelyconsidering the coupling of the hysteresis and dynamics, whereonly the input–output experimental data are sufficient withouttaking the physical behaviors into account. To this end, threeidentification steps are presented using the input–output exper-imental data under designed input signals. The experimentalresults on a piezoelectric-actuated stage show that both the hys-teresis and dynamics behaviors of the piezoelectric-actuatedstage are well predicted by the identified model, which clearlyverify the effectiveness of the development. In the future, themodel-based control techniques such as the H∞ [32], repeti-tive control [13] and active-damping control [33] will be in-vestigated to improve the motion-tracking performance of thepiezoelectric-actuated stages with the identified dynamic model.

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