Dynamic compensation and H1 control for piezoelectric actuatorsbased on the inverse Bouc–Wen model

Shunli Xiao a, Yangmin Li a,b,c,n

a Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira, Taipa, Macaob Autonomous Systems Research Group, King Abdulaziz University, Jeddah 21589, Saudi Arabiac School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300191, China

a r t i c l e i n f o

Article history:Received 18 April 2013Received in revised form19 August 2013Accepted 21 August 2013Available online 26 September 2013

Keywords:Piezoelectric actuatorHysteresisBouc–Wen modelControl

a b s t r a c t

Piezoelectric actuator, which is widely used in micro-/nano-manipulation, often demonstrates nonlinearrate-dependent hysteresis characteristics. This paper proposes a Bouc–Wen model based inversehysteresis compensator for rate-dependent hysteresis in PZT. The classical Bouc–Wen model forhysteresis is introduced firstly, then the identification of the parameters through the particle swarmoptimization (PSO) method is conducted. A novel modified inverse Bouc–Wen model is proposed andverified through experiment. The proposed modified inverse Bouc–Wen model with a parametricselector constructs a real-time online rate-dependent compensator for handling PZT's hysteresis. Duringexperiments of tracking multi-frequency composed of signals, the rate-dependent hysteresis of the PZTcan be consistently compensated. The experimental results show that the proposed open loop hysteresiscanceling method greatly improves the PZT's tracking control accuracy.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Piezoceramic actuators possess advantages of high output force,large bandwidth and fast response time, which enable them to findapplications in micro-/nano-technology in terms of atomic forcescanning microscope (AFM), biological cell handling, integratedcircuit chip assembly, micro-machine tool, and active vibrationisolation device [1–4]. Similar to many intelligent materials, thePZT also owns a natural hysteresis characteristics which brings asevere positioning error to the system and affects the performanceof the system in the case of requiring precise positioning operation.Even worse, the hysteresis is rate-dependent which means increas-ing the rate of input driving signals of the PZT, the hysteretic loopwill vary to a large and round one. These features not only bringerrors to the system, but also cause instability to the controller.

Hysteresis model and compensator can be simply classified asfollows: closed-loop displacement controller, compensator modelbased closed-loop controller and open-loop compensator with inversehysteresis model. In more detail, the research works can be dividedinto canceling stationary hysteresis problems, high dynamic rate-dependent hysteresis compensator [5–9] and other hysteresis asso-ciated with drift and creep control [10]. Although this hysteresis loopcan be predicted by curve-fitting methods using the measured setsof curves, the hysteresis loops with higher order reversal curves

cannot be predicted by curve-fitting methods. They can also bepredicted by the classical Preisach model, but the Preisach model isa time consuming and discrete approach and the identification of theμ�density function is also a difficult issue.

For a closed-loop controller, feedback sensors in terms of strain-gauge sensors, capacitive sensors, or optical sensors are required,which will depend on applications in choosing suitable types.A controller may be adopted with a simple PID controller or a slidingmode controller. A lot of commercial products such as PI and Thorlabsadopt the strain-gauge type feedback sensor and many researchesfocus on this closed-loop feedback control, such as perturbationestimation and PID sliding surface, which is successfully used formotion tracking of a piezo-driven micromanipulator with opticalsensor [11], a PD-lag controller associated with the inverse Preisachmodel is adopted in [12], a sliding mode controller is used to controlthe hysteresis problems of PZT in [13], and many adaptive controllersare designed to cater to the system with hysteresis characteristics[14,15]. Obviously, the feedback sensor increases the cost, and thecontroller will inevitably reduce the response speed of the actuator.Especially, for a parallel active vibration control system [16], three orsix PZTs will be used, if a feedback controller for each PZT is used, a lotof sensors are needed. Too many control loops will bring troubles forthe upper level controller and much computational load or eveninstability to the whole system.

The main idea of the inverse hysteresis model-based compensatoris to obtain a mathematical model that precisely describes thecomplex hysteretic behavior, then to implement an inverse feedforward compensator based on the inverse hysteresis model tolinearize the actuator response [6]. Various methods have been

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Robotics and Computer-Integrated Manufacturing

0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.rcim.2013.08.002

n Corresponding author at: Department of Electromechanical Engineering,Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira,Taipa, Macao. Tel.: þ853 8397 4462; fax:þ853 2883 8314.

E-mail address: ymli@umac.mo (Y. Li).

Robotics and Computer-Integrated Manufacturing 30 (2014) 47–54

developed to model the hysteresis problems, such as the Preisachmodel [10], the Maxwell model, the Duhem model, the Bouc–Wenmodel, and the Prandtl–Ishlinskii model [17].

Most of those researches are focused on canceling stationaryhysteresis problems, however many applications of the PZT are in anonstationary environment, just as mentioned before, parallel activevibration control and most of these modeling methods do not workfor nonstationary situations. A rate-dependent controller based on thePrandtl–Ishlinskii (PI) model is investigated in [6], but the flaw of thismodel is that singularities will occur at higher frequency, the turningpoint is around 40 Hz found in the experiment.

Therefore, this paper tries to find a model with characteristicsof less time consuming, rate-dependent and high accurate com-pensation for a real time control system. Firstly, the experimentalsetup is introduced in Section 2. A series of tests are conducted tostudy the hysteresis properties of the piezoceramic actuatorsystem and a series of hysteresis reversal curves are measuredand recorded through dSPACE in Section 3. Secondly, the para-meters for Bouc–Wen model and inverse Bouc–Wen model of eachreversal loop are identified via the particle swarm optimization(PSO) method, and the performances of the identified Bouc–Wenmodels are verified through experiments in Section 4. Thirdly, theH1 controller is designed in Section 5. Then the accumulation ofinverse Bouc–Wen model for each line-segment with a differentslope rate is used to adjust the hysteresis problems of the PZT inreal time. Lastly, the proposed method is applied to the PZT, theperformance is discussed and a conclusion is made in Sections6 and 7, respectively.

2. Experimental setup

The experimental setup is shown in Figs. 1 and 2. One 30�μmstroke PZT (from Physik Instrumente, Inc.) is adopted, and the PZT isactuated with a voltage of 0–100 V. The power amplifier (PhysikInstrumente Inc.) can amplify the input signal from 0–10 V to 0–100 Vto drive the PZT. The output motions of the PZT are measured by a capdisplacement sensor (D-510 from Physik Instrumente, Inc.) with aresolution of 1 nm and linearity better than 0.1% over a measuringrange of 50 μm with a bandwidth of up to 10 kHz. The commandsignal and the displacement output signal are generated and mea-sured through dSPACE which is a real time measurement and controlsystem platform.

3. Simulations and experimental results

To study the rate-dependent property of the PZT, firstly, twotriangle signals with the same amplitude and frequency of 40 Hzand 100 Hz are applied to the PZT, the relationship between theresponse and the input signals can be observed in Figs. 3 and 4.It shows that the higher the frequency, the larger the hysteresisloop is, which means that the hysteresis problem of the PZT israte-dependent. The same experiment under very low frequencycan also find the rate-dependent phenomena. To further investi-gate the rate-dependent characteristics, a series of triangle inputsignals with the same amplitude and increased frequencies areapplied to the PZT through dSPACE and the power amplifier, therelationship between the input signals and the responses indicatesmuch clearly the rate-dependent characteristics as shown in Fig. 5.

4. Bouc–Wen model and inverse Bouc–Wen model of the PZThysteresis

4.1. Bouc–Wen model and parameters identification

The static Bouc–Wen model adapted to piezoelectric actuatorsis described as follows:

yðtÞ ¼ _u�h; _h ¼ α _u�βj _ujh�γ _ujhj; ð1Þ

Fig. 1. System block diagram.

Fig. 2. Experimental setup.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–5448

where y is the output displacement, h is the hysteresis loop in termsof displacement whose magnitude and shape are determined byparameters α, β and γ as shown in Fig. 6. There are many optimiza-tion methods which can be used to identify the parameters of theBouc–Wen model. Since the particle swarm optimization (PSO)method is with great performance and easy to be implemented bythe MATLAB PSO toolbox, it is adopted to identify the parameters inthis research [18]. One of the key problems of the particle swarm

optimization (PSO) methods is the selection of the fitness function. Inthis research, the fitness function is chosen as

Fðα; β; γÞ ¼ 1n

∑n

i ¼ 1ðyi�yBWi Þ2; ð2Þ

where n is the total number of samples and ðyi�yBWi Þ denotes theerror between the ith sample data yi and the predicted data yBWi viathe Bouc–Wen model with parameters α, β and γ. Through theMATLAB PSO toolbox running in the MATLAB software environment,the parameters can be selected properly for the Bouc–Wen model.

4.2. Comparison of the simulated and experimental hysteresisbehavior of the PZT

In this section, the Bouc–Wen model is numerically implemen-ted to predict the displacement outputs of the PZT. After measur-ing and recording the input and output of the PZT, many ways canbe adopted to model and identify the hysteresis loops [19–21].Based on experimental data, the parameters of the Bouc–Wenmodel are identified through the particle swarm optimization(PSO) method. After the parameters are identified, the output ofthe plant can be predicted in real time by the Bouc–Wen model.

Fig. 3. Hysteresis loop at 40 Hz.

Fig. 4. Hysteresis loop at 100 Hz.

Fig. 5. Hysteresis loops from 20 Hz to 100 Hz.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–54 49

The selected predicted hysteresis loops and the experimentalloops with much larger deviation among a lot of experiments dataare compared in Figs. 7 and 8. It can be seen that they still matchvery well with each other. It can also be observed from Figs. 7 and8, for a set of identified parameters of the Bouc–Wen hysteresismodel under certain driving signal with constant frequency, notonly the major full range loops but also the smaller loops can bepredicted accurately. For these smaller loops, whose start pointand end point are randomly selected. In other words, the para-meters of the Bouc–Wen hysteresis model are just affected by thedifferent driving signals with different frequencies and have norelation with the start and end points of the hysteresis loops. It isalso one of the most important advantages of the Bouc–Wenhysteresis model to predict rate-independent hysteresis loopscompared with those curve fitting methods.

4.3. Classical rate-independent inverse Bouc–Wen model

The key idea of an inverse feed-forward compensator is tocascade the inverse hysteresis operator Γ�1 with the actualhysteresis, which is represented by the hysteresis operator Γ toobtain an identified mapping between the desired actuator outputbyðtÞ and the actuator response y(t) [6,12]. The mathematicalequation can be expressed by

yðtÞ ¼ Γ½Γ�1½by��ðtÞ ¼ I½by�ðtÞ ¼ yðtÞ: ð3ÞThe operation of the cascade inverse feed-forward controller is

given in Fig. 9(a) and (c).The advantages of the Bouc–Wen model are that it is a pure

mathematical function and a time saving model with just fewparameters. There is no direct relationship between the modelingparameters and the physical hardware system. Therefore, from themathematical point of view, the relationship between the inputand the output is reversible. Furthermore, it is not difficult to usethe experimental method to prove that the inverse Bouc–Wenloops of a virtual machine with displacement as an input signaland voltage as an output signal can be identified as normal Bouc–Wen loops at all. It is necessary for us to build up the inverseBouc–Wen model, the input can be the reference distance and theoutput can be the voltage value which is going to be used as thedriving signal by the following cascade real hardware system torecover the reference distance signal.

As shown in Eq. (3) and Fig. 9, it describes the whole modifiedinverse Bouc–Wen model based compensator. After the referencedisplacement signal is input through the inverse Bouc–Wenmodel, the output voltage signal, which is with the inversehysteresis characteristics, is sent to the real system to cancel thehysteresis problems.

4.4. Rate-dependent inverse Bouc–Wen hysteresis model

The basic idea of the rate-dependent Bouc–Wen hysteresismodel is to discretize the input signal and predict the discreteline segments one by one, then sum them together. Because thecontinuous driving signals can be described as a curve on a time–voltage plane, it can be approximated or fitted by broken-lines asshown in Fig. 10. Since the parameters of the Bouc–Wen hysteresismodel are just affected by the different driving signals withdifferent frequencies and have no relation with the start and endpoints of the hysteresis loops, for lines with different slope-rates,the increased or decreased output voltage can be predictedaccurately by the model with related parameters. If all thehysteresis loops with different slope-rates are measured andidentified, the hysteresis of each section of line can be predictedvia the Bouc–Wen model. Then the displacement output of the PZTdriven by a continuous signal can be predicted by accumulatingthe hysteresis predicted through those line segments.

Fig. 6. Bouc–Wen model for calculating the hysteresis term h from the input voltage u.

Fig. 7. Performance of the predicted loops through identified parameters at 40 Hzof triangle signal.

Fig. 8. Performance of the predicted loops through identified parameters at 100 Hzof triangle signal.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–5450

Just the same process as the prediction of the output of the PZTsystem, the inverse rate-dependent model can be constructed. Thereference displacement signal, which is the input of the inverseBouc–Wen model, can be fitted through broken-lines, and theinverse hysteresis of these displacement line segments can bepredicted via the inverse Bouc–Wen model. Then the voltagesignal, which is the output of the inverse Bouc–Wen model, canbe predicted via the accumulation of these hysteresis outputs.When the predicted voltage is fed to the real PZT system withhysteresis accordingly, the displacement output is adjusted, whocan follow the reference displacement signal as illustrated inFigs. 9 and 11.

5. Controller design for the PZT system and its experimentalverification

Since the hysteresis problems of the PZT are compensated bythe proposed feed-forward compensator, the system can betreated as a linear system approximately. The transfer functionof the PZT system can be identified through the AutoRegressiveeXogenous (ARX) method via the MATLAB toolbox with “arx”command conveniently. Eq. (4) shows the identified continuoustime domain open loop transfer function of the PZT system. Fig. 12shows that the identified system can match the measured stepresponse well. Based on the identified model, the state spaceequation of the PZT system is obtained. The H1 controller is

selected as it can be easily designed via the MATLAB toolbox

GðsÞ ¼ 1135sþ35 770s3þ3960s2þ5 046 000sþ145 200 000

: ð4Þ

The time-domain transfer function can be converted into statespace equation as

_x ¼ AxþBu;y¼ CxþDu; ð5Þwhere

A¼�3960 �5 046 000 �145 200 000

1 0 00 1 0

264

375;

B¼ ½1;0;0�T ; C ¼ [0, 1135, 35 770], D ¼ 0.When the state space equation is identified, the H1 controller

can be designed by the MATLAB robust control toolbox usingfunction “hinf”. After the control algorithm is realized and com-plied in MATLAB Simulink with RTI environment, it can be down-loaded into dSPACE to control the PZT system in real time.

6. Compensator performance and discussion

After all parameters of the inverse Bouc–Wen model areidentified through the particle swarm optimization (PSO) method,the rate-dependent inverse Bouc–Wen model can be constructedas shown in Fig. 13. Motion tracking experiments can be per-formed. The first experiment is to investigate the performance ofthe inverse-Bouc–Wen model with the driving signal of mono-frequency triangle wave. The response curves, tracking rms errorand maximum error at different frequencies are measured andsummarized. The second experiment is to investigate the perfor-mance of the proposed control strategy under random complexinput signals.

In the first experiment, tracking 30�μm p–p mono-frequencytriangle signal, both the response performances with or withoutcontrol models are measured and summarized in Table 1 andFig. 14. As the Bouc–Wen model has just three more parametersand continuous mathematical expression, it is with great advan-tages in less time-consuming and continuous output when

Fig. 9. Inverse Bouc–Wen model and the rate-dependent compensator: (a) the compensating process; (b) the mechanism of the compensator via selecting inverse Bouc–Wen model parameters; (c) the mechanism of the inverse Bouc–Wen model compensator.

Fig. 10. Broken-line approximation of continuous signal.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–54 51

predicting the well-regulated and symmetric hysteresis loops. Inthis research, the hysteresis loops are almost symmetric and canbe fitted well with the Bouc–Wen model. So, the proposed Bouc–Wen model based method is appropriate for the inverse rate-dependent compensator. Table 1 and Fig. 14 show clearly that theproposed controller significantly reduces the tracking error due tothe hysteretic nonlinearity of the piezoelectric actuator. FromFig. 14, the rms error and the max of the experiment withoutcompensator are going up according to the driving frequencies asthe hysteresis problems getting severer and severer. The rms errorand the max error are going up to 10 μm and 12 μm respectively,when the frequency is going high to 60 Hz which is really a veryserious problem and the PZT is limited to be used in a dynamicenvironment. The performance of the proposed Inverse-Bouc–Wen model is shown clearly in Table 1 and Fig. 14. Although themax error is going up according to the frequency of the drivingsignal, they are less than 6 μm which is improved a lot comparedwith the system without compensator model. The rms error islimited in 5 μm throughout the frequency bands from 0 to 100 Hz,which are measured in the experiment, this good performancemainly benefits from the accurate modeling ability of the Bouc–Wen model. It can be seen that, even on the high frequency bandsuch as 100 Hz or above, the Bouc–Wen model can be measured,identified and predicted by the model perfectly. It is worth tomention that the rms errors are limited within 2 μm from 0 Hz upto 40 Hz which show a good performance at the lower frequencyband. The compensator performance at the high frequency bandstill sounds good, although the hysteresis is getting severer andseverer.

For an open loop system with inverse rate-dependent hyster-esis compensator as shown in Fig. 11 and without compensator,two mono-frequency tracking performances can be observed inFigs. 15 and 16. The max errors always occur when the value of thedriving signal reaches to the top or the bottom of the range, inother words, they always occur at the place when the

displacement reaches to zero or to full output. As shown inFig. 8, near the same place we can observe that the biggestprediction error occurs, so the maximum tracking error may comefrom the prediction error when the driving signal reaches the topor the bottom of the limited range scope. That is a drawback of theexperiment, but for most application cases, the working range ofthe PZT is mainly limited in the middle range, such as our futureapplication is mainly aimed for micro-active vibration isolationsystem, the useful working scope may mainly be limited in themiddle range of the whole working scope. It is obvious that whenthe working range is limited within the middle range, both therms errors and the max errors will be decreased significantly.

In the second experiment, a test signal combined with twoor three sinusoids with arbitrary selected different frequenciesis amplified and used to drive the open loop system withoutcompensator. It can be observed in Fig. 17 that the referencedisplacement signal is linearly scaled from input voltage and thescale-factor is 30 μm=100 V according to the user's manual of thePZT and the amplifier. The tracking error without compensator isvery intolerable. When the compensator is added to the open loopsystem as shown in Fig. 11, the reference, response and trackingerror curves are shown in Fig. 18. It demonstrates that theresponse has matched the reference signals well.

In the third experiment, the same test signal with the secondexperiment is used to drive the PZT system with inverse rate-dependent hysteresis compensator and H1 closed-loop controller.As shown in Fig. 19, the tracking error can be limited within 1 μm.It can be seen that the H1 controller associated with the inverseBouc–Wen model considerably reduces the tracking error of thePZT system in a high dynamic application. Moreover, a randomlycombined signal with very low frequencies of sinusoids is used totest the tracking ability of the proposed compensator basedclosed-loop H1 controller in the lower frequency band. As shownin Fig. 20, the tracking error can be limited within 0:05 μm underthe driving signal with about 1–5 Hz sinusoid. If much lowerfrequency such as quasi-static signal is used, better performancecan be achieved definitely.

7. Conclusion

Errors caused by the hysteresis issue of a PZT bring muchtrouble to its applications in high precision and dynamic controlfield, especially the hysteresis problems are rate-dependent whichwill further limit its performance in micro-dynamic trajectorytracking. We have presented a rate-dependent inverse Bouc–Wenmodel to compensate the trajectory error under dynamic situation.Based on the hysteresis loops of mono-frequency triangle drivingsignals, the parameters for inverse Bouc–Wen model on all kindsof slope rates are identified. The input signal is discretized and

Fig. 11. Open loop compensator model in dSPACE for the system.

Fig. 12. Measured step response and the step response of the identified model.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–5452

Fig. 15. Tracking performance with inverse Bouc–Wen model at about 40 Hz oftriangle signal.

Fig. 13. Closed-loop H1 controller for dSPACE in MATLAB Simulink environment.

Table 1Measured performance of the rate-dependent inverse feed-forward compensator.

Freq. (Hz) Without compensator model With inverse-Bouc–Wen model

rms err ðμmÞ max err ðμmÞ rms err ðμmÞ max err ðμmÞ

10 4.4810 5.7825 1.2084 1.442820 4.8418 7.0556 1.5451 2.640430 5.3738 8.4159 1.5285 3.338740 5.9573 9.3596 1.8950 4.063750 6.5253 10.0506 2.3403 4.107760 7.1025 10.9270 2.0906 5.006870 7.6944 11.2938 2.9550 5.088680 8.3190 11.8209 3.5554 5.548990 8.8429 12.0198 3.0254 5.4106

100 9.3656 11.8839 3.0701 5.8218

Fig. 14. Performance of the compensators about rms error and max error from10 Hz to 100 Hz.

Fig. 16. Tracking performance with inverse Bouc–Wen model at about 100 Hz oftriangle signal.

Fig. 17. Tracking performance without compensator for a random reference signal.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–54 53

fitted through line-segment. The proposed accumulation ofinverse Bouc–Wen model on line-segment method is proven byexperimental studies, which is easy to design and implement.Experimental results show that the proposed open loop rate-dependent inverse Bouc–Wen compensator model has enhancedthe performance of the PZT significantly in a large frequency band,which can be widely used in the field of parallel micro-robot field,such as active vibration isolation. In the experiment, only oneaspect of the influence of the hysteresis problems taken intoconsideration is the rate of variation, many other aspects such astemperature, moisture, creep, and drift can be included in themodel to further increase the accuracy. Furthermore, if feedbackclosed-loop controller is added on the whole cascade compensatorand the PZT system, the accuracy can be further increased. Then,

the proposed compensator can be further applied in all kinds ofsituations with rate-dependent hysteresis problems.

Acknowledgments

This work was supported by Macao Science and TechnologyDevelopment Fund (Grant no. 016/2008/A10), National Natural ScienceFoundation of China (Grant no. 61128008), Research Committee of theUniversity of Macau (Grant nos. MYRG203(Y1-L4)-FST11-LYM andMYRG183(Y1-L3)-FST11-LYM).

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Fig. 19. Closed-loop H1 tracking control performance with modified rate-depen-dent inverse Bouc–Wen model for a high dynamic random reference signal.

Fig. 20. Closed-loop H1 tracking control performance with modified rate-depen-dent inverse Bouc–Wen model for a lower frequency random reference signal.

Fig. 18. Tracking performance with modified rate-dependent inverse Bouc–Wenmodel for a random reference signal.

S. Xiao, Y. Li / Robotics and Computer-Integrated Manufacturing 30 (2014) 47–5454