Feedback/feedforward control of hysteresis-compensated piezoelectric actuators forhigh-speed scanning applications

Yanfang Liu1,4, Jinjun Shan1,2,5 and Ulrich Gabbert3,6

1Department of Aerospace Engineering, Harbin Institute of Technology, Harbin, Peopleʼs Republic ofChina2Department of Earth and Space Science and Engineering, York University, Toronto, Canada3 Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

E-mail: liu-yanfang@hotmail.com, jjshan@yorku.ca and ulrich.gabbert@ovgu.de

Received 4 July 2014, revised 12 October 2014Accepted for publication 4 November 2014Published 28 November 2014

AbstractThis paper presents the control system design for a piezoelectric actuator (PEA) for a high-speedtrajectory scanning application. First nonlinear hysteresis is compensated for by using theMaxwell resistive capacitor model. Then the linear dynamics of the hysteresis-compensatedpiezoelectric actuator are identified. A proportional plus integral (PI) controller is designed basedon the linear system, enhanced by feedforward hysteresis compensation. It is found that thefeedback controller does not always improve tracking accuracy. When the input frequencyexceeds a certain value, feedforward control only may result in better control performance.Experiments are conducted, and the results demonstrate the effectiveness of the proposed controlapproach.

Keywords: piezoelectric actuator (PEA), feedback control, feedforward control, hysteresis,linearization, dynamic effect, trajectory scanning

(Some figures may appear in colour only in the online journal)

1. Introduction

Due to many advantages such as high output force, highresolution, fast response, high stiffness, no backlash, and nofriction, piezoelectric actuators (PEAs) have been increas-ingly used in high-precision scanning applications, e.g.,scanning tunneling microscopy [1–3]. A common scanningtrajectory is the triangular waveform [4–7]. In recent years,demand for high-throughput systems such as imaging thetime-varying nature of the specimen under investigation, e.g.,movements of biologic cells or the formation of surface fea-tures during materials processing, have posed new challengesfor triangular trajectory scanning using PEAs [2, 3]. Thenonlinearities and dynamic behavior of the PEA-based

positioning system are the major factors that limit speed andaccuracy.

Hysteresis in PEAs often refers to rate-dependent vol-tage-displacement nonlinearity of a non-smooth and non-memoryless nature. Actual observed hysteresis shows ratedependence due to the coupled dynamic effect [8]. Hysteresiscauses amplitude-dependent frequency responses or ampli-tude-dependent gains. Hysteresis limits the performance ofPEAs, such as tracking precision and stability [2]. Driving aPEA by a charge amplifier can reduce hysteresis [2, 9–11];however, charge amplifiers are not widely adopted [12].Operating a PEA in a linear range, e.g., 10% of the maximumtravel range [13], and keeping the excitation signal as small aspossible can also minimize hysteresis effects, but this restrictsthe usefulness of a PEA. Thus, elimination of hysteresis hasattracted significant attention in the past.

Creep is a rate-dependent material nonlinearity in whichthe material continues to deform under a constant load. It is

Smart Materials and Structures

Smart Mater. Struct. 24 (2015) 015012 (9pp) doi:10.1088/0964-1726/24/1/015012

4 Visiting Ph.D. student at York University; currently lecturer at HarbinInstitute of Technology.5 Associate professor of space engineering.6 Professor.

0964-1726/15/015012+09$33.00 © 2015 IOP Publishing Ltd Printed in the UK1

the effect of the remanent polarization, which continues tochange over time even though the applied voltage has reacheda constant value. Creep is more prominent in low-frequencyapplications and affects absolute positioning in slow or staticapplications. It is negligible when a PEA is operated fastenough and over short duration and can also be removedusing feedback techniques. Several models have been pro-posed for modeling of creep, for example, fractional-orderdynamics [8, 14].

Dynamic effect is caused by amplifier and mechanicalfixtures. This behavior becomes more obvious when a PEA isoperated at high frequencies. It can be represented by a linearsystem of an appropriate order.

Many feedback techniques, such as integral control [2],proportional-integral-derivative (PID) [15, 16], robust linearcontrol [17], and sliding mode control [18–21], have beendeveloped to deal with hysteresis nonlinearity and improvethe accuracy and stability of nano-positioning systems.Feedback controllers usually employ high gains at low fre-quencies to overcome creep and hysteresis effects. However,feedback control alone is insufficient, especially at high fre-quencies, due to the reduced open-loop gain at high fre-quencies and the effects of nonlinearities.

Feedforward compensation strategy has also been widelyused to deal with hysteresis and dynamic behavior in PEAsvia generating a preshaped input by passing the desired tra-jectory through the inverse model. Preisach [13, 22], Prandt-Ishlinskii [12, 23], and Maxwell resistive capacitor (MRC)[24–26] models, as well as their extended versions that arecapable of representing the creep effect simultaneously[8, 27–29], have been used for hysteresis modeling andcompensation. To handle the high-frequency dynamic beha-vior and the hysteresis effect, a PEA is also modeled as acascade connection of a linear system and a hysteresis block[3, 11, 13], and then both effects are compensated for byusing the respective inverse models [13]. Feedforward com-pensation can be implemented without additional sensors andis quite suitable for applications where reference trajectoriesare known. However, it is challenging to develop a prioriaccurate model since model parameters can hardly be knownaccurately, can change over relatively long time intervals dueto aging effects, and are sensitive to temperature change.Disturbances, unmodeled dynamics, and noises also affectpositioning performance.

Thus, feedforward compensation is usually combinedwith feedback techniques to increase bandwidth, improveprecision at high frequencies, and deal with modeling errors,uncertainties, noises, and disturbances [2, 11]. Feedforwardand feedback can be integrated in different ways such as (a)feedback linearization first and then feedforward compensa-tion based on the inversion of the resultant closed-loop systemto improve high-frequency performance [13, 30]; (b) aug-menting the feedback controller with a feedforwardloop [16, 31].

The frequency response of the PEA has the followingfeatures: (a) the magnitude decreases with frequency slowly atlow frequencies and rapidly at high frequencies, and (b) themagnitude is amplitude-dependent. The slow decrease in

magnitude at low frequencies is caused by creep and can behandled by feedback techniques easily. The rapid decrease inmagnitude at high frequencies is caused by the dynamic effectof the nanopositioning system, mainly the bandwidth limita-tion of the amplifier and the low-damping vibrationaldynamics of the PEA positioning system. The amplitudedependence of magnitude is due to hysteresis. The open-loopgain is low if the input amplitude is low and vice versa. Thisfeature gives rise to difficulties in feedback controller design.A high-gain feedback controller may lead to instability whenthe input has large amplitude. Reducing the controller gainmay cause large tracking errors.

Considering the foregoing features, this paper presents anew approach, as shown in figure 1 for high-speed triangulartrajectory scanning using a PEA. First the hysteresis iscompensated for to remove the amplitude dependence ofmagnitude. Then a feedback controller is designed for thehysteresis-compensated PEA (HPEA) to deal with the com-pensation errors and model uncertainties. Since the hysteresishas been compensated for, simple feedback controllers can bedesigned. A feedforward controller is further designed basedon the frequency response of the HPEA to increase the systembandwidth. A Maxwell resistive capacitor (MRC) model isused to model and compensate for the hysteresis because it isquasi-static. The compensation results in little loss in the fastresponse of PEAs and requires little computational effort [8].This approach is different from the approaches presented in[13, 30]. First, in [13, 30], the linear vibrational dynamics of aPEA were obtained with hysteresis neglected by using asmall-amplitude input. Then the system was linearized usinghigh-gain feedback with a notch filter to increase the gainmargin. For the approach proposed in this paper, the PEA islinearized by compensating for the hysteresis by using theinverse MRC model. Then the remaining dynamics areidentified as a linear system. Second, in [13, 30], the feed-forward controller is designed based on the inverse dynamicsof the closed-loop system. In our approach, the feedforwardloop is designed based on the HPEA to augment the feedbackcontroller and enhance performance at high frequencies. Withthis approach, a precise linear result can be obtained with littleloss in bandwidth, which enables simple feedforward andfeedback controller design.

This paper is organized as follows. Section 2 describeshysteresis modeling and compensation. Section 3 presentsfeedback/feedforward controller design. In section 4, experi-mental results are included to demonstrate the effectiveness ofthe proposed approach. Section 5 concludes this paper.

2. System modeling and identification

As mentioned earlier, a PEA can be modeled as a cascadeconnection of a linear system and a hysteresis block[3, 11, 13]. As shown in figure 2, the PEA-based nano-positioning system consists of an amplifier, a piezoactuator,and a sensor. The amplifier and the sensor can be representedby linear systems Ga(s) and Gs(s), respectively. The piezo-actuator is modeled as a cascade connection of a linear system

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Smart Mater. Struct. 24 (2015) 015012 Y Liu et al

Gp(s) and a hysteresis block . Thus, the PEA-based nano-positioning system can be represented by a cascade connec-tion of the hysteresis block and a linear system G(s), where

=G s G s G s G s( ) ( ) ( ) ( )a p s .

2.1. Modeling, identification, and compensation of hysteresis

Hysteresis in PEAs can be modeled by a modified MRCmodel [8], which is described by

⎪

⎪

⎧⎨⎩

=<⩾

[ ][ ]

q t

T

x t u t q t u

u t q t u

˙ ( ) ˙ ( ) ( ) sgn ˙ ( )

0 ( ) sgn ˙ ( )(1)i i S i

i S i

,

,

κ=q t

Tu t

( )( ) (2)i

i i

∑==

+

u t u t( ) ( ) (3)i

n

i

1

1

where qi and ui are the charge stored in the voltage-limitedcapacitor SCi and the voltage applied on it, κ=C Ti i and uS i,

are the capacitance and saturated voltage of SCi, u and x arethe voltage applied on a PEA and its displacement output, andT is the effective displacement-to-charge electro mechanicaltransfer ratio. To avoid singularity in simulation, SCn+1 is apure capacitor or a voltage-limited capacitor with sufficientlylarge saturation voltage that it will never be saturated. Thus,

= =+q t q t Tx t( ) ( ) ( ) (4)n 1

The principle of this model is as follows. When theapplied voltage increases or decreases monotonically, thevoltage/displacement gain of this model is piecewise linear,which is used to fit the nonlinear input/output gain of thePEA. When the direction of the applied voltage switches, thevoltage-limited capacitorʼs saturation feature causes a sudden

switch in voltage/displacement gain. This is used to char-acterize the sudden switch of the input/output gain of the PEAat the endpoint of the hysteresis loop. A detailed physicalinterpretation of this model can be found in [8]. When avoltage is applied to the MRC model, it is distributed to theunsaturated voltage-limited capacitors inversely proportionalto their capacitance. Thus an iteration is needed to calculatethe distribution and consequently the charge of each voltage-limited capacitor. Then the displacement is obtained byequation (4).

The model parameters are identified through the fol-lowing two steps. First, a set of initial parameters is calculatedby fitting the initial rising curve with +n 1 linear elements[24, 25, 32]. Then the optimal parameters are obtained bysolving the following minimization problem:

∑ −κ κ… … =−

[ ]x t MRC u tmin ( ) [ ]( ) (5)u u

k

N

k k, , , , ,

1

2

n S S n1 ,1 , 1

where N is the number of samples and MRC refers to themodified MRC model operator. A triangular excitation withstepwise decreasing amplitude is usually employed to excitethe PEA. To minimize the coupled dynamic effects, a lowfundamental frequency should be selected.

Hysteresis is compensated for by connecting the inverseMRC model with the PEA in series. Since equations (1)–(3)describe a general bidirectional relationship between the inputvoltage and the output displacement, the inverse MRC modelcan be constructed directly: each elementʼs output voltage iscalculated directly when the desired displacement serves asinput, and the sum of each elementʼs voltage gives the totaloutput voltage. The inverse MRC model shares the sameparameters with the MRC model.

2.2. Modeling and identification of dynamic effects

In the literature, linear systems are usually identified directlyfrom a PEA with small-amplitude excitation [3, 11, 13]; seefigure 3(a). This ignores the hysteresis-induced amplitudedependence of magnitude. In this paper, a different approachis proposed as shown in figure 3(b). First the hysteresis iscompensated for. Then a linear system is identified from thehysteresis-compensated PEA (HPEA). The hysteresis effect isfully taken into consideration by measuring the frequencyresponse of the HPEA. The hysteresis compensation error istreated as model uncertainty.

Goldfarb and Celanovic used a second-order system torepresent the dynamic effect of the mechanical part [24]. Inaddition, considering the first-order dynamics of the sensorand the second-order dynamics of the amplifier, the following

Figure 1. Feedback/feedforward control strategy.

Figure 2. Illustration of modeling of a piezoelectric positioningsystem.

Figure 3. Illustration of the proposed identification procedure.

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Smart Mater. Struct. 24 (2015) 015012 Y Liu et al

fifth-order transfer function is used to identify the remainingdynamics of the HPEA from the measured frequencyresponse:

τ ζ ω ω ζ ω ω=

+ +

+ + + + +

( )( )( )

G s

k s b s b

s s s s s

( )

( 1) 2 2(6)

21 2

21 1 1

2 22 2 2

2

The multiplicative model uncertainty is calculated fromthe measured frequency response of the hysteresis-compen-sated PEA, ωG (i )m , as

Δ ωω ω

ω=

−G G

G(i )

(i ) (i )

(i )(7)m

m

Then a candidate uncertainty boundary W(s) can beobtained and used to check the stability of the designedfeedback controller.

3. Control system design

In this section, a feedback controller is designed for theidentified linear system and then augmented by an inverse-based feedforward compensation. The control performance isanalyzed under plant uncertainty.

3.1. Feedback controller design

To demonstrate that a simple controller can achieve goodtracking results, a proportional plus integral (PI) controller isdesigned:

= +K s kk

s( ) (8)P

I

The system G(s) is type 0; therefore, a proportional termwill cause a large error when the input signal is ramp. That isthe reason why an integral term is employed.

Using the frequency-response method, the proportionaland integral gains are chosen so that the open-loop system hassufficiently large crossover frequency. Then these parametersare optimized by minimizing the integral of the time-weightedabsolute error (ITAE):

∫= t e t t( ) d (9)t

0

f

The robust stability is guaranteed if

∥ ∥ <∞W s T s( ) ( ) 1 (10)

where = +T s( ) K s G s

K s G s

( ) ( )

1 ( ) ( )is the complementary sensitive

function [33]. Thus, the designed parameters will be checkedbased on whether the condition in equation (10) holds.

3.2. Feedforward compensation

Due to bandwidth limitation, at high frequencies, there is anobvious phase lag and the magnitude decreases quickly. Thisis caused mainly by the bandwidth limitation of the amplifierand the low-damping vibrational dynamics of the PEA

positioning system. Thus, the performance of tracking a tri-angular waveform with high fundamental frequency may notbe achieved. The active damping controller can be developedto improve the tracking bandwidth [2, 11, 34, 35]. In thispaper, the feedforward control is employed to improvetracking performance at high frequencies.

For a desired periodic triangular scanning trajectoryy t( ),d the Fourier series is

∑ ω==

∞

y t A t( ) sin ( ) (11)k

k kd1

where =π

πA sin ( )k k

k8

22 2and ω π= kf2k with f being the

fundamental frequency of the triangular waveform. Then thefollowing inverse-based feedforward input generates exact-output tracking.

∑ω

ω ω= − ∠=

∞

( )( )

( )u tA

Gt G( )

isin i (12)

k

k

kk kff

1

However, in practice, the plant G cannot be knownexactly due to the model errors. Even if ωG| (i ) |k and

ω∠G (i )k are measured directly from the frequency responseof the system, they cannot be obtained precisely due to noises,disturbances, and uncertainties. Meanwhile, the identifiedmodel is accurate for only a finite bandwidth. Therefore, thefeedforward input is approximated by the first m harmonics ofthe desired trajectory:

∑ω

ω ω≈ − ∠=

( )( )

( )u tA

Gt G( )

isin i (13)m

k

mk

kk kff,

1

Finally the feedforward input is added to the feedbackcontrol loop, as shown in figure 1. The feedforward generatesa rapid input and does not share the bandwidth with the closedloop. Thus, with the feedforward enhancement, the systemresponse becomes faster and the feedback controller reducesthe error caused by noise, uncertainty, and disturbance.

3.3. Tracking performance analysis

Using the feedforward compensation alone, the tracking erroris given by

ω ω ω Δ ω ω= − = −E Y Y Y(i ) (i ) (i ) (i ) (i ) (14)dff d ff

With the feedforward/feedback control, the tracking errorcan be obtained as

ω ω ω Δ ω ω ω= − = −E Y Y Y S(i ) (i ) (i ) (i ) (i ) (i ) (15)dfbff d fb

where the sensitivity function is

ωω ω

=+

SK G

(i ) :1

1 (i ) (i )(16)

Comparing equation (14) with (15), one can concludethat

ω ω<E E(i ) (i ) (17)fbff ff

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Smart Mater. Struct. 24 (2015) 015012 Y Liu et al

if

ω <S (i ) 1 (18)

That is to say, the feedback loop enhances the trackingperformance when the magnitude of the sensitive function issmaller than 1. Equation (18) is referred to as the feedbackacceptable condition.

4. Experimental study

4.1. System setup

A piezoelectric nano-positioning system, shown in figure 4,has been acquired by the Spacecraft Dynamics Control andNavigation Laboratory (SDCNLab) of York University forresearch on the dynamics and control of piezoelectric actua-tors. It consists of a piezoelectric nano-positioner P-753.1CDfrom Physik Instrumente, which is integrated with a capaci-tive sensor and driven by an E-625 piezo amplifier/controller.A host–target rapid prototyping system is employed withMATLAB Real-Time Workshop and xPC Target. The targetcomputer is equipped with a data acquisition and control cardPCI-6289 from National Instruments Corporation, having 3218-bit A/D channels and 4 16-bit D/A channels. The PCI-6289 acquires the output displacement of the PEA from thecapacitive sensor and converts the control command into ananalog signal, which is further amplified by the E-625 to drivethe PEA. Experiments are carried out with a sampling rate ofupto 100 kHz at room temperature.

4.2. Open-loop performance

The hysteresis loop shown in figure 5 is generated by excitingthe PEA with a triangular waveform with stepwise decreasingamplitudes. The ascending curves do not coincide with thedescending ones for a given input. The input nonlinearitydenotes the maximum difference as δu (shown in the figure),normalized by the the peak-to-peak value of the reference

input. Similarly, the maximum difference in outputs betweenthe ascending and descending curves (δy) is normalized andreferred to as the output nonlinearity. As shown in figure 5,the input and output nonlinearities are up to 11.84% and13.14%, respectively.

A frequency response of the PEA is also obtained forsignals with different amplitudes, as shown in figure 6. ThePEAʼs frequency behavior is complicated: (a) the magnitudedecreases with frequency slowly at low frequencies andrapidly at high frequencies, and (b) the magnitude is ampli-tude-dependent. The slow decrease in magnitude at low fre-quencies is caused by creep and can be handled by feedbacktechniques easily. The rapid decrease in magnitude at highfrequencies is caused by dynamic effects of the nanoposi-tioning system, mainly the bandwidth limitation of theamplifier and the low-damping vibrational dynamics of thePEA positioning system. These greatly restrict the scanningfrequency. The amplitude-dependent magnitude is due to thehysteresis. The open-loop gain is low if the input amplitude islow and vice versa.

Figure 4. Piezoelectric nano-positioning system at SDCNLab, YorkUniversity. Figure 5. Hysteresis loops of the uncompensated and compensated

systems.

Figure 6. Frequency response of the PEA.

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Smart Mater. Struct. 24 (2015) 015012 Y Liu et al

4.3. Hysteresis identification and compensation

A triangular excitation with stepwise decreasing amplitude isapplied to the PEA. The fundamental frequency is 1 Hz andthe sampling rate is 10 kHz. Following the procedure insection 2.1, the parameters are identified and given in table 1.

The inverse MRC model is built directly fromequations (1)–(3) with parameters in table 1. The hysteresisis compensated for by connecting the inverse model and thePEA in series. Figure 5 also shows the hysteresis loop of thecompensated PEA, as compared with the uncompensatedone. The input and output nonlinearities of the HPEA are0.4% and 0.38%, respectively. Compared with the uncom-pensated ones, they have been reduced by 96.96% and96.78%, respectively. The frequency response of the HPEAis illustrated in figure 7. Compared with the frequencyresponse of the PEA in figure 6, the amplitude-dependentfeature has been removed at low frequencies (up to 1 kHZ),and an excellent linear result has been obtained. As MRCmodel is quasi-static; the compensation causes little loss inbandwidth.

4.4. Identification of dynamic effect

The dynamic effects are identified from the frequencyresponse of the HPEA with the model given in equation (6).The parameters are given in table 2. Figure 7 shows the fre-quency responses of the identified model G(s). It can be seenthat the identified result gives a good fit with the experiment.

As mentioned earlier, the compensation error is treated asmultiplicative model uncertainty. It is measured for differentoperation conditions, i.e., DC offset from −1.5 V to 1.5 Vand amplitude from 0.5 V to 2.5 V, and the results areshown in figure 8. To design the feedback controller, thefollowing uncertainty bound candidate is employed:

=+ ×

+ × + ×

( )( )( )

W ss

s s( )

1.075 1.885 10

4.712 10 1.257 10(19)

3 2

3 4

which is also shown in figure 8.Using the method given in section 3.1, the parameters of

the feedback controller are designed and are listed in table 2.The robust stability of the designed controller is guaranteedsince ∥ ∥ <∞W s T s( ) ( ) 1. The simulated and experimentalfrequency responses of the closed-loop system are shown in

figure 9. The simulated results can predict the system per-formance perfectly, which demonstrates the effectiveness ofthe hysteresis compensation–based system linearization. Anexcellent tracking performance can be achieved at low fre-quencies. However, as mentioned earlier, due to the band-width limitation of the closed-loop system, there is an obviousphase lag and the magnitude decreases quickly at highfrequencies.

4.5. Feedback acceptable condition

Based on the identified model, the feedback conditionobtained from equation (18) approximates to 425 Hz. Toverify these results, sinusoidal signals with amplitude of μ0.5m and different frequencies are applied to the system.

Table 1. Identified model parameters.

i uS i, (V) κi ⎜ ⎟⎛⎝

⎞⎠

μm

Vi uS i, (V) κi ⎜ ⎟⎛

⎝⎞⎠

μm

V

1 0.0162 3.5859 7 0.0650 32.51522 0.0340 8.5447 8 0.0434 41.39303 0.0494 11.4930 9 0.0123 180.16524 0.0335 34.5231 10 0.1048 24.69525 0.0567 15.7278 11 — 1.28086 0.0630 22.6789

Figure 7. Frequency response of the compensated PEA.

Table 2. Parameters of identified transfer function and controllers.

ζ1 0.632 ζ2 0.113ω1 ×1.529 104 ω2 ×4.573 104

b1 − ×3.470 104 b2 ×1.193 109

τ × −1.5997 10 4 k ×2.531 1012

kPPI × −2.39 10 1 kI

PI ×2.63 103

Figure 8. Measured model uncertainty and its boundary.

6

Smart Mater. Struct. 24 (2015) 015012 Y Liu et al

Figure 10 shows the experimental feedback acceptable con-dition, where NRMSE represents the normalized root meansquare error. The feedback loop improves tracking precisionwhen the input frequency is lower than 450 Hz and causes thetracking errors to increase when the input frequency is higherthan 500 Hz. These results validate the correctness of theacceptable condition obtained in equation (18). However, dueto the uncertainties, noise, and disturbances in the real system,the acceptance condition is not a fixed number but varies fordifferent operation conditions. By applying signals with dif-ferent amplitudes, table 3 gives the frequencies of theexperimental feedback acceptable conditions, under which thefeedforward and feedback/feedforward control schemes havesimilar performance.

4.6. Tracking performance

The tracking results for a 300-Hz triangular waveform withfeedback control, feedforward control, and feedback/feed-forward control are shown in figure 11. In these experiments,the first 9 harmonics are used to calculate the feedforwardinput. As mentioned earlier, with feedback alone, the phaselag and gain reduction of a PI controller cause large trackingerrors. Faster response and smaller tracking errors areachieved with feedforward only or feedback/feedforwardcontrol schemes. The NRMSEs for both cases are given intable 4, and the maximum errors are μ0.22 m and μ0.12 m,respectively. Thus, the feedback controller improved thetracking performance. More results for different operationconditions are given in table 4. The tracking errors tend toincrease with frequency due to large uncertainty at high fre-quencies, and robust performance is obtained for differentamplitudes. However, the feedback loop does not alwaysimprove performance. At sufficiently large frequencies, i.e.,700 Hz and 800 Hz, the performance with the feedback/feedforward scheme is worse than that with feedforward only.Actually, these frequencies have exceeded the bandwidth ofthe closed-loop system.

4.7. Discussion

The effectiveness of the linearization of the nano-positioningsystem using hysteresis compensation enables the design of asimple linear feedback controller and an inverse-based feed-forward compensator. The feedforward alone improves theresponse speed, but it can hardly handle the remaininguncertainties and disturbances. At low frequencies, the feed-back alone can achieve a satisfactory performance. The

Figure 9. Frequency response of the closed-loop system.

Figure 10. Feedback acceptable condition.

Figure 11. Triangular waveform tracking performance.

Table 3. Feedback acceptable condition.

Amplitude (μm) Frequency (Hz)

0.5 450 ∼ 5001.0 850 ∼ 5501.5 650 ∼ 7502.0 6502.5 550

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Smart Mater. Struct. 24 (2015) 015012 Y Liu et al

combination of feedback and feedforward can achieve a betterperformance at high frequencies compared with those caseswith feedback or feedforward alone. However, the feedbackloop does not always reduce tracking errors. When the fre-quency of the reference signal is sufficiently high, the feed-back loop may cause the tracking performance to deteriorate.In this case, it is better to use feedforward only. Consideringthe remaining uncertainties in the system, there is not a fixedfrequency value beyond which feedforward only will result inbetter performance. Thus, experiments are required to deter-mine this frequency for a specific operation condition. At thesame time, truncation of input preshaping also introduceserrors. Increasing the harmonics number can reduce thetruncation error; however, it will introduce a new error due tothe model uncertainty at the frequencies of the added har-monics. Also, this error reduction will get smaller as theharmonics number increases; see equation (11). The newlyintroduced error may exceed the reduced error since themodel uncertainty increases with frequency; see figure 8. Theincrease in the harmonics number does not always reduce thetracking error [36]. Therefore, the harmonics number is alsorequired to be determined experimentally.

For scanning applications, the desired trajectory is aperiodic signal; then the proposed inverse-based feedforwardutilizes the Fourier transformation of the desired trajectory.However, for other applications, the desired trajectory may bearbitrary. Therefore, the feedforward can be replaced by otherinverse-based feedforward approaches [2, 7, 9, 11].

5. Conclusions

A control scheme is proposed for high-speed triangular tra-jectory tracking using piezoelectric actuators. A linearizedsystem is obtained through hysteresis compensation with aninverse Maxwell resistive capacitor model. This allows thedesign of simple linear feedback controllers and inverse-basedfeedforward compensation. Feedback and feedforward arecombined to improve tracking performance. The feedforwardcontroller increases response speed, whereas the feedbackcontroller improves tracking accuracy. It is also found thatfeedback control does not always improve tracking perfor-mance. It may cause the performance to deteriorate when thefrequency is sufficiently high. Thus, if the frequency exceeds

a certain value, feedforward only may result in better trackingperformance.

Acknowledgments

This research has been supported by a discovery grant fromthe Natural Sciences and Engineering Research Council ofCanada (NSERC) and by the National Natural ScienceFoundation of China (Project No. 11472090).

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Table 4. NRMSE of tracking triangular waveform (%).

Amplitude

Frequency

300 Hz 400 Hz 500 Hz 600 Hz 700 Hz 800 Hz

FF FBFF FF FBFF FF FBFF FF FBFF FF FBFF FF FBFF

0.5 μm 1.85 1.45 2.49 2.07 2.30 2.79 3.12 3.10 3.18 3.50 3.71 4.031.0 μm 1.99 1.23 2.39 1.81 2.65 2.18 2.81 2.45 2.99 2.94 3.32 3.441.5 μm 2.10 1.19 2.46 1.72 2.78 2.03 2.71 2.31 2.69 2.73 3.02 4.202.0 μm 2.10 1.22 2.51 1.70 2.71 2.07 2.62 2.47 3.05 4.26 4.39 9.732.5 μm 2.19 1.29 2.63 1.79 2.63 2.18 2.57 3.43 4.12 8.75 8.70 10.2

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