A novel similarity-based hysteresis empirical model for piezoceramic actuators

Accepted Manuscript

Title: A Novel Similarity-Base Hysteresis Empirical Modelfor Piezoceramic Actuators

Author: <ce:author id="aut0005" biographyid="vt0005">Zhi-Lin Lai<ce:author id="aut0010" biographyid="vt0010">Zhen Chen<ce:author id="aut0015" biographyid="vt0015">Xiang-Dong Liu<ce:author id="aut0020"biographyid="vt0020"> Qing-He Wu

PII: S0924-4247(13)00159-3DOI: http://dx.doi.org/doi:10.1016/j.sna.2013.04.002Reference: SNA 8289

To appear in: Sensors and Actuators A

Received date: 13-9-2012Revised date: 1-4-2013Accepted date: 1-4-2013

Please cite this article as: Zhi-Lin Lai, Zhen Chen, Xiang-Dong Liu, Qing-He Wu, ANovel Similarity-Base Hysteresis Empirical Model for Piezoceramic Actuators, Sensors& Actuators: A. Physical (2013), http://dx.doi.org/10.1016/j.sna.2013.04.002

This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.

http://dx.doi.org/doi:10.1016/j.sna.2013.04.002

http://dx.doi.org/10.1016/j.sna.2013.04.002

of 39

Accep

ted

Man

uscr

ipt

A Novel Similarity-Base Hysteresis Empirical Model for

Piezoceramic Actuators

Zhi-Lin Laia,b, Zhen Chen∗a,b, Xiang-Dong Liua,b, Qing-He Wua,b,

aSchool of Automation Beijing Institute of Technology, Beijing 100081, ChinabKey laboratory for Intelligent Control & Decision of Complex Systems Beijing Institute of

Technology

Abstract

This paper presents a novel dynamic hysteresis model for the Piezoceramic

actuators. The model is based on two new phenomena discovered in this paper,

the geometric similarity and the time scale similarity. In order to improve the

accuracy of the model, the method of virtual extremum is proposed to amend

the similarity. In the experiments, four kinds of input signals with different

frequencies are used to test the proposed model. And the outputs predicted

from the proposed model are compared with those obtained from the Preisach

model. The results show that the proposed model gives good accuracy and

performs better than the Preisach model. Especially, with the help of the time

scale similarityit is much easier to build a dynamic model.

Keywords: Piezoelectric actuator; Dynamic hysteresis model; Similarity

model

1. INTRODUCTION

Piezoceramic actuators(PZTs) with advantages of high precision and fast

response have been widely used in nano-positioning technology. Nevertheless,

the hysteresis behavior of PZTs, becomes a major difficulty of the high precision

positioning technology. Many methods have been studied to characterize the

hysteresis phenomenon of PZTs. Among them, the earlier studies of the hys-

teresis modeling focused on the description of the single-loop hysteresis curve,

Preprint submitted to Sensors & Actuators: A. Physical February 7, 2013

*Manuscript(includes changes marked in red font for revision documents)

http://ees.elsevier.com/sna/viewRCResults.aspx?pdf=1&docID=11346&rev=2&fileID=394724&msid=87DA378F-4173-46F5-B0A6-66C8C59A7DD0

of 39

Accep

ted

Man

uscr

ipt

for example, using polynomials to capture the major loops[1]. As research con-

tinues, many different models have been proposed to describe the hysteresis of

the piezoceramics. The most widely used model is the Preisach model[2−6], the

Maxwell resistive capacitor (MRC) model [7] and the Bouc-Wen model[8−9].

Besides, several approaches for modeling the hysteresis based on empirical

observations have been developed[10,11,12,13,14], most of which provide physical

insights into the hysteresis phenomenon. Giving several empirical observations,

Jung[10] built reference-models of the PZTs and got satisfactory results. Sun

Lining[11] proposed a turning voltage based mathematical model to describe

hysteresis, in which assumes that the reference curves have an approximately

linear relationship with other curves. By introducing the properties effects at

the turning points (called ”targeting the turning points”, ”curve alignment”,

and ”wiping out” ), Bashash[12] proposed a constitutive memory-based math-

ematical modeling framework. But a large number of memory units must be

used to record the key points of the hysteresis past trajectory that are required

for the prediction of its future response. Then in the Ref [13-14], Bashash gave

the improvement of the proposed model. However, these empirical models are

not rate-dependent and the memory-based hysteresis model still needed many

memory units.

Using the empirical modeling methods, a rate-dependent empirical model

without using many memory units has been proposed for the PZTs in this

paper. Two new phenomena, geometric similarity and time scale similarity, are

introduced to constitute the similarity of hysteresis. The geometric similarity

describes the similarity among the output displacement curves in static; and the

time scale similarity which discloses the relationships of the output displacement

curves with different frequencies is to make the model dynamic. Using the two

similarities, a rate-dependent empirical model can be built. However, large

errors may generate in predicting the outputs of the higher-order sub-loops.

The virtual extremum is proposed to amend the similarity of the displacement

2

of 39

Accep

ted

Man

uscr

ipt

curves to reduce the modeling errors. Finally, a rate-dependent empirical model

based on the amended similarity is built and verified in the experiments.

The structure of the article is arranged as follows: In section 2, the exper-

imental platform has been introduced; In section 3, the geometric similarity

and the time scale similarity is discovered to describe the similarity relation-

ship of hysteresis of the PZTs; Then, a hysteretic model modeling by the two

similarities is established and amended in section 4. Also in this section, the

wiping-out property and congruency property of the model has been verified;

In section 5, the model has been tested by different kinds of inputs. And the

proposed model is also compared with the Preisach model; Finally, Section 6

concludes the researches.

2. THE EXPERIMENTAL SYSTEM

The researches in this paper are carried out on a PZT experimental system.

Figure 1 shows the structure of the experimental system. The controller can

be seen in figure 2. Where, the cpu is DSP(TMS320LF2407); a 16-bit D/A

converter (AD669) and a 16-bit A/D converter(AD976) are used here. Figure

3 illustrates some of the equipment used in the experiment. The high voltage

power amplifier (HPV series) is used to drive the PZTs(MPT-1JRL/I002); the

output voltages of the power amplifier range from 0V ∼ 150V, and the resolution

of which is 5mV. The input voltages which the PZTs can withstand are -30V

∼ 120V. Considering the safety margin and the driving ability of the power

amplifier, the input voltages of the PZTs are limited in the range of 0V to

100V. A resistance strain gauge sensor has been installed within the platform as

a micrometer to get the displacement signals. The resolution of the resistance

strain gauge sensor is 0.01 µm. And the actual output displacements of the PZTs

are calibrated by a laser interferometer(Agilent 5529A Dynamic Calibrator)

whose resolution is 1nm. Figure 4 is the Laser interferometer.

3

of 39

Accep

ted

Man

uscr

ipt

Figure 1: The PZT control system Figure 2: The controller

Figure 3: The PZT experiment devices Figure 4: The laser interferometer

4

of 39

Accep

ted

Man

uscr

ipt

3. THE SIMILARITY OF THE HYSTERESIS NONLINEARITY

In the figure 5, the voltage signal is the input of the PZTs; and the dis-

placement curve is the output of the PZTs. The displacement curve consists of

ascending curve and descending curve. The horizontal axis of the chart which

is labeled by the sample number is time line. The input voltage and the output

displacement can form a hysteresis loop, as shown in figure 6. The maximum

and the minimum point (in figure 5) is the turning point in the hysteresis loop.

In Ref.[15] Madelung proposed that the hysteresis curve can only determine

by the turning point. And how the turning point (maximum/minimum point)

determines the hysteresis curve is what will be study here.

In this section, isosceles triangular input signals (input voltage sequences)

have been used to drive the PZTs. The input voltage sequences have the same

voltage difference (νV) between adjacent points(That’s to say the slope is fixed.).

Define reference curve Dr[n] is the displacement curve driven by the low fre-

quency triangular input with the maximum voltage of 100V and minimum volt-

age of 0V. The geometric similarity does not consider the frequency of the

triangle waves, and the sampling period of the displacement curve keeps the

same. When discussing the time scale similarity, the frequencies of the triangle

waves and the sampling periods are different, but the sampling times are the

same in one cycle of the triangle waves.

!

"#$%&'()* !

+)&,#-'./-0#&1.2'03/0-3/.%&#2'$'0,2456' 7'.2'03/0-3/.%&#2'$'0,2456' 8/0/$4$%)/0,8#9/$4$%)/0,

Figure 5: Some basic concepts

:;<=>?@AB@C>DEF?

G?=>?@AB@C >DEF?H IJKL

MDE@B@CNOB@PQORPSC?TQU

Figure 6: The hysteresis loop

5

of 39

Accep

ted

Man

uscr

ipt

3.1 The geometric similarity of the displacement curves

1 The relationship between the ascending displacement curves of the PZTs

First, consider the triangle input signals with the same minimum voltages

but different maximum voltages. Figure 7 shows three sets of triangle signals

with the same minimum voltage (0V), and the maximum voltage of the three

signals is 80V, 70V, 60 V, respectively. Figure 8 shows the displacement curves

of the PZTs. And the hysteresis loops can be seen in figure 9. We can see

from the displacement curves that the three ascending curves include by the

rectangular box have the same path. The errors of the three paths shown in

figure 10 are ranged from -0.02 µm to 0.02µm. That means the ascending curves

in the rectangular box have the same track.

0 100 200 300 400 500 600 700 8000

10

20

30

40

50

60

70

80

Sample No.

Vo

lta

ge

(V)

V1

V2

V3

Figure 7: Voltages signals

0 100 200 300 400 500 600 700 800−2

0

2

4

6

8

10

Sample No.

Dis

pla

ce

me

nt(

µm

)

D1

D2

D3

Figure 8: Displacement curves

Then, consider the triangle inputs with the same maximum voltages but

different minimum voltages. Figure 11 shows the driving voltage signals. The

maximum voltage of triangle input signals is 100V, while the minimum voltages

range from 0V to 95V. Figure 12 and figure 13 are the displacement curves and

hysteresis loops, respectively. In Ref.[11], the relationship between the ascending

lines in hysteresis loops is regarded as linear(the line OA,OB,..., in figure 13).

Here, we are interested in the ascending displacement curves in figure 12. Use

D0[n] to describe the ascending displacement curve of the reference curve (the

6

of 39

Accep

ted

Man

uscr

ipt

0 10 20 30 40 50 60 70 80−2

0

2

4

6

8

10

Voltage(V)

Dis

pla

se

me

nt(

µm

)

L1

L2

L3

Figure 9: Hysteresis loops

0 50 100 150 200 250−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Sample No.

Dis

pla

ce

me

nt(

µm

)

Error1Error2Error3

Figure 10: Ascending errors

red curve OdO′

d in the figure 12); and use DVmin[n] to describe the ascending

displacement curve driven by the triangle signal with the minimum voltage of

VminV. We can see in figure 12, when Vmin = 50V, the displacement curve

DVmin[n] is the curve JdJ

′

d (in figure 12), and the driving voltage is JvJ′

v in

figure 11. Define a displacement sub curve D0Vmin

[n] (the curve OdJ) which is a

portion of the D0[n] to correspond the curve JdJ′

d. The length of the curve OdJ

is determined by the driving voltage: In figure 11, VJ is the voltage of the point

J, and VJ − VOv= VJ′

v− VJv

. Actually, what we study here is the relationship

between JdJ′

d and OdJ .

Suppose the curve JdJ′

d similar to OdJ , so the two curves meet the Eq.(1).

DVmin[n] = λa ×D0

Vmin[n] + ωa (1)

where, λa ∈ R is the slope of the similarity, ωa ∈ R is the intercept of the

similarity. Because the similarity between the output displacement curves is

relative to the extreme value of the input voltage, the slopes and the intercepts

vary with the extreme value. Here, define ∆Va = Vmin − 0 (the difference of

the minimum voltages between VminV and 0V) to describe the variation of the

extreme. And the driving voltage of JdJ′

d is JvJ′

v (in figure 11), which can be

expressed by V amin[n] = Vmin + n× ν.

7

of 39

Accep

ted

Man

uscr

ipt

Different PZTs have different characteristics of the hysteresis, so the identifi-

cation of the slopes and the intercepts should be completed through experiments.

According to least-square criterion, we can get the values of λa and ωa corre-

sponding to each extremum voltage. Then, higher order polynomials are used

to fit the variations of parameters (λa and ωa), whose variations follow ∆Va.

The relationship between ∆Va and λa can be seen in figure 14; and the

relationship between ∆Va and ωa can be seen in figure 15. The two curves can

be fitted by the polynomial equations (2) and (3), respectively. The coefficients

of the polynomials can be seen in table 1.

λa(∆Va) =

6∑

i=0

ai × (∆Va)6−i (2)

ωa(∆Va) =

3∑

i=0

bi × (∆Va)3−i (3)

Table 1: The coefficients of the polynomial λa(∆Va) and ωa(∆Va)

Polynomial Coefficient Value

λa(∆Va)

a0 5.669× 10−12

a1 -1.356× 10−9

a2 1.198× 10−7

a3 -4.809× 10−6

a4 0.0001

a5 -0.0047

a6 1.0008

ωa(∆Va)

b0 -2.333× 10−7

b1 -0.0004

b2 0.1573

b3 -0.0094

The above analysis shows: when giving an ascending input, the geometric

8

of 39

Accep

ted

Man

uscr

ipt

0 200 400 600 800 1000

0

20

40

60

80

100

Sample No.

Vol

tage

(V)

Ov

O’v

Av

A’vJ’v

JvJVj

Vov

Vjv

Vj’v

...

...∆Va

...

Figure 11: 20 sets ascending voltage signals

0 200 400 600 800 10000

2

4

6

8

10

Sample No.

Dis

plas

emen

t(µm

)

O’dJ’d A’d

OdAd

Jd

J

...

...

...

Figure 12: 20 sets ascending displacement curves

0 20 40 60 80 1000

2

4

6

8

10

Voltage(V)

Dis

plac

emen

t(µm

)

A

BC

O

...

...

Figure 13: 20 sets hysteresis loops

9

of 39

Accep

ted

Man

uscr

ipt

0 20 40 60 80 1000.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

∆ Va(V)

λ a

Actual points The fitting curve

Figure 14: Relationship between λa and ∆Va

0 20 40 60 80 100−2

0

2

4

6

8

10

12

∆ Va(V)

ωa

Actual pointsThe fitting curve

Figure 15: Relationship between ωa and ∆Va

similarity between DVmin[n] and D0

Vmin[n] can be described by Eq.(4).

DVmin[n] = λa(∆Va)×D0

Vmin[n] + ωa(∆Va) (4)

2 The relationship between the descending displacement curves of the PZTs

First, giving 20 sets of the driving voltage signal as shown in figure 16, which

have the same minimum voltages of 0V and the different maximum voltages

range from 100V to 5V. Figure 17 is the displacement curves driven by the

signals. Just like the ascending curves we study above, We use D0[n] to describe

the ascending displacement curve of the reference curve (the red curve OO′ in

the figure 17); and use DVmax[n] to describe the ascending displacement curve

driven by the triangle signal with the maximum voltage of VmaxV. We also

define a displacement sub curve D0Vmax

[n] (the curve OA in figure 17) which is a

portion of the D0[n] to correspond the curve BB′ in figure 17. Here, the point

A (in figure 17) is determined by the driving voltage: VO − VA = VB − V ′

B(in

figure 16). So, what we study here is the relationship between OA and BB′.

And suppose the displacement curve OA similar to BB′, as shown in figure 17.

So the two curves meet the Eq.(5).

DVmax[n] = λd ×D0

Vmax[n] + ωd (5)

where, λd ∈ R is the slope of the similarity, ωd ∈ R is the intercept of the

10

of 39

Accep

ted

Man

uscr

ipt

similarity, and they are both related to the ∆Vd = 100 − Vmax(the difference

of the maximum voltages between 100V and VmaxV). And the driving voltage

of displacement curve BB′(in figure 17) is the voltage curveBB′(in figure 16),

which can be expressed by V dmax[n] = Vmax − n× ν.

0 100 200 300 400 500 600 700 800 900

0

20

40

60

80

100

Sample No.

Vo

lta

ge

(V)

BA

OVo

Vb

Va

Vb’ B’ O’

∆Vd

Figure 16: 20 sets descending voltage signals

0 200 400 600 800

0

2

4

6

8

10

Sampling No.

Dis

pla

ce

me

nt(

µ m

)

A

O

O’

B

B’

Figure 17: 20 sets descending displacement curves

According to least-square criterion, we can get the values of λd and ωd

corresponding to each extremum voltage. The relationship between λd and

∆Vd can be seen in Figure 18; and the relationship between ωd and ∆Vd can be

11

of 39

Accep

ted

Man

uscr

ipt

seen in Figure 19. The two curves can be fitted by the polynomial equations (6)

and (7), respectively. The coefficients of the polynomials can be seen in table 2.

λd(∆Vd) =

7∑

i=0

ci × (∆Vd)7−i (6)

ωd(∆Vd) =

6∑

i=0

di × (∆Vd)6−i (7)

0 20 40 60 80 1001

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

∆Vd(V)

λd


Figure 18: Relationship between λd and ∆Vd

0 20 40 60 80 100−18

−16

−14

−12

−10

−8

−6

−4

−2

0

∆ Vd(V)

ωd


Figure 19: Relationship between ωd and ∆Vd

The above analysis shows: when giving an descending input, the geometric

similarity between D0Vmax

[n] and DVmax[n] can be described by Eq.(8).

DVmax[n] = λd(∆Vd)×D0

Vmax[n] + ωd(∆Vd) (8)

3.2 The time-scale similarity of the displacement curves

The hysteresis of the PZTs is rate-dependent. In order to fully describe the

hysteresis of PZTs, the frequency must be considered.

First, two sets of triangular signals whose only difference is the frequency

will be used here, as shown in figure 20. The different frequencies are labeled

as f0 and f (f0 < f), and the cycle are T0 and T . The displacement curves

are shown in figure 21. And use the DT0[n] and DT [n] to describe the two

displacement curves. In order to compare two displacement curves, the time

scale amplification needs to be carried out. If the cycle of DT0[n] is amplified

12

of 39

Accep

ted

Man

uscr

ipt

Table 2: The coefficients of the polynomial λd(∆Vd) and ωd(∆Vd)


λd(∆Vd)

c0 5.233× 10−13

c1 -1.535×10−10

c2 1.773× 10−8

c3 -1.015× 10−6

c4 2.957× 10−5

c5 -0.0004

c6 0.00626

c7 0.999

ωd(∆Vd)

d0 -2.398×10−10

d1 5.903× 10−8

d2 -5.439× 10−6

d3 0.0002

d4 -0.0047

d5 -0.125

d6 -0.0245

13

of 39

Accep

ted

Man

uscr

ipt

VWXYZ[\]^_`abc ad^_`abc ad efg

Figure 20: Triangular voltage signals with different frequency

hijklmnopqrst ruopqrst ru vwx

Figure 21: Triangular displacement outputs with different frequency

14

of 39

Accep

ted

Man

uscr

ipt

NT (NT < 1) times to make NT × 1/f0 = 1/f , then the curves after time scale

(labeled DNT×T0[n]) has the same cycle with DT [n]. For example, three sets

of displacement curves with different frequencies are shown in figure 22. The

cycle of the displacement curve in red is 0.01s; that in black is 1s; and in blue

is 100s. Then the time scale amplification is carried out: to make the cycle

of the displacement curve with the frequency of 0.01Hz magnify 1/1000 times,

the cycle of the displacement curve with the frequency of 1Hz magnify 1/100

times, then they can be drawn in the same figure with the displacement curve

of 100Hz, as the figure 23 shown (Also, the time is labeled by the sampling

number.). From figure 23 we can see that the three curves are similar. So, we

suppose DNT×T0[n] similar to DT [n]. And the similarity can be expressed by

the following Equation:

DT [n] = λf ×DNT×T0[n+m] + ωf (9)

where, λf ∈ R is the magnification; ωf ∈ R is to move the curves in vertical

axis direction. m is the phase difference of two sets of curves, which is to move

the curves in horizontal axis direction. In order to obtain these coefficients, the

genetic algorithm is used. The number of species is 100. the genetic generation

is 4000. And the search interval of λf is [1 1.5]; The search interval of ωf is

[-1 1]; The search interval of m is [-10 10], and m is an integer. The indicator

function is the fitting errors. The results are shown in table 3.

Figure 24 and 25 shows the λf and ωf changes with lg(f/f0). The two

curves can be fitted by the polynomial equations (10) and (11), respectively.

The coefficients of the polynomials can be seen in table 4.

λf (lg(f/f0)) =3

∑

i=0

xi × (lg(f/f0))3−i (10)

ωf (lg(f/f0)) =3

∑

i=0

yi × (lg(f/f0))3−i (11)

15

of 39

Accep

ted

Man

uscr

iptyz |~ ~ |~ ~ |~ ~

z

Figure 22: Curves before scale amplification

0 200 400 600 800 1000 1200 1400 1600 18000

2

4

6

8

10

Sample No.

Disp

lace

men

t(¦Ìm

)

1180 1200 1220 1240

8.68.8

99.2 Curve of 100Hz

Curve of 1HzCurve of 0.01Hz

Figure 23: Curves after scale amplification

100

101

102

103

104

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Frequency(Hz).

λ f


Figure 24: Relationship between λf and

lg(f/f0)

100

101

102

103

104

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency(Hz).

ωf


Figure 25: Relationship between ωf and

lg(f/f0)

16

of 39

Accep

ted

Man

uscr

iptTable 3: The coefficients λf , ωf and m

f/f0 λf ωf m

2 0.9905 0.0041 0

3 0.9889 0.0268 0

4 0.9866 0.0305 0

5 0.9852 0.0339 0

6 0.9836 0.0391 0

7 0.9820 0.0477 0

8 0.9806 0.0558 0

9 0.9800 0.0569 0

10 0.9786 0.0655 0

50 0.9665 0.1087 1

100 0.9589 0.1850 1

1000 0.9338 0.2945 6

10000 0.9252 0.3283 9

Table 4: The coefficients of the polynomial λf (lg(f/f0)) and ωf (lg(f/f0))


λf (lg(f/f0))

x0 0.000936

x1 -0.00468

x2 -0.014635

x3 0.99748

ωf (lg(f/f0))

x0 -0.011632

x1 0.061259

x2 0.02523

x3 -0.005604

17

of 39

Accep

ted

Man

uscr

ipt

The time scale similarly has been defined to the displacement curves which

driven by the different frequencies. If the cycle of the displacement curves with

the frequency of f0 is amplified NT times to make DNT ∗T0[n] and DT [n] have

the same cycle, then, the two curves meet the following equation:

DT [n] = λf (lg(f/f0))×DNT×T0[n+m] + ωf (lg(f/f0)) (12)

4. A NOVEL SIMILARITY-BASED MODEL

4.1 Hysteresis model based on the similarity

After giving the geometric similarity and the time scale similarity, a novel

similarity-based model will be built in this section. The new model need the

reference curve Dr[n], which is driven by a triangle inputs with the frequency

of f0, the maximum voltage of Vmax (that’s 100V in this paper.), the minimum

voltage of Vmin (that’s 0V in this paper.). Then, a displacement curve Dc[n]

can be get through the similarities introduced above. Here, Dc[n] is the any

curve with the frequency of f > f0, the maximum voltage of V ′

min > Vmin, the

minimum voltage of V ′

max < Vmax. The ascending and descending curve of Dr[n]

is respectively Dra[n] and Dr

d[n]. In the same way, the ascending and descending

curve of Dc[n] is respectively Dca[n] and Dc

d[n].

The schematic diagram of the modeling is shown in figure 26. Suppose the

frequency of the input signal is f ; the minimum voltage is V ′

min; the maximum

voltage is V ′

max.

Step 1: using the time scale similarly Eq.12, transfer the reference curve into

the curve with the frequency of f :

DrT [n] = λf (lg(f/f0))×Dr

NT×T0[n+m] + ωf (lg(f/f0)) (13)

where, by looking up table, the value of m can be get. Here, we use DrT [n] to

describe the new reference curve of the frequency of f . So, the D0T V ′min[n] to

correspond the D0Vmin[n] , and D0

T V ′max[n] to correspond the D0V max[n].

18

of 39

Accep

ted

Man

uscr

ipt

Figure 26: Schematic diagram of modeling

Step 2: When meets the minimum/maximum input voltage, using the geo-

metric similarity to obtain the ascending/descending displacement curves driv-

ing by triangular voltage input:

Dca[n] = λa(V

′

min − Vmin)×D0T V ′min[n] + ωa(V

′

min − Vmin) ascending

Dcd[n] = λd(Vmax − V ′

max)×D0T V ′max[n] + ωd(Vmax − V ′

max) descending

(14)

Step 3: Using the displacement curve got in step 2 and their triangular

driving voltage signals, a hysteresis loop can be get, as shown in figure 27.

NOTE: The triangular driving voltage signals are the isosceles triangle sig-

nals, and whose maximum value and minimum value is determined by the input

signal. And the slope of the isosceles triangle is fixed. The triangular voltage

19

of 39

Accep

ted

Man

uscr

ipt

can be expressed by Eq.(15):

V Ta [n] = V ′

min + n× ν ascending

V Td [n] = V ′

max − n× ν descending(15)

where, the ν is the voltage difference between adjacent points in triangular

voltage sequences.

¡¢¢£¤ ¥¤ ¦¤ §¤ ¨¤ ©¤ ª¤ «¤ ¬¤¦ª¬

¥¤¢¡®¯°±

²³³ ´³³ µ³³ ¶³³ ·³³³³²³´³µ³¶³ ¸ ¹¢¡®¯ º®»£¡ ¹¯ ¼¢½¸®¾¢¡®¯

²³³ ´³³ µ³³ ¶³³ ·³³³³·²¿ÀµÁ¶Â

º®»£¡ ¹¯ ¼¢½

Ã ®¹¯Ä¡®Å¢¡®¯ ¯¹®¡Ã ®¹¯Ä¡®Æ £¡®Ç»¹ÇÄÅ

Figure 27: The establishment of intermediate hysteresis loop

Step 4: Using the hysteresis loop consists of Eq.(14)and Eq.(15) to get the

output displacements.

4.2 The amending of the similarity model

In actually, the geometric similarity is just the total describe of the rela-

tionship of the hysteresis loops between the reference loop (Corresponding to

the reference curve.) and the internal loops having the turning points(like the

points A, B, C, D in figure 28.) on it , which is called first-order segment. For

the other internal hysteresis loops whose turning points are not in the reference

20

of 39

Accep

ted

Man

uscr

ipt

ÈÉÊËÌËÍËÎÏËÐÑÏËÎÒÓÎÔÏÕÍÖË

ÊËÌËÍËÎÏË ÒËÑÏËÎÒÓÎÔÏÕÍÖË×Ø ÙÚÛÜ

ÝÞ ßà áâãäåÐÔËæâç

Figure 28: Relationship of turning points and the hysteresis loop

loop (Like the points E and F. ), the similarity Eq.(14) may bring some errors.

Therefore, the proposed displacement curve similarity should be amended as

follow:

Dma [n] = La(N1)× Dc

a[n] + Ca(N1) ascending

Dmd [n] = Ld(N2)× Dc

d[n] + Cd(N2) descending(16)

where, Dm[n] is to describe the displacement curves being modified. La(N1)

and Ld(N2) is the correction factor of the slope; and Ca(N1) and Cd(N2) is

the correction factor of the displacement migration. All of the four factors are

relative to the max/min voltage (the turning points). N1 is the latest sam-

pling number of the minimum voltage; N2 is the latest sampling number of the

maximum voltage.

1. La(N1) and Ld(N2)

The hysteresis loop has a phenomenon: the ascending curves will converge

to one point; and the descending curves will also converge to the other point

[11]. As shown in figure 29, four different lines (La, Lb, Lc, Ld) are excited by

different input signals which are all have a maximum voltage of 100V. We can

see from the displacement curves that all the curves have the same maximum

displacement. The minimum voltages (0V) also have the same property. With

21

of 39

Accep

ted

Man

uscr

ipt

the help of those properties, a method can be proposed to get the La(N1) and

Ld(N2).

Here a concept of virtual extrimum (include the virtual upper limit and the

virtual lower limit. ) has been introduced. Suppose that all the ascending

voltage signals will rise to the virtual upper limit; and all the descending volt-

age signals will down to 0 V (virtual lower limit). According the properties

mentioned above, all the virtual maximum displacements and virtual minimum

displacements converge to one point, respectively. As shown in figure 30, the

virtual upper limit predicted by the model is the point B (The displacement

is r′), however, it should be A (The displacement is r). So we use La(N1) to

modify the ascending curve O1B to the curve O1A. Support the displacement

of point O1 is ro, then we can get:

La(N1) = (r − ro)/(r′− ro) = (r −Dm

d [N1 − 1])/(r′ −Dmd [N1 − 1]) (17)

And in the descending process, the same conclusion can be get. In the

figure 30, the virtual lower limit predicted by the model is the point D (The

displacement is d′), however, it should be C (The displacement is d ). So we

use Ld(N2) to modify the descending curve O2D to the curve O2C. Support

the displacement of point O2 is d0, then we can get:

Ld(N2) = (d− do)/(d′− do) = (d−Dm

a [N2 − 1])/(d′ −Dma [N2 − 1]) (18)

2. Ca(N1) and Cd(N2)

As we know, if the input signals are continuous, then the output displace-

ments are also the continuous. So, the displacements in turning points should

be continuous. Then, we can get:

Ca(N1) = Dma [N1 − 1]− La(N1)×Dc

a[N1]

Cd(N2) = Dmd [N2 − 1]− Ld[N2]×Dc

d[N2](19)

4.3 Wiping-Out Property and Congruency property verification of the simi-

larity model

22

of 39

Accep

ted

Man

uscr

ipt

0 200 400 600 800 1000 1200 1400 1600 1800 2000−2

0

2

4

6

8

10

12

Sample No.

Dis

pla

se

me

nt(

¦Ìm

)

LaLbLcLd

100V

Figure 29: Inputs with the same maximum

voltage

0 200 400 600 800 1000 1200 1400 1600 1800 2000−2

0

2

4

6

8

10

12

Sample No.

Dis

pla

se

me

nt(

¦Ìm

)

100VB

CD

0V

A

O2

O1

N1 N2

Figure 30: Sketch of amendment

Wiping-Out Property and Congruency property is two important properties

of the Hysteresis mentioned by Mayergoyz [3].

Wiping-Out property means: the input maximum will wipe out the history

maximums which are smaller; and the input minimum will wipe out the history

minimums which are bigger. The wiped out historical maximums or minimums

will exert no influence to the new outputs anymore.

Using the input signals S1 and S2, as shown in figure 31(a) (after the point

H , the input signals are the same.), we can get the model output curves S1 and

S2 in figure 31(b). We can see from the figure 31(b), curve S1 and S2 have the

same displacement in point C, that’s to say: the extremum value A does not

exert influence on C (extremum value A is wiped out); and after the point H ,

S1 and S2 have the same path, that’s to say: after the point H , those points

A,B,D,E, F,G are all wiped out. In the figure 31(b) also has curve S2′, which

is the output of the model does not amend. We can see from the figure, S2′ and

S1 does not overlap after point H . That’s to say, the model without amending

dose’t have the wiping out property.

Congruency property: if the inputs have the same two consecutive extremum

values, then the two minor (internal) hysteresis loops are congruent. As shown in

23

of 39

Accep

ted

Man

uscr

ipt

0 500 1000 1500 20000

20

40

60

80

100

120

Sample No.

Volta

ge(V

)

S1S2

A

B

C

D

E

F

G

H

I

(a) Input signals

200 400 600 800 1000 1200 1400 1600 1800 2000

0

2

4

6

8

10

12

Sample No.

Disp

lase

men

t(µm

)

1800 180510.52510.53

10.535

S1S2S2’

1500 16002

2.5

FB

D

E

G

IA

C

H

(b) Displacement curves

Figure 31: Wiping-Out Property

24

of 39

Accep

ted

Man

uscr

ipt

figure 32(a), the extremum values of the model inputs are (20, 80, 20, 100, 20, 80, 20V ).

There are two consecutive extremum values (20, 80, 20V )(labeled 1, and 2 in fig-

ure 32(a).), which have the two minor (internal) hysteresis loops are congruent

(labeled 1,and 2 in figure 32(b) ).

0 500 1000 1500 20000

20

40

60

80

100

Volta

ge(V

)

Sample No.

1 2

(a) Input signals

20 40 60 80 1002

4

6

8

10

12

Voltage(V)

Disp

lacem

ent(u

m)

78 808

8.2

8.4

12

(b) Hysteresis loops

Figure 32: Congruency Property

4.4 The memory space and the computation complexity of the model

Figure 33 is an implementation of the modeling. From the Figure, we can

see that three important steps should do in the implementation.

1. Update the parameters of the time scale similarly when the frequency of

the input signal is changed.

2. Update the parameters of the geometric similarity when meets the mini-

mum/maximum input voltage.

25

of 39

Accep

ted

Man

uscr

ipt

Get, , ,a a a aL C !

Update, ,f f m !

Ascending

hysteresis

loop

Get, , ,d d d dL C !

Descending

hysteresis

loop

Output

Figure 33: An implementation of the modeling

3. Using the reference hysteresis loop (Created by the reference curve and

its driving voltage.) to find the output displacement corresponded to the input

voltage. And calculate the output of the model with the help of the parameters

get in step 1 and 2.

And now, we analyze the implementation in two aspects: memory space and

the computation complexity.

1. Memory space:

Because the parameters in step 1 are got from the Eq.10, Eq.11; the param-

eters in step 2 are got from the Eq.2, Eq.3, Eq.6, Eq.7, Eq.17, Eq.18 and Eq.19.

And in the step 3, the reference hysteresis loop is stored by two polynomials.

Therefore, a low number memory spaces are enough in this implementation.

2. Computation complexity:

Computational complexity is a measure of the resource needed to perform a

computation. Often, it used to reflect the run-time of an algorithm. It can be

seen clearly in the Figure 33: time is consumed in the three steps mentioned in

this section. The step 1 will be run when the frequency changed; and the step 2

will be run when encounters extremums. So, step 1 and step 2 is executed in a

26

of 39

Accep

ted

Man

uscr

ipt

low frequency. As a result, the run-time of the implementation is determined by

step 3 which will be executed in every cycle. In step 3, a polynomial about the

reference hysteresis loop is executed. And the parameters get in step 1 and step

2 are involved in the calculation according the Eq.14 and Eq.16. Here, we can

find that the computational complexity of the implementation is determined by

the polynomials of the hysteresis loop.

In figure 34, two polynomials are used to fitting the hysteresis loop (Divided

into the ascending part and the descending part). The order of the two polyno-

mials is 3. And the polynomials shown in Eq.20, where La and Ld is the output

displacement of the hysteresis loop, V ol is the input voltage. And the coeffi-

cients of the two polynomials are in table 5. So the computational complexity

of the implementation is O(3).

La(V ol) =3∑

i=0

Lai × (V ol)3−i ascending

Ld(V ol) =3∑

i=0

Ldi × (V ol)3−i descending

(20)

−20 0 20 40 60 80 1000

2

4

6

8

10

12

Voltage(V)

Dis

plac

emen

t(µm

)

48.5 49 49.56.92

6.94

6.96

6.98

7

7.02

Ascending hysteresis curve

Descending hysteresis curve

Figure 34: Fitting the hysteresis loop

27

of 39

Accep

ted

Man

uscr

ipt

Table 5: The polynomial coefficients of hysteresis loop


La(V ol))

La0 -4.4656×10−6

La1 0.0008

La2 0.0825

La3 0.1806

Ld(V ol))

Lb0 -1.1201×10−6

Ld1 -0.00025

Ld2 0.1496

Ld3 0.3613

5. EXPERIMENT

Although the similarities are concluded using the triangular input voltage,

the model also can be used when giving other input signals. In this section, 4

kinds of input signals are used to verify the model we proposed. The reference

curve with a frequency of 0.01Hz is used to establish the model. And a Preisach

model has been built to compare with the new model. Here, the Preisach model

is built according to the Eq.(21) as follows [2]:

f(t) =n−1∑

k=1

[F (αk, βk−1)− F (αk, βk)] + F (u(t), βk) ascending

f(t) =n−1∑

k=1

[F (αk, βk−1)− F (αk, βk)] + [F (αn, βn−1)− F (αn, u(t))] descending

(21)

where f(t) is the output displacement of the system and u(t) is the input voltage.

F (α, β) = fα− fαβ. ( fα presents the output of the first-order hysteresis curves

when u(t) = α . fαβ presents the output of the first-order hysteresis curves

when the input u(t) decline down to β from α .)

1 The triangular signals

The maximum and the minimum voltage of the input triangular are respec-

28

of 39

Accep

ted

Man

uscr

ipt

tively 80V and 20V. The results of this experiment with the frequency of 0.1Hz

are shown in figure 35(a-b). The results of this experiment with the frequency

of 100Hz are shown in figure 35(c-d). The specific data can be seen in table 6

and 7.

0 5 10 15 20 25 300

2

4

6

8

10

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of 0.1HzActual output of 0.1HzPreisach model output

(a) Displacement curves of low fre-

quency

0 5 10 15 20 25 30 34−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of 0.1HzPreisach error of 0.1Hz

(b) Error curve of low frequency

0.005 0.010 0.015 0.020 0.025 0.030 0.034

2

3

4

5

6

7

8

9

10

11

Time(s)

Dis

pla

se

me

nt(

µm

)

New model out put of 100HzActual output of 100HzPreisach model output

(c) Displacement curves of high fre-

quency

0 0.005 0.010 0.015 0.020 0.025 0.030 0.034−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of 100HzPreisach error of 100Hz

(d) Error curve of high frequency

Figure 35: Triangle inputs

2 The sinusoidal signals

The maximum and the minimum voltage of the sinusoidal input are 65V and

5V, respectively. The results of this experiment with the frequency of 0.1Hz are

shown in figure 36(a-b). The results of this experiment with the frequency of

100Hz are shown in figure 36(c-d). The specific data can be seen in table 6 and

29

of 39

Accep

ted

Man

uscr

ipt

7.

5 10 15 20 25 30 34

1

2

3

4

5

6

7

8

9

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of 0.1HzActual output of 0.1HzPreisach model output


quency

5 10 15 20 25 30 34

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of 0.1HPreisach error of 0.1Hz


−0 0.005 0.010 0.015 0.020 0.025 0.030

1

2

3

4

5

6

7

8

9

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of 100HzActual output of 100HzPreisach model output


quency

0 0.005 0.010 0.015 0.020 0.025 0.030 0.034−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of 100HPreisach error of 100Hz


Figure 36: Sinusoidal inputs

3 The attenuate sinusoidal signals

The results of this experiment with low frequency are shown in figure 37(a-

b). The results of this experiment with high frequency are shown in figure

37(c-d). The specific data can be seen in table 6 and 7.

4 The mixed signals

The results of this experiment with low frequency are shown in figure 38(a-

b). The results of this experiment with high frequency are shown in figure

38(c-d). The specific data can be seen in table 6 and 7.

30

of 39

Accep

ted

Man

uscr

ipt

0 5 10 15 20 25 30 34−1

0

1

2

3

4

5

6

7

8

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of low frequencyActual output of low frequencyPreisach model output


quency

5 10 15 20 25 30 34

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of low frequencyPreisach error of low frequency


0 0.005 0.010 0.015 0.020 0.025 0.030 0.034−1

0

1

2

3

4

5

6

7

8

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of high frequencyActual output of high frequencyPreisach model output


quency

0 0.005 0.010 0.015 0.020 0.025 0.030 0.034−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Dis

pla

se

me

nt(

µm

)

Time(s)

New model error of high frequencyPreisach error of high frequency


Figure 37: Attenuate sinusoidal inputs

Table 6: The performance of the new model

Frequency Triangular Sinusoidal Attenuate Mixed

Low

Max output(µm) 9.12 7.26 6.91 10.41

range(µm) −0.15 ∼ 0.0 −0.1 ∼ 0.1 −0.07 ∼ 0.1 −0.17 ∼ 0.17

Relative error(%) 1.64 1.38 1.45 1.63

High

Max output(µm) 8.86 7.06 6.79 10.0

range(µm) −0.16 ∼ 0.03 −0.03 ∼ 0.18 −0.08 ∼ 0.14 −0.01 ∼ 0.24

Relative error(%) 1.81 2.55 2.06 2.4

31

of 39

Accep

ted

Man

uscr

ipt

0 5 10 15 20 25 30 34

0

2

4

6

8

10

12

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of low frequencyActual output of low frequencyPreisach model output


quency

0 5 10 15 20 25 30 34

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of low frequencyPreisach error of low frequency


0 0.005 0.010 0.015 0.020 0.025 0.030 0.034

0

2

4

6

8

10

12

Time(s)

Dis

pla

se

me

nt(

µm

)

New model output of high frequencyActual output of high frequencyPreisach model output


quency

0.005 0.010 0.015 0.020 0.025 0.030 0.034

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time(s)

Dis

pla

se

me

nt(

µm

)

New model error of high frequencyPreisach error of high frequency


Figure 38: Mixed inputs

Table 7: The performance of the Preisach model

Frequency Triangular Sinusoidal Attenuate Mixed

Low

Max output(µm) 9.39 7.56 7.22 10.58

range(µm) −0.35 ∼ 0.0 −0.38 ∼ 0.0 −0.53 ∼ 0.17 −0.35 ∼ 0.22

Relative error(%) 3.73 5.03 7.34 3.3

High

Max output(µm) 9.39 7.56 7.22 10.58

range(µm) −0.62 ∼ 0.18 −0.63 ∼ 0.17 −0.65 ∼ 0.57 −0.91 ∼ 0.46

Relative error(%) 6.6 8.33 7.89 8.6

32

of 39

Accep

ted

Man

uscr

ipt

6. CONCLUSION

A novel similarity-base hysteresis empirical model for PZTs has been in-

troduced in this paper. First, using the triangle inputs with different extreme

points, the geometric similarity of the displacement curves has been presented;

and using the triangle inputs with different frequencies, the time scale similarity

of the displacement curves has been presented. Then, the similarity-based hys-

teresis model can be obtained. However, the similarity is just the total describe

of the relationship between the reference curve and the displacement curves

having the turning points on it. For others, big errors may be caused. So, some

methods have been proposed to amend the model. The proposed model is easily

built; saving memory space; and with the appropriate time complexity; more

important, the dynamic of the model is solved in a new and easy way.

It can be seen from the modeling process, the hysteresis loops corresponding

to each ascending process and decreasing process have been given (in Figure

27). As we know, the hysteresis loops are the input-output relationship of the

PZTs, so using those loops the inverse model can also be built. And that’s what

we will do next.

However, Problems are still exist: It is found that the trajectory in initial

stage is different from the later stage, although the input signal is the same (It

can be seen clear in figure 31(c-d)); The errors mainly caused by the approx-

imation of the higher-order sub-loops, so methods should be used to improve

it.

References

1. R. V. Lapshin, O. V. Obyedkov. Fast-acting Piezoactuator and Digital

Feedback Loop for Scanning Tunneling Microscopes.Review of scientific

instruments, 1993, 64 (10): 2883 ∼2887

33

of 39

Accep

ted

Man

uscr

ipt

2. I. D. Mayergoyz. Mathematical model of hysteresis [M]. NewYork: Springer,

1991.

3. M. Goldfarb, N. Celanovic. A lumped parameter electromechanical model

for describing the nonlinear behavior of piezoelectric actuators. Journal

of Dynamic Systems, Measurement, and Control 1997, 119: 478-485.

4. P. Ge, M. Jouaneh. Modeling hysteresis in piezoceramic actuators. Pre-

cision engineering 1995, 17: 211-221.

5. R. B. Mrad, H. Hu. A model for voltage to displacement dynamics in

piezoceramic actuators subject to dynamic voltage excitations. IEEE/ASME

Transactions of Mechatronics . 2002, 7(4): 479-489.

6. F. Wolf, A. Sutor, S. J. Rupitsch, R. Lerch. Modeling and measurement

of creep- and rate-dependent hysteresis in ferroelectric actuators. Sensors

and Actuators A: Physical 2011, 172(1) :245C252.

7. S. H. Lee, T. J. Royston. Modeling piezoceramic transducer hysteresis in

the structural vibration control problem. Journal of the Acoustical Society

of America. 2000, 108(6): 2843-2855.

8. R Bouc Modle mathmatique dhystrsis. Acustica 1971, 21:16C25.

9. W. Zhua, D.H. Wang. Non-symmetrical Bouc Wen model for piezoelectric

ceramic actuators. Sensors and Actuators A: Physical. 2012, 181:51-60.

10. S. B. Jung, S. W. Kim.Improvement of Scanning Accuracy of PZT Piezo-

electric Actuators by Feed - forward Model - reference Control.Precision

Engineering,1994,16(1):49∼55

11. Sun Lining, Ru Changhai, RongWeibin, Chen Liguo and Kong Minxiu.

Tracking control of piezoelectric actuator based on a new mathematical

model. Journal Of Micromechanics And Microengineering. 2004, (??):

1439-1444

12. S. Bashash, N. Jalili, Underlying Memory-Dominant Nature of Hysteresis

in Piezoelectric Materials. Journal Applied Physics, 2006, 100(1): 014103-

014109.

34

of 39

Accep

ted

Man

uscr

ipt

13. S. Bashash, N. Jalili, A polynomial-based linear mapping strategy for feed

forward compensation of hysteresis in piezoelectric actuators, Journal of

Dynamic Systems, Measurement and Control 130 (3) 2008: 1-10.

14. S. Bashash, N. Jalili, P. Evans and M. J. Dapino Recursive memory-based

hysteresis modeling for solid-state smartactuators Journal of intelligent

material systems and structures 2009, 20: 2161-71

15. E. Madelung. Uber magnetisierung durch schnellverlaufende strome und

die wirkungsweise des rutherford-marconischenmagnetdetektors. Annalen

der Physik. 1905: 17, 861-890.

35

of 39

Accep

ted

Man

uscr

ipt

Biography of the authors Zhilin Lai born in 1984, he received B.S. degree from Nanchang University in 2006 and M.S. degree from Beijing Institute of Technology (BIT) in 2009. Now he is a Ph.D. candidate in BIT. His main research interest lies in hysteresis modeling and compensation. Zhen Chen born in 1976, he received M.S. and Ph.D. degrees from BIT in 2005 and 2008, respectively. He is currently a lecturer with Automation College in BIT. His research interest is servo system design and control. Xiangdong LIU born in 1971, he received M.S. and Ph.D. degrees from Harbin Institute of Technology (HIT) in 1995 and 1998, respectively. From 1998 to 2000, he did his post doctor research in mechanical postdoctoral research center in HIT. He is currently a professor with Automation College in Beijing Institute of Technology (BIT). His research interests include high-precision servo control, spacecraft attitude control, Chaos theory. Qinghe Wu received the B.S. degree in electrical engineering from Huazhong University of Science and Technology, in 1982, the post-graduate diploma and the Dr. Tech. Sci. degrees from the Swiss Federal Institute of Technology (ETH), in 1984 and 1990, respectively. He is currently a professor with Automation College in Beijing Institute of Technology (BIT). His research interest mainly covers robust control theory and multidimensional systems.

Biography of the authors

of 39

Accep

ted

Man

uscr

ipt

Figure List

Figure 1 The PZT control system

Figure 2 The controller

Figure 3 The PZT experiment devices

Figure 4 The laser interferometer

Figure 5 Some basic concepts

Figure 6 The hysteresis loop

Figure 7 Voltages signals

Figure 8 Displacement curves

Figure 9 Hysteresis loops

Figure 10 Ascending errors

Figure 11 20 sets ascending voltage signals

Figure 12 20 sets ascending displacement curves

Figure 13 20 sets hysteresis loops

Figure 14 Relationship between a and aV

Figure 15 Relationship between a and aV

Figure 16 20 sets descending voltage signals

Figure 17 20 sets descending displacement curves

Figure 18 Relationship between d and dV

Figure 19 Relationship between d and dV

Figure 20 Triangular voltage signals with different frequency

Figure 21 Triangular displacement outputs with different frequency

Figure 22 Curves before scale amplification

Figure 23 Curves after scale amplification

Figure 24 Relationship between f and 0lg( / )f f

Figure 25 Relationship between f and 0lg( / )f f

Figure 26 Schematic diagram of modeling

Figure 27 The establishment of intermediate hysteresis loop

Figure 28 Relationship of turning points and the hysteresis loop

Figure 29 Inputs with the same maximum voltage

Figure 30 Sketch of amendment

Figure 31 Wiping-Out Property

(a) Input signals

(b) Displacement curves

Figure 32 Congruency Property

(a) Input signals

(b) Hysteresis loops

Figure 33 An implementation of the modeling

Figure List

of 39

Accep

ted

Man

uscr

ipt

Figure 34 Fitting the hysteresis loop

Figure 35 Triangle inputs

(a) Displacement curves of low frequency


(c) Displacement curves of high frequency


Figure 36 Sinusoidal inputs





Figure 37 Attenuate sinusoidal inputs





Figure 38 Mixed inputs





of 39

Accep

ted

Man

uscr

ipt

1. Two new phenomena have been discovered in this paper, the geometric similarity and the time scale similarity. 2. The use of the time scale similarity can make the hysteresis model dynamic in an easy way. 3. ‘Virtual extremum’ is a good way to improve the precision of the model. 4. The proposed model are easily built; and saving computational time and memory space.

*Highlights (for review)

A novel similarity-based hysteresis empirical model for piezoceramic actuators

Documents

Transcript of A novel similarity-based hysteresis empirical model for piezoceramic actuators