Online parameter identification of the asymmetrical Bouc–Wen model for piezoelectric actuators

Precision Engineering 38 (2014) 921–927

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Online parameter identification of the asymmetrical Bouc–Wen model for piezoelectric actuators Zhu Wei ∗ , Bian Lei Xiang, Rui Xiao Ting Institute of Launch Dynamics, Nanjing University of Science and Technology, Nanjing 210094, China

a r t i c l e

i n f o

Article history: Received 9 March 2013 Received in revised form 4 June 2014 Accepted 10 June 2014 Available online 18 June 2014 Keywords: Piezoelectric actuator Asymmetrical Bouc–Wen model Online parameter identification Recursive least-squares method Limited memory method

a b s t r a c t The hysteresis of piezoelectric actuators (PAs) possesses the asymmetrical and frequency-dependent characteristics. In order to accurately model the hysteresis of a PA, an asymmetrical Bouc–Wen model is proposed and established in this paper. The recursive least-squares online identification method is used to real-time identify the parameters of the proposed model. Meanwhile, in order to avoid the data saturation phenomenon, the limited memory method is used to limit the number of the data sets. The experimental system is setup and the performance of this method is experimentally verified. Experimental results show that the proposed online identification method can effectively improve the modeling accuracy. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Piezoelectric actuators (PAs) based on inverse piezoelectric effect are widely used in precision positioning due to their small size, high energy density, high positioning accuracy, high resolution, and quick frequency response [1–4]. However, the hysteresis between the output displacement and the applied voltage from and to a PA seriously affects the positioning accuracy. In the current literatures, the hysteresis models can be categorized into the physical and mathematical models. Through analyzing the inherent mechanism of hysteresis of piezoelectric layers, the physical models [5,6] which could provide a theoretical basis for designing and controlling PAs were established. However, the physical models are relatively complicated because their inherent mechanism is quite complex. The mathematical models include the Preisach model [7,8], the Prandtle-Ishlinskii model [9,10], the polynomial approximation [11], the Duhem model [12], the Maxwell slip model [13], the Jiles–Atherton model [14], the LuGre model [15], and the Bouc–Wen model [16–19]. Because the Bouc–Wen model can match the behavior of a wide class of hysteresis systems [20], it has been extensively adopted in many engineering fields to represent the hysteresis behavior of engineering elements and structures [21–24]. However, there are some difficulties in the application on the PA. For example,

∗ Corresponding author. Tel.: +86 25 8431 5901; fax: +86 25 8431 2400. E-mail address: [email protected] (Z. Wei). http://dx.doi.org/10.1016/j.precisioneng.2014.06.002 0141-6359/© 2014 Elsevier Inc. All rights reserved.

existing Bouc–Wen models cannot describe the highly asymmetric and the frequency-dependent hysteresis of a PA. To account for the strong asymmetry, Zhu and Wang [16] put forward an asymmetrical Bouc–Wen model by introducing an asymmetrical formula into the Bouc–Wen hysteresis operator. To avoid the large modeling error, Ha et al. [17] proposed the Bouc–Wen model to model a piezo-actuated positioning mechanism which only works in the single frequency. In order to improve the control accuracy, Gomis-Bellmunt et al. [18] offline identified the parameters of the Bouc–Wen model in a certain frequency range which can make the Bouc–Wen model adapt the certain frequency. In this paper, an asymmetrical Bouc–Wen model for a PA will be proposed and established. The recursive least-squares online identification method will be used to real-time identify the parameters of the asymmetrical Bouc–Wen model. 2. Asymmetrical and frequency-dependent characteristics of the hysteresis of a PA 2.1. Asymmetrical characteristics of the hysteresis and the asymmetrical Bouc–Wen model Fig. 1 shows the measured hysteresis curve of a PA under a 1 Hz sinusoidal voltage. The output displacement in the stable period is different from that in the initial period due to the memory of piezoelectric materials. Considering the fitted line in least-squares sense as the linear component, the hysteresis curve shown in Fig. 2 can be decomposed into a linear component (X(t)) and a hysteretic

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Z. Wei et al. / Precision Engineering 38 (2014) 921–927

20

Hysteresis Fitted

20 15

Displacement (µm)

Displacement (µm)

25

h(t) Stable period

10

X(t) x(t) 5

1 Hz 10 Hz 100 Hz 200 Hz

15

10

5

Initial period 0 0

20

40

60

80

0

100

Voltage (V)

0

20

40

60

80

100

Voltage (V)

Fig. 1. Measured hysteresis curve of a PA under a 1 Hz sinusoidal input voltage. Fig. 3. Frequency-dependent hysteresis of the PA.

component (h(t)), which are plotted in Fig. 2(a) and (b), respectively. Observing Fig. 2, the hysteresis curve can be considered as the superposition of the linear component and the hysteretic component. The burr in the hysteretic component as shown in Fig. 2(b) is mainly attributable to the external disturbances in the measurement process. Obviously, the curve shown in Fig. 2(b) demonstrates the significant asymmetry in the hysteretic component of the PA. From Fig. 2, the output displacement from the PA can be given by x(t) = X(t) + h(t)

(1)

hysteresis operator. Eq. (4) can be rewritten as [16] n−1 n ˙ ˙ ˙ ˙ ˙ h(t) = Au(t) − ˇ|u(t)||h(t)| h(t) −  u(t)|h(t)| + ıu(t)sgn(u(t))

(5) ˙ where ı is the asymmetry factor; ıu(t)sgn(u(t)) is the asymmetrical formula;

˙ sgn(u(t)) =

⎧ ⎨ 1 ⎩

X(t) is governed by X(t) = kv u(t) + x0

(2)

where u(t) denotes the input voltage; kv is a constant representing the ratio between the output displacement and the input voltage; x0 is the initial displacement. Substituting Eq. (2) into Eq. (1), then x(t) = kv u(t) + x0 + h(t)

˙ =0 u(t)

−1

˙ u(t) <0

Eqs. (3) and (5) define the asymmetrical Bouc–Wen model. In order to use the asymmetrical Bouc–Wen model to simulate the hysteresis of a PA, the parameters kv , x0 , A, ı, ˇ, n, and  need to be identified. 2.2. Frequency-dependent behavior of the hysteresis

(3)

Using the Bouc–Wen hysteresis operator [25] to simulate the hysteretic component n−1 n ˙ ˙ ˙ ˙ h(t) = Au(t) − ˇ|u(t)||h(t)| h(t) −  u(t)|h(t)|

(4)

where the dot at the top of variables represents the first order derivative with respect to time; A, ˇ, , and n are the parameters of the Bouc–Wen hysteresis operator. The model for the PA defined by Eqs. (3) and (4) is called the Bouc–Wen model. To model the asymmetrical hysteresis, a formula to represent the asymmetrical characteristic is introduced into the Bouc–Wen

We apply a set of sinusoidal voltages with different frequencies to the PA, and the hysteresis of the PA is frequency-dependent, which becomes more evident with the increase of input frequencies as shown in Fig. 3. The hysteresis curves of the asymmetrical Bouc–Wen model under different input frequencies are shown in Fig. 4, which indicates the asymmetrical Bouc–Wen model cannot characterize the frequency-dependent hysteresis of the PA. In order to improve modeling accuracy, the parameters of this model are identified by an online identification method (in Section 3), which make the asymmetrical Bouc–Wen model can describe the frequency-dependent hysteresis.

25

4

20

2

h(t) (µm)

X(t) (µm)

˙ u(t) >0

0

15 10

0

o

−2

5 0

Stable period

Initial period 0

20

40

60

u(t) (V)

(a)

80

100

−4

0

20

40

60

80

100

u(t) (V)

(b)

Fig. 2. Linear and hysteretic components of the hysteresis curve of the PA: (a) the linear component and (b) the hysteretic component.

Z. Wei et al. / Precision Engineering 38 (2014) 921–927

Limit the number of the maximum memory be N. When the sampling point is k + 1 (k ≥ N), N sets of data ([ uk+1−N xk+1−N ], . . ., [ uk+i−N xk+i−N ], . . ., [ uk+1 xk+1 ]) in the hysteresis curve are used. According to the limited memory recursive least-squares method and Eq. (3), there is

Displacement (µm)

20

1 Hz 10 Hz 100 Hz 200 Hz

15

⎧ ˆ k+1 = ˆ k + Kk+1 (k+1 xk+1 − k+1−N xk+1−N ) ⎪ ⎪ ⎪ ⎨

10

⎪ ⎪ ⎪ ⎩

5

0 0

20

923

40

60

80

Kk+1 = Pk+1 +

Pk+1 k+1−N

T 1 − k+1−N Pk+1 k+1−N

T k+1−N Pk+1

T K  Pk+1 = Kk − Kk k+1 (1 + k+1 k k+1 )

−1

(6)

k+1 Kk

T where k ≥ N; ˆ k+1 = [ kˆ v,k+1 xˆ 0,k+1 ] ; kˆ v,k+1 and xˆ 0,k+1 are the estimated values of kv and x0 in the sample point k + 1, respectively; the superscript ˆ expresses the estimated value of the parameter;

100

Voltage (V) Fig. 4. Hysteresis curves of the asymmetrical Bouc–Wen model under different input frequencies.

i = [ui 1]; Kk = (˚Tk ˚k )



−1

T k+i−N = T ˆ k

T ; ˚k = [ k−N

···



···

T

kT ] ;

xˆ k+1 k+1 . According T xˆ k+1−N = k+1−N ˆ k to Eq. (6), the estimated values of kv and x0 can be obtained. The initial values can be given by xk+1 = xk+1 − xˆ k+1 ; xk+1−N = xk+1−N − xˆ k+1−N

−1 ˆ N = (˚TN ˚N ) ˚TN [ x1

···

xi

···

xN ]

T

(7)

The initial values can also be obtained by the off-line identification method [16–18]. Consider N1 sets of points ([ u0,j x0,j ], j = 1, · · ·, N1 ) in the N sets of the sampling points, which ensure that u˙ 0,j > 0 and the corresponding hysteretic component h0,1 , . . ., h0,j , . . ., h0,N1 are equal to zero. According to Eq. (5), we can obtain x˙ 0,j = (A + kˆ v,k+1 )u˙ 0,j + ıu0,j

(8)

According to the least-squares method, Eq. (8) can be rewritten as



ˆ k+1 − kˆ v,k+1 A

 = (˚TN ˚N1 )

−1

1

ıˆ k+1

˚N1 =

where

(9)

1

··· ···

u˙ 0,1 u0,1

˚TN x˙ +,N1

u˙ 0,j u0,j

· · · u˙ 0,N1 · · · u0,N1

T ;

x˙ 0,N1 =

T

[ x˙ 0,1 · · · x˙ 0,j · · · x˙ 0,N1 ] . The estimated values of A and ı can be identified by Eq. (9). Consider N2 sets of points ([ u+,j x+,j ], j = 1, · · ·, N2 ) in the N sets of the sampling points, which ensure that u˙ +,j > 0 and the corresponding hysteretic component h+,j ≥ 0 are larger than zero. According to Eq. (5), the following equation can be obtained

Fig. 5. Experimental setup for PAs: (a) the schematic and (b) the photograph.

3. Online parameter identification method In order to make the asymmetrical Bouc–Wen model possess the frequency-dependent characteristics, the parameters of this model are real-time updated by recursive least-squares online identification method [26]. Meanwhile, in order to avoid the data saturation phenomenon, the limited memory method is used to limit the number of the data sets. In other words, if a new set of data is sampled, the oldest set of data will be removed.

N2 nˆ k+1 =

N2 j=1

ln h+,j ln

⎧ ⎪ h˙ +,1 − ıˆ k+1 u+,1 ⎪ ˆ k+1 − (ˇ + )hn ⎪ =A +,1 ⎪ u˙ +,1 ⎪ ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎨ h˙ − ıˆ u +,j

k+1 +,j

ˆ k+1 − (ˇ + )hn =A

(10)

+,j u˙ +,j ⎪ ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ ˆ ˙ h+,N2 − ık+1 u+,N2 ⎪ ⎪ ˆ k+1 − (ˇ + )hn =A ⎩ +,N2

u˙ +,N2

ˆ k+1 + ˆ k+1 > 0. Using the least-squares method, Eq. (10) Let ˇ can be rewritten as ˆ k+1 − A N2

h˙ +,j −ıˆ k+1 u+,j u˙ +,j

N2

j=1

  N2 −

2

(ln h+,j ) −

ln h+,j i=1



N2 j=1

N2

ln h+,j

ln j=1

2

ˆ k+1 − A

h˙ +,j −ıˆ k+1 u+,j u˙ +,j



(11)

924

Z. Wei et al. / Precision Engineering 38 (2014) 921–927

Experimental Tradition method Online identification

Displacement (µm)

20

15

10

5

0 0

0.5

1

1.5

2

2.5

3

3.3

2.5

3

3.3

Time (s)

(a)

Displacement error (µm)

1

0.5

0

−0.5

Tradition method Online identification −1 0

0.5

1

1.5

2

Time (s)

(b) Fig. 6. Time histories of the output displacements under the sinusoidal excitation whose frequency changes from 1 Hz to 10 Hz: (a) the output displacements and (b) the modeling errors.

⎛ N

2

ˆ k+1 + ˆ k+1 = exp ⎝ ˇ

j=1

ln

ˆ k+1 − A

h˙ +,j −ıˆ k+1 u+,j



u˙ +,j

− nˆ k+1

N2 j=1

ln h+,j

N2

According to Eqs. (12) and (14), the estimated values of the parameters ˇ and  can be given by

⎞

Consider N3 sets of points ([ u−,j x−,j ], j = 1, · · ·, N3 ) in the N sets of the sampling points, which ensure u˙ −,j < 0 and the corresponding hysteretic component h−,j ≥ 0. According to Eq. (5), we can obtain

⎧ ⎪ h˙ −,1 + ıˆ k+1 u−,1 ⎪ ˆ k+1 − (−ˇ + )hn ⎪ =A −,1 ⎪ u˙ −,1 ⎪ ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎨ h˙ + ıˆ u −,j

k+1 −,j

⎡

⎠ (12)

⎢ ˆ k+1 = 1 exp ⎢ ˇ 2 ⎣ ⎡ −

ˆ k+1 − (−ˇ + )hn =A

(13)

ˆ k+1 =

u˙ −,N3

+

ˆ k+1 + ˆ k+1 > 0. Using the least-squares method, accordLet −ˇ ing to Eq. (13), the following equation can be obtained 3

ˆ k+1 + ˆ k+1 = exp ⎣ −ˇ

j=1

ln

ˆ k+1 − A

h˙ −,j +ıˆ k+1 u−,j u˙ −,j

N3



− nˆ k+1

N3 i=1

ln h−,j

⎤ ⎦ (14)

⎢ 1 exp ⎢ 2 ⎣

ˆ k+1 − A

ln

h˙ +,j − ıˆ k+1 u+,j

 − nˆ k+1

u˙ +,j

N2 j=1

⎤ ln h+,j

N2



j=1

ˆ k+1 − A

ln

h˙ −,j + ıˆ k+1 u−,j

 − nˆ k+1

u˙ −,j



N2

N3 j=1

j=1

ln

ˆ k+1 − A

h˙ +,j − ıˆ k+1 u+,j

i=1

u˙ +,j

 − nˆ k+1

ln h−,j

N2 j=1

 ˆ k+1 − A

h˙ −,j + ıˆ k+1 u−,j u˙ −,j N3

 − nˆ k+1

N3 i=1

(15)

⎥ ⎥ ⎦ ⎤

ln h+,j

N2

ln

⎥ ⎥ ⎦

⎤

N3

N3

⎢ 1 exp ⎢ 2 ⎣ ⎡

j=1

N3

⎡

−,j u˙ −,j ⎪ ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ ˆ ˙ h−,N3 + ık+1 u−,N3 ⎪ ⎪ ˆ k+1 − (−ˇ + )hn =A ⎩ −,N3

⎡ N

⎢ 1 exp ⎢ 2 ⎣



N2

⎤ ln h−,j

⎥ ⎥ ⎦ (16)

⎥ ⎥ ⎦

According to Eqs. (6), (9), (11), (15) and (16), the parameters kv x0 , A, ı, n, ˇ, and  of the asymmetrical Bouc–Wen model can be online identified if the voltage u(t) applied to the PA and the corresponding output displacement are real-time sampled.

Z. Wei et al. / Precision Engineering 38 (2014) 921–927

Experimental Tradition method Online identification

20

Displacement (µm)

925

15

10

Steady state

Adaptive state

5

0 3

3.05

3.1

3.15

3.2

3.25

3.3

3.33

Time (s)

(a) Displacement error (µm)

1 0.5 0 −0.5 −1 −1.5

Tradition method Online identification 3

3.05

3.1

3.15

3.2

3.25

3.3

3.33

Time (s)

(b) Fig. 7. Time histories of the output displacements under the sinusoidal voltage whose frequency changes from 10 Hz to 100 Hz: (a) the output displacements and (b) the modeling errors.

4. Experimental verification 4.1. Experimental setup In order to experimentally validate the proposed asymmetrical Bouc–Wen model with the corresponding parameter identification method, the schematic and photograph of the developed experimental setup are shown in Fig. 5(a) and (b), respectively. According to Fig. 7, the experimental setup is composed of a power amplifier for PAs (type: P&I-2), laser Doppler vibrometer (LDV, type: OFV-505/5000, the Polytec GmbH, Germany, range: −40 to +40 ␮m, resolution: 2 nm, non-linearity: 0.05%), and real-

time simulation system (type: DS1103 (the dSPACE GmbH) with the MATLAB/Simulink (the MathWorks, Inc.), 16-Bit A/D converter). When conducting experiments, the PA (type: P885.51, applied voltage: 0–200 V, output displacement: 0–20 ␮m, PI Corporation, Germany), whose output displacement is monitored by the LDV, is driven by the power amplifier. The voltage applied to the power amplifier and the output from the LDV are acquired to the host computer by the real-time simulation system. 4.2. Experimental results Under the sinusoidal excitation of u (t) = 50 + 50 sin (2ft − /2), where

Table 1 Maximum absolute errors of the two identification methods under different input frequencies in the steady state (␮m). Method

1 Hz

10 Hz

100 Hz

200 Hz

Tradition method Online identification

0.47 0.47

0.75 0.48

1.25 0.45

2.91 0.49

f =

⎧ 1 ⎪ ⎪ ⎨ 10

0≤t<3 3 ≤ t < 3.3

100 3.3 ≤ t < 3.33 ⎪ ⎪ ⎩ 200

,

t ≥ 3.33

Table 2 Stable values of the parameters identified by the online identification method under different input frequencies. Frequency

kv (␮m/V)

x0 (␮m)

A

ˇ



n

ı

1 Hz 10 Hz 100 Hz 200 Hz

0.19 0.18 0.18 0.16

0.015 0.017 0.021 0.013

−0.072 −0.072 −0.072 −0.10

0.020 0.025 0.023 0.015

−0.019 −0.021 −0.018 −0.005

1.52 1.53 1.51 1.68

0.0075 0.0067 0.0063 0.0045

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Z. Wei et al. / Precision Engineering 38 (2014) 921–927

Experimental Tradition method Online identification

Displacement (μm)

20

15

10

Adaptive state

Steady state

Adaptive state

Steady state

5

0 3.3

3.305

3.31

3.315

3.32

3.325

3.33

3.335

3.34

3.345

Time (s)

(a) Displacement error (μm)

1 0 −1 −2 −3

Tradition method Online identification

−4 3.3

3.305

3.31

3.315

3.32

3.325

3.33

3.335

3.34

3.345

Time (s)

(b) Fig. 8. Time histories of the output displacements under the sinusoidal excitation whose frequency changes from 100 Hz to 200 Hz: (a) the output displacements and (b) the modeling errors.

20

Experimental Online identification

Displacement (μm)

Displacement (μm)

20

15

10

5

0

0

20

40

60

80

100

15

10

5

0

0

20

(a) 1 Hz

80

100

20

Displacement (μm)

Displacement (μm)

60

(b) 10 Hz

20

15

10

5

0

40

Voltage (V)

Voltage (V)

0

20

40

60

Voltage (V)

(c) 100 Hz

80

100

15

10

5

0

0

20

40

60

80

100

Voltage (V)

(d) 200Hz

Fig. 9. The hysteresis curves under different input frequencies measured by the LDV and predicted by the asymmetrical Bouc–Wen model with the parameters listed in Table 2.

Z. Wei et al. / Precision Engineering 38 (2014) 921–927

the measured output displacements of the PA are shown in Figs. 6(a), 7(a) and 8(a). The experimental data under the 1 Hz sinusoidal excitation is used to identify the parameters of the asymmetrical Bouc–Wen model using the traditional identification method [18], and the obtained parameters are set to the initial values of the online identification method. The corresponding output displacements predicted by the asymmetrical Bouc–Wen model with the traditional parameter identification method and the online identification method are also shown in Figs. 6(a), 7(a) and 8(a), respectively. The corresponding modeling errors are plotted in Figs. 6(b), 7(b) and 8(b), respectively. It can clearly be seen from Figs. 6–8, when modeling the experimental data which is used to identify parameters by the traditional identification method, both of the two identification methods can portray the measured output displacement well. However, under different input frequencies, the modeling errors of the traditional identification method increase sharply, but after an adaptation state, the modeling errors of the online identification method remain unchanged. Without considering the adaptive state, the maximum absolute errors of the two identification methods are listed in Table 1. Observing Table 1, under the 200 Hz sinusoidal excitation, the maximum absolute error of the traditional identification method is 2.91 ␮m. However, that of the online identification method is still about 0.48 ␮m, which may result from the structural error of the asymmetrical Bouc–Wen model. Improving the asymmetrical Bouc–Wen model, such as introducing a creep factor, may be a good way to reduce this error. Therefore, the proposed online identification method can effectively improve the modeling accuracy and make the asymmetrical Bouc–Wen model possess the frequency-dependent characteristics. The stable parameters values identified by the online identification method under different input frequencies are listed in Table 2. The asymmetrical Bouc–Wen model with the parameters listed in Table 2 is used to predict the hysteresis curves shown in Fig. 1, and the predicted results are plotted in Fig. 9. Obviously, the asymmetrical Bouc–Wen model with the online identification method can accurately model the hysteresis curves under different input frequencies 5. Conclusion In order to accurately describe the asymmetrical and frequencydependent hysteresis of a PA, the asymmetrical Bouc–Wen model was proposed and established. The recursive least-squares online identification method was used to real-time identify the parameters of the asymmetrical Bouc–Wen model. Meanwhile, in order to avoid the data saturation phenomenon, the limited memory method was used to limit the number of the data sets. Experimental results showed that, under different input frequencies, the maximum absolute error of the traditional identification method increase sharply, and the maximum value is 2.91 ␮m, but the maximum absolute error of online identification methods is still 0.49 ␮m which may be reduced by introducing a creep factor to the Bouc–Wen model. The online identification method proposed by this paper can effectively improve the modeling accuracy and make the asymmetrical Bouc–Wen model possess the frequencydependent characteristics. Therefore, to further reduce the modeling error, the future work can research the generalized Bouc–Wen model which can

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