robust model reference adaptive control for a two dimensional piezo driven micro displacement scanning platform based on the asymmetrical bouc wen model

Robust model reference adaptive controlfor a two-dimensional piezo-driven micro-displacement scanning platform based on the asymmetrical Bouc-Wen model Haigen Yang,1, aWei Zhu,2and Xiao

displacement scanning platform based on the asymmetrical Bouc-Wen model

Haigen Yang, Wei Zhu, and Xiao Fu

Citation: AIP Advances 6, 115308 (2016); doi: 10.1063/1.4967428

View online: http://dx.doi.org/10.1063/1.4967428

View Table of Contents: http://aip.scitation.org/toc/adv/6/11

Published by the American Institute of Physics

Robust model reference adaptive control

for a two-dimensional piezo-driven micro-displacement scanning platform based on the asymmetrical

Bouc-Wen model

Haigen Yang,1, aWei Zhu,2and Xiao Fu1

1Engineering Research Center of Wider and Wireless Communication, Technology,

Ministry of Education, Nanjing University of Posts and Telecommunications,

Nanjing 210094, China

2Institute of Launch Dynamics, Nanjing University of Science and Technology,

Nanjing 210094, China

(Received 28 June 2016; accepted 24 October 2016; published online 3 November 2016)

The hysteresis characteristics resulted from piezoelectric actuators (PAs) and the residual vibration in the rapid positioning of a two-dimensional piezo-driven micro-displacement scanning platform (2D-PDMDSP) will greatly affect the positioning accuracy and speed In this paper, in order to improve the accuracy and speed of the positioning and restrain the residual vibration of 2D-PDMDSP, firstly, Utilizing an online hysteresis observer based on the asymmetrical Bouc-Wen model, the PA with the hysteresis characteristics is feedforward linearized and can be used as a linear actua-tor; secondly, zero vibration and derivative shaping (ZVDS) technique is used to elimi-nate the residual vibration of the 2D-PDMDSP; lastly, the robust model reference adap-tive (RMRA) control for the 2D-PDMDSP is proposed and explored The rapid control prototype of the RMRA controller combining the proposed feedforward linearization and ZVDS control for the 2D-PDMDSP with rapid control prototyping technique based on the real-time simulation system is established and experimentally tested, and the corresponding controlled results are compared with those by the PID control method The experimental results show that the proposed RMRA control method can significantly improve the accuracy and speed of the positioning and restrain the

resid-ual vibration of 2D-PDMDSP © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4967428]

I INTRODUCTION

In recent years, piezoelectric actuators (PAs) based on the inverse piezoelectric effect have been widely used in precision positioning,1 , 2due to their small size, high energy density, high resolution, and quick frequency response However, on the one hand, owing to the hysteresis characteristics of PAs, piezo-driven positioning systems also has a strong nonlinearity, so how to effectively control piezo-driven positioning systems is difficult and emphasized in positioning systems;3 on the other hand, in the rapid positioning, the existing residual vibration of piezo-driven positioning systems slow down the positioning speed Therefore, how to eliminate the residual vibration and improve the positioning speed of positioning systems is also one of the problems which must be urgently solved

In order to reduce the influence of the hysteresis characteristics of PAs on the positioning precision

of PAs and piezo-driven positioning systems, the best way is the feedforward compensation for the hysteresis behavior of PAs and piezo-driven positioning systems as realized by using the established inverse hysteresis model4 8 to track the desired displacement.9 12 Because of exist of the inverse

a Electronic mail: yhg@njupt.edu.cn

2158-3226/2016/6(11)/115308/14 6, 115308-1 © Author(s) 2016

modelling error of the hysteresis behavior, the closed-loop control to compensate for the model error and restrain the outside disturbances was introduced.13–17Although the closed-loop control method could restrain the hysteresis of PAs and piezo-driven positioning systems, the inverse of the hysteresis model enlarged the model error to some extent

In the existing literature, it is few reported that how to eliminate the residual vibration The general methods include increasing the damping, increase stiffness, and using complex control algorithms However, increasing damping and increase stiffness affect the response speed of positioning systems and increase energy consumption, and using complex control algorithms is difficult to realize, further exacerbate the complexity of the precision positioning control, and reduce the real-time of the systems the piezo-driven positioning systems.18,19

Therefore, in this paper, the robust model reference adaptive (RMRA) control including feedfor-ward linearization, zero vibration and derivative shaping (ZVDS) for a two-dimensional piezo-driven micro-displacement scanning platform (2D-PDMDSP) is proposed and experimentally verified in SectionII, the structure of 2D-PDMDSP is analyzed and the dynamic model is put forward; in Section III, utilizing an online hysteresis observer based on an asymmetrical Bouc-Wen model, the PA with the hysteresis characteristics is feedforward linearize to a linear actuator; in SectionIV, zero vibration and derivative shaping (ZVDS) is used to eliminate the residual vibration of the 2D-PDMDSP in the rapid positioning; in SectionV, a robust mode reference adaptive (RMRA) control is put forward

to control the 2D-PDMDSP, and the stability and robustness of the RMRA control is theoretically proved; in SectionVI, the rapid control prototype of the RMRA controller combining the proposed feedforward linearization, ZVDS, and RMRA control for the 2D-PDMDSP with rapid control proto-typing technique based on the real-time simulation system are established and experimentally tested, and the corresponding controlled results are compared with those by the PID control method; in SectionVII, some conclusions are drawn

II 2D-PDMDSP

Figure1shows the photograph of a 2D-PDMDSP From figure1, the 2D-PDMDSP is composed of: two PAs I and II, which are used to generate output displacements and forces, and promote the X-axis and Y-axis directions of the 2D-PDMDSP respectively; two oval amplification mechanisms, which can enlarge the output displacements of PAs; a flexure hinge transmits the displacements and forces, and promote the movement of a two-dimensional position stage and a load The relationship between the X axis and Y axis directions is interrelated and independent of each other, therefore the

FIG 1 Photograph of the 2D-PDMDSP.

X axis and Y axis directions can be separately controlled In this paper, only the X axis direction examples for controlling are presented

Because the long axis of the oval amplification mechanisms is much larger than the maximum output displacement of the PAs, can be approximated as a linear amplification mechanism The dynamics model for the X axis direction of the 2D-PDMDSP can be expressed as

F=Fp

where F is the force transmitted by the flexure hinge; M is the magnification of the oval amplification mechanism; F p is the output force of the PA-I; X is the output displacement of the X axis direction

of the two-dimensional position stage and load; m0, c0, and k0are the mass, damping, and stiffness

of the X axis direction

Appling a 10 Hz sinusoidal voltage, as showed in figure 2, to the PA-I, the measured input-output relationship curve of the 2D-PDMDSP is shown in figure3 Observing from figure3, due

to the hysteresis characteristics of the PA-I, the input-output relationship of the 2D-PDMDSP also presents the hysteresis behavior

Appling a 1 Hz square voltage, as showed in figure4, to the PA-I, the measured output displace-ment of the 2D-PDMDSP is shown in figure5 Observing from figure5, the large amplitude and the long duration of the residual vibration of the 2D-PDMDSP limit its positioning speed, and even affect its life

Therefore, the hysteresis characteristics and the residual vibration in the rapid positioning of the 2D-PDMDSP greatly affect the positioning accuracy and speed In order to improve the positioning

FIG 2 Time histories of a sinusoidal applied voltage to the 2D-PDMDSP.

FIG 3 Measured hysteresis curve of the 2D-PDMDSP with the sinusoidal applied voltage.

FIG 4 Time histories of a square applied voltage to the 2D-PDMDSP.

FIG 5 Time histories of the output displacement of the 2D-PDMDSP with the square applied voltage.

FIG 6 Measured hysteresis curve of the PA-I with the 10 Hz sinusoidal voltage.

accuracy and reduce the complexity of the positioning control algorithm of the 2D-PDMDSP, the PA

is feedforward linearized, which make the input-output relationship of the 2D-PDMDSP be linear

At the same time, in order to increase its positioning speed, the ZVDS technique is used to eliminate residual vibration

III LINEARIZATION CONTROL FOR THE PA-I BASED ON THE ASYMMETRICAL

BOUC-WEN MODEL

Figure6shows the measured hysteresis curve of the PA-I under the no-loading condition with the 10 Hz sinusoidal voltage as shown in figure2 The output displacement in the stable period is different from that in the initial period due to the memory of piezoelectric ceramic materials Considering the fitted line in least-squares sense as the linear component, the hysteresis curve shown in figure6can be decomposed into the linear component (x (t)) and the hysteresis component

(xH(t)) The experimentally measured hysteresis curve can be considered as the superposition of a

linear component and a hysteresis component When modelling the hysteresis characteristics of the PA-I, the linear and hysteresis components can be predicted separately The output displacement from the PA-I can be given by

where x(t) is the output displacement of the PA-I The output force of the PA-I is

Considering that the hysteresis of PSAs is asymmetrical,16 utilizing a linear function and the asymmetrical Bouc-Wen hysteresis operator proposed by Zhu and Wang16to respectively simulate the linear component and the hysteresis component The asymmetrical Bouc-Wen model can be expressed as

˙

FH(t) = A˙u (t) − β |˙u (t)| |FH(t)| n−1 FH(t) − γ ˙u (t) |FH(t)| n + δu (t) sgn (FH(t)) (6)

where kvis a constant representing the ratio between the output force and applied voltage; F0is the

initial force without applied voltage; FH(t) is the hysteresis force; A, β, γ, and n are the undetermined parameters; sgn (x)= 1 x > 0

−1 x < 0; δ is the asymmetrical factor, and δ < 0 when modeling PSAs.

According to equation (5), if the hysteresis force can be compensated, the relationship between the output displacement and the applied voltage will be linear However, because the hysteresis force

of PSAs cannot be directly measured by sensors, the hysteresis observer is established to estimate the hysteresis force According to equation (6), the hysteresis observer can be expressed as

ˆ˙FH(t) = A˙u (t) − β |˙u (t)| FˆH(t)

n−1

ˆ

FH(t) − γ ˙u (t) FˆH(t)

n

+ δu (t) sgn ˆFH(t)

(7) where ˆFH(t) is the estimated value of FH(t).

According to equations (5) and (7), the linearization control for the PA-I can be expressed as









uFF(t) = u (t) − FˆH(t)

kv

ˆ˙FH(t) = A˙u (t) − β |˙u (t)| FˆH(t)

n−1

ˆ

FH(t) − γ ˙u (t) FˆH(t)

n

+ δu (t) sgn ˆFH(t) (8) According to equation (8), the block diagram of the linearization control for the PA-I based on the asymmetrical Bouc-Wen model is shown in figure7

By using the parameter identification method proposed by reference17, the parameters of the linearization control given by equation (8) are identified as follows

kv= 1.2∗

10−7, A= 4.5∗

10−8, β= 0.059, γ = −0.023, n = 1.055, and δ = 2.1∗

10−8 (9) Utilizing the values of the parameters given by equation (9), the hysteresis curves between the output displacements and the applied voltages from and to the PA-I without and with the feedforward linearization control are shown in figure8 In figure8, the fitted line according to the relationship between the output displacement and applied voltage of the PA-I with the feedforward linearization control in least-squares sense is also presented From figure8, with the feedforward linearization control, the input-output curve of the PA-I is almost linear, and the PA-I can be used as a linear actuator

FIG 7 Block diagram of the linearization control for the PA-I.

FIG 8 Relationships between the output displacement and the applied voltage of the PA-I with and without the linearization control.

IV ZERO VIBRATION AND DERIVATIVE SHAPING

In order to eliminate the residual vibration of the 2D-PDMDSP in the rapid positioning, the input shaping technique is used The idea of input shaping technique is a step signal, which be sent to a system at time t1, will stimulate a vibration response of the system, and at time t2, another step signal sent to the system will stimulate another vibration response If these two vibration responses have the same amplitude with the opposite phase, they can cancel each other out, and thus the vibration is eliminated at time t2

The 2D-PDMDSP can be considered as a linear system with the linearization control for the PA-I According to equation (1), the transfer function of the 2D-PDMDSP can be written as

G (s) = k ω2

where k is the magnification; ξ is the damping ratio; and ω nis the natural frequency According to equation (10), assume that at time t = 0, after the linearization control, the amplitude of the step

voltage to the ZVDS is UFF The output voltage upof the ZVDS is

up=





















1

1+ 2K + K2UFF 0 ≤ t < T2

1+ 2K

1+ 2K + K2UFF T2 ≤t < T

1+ 2K + K2

1+ 2K + K2UFF t ≥ T

(11)

where

K= exp * ,

π

ωnp

1 − ζ2

+

-; T= 2π

ωnp

According to equation (11), in order to use the ZVDS method to eliminate the residual vibration

of the 2D-PDMDSP, the parameters K and T need to be identified.

The measured step response curve of the 2D-PDMDSP with the linearization control is shown

in figure9 According to equation (10), we have

A= exp * ,

−πζ p

1 − ζ2

+

-(13)

ωnp

where A is the overshoot of the step response According to equations (12)–(14) and figure9, the

parameters K and T are

FIG 9 Step response of the 2D-PDMDSP with the linearization control.

FIG 10 Time histories of the output displacements of the 2D-PDMDSP without and with ZVD shaping applied to the square voltage.

Utilizing the values of the parameters given by equation (15), the time histories of the output displacements of the 2D-PDMDSP without and with the ZVDS applied to the square voltage are shown in figure10 It can be seen from figure10, with the ZVDS, the residual vibration of the 2D-PDMDSP becomes very small, the 2D-2D-PDMDSP can be quickly stabilized, and its positioning speed

is increased It should be noted that, according to equation (11), the ZVDS delay a vibration period Therefore, in that period the control error may be large

V RMRA CONTROL FOR THE 2D-PDMDSP

By considering the influence of external interference and modelling error, the error between the

dynamic model and the actual model of the 2D-PDMDSP is fd According to the equations (1) and (2), the actual model of the 2D-PDMDSP can be expressed as

¨

X+ α1X˙ + α0X= β0u+ β1(FH+ fd) (16)

˙

FH(t) = A˙u (t) − β |˙u (t)| |FH(t)| n−1 FH(t) − γ ˙u (t) |FH(t)| n + δu (t) sgn (FH(t)) (17) where α1=c0

Assume that the linearization control error is ∆FH Equations (16) and (17) can be rewritten as

¨

where f = (∆FH+ fd) Equation (18) is the linear model of the 2D-PDMDSP

For the linear model given by equation (18), an introduced stable reference model is

¨

where Xmis the output of the reference model, and also is the desired output of the 2D-PDMDSP; r

is the control signal; a , a , and b are positive real numbers.

Define the error signal as

According to equations (18) and (19), the dynamic error is

¨

Xm+ a1X˙m+ a0Xm− ¨X − α1X − α˙ 0X = br − β0u − β1f (21) According to equation (21), we have

¨e + a1˙e + a0e = br − β0u − β1f + (α1−a1) ˙X+ (α0−a0) X (22)

Define the error vector is X=f

e ˙egT, the state space of the equation (22) can be expressed as

˙

X = AX −

"

0

β0

#

u+" 0∆

#

where A=−0a 1

0−a1



; ∆= br − β1f + (α1−a1) ˙X+ (α0−a0) X; Z= −β0

0



u+ 0∆

All of the characteristic values of the matrix A are negative real parts According to the Lyapunov’s

first method, the system given by equation (23) is asymptotically stable Therefore, there are the

positive definite matrices P and Q, which make the following formula is established

Define the auxiliary control signal ˆe as

ˆe=f

0 1gPX=f

0 1g" p1p2

p2p3

# " e

˙e

#

= p2e + p3˙e (25)

The robust adaptive control law u is

u= φ0(ˆe, r) r+ φ1(ˆe, X) x+ φ2



ˆe, ˙ X ˙X + R (f ) (26) where φ0(ˆe, r), φ1(ˆe, X), and φ2



ˆe, ˙ X are the adjustable gain coefficients; R(f ) is the robust

compensation for the linearization error and the external disturbance

Theorem 1: use the robust adaptive control law given by equation (26) to control the system given

by equation (23) if the gain coefficients of the adaptive law and robust compensation are designed as

˙

˙

˙

R (fh)= −2β1|f |Msgn (ˆe) / β0 (30) where λ0, λ1, and λ2are the positive real numbers; |f |Misis the absolute maximum of f.

Then, for arbitrary α0, α1, f, and arbitrary initial conditions, when the disturbance is less than

2 |f |M, the error e is bounded and asymptotic convergence to zero.

Proof: define the Lyapunov function as

V (t)=1

2X

TPX+ 1 2λ0β0(b − β0φ0)2+ 1

2λ β (α0−a0−β0φ1)2+ 1

2λ β (α1−a1−β0φ2)2 (31)

V (t) is clearly positive definite.

1

2X

TPX

!0

=1 2

˙

XTPX+1

2X

TP ˙ X

=1

2(AX + Z)TPX+1

2X

TP (AX + Z)

=1 2



XTAT+ ZT

PX+1

2X

TP (AX + Z)

=1

2x

T

ATP + PA

X+1 2



ZTPX + XTPZ

= −1

2X

TQX+1 2



ZTPX=f

0 ∆ − β0ug" p1 p2

p2 p3

# " e

˙e

#

= ˆe (∆ − β0u)= XTPZ (33) According to equation (21), equation (33) can be rewritten as

ZTPX= ˆef

br − β1f + (α1−a1) ˙X+ (α0−a0) X − β0φ0r+ φ1x+ φ2X˙+ R g

= ˆer (b − β0φ0)+ ˆeX (α0−a0−β0φ1)+ ˆe ˙X (α1−a1−β0φ2)+ ˆe (−β1f − β0R) (34) According to equations (31), (32), and (34), the derivative of V (t) is

˙

V= −1

2X

TQX+ ˆe (∆ − β0u) −φ˙0

λ0(b − β0φ0) −φ˙1

λ1 (α0−a0−β0φ1) − φ˙2

λ2 (α1−a1−β0φ2)

= −1

2X

TQX+ ˆer (b − β0φ0)+ ˆeX (α0−a0−β0φ1)+ ˆe ˙X (α1−a1−β0φ2)

+ ˆe (−β1f − β0R) −φ˙0

λ0(b − β0φ0) − φ˙1

λ1 (α0−a0−β0φ1) − φ˙2

λ2(α1−a1−β0φ2)

= −1

2X

TQX+ ˆer − φ˙0

λ0

!

(b − β0φ0)+ ˆeX − φ˙1

λ1

! (α0−a0−β0φ1) + ˆe ˙ X − φ˙2

λ2

! (α1−a1−β0φ2)+ ˆe (−β1f − β0R) (35) According to equations (27) -(30), equation (35) can be rewritten as

˙

V= −1

2X

TQX+ ˆe −β1f − 2 β1|f |Msgn (ˆe)

(36)

When ˆe > 0, we have

−β1f − 2 |f |Msgn (ˆe)= −β1f − 2 β1|f |M≤0 (37)

When ˆe < 0, we have

−β1f − 2 |f |Msgn (ˆe)= −β1f + 2β1|f |M≥0 (38) According to equations (37) and (38), we have

ˆe − β1f − 2 β1|f |Msgn (ˆe)
≤ 0 (39) According to equations (36) and (39), we have

˙

V ≤ −1

2X

In other word, ˙V (t) is a negative definite matrix.

Therefore, according to the Lyapunov’s second method, when t ≥ 0, for arbitrary α0, α1, f, and arbitrary initial conditions, when the disturbance is less than 2 |f |M, the error e is bounded and

asymptotic convergence to zero

The block diagram of the RMRA control method including the linearization control based on the asymmetrical Bouc-Wen model proposed in SectionIIIand the ZVDS proposed in SectionIVis shown in figure11

By using the parameter identification... with the feedforward linearization control, the input-output curve of the PA-I is almost linear, and the PA-I can be used as a linear actuator

FIG Block diagram of the linearization... phase, they can cancel each other out, and thus the vibration is eliminated at time t2

The 2D-PDMDSP can be considered as a linear system with the linearization control for the PA-I According