stable adaptive fuzzy control with hysteresis observer for...

Intelligent Control and Automation, 2012, 3, 390-403 http://dx.doi.org/10.4236/ica.2012.34043 Published Online November 2012 (http://www.SciRP.org/journal/ica)

Stable Adaptive Fuzzy Control with Hysteresis Observer for Three-Axis Micro/Nano Motion Stages

Lih-Chang Lin, Bor-Yih Chang, Biing-Der Liaw Department of Mechanical Engineering, National Chung Hsing University, Taichung, Chinese Taipei

Email: [email protected]

Received July 31, 2012; revised August 31, 2012; accepted September 7, 2012

ABSTRACT

This paper considers the analytical dynamics with simplified Dahl hysteresis model for a three-axis piezoactuated micro/ nano flexure stage. An adaptive controller with nonlinear dynamic hysteresis observer is proposed using Lyapunov stability theory. In the controller, a fuzzy function approximator with parameters update law is included to compensate for the identification inaccuracy, model uncertainty, and flexure coupling effects. Simulation results are used to demon- strate the control performance. Keywords: Micro/Nano Stage; Adaptive Fuzzy Control; Hysteresis Observer; Fuzzy Function Approximator

1. Introduction

Recently, control of micro/nano stages considering the piezoactuator hysteresis effects has found great interests in the literature. Effective ultrafine-resolution trajectory tracking performance of stages is limited by the intrinsic hysteretic behavior of the piezoceramic material and the structural vibration of the devices [1].

Many efforts were trying to decrease the hysteresis effect of piezoactuators. Newcomb and Flinn [2] found that the relationship between the extension of a piezoceramic actuator and its applied electric charge has significantly less hysteresis nonlinearity than that between deforma- tion and applied voltage. Furutani et al. [3] proposed an induced charge feedback control for the piezoactuators. The approach needs measurement of the induced charge and a specially designed charge drive amplifier, and will cause an increase in the response time of the actuator.

In order to linearize the control system, many re- searches focused on the inverse feedforward compensation based on some inverse hysteresis model. Several models have been suggested for describing the complex hysteretic behavior, for example, the Preisach model in Ge and Jouaneh [4,5], Yu et al. [6], and Liu et al. [7], the generalized Preisach model in Ge and Jouaneh [8], the dynamic Preisach model in Yu et al. [9]; the generalized Maxwell elasto-slip model in Goldfarb and Celano- vic [10]; the variable time-relay hysteresis model in Tsai and Chen [11]; the Prandtl-Ishlinskii (PI) model (a sub- class of the Preisach model) in Ang. et al. [1] and Has- sani and Tjahjowidodo [12]; the Duhem model in Ste- panenko and Su [13]; the polynomial approximation

method in Croft and Devasia [14]; and the Jiles-Atherton model in Dupre et al. [15]. Ge and Jouaneh [5] proposed a PID feedback control using the classical Preisach model for the hysteresis. Song et al. [16] proposed a cascaded PD/lead-lag feedback controller based on a linear model for the piezoactuator with hysteresis being compensated via the feedforward cancellation using the inverse classical Preisach model. Recently, Maslan et al. [17] presented a discrete-time transfer function and its inverse for a highly nonlinear and hysteretic piezoelectric actuator, and traditional PID controller and PID with active force control were considered.

To mitigate the effects of the unknown hysteresis, Wang et al. [18] suggested a model reference control for linear systems with unknown input hysteresis using an inverse KP (Krasnosel’skii-Pokrovskii) hysteresis model [19]. Hwang et al. [20] proposed a neural-network nonlinear model for learning the hysteretic behavior of a piezoelectric actuator, and suggested a discrete-time variable-structure control for enhancing the nonlinear model-based feedforward control performance. Based on the learned nonlinear model of piezoelectric actuator systems in [20], Hwang and Jan [21] proposeed a controller including a nonlinear inverse control and a discrete neuro- adaptive sliding mode control using a recurrent neural network to compensate for the residue dynamic uncertainty. Wai and Su [22] presented a supervisory genetic algorithm (SGA) control system for a piezoelectric ceramic motor. The controller consists of a GA control to search an optimum control effort online via gradient de- scent training process and a supervisory control to stabi- lize the system states around a predefined bound region.

Copyright © 2012 SciRes. ICA

L.-C. LIN ET AL. 391

Recently, Ronkanen et al. [23] presented a two-input (velocity and voltage) one-output (current) feedforward backpropagation network to model the inverse nonlinear velocity-current relation of a piezoelectric actuator, and then introduced a feedforward charge control scheme.

Other analytical types of nonlinear differential hysteresis models include the simplified Dahl model used in Lyshevski [24], Sun and Chang [25], Sain et al. [26], and the Bouc-Wen model in Low and Guo [27], Chen et al. [28], and Gomis-Bellmunt et al. [29]. Chen et al. [28] proposed an H∞ almost disturbance decoupling robust control based on the Bouc-Wen hysteretic model. Shieh et al. [30] proposed an adaptive displacement control for a piezopositioning mechanism with the LuGre (hysteretic) friction model suggested by De Wit et al. [31]. Gu and Zhu [32] suggested a new mathematic model to describe the frequency-dependent and amplitude-dependent hysteresis in a piezoelectric actuator using a family of ellipses. These analytical hysteresis models will be much easier for precision positioning control design.

In this work, we consider the precision control of a three-axis piezoactuated micro/nano stage. An adaptive controller with simplified Dahl model-based hysteresis variables observer is designed using the Lyapunov stability theory. In the adaptive controller, a fuzzy function approximator with parameters update law is included to compensate for the identification inaccuracy, model uncertainty, and flexure coupling effects. Simulation results are used for illustrating the possible control performance.

2. Dynamic Model for a Three-Axis Micro/Nano Motion Stage

The dynamic model for a single-axis piezoactuated flexure stage with analytic simplified Dahl hysteresis model is as below [24]:

2 31 2mx k x k x k x k x 3x u fk u k f (1)

1

fx

f x x fk

(2)

where x is the output displacement of the flexure stage; is the mass of the flexure mover; m xk

,k kk

is the damping coefficient; 1 , 2 and 3 are the stiffness constants; u is the input voltage of the piezoelectric actuator; u is the input gain;

k

f is the hysteresis variable; f and govern the scale and the shape of the hysteresis loop.

k fxk

Consider a

Figure 1. Three-axis flexure stage.

2 31 2 3x u f Mx K x K x K x K x K u K f (3)

xyz three-axis flexure micro/nano stage (P-517.3CL, Physik Instrumente, PI) [33] driven by piezoelectric actuators shown in Figure 1. The hysteresis phenomena and the coupling effects among the three axes induced by the flexure structure, can be taken into account via the following complete matrix-vector model:

1fv f x K x f

T

(4)

x y zx

2 2 2 2 ;T

x y z

is the output displacements vector; where

x

3 3 3 3 ;T

x y z

x

diag ;x y z x

T

x y zD D D

diag , ,T

x y z x y zm m m u u u

is used to consider the coupling effects among the axes and the model uncertainty;

M u

1 1 1 1diag , diag ,x x y z x y zk k k k k k

K K

2 2 2 2 3 3 3 3diag , diag ,x y z x y zk k k k k k

K K

, diag ,T

x y z u ux uy uzf f f k k k

f K

, , ,diag , diag .f fx fy fz fv fx x fx y fx zk k k k k k

K K

For ease of numerical simulation and implementation, the system parameters in SI units could be scaled in terms of more suitable units: displacement in nm, mass in g, time in ms, and input voltage in mV. After scaling, the scaled models keep the same forms as Equations (3) and (4). The parameters of the stage are identified, based on input/output data pairs via genetic algorithms by Chang [34], and are given as follows:

30.2903 10 g; 249.27g ms;x y x ym m k k

2 21 1 4.579 10 g ms ;x yk k



3 2g nm ms ;2 2 1.6958 10x yk k

9 2 2g nm ms ;3 3 8.6767 10x yk k

3 2m mV ms ; 0.4716 10 g nux uyk k

2 2g ms ;3.6339 10k k fx fy

2 218.783 10 g ms ;fz zk k 3 2 1.8344 10 g ms ;

2 2ms ;2 1.9910 10 g nmzk

7 2 2g nm ms ;3 2.4296 10zk

3 2nm mV ms ;

4 110 nm

1 4 110 nm .

x x

0.4653 10 guzk

1 1, , 1.6242fx x fx yk k

, 4.9758fx zk

After defining the state vector as 1 , 2 x x

, the stage’s dynamic model can be written in the following vector state equations:

1 2

1 2 32 3

x x

x M K x K x K x K x2 2 1 1

1 1

12

x

u f

f

fv

M K u M K f

A Bu B f

f x k x f

(5)

where

1 2 31 1 2 3x

2A M K x K x K x K x

1 1, f u fB M K B M K

1x 1 2x x

1 1x ν

1 1 d e x x

x

1 1 1d d e x x ν x

3. Stable Fuzzy Approximator-Based Adaptive Control for Micro/Nano Stages

3.1. Control Design Using Backstepping Method

Based on the nonlinear dynamics model (5), this subsection considers the backstepping-based stable control law design for the three-axis flexure stage.

First consider the subsystem, . Let

(6)

where 1 is a virtual input. Define the tracking error signal as

ν

(7)

where d is the desired trajectory for the three-axis motion. Differentiating Equation (7), we have

(8)

Considering the Lyapunov function candidate

1 1 1 1

1

2TV e P e

3 3

(9)

where 1 P R

1 1 1 1 1 1 1T T

dV e P e e P ν x

1ν

1 1d

is symmetric and positive definite, and differentiating Equation (9), we have

(10)

Thus, we can choose the virtual input as

ν x κe (11)

with positive definite feedback gain matrix

1 2 3diag κ κ κκ

1 1 1 1 0TV

,

such that

e Pκe

1lim ( ) 0,t

t

(12)

and e

1 2

2 f

that is, the subsystem is asymptoti-

cally stable. Further, the actual whole nonlinear system is consid-

ered:

x x

x A Bu B (13)

f

After introducing new error signal

2 2 1, e x ν

1 1 2 1

22 2 1 2 1

d

f d

(14)

we can obtain

e x x e κe

e x ν A Bu B f x κe κ e (15)

Then by considering the Lyapunov function candidate as

1 2 2 2

1 1

2 2T T

sV V + e P e e Pe

1 2

TT T

(16)

where e e e , 1 2diag ,P P P 3 32, P R

is symmetric and positive definite, and taking the time derivative of Equation (16), we have

1 1 2 1

22 2 2 1

Ts

Tf d

V

e P e κe

e P A Bu B f x κe κ e

1 2 12 1 2 1 12

s

d f

u u

B x κe κ e A B f P P e

1 1 1 2 2 2T T T

sV e Pκe e P κe e PΚe

diag[ ]

(17)

Thus we can choose the nonlinear control law as follows:

(18)

and obtain

(19)

where Κ κ κ .

ΚIf further choose I 0 with , then we can



have

2 0sV e

TsV e P (20)

and lim 0t

t

e 0. Thus, the equilibrium point e

f

of

the closed-loop system is exponentially stable. The internal state variables can also be shown to

be bounded. Consider the Lyapunov function

1

2T

fV f f (21)

By choosing the class- K functions

1 2f f 1

2Tf f f f ,

then since

2f f f f f1 V , (22)

we know that fV is positive definite, decrescent, and radially unbounded [35]. Differentiating Equation (21) and substituting in the internal dynamics

12 ,fv

k x ff x (23)

we have

12 fv

T TfV f f f x k x f

2 dt tx x

0d xlim 0ft

V

0 0x

0f

(24)

Since limt

if the desired trajectory satisfies , then we can have Thus, f is bounded and the overall .

closed-loop system is stable. Let the output vector 1 , 2 , we can obtain

the system’s zero dynamics as follows: x

(25)

That is, the hysteresis variables will become constants when the flexure mover returns to the origin and remains there.

In order to further enhance the system’s active damping capability, we can introduce a nonlinear damping term

s

TV

e

0 B

where into the control law (18). That is, the control law can be modified as

TT

1 22 1

2 1 1

1 12 1 1 2 2

ˆ ˆ 2

ˆ ˆ ˆ ˆ

ˆ ˆ

T

sa s

u d

f x

E u

Vu u

e

K M x κe κ e

M K f K x K x

P P e M K P e

1 2 32 3

1

ˆ ˆ

ˆu

K x K x

K

1ˆ ,

(26)

where M ˆ ,fˆ ,u

ˆ , 1ˆ , 2

ˆ , 3ˆK K K K K and x K are the

nominal matrices for 1,M ,fK ,uK ,xK 1,K 2 ,K and

3K , respectively, obtained by substituting in the estimated parameters, and E represents the discrepancy due to the estimate error. Let t E u be the integral uncertainty, we can further design a fuzzy function approximator fs

1ˆ + K

to compensate for its effect. The modified control law can be written as follows:

1 22 1

1 2 32 1 1 2 3

1 12 1 1 2 2

ˆ

ˆ ˆ 2

ˆˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

u d

f x

u fs

u z,θ

K M x κe κ e

M K f K x K x K x K x

P P e M K P e

ˆ

(27)

f is the observed hysteresis vector for where f , z is the input vector of the controller, and is the parameters vector to be updated for the fuzzy compensator. Here

ˆ tθ

ˆ ˆdiag , , , diag , , ,x y z u ux uy uzm m m k k k M K

ˆ ˆdiag , diag ,f fx fy fz x x y zk k k k k k

K K

1 1 1 1 2 2 2 2ˆ ˆdiag , diag ,x y z x y zk k k k k k

K K

3 3 3 3ˆ diag

and

x y zk k k K

are used in Equation (27). The hysteresis observer and the fuzzy compensator design will be considered in the sequel.

3.2. Hysteresis Observer Design

Since the hysteresis variables are difficult to measure for feedback, a nonlinear observer can be suggested as:

1ˆ ˆ , 0, , ,i i i i oi fi oifii

f x x f k e k i x y zk

ˆ

(28)

where if are the estimated hysteresis variables, fi are the observer’s input variables to be defined later in the derivation of the stable control law and parameters update law, and are the input gains. Define estimate errors as

e

0oik

ˆi i if t f t f t ,

we have

1, 0, , ,i i i oi fi oi

fii

f x f k e k i x y zk

(29)

And Equation (29) can be written in the following vector form:



f o ff K f + K e

T

zt f t

(30) 1,1 1,2 1,3 1 ,T

e e e

where

x yt f t ff

is the estimate error vector, and

=diagf fxxx k yK fyy fzzk z k

,ox oy ozk k k

T

fy fzt e t

, , and

,

diago K

f fxt e t ee .

3.3. Fuzzy Function Approximators Design

This subsection will construct the fuzzy function approximators using T-S fuzzy systems to compensate for the modeling errors and coupling effects among the three axes. The tracking errors 1, 1, 1,x y z are chosen respectively as the input variable of the fuzzy approximator for each axis, and the compensating voltage of each axis is the output variable. In the universe of discourse of each input variable, five fuzzy sets are defined as in Figure 2. The rule base of the fuzzy approximator for the i-th ( i ) axis is considered as follows: Rule j: If is

e e e

1, 2,31,ie jA

Then

, , 1,1 , 1,2 , 1,3 ,i j i j i j i j i j

s

, 1,2, ,5e d j y a e b e c (31)

where jA are the fuzzy sets defined over the universe of discourse of each input variable 1,i , , stands for the

e 1, 2,and 3i x , y, and axis, respectively. z

Using singleton fuzzifier, product inference engine, and center average defuzzifier [36], the mapping of the fuzzy approximator for the i-th axis is

5

1,3i, j i, jc e + d

1,1 1,21

5

1

i, j i, j i, jj

i

i, jj

a e +b e +

y =

(32)

where , 1,i j j iA e is the degree of firing of the j-th rule’s antecedent. Let

Figure 2. Membership functions for each axis.

ψ

T

, , , , , ,i j i j i j i j i ja b c d

θ

,1 ,2 ,5 ,1 ,2 ,5μ μ μ TT T T T T T

i i i i i i iy

then

ψ ψ ψ θ θ θ

,1 ,2 ,5μ μ μTT T T

i i i i

(33)

Defining the regressor vector

φ ψ ψ ψ

,1 ,2 ,5

TT T Ti i i i

and the unknown parameter vector

θ θ θ θ

Ti iy

,

Equation (33) can be written as

i

ˆfs f

(34)

And the fuzzy approximators for the three axes can be written in the vector form as

1 2 3diag ,T T Tf

(35)

where

1 2 3ˆ ˆ ˆ ˆ .

TT T T

T

f fx fy fze e e

3.4. Derivation of Parameters Update Law and Stability of Overall System

In this subsection, the input signals

e

of the hysteresis observer, and the parameters update laws of the fuzzy function approximators will be selected in the stability consideration of the overall adaptive feedback control system for a three-axis piezoelectric flexure stage.

Consider the following Lyapunov function candidate,

1 1

1 11 1 1 2 2 2

1 1

2 21 1 1 1

2 2 2 2

T Ta s o

T T T To

V V

Γ

Γ

+ θ θ + f K f

e P e + e P e + θ θ + f K f

Γ ˆ .

(36)

where is symmetric and positive definite, θ θ θ

1 11 1 1 2 2 2

1 1 2 1

22 2 2 1

1 1

ˆ

ˆ+

T T T Ta o

T

Td

T To f o f

V

Γ

e P e e P e θ θ f K f

e P e κe

e P A Bu x κe κ e

θ Γ θ f K K f K e

Taking the time derivative, we have

(37)

After substituting in Equations (30) and (17), Equation


L.-C. LIN ET AL.

ICA

395

ˆ ˆ ,M I + εI

1 1ˆ ,m M M + ε I

1 2 3

Copyright © 2012 SciRes.

(37) becomes (38). Let

1 1u u

M K K

where

diag

ε

and

1 2 3 m m m m=diag

Iˆf f +

ˆ ˆ ˆ[ ]Tx y z

ε

are the error matrices, is the identity matrix. Since , where f

ˆ f f ff ,

we have

ˆ

f

f f f

ˆˆ ˆ

ˆ

f f f

K f K f K f f K f

K K f K f

1

1 12 2

1 1

ˆ

ˆ ˆ

Ta o f

u u

T T

,

and Equation (38) can be written as

1 1 1 2 2 2

2 2

12 2

2 2

ˆ ˆ ˆ

ˆˆ ˆ

ˆ ˆ

T T

T

Tf f

u f

V

e Pκe e P κe

f K K f

e P M K M

e P M K f e

e P M K Φ θ

K P e

f

θ Γ θ

(39)

where is defined as (40). Choosing the input vector of the hysteresis observer

fe1

2 2ˆ ˆT T

as:

f fe e P M K

12 2

ˆ ˆf f

e K M P e

11 1 1 2 2 2

1 12 2 2 2

1 12 2

ˆ ˆ ˆ ˆ

ˆ ˆˆ ˆ

T T Ta o f

Tu u

T Tu f

V

(41)

That is,

(42)

we can obtain

e Pκe e P κe f K K f

e P M K M K P e

e P M K Φ θ θ Γ θ

(43)

By further representing the uncertainty as:

12 2

ˆ ˆ ,Tu

e P M K

and substituting

ˆf f f f Φ θ Φ θ + θ Φ θ +Φ θ

11 1 1 2 2 2

1 12 2 2 2

1 12

12 2

ˆ ˆ ˆ ˆ

ˆˆ ˆ

ˆ ˆ

T T Ta o f

Tu u

T Tu f

Tu f

V

Γ

2

e Pκe e P κe f K K f

e P M K M K P e

e P M K Φ θ θ θ

e P M K Φ θ

in Equation (43), we have

(44)

Thus, we can choose the parameters adaptation law of the fuzzy approximators as:

1 02 2

ˆ ˆˆ ˆT

Tu

Γ

fθ e P M K Φ θ θ (45)

If further choose κ I , η I f

, and assume the approximation error fs Φ θ ω θ be bounded, i.e., ω W , then we can obtain

11 1 1 2 2 2

21 1

2 2 2 2 2 2

1

0 0

ˆ ˆ ˆ ˆ

ˆ

22

2

T T Ta o f

T Tu u

T Ts o f

T

V

W

WV

4

0

e P e e P e f K K f

e P M K P e e P M K

θ θ

f K K f θ θ

θ θ θ θ

1 2 32 1 1 2 3

1 1 2 2 3 12 1 2 1 1 2 3 2 1 1

1 2 1 12 2 2 1

ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆˆ ˆ

a f x

u u d f x

T T Tu fs d o f f

1

K f K x K x K x K x

K M x κe κ e M K f K x K x K x K x P P e

M K P e x + κe κ e θ Γ θ f K K f + f e

(46)

1 1 2 1 2 2+

T TV

e P e κe e P M

M K

(38)

2 32 1 1 1 2 2 3 3

2 3 2 12 1 1 2 3 2 1 2 1 1

1 2 3 1 1 12 1 1 2 3 2 2

ˆ ˆ ˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

f f x x

m f x d

f x u u u f

K f K K x K K x K K x K K x

K x K x K x K x ε x 2κe κ e εP P e

f K x K x K x K x ε M K M K P e ε M K Φ θ

2 2ˆ ˆ

T

1e P M K

ε K f

(40)

ˆˆ ˆ+εM K


Letting θ 2d

0 0T θ θ

class-

2

Wd

4

θ θ (47)

and defining K functions:

1 1

2e sV Te e Pe ,

and 1f f f

1To fK K f ,

Equation (46) can be rewritten as

1 12a e f

V e f

2T d

θ θ (48)

Hence, when 11 2e d e

or

or 11f

d f 0aV

, , and

, ,

and thus the overall adaptive control system is boundedly stable.

4. Results and Discussion

In this section computer simulation will be used to illustrate the performance of the proposed adaptive fuzzy control with hysteresis observer for a three-axis flexure stage. Triangular uncertainties for the x, y, and z axes ( x y z ) shown in Figure 3 are selected in the simulation. The desired trajectories for the x, y, and z axes are selected as follows (t in ms):

D D D

0 1000 2000 3000 4000 5000 6000 7000 8000t (ms)

8000

6000

0

-2000

-4000

-6000

-8000

4000

2000

ms2 )

Dx (

nm/

(a)

0 1000 2000 3000 4000 5000 6000 70t (ms)

00 8000

8000

6000

4000

2000

0

-2000

-4000

-6000

-8000

Dy (

nm/m

s2 )

(b)

0 1000 2000 3000 4000 5000 6000 700t (ms)

0 8000

6000

4000

2000

0

-2000

-4000

-6000

Dz (

nm/m

s2 )

(c)

Figure 3. Triangular uncertainties for the x, y, and z axes. (a) Dx, (b) Dy, (c) Dz.



π 2 nm5000 5000sin 2π 1000

d dx t y t

t

π 2 nm

3000 3000sin 2π 1000dz t t

Controller parameters are selected as follows:

1 diagP

,P I

15 15 50 ,

30.5 ,

2 3 κ I

30.4 , Iη 1 2 3diag , , ,Γ Γ Γ Γ

7 4 5 5

7 4

5 5 7

0 10 10 10

10

10 10

5 4 5

7 5

4 5 7

10 10

10

10 10

5 5 4

7 5

5 4 7

10 10

10

10 10

3ˆ, 0.1 and 0 0, , , .o i i x y z K I θ

, , and

4 5 5diag 10 10 10 1 1Γ

7 4 5 5

5 5 7 4

10 10 10 10 10

10 10 10 10 10

5 4 5 72 diag 10 10 10 10 10 Γ

7 5 4 5

4 5 7 5

10 10 10 10 10

10 10 10 10 10

5 5 4 73 diag 10 10 10 10 10 Γ

7 5 4 4

5 4 7 5

10 10 10 10 10

10 10 10 10 10

The simulation results are shown in Figure 4. From Fig- ures 4(a)-(c), we know that the tracking performances are very good. The tracking errors of x- and y-axes are within –2.5 nm - 2.2 nm, and the tracking error of z-axis is within ±2 nm. From Figures 4(d)-(f), the hysteresis- variable estimate errors of x- and y-axes are within ±0.5 nm, and the estimate error of z-axis is within ±1 nm. The control voltages x y z are shown in Figure 4(g), and the fuzzy compensation voltages ,fs x , ,fs y

u u u , and

,fs z are shown in Figure 4(h). And the parameters update processes of the function approximators for x-, y-, and z-axes are shown in Figures 4(i)-(k), respectively. The parameters of the first and fifth rules are not updated since the tracking errors are small and they are nearly not fired. Although the persistent exciting of the system signals of this considered simulation case are not sufficient enough to let the other parameters converge to constants, the adaptive control system can guarantee the tracking control performance to be still very good.

5. Conclusion

In this work, a stable adaptive control law with nonlinear dynamic hysteresis observer for a three-axis flexure stage

0

4

3

1

0

-1

-2

-3

-4

2

e 1,x (

nm/m

s2 )

1000 2000 3000 4000 5000 6000 7000 8000t (ms)

(a)

0 1000 2000 3000 4000 5000 6000 7000 8000t (ms)

4

3

2

1

0

-1

-2

-3

-4

e 1,y (

nm/m

s2 )

(b)



0 1000 2000 3000 4000 5000 6000 700t (ms)

0 8000

4

3

2

1

0

-1

-2

-3

-4

e 1,z (

nm/m

s2 )

(c)

0 1000 2000 3000 4000 5000 6000 700t (ms)

0 8000

0.6

0.4

0.2

0

-0.2

-0.4

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0 1000 2000 3000 4000 5000 6000 70t (ms)

00 8000

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m)

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t (ms) t (ms) t (ms) 5000 10000t (ms)

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t (ms) t (ms) t (ms) 5000 10000t (ms)

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0 5000 10000 0 5000 10000 0 5000 10000 0 t (ms) t (ms) t (ms)

5000 10000t (ms)

0 5000 10000 0 5000 10000 0 5000 10000 0 t (ms) t (ms) t (ms)

5000 10000t (ms)

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5000 10000t (ms)

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-1

(k)

Figure 4. Control results. (a) x-axis, (b) y-axis, (c) z-axis tracking errors; Hysteresis es erro ) timate rs, (d xf y; (e) f ; (f) zf ;

ages , yu and zu ,fs , y and ; Param process

(i) x-axis; (j) y-axis; (k) z-axis. is proposed. Fuzzy function approximators are included in the control law to compensate for the identification inaccuracy, model uncertainty, and flexure coupling effect. The stability of the overall closed-loop system is guaranteed using the Lyapunov theory. Simulation results are shown to illustrate the effectiveness of the suggested control approach. In the future study, actual implementation can be considered for the development of a precision stage for testing the control performance.

REFERENCES [1] W. T. Ang, P. K. Khosla and C. N. Riviere, “Feed-

forward Controller with Inverse Rate-Dependent Model for Piezoelectric Actuators in Trajectory-Tracking Appli- cations,” IEEE/ASME Transactions on Mechatronics, Vol. 12, No. 2, 2007, pp. 134-142. doi:10.1109/TMECH.2007.892824

(g) Control volt xu ; (h) Compensating voltages x , fs ,fs z eters adaptation

[2] C. Newcomb and I. Flinn, “Improving the Linearity of Piezoelectric Ceramic Actuators,” Electronics Letters, Vol. 18, No. 11, 1982, pp. 442-444. doi:10.1049/el:19820301

[3] K. Furutani, M. Urushibata and N. Mohri, “Displacement Control of Piezoelectric Element by Feedback of Induced Charge,” Nanotechnology, Vol. 9, No. 2, 1998, pp. 93-98. doi:10.1088/0957-4484/9/2/009

[4] P. Ge and M. Jouaneh, “Modeling Hysteresis in Piezo- ceramic Actuators,” Precision Engineering, Vol. 17, No. 3, 1995, pp. 211-221. doi:10.1016/0141-6359(95)00002-U

[5] P. Ge and M. Jouaneh, “Tracking Control of a Piezocer- amic Actuator,” IEEE Transactions on Control Systems Technology, Vol. 4, No. 3, 1996, pp. 209-216. doi:10.1109/87.491195

[6] Y. Yu, N. Naganathan and R. V. Dukkipati, “Preisach Modeling of Hysteresis for Piezoceramic Actuator Sys- tem,” Mechanism and Machine Theory, Vol. 37, No. 1, 2002, pp. 49-59. doi:10.1016/S0094-114X(01)00065-9

[7] L. Liu, K. K. Tan, A. S. Putra and T. H. Lee, “Com- pensation of Hysteresis in Piezoelectric Actuator with Iterative Learning Control,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Sun- tec Convention and Exhibition Center, Singapore City, 14-17 July 2009, pp. 1300-1305.

[8] P. Ge and M. Jouaneh, “Generalized Preisach Model for Hysteresis Nonlinearity of Piezoceramic Actuators,” Pre-


cision Engineering, Vol. 20, No. 2, 1997, pp. 99-111. doi:10.1016/S0141-6359(97)00014-7

[9] Y. Yu, Z. Xiao, N. Naganathan and R. V Dukkipati, “Dy- namic Preisach Modeling of Hysteresis for the Piezo- ceramic Actuator System,” Mechanism and Machine Theory, Vol. 37, No. 1, 2002, pp. 75-89. doi:10.1016/S0094-114X(01)00060-X

[10] M. Goldfarb and N. Celanovic, “Modeling Piezoelectric Stack Actuators for Control of Micromanipulation,” IEEE Control Systems Magazine, Vol. 17, No. 3, 1997, pp. 69- 79. doi:10.1109/37.588158

[11] M.-S. Tsai and J.-S. Chen, “Robust Tracking Control of a Piezoactuator Using a New Approximate Hysteresis Mo- del,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 125, No. 1, 2003, pp. 96-102. doi:10.1115/1.1540114

[12] V. Hassani and T. Tjahjowidodo, “Integrated Rate and Inertial Dependent Prandtl-Ishlinskii Model for Piezo- electric Actuator,” IEEE 2nd International Conference on Instrumentation Control and Automation, Bandung, 15- 17 November 2011, pp. 35-40.

[13] Y. Stepanenko and C.-Y. Su, “Intelligent Control of Pie- zoelectric Actuators,” Proceedings of IEEE Conference on Decision and Control, Tampa, 16-18 December 1998, pp. 4234-4239.

[14] D. Croft and S. Devasia, “Hysteresis and Vibration Compensation for Piezoactuators,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 5, 1998, pp. 710- 717.

[15] L. Dupre, R. van Keer and J. A. A. Melkebeek, “Identi-

of Applie ic

fication of the Relation between the Material Parameters in the Preisach Model and in the Jiles-Atherton Hysteresis Model,” Journal d Phys s, Vol. 85, No. 8, 1999, pp. 4376-4378. doi:10.1063/1.369789

16] G. Song, J. Zhao, X. Zho[ u and J. A. De Abreu-García, “Tracking Control of a Piezoceramic Actuator with Hys-teresis Compensation Using Inverse Preisach Model,” IEEE/ASME Transactions on Mechatronics, Vol. 10, No. 2, 2005, pp. 198-209. doi:10.1109/TMECH.2005.844708

[17] M. N. Maslan, M. Mailah and I. Z. M. Darus, “Identi- fication and Control of a Piezoelectric Bender Actuator,” IEEE 3rd International Conference on Intelligent Systems Modeling and Simulation, Kota Kinabalu, 8-10 February 2012, pp. 461-466. doi:10.1109/ISMS.2012.100

[18] Y. Wang, C. Y. Su and H. Hong, “Model Referencetrol Including Adaptive Inverse Hysteresis for Sy

Con-stems

with Unknown Input Hysteresis,” Proceedings of International king, Sensing and

IEEE Conference on Networ

Control, London, 15-17 April 2007, pp. 70-75. doi:10.1109/ICNSC.2007.372935

[19] M. A. Krasnosel’skii and A. V. Pokrovskii, “Systems with Hysteresis,” Springer-Verlag, Berlin, 1983.

[20] C. L. Hwang, C. Jan and Y. HUsing Intelligent Variable-Structu

. Chen, “Piezomechanicsre Control,” IEEE

Transactions on Industrial Electronics, Vol. 48, No. 1, 2001, pp. 47-59. doi:10.1109/41.904550

[21] C. L. Hwang and C. Jan, “A ReinNeuro-Adaptive Control f

forcement Dor Unknown Piezoelectric Ac-

tuator Systems with Dominant Hysteresis,” IEEE Tran- sactions on Neural Networks, Vol. 14, No. 1, 2003, pp. 66-78. doi:10.1109/TNN.2002.806610

[22] R. J. Wai and K. H. Su, “Supervisory Control for Linear Piezoelectric Ceramic Motor Drive Using Genetic Algorithm,” IEEE Transactions on Industrial Electronics, Vol. 53, No. 2, 2006, pp. 657-673. doi:10.1109/TIE.2006.870661

[23] P. Ronkanen, P. Kallio, M. Vilkko and H. N. Koivo, “Displacement Control of Piezoelectric Actuators Using Current and Voltage,” IEEE/ASME Transactions on Me- chatronics, Vol. 16, No. 1, 2011, pp. 160-166. doi:10.1109/TMECH.2009.2037914

[24] S. E. Lyshevski, “MEMS and NEMS: Systems, Device, and Structures,” CRC Press, New York, 2002, pp. 260- 262.

[25] X. Sun and T. Chang, “Control of Hysteresis in a Mono- lithic Nanoactuator,” Proceedings of American Control Conference, Vol. 3, Arlington, 25-27 June 2001, pp. 2261-2266.

[26] P. M. Sain, M. K. Sain and B. F. Spencer, “Models for Hysteresis and Application to Structural Control,” Pro- ceedings of American Control Conference, Vol. 1, Albu- querque, 4-6 June 1997, pp. 16-20.

[27] T. S. Low and W. Guo, “Modeling of a Three-Layer Piezoelectric Bimorph Beam with Hysteresis,” Journal of Microelectromechanical Systems, Vol. 4, No. 4, 1995, pp. 230-237. doi:10.1109/84.475550

[28] B. M. Chen, T. H. Lee, C.-C. Hang, Y. Guo nd S. Weerasooriya, “An H∞ Almost Disturbance Decoupling

rp

Systems Technology, Vdoi:10.1109/87.748143

iscrete

a

Robust Controller Design for a Piezoceramic Bimo h Actuator with Hysteresis,” IEEE Transactions on Control

ol. 7, No. 2, 1999, pp. 160-174.

[29] O. Gomis-Bellmunt, F. Ikhouane, D. Montesinos-Miracle, S. Galceran-Arellano and J. Rull-Duran, “Control of a Piezoelectric Hysteretic Actuator,” 13th European Con- ference on Power Electronics and Applications, Barce- lona, 8-10 September 2009, pp. 1-6.

[30] H. J. Shieh, F. J. Lin, P. K. Huang and L. T. Teng, “Adaptive Displacement Control with Hysteresis Model- ing for Piezoactuated Positioning Mechanism,” IEEE Transactions on Industrial Electronics, Vol. 53, No. 3, 2006, pp. 905-914. doi:10.1109/TIE.2006.874264

[31] C. C. De Wit, H. Olsson, K.J. Åström and P. Lischinsky, “A New Model for Control of Systems with Friction,” IEEE Transactions on Automatic Control, Vol. 40, No. 3, 1995, pp. 419-425. doi:10.1109/9.376053

[32] G. Y. Gu and L. Zhu, “Modeling Piezoelectric Actuator Hysteresis with a Family of Ellipses,” IEEE/ASME Inter- national Conference on Advanced Intelligent Mecha- tronics, Montréal, 6-9 July 2010, pp. 878-883.

[33] Physik Instrumente (PI), “Piezo Tutorial: Nanoposition- ing with Piezoelectrics.” http://www.pi.ws

[34] B. Y. Chang, “Stable Adaptive Control for a Three-Axis Nanopositioner: Implementation Using ALTERA DSP Development Board,” Master Thesis, Department of Mechanical Engineering, National Chung Hsing Uni-


http://dx.doi.org/10.1109/TMECH.2007.892824





http://dx.doi.org/10.1049/el:19820301

http://dx.doi.org/10.1049/el:19820301

http://dx.doi.org/10.1049/el:19820301

http://dx.doi.org/10.1109/87.491195

http://dx.doi.org/10.1109/87.491195

http://dx.doi.org/10.1109/87.491195

http://dx.doi.org/10.1016/S0094-114X(01)00065-9

http://dx.doi.org/10.1016/S0094-114X(01)00065-9

http://dx.doi.org/10.1016/S0094-114X(01)00065-9

L.-C. LIN ET AL.


403

ñez and K. M. Pa-

versity, Chung Hsing, 2005.

[35] J. T. Spooner, M. Maggiore, R. Ordó

niqu

ssino, “Stable Adaptive Control and Estimation for Nonli- near Systems: Neural and Fuzzy Approximator Tech-

es,” Wiley, New York, 2002. doi:10.1002/0471221139

[36] L.-X. Wang, “A Course in Fuzzy Systems and Control,” Prentice-Hall, Upper Saddle River, 1997.

http://dx.doi.org/10.1109/TNN.2002.806610

http://dx.doi.org/10.1109/TNN.2002.806610

stable adaptive fuzzy control with hysteresis observer for...

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