Model reference adaptive control for a piezo-positioning system

Precision Engineering 34 (2010) 62–69

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Precision Engineering

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / p r e c i s i o n

Model reference adaptive control for a piezo-positioning system

Yung-Tien Liu

_{, Kuo-Ming Chang}

_{, Wen-Zen Li}

a_{Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, No. 1, University Rd., Yen-Chau 824, Kaohsiung, Taiwan, ROC} b_{Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaoshiung 807, Taiwan}

a r t i c l e i n f o

Article history:

Received 6 October 2006

Received in revised form 23 February 2009 Accepted 11 March 2009

Available online 27 June 2009

Keywords: Precision positioning Hyperstability Adaptive control Hysteresis Piezoelectric actuator

a b s t r a c t

Piezoelectric (PZT) actuators having the characteristic of infinitely small displacement resolution are popularly applied as actuators in precision positioning systems. Due to its nonlinear hysteresis effect, the tracking control accuracy of the precision positioning system is difficultly achieved. Hence, it is desirable to take hysteresis effect into consideration for improving the trajectory tracking performance. In this paper, a model reference adaptive control scheme based on hyperstability theory is developed for a moving stage system driven by a PZT actuator. It is worth emphasizing that the controller can be constructed without a nonlinear hysteresis dynamic equation to compensate the hysteresis effect. According to simulation results, the tracking error was only nanometer order. Through experimental examinations, the tracking performance was obtained as precision as ten nanometers order which is the resolution limitation of the measurement system. The effectiveness of the proposed adaptive control scheme was validated.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Along with the fast growing of semiconductor and precision-manufacturing industry, high-precision and high performance devices are becoming more necessary. Piezoelectric (PZT) actuators are popularly applied as actuators in precision positioning sys-tems due to its advantages of infinite displacement resolution, high speed, high bandwidth, high electrical–mechanical transformation efficiency, and little heat generation. In recent years, PZT actuators have been used in many applications that require precise position-ing. These application examples include optical fiber alignment, mask alignment, high-precision machining, scanning tunneling microscopy, hard disk drive, and diamond turning machines[1–4]. Since the materials of PZT actuators are generally ferroelectric, non-linear hysteresis behaviour in response to an applied electric field is always present. It was shown that the maximum tracking error caused by the nonlinear hysteresis phenomena can be as much as 10–15% of the travelling path if the PZT actuators are operated in the open voltage driving loop system[2,5]. To improve the linear-ity of PZT actuator, the approach using a capacitor to compensate the hysteresis and creep phenomena was verified as effective[6]. Although an electric charge control approach can reduce the hys-teresis effect[7,8], this approach requires the use of a specially designed charge drive amplifier and cause a reduction in the sen-sitivity of the displacement. Lately, much work has been done on

∗ Corresponding author. Tel.: +886 7 601 1000x2220; fax: +886 7 601 1066.

E-mail address:[email protected](Y.-T. Liu). 1_{JSPE member (No. 2970378).}

feedback or feedforward–feedback control strategies for hysteresis compensation. The feedback control techniques usually did not use a precise hysteresis model, but the feedforward–feedback control scheme is designed to utilize the developed hysteresis models such as Preisach model, Maxwell model, Bouc–Wen model, polynomial model, etc. Okazaki[9]used a notch filter and a state feedback con-troller with a state observer. In this control design, the unmodelled phase lag resulting from neglecting the hysteresis effect in the PZT actuator control system will cause instability in a closed-loop sys-tem if sufficient phase margin is not provided. Rasmusesen et al. [5]proposed a repetitive control for a piezo tool system. Since the hysteresis is not considered in the control scheme, the repetitive control with fixed control parameters may not work if there are changes in the magnitude of the reference signal or external distur-bances. Jung and Kim[10]proposed a feedforward model reference control method to improve the scanning accuracy of PZT actua-tors in a scanning tunneling microscope. Their hysteresis model has either a local memory or symmetrical behaviour that can-not exactly reflect the hysteresis behaviour. Ge and Jouaneh[11] proposed a control technique incorporating the inverse linearized Preisach model in a feedforward loop and a PID feedback con-troller to improve the tracking accuracy of a PZT actuator. Croft and Devasia[12]applied an inverse polynomial model in the feed-forward loop to cancel the hysteresis and designed a PD feedback controller to achieve the tracking task. Choi et al.[13]presented a PID control augmented with feedback linearization loop in which the feedback linearization loop used a plant model drawn from the Maxwell slip model. It indicated that the tracking performance can be further improved by adding repetitive controller when the PZT actuator is subject to the periodic reference input signal. Tsai 0141-6359/$ – see front matter © 2009 Elsevier Inc. All rights reserved.

Y.-T. Liu et al. / Precision Engineering 34 (2010) 62–69 63

Fig. 1. The experimental setup of precision moving stage.

and Chen [14] introduced an approximate model with variable gains and variable time-delay to represent the model of the PZT actuator with hysteresis phenomena. According to the proposed approximate model, a Smith predictor-based robust H_∞controller is developed to achieve high-precision tracking control of a piezoac-tuator. Abidi and ˇSabanovic[15]constructed a disturbance observer based on sliding-mode control framework to attenuate the effect of piezostage hysteresis and then to achieve high accuracy in the actu-ator trajectory tracking. Huang et al.[16]developed an equivalent model including the hysteresis friction force dynamics to describe the motion dynamics of the piezo-positioning stage and further derived a sliding-mode controller to improve system transient per-formance.

This paper presents a model reference adaptive control based on hyperstability theory[17]for compensating the hysteresis effect of the PZT actuator in the moving stage system to achieve the precise trajectory tracking objective. Employing the Bouc–Wen model[18] for the PZT hysteresis, some experimental results are given to val-idate the output-tracking performance of the proposed controller. In addition to numerical simulations, the proposed adaptive control algorithm was applied to an experimental moving stage driven by a PZT actuator and the experimental results indicate the trajectory tracking performance can be improved.

2. Piezo-positioning system

In this paper, a moving stage driven by a PZT actuator is con-structed as shown inFig. 1. One end of the PZT actuator is fixed to the wall and the other end is connected to the moving stage slid-ing on the horizontal surface. If the frictional force is very small compared to the generating force of the PZT actuator, the physical model of the moving stage system can be depicted inFig. 2. Apply-ing an input voltage to the PZT actuator, an elongation is produced and then results in a force Fhacting on the imaginary wall. There-fore, based on the Bouc–Wen model[18], the dynamical equation is represented in the following form:

m¨x1+ b ˙x1+ kx1= Fh= k(deu − h) (1)

Fig. 2. The schematic diagram of precision moving stage driven by a PZT actuator.

Fig. 3. The block diagram of the moving stage system.

Fig. 4. Random input voltage.

where m is the equivalent mass of the PZT actuator and the moving stage, b is the equivalent damping coefficient, k is the equivalent spring coefficient, u is the input voltage, which is applied to the PZT actuator to drive the stage, x1is the displacement of the stage, deis the effective piezoelectric coefficient of the PZT actuator, and h is a variable used for describing the hysteresis effect, respectively.

In this paper, the Bouc–Wen model is used to describe the hys-teresis behaviour in the piezoelectric actuator. For the Bouc–Wen model, when the material is uniform elastic, the state variable h forms the hysteresis nonlinear dynamics and is governed by the following equation.

˙h = ˛de˙u − ˇ| ˙u|h −  ˙u|h| (2)

where ˛, ˇ, and  are the parameters adjusting the shapes of the hysteresis loop. Therefore, the block diagram of the moving stage system can be illustrated inFig. 3.

The system parameters of the moving stage system can be iden-tified using MATLAB identification tool (ARX model) and Simulink software. Generating an 1 kHz unit voltage randomly shown inFig. 4 from MATLAB software to a voltage amplifier, this input voltage is amplified 10 times to be 0–50 V and then it is sent to the PZT actu-ator to drive the stage. The displacement response of the stage is measured using a capacitance-type gap sensor having a resolution of 10 nm and is shown inFig. 5. The system parameters are thus obtained through the system identification of ARX model and listed

64 Y.-T. Liu et al. / Precision Engineering 34 (2010) 62–69

Fig. 6. Simulation and experimental hysteresis loops.

below.

m = 0.28 kg, b = 1302.28 N s m−1, k = 53, 452 N/m, de= 0.1027 nm/V, ˛ = 0.5136, ˇ = 0.124

and  = −0.073.

In order to confirm the validity of system parameters, a sinu-soidal input with amplitude =2 V and frequency = 1 Hz is selected and amplified to actuate the PZT actuator. From the hysteresis loops shown inFig. 6, it can be seen that experimental hysteresis loops matches the ones of the computer simulation closely.

3. Model reference adaptive control

In this section, a model reference adaptive control will be devel-oped for compensating the hysteresis effect of the PZT actuator to achieve the precise output-tracking aim of the moving stage. Con-sider an uncertain system with nonlinear input described in the following form:

˙x = Ax(t) + BF(u(t)) + d(x, t) (3)

y = Cx(t) (4)

where x(t)_{∈ R}n_{is the system state vector, u(t)∈ R}m_{, m≤ n, is the} sys-tem control vector, y(t)∈ Rp_{is the system output vector, A, B, and C} are system parameter matrices of appropriate dimensions, F(u(t)) =



f1(u) . . . fm(u)



∈ Rm_{is a continuous nonlinear function}

vec-tor and F(0) = 0, and d(x,t)∈ Rn_{represents system unmodelled errors} and external disturbances. In this paper, referring to the definition of a sector bounded function in Ref.[19], a so-called sector-like bounded function is defined as follows:

Definition 1. [20]. A continuous function fi(u(t)) with fi(0) = 0 is said to belong to the sector-like bounded [ci1, ci2] by ui, if there exist two positive constants ci1and ci2such that ci1≤ (fi(u)/ui)≤ ci2 for ui /= 0 and u =



u1 u2 . . . um



.

Assumption 1. Nonlinear input functions fi(u(t)), i = 1, . . ., m, are sector-like bounded by ui, i = 1, . . ., m, respectively. It yields that there exist positive constants ci1, i = 1, . . ., m, and ci2, i = 1, . . ., m, such that the following conditions are satisfied.

ci1≤fi_u(u)

i ≤ ci2, i = 1, . . . , m.

Assumption 2. Uncertain system disturbance vector d(x,t) is norm

bounded. It means that there exists one positive time function ı(t) such that||d(x,t)|| ≤ ı(t).

Assumption 3. Input parameter matrix B has full rank, i.e.

Rank(B) = m.

FromAssumption 1, it straightly gives the following results: ci1u2i ≤ uifi(u) ≤ ci2u2i, i = 1, . . . , m.

Then, we have

c11u21+ c21u22+ · · · + cm1u2m≤ u1f1(u) + u2f2(u) + · · ·

+ umfm(u) ≤ c12u21+ c22u22+ · · · + cm2u2m.

It yields that from the above inequality

c1uTu ≤ uTF(u) ≤ c2uTu (5)

where c1and c2are two positive constants with c1= min{ci1|i = 1, . . .,

m} and c2= min{ci2|i = 1, . . ., m}. FromAssumption 1, it also yields that

c2

1uTu ≤ FT(u)F(u) ≤ c22uTu or c1||u|| ≤ ||F(u)|| ≤ c2||u|| (6)

The control objective is to find an appropriate control input u(t) such that the system output vector y(t) can asymptotically follow the desired output vector ym(t) and all the signals in the controlled system are bounded in the presence of system parameter variation, unmodelling error, and external disturbance uncertainties. Here, a stable reference system is given by the following dynamic equation.

˙xm(t) = Amxm(t) + Bmr(t) (7)

ym(t) = Cxm(t) (8)

where x_m(t)∈ Rn_{is the reference model state vector, r(t)∈ R}m_{is the} reference model input vector, which is piecewise continuous and bounded, y_m(t)_{∈ R}p_{is the reference model output vector, A}

mand

B_mare matrices of appropriate dimension.

To achieve the control objective of output-tracking, in this sec-tion we use the hyperstability theory to design a robust adaptive control scheme for the uncertain nonlinear system expressed in Eqs.(3) and (4). Define a state error vector as

e(t) = x(t) − xm(t). (9)

It yields that the dynamics of the state error vector can be repre-sented as

˙e = Ame + (A − Am)x + BF(u) + d − Bmr = Ame − B1ω (10)

where B1∈ Rn×nis a designed nonsingular constant matrix and

ω = −B−11 [(A − Am)x + F1(u) + d − Bmr] (11)

F1(u) = BF(u). (12)

We then define a linear combination of state error vector as R = He

where H∈ Rn×n_{is also a designed constant matrix.}

Hence, it follows that a linear time invariant system with output

E is given by the following dynamic equation.

˙e = Ame − B1ω (13)

E = He (14)

Lemma 1. [21]. Let G(s) be a matrix of rational function such that

G(∞) = 0 and G(s) has poles only in Re[s] < −,  > 0. Let (H, A, B) be

a minimal realization of G(s). Then, G(s) is strictly positive real if and only if there exist a symmetric, positive definite matrix P and a matrix L such that

AT_{P + PA = −LL}T_{− 2P = −Q (15)}

Y.-T. Liu et al. / Precision Engineering 34 (2010) 62–69 69

of measurement system and environmental noise. Through both numerical and experimental examinations, the proposed adap-tive control algorithm can be effecadap-tively applied to the positioning devices using PZT actuators to obtain ultraprecision tracking per-formance.

Acknowledgement

The authors would like to acknowledge the support from National Science Council, Taiwan, Republic of China for this work, under Grant NSC 93-2212-E-151-006.

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