Robust Control of a Piezoelectric Stage under Thermal and External

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ed to linearize the actuator sliding mode based ation estimation is used to ternal load disturbances in of the proposed controller is promise as smart sensors and.
2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009

ThA09.3

Robust Control of a Piezoelectric Stage under Thermal and External Load Disturbances Disturbances Mohammad Motamedi, Seyed Mehdi Rezaei, Mohammad Zareinejad, Mohammad Zareinejad and Mozafar Saadat

Abstract— Piezoelectric actuators are widely used in micromanipulation tasks such as Atomic Atomic Force Microscopy and Cell Manipulation. However, the hysteresis nonlinearity and the creep reduce their fidelity and cause difficulties in the micromanipulation control procedure. Besides, variation of temperature and external loads could change the model mode parameters identified for the piezo actuator. In this paper, a novel feedforward-feedback feedforward feedback controller is proposed. The Prandtl Ishlinskii model is utilized to linearize the actuator Prandtl-Ishlinskii hysteresis in feedforward scheme and a sliding mode based impedance control control with perturbation estimation is used to cancel out the thermal and external load disturbances in feedback scheme. The efficiency of the proposed controller is verified by experiments.

I. INTRODUCTION

P

iezoceramics hold great great promise as smart sensors and actuators ctuators in a variety of applications. Micromanipulation, aerospace and bio-medical bio medical systems for improving performance and to augment stability are such examples. examples. It is well known that the piezoelectric actuator has many advantages such as: (1) there are no moving parts; (2) the actuators can produce large forces; (3) they have almost unlimited resolution; (4) the efficiency is high; (5) response is fast. The major limitation of piezoceramic actuators is their nonlinear hysteretic behavior that leads to performance performance degradation in precision precision positioning applications [1]. The maximum error due to hysteresis is found to be as much as 1010-15% 15% of the path covered if the actuators are run in an open-loop open loop fashion. This error affects the system performance. performan To deal with with the effect effect of hysteresis, feedforward and feedback techniques have been proposed. In the open-loop open loop technique (feedforward), a model with high precision is needed in order to model the hysteresis. The key idea of a feedforward controller is to cascade the the inverse of the hysteresis model with the actual hysteretic plant. In this manner, an identity mapping between the desired output and actuator response can be provided. Preisach [2]-[5], [2 [5], and Prandtl--Ishlinskii Ishlinskii (PI) [6] [ are the wellwell known feedforward models. models. Implementation complexity is the major setback of Preisach Preisach model. PI is less complex and its inverse inverse can be computed analytically. Identification of PI model is performed for a single loop. Therefore, Therefore in Manuscript received September 22, 2008. Mohammad Motamedi is with the Department of Mechanical Engineering, Amirkabir Amirkabir University of Technology, Tehran, Iran (corresponding author to provide phone: +989123331016; e-mail: e mail: [email protected]).

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feedforward scheme, scheme any deviation from the identified identi loop leads to hysteresis compensation error. In this study a modified PI model [7] is applied and its inverse is used to cancel out the hysteresis effect. The nonlinear piezoelectric actuator is linearized using feedforward rward inverse hysteresis. The linearized nearized uncertain model is used to design the controller [8]. To deal with the influence of parametric uncertainties, external disturbance effects and PI identification error, a perturbation term is considered in linearized model. mo For proper trajectory tracking, tracking, a sliding mode based impedance control with perturbation estimation is proposed. In order to evaluate the proposed approach, performance of the piezoelectric actuator in trajectory tracking under thermal and load disturbances is investigated. II. PIEZO STAGE AND HYSTERESIS MODELING A. Dynamic Modeling odeling for the Piezo Stage The piezo stage consists of a 1-DOF 1 DOF stage actuated by a piezo stack actuator. In many investigations, a second-order second linear dynamics has been utilized for describing the system dynamics. As shown in Fig. 1, this model combines massmass spring-damper damper ratio with a nonlinear hysteresis function appearing in the input excitation to the system.

Fig. 1. Piezoelectric actuator

The following equation defines the model:              

(1)

where   is the stage position.  , and  are stage mass, viscous coefficient and stiffness, respectively.   denotes the hysteretic   hysteretic relation relation between input voltage and excitation force. Piezoelectric actuators have very high stiffness, and consequently possess posses very high natural frequencies. In low-frequency low frequency operations, the effects of actuator damping and inertia could be safely

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neglected. Hence, the governing equation of motion is reduced to the following static hysteresis relation between the input voltage and actuator displacement:  





      

          

(2)

Equation (2) facilitates the identification of the hysteresis function    between the input voltage and the excitation force. This is performed by first identifying the hysteresis map between the input voltage and the actuator displacement,   . It is then, scaled up to  to obtain   .             

(3)

To consider the influence of parametric uncertainties, unmodeled dynamics, and identification error, a perturbation term  is added to the stage model. Thus the stage model (1), can be rewritten in the following form:                   

(4)

To consider the interaction with environment, the force  exerted by the environment is inserted into the model. Therefore, the dynamic model of piezo stage can be written as follows:                

where  is the control input,  is the actuator response, % is the control input threshold value or the magnitude of the backlash and ( is the sampling period. The initial consistency condition of (6) is given by: (7)

where " is usually but not necessarily initialized to zero. Multiplying the backlash operator  by a weight value *+ , the generalized backlash operator is obtained: (8)

  ,.- /0 , 12 #

/0 , 12 #  " , "" # … 4 , "5 # #6

(9)

(10)

With the weight vector ,.-  *+" … *+5 #, the threshold vector 0  %" … %5 #6 where 0  %" 7 8 7 %5 and the initial state vector 12  00 … 0' #( . The control input threshold values %5 are usually chosen to be of equal intervals between maximum and minimum of piezoelectric actuator displacement. 2) Modified PI Operator The PI operator inherited the symmetry property of the backlash operator about the center point of the loop formed by the operator. The fact that most real actuator hysteretic loops are not synonymic weakens the model accuracy of the PI operator. To overcome this restrictive property, a saturation operator is combined in series with the hysteresis operator. A saturation operator is a weighted superposition of linear-stop or one-sided dead zone operators. A dead zone operator is a non-convex, non-symmetrical and memory free nonlinear operator given by: $