Tracking control of piezoelectric actuator system using inverse hysteresis model

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1 International Journal of Applied Electromagnetics and Mechanics 33 (21) DOI /JAE IOS Press Tracking control of piezoelectric actuator system using inverse hysteresis model Yuanseng Chen, Jinhao Qiu, Hongli Ji and Kongjun Zhu The Aeronautic Key Laboratory for Smart Materials and Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 2116, China Abstract. The intrinsic hysteresis behavior of piezoelectric material limits the tracking control accuracy of the actuators. This paper describes a tracking control method for piezoelectric actuators based on the combination of feedforward and feedback loops. The hysteresis of piezoelectric actuators is linearized in feedforward loop with an inverse hysteresis model based on a neural network. The number of parameters in the neural network based inverse model is relatively small and the parameters are easy to identify. The feedback loop is used to reduce the accumulated error produced by the inverse hysteresis model. The experimental results show that the accuracy is significantly improved in tracking control by using the combination of feedback and feedforward control. Keywords: Hysteresis, piezoelectric actuators, tracking control 1. Introduction Due to their characteristics of high displacement resolution, high stiffness, and high frequency response, piezoelectric actuators have been widely used in high-precision positioning devices and tracking systems, such as scanning tunneling microscopy, and diamond turning machines. However, the piezoelectric actuators have the drawback of hysteretic behavior, which severely limits system precision. The mechanism of hysteretic behavior of piezoelectric actuators is very complex, and many physical models have been proposed for explanation of the mechanism. Many studies have reported to improve the hysteretic behavior of piezoelectric actuators. The control schemes can be broadly classified into three categories: 1) Electric charge control; 2) Feedback (closedloop) displacement control; 3) Feedforward control with a hysteresis model. In the first category, Newcomb and Flinn [1] used electric charge instead of applied voltage to drive piezoelectric actuators, because the relationship between the strain of piezoelectric actuators and applied electric charge exhibits less hysteretic behavior than the relationship between the strain and applied voltage. However, this approach requires special electric circuits to measure and amplify the charge, which increases the complexity of the control system. In the second category, various types of feedback control systems were developed. If hysteresis effect is not incorporated into the piezoelectric actuator control system, it will act as an unmodeled phase lag whose presence will cause overshot or even instability in the feedback control system if sufficient phase margin is not provided [2]. Therefore, feedback control methods are usually employed with feedforward control. In the third category, use of the feedforward control method Corresponding author. Tel.: ; qiu@nuaa.edu.cn /1/$ IOS Press and the authors. All rights reserved

2 1556 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model Fig. 1. Feedforward neural network. is the most popular way to improve the hysteretic behavior of the piezoelectric actuator [2 4]. Despite these models can approximately describe hysteresis nonlinearity, they have the following disadvantages: 1) Due to variations of material properties induced by the manufacturing process, actuators with the same specifications (composition and geometry) may have slight different properties, so that the parameter of a model is valid only for a specific actuator [5]. 2) The accuracy of a model is closely dependent on the total amount of stored parameters, and these parameters are difficult to identify. This paper presents a tracking approach for piezoelectric actuator, which is a combination of a feedforward loop and a feedback loop. In the feedforward loop, hysteresis nonlinearity is compensated by inverse neural network model. Feedback loop is used to reduce the static error and possible creep in the piezoelectric actuator. The performance of proposed controller is evaluated through experiments. 2. Modeling hysteresis of piezoelectric actuator 2.1. Neural network model Theoretically multilayer feedforward neural can approximate virtually any function of interest to any degree of accuracy [6]. However, neural networks can only approximate one-to-one and multi-to-one mapping and cannot directly approximate multi-value mapping such as hysteresis nonlinearity [6]. D. Makaveev [8] proposed an innovative mapping and used a feedforward neural network (FFNN) to model the magnetic hysteresis. In this paper, a feedforward neural network based on the mapping approach proposed by Makaveev is used in modeling and control of the hysteresis nonlinearity of piezoelectric actuators. Two important properties of the classic scalar hysteresis model are wiping-out property and congruency property [9]. Combining the two properties, the hysteresis model can be expressed mathematically as [8]: y (t) = y extr (t) + f [u extr (t),u(t),flag(t)], (1) where u(t) is the voltage applied on the piezoelectric actuator at time t, y(t) is the displacement of the actuator, u extr (t) is dominant extremum of the input voltage, which complies with the wiping-out property, y extr (t) is the output displacement at the latest extremum corresponding to u extr (t) and f

3 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model 1557 is a nonlinear function that can be approximated by FFNN. The variable FLAG(t), which is used to distinguish the virgin curve (no extremum values present in memory) from hysteresis branches, is not necessary for modeling piezoelectric actuators, because they have been poled before they are used. Therefore, FFNN with two inputs (Fig. 1) can describe the hysteresis behavior of piezoelectric actuator: or y (t) = y extr (t) + FFNN [u extr (t),u(t)] (2) y (t) = y (t) y extr (t) = FFNN [u extr (t),u(t)]. (3) 2.2. Error accumulation With suitable training set from measured data, the network parameters (weights wi,j k and bias bk i ) can be determined by Levenberg-Marquardt training algorithm. Although this neural network model operates more conveniently than classic hysteresis model, the model error will be accumulated in each hysteresis loop. For example, the piezoelectric actuator is driven by sinusoidal wave, that is u(t) =.5sin ( 2πt π 2 and the initial states are assumed to be ) +.5, (4) u extr () =,y extr () = y a (). (5) where y a () is the real displacement of the actuator. The neural network model can be expressed as { y (K) + FFNN [,u(t)], K < t K +.5 y (t) =,K =,1,2,3,, (6) y (K +.5) + FFNN [1,u(t)], K +.5 < t K + 1 where y(k) is the last local minimum and y (K+.5) is the last local maximum of output. When t >.5, y(k) and y (K+.5) can be calculated with history input and Eq. (6): y (K) = K FFNN (,1) + K FFNN (1,) + y(), y (K +.5) = (K + 1) FFNN (,1) + K FFNN (1,) + y(). (7) If the real displacement of the actuator is denoted by y a (t), the real outputs of the actuator at the minimum input and maximum input are y a (K) and y a (K+.5), respectively, which are constant for all K. The model output error is defined as e(t) = y (t) y a (t), (8) where y(t) is the output of the neural network model. According to the definition of the function FFNN, FFNN (,1) should be y a (K +.5) y a (K) and FFNN (1,) should be y a (K) y a (K+.5) if there is no model error. In a real neural network model, there are always model errors. The errors of the neural network for these special input conditions are defined as e 1 = FFNN (,1) [y a (K +.5) y a (K)], e 1 = FFNN (1,) [y a (K) y a (K +.5)]. (9)

4 1558 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model Inverse hysteresis Hysteresis of mocel piezoelectric actuator u ( t ) H -1 y ( t ) x ( t ) H Desired displacement Output displacement 1 Fig. 2. Inverse control principle. In general the error of the neural network is defined as e u = FFNN [,u(t)] [y a (t) y a (K)] for K < t K +.5, e 1u = FFNN [1,u(t)] [y a (t) y a (K +.5)] for K +.5 < t K + 1. (1) By substitution of Eqs (6), (7), (9)and (1) into Eq. (8) gives { eu + K(e e(t) = 1 + e 1 ), K < t K +.5 K =,1,2,3,. (11) e 1u + e 1 + K(e 1 + e 1 ) K +.5 < t K + 1, Obviously the model error increases with the cycles of actuation, K, if e 1 + e 1. Equation (9) shows that the condition e 1 + e 1 is equivalent to FFNN(,1) + FFNN(1,) =. In general, this condition is not satisfied. It means that the model error of the neural network accumulates as the number of actuation cycle increases if only the inverse model is used in hysteresis cancellation. Hence in the real-time control system a feedback controller is necessary to suppress the accumulation of modeling error. 3. Controller structure design 3.1. Feedforward tracking controller The feedforward controller for a piezoelectric actuator is based on compensating hysteresis nonlinearity using the inverse hysteresis model. Figure 2 shows the inverse control principle for piezoelectric actuator, where u(t) is the desired displacement, y(t) is the driving voltage, x(t) is the measured displacement, H is the hysteresis operator to describe the hysteretic behavior of the piezoelectric actuator and H 1 is the inverse hysteresis operator. Since the effects of H 1 and H cancel out each other, a unitary operator is obtained. That is, the output displacement is equal to the desired displacement. The inverse hysteresis is modeled by a FFNN: y (t) = y extr (t) + FFNN [u extr (t),u(t)] (12)

5 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model 1559 where u extr (t) is the dominant extremum of the desired displacement, which complies with wiping-out property, and y extr (t) is corresponding driving voltage at the relevant extremum. Since this controller needs to be implemented in discrete fashion, conversion from continuous time to discrete time is necessary. A discrete form of Eq. (12) is y (kt) = y extr (kt) + FFNN [u extr (kt),u(kt)] (13) or, equivalently y (kt) = y (kt) y extr (kt) = FFNN [u extr (kt),u(kt)] (14) where T denotes the sampling time interval. Figure 3 shows the feedforward tracking control algorithm which is composed of four steps: initialization, extrema detection, wiping-out and computing. In the initialization step, u(), y(), y extr (), u extr (), u b and y b are set to zero, num are set to 1, where u b and y b are arrays which store the dominant extrama of u(kt) and y(kt), variable num indicates how many dominant extrama stored in the arrays u b and y b. In the computing step, the voltage applied on the piezoelectric actuator is computed by FFNN. Tend is the time to end the tracking control Feedback tracking controller Due to the accumulated error of neural network model as explained in section 2.2, a feedback tracking controller is necessary to reduce the error. Figure 4 shows the block diagram of the tracking control system which is composed of a feedforward controller and feedback controller. In the feedforward loop, the driving voltage v(kt) is obtained from inverse hysteresis model. In the feedback loop, the desired displacement is compared with the measured displacement of piezoelectric actuator: e(kt) = ˆx(kT) x(kt) (15) where ˆx(kT)is desired displacement, x(kt) is measured displacement of the actuator. The error signal e(kt) is sent to a feedback controller to compute the difference of voltage extremum v extr (). The voltage applied on the actuator is v (kt) = v extr (kt) C g e(kt) + FFNN [ˆx extr (kt), ˆx(kT)] (16) where C g is coefficient of the feedback gain. 4. Experiments and results 4.1. Experimental setup To evaluate the quality of the new controller, a piezoelectric actuator system is used to test the performance. Figure 5 shows the structure of experimental system. A computer with dspace board (Model No. RTI 113) is used to generate desired position signal, acquire sensor data and implement the control procedure. A power amplifier is used to amplify the control signal before applied to the piezoelectric actuator. Because the displacement of piezoelectric actuator is very small, a lever mechanism is designed to increase resolution of the displacement. A laser displacement sensor is used to measure the motion of displacement amplifier.

6 156 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model 4.2. Open-loop tracking without control Fig. 3. Feedforward control algorithm. To get the training data for neural network and compare the tracking control performance, it is necessary to test the performance of an open-loop actuator system. The reference inputs are two kinds of decreasing sinusoidal wave, which amplitude reduced by 11% in every period. The open-loop response of the piezoelectric actuator system for the reference input is presented in Fig. 7. For first kind of reference input, the maximum tracking error is 39 µm, and the average of the absolute values of the error is 8.1 µm. For the second kind of reference input, the maximum tracking error is 19 µm, and the average of the absolute values of the error is 8.2 µm. Their obvious hysteresis loops between reference input and measured displacement indicate that the response of the open-loop system is poor.

7 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model 1561 Computer yyy y Fig. 4. A block diagram of feedforward and feedback controller. Laser Sensor dspace Board A/D D/A Power Amplifier Lever Mechanism Fig. 5. Experimental system. (a) (b) Fig. 6. Network training set: (a) network input; (b) target output.

8 1562 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model (a) (b) (c) (d) Fig. 7. Open-loop response of the piezoelectric actuator for two kinds of reference input. For reference input 1, (a) reference displacement and measured displacement, (b) tracking error, For reference input 2, (c) reference displacement and measured displacement, (d) tracking error Feedforward and feedback tracking control In order to use the neural network for inverse control, it must be trained using a suitable set of input and output pairs of the real system [8], which is derived from experimental data of section 4.2, as shown in Fig. 6. The target input is combined of measured displacement and its dominant extremum which comply with the wiping-out property, the target output is difference between the voltage applied on the piezoelectric actuator and its dominant extremum. The neural network with 8 neurons in input layer, 15 neurons in hidden layer and 1 neuron in output layer was trained for 5 epochs using the Levenberg-Marquardt training algorithm. After training the neural network, the same reference signal was used to test the performance of the feedforward controller that incorporates hysteresis modeling. The accumulated error is shown in Fig. 8, which indicates that the feedback loop is necessary to reduce error. Three sets of desired displacement signal were used to test the performance of the hybrid feedforward and feedback control strategy, two of which are the same as the reference signal in open-loop tracking without control. Figure 9 shows the tracking results. For the first set of desired displacement signal, the maximum tracking error is 8.8 µm, and the average of the absolute values of the error is 1. µm. For the second set of desired displacement signal, the maximum tracking error is 9.9 µm, and the average of the absolute values of the error is 3.4 µm. This result shows the tracking error caused by hysteresis nonlinearity is strongly reduced. Because the training set cannot cover all possible network input values, the feedforward controller exhibits low accuracy for the input values which are not included in the

9 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model Desired Measured Displacement (mm) (a) Error (mm) Fig. 8. Output response of the feedforward tracking control. (a) desired displacement and measured displacement, (b) tracking error. (b) 2 Reference Response Reference Response Displacement (mm) Displacement (mm) Reference Displacement (mm) Response (a) (c) (e) Error (mm) 4-4 Error (mm) -2-6 Error (mm) (b) (d) (f) Fig. 9. Output response of the proposed control. For set 1, (a) desired displacement and measured displacement, (b) tracking error; for set 2, (c) desired displacement and measured displacement, (d) tracking error; for set 3, (e) desired displacement and measured displacement, (f) tracking error.

10 1564 Y. Chen et al. / Tracking control of piezoelectric actuator system using inverse hysteresis model training set. For the third set of desired displacement signal, the maximum tracking error is 11.1 µm, and the average of the absolute values of the error is 2.2 µm. 5. Conclusion A tracking control scheme for piezoelectric actuator system which combines feedforward and feedback loop was proposed in this paper. In the feedforward controller, the neural network is used to model the inverse of the hysteresis and compute the control input. This control output is combined with the control input calculated from feedback loop. From the experimental results, the hysteresis model based on neural network is easier to be identified than classic Preisach model, and the proposed controller is effective for improving tracking accuracy. In real-time control, the neural network cannot perform accurately for all possible input value, and to reduce the disturbance will be discussed in the future. The extension of the technique to dynamic control of piezoelectric actuator will be discussed as well. Acknowledgments This research is supported by the Funds for Creative Scholars in Climbing Program of Jiangsu province (Grant No.BK292), and the Major Research plan of the National Natural Science Foundation of China (Grant No ). References [1] C. Newcomb, I. Flinn, Improving the Linearity of Piezoelectric Ceramic Actuators, Electron Lett 18(11) (1982), [2] G. Song, J. Zhao, X. Zhou and J.A.D. Abreu-García, Tracking Control of a Piezoceramic Actuator With Hysteresis Compensation Using Inverse Preisach Model, IEEE/ASME Transactions on Mechatronics 1(2) (25), [3] W.T. Ang, P.K. Khosla and C.N. Riviere, Feedforward Controller with Inverse Rate-Dependent Model for Piezoelectric Actuators in Trajectory-Tracking Applications, IEEE/ASME Transactions on Mechatronics 12(2) (27), [4] A. Badel, J. Qiu, G. Sebald and D. Guyomar, Self-sensing High Speed Controller for Piezoelectric Actuator, Journal of Intelligent Material Systems and Structure 19(3) (28), [5] J. Park, G. Washington and H. Yoon, A Hybrid Approach to Model Hysteretic Behavior of PZT Stack Actuator, Journal of Intelligent Material Systems and Structure 2(4) (29), [6] M.T. Hagan, H.B. Demuth and M. Beale, Neural Network Design, PWS Publishing Co. Boston, MA, USA, [7] C. Li and Y. Tan, A Neural Networks Model for Hysteresis Nonlinearity, Sensors and Actuators A 112 (24), [8] D. Makaveev, L. Dupré, M.D. Wulf and J. Melkebeek, Modeling of Quasistatic Magnetic Hysteresis with Feed-forward Neural Networks, Journal of Applied Physics 89(11) (21), [9] I.D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier Science Inc., New York, 23. [1] S. Bashash and N. Jalili, Intelligence Rules of Hysteresis in The Feedforward Trajectory Control of Piezoelectrically- Driven Nanostagers, J Micromech Microeng 17 (27), [11] R. Dong, Y. Tan, H. Chen and Y. Xie, A Neural Networks based Model for Rate-dependent Hysteresis for Piezoceramic Actuators, Sensors and Actuators A 143 (28), [12] D. Makaveev, L. Dupré, M.D. Wulf and J. Melkebeek, Dynamic Hysteresis Modelling Using Feed-forward Neural Networks, Journal of Magnetism and Magnetic Materials (23),

198 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL G. Song, Jinqiang Zhao, Xiaoqin Zhou, and J. Alexis De Abreu-García

198 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005 Tracking Control of a Piezoceramic Actuator With Hysteresis Compensation Using Inverse Preisach Model G. Song, Jinqiang Zhao, Xiaoqin