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

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1 198 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 Zhou, and J. Alexis De Abreu-García Abstract This paper presents the classical Preisach hysteresis modeling and tracking control of a curved pre-stressed piezoceramic patch actuator system with severe hysteresis. The actuator is also flexible with very small inherent damping. It has potential applications in active antennas. A series of tests are conducted to study the hysteresis properties of the piezoceramic actuator system. The numerical expressions of the classical Preisach model for different input variations are presented. The classical Preisach model is applied to simulate the static hysteresis behavior of the system. Higher order hysteresis reversal curves predicted by the classical Preisach model are verified experimentally. The good agreement found between the measured and predicted curves showed that the classical Preisach model is an effective mean for modeling the hysteresis of the piezoceramic actuator system. Subsequently, the inverse classical Preisach model is established and applied to cancel the hysteresis the piezoceramic actuator system for the real-time microposition tracking control. In order to improve the control accuracy and to increase damping of the actuator system, a cascaded PD/lead-lag feedback controller is designed with consideration of the dynamics of the actuator. In the experiments, two cases are considered, control with major loop hysteresis compensation, and control with minor loop hysteresis compensation. Experimental results show that RMS tracking errors are reduced by 50% to 70% if the hysteresis compensation is added in the feedforward path in both cases. Therefore, hysteresis compensation with the feedback controller greatly improves the tracking control accuracy of the piezoceramic actuator. Index Terms Hysteresis nonlinearity, piezoceramic, Preisach model, tracking control. I. INTRODUCTION PIEZOCERAMIC actuators have advantages of solid state actuation, high precision, and fast responses, and have found applications in many areas such as vibration controls [28], [29], shape controls [30], [31], and machine tool controls [21]. Like many other materials, piezoceramics possess Manuscript received January 29, 2004; revised November 10, The work of G. Song was supported by the National Science Foundation via CAREER Grant and by the National Aeronautics and Space Administration via a cooperative grant. G. Song is with Department of Mechanical Engineering, University of Houston, Houston, TX USA ( gsong@uh.edu). J. Zhao was with the Department of Electrical and Computer Engineering, University of Akron, Akron, OH USA. He is now with LHP Software Company, Columbus, IN USA. X. Zhou was with Department of Mechanical Engineering, University of Akron, Akron, OH USA. She is now with Cummins Engine Company, Columbus, IN USA. J. De Abreu-Garcia is with the Department of Electrical and Computer Engineering, University of Akron, Akron, OH USA. Digital Object Identifier /TMECH hysteresis, a form of nonlinearity containing memory of history. This nonlinearity considerably degrades a system s performance, especially in cases which require precision positioning, such as atomic force microscopes [17], [23], [24] and micro-manipulators [19]. Recently piezoceramics have found new applications in an active aperture antenna that is able to vary the direction and shape of its radiation pattern and to alter the shape of the reflector [25]. In such applications, curved pre-stressed piezoceramic patch actuators, which have the advantages of achieving larger displacement, longer life cycle, and greater flexibility than conventional piezoelectric actuators, are used [1], [26]. However, this type of curved pre-stressed piezoceramic patch actuator is flexible with litter inherent damping, and has a more severe hysteresis than a stack type piezo-actuator and thus, the hysteresis has to be studied and compensated for to ensure accurate control. Various methods have been developed to model hysteresis in piezoceramic actuators, such as the Ishlinskii hysteresis (IM) model [14], [16], the generalized Maxwell resistive capacitor-based lumped-parameter model [12], the variable time relay hysteresis model [20], the Jiles Atherton model hysteresis [15], and the Preisach model. The Preisach model is a phenomenological hysteresis model that offers mathematical generality, and although originated in magnetics, this model is applicable in many disciplines. Applications of the Preisach model to simulate the hysteresis of piezoceramics have been reported in [2] [10]. Hughes and Wen [2] applied the Preisach model for the hysteresis in piezoceramic and shape memory alloy materials. Ge and Jouaneh [3], [4] adapted the Preisach model to describe the nonlinear hysteretic behavior of piezoceramic actuators, and presented a modified generalized Preisach model that was used in a linearizing control scheme. Robert et al. [5], [6] derived new expressions for piezoelectric nonlinear behavior departing from the classical linear dependence and put forward a four-parameter distribution function that was experimentally refined using bias stress variations. Yu et al. [7] proposed a new dynamic Preisach model by introducing the dependence of the Preisach function on the input variation rate. Yu et al. [8] also presented a modified geometric interpretation and numerical implementation method for the Preisach model especially for the hysteresis modeling of piezoceramic actuator system. Ben Mrad and Hu extended the Preisach model to a recursive form which is more suitable for real-time applications [9]. Hu and Ben Mrad further studied the applicability of the classical Preisach model to piezoceramic actuator and found that this model remains accurate in applications where the load /$ IEEE

2 SONG et al.: TRACKING CONTROL OF PIEZOCERAMIC ACTUATOR 199 fuctuation is relatively small and the range of frequencies of the voltage excitation is limited [10]. Control of piezo-actuators with consideration of hysteresis also receives much attention in the literature. Most methods use the hysteresis model [3], [4], [19], [20], [22] or the inverse model [17], in addition to the feedback control portion, in their control design. Fuzzy control [26], sliding-mode-based robust control [27], control [23], [24], among others, have been used to deal with hysteresis in piezo-actuators without modeling hysteresis. Model-based control, though more computationally involved, has the advantage of canceling or partially canceling the hysteresis through feedforward terms. With the cancellation of the hysteresis, the dynamics of the piezo-actuator can be now considered linear and linear control design can be applied. This is important for performance-based design such as tracking control of the curved piezoceramic patch actuator, which requires specific damping increase to avoid exciting flexible modes. Few studies report real-time tracking control with hysteresis compensation, especially with minor loop compensation, of the curved pre-stressed piezoceramic patch actuator, which has more severe hysteresis than stack type piezo-actuators and is flexible with very small damping. This type of piezoceramic actuator has potential applications in active antenna, where high precision in shape control of the reflector subjected to thermal deformation is the top priority while high control bandwidth is appreciated but not specifically required [18]. This paper presents the classical Preisach modeling of the hysteresis and tracking control of the piezoceramic actuator with severe hysteresis. First the numerical expressions of the classical Preisach model are presented in details for different input variations. A series of tests are conducted to study the hysteresis properties of the piezoceramic actuator system. It is concluded that the wiping-out property holds for the piezoceramic actuator system, while the congruency property is not completely satisfied. For input signals at a frequency much less than the natural frequency of the system, the hysteresis behavior of the piezoceramic actuator system can be regarded as static. The classical Preisach model is then applied to simulate the static hysteresis behavior of the piezoceramic actuator system. To implement this model, a set of first-order hysteresis reversal curves is measured with the driving voltage starting from zero. Higher order hysteresis reversal curves predicted by the classical Preisach model are verified experimentally. The good agreement found between the measured and predicted curves shows that the classical Preisach model is an effective mean for modeling the hysteresis of the piezoceramic actuator system even though the congruency property does not completely hold for the system. Subsequently, the inverse classical Preisach model is established and applied to the piezoceramic actuator system to cancel its hysteresis for real-time microposition tracking control, in addition to the feedback controller. The Preisach model used here is not only for modeling, but also for control purposes. A cascaded PD/lead-lag feedback controller is designed based on a linear model of the piezo-actuator, assuming its hysteresis is compensated by the feedforward cancellation using the inverse classical Preisach model. Linear control design enables Fig. 1. Triangle T. the achievement of specific damping increase of the piezo-actuator to avoid exciting its lightly damped flexible modes, while ensuring high control accuracy. Tracking control experiments with and without hysteresis compensation are conducted and the results are compared. In the real-time micro-position tracking control, two cases are considered respectively, one is the cascaded PD/lead-lag feedback controller with major loop hysteresis compensator, and the other is the cascaded PD/lead-lag feedback controller with minor loop hysteresis compensation. Experimental results show that root mean square tracking errors are reduced from 50% to 70% if the hysteresis compensator is added in the feed forward path in both cases. Therefore, the tracking control accuracy with hysteresis compensation is greatly improved compared to that without hysteresis compensation. II. CLASSICAL PREISACH HYSTERESIS MODEL The Preisach hysteresis model is based on some hypotheses concerning the physical mechanisms of magnetization [10]. Though first regarded as a physical model of hysteresis, the Preisach model is in fact a phenomenological model that has mathematical generality. And thus it is applicable to many disciplines. Mathematically, the classical Preisach model can be written as where is the system output, is a weighting function, is the hysteresis operator having an output of or, and and correspond to up and down switching values of the input, respectively. The output of is set to or as the input exceeds the switching value or drops below the switching value. These differ from those values of 0 or used in [2] and [3]. The product of the weighting function and the operator is integrated over the triangle T, which corresponds to, as shown in Fig. 1, where and are determined by the system physical properties. From [1], the numerical expressions of the classical Preisach model, for situations in which the last variation of the input is decreasing or increasing, can be written, respectively, as for for (1) (2)

3 200 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005 Fig. 2. Rectangular and triangular cells. and (3) shown at the bottom of the page, where refers to the output when the input monotonically increases to from, and refers to the output when the input monotonically decreases to right after a monotonic increasing to from. If the experimental data and (, ) are known, the output can be predicted using (2) or (3). The output may be found without knowledge of and. When or do not belong to the measured data, they can be approximated by interpolation. In the plane, if a point (, ) belongs to a rectangular cell formed by (, ), (, ), (, ), and (, ), or a triangular cell formed by (, ), (, ), and (, ), for outputs at which points are experimentally measured already (Fig. 2), can be determined by III. EXPERIMENTAL RESULTS AND SIMULATION OF PIEZOCERAMIC ACTUATOR HYSTERESIS A. Experimental Setup The piezoceramic actuator studied in this paper is model TH8-R Thunder actuator, which has a weight of 2.1 g and a maximum applied voltage of 480 V. This is a curved, pre-stressed piezoceramic patch bonded on a stainless steel sheet. It has the advantages of achieving larger displacement, longer life cycle, and greater flexibility than conventional piezoelectric actuators. This type of actuator is a potential candidate for adaptive antennas. However it exhibits severe hysteresis and is flexible. Control design requirements include avoiding exciting its flexible mode, in addition to ensuring precision tracking. A dspace system (Model no. RTI1104) is used for digital data acquisition and real-time control. A power amplifier with (4) (5) Fig. 3. Piezoceramic actuator and the position sensor. a gain of 20 is used to drive the piezoceramic actuator. A noncontacting displacement sensor is used to measure the tip displacement of the piezoceramic actuator. This sensor is model kd2300-2s by Kaman, and has a measuring range of 2.5 mm and a resolution of mm. Fig. 3 shows the piezoceramic actuator and the sensor. The block diagram of the entire experimental system set-up is shown in Fig. 4. B. Experimental Hysteresis Loops and the Congruency and Wiping Out Properties of the Piezoceramic Actuator System The wiping-out and congruency properties are necessary and sufficient conditions for a hysteresis nonlinearity to be represented by the classical Preisach model [1]. To demonstrate these hysteresis properties, first a series of first-order reversal curves is tested for the piezoceramic actuator system of Fig. 4. The applied sinusoidal voltage has a frequency of 0.01 Hz, which is much less than the natural frequency of the system. Thus, these curves can be regarded as static hysteresis loops and therefore the classical Preisach model can be used. From Fig. 5, it can be seen that the ascending curves almost coincide, giving rise to a major ascending curve. It can also be observed that monotonically increases with respect to and. On the other hand, as the frequency of the input signals increases toward the natural frequency of the system, the hysteresis nonlinearity becomes more and more severe and the classical Preisach model no longer applies. For example, Fig. 6 is a set of hysteresis loops for 1-Hz input signals. The largest displacements do not appear at the highest input signals, but somewhere on the first-order reversal curve. Clearly, the classical Preisach model cannot model this kind of hysteresis phenomenon. In this paper, unless otherwise stated, 0.01 Hz input signals will be used. To investigate the wiping-out property of the piezoceramic actuator system, experiments using the input signals shown in Figs. 7 and 8, are performed. The corresponding displacement for for for (3)

4 SONG et al.: TRACKING CONTROL OF PIEZOCERAMIC ACTUATOR 201 Fig. 4. System block diagram. Fig. 7. A 0.01-Hz input signals. Fig. 5. Hysteresis loops for 0.01-Hz input signals. Fig. 8. Delayed 0.01-Hz input signals. Fig. 6. Hysteresis loops for 1-Hz input signals. outputs are compared in Figs. 9 and 10, respectively. From Fig. 9, it can be seen that the outputs overlap in the s time frame, indicating in the first experiment that the previous input extrema of 150 and 100 V do not have any influence on the output once the input reaches another extremum of 180 V. In other words, the previous extrema are wiped out. In Fig. 10, the displacement voltage curves overlap for input variations from 150 to 180 V, and then to 120 V. Clearly, the wiping-out property holds for the piezoceramic actuator system. To investigate the congruency property of the piezoceramic actuator system, the input signal is designed as shown in Fig. 11 (input extrema: 0, 120, 80, 120, 80, 180, 80, 120, 80, 120, 80 V). The corresponding displacement outputs are given in Fig. 12. The minor loops formed for the same reversal input values (80, 120, 80) are clearly not exactly congruent, however, the difference is not significant. Thus, the congruency property does not completely hold for the piezoceramic actuator system considered here. This may, to some extent, affect the simulation accuracy of the hysteresis of the piezoceramic actuator system using the classical Preisach model. However, it will be seen later that the accuracy of the classical Preisach model is still acceptable.

5 202 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005 Fig. 9. Displacement comparison 0.01-Hz input signals. Fig. 12. Output displacements congruency property. Fig. 10. Displacement comparison delayed 0.01-Hz input signals. Fig. 13. Predicted and experimental displacements. Fig. 11. Congruency property for 0.01-Hz input signals. C. Comparison of the Simulated and Experimental Hysteresis Behavior of the Piezoceramic Actuator System In this section, the classical Preisach model is numerically implemented to predict the displacement outputs of the piezoceramic actuator system. Based on experimental data (Fig. 5), the values of and (, ) are determined. The output is then predicted using (2) or (3). For a sinusoidal input signal, varying from 0 to 150 V, the hysteresis loop predicted by the classical Preisach model and the experimental loop are compared in Fig. 13. It can be seen that they match each other well. In fact, this output loop can be predicted by curve-fitting methods using the measured set of curves. As to the hysteresis loops with higher order reversal curves that cannot be predicted by curve-fitting methods, they can be predicted using the classical Preisach model based on the curves shown in Fig. 5. For example, the displacement outputs of the piezoceramic actuator system to the input signal shown in Fig. 14 are simulated using the classical Preisach model. The predicted hysteresis loop is verified experimentally and the results are compared in Fig. 15. The close match between the simulated curve and the experimental curve shows that the classical Preisach model works well for simulating higher order hysteresis curves of the piezoceramic actuator system for low-frequency input signals.

6 SONG et al.: TRACKING CONTROL OF PIEZOCERAMIC ACTUATOR 203 Applying (6) and (7), the input ( ) for a desired output can be written as (8) For a fixed,define the inverse of as Then, it follows that (9) Fig Hz input signals. (10) Applying (8) to (9), the input ( ) for a desired output can be written as (11) This numerical inverse Preisach model is converted into an -function in MATLAB using C language for real-time control purposes. Fig. 15. Displacements comparison. IV. TRACKING CONTROL OF A PIEZOCERAMIC ACTUATOR SYSTEM WITH HYSTERESIS COMPENSATION A. Inverse Preisach Model and its Implementation From Fig. 5, it can be seen that monotonically increases with respect to and for low-frequency input signals. Thus the inverse Preisach model can be derived from (2) and (3) as follows. For a fixed,define the inverse of as Moreover (6) B. Controller Design for the Piezoceramic Actuator System For controller design purposes, the piezoceramic actuator system is identified using total least squares. The identified model is then validated by comparing the step response of the identified model and the piezoceramic actuator system step response. Equation (12) shows the identified continuous time domain open loop transfer function of the piezoceramic actuator system (12) From the dominant poles of, it is found that the damping ratio of the piezoceramic actuator system is and the natural frequency is 328 rad/s (52 Hz). Since the damping ratio of the piezoceramic actuator system is very small, it is very difficult to achieve good tracking control performance. Therefore, it might be a desirable control goal to increase the damping ratio to 0.5 to avoid the excitation of the flexible modes of the actuator and reduce the steady state error to a unit step input to less than Based on the identified model, the controller is designed using the root locus approach to yield (7) (13) It can be found that the controller is composed of a PD cascaded with a lead-lag compensator. While the cascaded

7 204 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005 Fig. 16. Closed-loop system block diagram. PD/lead-lag compensator can be implemented using Simulink. the control output can be calculated using a numerical method such as the fourth-order Runge Kutta. Then, dspace can be used to compile the control block diagram and generate the C code that is downloaded to the hardware for real-time control purposes. The tracking control system is designed by incorporating the inverse Preisach model as a feedforward controller, and the cascaded PD/lead-lag compensator as a feedback controller (Fig. 16). Please note in Fig. 16 that a saturation operation is applied to the control output signal, and this is to limit the driving voltage range for the piezoceramic actuator for safety concerns. The numerical inverse Preisach model is converted to an -function using C language in Matlab, and then used as the feedforward loop in Simulink for real-time control. C. Tracking Results and Discussion For real-time microposition tracking control, two cases are considered, one is a cascaded PD/lead-lag feedback controller with major loop hysteresis compensation, the other is a cascaded PD/lead-lag feedback controller with minor loop hysteresis compensation. 1) Tracking Control With Feedforward Major Loop Hysteresis Compensation: To find the effectiveness of the hysteresis compensator, three different experiments are performed in tracking a 0.01-Hz, 0.1-mm sinusoid signal. Experiment 1: Tracking control with feedforward hysteresis compensator only. The feedforward voltage derived from the numerical inverse Preisach model is shown in Fig. 17(a). The control action in this case is the same as the feedforward voltage. Fig. 17(b) shows the reference trajectory and the actual tracking trajectory. The reference trajectory is a 0.01 Hz sinusoidal signal with amplitude of 0.1 mm. The maximum error is about 8.8 m, as shown in Fig. 17(c), indicating that the static property of the feedforward tracking controller is accurate. The foregoing clearly shows that the numerical inverse Preisach model is effective, albeit somewhat undesirable dynamic behavior. Introducing a lead-lag feedback controller should improve this behavior. Experiment 2: Tracking control with cascaded PD/lead-lag feedback controller only. In this experiment, the feedforward control output is zero. Fig. 18(a) shows the control output signal from the cascaded PD/lead-lag feedback controller. Fig. 18(b) shows the reference trajectory, a 0.01 Hz sinusoidal signal with amplitude of 0.1 mm, and the actual tracking trajectory. The maximum error is about Fig. 17. Feedforward tracking control. 6.5 m as shown in Fig. 18(c). It can be seen that the hysteresis nonlinearity adversely affects the piezoceramic actuator system performance. Experiment 3: Tracking control using cascaded PD/lead-lag feedback controller with feedforward compensator.

8 SONG et al.: TRACKING CONTROL OF PIEZOCERAMIC ACTUATOR 205 Fig. 18. Tracking control with cascaded PD/lead-lag controller only. Fig. 19. Tracking control with both feedforward and cascaded PD/lead-lag controller. The feedforward compensator output derived from the numerical inverse Preisach model is shown in Fig. 17(a), while Fig. 19(a) shows the total control signal. Fig. 19(b) shows the 0.01-Hz sinusoidal signal with amplitude of 0.1-mm reference trajectory and the actual tracking trajectory. Fig. 19(c) shows that the maximum error is about 2.5 m. Clearly, the combined cascaded PD/lead-lag feedback controller with feedforward compensator considerably reduces the maximum error. No flexible modes of the flexible actuator are excited. Table I gives a comparison of the root mean square tracking error in the above three experiments, clearly indicating that the tracking error is reduced by at least 50% in Experiment 3. 2) Tracking Control With a Feedforward Minor Loop Hysteresis Compensator: In the tracking control of the piezoceramic actuator system with minor loop hysteresis compensation, the desired trajectory [dashed line, Fig. 20(b)], is a 0.01-Hz TABLE I COMPARISON OF ROOT MEAN SQUARE TRACKING ERROR MAJOR LOOP signal which has input extrema of 0, 0.1, 0.02, 0.08, and 0.04 mm. This reference trajectory includes minor loop hysteresis. In this section, three experiments will be conducted to track the desired trajectory. Experiment (1): Tracking control with feedforward hysteresis compensator only. The feedforward voltage derived from the numerical inverse Preisach model is shown in Fig. 20(a). The control action in this

9 206 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005 Fig. 20. Tracking control with feedforward controller minor loop. case is the same as the feedforward voltage. Fig. 20(b) shows the reference trajectory and the actual tracking trajectory. The maximum error is about 8.8 m as shown in Fig. 20(c), which indicates that the static feedforward tracking control is accurate. The foregoing clearly shows that the use of the numerical inverse Preisach model is effective, although the system still exhibits some undesirable dynamic behavior. Experiment (2): Tracking control with cascaded PD/lead-lag feedback controller only. In this experiment, the feedforward control output is zero. Fig. 21(a) is the control output signal from the cascaded PD/lead-lag feedback controller. Fig. 21(b) presents the actual tracking trajectory compared with the reference, while the error is shown in Fig. 21(c). The maximum tracking error is about 6.5 m. From Fig. 21(c) it can be seen that the error includes hysteresis nonlinearity and that the conventional cascaded PD/lead-lag feedback controller cannot compensate Fig. 21. loop. Tracking control with cascaded PD/lead-lag compensator minor for it. Since the hysteresis nonlinearity adversely affects the performance, the cascaded PD/lead-lag compensator will be combined with the hysteresis compensator in the next experiment. Experiment (3): Tracking control using a cascaded PD/lead-lag feedback controller and a feedforward compensator. Fig. 22 shows the tracking control with both feedforward and cascaded PD/lead-lag controller. The feedforward compensator output derived from the numerical inverse Preisach model is shown in Fig. 22(a), while Fig. 22(b) shows the total control signal. Fig. 22(c) presents the reference and the actual trajectories, which almost overlap with each other. Note that the nonlinear effect has been cancelled in this case. Fig. 22(d)

10 SONG et al.: TRACKING CONTROL OF PIEZOCERAMIC ACTUATOR 207 TABLE II COMPARISON OF ROOT MEAN SQUARE TRACKING ERRORS MINOR LOOP PD/lead-lag feedback controller with the feedforward hysteresis compensator considerably reduces the maximum error. Table II shows the comparison of the root mean square tracking errors in the three experiments of tracking control with minor loop hysteresis compensator. Notice that the tracking error has been reduced by at least 70% in Experiment 3. 3) Discussion: Combining the cascaded PD/lead-lag feedback controller with the hysteresis compensator in the feedforward path, either for major loop or minor loop tracking control, substantially reduces the tracking error. From Tables I and II, it is seen that incorporating the hysteresis compensator in the feedforward path reduces the RMS tracking errors by 50% to 70%, as seen in Experiment 3. Also from Figs. 19(b) and 22(c), it can be seen that the tracking path almost overlaps the reference trajectory and that the nonlinear effect is cancelled. Therefore, the inverse numerical Preisach model is effective. Fig. 23 shows that tracking results are also satisfactory when the trajectory includes the wiping out property. Fig. 24 gives the Static hysteresis characteristic curves of the piezoceramic actuator system when the reference trajectory is applied to the system. It once again confirms the wiping-out phenomena of the system. Further investigation shows that this controller works well for small frequency changes of the reference signal. This is consistent with the findings by Hu and Ben Mrad [10]. However, if the frequency of the reference signal increases considerably, for example from 0.01 Hz to 0.1 Hz, the numerical inverse Preisach model would no longer be effective for compensating the nonlinear hysteresis of the piezoceramic actuator system. This is because the numerical inverse Preisach model is based on a fixed low frequency and cannot predict the hysteresis behavior at a different frequency. V. CONCLUSION Fig. 22. Tracking control with feedforward and cascaded PD/lead-lag controller minor loop. shows that the maximum error is about 2.5 m, which is 2.5% of the moving range, indicating that the combined cascaded This paper presents Preisach modeling of hysteresis and precision tracking control of a piezoceramic actuator system using inverse Preisach model and a feedback controller. The actuator is a curved, pre-stressed piezoceramic patch bonded on a stainless steel sheet. The actuator is flexible with very low damping. A series of tests are conducted to study the hysteresis properties of the piezoceramic actuator system. The classical Preisach model is then applied to simulate the static hysteresis behavior of this system. To implement this model, a set of first-order hysteretic reversal curves is measured. Higher order reversal curves are predicted based on experimental data and experimentally verified. The good agreement between the measured and predicted curves shows that the classical Preisach model is effective for modeling the hysteresis of the piezoceramic actuator

11 208 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 10, NO. 2, APRIL 2005 Fig. 24. system. Static hysteresis characteristic curves of the piezoceramic actuator the real-time microposition tracking control of the piezoceramic actuator system with both major and minor loop hysteresis compensations. The cascaded PD/lead-lag feedback controller considers the flexible dynamics of the actuator and increases the system s damping and accuracy through feedback control. Experimental results show that the root mean square tracking errors, in the case of hysteresis compensation, are reduced by 50% to 70%, compared in the case without hysteresis compensation. This demonstrates that control accuracy with hysteresis compensation is greatly improved compared to that without hysteresis compensation. Future work will involve designing other type controllers, such as loop shaping, for the curved pre-stressed piezo-actuator used in this paper and conducting comparative studies. ACKNOWLEDGMENT Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsors. Fig. 23. Tracking control with feedforward and cascaded PD/lead-lag controller minor loop with wiping out property. system. The inverse classical Preisach model along with a cascaded PD/lead-lag feedback controller is thereafter applied to REFERENCES [1] X. Zhou, J. Zhao, G. Song, and J. De Abreu-Garcia, Preisach modeling of hysteresis and tracking control of a Thunder actuator system, in Proc. SPIE, vol. 5049, 2003, pp [2] D. Hughes and J. T. Wen, Preisach modeling of piezoceramic and shape memory alloy hysteresis, Smart Mater. Struct., vol. 6, pp , [3] P. Ge and M. Jouaneh, Modeling hysteresis in piezoceramic actuators, Prec. Eng., vol. 17, pp , [4], Generalized Preisach model for hysteresis nonlinearity of piezoceramic actuators, Prec. Eng., vol. 20, pp , [5] G. Robert, D. Damjanovic, and N. Setter, Preisach modeling of piezoelectric nonlinearity in ferroelectric ceramics, J. Appl. Phys., vol. 89, no. 9, pp , [6], Preisach distribution function approach to piezoelectric nonlinearity and hysteresis, J. Appl. Phys., vol. 90, no. 5, pp , [7] Y. Yu, Z. Xiao, N. Naganathan, and R. V. Dukkipati, Dynamic Preisach modeling of hysteresis for the piezoceramic actuator system, Mech. Mach. Theory, vol. 37, pp , [8] Y. Yu, N. Naganathan, and R. V. Dukkipati, Preisach modeling of hysteresis for piezoceramic actuator system, Mech. Mach. Theory, vol. 37, pp , 2002.

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Park, and W. Hwang, Active shape control of a double-plate structure using piezoceramics and SMA wires, Smart Mater. Struct., vol. 10, pp , G. Song received the B.S. degree from Zhejing University, Hangzhou, China, in 1989, and the M.S. and Ph.D. degrees in mechanical engineering from Columbia University, New York, in 1991, and 1995, respectivley. He is an Associate Professor of Mechanical Engineering at the University of Houston, Houston, TX, He has research interests in smart materials and structures, structural vibration control, and advance control methods. He has developed two new courses in smart materials and published more than 100 journal and conference papers. Dr. Song is a co-inventor of a U.S. patent and a National Science Foundation CAREER Award recipient of Jinqiang Zhao received the Bachelor s degree in automation from Huazhong University of Science and Technology, Wuhan, China, and the Master s degree in electrical and computer engineering from the University of Akron, Akron, OH, in 1997, and 2003, respectively. As a graduate student, his research focused on designing control algorithms for the smart structures, and implementing algorithms using embedded systems. Currently, he works for the LHP Software Company, Columbus, IN, where he designs control algorithms for emission control for diesel engines and implements them using the electronical control module (ECM). Xiaoqin Zhou received the B.E. degrees in internal combustion engine and mechanical engineering, and marine engineering from Wuhan University of Technology, Wuhan, China, and the M.S. degree in mechanical engineering from the University of Akron, Akron, OH, in 1996, 1999, and 2003, respectively. Her research in the University of Akron focused on shape control and precision control using smart materials. She is currently a Structural Analysis Engineer with the Cummins Engine Company, Columbus, IN. J. Alexis De Abreu-Garcia (M 86) received the B.S. and Ph.D. degrees in electrical engineering from Queen s University at Kingston, Kingston, ON, Canada, in 1982 and 1986, respectively. From 1985 to 1986, he was a Visiting Scientist at Imperial College of Science and Technology, London, U.K. In 1987, he joined the faculty at The University of Akron, Akron, OH, where he is currently a Professor and Chair of the Electrical and Computer Engineering Department. From 1995 to 1996, he was a Controls Specialist in the Computer, Control and Electronic Technology Department at The Goodyear Tire & Rubber Company. His research interests include control system analysis, design, and algorithms; order reduction of multidimensional large-scale linear, nonlinear, descriptor, and distributed parameter systems; classical and modern robust control techniques; system simulation; fuzzy/neuro-fuzzy logic control system design, modeling and control of heart pumps, and smart materials. Prof. Abreu-Garcia served as Associate and Track Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS for over 14 years.