Novel multirate control strategy for piezoelectric actuators

Transcription

1 Novel multirate control strategy for piezoelectric actuators M Zareinejad 1 *, S M Rezaei 2, H H Najafabadi 2, S S Ghidary 3, A Abdullah 2, and M Saadat 4 1 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran 2 Department of Mechanical Engineering and New Technology Research Center, Amirkabir University of Technology, Tehran, Iran 3 Department of Computer Engineering, Amirkabir University of Technology, Tehran, Iran 4 Department of Mechanical and Manufacturing Engineering, University of Birmingham, Birmingham, UK The manuscript was received on 21 October 2008 and was accepted after revision for publication on 25 March DOI: / JSCE Abstract: In this article, a novel control method is proposed for feedforward compensation of hysteresis non-linearity in various frequency ranges. By integrating a multirate hysteresis compensator controller with PID feedback control, a combined controller is developed and experimentally validated for a piezoelectric micro-positioning system. Piezoelectric materials show non-linear hysteresis behaviour when they experience an electrical field. A fundamental study of a piezoelectric actuator (PEA) shows that the hysteresis effect deteriorates the tracking performance of the PEA. This paper presents a non-linear model which quantifies the hysteresis non-linearity generated in PEAs in response to the applied driving voltages. The tracking control method is based on multirate feedforward control. The proposed multirate control method uses an inverse modified Prandtl Ishlinskii operator to cancel out hysteresis non-linearity. The controller structure has a simple design and can be quickly identified. The control system is capable of achieving suitable tracking control and it is convenient to use and can be quickly applied to practical PEA applications. Experimental results are provided to verify the efficiency of the proposed method. Keywords: piezoelectric actuators, hysteresis, Prandtl Ishlinskii, multirate control 1 INTRODUCTION The properties of piezoelectric actuators (PEAs) such as their ability to directly convert electrical energy into mechanical energy and low power requirements allow their use in sub micrometre positioning systems [1]. Such systems are of interest due to their rapid response times (generally only microseconds). Also, PEAs have no moving parts in contact with each other to limit the resolution. Thus PEAs show no wear and tear effects that normally causes a decrease in life time and precision. Heavy duty PEAs can move or operate under high loads of up to several tons. These advantages make them suitable for electromechanical applications. Currently, there *Corresponding author: Department of Mechanical Engineering, Amirkabir University, Tehran, Iran. mzare@aut.ac.ir is considerable interest in the use of piezoelectric and ferroelectric materials in scientific and engineering applications. Examples include active vibration control [2], needle-valve actuation in precision machining in [3], atomic force microscopy (AFM) [4], and cell manipulation in medical technology [5]. However, PEAs suffer from the serious disadvantage of non-linear hysteresis behaviour which leads to tracking errors. Thus, hysteresis behaviour as a function of applied driving voltage is one of the most critical fields in the modelling of PEAs. The hysteresis is not a differentiable and neither can a one-toone non-linear mapping approach be applied. The system can be considered to be a non-linear operator with a local memory. This means that the output of the system depends not only on the instantaneous input value but also on the history of its operation. This is especially true for the case of returning values [6]. The non-linear hysteresis effect JSCE695 F IMechE 2009 Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering

2 674 M Zareinejad, S M Rezaei, H H Najafabadi, S S Ghidary, A Abdullah, and M Saadat can be corrected using charge control or can be compensated by hysteresis modelling. However, charge control is inherently bulky, costly, uncommon, and offers limited sensitivity. It may lead to drift and saturation problems and reduces the operating range and life of PEAs [7]. Consequently, hysteresis modelling strategies for PEAs prove to be a more promising, economic, and a commercially acceptable control method. Many investigations have been performed to model the dynamics of PEAs [2, 8]. To achieve a precise tracking control in a PEA system, a modelbased controller design is necessary, particularly in open-loop operation. The model should represent the PEA behaviour perfectly. Different models of the hysteresis have been proposed in the literature. The hysteresis models can be divided into mathematical and non-linear differential models. The Preisach model [4, 9, 10], the Prandtl Ishlinskii model [11], and the Maxwell slip model [12, 13] are examples of mathematical models. The Duhem model [14] and the Bouc Wen model [15] are examples of non-linear differential models. In practice, differential models are more sensitive to measurement noise. A comprehensive model of a PEA should account for the inherent hysteresis behaviour of the piezoelectric material. The aim of the present study is to compensate the hysteresis non-linearity and the effect of mechanical loading on PEA behaviour. In this paper multirate sampled-data control of PEA systems is considered. The control process sampling rate is faster than the control update rate. The hysteresis is modelled using a Prandtl Ishlinskii operator. It uses a multirate scheme to produce a desired control input without inversing the Prandtl Ishlinskii (P I) operator. The P I operator, while being able to accurately model the hysteresis behaviour of a PEA, has one major inadequacy: the inverse of the operator does not exist when the slope of the hysteretic curve is not positive definite [16]. The proposed approach combines feedforward inverse control with a PID controller to ameliorate the tracking of the PEA especially in the presence of variation in the input (rate) frequency. The PID feedback modifies the hysteresis model error in situations where the real hysteresis loop is affected by external effects such as mechanical loading. 2 RELATED WORK Croft applied an integrated inversed approach to compensate the three adverse effects of creep, hysteresis, and vibration using AFM [4]. A Preisach model was used to model the hysteresis behaviour and a linear high-order spring damper model was applied to model the creep and vibration of PEA. Bashash and Nader presented an on-line estimation strategy based on perturbation estimation [18]. A non-linear model was used with time-varying coefficients to approximate the hysteresis non-linearity in the PEA. Sliding mode control was used to achieve insensitivity against parameter uncertainties. Shieh et al. extended the LuGre friction model to represent the motion dynamics of a PEA system [19]. An adaptive displacement tracking control was proposed with the parameter adaptation of a parameterized hysteresis function. However, chattering effects were created when the frequency or amplitude of the input was increased. Preisach models and first-order reversal curves have been extensively utilized to approximate the non-linearity of the hysteresis. The Preisach model needs a large experimental database and a timeconsuming parameter estimation procedure. Also, considerable computation efforts are required during the control process [4, 9, 10]. In the Maxwell slip model, the hysteresis is approximated by using motion dynamics. It is constructed in terms of a force applied to one set of massless bodies parallel to the springs. In this model, relationships are in terms of the applied force, spring constants, and break forces to determine the hysteresis dynamics. The critical number of the springs and mass-free bodies necessary for accurate hysteresis estimation are very difficult to determine [12]. Hu and Ben Mrad [9] used experimental results to show that the classical Preisach model offers excellent modelling accuracy. This occurs when the actuator is subjected to an excitation voltage signal at a low frequency without any load. The accuracy of the Preisach model is shown to rapidly deteriorate as the applied load is increased or the range of frequencies contained in the voltage excitation signal gets wider. However, the classical Preisach model remains a good model for PEA hysteresis in applications where the load fluctuation is relatively small and the range of frequencies in the excitation is limited. Therefore, the classical Preisach model can be potentially used when the variation in the load applied to the actuator is small or when the load applied to the actuator itself is small. This is the case in numerous applications such as those in [4], [14], and [20]. The polarization in PEA is affected by both the applied voltage and external forces. When an Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering JSCE695 F IMechE 2009

3 Novel multirate control strategy for piezoelectric actuators 675 external force is applied to a polarized piezoelectric, the expansion of the PEA depends on the stiffness of the piezoelectric material and the change in remnant strain (caused by the polarization). PEAs produce an electrical response (charge) when mechanically stressed in dynamic operations such as imprint applications. The induced charge affects the driving voltage. Hence, there have been numerous studies which consider external load in hysteresis models of PEAs for dynamic measurement in simultaneous sensing, actuating, and precision positioning in machining [17, 21]. An external mechanical load affects the inclination of the voltage-to-displacement hysteresis curve in PEAs. The effect of the load on the voltage-to-displacement curve clearly increases as the load increases [17, 22]. Electromechanical models of PEAs have a prominent role in open-loop operations. They are less significant in close-loop controller models. Georgiou and Ben Mrad [12] demonstrated an electromechanical model for PEAs. It utilized the Maxwell slip model to represent the hysteretic nonlinearity of the PEA. The non-linear voltage-tocharge properties of the PEA were represented by a series of voltage-limited capacitors. The model contained five parameters and required experiments to be performed to determine parameter values. A first-order differential equation has been used to describe the hysteresis effect and a partial differential equation (PDE) to describe the mechanical behaviour of the PEA in [8]. However, there is no experimental result for this model. Furthermore it seems difficult to design a tracking control system based on the proposed model. Bashash and Nader proposed a model by integrating a modified P I hysteresis operator with a secondorder linear dynamics [23]. A reference model was obtained for both open-loop and closed-loop control techniques. This was used with an inverse feedforward controller to achieve trajectory tracking control in a PEA. 3 MODELLING OF PEAS 3.1 Dynamic modelling of PEAs The hysteresis effect observed for PEAs in the presence of an applied electric field is the main drawback in precise positioning applications. Therefore, the development of a dynamic model which describes the hysteresis behaviour is very important. Second-order linear dynamics have been previously utilized to describe the system dynamics. As shown Fig. 1 Piezoelectric actuator equivalent dynamic model in Fig. 1, this model combines mass-spring-damper ratio with a non-linear hysteresis function appearing in the input excitation to the system. The following equation defines the model m s x s ðþzb t s _x s ðþzk t s x s ðþ~h t F ðvt ðþþ ð1þ where x s (t) is the salve position, m s, b s, and k s are mass, viscous coefficient, and stiffness respectively, H F (v(t)) denotes the hysteretic relation between input voltage and excitation force. PEAs have very high stiffness values, and consequently possess a very high natural frequency. In low-frequency operations, the effects of actuator damping and inertia can be safely neglected. Hence, the governing equation of motion is reduced to the following static hysteresis relation between the input voltage and actuator displacement xt ðþ~ 1 H F ðvt ðþþ~h x ðvt ðþþ for k s m s x s ðþ%b t s _x s ðþ%k t s x s ðþ t ð2þ Equation (2) facilitates the identification of the hysteresis function H F (v(t)) 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, H x (v(t)). It is then scaled up to k s to obtain H F (v(t)) m s x s ðþz±b t s _x s ðþzk t s x s ðþ~k t s H x ðvt ðþþ ð3þ 3.2 The P I operator In this section hysteresis modelling using the P I operator is described. This model can be used to accurately approximate the hysteresis loop and its inverse can be obtained analytically which facilitates inverse feedforward control design. JSCE695 F IMechE 2009 Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering

4 676 M Zareinejad, S M Rezaei, H H Najafabadi, S S Ghidary, A Abdullah, and M Saadat Background to the P I operator There is a backlash operator in the P I hysteresis model (Fig. 2) that is defined by yt ðþ~h r ½x, y 0 ŠðÞ t ~maxfxt ðþ{r, min½xt ðþzr, yt{t ð ÞŠg ð4þ where x is the control input, y is the actuator response, r is the control input threshold value or the magnitude of the backlash, and T is the sampling period. The initial consistency condition of equation (4) is given by yð0þ~±max½xð0þ{r, minðxð0þzr, y 0 ÞŠ ð5þ where y 0 is usually, but not necessarily, initialized to zero. Multiplying the backlash operator H r, by a weight value w h, the generalized backlash operator is obtained yt ðþ~w h H r ½x,±y 0 ŠðÞ t ð6þ The weight w h defines the gain of the backlash operator and may be viewed as the gear ratio in a gear mechanical play analogy. Complex hysteresis non-linearity can be modelled by a linear weighted superposition of many backlash operators with different threshold and weight values yt ðþ~w T h H r x, y 0 ðþ t ð7þ where H r x, y 0 ðþ~ t ½ Hr0 ½x, y 00 ŠðÞ...H t m ½x, y 0n ŠðÞ t Š T ð8þ with the weight vector w T h ~ ½ w h0...w hn Š, the threshold vector r 5 [r 0 r n ] T where 0 5 r 0,, r n and the initial state vector y 0 5 [y 00 y 0n ] T. The control input threshold values r n are usually chosen to be of equal intervals between the maximum and minimum of PEA displacement Modified P I operator The P I operator inherits the symmetry property of the backlash operator about the centre 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 P I 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 non-linear operator given by S d ½xŠðÞ~ t max ½ xt ðþ{d, 0Š dw0 zt ðþ~w T s S d½yšðþ t xt ðþ d~0 ð9þ where y is the output of the hysteresis operator and z is the actuator response. w T s ~ ½ w s0...w sn Š, is the weight vector, S d [y](t) 5 [S d0 [y](t) S dm [y](t)] With the threshold vector d T 5 [d 0 d n ] T 0 5 d 0,, d m. Thus, the modified P I operator is defined as follows zt ðþ~h x ðþ~w t T s S d w T h H r x, y 0 ðþ t ð10þ d i is usually chosen to have equal intervals between the maximum and minimum of the hysteresis operator output The inverse P I operator The inverse P I operator is given by H x {1 ½x d ŠðÞ~w t T h H r w T: h S d : ½ xd Šy 0 ðþ t ð11þ Fig. 2 The backlash operator Cascading the inverse hysteresis model with the actual hysteresis model gives the identity mapping between the control input x d (t) and the actuator response x(t) xt ðþ~h x H {1 ½x d ðþ t Š ð12þ x The inverse model parameters can be calculated analytically as follows Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering JSCE695 F IMechE 2009

5 Novel multirate control strategy for piezoelectric actuators 677 w h0 ~ 1 w h0, w s0 ~ 1 = ws0 ð13þ w hi ~ P i j~0 w h j w si ~ P i j~0 w s j r i ~ Xi j~0 d i ~ Xi j~0 w hj w sj {w hi Pj{1, i~1,..., n ð14þ {w si Pj{1 j~0 w h j, i~1,..., n ð15þ j~0 w s j ±r i {r j, i~0,..., n ð16þ d i {d j, i~0,..., m ð17þ 3.3 Identification of the hysteresis model In this section the method for the identification of the hysteresis between the input voltage and the actuator displacement as defined by equation (10) is described. Weighting parameters are identified using the least-square optimization technique for error minimization. Static hysteresis is identified using a quasi-static triangular input. Appropriate values for the order of the backlash operator n, saturation function m, and threshold vectors r and d are selected for correct approximation of the hysteresis. The values for n and m can be set as 25 and 15 respectively. Figure 3 refers to the estimated hysteresis loop using the P I model compared to the actual hysteresis of the PEA. Identification of the P I parameters is performed for the measured actuator response subjected to 100 V peak-to-peak sawtooth control input with frequency of 0.5 Hz. y 0i ~ Xi j~0 w hj y 0i z Xn j~iz1 w hj y 0j, i~0,..., n ð18þ After setting the threshold parameters r and d as described in the previous section, the weight parameters w h and w s are estimated by performing a least square fit of equation (10). Graphically, the inverse is the reflection of the resultant hysteresis loop about the 45u line. 4 CONTROLLER DESIGN 4.1 Feedforward hysteresis compensation The structure of the inverse feedforward hysteresis compensation is shown in Fig. 4. The key idea of an inverse feedforward controller is to cascade the inverse hysteresis operator H x {1 with the actual hysteresis. This is represented by the hysteresis operator H x to obtain an identity mapping between Fig. 3 Estimated hysteresis loop using P I versus experimental result JSCE695 F IMechE 2009 Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering

6 678 M Zareinejad, S M Rezaei, H H Najafabadi, S S Ghidary, A Abdullah, and M Saadat Fig. 4 The feedforward inverse control the desired actuator output x d (t) and actuator response x(t). The inverse P I operator H x {1 uses x d (t) as its input and transforms it into a control input v H {1ðÞwhich t x produces x(t) in the hysteretic system that closely tracks x d (t). 4.2 Multirate control Fig. 5 Block diagram of multi-rate feedback/feedforward control strategy for PEA To deal with the influence of P I identification error, a feedback control is utilized. The multirate output feedback (MROF) concept consists of sampling the control input and sensor output of a system at different rates [24]. Dabroom and Khalil [25] implemented an output feedback controller that was designed under continuous-time state feedback. A discrete-time high-gain observer was used to estimate the system states. Ahrens and Khalil [24] used a MROF control scheme for a class of nonlinear systems based on discrete-time high-gain observers. The stability of a system under sampled data output feedback was studied. This was done while the control rate was fixed by the sampled data state feedback design and the output sampling rate was faster. This paper is motivated by applications to PEAs that utilize a computationally demanding control structure including hysteresis inversion algorithms [8]. Furthermore, a tracking control method that is based on a multirate feedforward approach has fewer difficulties in measuring the system states of the PEA. The block diagram shown in Fig. 5 schematically represents the multirate control strategy for a PEA. A multirate feedforward control approach is considered to update the feedforward input of a two-degree-of-freedom control system at a rate N- times faster than the output measurement sampling rate. The feedback loop of the system is closed at the measurement sampling rate. This method was proposed in [26] along with frequency domain interpretation of the improvement attained by the higher rate update of the feedforward input. Since the higher updating rate is applied to the feedforward input, the scheme does not influence the stability of the feedback loop system. Moreover, the fast feedforward scheme cancels out hysteresis non-linearity. Therefore, a PID controller is used for appropriate response of a second-order linearized system described by equation (3). The asymptotic convergence of the plant output to the desired output signal utilizing this scheme, was proved in [27]. 5 EVALUATION AND EXPERIMENTAL RESULTS The proposed strategy was investigated by a set of experiments on a Physik Instrumente nanopositioning stage with high resolution strain gauge position sensor. The multirate control structure was modelled in Simulink. It was then compiled and loaded into a data acquisition controller board (dspace1104) to produce the desired control input. Then, the control input was applied to the piezo stage via its amplifier. The displacement of the piezo stage was measured and fed back via a strain gauge sensor. To close the loop, a PID controller was used as a sampled data controller in parallel with the hysteresis compensation operator as shown in Fig. 5. In order to Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering JSCE695 F IMechE 2009

7 Novel multirate control strategy for piezoelectric actuators 679 decouple feedback and feedforward controllers effects, Fig. 6 depicts tracking of the system using a feedforward inverse control. Figures 7 and 8 compare the response of a single-rate PID controller with a constant sampling period of s, against the response of the multirate controller for which by considering N 5 10 the period of fast rate was T f 5 T/ s. As can be clearly seen from Figs 6 to 8, the multirate controller with the more accurate hysteresis estimation, was able to achieve more accurate tracking. The tracking errors shown in Figs 6 and 7 show that the feedforward multirate controller performs more accurate than the single-rate PID controller. Figure 9 compares the error signals for the three control approaches. Table 1 lists the measured performance of the PID and multirate controllers in tracking a sinusoidal input. In higher frequency trajectories as shown in Fig. 10, the plant output remains closer to the desired output by increasing the frequency of updating the feedforward control input. 6 CONCLUSIONS A multirate sampled-data controller is proposed for the control of PEAs. A multirate feedforward inverse control approach is considered to update the feedforward input of a two-degree-of-freedom control system. It operates at a rate N-times faster than the output measurement sampling rate. This scheme cancels out hysteresis non-linearity and does not influence the stability of the feedback loop system. Experimental results on a PEA demonstrate that the proposed scheme is more accurate in tracking than a Fig. 6 Multiple frequency trajectory tracking result for feedforward inverse control Fig. 7 Multiple frequency trajectory tracking result for single-rate PID control JSCE695 F IMechE 2009 Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering

8 680 M Zareinejad, S M Rezaei, H H Najafabadi, S S Ghidary, A Abdullah, and M Saadat Fig. 8 Multiple frequency trajectory tracking result for multirate-pid/feedforward control Fig. 9 Error signals Fig. 10 Multiple high-frequency trajectory tracking result for multirate-pid/feedforward control Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering JSCE695 F IMechE 2009

9 Novel multirate control strategy for piezoelectric actuators 681 Table 1 PID controller. Performance of the PEA in multifrequency trajectory tracking was improved by using the proposed new controller structure. The quick and simple identification procedure of the proposed controller structure makes it convenient and valuable in PEA practical applications. REFERENCES Measured performance of PID and multirate controllers in tracking 60 mm peak-to-peak multi-frequency input Control method RMS (mm) e max (mm) Multirate control Single-rate PID control Feedforward control Habibollahi Najafabadi, H., Rezaei, S. M., Ghidari, S. S., and Zareinejad, M. Hysteresis compensation of piezoelectric actuator under dynamic load condition. In Proceeding of the IEEE/RSJ International Conference on Intelligent robots and systems, 2007, pp (IEEE, Piscataway, New Jersey). 2 Caruso, G., Galeani, S., and Menini, L. Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators. Simul. Model. Pract. Theory, 2003, 11, Woronko, A., Huang, J., and Altintas, Y. Piezoelectric tool actuator for precision machining on conventional CNC turning centers. Precis. Engng., 2003, 27, Croft, D., Shed, G., and Devasia, S. Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application. Trans. ASME: J. Dyn. Syst. Meas. Control, 2001, 123, Thaomine, O., Ott, A., Cradoso, O., and Meister, J.-J. Microplates a new tool for manipulation and mechanical perturbation of individual cells. J. Biochem. Biophys. Methods, 1999, 39, Maygergoyz, I. Mathematical model of hysteresis, 1991 (Springer-Verlag, New York). 7 Hu, H., Georgiou, H. M. S., and Ben Mrad, R. Enhancement of tracking ability in piezoceramic actuators. IEEE/ASME Trans. Mechatron., 2005, 10, Adriaens, A., de Koning, W. L., Han, J. M. T., and Banning, R. Modeling piezoelectric actuators. IEEE/ASME Trans. Mechatron., 2000, 5(4), Hu, H. and Ben Mrad, R. On the classical Preisach model for hysteresis in piezoceramic actuators. Mechatronics, 2003, 13, Ge, P. and Jouaneh, M. Modeling hysteresis in piezoceramic actuators. Precis. Engng, 1995, 17, Kuhnen, K. and Janocha, H. Complex hysteresis modelling of a broad class of hysteretic nonlinearities. In Proceedings of the Eighth International Conference on New actuators, Bremen, Germany, 2002, pp Georgiou, H. M. S. and Ben Mrad, R. Electromechanical modeling of piezoceramic actuators for dynamic loading applications. Trans. ASME: J. Dyn. Syst. Meas. Control, 2006, 128, Goldfarb, M. and Celanovic, N. Modeling piezoelectric stack actuators for control of micromanipulation. IEEE Control Syst. Mag., 1997, 17, Stepanenko, Y. and Su, C. Y. Intelligent control of piezoelectric actuators. In Proceedings of the 37 th IEEE Conference on Decision and control, Tampa, Florida, 1998, pp (IEEE, Piscataway, New Jersey). 15 Lin, C.-J. and Yang, S. R. Precise positioning of piezo-actuated stages using hysteresis-observer based control. Mechatronics, 2006, 16, Tan, U. X., Win, T. L., and Ang, W. T. Modeling piezoelectric actuator hysteresis with singularity free Prandtl Ishlinskii model. In Proceedings of the IEEE International Conference on Robotics and biomimetics, Kunming, People s Republic of China, 2006, pp (IEEE, Piscataway, New Jersey). 17 Ling, S.-F., Hou, X., and Xie, Y. Decoupling loading effect in simultaneous sensing and actuating for dynamic measurement. Sens. Actuators A Phys., 2005, 120, Bashash, S. and Nader, J. A new hysteresis model for piezoelectric actuators with application to precision trajectory control. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Symposium on Vibration and Noise Control, Orlando, Florida, pp Shieh, H.-J., Lin, F.-J., Huang, K., and Teng, L.-T. Adaptive displacement control with hysteresis modeling for piezoactuated positioning mechanism. IEEE Trans. Ind. Electron., 2006, 53(3), Leang Kam, K. and Devasia, S. Design of hysteresis-compensating iterative learning control for piezo-positioners: application to atomic force microscopes. Mechatronics, 2006, 16, Cuttino, J. F., Miller, A. C., and Schinstock, D. E. Performance optimization of a fast tool servo for single-point diamond turning machines. IEEE/ ASME Trans Mechatron., 1999, 4, Dayu, Z. and Kamlah, M. Dielectric and piezoelectric performance of soft PZT piezoceramics under simultaneous alternating electromechanical loading. J. Eur. Ceram. Soc., 2005, 25, Bashash, S. and Nader, J. Robust multiple frequency trajectory tracking control of piezoelectrically driven micro/nanopositioning systems. IEEE Trans. Control Syst. Technol., 2007, 15(5), Ahrens, J. H. and Khalil, H. K. Multirate sampleddata output feedback using high-gain observers. In Proceedings of the IEEE Conference on Decision JSCE695 F IMechE 2009 Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering

10 682 M Zareinejad, S M Rezaei, H H Najafabadi, S S Ghidary, A Abdullah, and M Saadat and control, 2006, pp (IEEE, Piscataway, New Jersey). 25 Dabroom, A. M. and Khalil, H. K. Output feedback sampled-data control of nonlinear systems using high-gain observers. IEEE Trans. Autom. Control, 2001, 46, Gu, Y. and Tomizuka, M. High performance tracking control system under measurement constraints by multi-rate control. The 14th WAC World Congress, Beijing, July Gu, Y. and Tomizuka, M. Multi-rate feedforward tracking control for plants with nonminimum phase discrete time models. In Proceedings of the American Control Conference, San Diego, California, June 1999, pp Proc. IMechE Vol. 223 Part I: J. Systems and Control Engineering JSCE695 F IMechE 2009