2013 International Conference on Robotics, Biomimetics, Intelligent Computational Systems (ROBIONETICS) Yogyakarta, Indonesia, November 25-27,2013

Sliding-Mode (SM) and Fuzzy-Sliding-Mode (FSM) Controllers for High-Precisely Linear Piezoelectric

Ceramic Motor (LPCM)

Hendro Nurhadi Department of Mechanical Engineering, Institut Teknologi Sepuluh Nopember (ITS)

Keputih-Sukolilo, Surabaya 60112, Indonesia Email: hdnurhadi@me.its.ac.id

Abstract-In nanotechnologies, usages of piezoelectric

actuators are widely used. Since the properties of piezo-materials

itself are dependent on Curie temperature and they have a

reversible effect, a precise positioning using linear piezoelectric

ceramic motor (LPCM) is a subject to achieve in this work. A proposed controller mode by combining sliding mode (SM) and

fuzzy inference system, so called Fuzzy-Sliding Mode (FSM)

controller, is hereafter introduced. In this controller scheme, a

sliding part obtains to counter disturbances, while the fuzzy

eliminates uncertainties. The results show Excellencies in this

work that emphasizes performance improvement of minimizing

errors with steady-state errors. The circular motion is then

conducted to verify the controller's performance. The controller

(FSM) presented in this paper possessed excellent tracking speed

(adaptive) and robustness properties.

Keywords-component; Sliding control; fuzzy logic; linear motor; piezoelectric actuator

I. INTRODUCTION

The piezoelectric actuators have been widely used in precision positioning applications due to the requirements of high resolution in displacement [1], [2]. The two basic phenomena of piezoelectric materials permit them to be used as sensors and actuators in a control system. The piezoelectric effects of actuator are usually used to provide linear and rotational motion due to the capability of achieving fine motion without the use of moving mechanical systems [3]. The driving principles of linear piezoelectric ceramic motors (LPCM) are based on the ultrasonic vibration force of piezoelectric ceramic elements and mechanical frictional force. A linear piezoelectric ceramic motor (LPCM) is a new type of motor, which is driven by the ultrasonic vibration force of piezoelectric ceramic elements and the mechanical friction effect. Therefore, their mathematical models are complex and the motor parameters are time varying due to increasing in temperature and changing in motor drive operating condition [4], [5]. An artificial intelligent controller is then applied to overcome a precise positioning using LPCM, such as NN (Neural Network) which demonstrated by [6]-[ I 0]. An extended PD controller is also shown by [11], [12]. Similar methods for other applications can be found in following works, for instance a steam engine with two-inputs and two-outputs controlled by fuzzy logic controller

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is demonstrated by [13]. A linear motion table as a basic need of machine tools is proposed by [14]. A piezomechanics using an intelligent sliding controller mode is presented by [2]. Most of works on control designs are model-free. One example of works of model-based control design is introduced by [15]; a robust control system with the fuzzy adaptive controller and the additional compensator is presented.

In this paper, all above controller's drawbacks for precise positioning and high speed will be eliminated by developing Fuzzy Sliding controller mode (FSM). Hence, to convince this controller's performance, FSM will be compared to the single controller mode, a sliding controller mode (SM). The main characteristics of a linear piezoelectric ceramic motor are precise positioning and high speed positioning. Applying an FSM is therefore a solution for those problems. Sliding controller part will guarantee the enrichment of the speed, and fuzzy logic controller part will do the precise positioning. The main contribution of this work is developing a model-free based controller which can solve a unique problem in a precise positioning system of LPCM. Hybrid FSM then has improved the performance of a precise positioning of LPCM.

I I. LINEAR PIEZOELECTRIC CERAMIC MOTOR (LPCM)

Adriaens et al. (2000) presented an electromechanical piezoelectric model based on physical principles [16]. In this model, a first-order differential equation was adopted to describe the hysteresis effect encountered between supplied voltage and electric charge, and a partial differential equation was used to describe the mechanical behavior between force and elongation. However, a hysteresis effect is dynamic, nonlinear, and independent of the time scale, and there are infmite orders in this model practically because the piezoelectric actuator is a distributed parameter system. Thus, the number of parameters in this model is relatively large and the dynamic equation is more complicated such that this approximation model is not suitable for model-based control design purposes.

A. Driving principle

The entire piezoelectric ceramic device is constrained by four support springs and kept in close contact with the moving table by a preload spring.

x

B

spring Piezoelectric ceramic

spring

Fig. l. Principle of piezoelectrical ceramic actuator

Fig. 2. System design

Hence, the friction force between the fingertip and the moving table can be utilized to drive the linear motion of the table. A

command voltage of ± 10 V can be sent to the driver unit and converted to a high sine-wave voltage of 39.6 kHz that drives the motor. The longitudinal and bending modes are generated by the driver as shown in the right part of Fig. 1. Proper lengths of the piezoelectric ceramic device are selected so that the natural frequency of the two modes are identical, and a

circular or oval motion can be produced through the combination of the longitudinal and bending deformations of the device, in addition to the back-and-forth motion induced by the fingertip.

B. Characteristics of LPCM

The structure of the Nanomotion HR4 series LPCM is a large face of a relatively thin rectangular piezoelectric ceramic device, as depicted in Fig. 1.

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To generate the different moving direction, the electrodes are electrified by an AC voltage in the pairs of the diagonal electrodes. The movement of LPCM is constrained by four support springs with large stiffness. These support springs along a pair of long edges of the LPCM contact the piezoelectric ceramic at points of zero movement in the X­direction. A relatively hard ceramic spacer is attached with cement to the short edge of the piezoelectric ceramic at the center of the edge. In general, the moving table usually mounts in a V-flat way. A friction force exists on the surface between the moving table and the V-flat way, and it also exists on the contact surface between the moving table and the spacer. In order to transmit the motion of the spacer to the moving table, a preload spring is designed to supply pressure between the spacer and the moving table. The control configuration of the entire experiment is plotted in Fig. 2; a system design using LPCM.

III. CONTROLLER DESIGN

A. PID

As a default setting performance in this work, PID control is the major process control method. A general controller output of PID as shown in Fig. 3 (a) is:

f de

u = Kpe+KpK/ e·dt+KpKf)­dt

(1)

with gains In X and Y axis: Kp = 10, K/ = 0.2

Kf) = 0.1.

(a)

and

r + e Output

(b) ce

e

Sliding

Sensor (EneoderiDecoder)

(e)

s<O 0 s>O

Fuzzy

Fig. 3. Principles of (a) PID; (b) sliding; and (c) fuzzy

TABLE I RULE TABLE

ce\e

NVB NB NVM NM NS NVS

NVB -3 -3 -3 -3 -3 -3

NB -3 -3 -3 -3 -3 -3

NVM -3 -3 -3 -3 -3 -2.5

NM -3 -3 -3 -3 -2.5 -2

NS -3 -3 -3 -2.5 -2 -1.5

NVS -3 -3 -2.5 -2 -1.5 -I

ZO -3 -2.5 -2 -1.5 -I -0.5

PVS -2.5 -2 -1.5 -\ -0.5 0

PS -2 -1.5 -I -0.5 0 0.5

PM -1.5 -\ -0.5 0 0.5 I

PVM -\ -0.5 0 0.5 I 1.5

PB -0.5 0 0.5 I 1.5 2

PVB 0 0.5 I 1.5 2 2.5

B. Sliding mode (SM) controller

Since a sliding mode (SM) controller can be used to stabilize single input systems (reduce disturbances) and is no requirement on continuity, therefore, this SM is applied. The basic idea of SM are, flrstly will drive the system to stable manifold, then slides to equilibrium (sliding phase) which shown in Fig. 3 (b). The procedure of applying sliding controller is by reducing a gradually (0.6 0.5 0.4 ... ) to have a lower chattering or noise. Phase dynamic reaches for:

so that

s (t) = a· e (t ) + ce (t ) (2)

ce = -ae+s

is stable if s = 0 .

(3)

C. Fuzzy

Most cases use Mamdani or Sugeno fuzzy inferences which its principle is shown in Fig. 3 (c). In this work, it uses Mamdani-style fuzzy inference as an intuition and general usages in practice and most applications.

Widely used defuzziflcation methods are Centroid Average (CA); Maximum Center Average (MCA); Mean of Maximum (MOM); Smallest of Maximum (SOM; and Largest of Maximum (LOM). Centroid Average and Maximum Center Average methods belong to continuous ones and are frequently used in control engineering and process modeling. The rest

ZO PVS PS PM PVM PB PVB

-3 -2.5 -2 -1.5 -\ -0.5 0

-2.5 -2 -1.5 -\ -0.5 0 0.5

-2 -1.5 -I -0.5 0 0.5 I

-1.5 -\ -0.5 0 0.5 \ 1.5

-\ -0.5 0 0.5 I 1.5 2

-0.5 0 0.5 I 1.5 2 2.5

0 0.5 I 1.5 2 2.5 3

0.5 I 1.5 2 2.5 3 3

\ 1.5 2 2.5 3 3 3

1.5 2 2.5 3 3 3 3

2 2.5 3 3 3 3 3

2.5 3 3 3 3 3 3

3 3 3 3 3 3 3

represents discontinuous methods, which are mainly used in decision making and pattern recognition applications for selecting the alternative. A new developed defuzziflcation method called height method is applied, since this method does not synthesize the fuzzy set (membership function). This is better than the center of gravity method in respect of high­speed computing. Consequently, it is crucial to investigate the convergence of feedback laws constructed by fuzzy approximate reasoning method and the convergence of solutions of the nonlinear state equation in the fuzzy control system [17].

D. Fuzzy Sliding

Fuzzy sliding mode (FSM) controller is a sliding mode (SM) controller followed/continued by Fuzzy. The sliding mode controller can be applied to higher order systems with the

sliding manifold must be of order n -1. SM is restricted to single input systems and requires not being continuous system. SM also can be expanded to include integral action and combined with PO controllers. Principles of single sliding mode controller scheme (SM) and single fuzzy logic are shown by Fig. 3 (b) and (c), while combination among them as an FSM controller is illustrated in Fig. 4.

A sliding part obtains to counter the disturbances, while the fuzzy eliminates uncertainties. In this research, for X and Y

axis are a=O.25, G, =1.2 and Gu =4.

Fig. 4. The block diagram of fuzzy-sliding mode (FSM)

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IV. EXPERIMENTAL SETUP

The experimental setup of this study includes a table consisting of two HR4 piezoelectric ceramic motors from Nanomotion Limited, two LIES linear scales of 20-nm resolution, the independent X-axis table of 150-mm travel, the Y-axis table of 100-mm travel located above the X-axis table, two AB2 drivers, an Analog I/O card AIO-3320 and Motion control card MPC-3042, both of which are provided by JS Automation Corp. This study uses Borland C++ 6.0 language for the program design.

Initially, the "current position" of the LP AT is detected by linear scale and transmitted to a PC via the motion card, where will be compared to the "target position". The difference between two positions is then calculated. Proper control voltage corresponding to the position error can be sent to the driver according to the established control rule base, and then converted into a high sine-wave voltage of 39.6 kHz so as to drive the LPAT. The aforementioned positioning process continues until the position error becomes ± 0.02 flm.

V. RESULT AND DISCUSSION

In this section, all experimental results, according to performance improvement and critical area that SM and FSM affected are, will be discussed. Results are plotted in Fig. 5-6 and numerated in Table 3-4.

TABLE 2 RULE TABLE

Maximum Minimum Average Error

(all in mm) Error Error

of absolute value

SM X axis 0.09744 -0.09284 0.00919

Y axis 0.09849 -0.0835 0.00806

FSM X axis 0.05317 -0.04311 0.00766

Y axis 0.06084 -0.04981 0.00654

X axis Tracking Sin Wave (2mm) Y axis Tracking Sin Wave (2mm)

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X axis Tracking Sin Wave (20mm) 30

,--------;=_= __ =o'T"'=g=.et'il -SM -FSM

5 10 15 Time (sec)

Y axis Tracking Sin Wave (20mm) 30,-

----------,-,

Fig. 6. Tracking errors of sine wave on XY axes with SM and FSM at lOmm Amplitude

TABLE 3 TRACKING ERRORS OF SINE WAVE ON XY AXES WITH SM AND

FSM AT IOMM AMPLITUDE

Average

(all in mm) Maximum Minimum Error of

Error Error absolute value

SM X axis 0.15966 -0.17981 0.04792

Y axis 0.16928 -0.19512 0.03956

FSM X axis 0.12052 -0.09309 0.03095

Y axis 0.12798 -0.12341 0.02828

TABLE 4 CIRCLING AND ROUNDNESS ERROR PROPERTIES

(all in SM-Rlmm FSM- SM- FSM-mm) Rlmm RlOmm RIOmm

Actual 1.00015 1.00031 9.99927 9.99942

radius Roundne

0.110525 0.0532068 0.173594 0.148151 ss error

Maximu 0.042947 0.0254359 0.0864493 0.0782624

m error Minimu

-0.0675847 -0.0277709 -0.0871443 -0.0698889 m error

A validation case is hereafter necessarily performed to verify experimental results driven by sin-waves and step­responses. It is an experiment of circling. The results are then summarized in Table 5. The improvements of minimizing roundness-errors are achieved by FSM for Rlmm of O.llm to be 0.05mm (improved around 55%); and RIOmm of 0.17m to be 0.14mm (improved around 18-20%). Generally, FSM has improved the performance of higher precision positioning and higher accuracy than SM.

VI. CONCLUSION

Several considerations in developing SM and FSM are: sliding part can counter the disturbances, while the fuzzy eliminates uncertainties; main drawback of SM is a

chattering during a sliding-process; by combining sliding with fuzzy be a fuzzy-sliding mode controller (FSM), it can solve the chattering problem and can improve the performance.

The hysteresis or others characteristics of linear piezoelectric ceramic motors (LPCM) are hence covered by this proposed method.

Conclusively, the applicability of FSM controller to the fulfillment of high-precisely position response of adaptive signal input and reduction of disturbances at LPCM was shown. The performance comparison of both mode controllers also shows an improvement of performance. The controller (FSM) presented in this paper possessed excellent tracking speed (adaptive) and robustness properties.

REFERENCE

[1] R.J. Wai and K.H. Su (2006), Supervisory control for linear piezoelectric ceramic motor drive using genetic algorithm, IEEE Trans. Ind. Electron., 53(2), pp.657-673.

[2] C. L. Hwang, C. Jan, and Y. H. Chen (2001), Piezomechanics using intelligent variable-structure control. IEEE Trans. Ind. Electron., 4S(I), pp. 47-59.

[3] Y.F. Peng, R.J. Wai and C.M. Lin (2003), Adaptive CMAC model reference control system for linear piezoelectric ceramic motor, Proceed. IEEE Int. Symp. on Computational Intell. in Robotics and Automation, Kobe, Japan, pp. 30-35.

[4] T. Sashida and T. Kenjo (1993). An Introduction to Ultrasonic Motors, Oxford, U.K. : Clarendon.

[5] Y. Ting, J. S. Huang, F. K. Chuang, and C. C. Li (2003), Dynamic analysis and optimal design of a piezoelectric motor. IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 50(6), pp. 601-613.

[6] C.F. Hsu, B.K. Lee and T.T. Lee (200S), Robust Intelligent Motion Control for Linear Piezoelectric Ceramic Motor System Using Self­constructing Neural Network, Lecture Notes in Electrical Engineering, Volume 5, Book of Advances in Industrial Engineering and Operations Research. Chapter 27. Springer. US, pp 37S-390.

[7] M.Y. Sung and S.J. Huang (2004), Fuzzy logic motion control of a piezoelectrically actuated table, Proc. Isntn Mech. Engrs, Vol. 21S, Part 1: 1. Systems and Control Engineering. pp.3S1-397.

[S] Botta. B. Lazzerini, F. Marcelloni (200S), Context adaptation of mamdani fuzzy rule based systems, Int. J. Intell. Syst., 23 (4), pp.397-41S.

[9] y.E. Omurlu. S.N. Engin, 1. Yuksek (200S). Application of fuzzy PID control to cluster control of viaduct road vibrations, J. Vibration and Control, 14(S), pp.l201-121S.

[10] B. Shao. W. Rong. B. Guo, C. Ru and L. Sun (200S). Modeling and control of a novel piezoelectric actuated precision fast positioning system, Smart Mater. Struct. 17 (200S) 02S032 (7pp). doi: 10.1 OSS/0964-1726/17 /2/025032.

[11] D. Sun, J. Shan, Y. Su, H.H.T. Liu and C. Lam (2005), Hybrid control of a rotational flexible beam using enhanced PD feedback with a nonlinear diflerentiator and PZT actuators. Smart Mater. Struct. 14, pp.69-7S.

[12] K.L. Sorensen. W. Singhose, S. Dickerson (2007), A controller enabling precise positioning and sway reduction in bridge and gantry cranes, Control Engineering Practice, 15(7), pp.S25-S37.

[13] E. Mamdani. S. Assilian (1999). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Human-Computer Studies, 51(2), pp. 135-147.

[14] H. Nurhadi and YS. Tarng (2009). Experimental approached optimization of a linear motion performance with grey hazy set and taguchi analysis methods (GHST) for ball-screw table type, Int J Adv Manuf Tech, 44, pp.149-160.

[IS] K. Szabat. M. Kaminski. T. Orlowska-Kowalska (2007). Robust Control of an Electrical Drive using Adaptive Fuzzy Logic Control Structure with Sliding-Mode Compensator. EUROCON 2007 The

66

International Conference on Computer as a Tool, Warsaw, September 9-12, pp.1706-1711.

[16] H.1.M.T.A. Adriaens, W.L. de Koning, and R. Banning (2000). Modeling piezoelectric actuators, IEEE/ ASME Trans. Mechatronics, 5(4). pp. 331-341.

[17] T. Mitsuishi and Y Shidama (2009), Height defuzzification method on Loo space, C. Alippi et al. (Eds.): ICANN 2009, Part I, LNCS 576S. pp.S9S-607.

[IS] M. Mizumoto (1990), Improvement of fuzzy control (IV) - Case by product-sum-gravity method. In: Proc. 6th Fuzzy System Symposium, pp.9-13.