Robust Position Control of Ultrasonic Motor Considering Dead-Zone

Shogo Odomari1, Araz Darba1, Kosuke Uchida1, Tomonobu Senjyu1, and Atsushi Yona1 1Department of Electrical and Electronics Engineering, University of the Ryukyus, Japan E-mail: [email protected]

Abstract— Intrinsic properties of ultrasonic motor (high torque for low speed, high static torque, compact S1 D1 S3 D3 C L1 USM Load in size, etc.) offer great advantages for industrial ap- 1 Va plications. However, when load torque is applied, dead- E zone occurs in the control input. Therefore, a nonlinear Vb y controller, which considers dead-zone, is adopted for ul- C2 L 2 trasonic motor. The state quantities, such as acceleration, S2 D2 S4 D4 RE speed, and position are needed to apply the nonlinear controller for position control. However, rotary encoder causes quantization errors in the speed information. This − f MOS FET Micro ye paper presents a robust position control method for ultra- driver φ computer sonic motor considering dead-zone. The state variables y for nonlinear controller are estimated by a Variable r e Structure System(VSS) observer. Besides, a small, low Personal cost, and good response nonlinear controller is designed computer by using a micro computer that is essential in embedded system for the developments of industrial equipments. Fig. 1 Drive system of USM. Effectiveness of the proposed method is veriﬁed by the experimental results. 150 Va Vb 100 I. INTRODUCTION B V

, [V] 50 A

In recent years, ultrasonic motor(USM) is gaining V 0 attention as it has good characteristics and is small in size. The drive source of ultrasonic motor is ultrasonic -50 Control input -100 vibration of piezoelectric element. USM is expected to 0 5 10 15 20 be applied to robot actuator, high precision positioning Time [msec] and medical equipments[1]. The operating principle Fig. 2 Output voltages of two-phase inverter. of an USM has complicated speeds characteristics compared to a conventional electromagnetic motor which makes it a special kind of motor. control system at low cost, can upgrade response, and However, an USM has dead-zone in its control input can make the system design easily). with applied load torque[3]. Since H∞ controller is a This paper presents a digital implementation[8]-[10] linear controller[2], it cannot control the USM with un- of a nonlinear controller and a VSS observer by using known dead-zone. Therefore, nonlinear controller[4]- a micro computer for efficient position control of the [6], which is not used dead-zone inverse, is used for USM with unknown dead-zone. The state variables robust position control of USM with unknown dead- are estimated by the VSS observer and are used in zone. To apply nonlinear controller for position control nonlinear controller. The proposed nonlinear controller of USM, state variables such as acceleration, speed and is found satisfactory. position of the USM are needed. Speed information detected by a rotary encoder have quantization errors, II. SYSTEM CONFIGURATION especially in low speed region. Therefore, to estimate actual rotor speed accurately, a VSS observer[7] is pro- A. Driving system of USM posed with the possibility of decreasing quantization Configuration of the USM control system used in error. this study is shown in Fig. 1. The USM used in Essential industrial equipments are developed in the experiment is a traveling wave USM(SHINSEI small size, lightweight and power-saving technology, CORPORATION : USR-60). Output voltages of the by using embedded system. Usage of micro computer two-phase inverter are shown in Fig. 2. A traveling in embedded system, has advantages(such as it can dis- wave is formed on the stator surface when this voltage cretize all processing, can design stable and small size is applied to the stator, the rotor moves, and USM turns circuit in comparison to analog circuit, can construct to the opposite direction of the traveling wave. Table 1. Design specifications of USM. r Drive frequency 40 kHz + e Nonlinear u y USM RE ye Drive voltage 100 Vrms - Controller Rated current 53 mA/phase Rated torque 0.314 Nm Rated output power 3 W x Rated speed 9.0 rad/s VSS Observer Mass 0.240 kg Fig. 5 System configuration.

y(t) mr u(t) 6 5 4 mr 3 S1 S2 bl 2 0 u(t) 1 ml br 0 PWM Voltage [V] -1 ml u(t) 0 5 10 15 20 Time [μsec] Fig. 6 Dead-zone model. Fig. 3 PWM signal.

get sinusoidal voltage by making resonance with the equivalent circuit of the USM as shown in Fig. 4.

III. CONTROL ALGORITHM Fig. 4 Equivalent circuit of USM. System conﬁguration in this study is shown in Fig. 5. This section is designed nonlinear controller and VSS observer. Speciﬁcations of the USM is shown in Table 1. An electromagnetic brake of the load and the rotary A. Dead-zone model encoder are connected by a coupling. Electromag- Dead-zone model is shown in Fig. 6 and mathemat- netic brake is used to apply the load torque when ical equation is given as follows. voltage is applied. The rotary encoder is used for ⎧ ⎨ m u t − b b ≤ u t , detecting the produced pulse in proportion to angle r( ( ) r) r ( ) y t b

1 (3) T a3 u(t)=kds(t)+ bm (r (t)+Λv e(t)) + bm x¨ END a d u t 2 x − ( ( )) M s t , Fig. 7 Control algorithm. +bm ˙ m + sgn( ( )) (8)

where kd is a constant and M is the positive constant. We obtain the following equation when we rewrite Eq. State estimate error e is deﬁned by (7) using Eq. (8). e(t)=ˆx(t) − x(t). (16) s˙(t)=ud(t)+ a3x¨ + a2x˙ − bd(u(t)) and F estimation error α is deﬁned by u (t) −bm k s t d a3 x d ( )+ bm + bm ¨ α F {y t − y t }. a d u t = ˆ( ) ( ) (17) 2 x − ( ( )) M s t . + bm ˙ m + sgn( ( )) (9) where, xˆ(t) and yˆ(t) are the estimated value of x(t) and y(t). Using these equations, the design of the VSS u r(3) t ΛT e t where, d = ( )+ v ( ). observer can be obtained. Next, Lyapunov function is selected as follows. xˆ˙ (t)=A0xˆ + Ly + Bu + Bδ, 1 2 − α ρforα , V t s t . α =0 (18) c( )= bm ( ) (10) δ(t)= 2 0 for α =0. Differentiating the equation, we have where, ρ is a constant. The following error equation 1 can be obtained by differentiating Eq. (18). V˙c(t)= bm ss˙, uds(t) s(t) a x a x − d(u(t))s(t) = bm + bm ( 3 ¨ + 2 ˙) m ˙ e˙ = xˆ − x˙ ud a3 a2 −s(t) kds(t)+ + x¨ + x˙ = A0xˆ + Ly + Bu + Bδ − Ax − Bh − Bu bm bm bm = A0e + Bδ − Bh d(u(t)) − m + Msgn(s(t)) , α A0e − B α ρ − Bh for α =0 , −k s2 t − M|s t |≤−k s2 t . = (19) = d ( ) ( ) d ( ) (11) A0e − Bh for α =0.

Therefore if kd > 0 and Lyapunov function is negative, It selects Lyapunov function Vo as following equa- nonlinear controller can be stable. tion. 1 T C. Design of VSS observer Vo = e Pe. (20) 2 This section discusses the design of a VSS observer. By differentiating Vo with respect to t,wehave The continuous time system can be presented as 1 T 1 T V˙o = e˙ Pe+ e P e,˙ (21) x˙(t)=Ax(t)+Bh(t)+Bu(t), 2 2 y t Cx t . (12) ( )= ( ) where, by substituting Eq. (19), FC = B T P , T T T T T where, h is the nonlinear term or uncertainty param- e C F = α , and |α h|≤αρ,wehave eter. Since (C, A) is observable, constant matrix L exists, and conﬁguring to complex plane half, left- α =0 V˙ t 1 eT PA AT P e hand side of the following equation, A o can have stable o( )= 2 ( 0 + 0 ) −eT PB α ρ − eT PBh eigenvalue, α T T T T α T T T = −e Qe − e C F α ρ − e C F h Ao A − LC. (13) = −eT Qe −αρ − αT h = T T Therefore, there is ≤−e Qe −αρ + αρ = −e Qe, (22) T α PA0 + A0 P = −Q, (14) =0 V˙ 1 T T T o(t)= 2 e (PA0 + A0 P )e − e PBh P> A Q> T T satisfying 0 for o and positive matrix 0, = −e Qe − α h and F can be deﬁned as ≤−eT Qe, (23) T FC = B P. (15) which obtains e(t) → 0(t →∞) for V˙o. Table 2. Control parameters. S [−1.1 × 104 8.8×103 147.7 3.2] Kf 3.3 k 0.1 F 1 ρ 0.1 L [0.3 1.4×10−9 −7.1 × 10−5]T ζ 0.2 ωn 2200.8

Fig. 8 Conﬁguration of USM. position estimation. Figs. 10(f), and 11(f) provide that quantization error is reduced in estimated speed. Therefore, from the discussion of the experimental results, it can be said that the proposed nonlinear controller has robustness, provide effective control of the USM with unknown dead-zone.

V. C ONCLUSIONS The USM has an excellent performance and many other useful features. However, dead-zone occurs in Fig. 9 Micro computer and drive circuit. control input with applied torque. In this paper, we propose robust position control of the USM using the nonlinear controller considering dead-zone and the VSS observer. The nonlinear controller considering IV. EXPERIMENTAL RESULTS AND DISCUSSION dead-zone achieves robust position control of USM. A. Micro computer algorithm The VSS observer is a nonlinear observer, achieves reduction of quantization error and provides good Micro computer used in this research is SH7125 position estimation. of SH/Tiny series. It uses development environment Then, the dead-zone effect is reduced by the non- tool HEW(High-performance Embedded Workshop) linear controller with the VSS observer. Experimental of Renesas Technology Corp, and is written in C. results demonstrated good tracking performance and Features of SH7125 are PWM mode, timer interrupt, robustness of the proposed control and estimation and phase number count mode of MTU2. Using these scheme. features, we implemented nonlinear controller for the USM. REFERENCES The ﬂow chart of the control algorithm is shown in [1] T. Kenjyo, and T. Sashida, “An Introduction of Ultrasonic Motor”, Fig. 7, and the steps are given as following. Oxford Science Publications, 1993. [2] Tomohiro Yoshida, Tomonobu Senjyu, Mitsuru Nakamura, Atsushi Yona, STEP1 Initialize micro computer. Naomitsu Urasaki, and Hideomi Sekine, “Position Control of Ultrasonic Motors using Two-control Inputs H∞ Controller”, Electric Power Com- STEP2 Display opening message. ponents and Systems, vol. 35, pp. 741-755, 2007. STEP3 Wait for timer interrupt. [3] Tomonobu Senjyu, Mitsuru Nakamura, Naomitsu Urasaki, Hideomi Sekine, and Toshihisa Funabashi, “Mathematical Model of Ultrasonic STEP4 Determine the setting time. Motors for Speed Control”, Electric Power Components and Systems, STEP5 Decide the control input. vol. 36. pp. 637-648, 2008. [4] Hyonyong Cho, and Er-Wei Bai, “Convergence Results for an Adaptive STEP6 Estimate the state values. Dead Zone Inverse”, International Journal of Adaptive Control and STEP7 Display the measured value. Signal Processing, vol. 12, pp. 451-466, 1998. [5] Xing-Song Wang, Chun-Yi Su, and Heny Hong, “Robust adaptive control of nonlinear systems with unknown dead-zone”, Automatica, vol. 40, pp. B. Experimental Results 407-413, 2004. [6] Brice Beltran, Tarek Ahmed-Ali, and Mohamed EI Hachemi Benbouzid, USM experimental set up is shown in Fig. 8, and the “Nonlinear Power Control of Variable-Speed Wind Energy Conversion Systems”, IEEE Trans. Energy Conversion, vol. 23, pp. 551-558, 2008. micro computer and the drive circuit are shown in Fig. [7] Yi-feng Chen, and Tsutomu Mita, “Sliding Mode Control with Adaptive 9. Here, control cycle is 1 ms, data sampling time is VSS Observer”, T. IEE Japan, vol. 111-C, pp. 514-522, 1991. [8] Liping Guo, John Y. Hung, and R. M. Nelms, “Digital Implementation 20 ms, reference position of USM is sinusoidal wave, of Nonlinear Fuzzy Controllers for Boost Converters”, Applied Power reference position frequency is 0.2 Hz, drive frequency Electronics Conference and Exposition(APEC’06), pp. 1424-1429, 2006. f y . [9] Hsu-Chih Huang, and Ching-Chih Tsai, “FPGA Implementation of an is 41 kHz, initial position is =00radand they are Embedded Robust Adaptive Controller for Autonomous Omnidirectional implemented digitally. Control parameters are shown Mobile Platform”, IEEE Trans. Ind. Electron. , vol. 56, pp. 1604-1616, in Table 2. 2009. [10] Da Zhang, and Hui Li, “A Stochastic-Based FPGA Controller for an Experimental results by using the proposed nonlin- Induction Motor Drive With Integrated Neural Network Algorithms”, ear controller are shown in Figs. 10, and 11. Figs. IEEE Trans. Ind. Electron. , vol. 55, pp. 551-561, 2008. 10(a), and 11(a), provide good control for both no load and applied load. Figs. 10(d), and 11(d), provide good

1 r y 1 r y [rad]

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Rotor Position Position Rotor Position Rotor 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time[s] Time[s] (a) Reference position and measured position. (a) Reference position and measured position.

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Position error e [rad] Position error e [rad] -0.2 -0.2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time[s] Time[s] (b) Position error. (b) Position error.

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-45 -45 Control input input Control Control input input Control -90 -90 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time[s] Time[s] (c) Control input. (c) Control input.

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ω ω 1 e ω 1 e ω [rad/s] [rad/s] ω ω

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Estimate speed speed Estimate 0 1 2 3 4 5 6 7 8 9 10 speed Estimate 0 1 2 3 4 5 6 7 8 9 10 Time[s] Time[s] (f) Measured speed and estimate speed. (f) Measured speed and estimate speed. Fig. 10 Experimental result with nonlinear Fig. 11 Experimental result with nonlinear controller(τL=0.0Nm). controller(τL=0.2Nm).