Design of Adaptive Fuzzy Sliding Mode Control for a Traveling-wave Ultrasonic Motor

Design of Adaptive Fuzzy Sliding Mode Control for a Traveling-wave Ultrasonic Motor

Youssef Baba, Mostafa Bouzi, *Ismail Lagrat, Mounir Derri

Univ Hassan 1 ,Laboratory of Mechanical Engineering, Industrial Management and Innovation Faculty of Science and Technology, Settat, Morocco

*National School of Applied Sciences, Khouribga, Morocco

Abstract In this paper, we propose an adaptive fuzzy sliding mode control scheme for a traveling-wave ultrasonic motor. Because external disturbances and parameter variations, it is difficult to design a conformable model-based control scheme. In order to deal with this problem, the proposed control law is combined a fuzzy logic and the sliding mode control (SMC). Firstly, the equivalent controller is obtained by a fuzzy strategy. Secondly, we use a boundary layer approach to avoid chattering problem and satisfactory trajectory tracking. An on- line adaptive tuning algorithm for the consequent parameters in the fuzzy rules is designed. Simulation studies have shown that the presented adaptive design of fuzzy sliding mode controller performs very well in the presence of unknown disturbances.

KeywordsSliding Mode Control; Fuzzy Control; Traveling-

introduces the proposed adaptive fuzzy sliding mode controller. Simulation results are presented in Section 4. Section 5 offers our concluding remarks.

1. USR60 MODEL

   Traveling-wave ultrasonic motors are complex electromechanical devices in which a mechanical resonant vibration is excited in the stator through proper forcing piezoelectric ceramics. This stator vibration is transformed into a rotation through friction contact between the stator and rotor.

   The model of piezoelectric and stator can be described by the following equation

   Wave Ultrasonic Motor.

   M D C v Fd

   (1)

   I. INTRODUCTION

   The Piezoelectric traveling-wave ultrasonic motor (TUSM) has excellent performance and many useful features such as high holding torque, high torque at low speed, quiet operation, simple structure, compact size, and no electromagnetic interferences [1].

   Nevertheless, the control characteristics of TUSM are complex and no-linear. The motor parameters are time- varying due to increase in temperature and change of motor drive operating conditions [2-6]. In order to overcome these problems, several control systems have been proposed [6-8]. But the complexity of these algorithms is far beyond the fixed parameters PID control requiring higher on-line calculation ability and the resulting increase of the cost of hardware as well as software of the system. In [9-11] authors tray to use the nonlinear characteristic of fuzzy and neural control to deal with the nonlinear problems of TUSM control, except that we must find a compromise between the complexity of the algorithm proposed and its real time implementation.

   In this work, an adaptive fuzzy sliding mode control (AFSMC) scheme is proposed for a traveling-wave ultrasonic motor type USR60. The proposed control law is based to combine a fuzzy logic and the SMC. Firstly, in order to realize the control law without the model of system, a fuzzy logic

   with represents the modal amplitude of the vibrating system (ceramics and stator), M is the total mass matrix of system(ceramics and stator), D is the structural damping

   matrix assumed to be diagonal, and C is the total stiffness

   matrix. H is the electromechanical coupling matrix and v is

   the voltage excitation vector. The term Fd is a nonlinear modal force vector to consider the interaction between the stator/rotor-contact.

   TABLE I. SPECIFICATIONS OF USR60

   Drive frequency [kHz] 40.0

   Drive voltage [Vrms] 100

   Rated torque [Nm] 0,32

   Rated speed [rpm] 130

   Rotor inertia [Nms2] 7,2.10-6

   Rotor damping in spinning direction 0.05

   In dealing with the dynamics of the rotor, two degrees of freedom must be taken into account: first the rotation of the rotor and second the motion in z-direction. The motion in z- direction is represented by the quantity w . The dynamics of the vertical rotor motion is obtained by the following equation

   controller is designed to estimate the equivalent controller.

   mr w dz w Fz – Fn

   (2)

   The fuzzy parameters are estimated on-line by the adaptive laws. Secondly, a sliding mode control is used, but this control

   with mr is the mass of the rotor, d z is the damping of the

   scheme suffers from chattering problems. In order to

   vertical motion, and Fn is the applied axial force. The

   guarantee the stability of the sliding mode system, the boundary of the uncertainties has to be estimated.

   The paper is organized as follows. In section 2, a mathematical model of USR60 is presented and a reference model is proposed in order to control the motor. Section 3

   equation of rotational motion is calculated by

   Jr dr Tr -TL

   (3)

   where Jr is the rotor inertia, dr denotes the damping in spinning direction, and TL is the applied torque.

   In Fig.1 the Matlab-simulink model of USR60 is described. Fig.2 shows the relation between speed and torque measurements when the driving frequency is 40 kHz .The speed versus drive frequency for different applied load torques is represented in Fig.3. The speed of the TUSM has its maximum at the mechanical resonant frequency of the motor. So, any deviation from this frequency degrades the motor performance. However, this effect seems more serious for frequency decrements. The characteristic curves in Fig.2 and Fig.3 are derived from calculations using the Matlab-simulink model represented in Fig.1.

2. DESIGN OF ADAPTIVE FUZZY SLIDING MODE CONTROL

   1. Design of Sliding Mode controller

      In order to apply sliding mode control, let us write (3) as

      – a bu h(t)

      Fig. 2. Speed-torque characteristics

      where

      a, b 0

      h(t)- T J -1- J -1J -a bu

      (4)

      is the equivalent

      L r r

      disturbance torque,

      J ,

      a , and b

      are the parameter

      variations from normal value. Fig.4 shows the block diagram

      of motor speed control using AFSMC, where c denotes the

      given speed, denotes the actual speed. Considering e -c as speed tracking error, the time-varying surface of the sliding mode control is introduced as [12-13]

      s e e

      (5)

      with is a strictly positive constant, whose choice we shall interpret later. If sliding mode exists, the following condition satisfies

      s e e 0

      Therefore, the control law is obtained as [13]

      u ue us

      (6)

      (7)

      Fig. 3. Speed-frequency characteristics

      The term us (t) is called the hitting controller and obtained as

      where

      ue (t) is a solution of (6) and called the equivalent

      us – K0sgn(s)

      (8)

      controller.

      where sgn is the sign function

      sgn(s) 1 if s 0

      sgn(s) 1 if s 0

      In order to ensure the existence condition of sliding mode using (8), the condition ss 0 must be satisfied [12-13].

      ss s (e e)

      s(-a( – a) ( – a) d (t) – b( – a)K0 sgn(s)) where ( a)d (t) h(t) ( a)h(t) c c assumed to be bounded.

      So with

      (9)

      and

      bK0 | -a d (t) |

      (10)

      the value of s and s have opposite signs and the state reaches

      the sliding line s 0

      after a finite time interval. Inequality

      Fig. 1. Matlab-simulink model of USR60

      (10) determines the frequency needed for enforcing the sliding mode; as a result, the control error is steered to zero.

      While control law (7) achieves the target dynamics (5) exactly, the presence of the switching term K0 sgn(s) implies

      s

      s

      that in practice undesirable control chattering will occur. To suppress the chattering and obtain a band-width-limited

      where is the membership functions width.

      The local control of fuzzy system is defined as

      e

      1 2

      ui i s i s

      (13)

      controller that best approximates the exact behavior described

      The local control actions i

      and i

      which are contained in

      above, the switched action K0 sgn(s) is replaced by a smooth

      the parameters vector , T are calculated on-line by

      interpolation in a boundary layer neighboring the sliding surface as [13]

      1 2

      the following least squares algorithm

      us K0sat(s)

      The sat function is defined as

      sat(s) -1s if s

      (11)

      (i) (i 1) P(i1)* (i1)*((i) (i1)T *(i1))

      (i) (i1)T *P(i1)* (i1)

      P(i1)* (i1)* (i1)T *P(i1)

      sat(s) sgn(s)

      if otherwise

      P(i) P(i 1)

      (i) (i1)T *P(i1)* (i1)

      where is the boundary layer thickness.

      with P is a covariance matrix,

      is an estimation of the

   2. Adaptive Fuzzy Sliding Mode control

   According to the universal approximation of TS-fuzzy

   9

   systems [14], an optimal fuzzy controller u exists such that

   vector parameters , and denotes the forgetting factor. By fuzzification, the fuzzy inputs s and s are obtained. The fuzzy equivalent controller is obtained by defuzzification

   e

   i

   the approximation error of fuzzy controller can be defined as

   Ai Bi ue T

   ue ue

   (12)

   u

   i1

   (14)

   e 9 i i

   where is the approximation error and assumed to be bounded. In this work, there are nine rules in a fuzzy base and they have the following form

   where [s s]T .

   A B i1

   1 2

   Rule i : if s is Ai and s is Bi then ue is i s

   Finally the control law in (7) becomes

   T

   1 i 9

   u K0sat(s)

3. SIMULATION RESULTS

   (15)

   In order to evaluate the performance of our control scheme, the simulation results of the proposed controller are achieved. The simulation study of the system was implemented using Matlab. The specification of USR60 is shown in Table I. Choose the sliding surface as s e5e . The

   control parameters: K0 0.02 , and 0.01 . The initial

   Fig. 4. Speed control block diagram

   e

   where s and s are the inputs variables of the fuzzy system and u is its output variable. The linguistic terms Ai and Bi are defined as

   Ai , Bi{N (negative), Z (zero), P(positive)}

   and they are characterized by their corresponding membership functions

   values: (1) 0 , and P(1) 1000I . The unknown disturbances are modeled by the randn function from Matlab library. Fig.6 shows the speed tracking response by

   applying the control law (15) represented in Fig.5. The

   dynamic response is good. The speed characteristics of motor changed when the load torque is applied or small parameter variations occur, but the control gains of the fuzzy sliding mode controller also changed to compensate the parameter

   variations. Adaptive parameters 1 and 2 are represented in Fig.7 and Fig.8 respectively.

   N (s) exp(-0.5 -2 (s 1)2 ); N (s) exp(-0.5 -2 (s 1)2 )

   P (s) exp(-0.5 -2 (s 1)2 ); P (s) exp(-0.5 -2 (s 1)2 )

   Z (s) exp(-0.5 -2s2 ); Z (s) exp(-0.5 -2s2 )

   Fig. 5. Total control law u

   Fig. 8. Evolution of 2

   It is clear that the proposed AFSMC control scheme introduces excellent performance where the controller variables track their reference values exactly in a very short time.

4. CONCLUSIONS

In this paper, an adaptive fuzzy sliding mode structure has been proposed for speed control of a traveling-wave ultrasonic motor USR60. The strategy of control is based to combine the fuzzy logic and the sliding mode control to guarantee the stability and the tracking performance. The main advantages of the proposed speed controller are robustness to parameter variations and external load disturbances. Simulation results confirm the abovementioned claims for the control scheme in TUSM control drive.

Fig. 6. Speed tracking response

Fig. 7. Evolution of 1

REFERENCES

Temperature evaluation of traveling-wave ultrasonic motor considering interaction between temperature rise and motor parameters. Ultrasonics, 57, 159-166.

1. Mashimo, T., & Terashima, K. (2015). Dynamic analysis of an ultrasonic motor using point contact model. Sensors and Actuators A: Physical, 233, 15-21.

2. Bolborici, V., Dawson, F. P., & Pugh, M. C. (2014). A finite volume method and experimental study of a stator of a piezoelectric traveling wave rotary ultrasonic motor. Ultrasonics, 54(3), 809-820.

3. Maas, J., Ide, P., Fröhleke, N., & Grotstollen, H. (1995, October). Simulation model for ultrasonic motors powered by resonant converters. In Industry Applications Conference, 1995. Thirtieth IAS Annual Meeting, IAS'95., Conference Record of the 1995 IEEE (Vol. 1, pp. 111-120). IEEE.

4. Ou, W., Yang, M., Meng, F., Xu, Z., Zhuang, X., & Li, S. (2015). Continuous high-performance drive of rotary traveling-wave ultrasonic motor with water cooling. Sensors and Actuators A: Physical, 222, 220- 227.

5. Gencer, A. (2014, September). A new speed/position control technique for travelling wave ultrasonic motor under different load conditions. In Power Electronics and Motion Control Conference and Exposition (PEMC), 2014 16th International (pp. 65-70). IEEE.

6. Jingzhuo, S., Yu, L., Jingtao, H., Meiyu, X., Juwei, Z., & Lei, Z. (2014). Novel intelligent PID control of traveling wave ultrasonic motor. ISA transactions,53(5), 1670-1679.

7. Shi, J., Zhao, J., Cao, Z., Liang, Y., Yuan, L., & Sun, B. (2014). SELF- TUNING FUZZY SPEED CONTROLLER OF TRAVELLING WAVE ULTRASONIC MOTOR. International Journal on Smart Sensing & Intelligent Systems, 7(1).

8. Mu, S., Tanaka, K., Nakashima, S., Alrijadjis, D., & Tomimoto, H. (2014, October). Intelligent PID control based on PSO type NN for USM. In Consumer Electronics (GCCE), 2014 IEEE 3rd Global Conference on (pp. 612-615). IEEE.

9. Chen, T. C., Ren, T. J., & Lou, Y. W. (2013). Ultrasonic Motor Control Based on Recurrent Fuzzy Neural Network Controller and General Regression Neural Network Controller. In Computational Intelligence (pp. 291-305). Springer Berlin Heidelberg.

10. Utkin, V. (1993). Sliding mode control design principles and applications to electric drives. Industrial Electronics, IEEE Transactions on, 40(1), 23-36.

11. Slotine, J. J. E., & Li, W. (1991). Applied nonlinear control (Vol. 199, No. 1). Englewood Cliffs, NJ: Prentice-hall.

12. Yan, S., Sun, Z., & Li, Z. (2010, June). Universal approximation of TS fuzzy systems constructed dynamically-MISO cases. In American Control Conference (ACC), 2010 (pp. 6472-6479). IEEE.