Position control of ultrasonic motor based on two-degree-of-freedom control system with self-tuning PID type neuro-controller

IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, August 30 – September 1, 2004

POSITION CONTROL OF ULTRASONIC MOTOR BASED ON TWO–DEGREE–OF–FREEDOM CONTROL SYSTEM WITH SELF–TUNING PID TYPE NEURO-CONTROLLER Akihiro Naganawa ∗ Kanya Tanaka ∗∗ Masato Oka ∗∗∗ ∗

Department of Mechanical Engineering, Akita University, Akita, 010–8502, Japan ∗∗ Department of Electrical and Electronic Engineering, Yamaguchi University, Ube, 755–8511, Japan ∗∗∗ Department of Mechanical Engineering, Ube National College of Technology, Ube, 755–8555, Japan

Abstract: The ultrasonic motor (USM) is expected to be applicable to various fields, but it is difficult to precisely control the USM as it contains nonlinear properties caused by various frictions, changes of temperature, fluctuation of loadmass and so on. In this paper, we propose a position control method for the USM based on a two-degrees-of-freedom control system with a self-tuning PID type neuro controller to hold optimal tracking performance. The gains of the PID controller are self-tuned by a neural network which adaptively suppresses the influence of the plant perturbation and disturbance. The experimental results show the effectiveness of the proposed method. Keywords: Ultrasonic motor, Position control, Two-degrees-of-freedom control system, PID controller, Neural network

1. INTRODUCTION

according to drive conditions such as an increase in temperature of the USM by frictional heat, change in load and so on. For these reasons, it is thought that it is difficult to realize position control performance by the PID control technique with fixed gains.

The USM has many features, for example small size, light weight, low speed high torque and so on. Therefore, the applications as an actuator of small motion control are expected (The Robotics Society of Japan. Robotics Handbook., 1990). Moreover, the USM does not generate electromagnetic noise and is not influenced by it either as magnetic action is not the principle of its driving. Therefore, its use in environments such as medical treatment and the welfare field which cannot use an electromagnetic motor is also expected. However, the principle of driving of the USM is a friction drive, and essentially has a nonlinear property for its input-output relation. Also, the dynamics changes

So, in this research, a position control technique of the USM based on two-degrees-of-freedom control system which has a neural network (NN) to dynamically adjust the gains of the PID controller is discussed (K. Fukuda and Harada, 1998; K. Tanaka and Morioka, 2002). The control system is based on the MRAC (Model Reference Adaptive Control) system (Astrom and Wittenmark, 1995). Until now, a conventional MRAC

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Reference model r

e

Controller

u

v

the PC (Personal Computer). The control signal calculated within the PC is transmitted to a drive circuit via DO (Digital Output) board, which rotates the USM. The driving method which is used is the phase difference control method, since this method has a small hysteresis property and the input and output properties are linear in relation. The phase difference control circuit consists of digital circuits which used the shift register, and minimum phase difference is 0.0245 rad which is divided into 128 between -π/2 to π/2 rad. Moreover, the driving frequency used is 0.5 kHz higher than the resonance frequency, 35.5 kHz, of the USM. Table 1 shows the specifications of the experimental system.

y

Plant

Tuning algorithm

Fig. 1. Conventional MRAC system system was constituted as shown in Fig. 1. In this control system, the feedback controller must be tuned to enhance the characteristic for both the reference response and feedback properties in response to perturbation of the plant, influence of disturbance and so on. Therefore, the proposed MRAC system is made up of a two-degreesof-freedom control system. The feedback controller adaptively suppresses, effecting the nonlinear property and perturbation of the USM, by adjusting the gains of a PID controller using a NN (Omatu and Khalid, 1995; K. J. Astrom and Ho, 1993). Moreover, the feedforward controller with the desirable transient response characteristic is also realizable. The effectiveness of the proposed method can be verified by experimental results.

Fig. 3 shows the characteristic of phase difference verses the revolving speed of the USM. Fig. 3 has three plots. One; No-load 0s, which means data taken immediately after the motor is turned on with no load, two; No-load 600s, which means data taken 600s after the motor is turned on with no load and three; Load, which means data taken from adding a load of 0.125Nm. From Fig. 3, it can be seen that in the case of no-load, when the applied phase difference rotates the USM continuously, it is also able to rotate in the reverse direction. Also, with the progression of time, because in the highspeed rotation region of Fig. 3, the rotation speed starts to slow down, it is understood that the motor properties change as a function of time. This perturbation is considered to be similar to the characteristic of a piezoelectric element with change in temperature. On the other hand, in the case that a load is applied to the USM, it can be seen that a dead band is produced into the phase difference around the zero phase region as well as the slow down of the rotation speed in the high-speed rotation region. As mentioned above, it can be said that a big change is produced in the speed characteristic of the USM by the length of time which the USM is in operation. As this operation time is increased the frictional heat produced is also increased (frictional heat

2. CHARACTERISTICS OF USM Fig. 2 shows a block diagram of the experimental system. The USM, a rotary encoder and load are connected through coupling on the common axis. The signal containing position information from an encoder is inputted into the counter board on

PC Counter DO board board Rotary encorder

Drive circuit

Load

USM

Fig. 2. Experimental system of USM Table 1. Specification of experimental system USM Drive ciruit Encorder Load

No-load rotational speed Strating torque Self-holding torque Driving frequency Phase difference Minimun phase difference Resolution min ∼ max

15.708 rad/s 0.39 Nm 0.39 Nm 35.5kHz −π/2 ∼ π/2 rad 0.0245 rad 6.28 × 10−5 rad 0 ∼ 0.25 Nm

820

Fig. 3. Characteristic properties of USM

4. TUNING OF PID GAINS VIA A NEURAL NETWORK

change). The existence of a dead band by the addition of a load also produced a large change in the speed characteristic.

As has been mentioned so far, due to the error resulting from a strong non-linear characteristic and frictional heat produced by the USM, good performance cannot be obtained by using a PID controller with fixed gains. Therefore, a method of tuning the gain of the PID controller by a neural network (NN) is considered.

Since the USM has the characteristic as stated above, it is thought that control by a PID controller with fixed gains will not result in good performance.

Fig. 5 shows a three-layer structure NN, which is used in this research. In the input layer of the NN, the output v(k), v(k − 1) of the reference model and the output y(k − 1), y(k − 2) of the plant are inputted. The hidden layer has a number of 6 neurons. OP (k), OD (k) and OI (k) are outputted from the output layer of the NN. If the fixed gain of a PID controller is set to kP , kI and kD , the gain of a controller will be adjusted as follows by the NN,

3. TWO-DEGREES-OF-FREEDOM CONTROL SYSTEM Fig. 4 shows the structure of the two-degrees-offreedom control system with a NN. P˜ (z) is the USM, P (z) is the model for the USM and F (z) is the reference model. It is assumed that P (z) is a minimum phase system. C(z) is the PID controller and is described by the following formula, KI z −1 + KD (1 − z −1 ). C(z) = KP + 1 − z −1

KP (k) = kP + OP (k), KI (k) = kI + OI (k), KD (k) = kD + OD (k).

(1)

By this control system, e(k) in Fig. 4 becomes zero, when the characteristic of the USM does not change (P˜ (z) = P (z)) and disturbance does not exist. Then the output y(k) coincides with the output of the reference model v(k), the desired response can be obtained.

kP , kI and kD are set constant values greater than zero beforehand. The NN adjusts the value of OP (k), OI (k) and OD (k) so that the PID gain KP (k), KI (k) and KD (k) may become the optimal value to account for the characteristic change and disturbance. Self-tuning is performed so that this may become an ideal controller for the USM.

On the other hand, in the case that there is a perturbation of the USM (P˜ (z) = P (z)) and disturbance exists, e(k) does not go to zero so a feedback controller is needed to suppress these effects.

The output from an input layer uses a linear function, and the output from the hidden layer is the sigmoid function as shown by the following formula. 1 (5) f (x) = 1 + e−ax The output from an output layer uses an linear function so that the arbitrary gains of positive/negative can be taken out.

P -1(z)F(z) r

F(z)

v

w

e

C(z)

u

~

P(z)

y

Neural Network

A connection weight from a unit p at the input layer to a unit q at the hidden layer is denoted by hid . Similarly, a connection weight from a unit q wqp at the hidden layer to a unit r at the output layer out . Thus, we have at the output is denoted by wqr layer

Fig. 4. Block diagram of two-degrees-of-freedom control system

I1 I2 I3 In

(2) (3) (4)

Or (k) = Ornet (k),  out wrq Hq (k), Ornet (k) =

OP(k) OI(k) OD(k)

(6) (7)

q

and at the hidden layer Hq (k) = f (Hqnet (k)),  hid wqp Ip (k). Hqnet (k) =

(8) (9)

p

The criteria function used here is a squared error function E given by

Fig. 5. Neural networks

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hid Similarly, the renewal of weighting function wqp (k) of the hidden layer from the input layer is as follows.   ∂E ∂y(k) hid (k) = −η Δwqp ∂u(k − 1) ∂y(k) r net ∂u(k − 1) ∂Or (k − 1) ∂Or (k − 1)  × ∂Or (k − 1) ∂Ornet (k − 1) ∂Hq (k − 1) ∂Hq (k − 1) ∂Hqnet (k − 1) (23) × hid (k − 1) ∂Hqnet (k − 1) ∂wqp  out = αη (k − 1) δrout wrq

1 {v(k) − y(k)}2 , 2 1 (10) = e2 (k). 2 In learning of the NN, a back propagation algorithm is used to minimize the error function E. E=

According to the steepest method, out (k) = −η Δwrq

∂E out (k ∂wrq

− 1)

,

(11)

out where η is a learning rate and Δwrq (k) is the out , change to be made to the connection weight wrq that is, out out out (k) = wrq (k) − wrq (k − 1). Δwrq

r

× Hq (k − 1){1 − Hq (k − 1)} · IP (k − 1).

(12) 5. EXPERIMENTAL RESULTS

The chain rule yields ∂y(k) ∂u(k − 1) ∂E ∂y(k) ∂u(k − 1) ∂Or (k − 1) ∂Or (k − 1) ∂Ornet (k − 1) . (13) × out (k − 1) ∂Ornet (k − 1) ∂wrq

Experimental results show the validity of this research. Fig. 6 shows the model of the USM. u(k) represents the phase difference and y(k) represents the position of the USM. A transfer function G(s) derived from the step response when the phase difference was changed from 0 to π/2 rad is as follows, 10078.1 (25) G(s) = s + 5000.0

out (k) = −η Δwrq

On the other hand, the control input u(k) from NN-PID controller is represented as the following discrete-time representation, u(k) = u(k − 1) + KP (k){e(k) − e(k − 1)} + KI (k)e(k − 1) + KD (k){e(k) − 2e(k − 1) + e(k − 2)}. (14)

For the USM model in Fig. 6, a discrete-time plant model P (z) is derived from P (s) discretized at a sampling period of 4 ms of using the zeroorder hold method. The reference model F (z) was derived by discretized the following transfer function 2

m . (26) F (s) = s+m

As mentioned above, each element of Eq. (13) is expressed as follows. ∂E = −e(k) (15) ∂y(k) ∂y(k) = η (16) ∂u(k − 1) ∂u(k − 1) (17) ∂Or (k − 1) ⎧ ⎨ e(k − 1) − e(k − 2) (r = P ) = e(k − 2) (r = I) ⎩ e(k − 1) − 2e(k − 2) + e(k − 3) (r = D) (18) ∂Or (k − 1) =1 (19) ∂Ornet (k − 1) ∂Ornet (k − 1) = Hq (k − 1) (20) out (k − 1) ∂wrq

The fixed gains of the PID controller are kp = 0.3, ki = 0.00006 and kd = 0.006, respectively. Moreover, η in Eq. (21) is 0.01 and α in Eq. (24) is 0.3. The initial values of connection weights of the neural network were used as the random number of the range of 0 to 0.5. Fig. 7 – 9 show the experimental results for m = 10 in Eq. (26). Fig. 7 shows the result immediately after a drive. Fig. 9 shows result after 600 seconds. Fig. 8 shows the enlargement figure immediately after a drive. From these figures, the output y(k) of the USM coincides well with the output v(k) of the reference model, although the output y(k) is vibrating near around the 4 second mark. Moreover, a good response y(k) was obtained after 600 seconds.

In Eq. (16), ∂y(k)/∂u(k − 1) cannot be derived, since the plant is unknown. For this reason, it sets with constant η  and is a constituent of the learning rate η. Thus, the renewal of the weighting out (k) of the output layer from the function wrq hidden layer is as follows, out (k) Δwrq

=

(24)

ηδrout Hq (k

− 1),

P(s)

(21)

u

where δrout = e(k)

∂u(k − 1) . ∂Or (k − 1)

(22)

.

G(s)

y

Fig. 6. Model of the USM

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1 s

y

Fig. 12 – 14 show the results of changing the parameter m of the reference model. Fig. 12 shows the result of the output when m = 5 immediately after the start of experiment. It can be seen that the output does not vibrate and so a good response is obtained. Fig. 14 shows the result of m = 15. In this case, although vibration was seen immediately after the drive, a good response was obtained after that.

Fig. 10 and 11 show the time progress of the PID gains which were tuned by the NN. When the results of immediately after the experiment was started are compared with the results of those after 600 seconds, it turns out that good adjustment is performed in the amplitude of a gain to characteristic change (temperature change) of the USM.

6. CONCLUSION In this research, the method based on the MRAC (model reference adaptive control) system was proposed as the position control technique of the USM. The composition of this control system is a new technique based on a two-degrees-of-freedom control system which can consider independently both a reference response property and a feedback property. The validity of the proposed technique was verified by the experiment results. Fig. 7. Outputs (Immediately after start of experiment, m = 10)

In future work, the method proposed in this research will be compared with results obtained from the PID control technique which will be used to check the validity of the proposed method.

Fig. 8. Enlargement figure of outputs (Immediately after start of experiment, m = 10)

Fig. 10. PID gain (Immediately after start of experiment, m = 10)

Fig. 9. Outputs (600s after start of experiment, m = 10)

Fig. 11. PID gain (600s after start of experiment, m = 10)

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Moreover, the proposed control technique is due to be applied to a remote-control robot using the USM (J. Karasawa and Tanaka, 2004).

J. Karasawa, A. Naganawa and K. Tanaka (2004). Control performance of teleoperated robot using ultrasonic motor. SICE Symposium on Adaptive and Learning Control pp. 11–14. K. Fukuda, T. Kamano, T. Suzuki T. Yasuno and H. Harada (1998). High accuracy positioning system with ultrasonic motor using frequency and phase neural networks. The Japan Society of Applied Electromagnetics and Mechanics 6, 350–357 (in Japanese). K. J. Astrom, T. Hagglund, C. C. Hang and W. K. Ho (1993). Automatic tuning and adaptation for pid controllers – a survey. Contr. Eng. Practice 1, 699–714. K. Tanaka, M. Oka, A. Uchibori Y. Iwata and H. Morioka (2002). Precise position control of ultrasonic motors using pid controller combined with nn. Trans. IEE Japan 122– C, 1317–1324 (in Japanese). Omatu, S. and M. Khalid (1995). Neuro-control and its applications. Springer–Verlag. The Robotics Society of Japan. Robotics Handbook. (1990). Corona Publishing Co. Ltd. (in Japanese).

REFERENCES Astrom, K. J. and B. Wittenmark (1995). Adaptive control. Addison Wesley Publishing Company.

Fig. 12. Outputs (Immediately after start of experiment, m = 5)

Fig. 13. Outputs (Immediately after start of experiment, m = 15)

Fig. 14. Enlargement figure of outputs (Immediately after start of experiment, m = 15)

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