Servohydraulic cylinder position control using a neuro-fuzzy controller

Pergamon

0957-4158 (95) 00027-5

Mechatronics Vol. 5, No. 5, pp. 497-512, 1995 ~) 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved. 0957-4158/95 $9.50+0.00

SERVOHYDRAULIC CYLINDER POSITION CONTROL USING A NEURO-FUZZY CONTROLLER

MING-CHANG SHIH and CHUNG-PIN TSAI

Department of Mechanical Engineering, National Cheng-Kung University, Tainan, Taiwan 70101, R.O.C.

(Received 10 October 1994; revised 15 February 1995; accepted 15 February 1995)

Abstract--A neuro-fuzzy controller to control the position of a servohydraulic cylinder is designed. Bell-shaped membership functions and a look-up table are used to construct the network structure, and then one can train the learning data by the back-propagation method. The designed neuro-fuzzy controller is implemented in a microcomputer to control a servohydraulic cylinder.

1. INTRODUCTION

The hydraulic control system plays an important role in industrial automation, e.g. machine tools, plastic injection molding machine, testing machine, aircraft, auto- mobiles, etc., because the hydraulic system has the following advantages: high speed response, high power output, ease of control, etc. [1]. The servohydraulic control system is a nonlinear time variant system and is difficult to simulate with an accurate mathematical model because of the oil viscosity, friction forces between cylinder and piston, oil flow through the hydraulic servovalve and the variable loading on the system [2, 3]. Fuzzy control technology is based on the linguistic variables--human thought process; the input signal is fuzzified at first, then goes through the fuzzy reasoning process under operational experience and expert knowledge, and finally the control signal is defuzzified and sent out [4-9]. In the literature [10-12], servohydraulic system control has been executed by fuzzy controller successfully; however, which control parameters to use to satisfy the performance requirement is determined through trial and error method. Neuro-fuzzy control technology combines the operational experience, expert knowledge and neuro-network learning ability [13-17]. In the present paper, based on the bell-shaped membership function, look-up table fuzzy reasoning and network learning ability, the structure of the neuro-fuzzy controller is built up and the controller can then be designed. Therefore, the servohydraulic cylinder system can be controlled precisely, even under variable load. The servohydraulic cylinder system is also a so-called "mechatronic" system [16], which can be divided into the electronic part (controller design and sensor technology) and the mechanical part (hydraulic control valve and mechanical actuator system). The object of this paper is to design a neuro-fuzzy controller for implementa- tion in a microcomputer and to determine the robustness performance for position control of the hydraulic cylinder.

497

498 MING-CHANG SHIH and CHUNG-PIN TSAI

2. LAYOUT OF THE EXPERIMENTAL DEVICES

The layout of the servohydraulic cylinder control system is shown in Fig. 1. It consists of a symmetrical cylinder 1 rigidly connected to a load generator. The position of the cylinder is measured by a linear scale with a precision of 0.01 mm. The oil flow rate to the cylinder is controlled by a four-way electro-hydraulic servovalve, which is controlled by a 16-bit microcomputer with a 12-bit D/A converter. The load generator consists of cylinder 2 with the same dimensions as cylinder 1. The variable load will be generated by the variable electrical signal to the proportional pressure control valve. The system parameters are listed in Table 1.

3. FUZZY AND NEURO-FUZZY CONTROL THEORY

3.1. Fuzzy control theory

Fuzzy control theory has been widely applied in control systems in recent years [7, 9]. For the fuzzy control system, the input state variables become the linguistic variables through the fuzzification process and, based on the knowledge and adequate decision making logic rules, the fuzzy reasoning process is undertaken; finally, through the defuzzification process the control signal can be calculated, determined through the centre of area method (COA), and sent to the system.

3.2. The structure of neuro-fuzzy control

The network structure for neuro-fuzzy controller design is expressed as shown in Fig. 2; there are five layers. The typical neuron is shown in Fig. 3; if the input signal x = kx~ is multiplied by a weighting factor kw i and sent to another neuron, it will be managed by a function a ( . ) to obtain NET = a(x) , and excited by the function f ( - ), the output of the neuron is y = f(NET).

(a) The first layer. The input signals will be distributed to the next neuron according to the following relationships.

l w i = l, a = lxi and f = a, (1)

where l wq is the weighting factor of the first layer.

(b) The second layer. membership function, e.g. requirement.

The linguistic variables are normalized through the the bell-shaped function can be used to satisfy the

(2Xi- mq) 2 -wij = 1, a = ~ and f = e a, (2)

oij where 2wq is the weighting factor of the second layer, mq is the centre of the membership function and oq is the width of the membership function.

(c) The third layer. The degree of adaptation can be obtained with the minimum algorithm [18].

Servohydraulic position control using neuro-fuzzy controller 499

1

I I

I I _

13

( 5--

14 . . . . 15.~16 ( , ~ 0 6

18 I

61

2

slHi8

i! iI i i 'I I

9

ii 11

li:

9

1 Cylinder I 11 IBM compatible microcomputer 2 Cylinder 2 12 Hydraulic pump 3 Servovalve 13 Accumulator 4 Servo amplifier 14 Pressure control valve 5 Proportional pressure control valve 15 Filter 6 Linear pulse scale 16 Cheek valve 7 Decoder of pulse scale 17 Heat exchanger 8 Pressure sensor 18 Oil tank 9 Amplifier 19 Directional control valve

10 Pressure gauge

Fig. 1. Layout of the experimental devices.

Table 1. System parameter values

Component Brand Specifics

Flow servovalve Yuken (SVD-F11-10-7.5-11) Amplifier Yuken (AMS-7.5-S-20) Pressure servovalve Rexroth (3DSE2EH10) Cylinder Linear scale Futaba (SM-255A) Decoder Hewlett Packard (HCTL-2020) CLK Aukey Pressure sensor Kyowa (PGM-200KE) Presure amplifier Kyowa (DPM-612A) AD/DA card Flytech

Computer IBM compatible

Four-port three-way +7.5 mA limit +10 V/+7.5 mA +10 V/+70 bar Stroke 30 cm, area 30 cm 2 0.0005 era/bit 16-bit counter 1.8432 MHz 0-200 kgfcm -2 0-5V/0-200 kg f cm -2 12-bit A/D: 12-bit/0-5 V D/A: +-10 V/12-bit PC/AT-286, 8 MHz


O 1 0 2 • • • O m

I 5th Output layer

4th I Normalized layer

3rd [ Adaptation

layer

2nd Membership

layer

I st Input layer

- - U 1 U 2 • . . U n

Fig. 2. Network combination structure of neuro-fuzzy controller,

E

W

I/E

min

mem.

I

ki2 !

k x n k th layer

Fig. 3. Function of a typical neuron.

kyj

(d) The fourth layer. Using the centroid method to defuzzify the results of the fuzzy reasoning process, it is expressed as follows:

~li~= 14 xi a = 4x i and f = a. (3)

(e) The fifth layer. The defuzzification process in the fourth layer is only the


normalization of the degree of adaptation. It must be calculated as follows: n

5 5 a = ~ X i Wij and f = a. (4) i=1

3.3. Back-propagation learning process

If the output value Ok cannot satisfy the desired value dk, then it must be modified by the back-propagation method. The objective function is defined as

~k~__l lk~__ 1 E = (5)

where 6 k is the error between the desired and the output value. Assume the weighting parameter of the outside layer can be modified as the

following equation:

DE w(t + 1) = w(t) - rIDw (6)

DE _ DE Df Da, (7) Dw Df Da Dw

where r/is the learning rate factor. Using the look-up table and bell-shaped function, the adjustable parameters include

the value of the weighting factor wij, the centre of the membership function mrs, and the width of the membership function trrs. The weighting factor wij is adjusted from the outside layer to the inside layer. In the present paper, the weighting factor w~j is assumed to be 1. The adjustments of the membership function, mrs and ars, are as follows:

DE _ D E Df 5 Da____5___ 5 - ( d j _ o j ) . S x i j (8) Dw 0 Df 5 Da 5 DWij

wq(t + 1) = wij(t) + rl(dj - oj). 5xq (9)

m l J j~=l~[~=lJ 5 1} DE _ ~, ~_~_E I ~ DOk E ~xij D4Xp''~q Doxrs (10)

~mr s k=X O0 k ki=l ~'=l D4Xpq D3Xrs Dm~s

E m 1 j~Jl~[p~=l J 5 D3Xrs]l, (11)

where

aE - O k - - d ~ (12)

aOk

DOk _ 5W/j (13) DSxij


( t'~l [ li~=l '1]~=1 ( lp~=l ~ ~5Xij ~4Xpq ~3Xrs ) ] ) mrs(t + 1) = mr~(t ) + 17 (dk - Ok) Wq (14) q=l 94Xpq 33Xrs 3 m r s

{ nk~_z 1 [i~l ]. _~1 (~__1 ~o5xij ~4XpqO3Xrst] I (15) Ors(t + 1) = Ors(t ) + 17 (dk - Ok) wq q=l~axpq 93Xrs 30rs/I)

~Sx~j _ [ ~4Xpq

4 x 4 x

(~i~j4Xi] - 4Xi])/(~i~i4Xij)2

if/=/= p o r j 4= q

if i = p, a n d j = q

(16)

34xiy _ [ 1 if4Xpq : 3Xrs

33Xrs t 0 if4Xpq =/= 3Xrs (17)

33X's -- 2(2X" --~ m~s)e-i:x'-mr')/°~" (18)

D3x~ - 2(eXrs- mrs)ee -(2xr-m~')/'o2~'. (19) 3 ~Or.s Or. `

4. NEURO-FUZZY CONTROLLER DESIGN

According to the above theory, the neuro-fuzzy controller can be designed in the following steps.

4.1. Determination of the system state variables

For the electro-hydraulic cylinder position control system, the position error e(t) between the reference input and the actual output, the error change rate Ve(t) and the control signal u(t) are chosen as the state variables, so that the control system is defined as a two-input e(t), Ve(t), single-output, u(t), system.

4.2. Choice of the membership function

In the experimental process, the error e(t) and the velocity Ve(t) are normalized and defined as three linguistic variables, zero (ZO), small (S) and big (B), by using the bell-shaped membership function. The correlations of the membership function values vs the normalized error and the normalized velocity are shown in Fig. 4.

4.3. Choice of fuzzy reasoning rules

For the learning process, one can define the initial values based on the training data distribution in the phase plane to accelerate the convergence speed and avoid the


:zl.

1.0

0.5

0

(a)

ZE S B

O.5 1.0

ZE S B

::1" 1"0

0.5

0 0.5 1.0 (b)

Fig. 4. (a) Membership function value vs normalized error. (b) Membership function value vs normalized velocity.

danger of stopping the network combination. The fuzzy reasoning rules of every network can be expressed as the following equation (see Table 2):

RI: if ((e is ZO) and (Ve is ZO)) then Yl = Cl

R2: if ((e is S) and (Ve is ZO)) then Y2 = c2

R3: if ((e is B) and (Ve is ZO)) then Y3 = c3

R4: if ((e is ZO) and (Ve is S)) then Y4 = C4

Rs: if ((e is S) and (Ve is S)) then Y5 = c5

R6: if ((e is B) and (Ve is S)) then Y6 = c6

R7: if ((e is ZO) and (Ve is B)) then Y7 = c7

R8: if ((e is S) and (Ve is B)) then Y8 = c8

R9: if ((e is B) and (Ve is B)) then Y9 = c9.

(20)

4.4. Construction of the combination structure of the neuro-fuzzy controller

The combination structure of the neuro-fuzzy controller for the cylinder position control system is shown in Fig. 5. In the input layer, there are two inputs e(t) and Ve(t); the membership layer includes three different linguistic variables. The minimum membership function values are obtained in the adaptation layer, through the normalized layer according to Table 2 and using the centre of area method the control output value can be calculated.

Table2. Linguistic normalized error and velocity variable vs fuzzy reasoning values

Ve e

ZO S B

Z O ¢1 ¢2 ¢3 S ¢4 c5 ¢6 l ¢7 c8 £9


u (t+ 1)

5th Output layer

4th Normalized

layer

I F

nve (t) ne (t) Velocity Error

3r [ Adaptation layer

2nd Membership ]

layer L

I st I Input layer

Fig. 5. Network combination structure o[ neuro-fuzzy controller for servohydraulic cylinder.

4.5. Analysis of the training data

Owing to the different type of learning data in the small network, the training data must be separated to avoid conflict of the margin through boundary compromise. As the data approach the object, they are separated in more detail, into a fine table and a coarse table (see Fig. 6), as follows.

Fine table: range of error e(t) 0-2 .0 mm range of velocity Ve(t) 0-40.0 mm s 1 size of network ge x go 0.5 m m x 20.0 m m s -1.

Coarse table: range of error e(t) 0-8 .0 m m range of velocity Ve(t) -100 .0 -100 .0 m m s 1 size of the network ge x gv 2.0 mm x 20.0 m m s -1.

The total number in the small network is 46 and it has the anti-symmetry relationship if the value of e is f rom 0 to -0 .8 .

4.6. Network learning

From the above structure of the network (Fig. 5), one knows that the servohydraulic cylinder control system is a two-input, one-output system. The fuzzy reasoning rules are reduced owing to the data separation. One can obtain 46 small network results separately by using the back-propagat ion learning method.


Ve

100

-!00 -0.8

I 35 36 37 38 I

gv=2Omm/s 31 32 33 34

~ 1 ~ 28 29 30

24 25 26

" ~ " . . . . . . . ~ W 17 I 18 19 20 ----~e

13 14 15 16

I 9 I0 II 12

I 5 6 7 8 _ r l Coar~ table ge=2mm . . . .

Fine table ge=O.5mm 1 2 3 4 I I I

0 0.8

Fig. 6. Training data distribution in phase plane.

For example, in the No. 28 small network:

Da ta ranges: 2 .0-4 .0 m m and Ve = 60.0-80.0 m m s -1. Initial conditions are listed in Table 3a and shown by the broken line in Figs 7 and 8.

Table 3. (a) Initial values of No. 28 network before training. (b) Final values of No. 28 network after

training

(a)

V e e

ZO S B

ZO 2.0 4.0 7.0 S 2.0 3.5 5.5 B 1.5 3.0 4.5

(b)

V e e

ZO S B

ZO 2.0008 4.0229 7.0397 S 1.8658 3.5471 5.5040 B 1.4117 2.9744 4.2625


1.0

0.8

--t 0.6

0.4

0.2

0 0

- - After training . . . . . . Before training

ZE S B

0.2 0.4 0.6 0.8 1,0 Normalized error

Fig. 7. Membership function value vs normalized error in No. 28 network before and after training.

After training . . . . . . Before training ZE S B

1.0

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1.0

Normalized velocity

Fig. 8. Membership function value vs normalized velocity in No. 28 network before and after training.

Table 4 shows the relationship between the learning factor and the learning cycle for the convergence of the network. If the learning factor is too small, the converging speed is also too slow, e.g. the learning cycle is 307 for the learning factor )7 = 0.0025; if the learning factor is large, the network may be diverge; therefore we must choose an optimal value of learning factor. Figures 7 and 8 also show the membership function values of the error and the velocity before and after learning. The root mean

Table 4. Relation between learning factor and convergence cycle

~/ 0.1 0.075 0.05 0.025 0.01 0.005 0.0025

Ncycle Diverge Diverge 120 78 114 183 307


square error of the output converges to be 0.004 with the learning factor 0.025 after 78 learning cycles, which is shown in Fig. 9.

5. EXPERIMENTAL RESULTS

In the following experimental results, the system pressure is set to be 100 bar, and the sampling time is 20 ms.

(a) Cylinder under no load

The time response of the position control of the cylinder under no load is shown in Fig. 10. According to the characteristics of fuzzy control, if the position error is not smaller than 8.0 mm, the situation does not belong to the range of fuzzy reasoning, the servovalve is kept full open and the cylinder moves forward at full speed; then, according to the neuro-fuzzy control theory, the cylinder can be controlled to the required position. The difference of the steady state error of the input (in the case that it is a square wave input) may be due to the zero-shift characteristic of the servovalve.

(b) Effect of load change (disturbance)

Figure 11 indicates the time responses of the cylinder position, control signal, error and the velocity as the load changes. The cylinder position is controlled according to the command signal 50 mm at first, and then, at the time instant 3 s, the system load pressure is changed to 50 bar. One can see that the cylinder position control signal and the velocity have only a small change and return very quickly, wherever the steady state error is 0.09 mm, which is larger than that of the no load condition. One can see that the system has very good disturbance rejection with such a controller.

0.025

t Moo,olt

0~ 0

lUlllllllilllmllllllilllmllllllllllllllll 20 40 60 80

Leamfing cycle

Fig. 9. The relationship between root mean square error and learning cycle in No. 28 network.


80 [ Actual output L .......... Ref. input

60.~. .... essl =0.01 mm

0

-20 0 1 2 3 4 5

Time (s)

, ~ 10 ~ > o o -10

5O

-50

70

-70

f

I I I " " ' : I I

7*%'..**..°

..... °°°1. ~ °oo°O"

I I t I

. . . . . . . . . . . . .

%'1 .... " .° . '

I I [ I I 0 I 2 3 4 5 6

Time (s)

Fig. 10. Time responses of the cylinder position control using square wave input with neuro-fuzzy control without load.

(c) Comparison of results with those from the Ziegler-Nichols PID controller and fuzzy PID controller

Figure 12 shows the dynamic responses of the control system with the PID controller and fuzzy PID controller [10]. The controller parameters Kp, Ki and Kd of the PID controller are based on the Ziegler-Nichols law and are 1.875, 4.5 and 0.375, respectively. The parameters based on fuzzy PID control method are 1.0, 0.01 and 0.005, respectively. One can see that they have the following disadvantages: the time response of position with the PID controller shows large overshooting and it takes a long time to try to find the control parameter values with the fuzzy PID controller.


60

0=40

.~ 2o

0

80

o r.)

ess 1 = 0.01 ram i

t s 1 = 0 . 8 8 s e m ~ = O . 9 4 m m

t t r l = 0 . 6 4 s I I , v -- --

I I I I

Actual output . . . . . . . . . . R e f , i n p u t - - - Lo~l p~ure (bar)

ts2=0.36s ess2=0.9mm

-20 I I I I I 0 ! 2 3 4 5 6

Time (s)

lo [ ............ - o "" ~ .....

-|o I I I I I

5O

-50

;"~ 70 I ............ "%:" ~[

-7O I I I 0 1 2 3

Time (s)

I I 4 5 6

40~

Fig. 11. Time responses of the cylinder position control using step input with neuro-fuzzy control under load change.

(d) Comparison of results with those from the model reference adaptive controller

Some of the exper imenta l results o f the system with the mode l reference adapt ive control ler are shown in Fig. 13 [19]. T he t ime response o f the posit ion has also t racked the reference mode l very well under load change. H o w e v e r , the reference mode l must be chosen and designed. O n e can see that the t ime responses are faster and the settling t ime is shor ter with the neuro- fuzzy controller .

6. C O N C L U S I O N S

F r o m the above exper imenta l s tudy, the fol lowing conclusions can be made .

(a) Neuro- fuzzy control ler design does not require a system model , has learning

70

60

10

50

"= 30 o ¢x.

20

10

40

• - 4

e~ o 2

30 ~ . Fuzzy-PID

20

!

i I

( - - Fuzzy-PID . . . . PID

% %

i I I I 1 2 3 4

Time (s) (a) Dynamic responses of position

iI - - Fuzzy-PID

11 . . . . PID

I I

I 14 IIi 11 I

',i

I I L I l 2 3 4 5

Time (s)

(b) Dynamic responses of control input


-10 I 1 t I 0 l 2 3 4 5

Time (s) (c) Dynamic response of load

Fig. 12. T ime re sponses of the cy l inder pos i t ion cont ro l us ing s tep input wi th P ID and fuzzy P I D control .

ability and can obtain the strategy of the controller design through the adjust- ment of the controller parameter.

(b) With the neuro-fuzzy controller, the system has very good adaptivity and can reject the effect of uncertainties and external load.

40

30

O

"~ 20

0

1.0

10

2 4 6 8 10

50

50

- I . 0 I I I I 0 2 4 6 8 10

v 0

-50 0 2 4 6 8 10

10


o

-10 I I I I 2 4 6 8 10

Time (s)

Reference model A Response of cylinder position

Fig. 13. T ime responses of the cylinder position control using step input with model reference adaptive control.

(c) Using the look-up table of fuzzy reasoning rules in neuro-fuzzy controller design, one can set the initial values of the training data, obtain converging results earlier and avoid the stalling of the network.

(d) Further study of the application of the neuro-fuzzy controller in the servohydraulic control system includes the following: how to find a faster learning

512 M I N G - C H A N G S H I H and C H U N G - P I N T S A I

f ac to r , h o w to use a n o t h e r fuzzy r e a s o n i n g ru le ( in s t ead o f us ing a l o o k - u p

t ab l e ) , h o w to c o m b i n e the u n s u p e r v i s e d s e l f - o r g a n i z e d l e a r n i n g m e t h o d to f ind

the su i t ab le c o r r e l a t i v e p a r a m e t e r and e v e n to o r g a n i z e an o p t i m a l m o d e l

f o l l o w i n g n e u r o - f u z z y c o n t r o l l e r des ign m e t h o d .

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Cybern. 20(2), 404-435 (1990). 6. Klir G. J. and Folger T. A., Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Cliffs,

NJ (1992). 7. King P. J. and Mamdani E. H., The application of fuzzy control system to industrial process.

Automatica 13,235-242 (1977). 8. Bouslama F. and Ichikawa A., Fuzzy control rules and their natural control laws. Fuzzy Sets Syst. 48,

65-86 (1992). 9. Mamdani E. H. and Assllian S., An experiment in linguistic synthesis with a fuzzy logic controller.

Man-Machine Studies 7, 1-13 (1975). 10. Shih M. C. and Chen P. C., An experimental study on the position control of a hydraulic cylinder using

a fuzzy logic controller. JSME, Series III 34(4), 481-489 (1991). 1l. Shih M. C. and Liaw S. M., Hydraulic servocylinder position control using a fuzzy-logic controller.

Proceeding of 2rid JHPS International Symposium on Fluid Power, pp. 279-284, Tokyo, Japan, September (1993).

12. Zhao T. and Virvalo T., Fuzzy state controller and its application in hydraulic position servo. Proceeding of 2nd JHPS International Symposium on Fluid Power, pp. 417-422, Tokyo, Japan, September (1993).

13. Lin C. T. and Lee C. S. G., Neural network based fuzzy logic control and decision control and decision system. IEEE Trans. Comput. 40(12), 1320-1336 (1991).

14. Bouslama F. and Ichikawa A., Fuzzy control rules and their natural control laws. Fuzzy Sets Syst. 48, 65-86 (1992).

15. Horikawa S., Furhashi T. and Uchikawa Y., On fuzzy modeling using fuzzy neural networks with the backpropagation algorithm. IEEE Trans. Neural Networks 3(5), 801-806 (1992).

16. Waton J., Fluid power control system: some mechatronics issues. Proceeding of "Flucome '91", pp. 1-14, San Francisco, CA (1991).

17. Dransfield P., Stecki J. S. and Lin P., Neural nets--their potential for intelligent control of fluid power drives, lOth Aachner Fluid Technisches Kolloquium, Volume 2, pp. 407-418, Aachen, Germany (1992).

18. Li Y. F. and Lau C. C., Development of fuzzy algorithms for servo systems. IEEE Contr. Syst. Mag. 9(3), 65-72 (1989).

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Servohydraulic cylinder position control using a neuro-fuzzy controller