Solenoid valve

Mechatronics 15 (2005) 683–702

Intelligent switching control ofpneumatic actuator using on/off solenoid valves

Kyoungkwan Ahn a,*, Shinichi Yokota b

a School of Mechanical and Automotive Engineering, University of Ulsan, San 29, Muger 2dong,

Nam-gu, Ulsan, 680-764, Republic of Koreab Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta,

Midori-ku, Yokohama 226-8503, Japan

Received 15 November 2003; accepted 28 January 2005

Abstract

The development of a fast, accurate, and inexpensive position-controlled pneumatic actu-ator that may be applied to a variety of practical positioning applications with various externalloads is described in this paper. A novel modified pulse-width modulation (MPWM) valvepulsing algorithm allows on/off solenoid valves to be used in place of costly servo valves. Acomparison between the system response of the standard PWM technique and that of themodified PWM technique shows that control performance was significantly increased. Astate-feedback controller with position, velocity and acceleration feedback was successfullyimplemented as a continuous controller. A switching algorithm for control parameters usinga learning vector quantization neural network (LVQNN) has newly proposed, which classifiesthe external load of the pneumatic actuator. The effectiveness of this proposed control algo-rithm with smooth switching control has been demonstrated through experiments with variousexternal loads.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Pneumatic; Switching control; On/off solenoid valve; Pulse width modulation; Neural network;Intelligent control

0957-4158/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechatronics.2005.01.001

* Corresponding author. Tel./fax: +82 52 259 2282.E-mail address: [email protected] (K. Ahn).

mailto:[email protected]

684 K. Ahn, S. Yokota / Mechatronics 15 (2005) 683–702

1. Introduction

Pneumatic control systems play very important roles in industrial automation sys-tems owing to the advantages of low cost, easy maintenance, cleanliness, ready avail-ability and cheap power source, etc. [1]. A particularly well-suited application forpneumatic actuators is the position control of robotic manipulators, loading/unload-ing systems, air balance systems and grippers, where stiff and lightweight structuresare critical. Unfortunately, pneumatic actuators are subject to high friction forces,dead-band due to stiction, and dead-time due to the compressibility of air. Thesenonlinearities make accurate position control of pneumatic actuators difficult toachieve.As a result, a considerable amount of research has been devoted to the develop-

ment of various position control systems for pneumatic actuators [2–9]. Many ofthese systems, though successful, use expensive proportional servo valves and pres-sure sensor feedback loops and the external loads are also assumed to be constant orslowly varying.The object of this paper is to implement inexpensive on/off solenoid valves, rather

than expensive servo valves, to develop a fast, accurate, inexpensive and intelligentpneumatic control system taking account of the changes of external loads. However,with solenoid valves, fine motion control is difficult to achieve because of the limita-tion of valve response time and its discrete on/off nature. Previous studies [10–19]have tried to implement on/off solenoid valves for the position control of pneumaticactuators. Noritsugu has used the PWM method and solenoid valve to control thevelocity and the position of pneumatic cylinders [10,11]. Muto has used differentialPWM method to control hydraulic actuators [12]. These systems were successfulin addressing smooth actuator motion in response to step inputs. However, somelimitations still exist, such as deterioration of the performance of transientresponses due to cases of abrupt changes of external loads. To overcome thisproblem, the MPWM (modified pulse width modulation) on/off valve controlscheme and a switching control algorithm by LVQNN (learning vector quantizationneural network), have been newly proposed and control performance has beenexperimentally.

2. Pneumatic position control system

2.1. Experimental apparatus

A schematic diagram of the control system is shown in Fig. 1. An IBM-compat-ible computer (Pentium 1 GHz) was applied to control the on/off solenoid valve andto get experimental data. The stroke and the bore diameter of the rodless cylinder(SMC, MY1M32-1000L) were 1000 mm and 32 mm, respectively, and differentmasses could be attached to the table of the rodless cylinder. The displacement ofthe cylinder was measured by a linear scale (US Digital, resolution 0.05 mm) andthe air pressure were also measured by air pressure sensor (SMC, ISE 40-01-22L)

Counter

Chamber 1 Chamber 2

+

DC(5V/24V)Converter

Dig.Out

PressureSensor 1

PressureSensor 2

A/D

V1 V2 V3 V4 V5 V6 V7 V8

Fig. 1. Schematic diagram of pneumatic control system.

K. Ahn, S. Yokota / Mechatronics 15 (2005) 683–702 685

which was fed back to the computer through a 24-bit counter board (Advantech,PCL 833) and A/D board (Advantech, PCI 1731), respectively. The control signalwas changed into pulse width modulated signal through our proposed MPWM algo-rithm. The control signal was sent to control the solid state relay and eight pneu-matic on/off solenoid valves (MAC, 111B-872JD). The bandwidth of the valvesused in the experiments was 250 Hz. The experiments were conducted under a pres-sure of 0.5 MPa and all control software was coded in C. A photograph of the exper-imental apparatus was shown in Fig. 2.

2.2. Characteristics of the on/off solenoid valve

When using on/off solenoid valves to control the position of a pneumatic cylinder,it is necessary to understand the characteristics of the on/off solenoid valve. This in-cludes the dead-time and the rise time of valve, the maximum current, etc. Fig. 3shows the current of the solenoid valve when the ON signal was applied to theon/off solenoid valve for 100 ms. The current was measured from the voltage dropacross the resistance, which is serially connected to the solenoid valve.In the zoom graph of Fig. 3, the valve started to move after tBa and the valve was

fully opened after tBbB. This means that the valve started to move when the durationof the ON pulse to the valve was longer than 4 ms.

2.3. Modified PWM Algorithm

When using on/off solenoid valves to control the position of the cylinder, the con-trol input u must be converted into the pulse width modulation on/off signal in each

Fig. 2. Photograph of experimental apparatus.


solenoid valve. Previous studies have used (1) a conventional PWM scheme [10–12],where the valve opening time was proportional to the magnitude of the control inputand (2) a modified PWM scheme [20,21], where the dead-time of valve and othernonlinear terms were considered. In modified PWM scheme, the dead-time of on/off solenoid valves is important in the design of our PWM algorithm because thevalve failed to move if the duration time of ON signal to the solenoid valve wasshorter than the dead-time of the valve. To overcome this problem, we applied amodified PWM algorithm, where the minimum pulse width time was added to theoutput of the conventional PWM even if the control input was very small. However,the control input became zero within any allowable position error in order to preventthe actuator�s oscillation near the reference position. The limit of position error isexpressed by e in Eq. (1).

UMPWMðtÞ ¼signðUðkÞÞ � U 0; ðk � 1ÞT 6 t 6 ðk � 1ÞT þ tpðkÞ0; ðk � 1ÞT þ tpðkÞ 6 t < kT

(

tpðkÞ ¼tsðkÞ; 0 6 ts 6 ðkÞ < T

T ; tsðkÞ P T

(

tsðkÞ ¼0; jyref � yj 6 e

ts1; jyref � yj > e

(ð1Þ

0 50 100 150 200

0.0

0.2

0.4

0.6

Current

Ref

eren

ce V

olta

ge [

V]

Zoom

100 ms

Time (ms)

Cur

rent

(A)

0

1

2

3

4

5 Reference

Input

56 58 60 62

0.0

0.1

0.2

0.3

0.4

Zoom

tβ = 4.6 [ms]

tα = 4 [ms]

Cur

rent

[A]

Time [s]

Fig. 3. Measurement of valve opening time.


ts1ðkÞ ¼jUðkÞjUmax

T þ tDZ; 0 < jUðkÞj 6 Umax � ð1� tDZ=T Þ

T ; jUðkÞj > Umax � ð1� tDZ=T Þ

8<:

UMPWM(t) MPWM outputU0 valve opening signaltp ON duty ratio of valve for one MPWM cyclets modified duty ratio with dead-band regionts1 modified duty ratio of On/Off valvey position of rodless cylinderyref reference positione maximum error limittDZ dead-time of valve


T MPWM cycle timet continuous timeU(k) sampled control input of u(t)Umax saturated control input for MPWM modulatork discrete sequence

Our proposed modified PWM (MPWM) valve pulsing method is shown in Eq. (1)and one example of the MPWM output with respect to the continuous control signalis shown in Fig. 4 where the MPWM period is 10 ms. Particularly, the same size ofpaired on/off solenoid valves (V1 and V2, V3 and V4, V5 and V6, V7 and V8 in Fig.1) were used in order to improve the control performance of this pneumatic system.Fig. 5 shows a comparison of system responses according to four types of valvecombinations.From Fig. 5, it is understood that system responses of this pneumatic system de-

pended more on on/off valves in the exhaust part (V3 and V4, V7 and V8 in Fig. 1)rather than on/off valves in the supply part (V1 and V2, V5 and V6 in Fig. 1).To improve the response of pneumatic cylinder, (V1,V2) and (V5,V6) valves are

used as intake and exhaust valves, respectively, and ON duty ratio of each valve be-comes MPWM cycle time if the control input is positive and greater than UBmaxB. Ifthe control input is positive and less than UBmaxB, only V1 and V5 valves are used asintake and exhaust valves, respectively, where the opening time of each valve is deter-mined by the modified PWM algorithm (Eq. (1)).Conversely, (V7,V8) and (V3,V4) valves are used as intake and exhaust valves,

respectively, if the control input is negative and less than �UBmax. If the control

1.0 1.1 1.2

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-60

-40

-20

0

20

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cont

rol i

nput

[%]

Time(s)

control input MPWM signal

Fig. 4. Example of the MPWM signal.

0.0 0.2 0.4 0.6 0.8 1.00

200

400

600Po

sitio

n(m

m)

Time(s)

Intake Exhaust Valve Valve

V1 V5 (V1,V2) V5 V1 (V5,V6) (V1,V2) (V5,V6)

Fig. 5. Comparison of system responses according to valve combination.


input is negative and greater than �UBmaxB, only V7 and V3 valves are used asintake and exhaust valves, respectively, where the opening time of each valve is alsodetermined by the modified PWM algorithm (Eq. (1)).

3. Intelligent switching control algorithm of the pneumatic system

3.1. The overall control system

Design procedure of the proposed control system is shown in Fig. 6. In order toappropriately adjust the control parameters according to external loads, load condi-tions must be recognized using dynamic information of the pneumatic system in anon-line manner. Here we applied the LVQNN as a means of supervisor, which clas-sifies four typical external loads (0, 10, 20, 30 Kg). The overall control system with theLVQNN as supervisor was shown in Fig. 7. To control the pneumatic system, a three-state feedback control algorithm was applied in this paper, where the position, velo-city and acceleration of the rodless cylinder were used. It is reported that a three-statefeedback control method was effective in the pneumatic control system [18,19]. Thecontroller output at sampling sequence k becomes

uðkÞ ¼ kpðydðkÞ � yðkÞÞ � kvvðkÞ � kaaðkÞ ð2Þ

where u(k), y(k), v(k), a(k), yd(k), kp, kv and ka represent control input, position,velocity and acceleration of the rodless cylinder, reference position, proportionalgain, velocity gain and acceleration gain, respectively. The control input u is

ModifiedPWM

Pneumatic Actuatorwith various loads

yd +

-

u y

SwitchingAlgorithm

by LVQNN

3 State FeedbackController

kp

Fig. 7. Structure of newly-proposed control algorithm.

Selection of Optimal Control Parameterswith respect to Each External Inertia Load

by Experiment

Generation of Training Data for LVQNNwith respect to Each External Inertia Load

Off-Line Training of LVQNNusing Training data and Target Class

Application of Trained LVQNN and SmoothSwitching Algorithm to the Position Control

Fig. 6. Design procedure of the proposed control system.


calculated from position, velocity and acceleration of the rodless cylinder and isconverted into a pulse width modulation signal to operate the on/off solenoid valves.

3.2. Classification of load conditions by the LVQNN

The LVQNN was utilized here for classifying input patterns with inherent classintersections. The core of the LVQNN is based on the nearest-neighbor method


by calculating the Euclidean distance. During training stage, the values of weightsused to from the reference vectors are adjusted according to the patterns of inputsamples in order to match with desired classes. The distance of an input vector tothe weight vector of each node in competitive layer is computed.The node of a particular class which has the smallest distance is declared to be the

winner. The weights will be moved closer to the class if it is the expected winningclass, otherwise they will be moved away. After the training process is finished,the LVQNN is then ready for classifying unknown input vectors. During the evalu-ating stage, the distance of an unknown input vector to each of all classes� referencevectors is calculated again. After computing the distance to this input vector andeach reference vector, the node which has the closest distance to the input vectoris declared to be the winner. Then the unknown input vector will be assigned tothe class which the reference vector belongs to.

3.2.1. Structure of the neural classifier

Fig. 8 shows the architecture of the LVQNN, where P, y,W1,W2, R, S1, S2 and Tdenote input vector, output vector, weight of competitive layer, weight of linearlayer, the number of neurons in the input layer, competitive layer, linear and targetlayer, respectively. The LVQNN is a supervised learning algorithm and is composedof two layers. The first is the competitive layer which functions the learning of theclassification of the input vector. The second is the linear layer which classifies thecompetitive layer according to a designer�s intention.For the generation of training data, the valves were fully opened in order to move

the rodless cylinder in a forward direction. This is explained in the next section.For each input vector P(j), a winner neuron W1(i, j) is chosen to adjust its weight

vector:

kP ðjÞ � W 1ði; jÞk 6 kP ðjÞ � W 1ðk; jÞk; 8k 6¼ i ð3Þ

Fig. 8. Structure of the LVQNN.


The weight vector W1(i, j) is updated to the next step by the following Kohonen([22]) learning rule:

DW 1ði; jÞ ¼ k � a1ðiÞ � ðpðjÞ � W 1ði; jÞÞ ð4Þwhere k is the learning ratio and a1(i) is the output of competitive layer.The output of competitive layer a1(i) becomes 1 if the input pattern is classified

correctly and 0 if the input pattern is classified incorrectly.

3.2.2. Data generation for the training of the LVQNN

In this section, the training data for the LVQNN is explained in detail. The inputand output data of LVQNN are shown in Fig. 9. The input data for the LVQNN isthe velocity of the rodless cylinder, and the output of the LVQNN is a class amongfour discrete classifications, where four cases were classified according to the externalload. For example, class 3 indicates that the range of the external load was approx-imately between 15 kg and 25 kg, as shown in Table 1. To obtain training data forthe LVQNN, a series of experiments were conducted under 12 different load condi-tions, as shown in Table 1. In experiments for the generation of training data, theinitial position of rodless cylinder was set to 80 mm and the four valves (V1, V2,V5 and V6 in Fig. 1) were fully opened for 0.4 s in order to make the rodless cylindermove in a forward direction at full speed.Therefore, a total 12 cases of experiments were carried out to prepare for the

training data of the LVQNN, as shown in Fig. 10. As the load increased, the systemresponse became slower and had more of a time delay. From Fig. 10, the externalload may be classified by using the LVQNN. In the training stage of the LVQNN,the numbers of input vectors were adjusted from 27 to 32 and the number of neuronsin the competitive layer was adjusted from 20 to 40 in 10 steps in order to obtain theoptimal parameters of the LVQNN. From Table 2, it is understood that the optimalnumber of input vectors and the number of neurons of the competitive layer were 30and 40, respectively.

3.3. Proposition of the smooth switching algorithm

If the load condition is different from its load condition used in the training, theoutput of the LVQNN may not belong to the predefined class, but to mixed classeswith different ratios in each class (i.e. if the external load is 25 kg, it may belong to 3

LVQNN...

v(k)

v(k+1)v(k+2)

v(k+n-3)v(k+n-2)v(k+n-1)

Class

Fig. 9. Training data for the LVQNN.

Table 1Classification of external load

No. Class 1 (kg) Class 2 (kg) Class 3 (kg) Class 4 (kg)

1 0 8 18 282 2 10 20 30

3 4 12 22 32

0.0 0.1 0.2 0.3 0.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0kg 8kg 18kg 28kg 2kg 10kg 20kg 30kg 4kg 12kg 22kg 32kg

Velo

city

(m/s

)

Time(s)

Fig. 10. Experimental results for training data generation.

Table 2Training success rate of the LVQNN

NCL NIV

27 (%) 28 (%) 29 (%) 30 (%) 31 (%) 32 (%)

20 94.1 95.5 97.2 98.5 97.5 98.130 98.8 98.7 99.3 97.7 99.1 10040 98.8 98.1 99.3 100 100 100

Leaning rate: 0.04; training epoch: 10,000; NIV: number of input vector; NCL: number of competitivelayer.


or 4 class). Therefore, the following switching algorithm was proposed to apply toabrupt changes of external loads as follows:

Table 3Optimal parameters of 3-loop controller and MPWM

Class no. Kv Umax Kp Ka

1 0.07 60 1.5 10�3

2 0.12 1003 0.166 1504 0.195 200


classðkÞ ¼ k � classðk � 1Þ þ ð1� kÞ � classðkÞ ð5Þwhere k is discrete sequence, k is forgetting factor and class(k) is the output of theLVQNN at k time sequence. The selection of k is dependent on one cycle time ofMPWM signal and the convergence rate of each control parameter. In our experi-ment, one cycle time of MPWM signal and convergence time of classification wasset to be 10 ms and 0.1 s, respectively. The initial value of class was set to be 1and k was selected to be 0.53 in order to reach the target class within 1% error withrespect to the true class within 0.1 ms. By this setting of k, the calculated class be-comes 3.9901 (within 1% error) after 0.1 ms if the estimated classes all belongs to 4.The optimal parameters of the 3-loop controller and the MPWM were obtained

by a trial-and-error method through experiments, which are shown in Table 3. FromTable 3, it may be understood that velocity feedback gain (Kv) and saturated controlinput (Umax) increased according to an increase of external loads and had to be auto-matically switched according to the external load.The following equations are the updating laws of velocity feedback gain (Kv) and

saturated control input (Umax):

KvðkÞ ¼ KvðpÞ � ðq� classðkÞÞ þ KvðqÞ � ðclassðkÞ � pÞ ð6Þ

UmaxðkÞ ¼ UmaxðpÞ � ðq� classðkÞÞ þ UmaxðqÞ � ðclassðkÞ � pÞ ð7Þ

p ¼ floorðclassðkÞÞ; q ¼ ceilðclassðkÞÞ ð8Þ

where k is discrete sequence, class(k) is the output of the LVQNN at k time sequence,function floor(Æ) rounds to the nearest integer less than or equal to the input valueand function ceil(Æ) rounds to the nearest integer greater than or equal to the inputvalue. For example, floor(1.7) equals 1 and ceil(1.7) equals 2.

4. Experiments

Experiments without external loads are shown in Fig. 11. The reference positionof the rodless cylinder is set to be 600 mm, the experiment time for 3 s and the pro-portional, velocity and acceleration gains (kp,kv,ka) of the three-state feedback con-troller were set to be 1.5, 0.065 and 0.0001, respectively. These control parameterswere based on the following design specifications:

0 2

0.6 0.9 1.2 1.5 1.8596

597

598

599

600

601

598.25 (-1.75mm)

Zoom

Posi

tion(

mm

)

Reference Position

(a)

00

100

200

300

400

500

600

700

0

100

200

300

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700

1.2 1.4 1.6

599

600

601

602601.3mm

600.2mm

Zoom

Posi

tion(

mm

)

Time(s)

Time(s)

reference position

(b)

1 2 3

1 3

Fig. 11. Experimental results of rodless cylinder in the case of (a) conventional PWM and (b) proposedMPWM.


• maximum overshoot: within 0.3%,• settling time: less than 1.5 s.

The steady state error was about 1.75 mm in the case of the conventional PWMalgorithm and about 0.2 mm in the case of our proposed MPWM algorithm as seen


in Fig. 11. From the experimental results, it was verified that the proposed MPWMmethod was very effective in the accurate position control of the pneumatic system.Fig. 12 shows the experimental results of position control with different externalloads (10 kg, 20 kg and 30 kg), where the control gains were the same as that ofthe unloading condition. From Fig. 12, it may be understood that the system re-sponse became more oscillatory according to an increase of external load and thissuggested that the control parameters must be adjusted according to the variationsof the external load. Next, experiments were carried out to verify the effectivenessof our proposed MPWM and switching algorithm by the LVQNN. These experi-mental results are shown in Figs. 13 and 14, which correspond to 20 kg and 30 kgin the external load, respectively. In the figure, position, velocity and accelerationof the rodless cylinder and control input, the output of the LVQNN and the MPWMpulsing signal are shown respectively in each figure. The output of the LVQNNstarted to take effect only after the magnitude of velocity of rodless cylinder wasgreater than 0.5 m/s. From these experimental results, it was verified that the exter-nal load condition was correctly recognized by class number 3 and 4, respectively,and that the accurate position control was also realized with a steady state errorof 0.2 mm.Experimental results with a 25 kg load condition are shown in Fig. 15. This load

condition corresponded to the class 3 and 4. The class number calculated from theoutput of the LVQNN was 3.45, which proved that the external load was between20 and 30 kg. The velocity feedback gain (Kv) and saturated control input (Umax)were updated using the updating laws of Eqs. (5) and (6).

0 20

100

200

300

400

500

600

1.0 1.2 1.4 1.6

560

580

600

620

640

660

Zoom

Posi

tion(

mm

)

Time(s)

Ref 0kg 10kg 20kg 30kg

1 3

Fig. 12. Experimental results of rodless cylinder without switching control with respect to four differentexternal loads.

0.00.2

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MO

utpu

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lass

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trol

Inpu

t[x10

3 ]Ac

cel.

[m/s

2 ]ve

loci

ty[m

/s]

Posi

tion

[m] Reference

Position

Time(s)

2:Valve(1,2,5,6) open 1:Valve(1,6) open 0:Valve(1,2,7,8) open-1:Valve(3,7) open

Fig. 13. Experimental results when external load is 20 kg.


In Fig. 16, experiments were conducted to compare system responses with respectto four different weight conditions (0, 10, 20 and 30 kg) with and without the pro-posed switching algorithm by the LVQNN. From these experimental results, itwas found that the system response became oscillatory according to an increase ofthe external load. On the contrary, the system response was almost the same andthe steady state error was within 0.4 mm in any case when using our proposedswitching algorithm by the LVQNN.In Fig. 16, experiments were conducted to investigate the performance of our pro-

posed control system with respect to abrupt changes of external load. The external

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t[x10

3]

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l.[m

/s2]

velo

city

[m/s

]

Posi

tion

[m]

Reference Position

Time(s)




load was abruptly changed from 20, 0 to 30 kg and the experiment was executed for5 s in each load condition. From the results of class classification (second row frombottom in Fig. 17), it was verified that our proposed switching algorithm by theLVQNN was very effective in the classification of abrupt changes of external loadand that the parameters of the controller were automatically adjusted by using asmooth switching algorithm and that the control performance of transient responseswas greatly increased with respect to abrupt changes of external load.

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t[x10

3]

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l.[m

/s2 ]

velo

city

[m/s

]Po

sitio

n[m

] Reference Position

Time(s)




One of the application fields of this research is the loading/unloading task and apoint to point control is important in such a task. This proposed algorithm can onlywork with step inputs because LVQNN was trained by the experimental data of stepresponses. Therefore LVQNN cannot recognize or recognize poorly the externalload conditions in the case of sinusoidal input.

5. Conclusion

A fast, accurate, inexpensive and external load independent pneumatic actuatorthat may be applied to a variety of practical positioning applications was developed.

0 1 2 30

50100150200250300350400450500550600650700750

Zoom

1.0 1.2 1.4 1.6 1.8 2.0550

600

650

Posi

tion(

mm

)

Time(s)

reference(600mm) [10kg,Class 2]After tunning [10kg,Class 2]Before tunning

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50100150200250300350400450500550600650700750

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reference(600mm) [0kg,Class 1]

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1.0 1.2 1.4 1.6 1.8 2.0550

600

650

Posi

tion(

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)

Time(s)


Fig. 16. Comparison of experimental results with and without the LVQNN.


The position control was successfully implemented using on/off solenoid valves in-stead of expensive servo valves. The valves were pulsed using a modified PWM algo-rithm which compensated for the dead-time of on/off valves. This was verified byexperiments of position control in which steady state errors were reduced to within0.2 mm.The second contribution of this paper was to apply the learning vector quantiza-

tion neural network (LVQNN) as a supervisor of switching controllers in pneumaticservo systems with an on/off valve, where the LVQNN functions to classify the con-dition of the external load and selects suitable gains of controller for each weightcondition.From the experiments of position control of pneumatic cylinder, it was

verified that the proposed MPWM and smooth switching algorithm were veryeffective to overcome deterioration of the control performance of transientresponses due to four different load conditions and abrupt changes of externalloads.

0.00.20.40.60.8

-0.30.00.30.60.91.2

-10

0

10

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-2

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Load=30KgLoad=0KgLoad=20Kg

MPW

MO

utpu

tC

lass

No.

Con

trol

Inpu

t[x10

3 ]Ac

cel.

[m/s

2 ]ve

loci

ty[m

/s]

Posi

tion

[m]

Reference Position

Time(s)


Fig. 17. Experimental results when external load changes abruptly.


Acknowledgment

This work was partly supported by the University of Ulsan Research Fund 2003at University of Ulsan.

References

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