Two-Degree-of-FreedomControlwithAdaptiveDeadZone ...€¦ · Two-Degree-of-Freedom Control with Adaptive Dead Zone Compensation for Pneumatic Valves (Yui Shirato et al. ) Table 1

IEEJ International Workshop on Sensing, Actuation, Motion Control, and Optimization

Two-Degree-of-Freedom Control with Adaptive Dead ZoneCompensation for Pneumatic Valves

Yui Shirato∗a) Student Member, Wataru Ohnishi∗b) Member

Takafumi Koseki∗c) Member

As a pneumatic cylinder has several advantages, it is expected to be implemented for a long stroke coarse stage forthe high-precision positioning system. It, however, cannot achieve the high precision and high speed of positioning.One of the problems originates from that the bandwidth of pressure control is limited due to the time delay and thenonlinearity such as a dead zone of a valve. In this study, we propose a 2-DOF (two-degree-of-freedom) mass flowrate control system inside the pressure control system to address the nonlinearity. In order to compensate for the deadzone of a valve, we first propose to make an inverse model of a valve whose parameters are set by adaptive identifica-tion. We also propose the feedback control in the mass flow rate control system so as to eliminate a static error. Theexperimental result indicates that the mass flow rate tracking error decreases by 2-DOF control.

Keywords: pneumatic cylinder,pneumatic valve, 2-DOF mass flow rate control, dead-zone compensation, adaptive identification

1. Introduction

A high-speed, high-precision and large stage is requiredfor high-precision industry. Especially, the size of the stageof flat panel manufacturing machines becomes larger andlarger (1). In the same time, the more precise positioning ofa stage is required due to the high resolution of flat panels.In addition, as flat panel manufacturing machine is requiredto have a high throughput that can produce panels at a highspeed in order to reduce the manufacturing cost, high-speedpositioning is also required for the stage.

To improve the performance of the machine, a new struc-ture of the motion system is required. In order to managehigh throughput and high precision positioning, a dual stagestructure composed of a long stroke coarse stage and a shortstroke fine stage (2) are widely used. Generally, linear motorsand ball screws capable of high precision positioning are usedfor both the coarse stage and the fine stage. However, the in-crease in the size of the stage causes the generation of heatthat deteriorate the positioning accuracy (3)∼(6). Therefore, weaim to replace the electric motors implemented in the coarsestage with the pneumatic actuators which have the followingadvantages.

The advantages of the pneumatic cylinder compared to theelectric motors as follows.

( 1 ) Lightweight　It is possible to reduce the mass of the moving parteven though the stage is heavy. Additionally, the stagereaction force is decreased with the light weight stage.

( 2 ) Low heat generation

a) Correspondence to: [email protected]) Correspondence to: [email protected]) Correspondence to: [email protected]∗ The University of Tokyo

7-3-1, Hongo, Bunnkyoku, Tokyo 113-8656, Japan

As a compressor, which is a pressure source, is placedapart from the stage, the measurement system of theposition is less susceptible to heat.

( 3 ) Low cost (7)

( 4 ) Nonmagnetic (4) (8) 　A pneumatic cylinder is applicable to the process suchas electron beam exposure, etching.

However, for high-speed precision positioning control bya pneumatic drive, there are problems of nonlinearity, timedelay, and acoustic resonance. In particular, the nonlinearityof a valve due to the current dead zone and the time delayfrom the input of a current command to the output of air flowdeteriorate the precision and the feedback bandwidth in thedimension of the pressure, which limits the positioning accu-racy.

Because the piston and load of the pneumatic cylinder aremoved by the pressure difference between the two chambers,a structure having a pressure control system as an inner loopand a position control system as an outer loop is proposed(9) (10). With this structure, it has been shown that by improvingthe responsiveness of the pressure in the chamber, high-speeddriving is possible (10). In this research as well, for high-speeddriving and compensation of pressure disturbance, a controlstructure with a pressure control system inside the positioncontrol system is adopted.

To enhance the pressure control accuracy, this paper con-structs a mass flow rate control system inside the pressurecontrol system and proposes a control structure consistingof position control system, pressure control system, massflow rate control system. The block diagram of the proposedmethod is shown in Fig. 2. Although it is assumed that theaccuracy and the bandwidth of pressure control improve bynonlinear compensation of a valve, the mass flow rate feed-back control system inside the pressure control system is notwidely employed (11) (12). The reason is considered that feed-

c⃝ 2019 The Institute of Electrical Engineers of Japan. 1

TT8-4

Two-Degree-of-Freedom Control with Adaptive Dead Zone Compensation for Pneumatic Valves (Yui Shirato et al.)

Table 1. Characteristics of Each Method and Section

method inverse model FB control FF control section

Conv1 fixed × ✓ 3

Prop1 adaptive × ✓ 4.1Prop2 adaptive ✓ × 4.2Prop3 adaptive ✓ ✓ 4.3

back control causes a large overshoot of mass flow rate re-sponse because the response time of the thermal flow meterwidely used for gas measurement is long compared to theaiming response time of a valve (13).

Therefore, in the field of the positioning of pneumaticcylinders, conventional nonlinearity compensation of a valveis based on the mathematical modeling of valve dead zonesand compressibility of air (14) (15). In this study, we treat thecase of nonlinearity compensation using an inverse model ofa valve as conventional method Conv1. The block diagramof Conv1 is shown in Fig. 1. NL−1 is the inverse model of avalve for nonlinear compensation. Because the input-outputcharacteristic of a valve changes depending on the operat-ing condition, a stationary error remains when nonlinearitycompensation is performed by an inverse model using fixedparameters. In the pneumatic drive system, as modeling thevalve is difficult due to the strong nonlinearity of pressure andmass flow rate, and identification of the frictional force of acylinder is complicated, adaptive control has been studied (11).Thus, we proposed the nonlinear compensation in the massflow rate control system by using the parameters estimatedby adaptive identification. On the other hand, a flow meterwith a time constant of 5 ms which is faster than the com-mon flow meter in the market has released. Additionally, itis faster than the aiming response time of a valve. For thisreason, this paper proposes a mass flow rate feedback con-trol with the fast flow meter. Additionally, we propose tocombine adaptive identification method for the valve modelinversion and the mass flow rate feedback control. We namethis proposal method Prop3. The block diagram of Prop3 isshown in Fig. 2. The series of proposed methods are listed inTable 1.

2. Experimental Setup for Mass Flow Rate Con-trol

2.1 Structure of Pneumatic Drive System Thestructure of the pneumatic drive system is shown in Fig. 3.The power source of a pneumatic circuit is a compressor. Theair compressed by the compressor passes through an air filterand a pressure regulator and then it is supplied to the valvesvalve1sup and valve2sup in Fig. 3. On the contrary, the air isexhausted through the valves valve1exh and valve2exh. The airpressure inside a chamber is determined by the difference ofsupplied mass flow rate and exhausted mass flow rate. Thegenerated force by the pneumatic cylinder is determined bythe pressure difference of the two chambers.

In order to improve the nonlinear compensation of massflow rate, we use a single valve for this paper. Experimentalsetup for mass flow rate control is shown in Fig. 4.

2.2 Input-Output Characteristics of a Valve Al-though various studies on modeling of nonlinearity of a valvehave been investigated (14) (16) (17), the complication of the model

implies an increase in parameters to be identified. The in-put of a valve is the current command, and the output is themass flow rate of air which passes through a valve. Contraryto this, the input of an inverse model of a valve is the massflow rate command, and the output is the current commandto a valve. The input-output characteristics of a valve and aninverse model depend on the pressure difference between anentrance port and exit port of a valve.

A measurement result of current-mass flow rate charac-teristics under 6 patterns of constant pressure difference isshown in Fig. 5(a). This figure indicates that the current deadzone has a dependency on the pressure difference. The mea-surement is performed in the degree of rated current value,0.165 A.

Moreover, the current-mass flow rate characteristic varieseven under the constant pressure difference condition. Mea-surement results of current-mass flow rate characteristic for4 times experiments are shown in Fig. 5(b). In this measure-ment, the pressure difference is set as 0.3 MPa, and the valveinput current is increased from 0 A to 0.09 A.

3. Conventional Mass Flow Rate FeedforwardControl

The block diagram of the mass flow rate control system inthe case of the conventional method using an inverse modelwith fixed parameters (Conv1) is shown in Fig. 6. In thiscase, the variation of the mass flow rate is compensated bya pressure feedback control system which is an outer loop ofmass flow rate control system. The inverse model is designedby approximating a valve input current as a cubic function ofmass flow rate (15).

4. Proposal of 2-DOF Mass Flow Rate Control4.1 Adaptive Identification of Inverse Model Since

the current- mass flow rate characteristic of a valve has largevariations, there is a problem that a large error occurs in thecase of using an inverse model with fixed parameters. For thisreason, we propose a method to estimate parameters of theinverse model by using the recursive-least-squares method(RLS method) (18). We named the proposed method using theinverse model whose parameters is tuned by adaptive iden-tification Prop1. The block diagram of the mass flow ratecontrol system in the case of Prop1 is shown in Fig. 7.

The inverse model is designed by modeling the valve inputcurrent is approximated as a first order model

i = amre f + b (1)

and by setting the parameters a and b by means of adap-tive identification. Equation (2) is an equation of the inversemodel. θ, the estimated parameter, is set as (3). ϕ is set as(4). Since the current-mass flow rate characteristic is dependson the pressure difference and the pressure difference is time-varying, we introduce the RLS method with a forgetting fac-tor (18) in order to address a change of the pressure difference.A recurrence formula of the RLS method with a forgettingfactor is shown in (6),(7), and (8). The initial value of θ isset as a and b, which was determined by least-square methodwith the input-mass flow rate data shown in Fig. 5. P0, theinitial value of P is expressed as (5) with γ. Consideringthe speed of estimation and the noise, γ is set as 1000. In this

2


positionreference

fref

pressurereference

prefpd

f

pressuredisturbance

forceCxff

12A pref

Cxfb

Cpff

Cpfb

−

+ +

+

+ +

+−1

s

ppressure

AA1

mcs2

+

+

+

fd

p

+

forcedisturbance

xref

forcereference

mref

G1

mG2

mass flow ratereference

currentreference

xNL−1

iref

valvechamber

Fig. 1. Block Diagram of Pneumatic Drive System in the Case of Conv1. Cxf f - position feedforward controller;

Cxf b - position feedback controller; Cp

f f - pressure feedforward controller; Cpf b - pressure feedback controller; A

- area of a cylinder; mc - mass of a cylinder; G1 =pV+pV

RT ; G2 =mRT−pV

V ; NL−1 - inverse model of a valve fornonlinear compensation.

positionreference

fref

pressurereference

prefpd

f

pressuredisturbance

forceCxff

12A pref

Cxfb

Cpff

Cpfb

−

+ +

+

+ +

+−1

s

ppressure

AA1

mcs2

+

+

+

fd

p

+

forcedisturbance

xref

forcereference

mref

G1−

+ +

+

Cmff

Cmfb

mG2


currentreference

xNL−1

iref

valvechamber

e−τs

Fig. 2. Block Diagram of Pneumatic Drive System in the Case of Prop3. Cmf f - mass flow rate feedforward

controller; Cmf b - mass flow rate feedback controller.

x

p1 p2

compressor

dryer

filter+regulator

cylinder

pressuresensor

i1supi1exh

i2exh

i2sup

m2sup

m2exh

m1sup

m1exh

m1 = m1sup − m1exh

m2 = m2sup − m2exh

p1exh

p2sup

p2exh

p1sup

mass

valve1exh

valve2exh

valve2sup

valve1sup

Fig. 3. Structure of Pneumatic Drive System for Posi-tioning.

study, λ, a forgetting factor, is set as 0.94 based on the aimingpressure feedback bandwidth and the sampling frequency.

y(k) = θTϕ(k) (2)

θ =

(ab

)(3)

ϕ =

(m1

)(4)

P(0) =(γ 00 γ

)(5)

air

compressor dryer regulator valve flowmeter

pressure sensor

controller current amplifier

air air

m: flow rate valuep: pressure value

i:valveinput current

current commandto an amplifier

Fig. 4. Experimental Setup for Mass Flow Rate Control.(blue:air flow, green: signals)

　

θ(k) = θ(k − 1) +P(k − 1)ϕ(k)

λ + ϕT(k)P(k − 1)ϕ(k)ϵ(k) (6)

ϵ(k) = y(k) − ϕT(k)θ(k − 1) (7)

P(k) =1λ

{P(k − 1) − P(k − 1)ϕ(k)ϕT(k)P(k − 1)

λ + ϕT(k)P(k − 1)ϕ(k)

}(8)

As a condition of adaptive identification, the persistent ex-citation (PE) index is introduced (18). The input signal foridentification is required to be persistently exciting of orderat least with the order of 2 because the inverse model is de-signed to output the valve input current as a first order func-tion of mass flow rate command. Since a constant signal isPE of order 1, identification is not performed when the valveinput current is a constant value.

There exists a delay from the input of the current command

3


0 0.04 0.08 0.12 0.16 0.2

0

10

20

30

40

50

(a)

-2 -1 0 1 2 3 4 5

time [s]

0.07

0.075

0.08

0.085

0.09

Cu

rren

t [A

]

20

30

40

50

60

Mass

Flo

w R

ate

[L

/min

]

current

mass flow1

mass flow2

mass flow3

mass flow4

(b)

Fig. 5. Measurement Result of Current-Mass Flow RateCharacteristic. (a) current-mass flow rate characteris-tic(color - pressure difference between two ports of avalve ∆p) (b) variations of current-mass flow rate char-acteristic. (black: valve input current, red, green, cyan,pink: measured mass flow rate)

NL−1

imref m


valve

∆

Fig. 6. Block Diagram of Mass Flow Rate Control Sys-tem in the Case of Conv1.


NL−1 imref

m

valve

G(jω)

adaptiveidentification

Fig. 7. Block Diagram of Mass Flow Rate Control Sys-tem in the Case of Prop1.

to the output of air flow. To compensate for the delay effect onthe adaptive identification, the input current data are shiftedto match the input-output data defined in (1).

4.2 Feedback Controller Design We named the


NL−1 imref

m

valve

+

−Cm

fb

G(jω)


mass flow ratefeedback

Fig. 8. Block Diagram of Mass Flow Rate Control Sys-tem in the Case of Prop2．

transfer function from the input of the valve inverse modelto the output of a valve G( jω). The inverse model is de-signed by adaptive identification. We named the methodwith feedback control of G( jω) Prop2. The block diagramof Prop2 is shown in Fig. 8. The feedback controller is aproportional–integral–derivative (PID) controller and it wasdesigned by pole placement (19).

To impose stability margins, we introduced circle condi-tions (20). The gain margin is set as 4 dB and the phase mar-gin is set as 45 deg. The circle which indicates the stabil-ity is specified to have the center of (−σs, 0) and the ra-dius of rs. σs and rs are given by (9) with the gain marginGm(gm = 20 log10 Gm[dB]) and the phase margin Φm(ϕm =

180Φm/π[deg]). To satisfy the circle condition means theNyquist trajectory of L( jω) does not pass through the insideof the circle with the center of (−σs, 0) and the radius of rs.

σs =Gm

2 − 12Gm(Gm cosΦm − 1)

rs =(Gm − 1)2 + 2Gm(1 − cosΦm)

2Gm(Gm cosΦm − 1)(9)

We named the maximum frequency that satisfies the circlecondition ωp.

As G( jω) is identified as Ke−Ls with multisine mass flowrate command, the controller is designed with Gpade(s) whichis a pade approximation of G( jω). Here, a constant K is thestatic gain of G( jω) and L is the dead time of a valve, respec-tively.

Gpade(s) = K12 − 6Ls + L2s2

12 + 6Ls + L2s2 (10)

The PID feedback controller is expressed as

C =b2s2 + b1s + b0

a2s2 + a1s. (11)

As shown in (12), poles of the closed loop is set as(s + ωp)4 with ωp. Here, Dg is the denominator of Gpade(s),Ng is the numerator of Gpade(s), Dc is the denominator ofC( jω) and Nc is the numerator of C( jω). Equation (12) isrepresented as (13) using the Sylvester matrix. By normaliza-tion of a2 = 1, a2, a1, b2, b1, b0 is uniquely determined. Fromhere, the gain of the PID controller Kp,Ki,Kd and the timeconstant of pseudo differential controller τd are determinedas (14).

DgDc + NgNc = (s + ωp)4 (12)

4



NL−1 i

mref

m

valve−

Cmfb

G(jω)Cm

ff

+

+


mass flow ratefeedback

mass flow ratefeedforward

+

e−Ls

Fig. 9. Block Diagram of Mass Flow Rate Control Sys-tem in the Case of Prop3．

L2 0 KL2 0 06L L2 −6KL KL2 012 6L 12K −6KL KL2

0 12 0 12K −6KL0 0 0 0 12K

a2a1b2b1b0

=

14ωp6ωp

2

4ωp3

ωp4

(13)

Kp = τd(b1 − Ki)Ki = b0τd

Kd = τd(b2 − Kp)τd = a2 (14)

In the high-frequency region where the influence of non-linearity and uncertainty is large, gain stabilization was per-formed using a second-order low-pass filter. The cut-off fre-quency of the low-pass filter is set as 160 Hz where the signalto noise ratio becomes low.

4.3 Design of 2-DOF Mass Flow Rate Control UsingTuned Inverse Model Fig. 9 shows the block diagram ofthe proposed 2-DOF mass flow rate control, which is namedas Prop3. To improve the speed of tracking, a feedforwardcontroller is added to the block diagram in the case of Prop2.As shown in Fig. 9, a nominal valve delay is implementedto match the response of the feedforward and feedback con-trollers.

5. Experimental ResultThe response to the mass flow rate step command, the mass

flow rate error and the valve input current are shown in Fig.10. Until t < 0 the current (0.09 A) is input to a valve, andafter t = 0 the mass flow rate command is input to the inversemodel of a valve. Fig. 10(a) indicates that the mass flow rateerror decreases but it still remains by introducing the adaptiveidentification. It is assumed that the error in the case of Prop1originates from the parameter estimation error because theestimated parameters contain certain noise. As indicated inFig. 10(c), the settling time of valve input current in the caseof Prop1 is same as that in the case of Conv1, that is, there isno delay of mass flow rate response except for the valve deadtime. Next, in the case of Prop2 with PID control, there is nosteady state error as shown in Fig. 10(e). On the other hand,the settling time in the case of Prop2 is about 100 ms, whichis longer than that of Conv1 because the feedback bandwidthof Prop2 is about 10 Hz. And the figure Fig. 10(e) indicatesthat there is a saturation because the mass flow rate commandto the inverse model must be a positive value or zero. Finally,

Table 2. Experimental Result of Each Method

method inverse model FB control FF control static error overshoot

Conv1 fixed × ✓ −

Prop1 addaptive × ✓ +

Prop2 addaptive ✓ × ++ −Prop3 addaptive ✓ ✓ ++ +

in the case of Prop3, added the feedforward control system toProp2, the mass flow rate error decreases at the moment afterthe dead time (at 8 ms) because of the feedforward control.And the mass flow rate response follows the mass flow ratecommand with because of the feedback control. As the er-ror is already reduced by the feedforward control, there is nosaturation and the overshoot is small in the case of Prop3.

The characteristics of the experimental result are listed inTable 2.

6. ConclusionIn this research, we proposed the compensation method of

the nonlinearity of valves which limit the accuracy and thespeed of positioning, in order to utilize a pneumatic cylinderto a coarse stage of the high-precision positioning stages. Thedead zone of the valve input current is a main nonlinearityof valves, and there is a dynamics due to the air compress-ibility. As the nonlinearity in the mass flow rate degradesthe coherence of the plant for the pressure control system, itlimits the bandwidth of the pressure control, in consequence,the accuracy and the speed of positioning are limited. Toaddress the nonlinearity, we proposed the inverse model ofvalves based on adaptive identification and the feedback con-trol of the mass flow rate.

The experimental result shows that using the proposedinverse model based on adaptive identification reduces themass flow rate error only with the time delay of valve deadtime. Thus the effectiveness of feedforward control has im-proved. On the other hand, mass flow rate response followsthe mass flow rate command in 100 ms by means of feedbackcontrol with the flow meter whose response is faster than thevalve dead time. Therefore, the mass flow rate response fol-lows the mass flow rate command with small overshoot by2-DOF control based on adaptive identification.

As a future work, the effectiveness of the proposed methodProp3 will be verified in the pressure control system.

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5


0 0.05 0.1 0.15

Time [s]

0

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100M

ass

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ate[

L/m

in]

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prop1

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(b) Mass Flow Rate Error（Conv1, Prop1）

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Time [s]

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Time [s]

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Mass

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PROP2

PROP3

(d) Mass Flow Rate（Prop2, Prop3）

0 0.05 0.1 0.15

Time [s]

-60

-40

-20

0

20

Mass

Flo

w R

ate

Err

or

[L/m

in]

PROP2

PROP3

(e) Mass Flow Rate Error（Prop2, Prop3）

0 0.05 0.1 0.15

Time [s]

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Valv

e C

urr

ent[

A] PROP2

PROP3

(f) Valve Input Current（Prop2, Prop3）

Fig. 10. Response to the Mass Flow Rate Command, Mass Flow Rate Error and Valve Input Current. (black:mass flow rate command, red: Conv1, cyan: Prop1, pink: Prop2, blue: Prop3)

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Two-Degree-of-FreedomControlwithAdaptiveDeadZone ...€¦ · Two-Degree-of-Freedom Control with Adaptive Dead Zone Compensation for Pneumatic Valves (Yui Shirato et al. ) Table 1