Synchronous Position Control Strategy for Bi-cylinder Electro-pneumatic Systems

Hong Zhao and Pinhas Ben-Tzvi

International Journal of Control, Automation and Systems 14(6) (2016) 1501-1510

ISSN:1598-6446 (print version)eISSN:2005-4092 (electronic version)

To link to this article:http://dx.doi.org/10.1007/s12555-014-0506-5

International Journal of Control, Automation and Systems 14(6) (2016) 1501-1510http://dx.doi.org/10.1007/s12555-014-0506-5

ISSN:1598-6446 eISSN:2005-4092http://www.springer.com/12555

Synchronous Position Control Strategy for Bi-cylinder Electro-pneumaticSystemsHong Zhao and Pinhas Ben-Tzvi*

Abstract: Pneumatic systems have been widely used in industrial applications because of their well-known advan-tages. However, pneumatic systems possess several disadvantages that include strong non-linearity and low naturalfrequency. These drawbacks make it difficult to obtain satisfactory control performances in comparison to hydraulicand electromechanical systems. In this paper, the fundamental characteristics and nonlinear synchronous controlstrategy of pneumatic systems are studied. A two-layer sliding mode synchro-system based on friction compen-sation is applied to electro-pneumatic cylinders and a synchro-PID controller is utilized for position tracking. Tovalidate the developed strategy, experiments with bi-cylinder electro-pneumatic systems were performed. The ex-perimental results demonstrate that the synchronous position control scheme is effective in terms of accuracy androbustness.

Keywords: Friction compensation, pneumatic servo system, sliding mode control, synchro-PID controller.

1. INTRODUCTION

With the development of aerospace technologies andmodern machine manufacturing, pneumatic actuatorshave been widely used in numerous engineering areas.The advantages of pneumatic actuators include cleanoperation and easy serviceability. In most cases, theirapplication is limited to point to point control. The dis-advantages of the pneumatic system mainly include aircompressibility, strong non-linearity and large mechani-cal friction. These drawbacks restrict their applications inservo-controlled systems.

More effort has been dedicated to improving this re-search field. Numerous researchers focused their effortson studying different aspects of pneumatic servo systems,such as air flow, thermodynamics, linear and non-lineardynamics, valve modelling and friction modelling [1, 2].Due to high friction in pneumatic systems [3], distur-bances lead to system uncertainties. This is usually con-sidered to be the main problem in pneumatic control.

In recent years, the development of pneumatic systemsand the improvement of control theories has led to theimplementation of modern control laws in pneumatic de-vices [4–11]. Sliding mode control has been used as arobust nonlinear control technique in many applicationsrequiring insensitivity to parametric uncertainty [12–14].

Manuscript received November 16, 2014; revised June 28, 2015 and November 3, 2015; accepted December 27, 2015. Recommended byAssociate Editor Won-jong Kim under the direction of Editor Hyouk Ryeol Choi. This work was in part supported by the National NaturalScience Foundation of China (Grant No. 51575528).

Hong Zhao is with the College of Mechanical and Transportation Engineering, China University of Petroleum, Beijing 102249, China (e-mail:hzhao_cn@163.com). Pinhas Ben-Tzvi is with the Department of Mechanical Engineering and the Department of Electrical and ComputerEngineering, Virginia Tech, Blacksburg, VA 24061, USA (e-mail: bentzvi@vt.edu).* Corresponding author.

Over the last decade, multi-surface sliding mode controlreceived a lot of attention [15–19], mainly due to the ad-vantages this tool presents, from higher accuracy and thepossibility of using continuous control laws, to the possi-bility of utilizing the Coulomb friction model in controlalgorithms [20] and the finite time convergence for sys-tems with arbitrary relative degrees [21].

However, new requirements in the performance ofpneumatic systems are continually being proposed. Theneed for motion synchronization between multiple pneu-matic actuators for lifting and rolling applications hasbeen the focus of the research efforts [22, 23].

In order to eliminate the synchro error and improve thecontrol performance with one air source - especially re-ducing the effects of static friction - a bi-cylinder con-troller design is proposed. In this paper, the electro-pneumatic motion synchronization servo control systemwith one air source is introduced, and the fundamentalcharacteristics and the nonlinear control strategy of thepneumatic system are studied. A bi-cylinder controlleris adopted for the control of the electro-pneumatic syn-chro position system, and the effectiveness of the pro-posed technique is demonstrated through simulations andexperiments.

c⃝ICROS, KIEE and Springer 2016

1502 Hong Zhao and Pinhas Ben-Tzvi

2. SYSTEM DESCRIPTION

2.1. Configuration of the electro-pneumatic positionsynchro servo control system

The proposed servo control system consists of two rodlesspneumatic cylinders controlled by two proportional direc-tional valves as shown in Figs. 1(a) and (b). The exper-imental sampling rate is 2.5 KHz. The main componentsof the system are as follows:Two Festo electro-pneumaticfive-port proportional valves: MPYE-5-1/4-010B; maxi-mum flow rate 1400 l/min, operating voltage DC 24 V,voltage variant: 0-10 V, Nominal width 8mm; Two rodlesspneumatic cylinders with 600 mm stroke length: DGP-40-600-PPV-A-B; One air source: maximum pressure0.6 MPa, maximum flow rate 1500 l/min; Four pressuresensors: PT351-0.6MPa-0.3, resolution 0.3%; Two posi-tion sensors: MLO-POT-600-TLF, resolution 0.01 mm;Mass load A (5.75 kg) and B (5.85 kg) are mounted onsliding parts, which are connected to the pistons of cylin-ders A and B, respectively. External forces of 17 N and34 N are applied to cylinder A to simulate external loaddisturbances.

2.2. Computer softwareThe pneumatic control software that we developed for

the purpose of this study was the key component in obtain-ing experimental results. It includes the human-machineinterface built in LabWindows/CVI [24], the control algo-rithm as well as the data processing routines.

3. MATHEMATICAL MODEL

3.1. Subsystem 1 (with cylinder A)A nonlinear state-space model of the system is derived

based on a single cylinder pneumatic model from refer-ence [5], as follows:

x1 = x = x2, (1)

x2 =Ap

M1(px1 − px2)−

Ff 1

M1− F1

M1, (2)

x3 = px1

=

kRCdC0√R/T

[A1s(u1) · ps · f

(px1ps

)−A1e(u1) · px1 · f

(pepx1

)]V10 +

12 ApL+Apx1

− kAp px1x2

V10 +12 ApL+Apx1

, (3)

x4 = px2

=

kRCdC0√R/T

[A2s(u1) · ps · f

(px2ps

)−A2e(u1) · px2 · f

(pepx2

)]V20 +

12 ApL−Apx1

+kAp px2x2

V20 +12 ApL−Apx1

. (4)

Definitions for other parameters are provided in thenomenclature.

control algorithm as well as the data processing routines.

Weight

Mass load A

Cylinder A

Mass Load B

Cylinder B

Computer

Pressure

sensor

Displacementsensor

Pressuresensor

A/D

Displacementsensor

Pressure

sensor

A/D D/AA/D A/D

Pressure

sensor

D/A

Proportional

directional

control Valve

Proportional

directional

control Valve

(a)

(b)

Fig. 1. (a) Scheme of the experimental system, (b) photosof the experimental setup.

3.2. Subsystem 2 (with cylinder B)

z1 = z = z2, (5)

z2 =Ap

M2(pz1 − pz2)−

Ff 2

M2− F2

M2, (6)

z3 = pz1

=

kRCdC0√R/T

[A1s(u2) · ps · f

(pz1ps

)−A1e(u2) · p1 · f

(pepz1

)]V10 +

12 ApL+Apz1

− kAp pz1z2

V10 +12 ApL+Apz1

, (7)

z4 = pz2

=

kRCdC0√R/T

[A2s(u2) · ps · f

(pz2ps

)−A2e(u2) · pz2 · f

(pepz2

)]V20 +

12 ApL−Apz1

+kAp py2z2

V20 +12 ApL−Apz1

, (8)

Synchronous Position Control Strategy for Bi-cylinder Electro-pneumatic Systems 1503

where,

f(

pd

pu

)=

√√√√ 2k−1

·(

k+12

)k+1/k−1

×

√√√√√( pd

pu

)2/k−(

pd

pu

)k+1/k

if patm/

pu ≤ (pd/

pu)≤Cr

1 if Cr ≤ (pd/

pu)< 1.

A1s, A2s, A1e and A2e are the valve geometric orificearea function, and they are the nonlinear functions of u[1] (u-valve input opening size).Due to the complexity ofthe equations, the following linear functions are used forapproximation:

A1s(ui) = πDui, (9)

A2s(ui) = πDδ = A f , (10)

A1e(ui) = πDδ = A f , (11)

A2e(ui) = πDui, (12)

where, δ is the valve’s clearance, A f is valve’s leakagemeasurement, D is the piston’s valve diameter, cylinder Asystem, i = 2 for cylinder B system.

3.3. Friction model of the system

Friction has an accentuated influence on the step ortracking tasks, mainly when the piston velocity is small.From the experimental position step-input response shownin Fig. 2, the dithering effect at the end of the piston strokeis observed.

Friction forces are the main factor that affects the per-formance of the pneumatic servo system controller. Thoseforces are mainly generated by contact between the pis-ton seals and the cylinder walls. This nonlinear effectbecomes even more dominant at lower sliding velocities.Typically, friction forces in the cylinder include static anddynamic components. Stribeck model [25] is one of thewidely used friction models, but for our experimental ap-plication, we simplified the switching process between theCoulomb static friction and the Coulomb dynamic fric-tion. The chosen model is shown in Fig. 3.

When v > 0,

Ff = Fs1 − k2 · v for v < vd ,

Ff = Fs1 −Fd1 + k1 · v for v ≥ vd , (13)

where vd is the velocity cut-off point. When v<0, the restcan be deduced by analogy. From experimental tests onthe pneumatic cylinders, the friction parameters were es-timated and summarized in Table1.

Fig.2. Open-loop system response to a position step input.

Fig. 2. Open-loop system response to a position step in-put.

fF

v

1sF

2sF

1dF

2dF

1k

4k

3k

2k

dvdv

Fig. 3. Simplified friction force model.

Table 1. Friction parameters summary.

Cylinder k1 k4 Fd1(N) Fd2(N) Fs1(N) Fs2(N)

A 91.83 89.33 54.97 −57.43 84.87 −89.22B 85.03 86.82 55.06 −63.1 75.52 −75.13

4. CONTROL DESIGN STRATEGY

The block diagram of the proposed controller is shownin Fig. 4. It consists of two cylinder controllers, and asynchro PID controller. Each of the cylinder controllersincludes a second order sliding mode controller and afriction compensator. The synchro PID controller coor-dinates the operation of the two cylinder controllers andimproves the synchro motion control of the cylinders. Thebi-cylinder synchro controller is required to maintain thedisplacement error between subsystems 1 and 2 to lessthan a specified tolerance, keeping their correspondingdisplacement values to approximately the same. This re-quirement can be expressed mathematically as follows:

max |x(t)− z(t)| ≤ et ∀t ∈ [a,b], (14)

where et is the permitted position tolerance (it is chosento eliminate the large error in position tracking), x(t) is

1504 Hong Zhao and Pinhas Ben-Tzvi

the real position of cylinder A, z(t) is the real position ofcylinder B, and [a,b] is the time domain.

Let esy = x(t)− z(t), t ∈ [a,b]. The synchro PID con-troller is given by:

uad1 =

Ksy1 |esy|+Ksy2

∣∣∣∣∫ esydt∣∣∣∣

+Ksy3

∣∣∣∣desy

dt

∣∣∣∣ , esy < 0

0, esy ≥ 0,(15a)

uad2 =

Ksy1esy +Ksy2

∫esydt

+Ksy3desy

dt, esy > 0

0, esy ≤ 0.(15b)

As the synchro PID controllers are made to adjust thedifference between two cylinder’s displacements, uad1 isadded in cylinder A control signal and uad2 is added tocylinder B’s control signal. Whenesy = x(t)− z(t) < 0,cylinder A should be adjusted to keep up with cylinder B’sdisplacement and uad1 should be positive and uad2 is zero.When esy = x(t)− z(t)> 0, cylinder B should be adjustedto keep up with cylinder A’s displacement and uad2 shouldbe positive and uad1 is zero.

In order to adjust the parameters of the synchro PIDcontroller, the integral absolute error (IAE) performanceminimization measure is considered. The synchro PIDcontroller’s parameters are set as follows: Ksy1 = 2, Ksy2 =0.0015, Ksy3 = 0.001. Based on the mathematical modelpresented in the previous section, it is clear that pressurefunctions in the pneumatic cylinder are nonlinear; hencethe pressure drop is taken as the sliding mode layers. Thebi-layer sliding mode control strategy with friction com-pensation is proposed in order to overcome friction distur-bances in the cylinders. Inner pressure loop control en-hances the speed of the system response and attenuatesfluctuations in the pressure, hence improving the systemtracking accuracy. The control approach is developed asfollows.

4.1. Control rule in cylinder AFor the purpose of controller design, if the nonlinear

pneumatic model is rewritten in the affine function formwith (1)-(4), we can get

x1 = x2

x2 =∆p ·Ap −K f 1x2

M1− Fkcsgn(x2)

M1

∆ p = (px1 − px2) = f1(x)+g1(x) ·u,

(16a)

From (16a),

x = A1x+B1p+q, (16b)

∆ p = (px1 − px2) = f1(x)+g1(x) ·u (16c)

control approach is developed as follows.

Two-layer sliding

mode controller

Friction

compensation

Cylinder

A plant

Synchro PID

controller

Cylinder

B plant

Two-layer sliding

mode controller

Friction

compensation

yd

x

z

u1

u2

xv

xv

uad1

uad2

esy

Synchro PID

controller

ua1

ub2

d/dt

d/dt

Fig. 4. Block diagram of the integrated control strategy forthe bi-cylinder actuator.

with x = [x1, x2]T , A1 =

[0 10 −K f 1

M1

], B1 =

[0Ap

M1

],

q =

[0

−Fkcsgn(x2)M1

],

f1(x) =−kApxpx1

V10 +12 ApL+Apx

−kApxpx2

V20 +12 ApL−Apx

− kRCdC0√R/

T·

(A f · px1 · f ( pe

px1)

V10 +12 ApL+Apx

+A f · ps · f ( px2

ps)

V20 +12 ApL−Apx

),

g1(x) =kRCdC0πD√

R/

T

(ps · f ( px1

ps)

V10 +12 ApL+Apx

+px2 · f ( pe

px2)

V20 +12 ApL−Apx

).

It should be noted that the ccontrollability of the plantis proven using (16b) and the controllability matrix.

px1, px2 cannot be in excess of the range between thesupply pressure ps and the exhaust pressure pe, thereforethe function f1(...) is always positive and g−1

1 does exist.The maximum value of g−1

1 satisfies max∣∣g−1

1

∣∣≤ gm.Generally, the static friction force Fkcsgn(x2) cannot be

measured accurately but it is assumed that the maximumvalue of static friction is known. Since xd and ∆pd arebounded, q in (16b) can be treated as a bounded uncer-tainty.

The simplified diagram for pressure drop is shown inFig. 5. Let the inner loop tracking error be ep = ∆p−∆pd

Synchronous Position Control Strategy for Bi-cylinder Electro-pneumatic Systems 1505

Fig. 5. The block diagram of the single cylinder systemwith simplified pressure control strategy.

and choose the control input u f l1 as follows:

u f l1 =1

g1(x)(− f1(x)+∆ pd − kep(∆p−∆pd)). (17)

Then we can get

∆ p = ∆pd − kep(∆p−∆pd). (18)

A simplified control approach can be adopted by mak-ing ∆p → ∆pd as x → yd . For the inner loop (pressureloop), k is chosen to make the function

ep + kepep → 0. (19)

Then ∆p → ∆pd .Let’s define the sliding surface by σ = ∆p−∆pd . Then:

σ1 = ∆ p−∆pd = f1 +g1u−∆ pd . (20)

If the Lyapunov function is chosen to be Vlya1 = 0.5σ 21 ,

then Vlya1 = σ1σ1.To ensure that Vlya1 = σ1σ1 is a negative-definite func-

tion, we choose the control law us.When σ1 < 0 and σ1 > 0

uσ1 >∣∣g−1

1 f1 −g−11 ∆pd

∣∣ . (21)

When σ1 > 0 and σ1 < 0

uσ1 <−∣∣g−1

1 ( f1 −∆pd)∣∣ . (22)

However, by the triangle inequality∣∣g−11 f1

∣∣+ ∣∣g−11 ∆ pd

∣∣> ∣∣g−11 ( f1 −∆pd)

∣∣ . (23)

Then∣∣g−1

1 f1∣∣+gmα >

∣∣g−11 ( f1 −∆pd)

∣∣.Therefore, the system can be stabilized by means of

control law

uσ1 =

{ ∣∣g−11 f1

∣∣+gmα , σ1 < 0

− (∣∣g−1

1 f1∣∣+gmα), σ1 > 0,

(24)

and

uσ1 =−κ1sgn(σ1), κ1 > 0, (25)

Vlya1 = σ1σ1 < 0, (26)

and

ucy1 = u f l1 −κ1sgnσ1. (27)

For the outer loop (position loop), ∆pd is chosen to makethe function

(x− yd)+ξ1(x− yd)+ξ2(x− yd) = 0. (28)

The control objective is to obtain the state X in orderto track a desired state Yd = (yd , yd , yd), in the presence ofmodel uncertainties and unknown disturbances.

∆p =M1

Apx+

1Ap

(K f 1x+Fkcsgn(x)) . (29)

From (28), we get x = yd − ξ1e1 − ξ2e1, where e1 =x− yd .

The above control law is based on the exact cylinderfriction model. This means that there are no other dis-turbances on the friction factors K f 1 andFkc. In fact, ∆pd

should be modified to improve the robustness of the sys-tem. Therefore, a pseudo control input w1 is added in or-der to eliminate the changing effects of the friction factors.The expression for ∆pd then becomes

∆pd =M1

Ap[yd −ξ1(x− yd)−ξ2(x− yd)]

+1

Ap(K f 1x+ Fkcsgnx+M1w1)+

1k

ep, (30)

and

∆p−∆pd =M1

Ap(e1 +ξ1e1 +ξ2e1)

+1

Ap(∆K f 1x+∆Fkcsgn(x)−M1w1)

− 1k

ep. (31)

External force F1is eliminated from equation (31) sincethis controller is insensitive to external force disturbances.For ∆p → ∆pd , equation (31) becomes:

e1 +ξ1e1 +ξ2e1 +1

M1(∆K f 1x+∆Fkcsgnx)−w1 = 0,

(32)

where ∆K f 1 = K f 1 − K f 1 and ∆Fkc = Fkc − Fkc are the fric-tion factors differences, and K f and Fkc are the exact fric-tion factors.

The second sliding surface is chosen ase1 = x − yd .Choosing the sliding mode

s1 = e1 + c1e1 = (x− yd)+ c1(x− yd),c1 > 0, (33)

then

s1 = e1 + c1e1 = (x− yd)+ c1(x− yd)

= A1x+B1∆ p+ q− yd − c1x− c1yd ,

s1 = (A1 − c1I)(A1x+B1∆p+q)+B1( f1 +g1u)+ q− yd − c1yd . (34)

1506 Hong Zhao and Pinhas Ben-Tzvi

Let the Lyapunov function be Vlya2 =12 s2

1, then

Vlya2 = s1s1.

When s1 < 0 and s1 > 0,

us1 >

∣∣∣∣ B−11 ·g−1

1 [(A1 − c1I)(A1x+B1∆p+q)+B1 f1 + q− xd − c1xd ]

∣∣∣∣ . (35)

When s1 > 0 and s1 < 0,

us1 <

∣∣∣∣ B−11 ·g−1

1 [(A1 − c1I)(A1x+B1∆p+q)+B1 f1 + q− xd − c1xd ]

∣∣∣∣ . (36)

So

us1 =−ε1sgn(s1), (37)

and

Vlya2 = s1s1 < 0. (38)

Finally, the variable structure control is

ucy1 = w1 − ε1sgn(s1) ,ε1 > 0. (39)

Combining functions (27) and (39) into one controlfunction, the integrated control law becomes

ua1 = u f l1 +w1 −κ1sgnσ1 − ε1sgn(s1) ,

κ1 > 0, ε1 > 0, (40)

where K f 1 is the coefficient of viscous friction; Fkc isCoulomb friction in cylinder A; σ1 and s1 are the first layerof sliding mode and synthetic sliding mode in cylinder A,respectively.

4.2. Control rule in cylinder BThe second layer of the sliding surface is:

s2 = e2 + c2e2, c2 > 0. (41)

The control rule is:

ub2 = u f l2 +w2 −κ2sgnσ2 − ε2sgn(s2) ,

κ2 > 0, ε2 > 0, (42)

where e2 = z− yd . Parameter definitions in this sectionare similar to the previous sections for cylinder A with theonly difference that subscript 2 in this section pertains tocylinder B.

4.3. Friction compensationAs mentioned in Table 1, the maximum static friction

is 89.22 N. When the pressure in the cylinder is below0.2 MPa, the force is 251.2 N and as such friction cannotbe neglected. Therefore, this friction needs to be compen-sated, and the adopted compensation rule was chosen asfollows:

xv =

{∆1 |e|> ed and u1, u2 > 0,∆2 |e| ≤ ed or u1, u2 = 0,

(43)

where ed is the error tolerance; u1, u2 are the control valuebefore friction compensation; ∆1 = 0.02, ∆2 = 0.04 are thefriction compensation values.

5. EXPERIMENTAL RESULTS

5.1. Fundamental performance analysis of the pneu-matic position synchro system

Experiments were conducted to thoroughly study the per-formance of the pneumatic synchro system control. Ex-perimental results for an open-loop system response to astep input are shown in Figs. 2, 6, 7, and 8.

Fig. 2 shows the displacement response of each pistonmoving from the initial position (0 [mm]) to the final posi-tion of 600 [mm]. The chamber diameter is 40 [mm], andthe supply pressure is set to 0.6 [MPa]. Fig. 6 shows thepressure in each cylinder. Fig. 7 shows the displacementof each piston moving from the final to the initial position.Fig. 8 shows the pressure variations in each cylinder.

It can be seen from Figs. 2, 6, 7, and 8 that air com-pressibility introduces time delay in the piston response.Figs. 2 and 7 show that the two cylinders are operatingunder two different pressures. The cylinder with the lowerpressure exhibited more delay in the response and was af-fected by dithering at the end of the stroke – from 0 [mm]to 600 [mm]. One reason is due to the friction in the cylin-ders and the other may be attributed to the structure of thecylinder. The inlet and outlet of the rodless pneumaticcylinders are located on the same side. Because air mustflow through a long outlet hole in the cylinder, the resis-tance is relatively larger and hence the flow is not smooth.

According to Fig. 7, when the pistons displace from600 [mm] to 0 [mm], the outlet hole becomes the inlethole, and hence pressure dithering is no longer produced.But because the two cylinders operate under one commonair source, the pressure in each cylinder is practically notthe same. The pressure in cylinder B is lower than thatof cylinder A. Therefore, “chattering” is produced in thechamber of cylinder B as shown in Fig. 8(b).

The static friction and dynamic friction in the cylin-der and air compressibility made the left chamber in thecylinder show the “chattering” phenomenon, especially incylinder B. The pressure of the left chamber is low andbelow a certain value. If the pressure in the left chamberof cylinder B is estimated as 0.15 MPa, the force in theleft chamber becomes 188.4 N and the maximum staticfriction is 75.52 N. The static friction and dynamic fric-tion affect the piston displacement and the pressure in thechamber becomes oscillatory. The load masses, friction,leakage and other effects in both cylinders A and B are notcompletely identical. In addition, pneumatic systems havethe inherent disadvantage of strong nonlinearity and lownatural frequency. For all these reasons, it is usually dif-ficult to obtain satisfactory synchro control performancesusing traditional control methods.

5.2. Performance of PID controller in trackingFig. 9 shows the experimental results with an ideal

sinusoidal displacement input yd(t) = [300 + 100sin

Synchronous Position Control Strategy for Bi-cylinder Electro-pneumatic Systems 1507

(a) Pressure in Cylinder A.

(b) Pressure in Cylinder B.

Fig. 6. Pressure in cylinders A and B.

x,z

[mm

]

0 5 10 15 20

0

200

400

600

12

1 - Cylinder B

2 - Cylinder A

t [sec]

Fig. 7. Open-loop system response to a step input from theother direction.

(0.314t)] [mm] and two displacement output responsesx, z, for cylinders A and B, respectively. The systemis controlled using a classical PID controller with thefollowing control parameters: For cylinder A: kp = 6,ki = 0.0015, kd = 0.01; and for cylinder B; kp = 7,ki = 0.004, kd = 0.015.

It is clear from Fig. 9 that the response curves are dis-torted and fail to track the sinusoidal input due to non-linear friction effects. The experimental results show thatthe displacement synchro relative error between cylinderA and cylinder B is 15%.

5.3. Simulation of the synchronous control strategyIn order to compare the proposed controller with the

PID controller, the response simulation is performed. Fig.10 shows the simulation results with an ideal sinusoidaldisplacement input yd(t)= [0.1sin(0.314t)] [mm] and twodisplacement output responses x, z, for cylinders A and

0 5 10 15 200

200

400

600

t [sec]

p[kPa]

2

1

1 - Right Chamber

2 - Left Chamber

(a) Pressure in Cylinder A.

t [sec] 0 5 10 15 20

200

300

400

500

p[kPa]

1

2

1 - Right Chamber

2 - Left Chamber

(b) Pressure in Cylinder B.

Fig. 8. Pressure in cylinders A and B from the other direc-tion.

0 5 10 15 20 25 30 350

100

200

300

400

500

600

t [sec]

x,z[mm]

z

xIdeal input

Fig. 9. Tracking synchro control with PID controller.

0 5 10 15 20 25 30 35 40 45 50-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Displacement simulation in cylinder A

Displacement simulation in cylinder B

The ideal input

t [sec]

x,z

[m

]

Fig. 10. Tracking synchro control with the proposed con-troller.

B, respectively. From Fig. 10, the proposed controllershows better performances in synchro-tracking. The slid-ing mode control signal is shown in Fig. 11 and the totalcontrol signal is shown in Fig. 12.

1508 Hong Zhao and Pinhas Ben-Tzvi

t [sec]

slid

ing

mo

de

sig

na

l[V

]

Fig. 11. The proposed sliding model signal.

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

t [sec]

contr

ol

signal[V]

Fig. 12. The proposed control signal.

5.4. Experimental performance of the proposed syn-chronous control strategy

Fig. 13 shows the experimental results of the displace-ment output of cylinders A and B performed under differ-ent displacement inputs. In this experiment, we used thesynchronous control strategy and two-layer sliding modecontrol with feedback linearization based on friction com-pensation as proposed earlier. No external load was ap-plied to cylinder A. Fig. 10 shows the step response (with300 [mm] step size), with maximum relative synchro errorof 3%.

Fig. 14 shows the experimental results for the syn-chro tracking sinusoidal displacement curve without anexternal force. For the ideal displacement input yd(t) =[300+100sin(0.314t)] [mm], the maximum relative syn-chro error was 5%. The ideal displacement input is labeledas number 3 in Figs. 14-16. Since the cylinder stroke is600 mm, it is straightforward to choose the tracking of thesinusoidal displacement from the middle of the cylinder.The input signal frequency was based on the fact that typ-ically pneumatic system’s frequency is low. The system’sresponse to step and sinusoidal signal and the choice ofthe signal’s excitation frequency was based on what otherresearchers usually studied in analysis of single cylindersystems.

Fig. 15 shows the synchro tracking results with the 17

Time/s

t [sec]

x,z

[mm

]

0 0.5 1

0

100

200

300

1

21-Displacement response of cylinder A

2-Displacement response of cylinder B

Fig. 13.The response of the synchro control without external force Fig. 13. The response of the synchro control without ex-ternal force using the proposed control strategy.

0 10 20 30 40 50-8

-6

-4

-2

0

2

t [sec]

x,z,e[mm]

2

1

1-Displacement response of cylinder A

2- Displacement response of cylinder B

3-The ideal input

4-The Synchro error between cylinder A and B

0 20 40 60 80

0

100

200

300

400

500

600

4

32

4

0

-

2-

Fig. 14. Tracking of the synchro control without externalforce using the new control strategy.

0 10 20 30 40 50

0

100

200

300

400

500

600

t [sec]

x,

z,

e[m

m]

2

3

1

4

1-Displacement response of cylinder A

2-Displacement response of cylinder B

3- The Ideal input

4-The synchro error between cylinder A and B

5

0

5-

Fig. 15. Tracking of synchro control with 17 N externalforce.

N external force applied to cylinder A, for which the max-imum relative synchro error was 5%. The results showthat the control strategy possesses certain robustness toexternal disturbances. Fig. 16 shows the synchro track-ing results with the 34 N external force applied to cylinder

Synchronous Position Control Strategy for Bi-cylinder Electro-pneumatic Systems 1509

0 10 20 30 40 50 60 70 80-20

-15

-10

-5

0

5

10

t [sec]

x,z,e

[mm

]

0 20 40 60 80

0

100

200

300

400

500

600

3

2

1

4

1-Displacement response of cylinder A

2-Displacement response of cylinder B

3-The ideal input

4-The Synchro error between cylinder A and B

5

0

5-

Fig. 16. The tracking of synchro control with 34 N exter-nal force.

A, for which the maximum relative synchro error was 8%.According to the experimental results, the proposed con-trol strategy performed much better compared to the PIDcontroller. The synchro error was reduced and the controlaccuracy was enhanced.

6. CONCLUSION

In this paper, the system model was developed us-ing both theoretical and experimental methods. Asynchronous position control strategy was proposed toachieve good position tracking performances in the pres-ence of load disturbances and system uncertainties. Sinceelectro-pneumatic systems are highly nonlinear – espe-cially at low velocities – a bi-cylinder sliding mode con-troller was used to eliminate ‘chattering’ in the cylinderand improve the system robustness. Experimental resultsshowed the controller’s insensitivity to external distur-bances and ability to reduce the synchro position error.This technique can be readily applied to bi-cylinder sys-tems by choosing suitable PID parameters for the synchroPID controllers. As condition limits, the proposed con-trol strategy only used two cylinders with one air source.Since in many industrial applications there is only one airsource for many cylinders, the nonlinearity resulting fromlow pressure makes the control condition more serious interms of air compressibility and friction effects. Futurework will study the performance of multi-cylinder systemswith low pressure conditions.

NOMENCLATURE

Ap - Piston areau - Valve input opening sizeM1 - Mass load of cylinder AM2 - Mass load of cylinder Bx - Piston output displacement of cylinder Ax1 - Piston output displacement of cylinder A

x2 - Piston output velocity of cylinder Apx1(x3) - Left chamber pressure of the cylinder Apx2(x4) - Right chamber pressure of the cylinder Az - Piston output displacement of cylinder Bz1 - Piston output displacement of cylinder Bz2 - Piston output velocity of cylinder Bpz1(z3) - Left chamber pressure of the cylinder Bpz2(z4) - Right chamber pressure of the cylinder Bps - Supply pressurepe - Exhaust pressurePd - Downstream pressurePu - Upstream pressurePatm - Atmospheric pressureFf 1 - Friction force in cylinder AF1 - External force in cylinder Ak - Specific heat k = 1.4Cr - Critical pressure ratio = 0.528Cd - Valve orifice discharge coefficient = 0.8

C0 =

√k ·( 2

k+1

)(k+1)/(k−1)= 0.685 (constant)

V10 - Initial volumes of the chamberV20 - The volumes of the long hole in the cylinderL - Piston strokeR - Gas Constant [J/kg·K]δ - Valve’s clearanceA f - Valve’s leakage measureT - Absolute temperature T = 294 KFf 2 - Friction force in cylinder BF2 - External force in cylinder BD - Piston of valve’s diametervd - Velocity cut-off pointK f - Exact viscous friction factorFkc - Exact Coulomb friction factor

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Hong Zhao received her Ph.D. degreefrom Xi’an Jiaotong University, Xi’an,Shannxi Province, China in 2003. She iscurrently an Associate Professor in ChinaUniversity of Petroleum (Beijing). Herresearch interests include mechanical andelectrical transmission control, virtual in-strument measurement and control tech-nology.

Pinhas Ben-Tzvi received his B.S. degree(summa cum laude) in mechanical engi-neering from the Technion–Israel Instituteof Technology, Israel and his M.S. andPh.D. degrees in mechanical engineeringfrom the University of Toronto, Canada.He is currently an Associate Professorof Mechanical Engineering and Electricaland Computer Engineering, and the found-

ing Director of the Robotics and Mechatronics Laboratory at Vir-ginia Tech. His current research interests include robotics and in-telligent autonomous systems, human-robot interactions, mecha-tronics, mechanism design and system integration, dynamic sys-tems and control, and novel sensing and actuation. Applicationareas are varied and range from search & rescue on rough terrainto medical diagnostics, surgery, and therapy. Dr. Ben-Tzvi is asenior member of IEEE and a member of ASME.