A model reference robust multiple-surfaces design for tracking control of radial pneumatic motion systems

Nonlinear Dyn (2012) 67:2585–2597DOI 10.1007/s11071-011-0171-7

O R I G I NA L PA P E R

A model reference robust multiple-surfaces designfor tracking control of radial pneumatic motion systems

Chia-Hua Lu · Yean-Ren Hwang

Received: 4 October 2010 / Accepted: 18 July 2011 / Published online: 3 September 2011© Springer Science+Business Media B.V. 2011

Abstract A robust model reference backstepping(multiple-surfaces) controller is proposed for radialpneumatic motor motion systems with variable inletpressure and mismatched uncertainties (time-varyingpayload). A radial pneumatic motor is first modeledby a non-autonomous equation with consideration ofa ball screw table. A practical integral action and ro-bust action are included in the backstepping designto compensate for the disturbance, mismatched un-certainty, and to eliminate the steady state error. Themotion system is proved to have asymptotically stableperformance and the experimental results show thatthe proposed controller is able to track the referencemodel output signal and maintain steady-state error.

Keywords Radial pneumatic motor · Modelreference · Backstepping control

C.-H. Lu (�)Department of Mechanical Engineering, National CentralUniversity, Chung-Li 320, Taiwane-mail: [email protected]

Y.-R. HwangDepartment of Mechanical Engineering and the Institute ofOpto-Mechatronics Engineering, National CentralUniversity, Chung-Li 320, Taiwan

1 Introduction

Pneumatic motors offer remarkable advantages: in-difference to overloading, high power-to-weight ratio,cool operation, very high rotational speed, etc. In addi-tion, air does not generate sparks and is easy to store.A simple tank can store enough to produce pneumaticenergy for a long time, even under extreme tempera-ture conditions. The pneumatic motor can operate ata high number of cycles per workday and has alreadybeen applied in many industrial settings where a no-spark enviroment is required (for example, in chem-ical plants). Their torque versus speed characteristicsmake them very well suited for many applications andthey are easily reversible. There are many different de-signs for air motors, but two types of pneumatic mo-tors dominate: the general vane-type air motor [1–6]and the radial piston air motor [7]. There have been anumber of investigations of the dynamics of vane-typeair motors. However, the high compressibility of airmakes control of the motor’s velocity very difficult andthis type of motor is typically characterized by non-autonomous dynamics [1]. This in turn means that aconventional PID controller may not provide satisfac-tory performance. In this study, a vane-type air mo-tor was mounted on a wall-climbing machine. A first-order dynamic is considered for implementation of aPI controller for velocity regulation. Even though thePI control theory is highly reliable when the values arewell adjusted, the PI controller cannot maintain per-formance all the times. There is large overshoot and

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2586 C.-H. Lu, Y.-R. Hwang

dead-zones during the experiments. The mathematicalmodel of a vane-type air motor non-linear system isconsidered a fourth-order model [4]. The parameterswere identified from the experiments and the modelwas validated through a simulation and actual sys-tem velocity output. A different approach based on theneural network techniques is proposed in [5]. Insteadof developing a high order non-linear model, a neural-model reference control based on neural network tech-niques is utilized to control the rotational speed of theair motor. The ball screw table with vane-type air mo-tor is described in [6]. The controller is a hybrid con-cept, combining sliding mode control with fuzzy rules.When the system approaches the sliding surface, thefuzzy controller adjusts the surface in order to reducechattering phenomenon and accomplish accurate posi-tion control. The applications of vane-type air motorsare discussed in the above studies. The radial pistonair motor has been applied for velocity control withflywheels [7]. Conventional PID control has been pro-posed for use in this type of non-linear system butthere is no mathematical model. Dut to the high com-pressibility of air and lack of modeling, the experi-mental results show large overshoot, a chattering phe-nomenon and serious time-delay for motion control.Clearly, mathematical modeling of a radial piston airmotor based on advanced control theory is needed inorder to develop more sophisticated controllers. Ad-vances in electronics have helped to develop controlsystems for electric drives that allow for superior airmotors with enhanced performance. Examples includeopen or closed-loop controlled cylinders for manufac-turing, pressure controlled chambers in lorry brakingcircuits or position controlled actuators for processvalves [8, 9].

In most motion control systems, the PI controllerpresents apparently acceptable performance. However,when parameter uncertainties become significant, forinstance because of, large variations in inertia and un-known but bounded time-varying loading, it is diffi-cult to achieve satisfactory performance based on thePI scheme. For pneumatic motion systems, the biggestissue for tracking control is the compressibility of air.Some researchers [6, 8, 9] have mentioned that a slid-ing mode controller can prove very robust for pneu-matic motion applications but there could be a chat-tering phenomenon when the system is approaching asliding surface. In the last decade, various backstep-ping non-linear design methodologies have been pro-posed [10–17] in order to reduce serious chattering

in a pneumatic non-linear system and there has beenincreased effort attempting to use backstepping con-trol for industrial applications. In this study, we pro-pose a new controller that utilizes a non-linear robustbackstepping design with a reference model for in-dustrial tracking position control on a ball screw ta-ble: Fig. 1. The objectives of this study are to de-velop a normal mathematical model of a radial pis-ton air motor, of which there are two major types(three or five radial pistons). The aim is to reduce thechattering phenomenon and to achieve robust perfor-mance for different inlet pressures, parameter varia-tions, load disturbances, and frictions present in theradial piston air motor drives. A practical integral ac-tion is included in the backstepping design for thispneumatic motion system. It is necessary to compen-sate for disturbances, mismatched uncertainties (dif-ferent inlet pressure and time-varying payload) and toeliminate the steady state error. The piston air motormodel will be represented in formal form. The systemstability is proven to be asymptotically stable accord-ing to the Lyapunov method. The experimental resultsshow that model reference robust backstepping algo-rithm presents very good position tracking responseregardless of the uncertainties.

2 Modeling of radial pneumatic motor system

The principles of radial piston air motor operation areillustrated in Fig. 2. The application of the air motoris very widespread because of its advantages of smallvolume, light weight, low speed, high efficiency, andlong service life. As seen in Fig. 1, the pistons (a)and the outgoing shaft (c) are rotated by connectingrods (b). When the air motor is powered by air fromport A (e), port B (d) acts as the exhaust port. The pis-tons and the connecting rods propel the outgoing shaftin an anti-clockwise direction. The velocity of the airmotor can be adjusted by the inlet pressure. High ve-locity is attained with incremental pressure and viceversa.

2.1 Airflow dynamics

The airflow through the orifice of the valve can beconsidered as a one-dimensional compressible flowin a nozzle. Depending on the downstream pressure(outlet pressure) Pd (bar) and the upstream pressure

A model reference robust multiple-surfaces design for tracking control of radial pneumatic motion 2587

Fig. 1 Schematic diagramof the air motor system

Fig. 2 Principles of radial piston air motor operation

(supply pressure of the valve) Pu (bar), the mass flowrate through one orifice of the valve can be describedas [18]

m = CdCoAof (Pr), (1)

where

f (Pr) = Pu√Tu

{1, Patm

Pu< Pr < Cr ;

Ck[P2K

r − PK+1K

r ] 12 Cr < Pr < 1;

Pr = Pd

Pu

, Co = Rg

[(K + 1

2

)]K+1K−1 = 0.0404,

Cr =(

2

K + 1

) KK−1 = 0.528, Ck = 3.864,

Cd = 0.8.

The symbol m (kg/s) is the mass flow rate; Cd is thedischarge coefficient; Co is the flow constant; Pr is theratio between the downstream and upstream pressureacross the orifice; K is the specific heat ratio; Ao (m2)is the area of the orifice; Rg (J/kg K) is a gas constant;and Tu (K) is temperature upstream.

Consider the case when flow enters the piston. Thisnow becomes a drive chamber and builds pressure(i.e., control signal u > 0) under the zero-lap valve as-sumption, as shown in Fig. 3. Since the orifice areais proportional to the control signal uc (V), we would


Fig. 3 Illustration of pressure in the piston cylinder

rewrite (1) as

ma = CdCof (Pr)

(Ao max

umax

)uc,

f (Pr) = Ps√Ts

{1, Patm

Ps< Pr < Cr,

Ck[P2K

r − PK+1K

r ] 12 Cr < Pr < 1,

Pr = Pa

Ps

. (2)

Subscript a represents the status when the flow is en-tering the piston; ma is the mass flow rate into the pis-ton; Pa is the pressure in the piston; Ps is the supplypressure; Ts is the supply temperature; Ao max is thelargest area of the orifice; and umax (V) is the maxi-mum control signal of the proportional valve.

In contrast, when the piston is exhausting pressure,we have

mb = CdCof (Pr)

(Ao max

−umax

)(−uc),

f (Pr) = Pb√Tb

⎧⎨⎩

1, PatmPb

< Pr < Cr,

Ck[P2K

r − PK+1K

r ] 12 Cr < Pr < 1,

Pr = Pe

Pb

. (3)

Subscript b represents the status when the flow isexhausting from the piston; mb is the mass flow rateout the piston; Pb is the pressure in the piston; Pe isthe exhaust pressure; and Tb is the piston temperaturewhich can be considered the same as the supply tem-perature.

Under the simple compressible system assumption,the process in the piston cylinder is adiabatic and leak-

age of the air motor can be neglected. The mass flowrate in the piston cylinder can be described as follows:

ma = APa

RgTs

d + Va

KRgTs

Pa,

mb = − APb

RgTs

d + Vb

KRgTs

Pb,

(4)

where A = πD2/4 is the area of the piston; D is thediameter of the piston; Ts is the supply temperature;Pa is the inlet pressure; Pb is the outlet pressure; d isthe displacement of the piston; Va is the volume whenthe flow enters the piston; Vb is the volume when flowexhausts to the atmosphere; and K is the specific heatconstant.

The volume in the piston cylinder is V = Ad , there-fore, substituting (4) into (2) and (3) we get

Pa = KRgTs

AdCdCoAof

(Pa

Ps

)(Ao max

umax

)uc

− KPa

dd,

Pb = KRgTs

AdCdCoAof

(Pe

Pb

)(Ao max

−umax

)(−uc)

+ KPb

dd.

(5)

In most pneumatic systems, a proportional directionalvalve will be adopted for position control. The systemneeds a pressure sensor to feedback the signal, but theproblem is that this pressure sensor signal will includea lot of noise. This could affect the controller and re-duce the system performance. Hence, a proportionalpressure valve is adopted to generate stable pressure.The derivation of the pressure can be treated as 0, sothe system satisfies P = 0 and (5) can be rewritten asfollows:

Pa = RgTs

AdCdCof

(Pa

Ps

)(Ao max

umax

)uc,

Pb = −RgTs

AdCdCof

(Pe

Pb

)(Ao max

−umax

)(−uc).

(6)

2.2 Piston dynamics

A free body diagram of the piston is shown in Fig. 4,where D is the diameter of the piston; L is the lengthof the connecting rod; R is the radius of the crank; y is


Fig. 4 Free body diagramof a piston

the displacement; Fp is the gas load along the cylindercenterline; Fc is the force on the connecting rod; andFt is the tangential force on the crank.

The force on the piston is

Fp = π

4D2P, (7)

where P is the pressure in the cylinder.The force on the connecting rod or thrust force is

Fc = Fp

cos(ϕ). (8)

The tangential force on the crank is

Ft = Fc sin(ϕ + θ)

= Fp

sin(ϕ + θ)

cos(ϕ)

= Fp

sinϕ cos θ + cosϕ sin θ

cos(ϕ)

= Fp

(tan(ϕ) cos(θ) + sin(θ)

). (9)

The motor torque, M is

M = FtR = πD2

4P


)R. (10)

The relationship between ϕ and θ is

sin(θ)

sin(ϕ)= L

R⇒ L sin(ϕ) = R sin(θ). (11)

The torque can be described as

M = π

4D2PR


)

= π

4D2PR

(sin(θ)

+ cos(θ) tan

(sin−1

(R sin(θ)

L

)))(12)

There are three pistons in this air motor. If Pa is theinlet pressure needed to propel one of the three pistons,Pb is the outlet pressure needed to exhaust air from theother two pistons.

The total torque output Mtol is

Mtol = πD2

4R

{sin(θ)

+ cos(θ) tan

[sin−1

(R sin(θ)

L

)]}× (Pa − Pb1 − Pb2), (13)

where the subscripts b1 and b2 indicate other two pis-tons to exhaust airflow,

Pa = RgTs

AdCdCof

(Pa

Ps

)(Ao max

umax

)uc,

π

36≤ θ <

2π

3,

Pb1 = −RgTs

AdCdCof

(Pe

Pb1

)(Ao max

−umax

)(−uc),

2π

3≤ θ <

4π

3,

Pb2 = −RgTs

AdCdCof

(Pe

Pb2

)(Ao max

−umax

)(−uc),

4π

3≤ θ < 2π,

(14)

where

d(θ) = R

{1 − cos(θ) + R

4L

[1 − cos(2θ)

]}. (15)

Assume that πD2

4 R{sin(θ) + cos(θ) ×tan[sin−1(

R sin(θ)L

)]}(Pb1 + Pb2) is the disturbance andthat Mtol can be

Mtol = πD2

4R

{sin(θ)

+ cos(θ) tan

[sin−1

(R sin(θ)

L

)]}Pa − Ddis,

(16)


where Ddis is the disturbance caused from the upwardmovement of the two pistons.

Considering friction and different payloads, and us-ing Newton’s second law of angular motion, we get

Mtol − McS(θ) − Mf θ = J θ, (17)

where θ is the angle velocity; θ represents the angularacceleration; Mc is the stiction coefficient; Mf is thefriction coefficient, and S(θ) is described as below:

S(θ) ={

0, θ = 0,

sgn(θ), θ �= 0.(18)

The system can be presented as follows:

J θ = πD2

4R

{sin(θ)

+ cos(θ) tan

[sin−1

(R sin(θ)

L

)]}Pa

− McS(θ) − Mf θ − Ddis. (19)

Assume that x is the displacement of the ball screwtable and ρs is the screw pitch. The displacement x =ρsθ and velocity x = ρsθ .

Equation (17) can be rearranged as follows:

Mtol − McS(θ) − Mf θ − Tl = Jmθ, (20)

where Jm is the total system inertia (motor and ballscrew); Tl is the load torque. The dynamics of the sys-tem can be described by the following equation con-sidering friction:

x = π

4D2R

(ρs

Jm

){sin

(x

ρs

)

+ cos

(x

ρs

)tan

[sin−1

(R sin( x

ρs)

L

)]}Pa

− Mf

Jm

x − ρs

Jm

[McS

(x

ρs

)+ Ddis + Tl

]. (21)

Assuming that x = x1, x2 = x1, the state-space dy-namic model

x1 = x2,

x2 = Ax1 + B(x)u + F,(22)

where

A = −Mf

Jm

,

Table 1 All parameters of the experimental system

Parameter Value Unit Parameter Value Unit

L 67 mm Jm 8.77 × 10−4 kg m2

D 24 mm Mf 0.34 N

R 37.5 mm Mc 0.06 Ns

ρs 0.795 mm/rad Ao max 6.28 mm2

B = πD2

4R

(ρs

Jm

){sin

(x

ρs

)

+ cos

(x

ρs

)tan

[sin−1

(R sin( x

ρs)

L

)]},

u = RgTs

AdCdCof

(Pa

Ps

)(Ao max

umax

)uc,

F = − ρs

Jm

[McS

(x

ρs

)+ Ddis + Tl

].

Mathematical model validation has been verifiedby comparing the open loop step dynamic of all ballscrew table system responses which are obtained fromboth experiments and simulation. Specifications of theair motor used for the experiment and simulation arelisted in Table 1. The system response for an open loopair motor system is shown in Fig. 5 and the simulatedresults based on (22). The simulated and experimentalresults are consistent.

3 Model reference robust sliding controller design

Before designing the controller, the following assump-tions are significant for deriving controller:

Assumption

1. Jm is bounded by Jmin ≤ Jm ≤ Jmax.2. A is bounded by Amin ≤ A ≤ Amax.3. B is bounded by βmin ≤ B−1 ≤ βmax.4. The uncertain force is bounded by F ≤ Fmax.

The reference model is described as

xm1 = xm2,

xm2 = Amxm2 + Bmum,(23)

where xm is the reference state, Am < 0, Bm > 0, andum is the reference control input. Our task is to design


Fig. 5 Experimental andsimulated results withvarious inlet pressures

a controller to track a bounded position reference sig-nal xr . By assumption, the given reference signal is acontinuous, differentiable smooth signal. In this con-troller design, integral action is implemented to helpthe controller to deal with the disturbances existing inthe tracking control systems and enhance the systemtransient and steady state performance. First, we de-fine the reference tracking error er1 = xr − xm1 andthen

er1 = xr − xm1 = vr − vm, (24)

where vr = xr , vm = xm1 = xm2.In order to minimize the tracking error er1, asymp-

totically converging to zero, the xm2 can be designedas xm2 = c1er1 + xr (c1 is a positive constant at thedisposal of the designer and this generates the desiredexponential behavior for the tracking error er1); andthen

er1 = xr − (c1er1 + xr ) = −c1er1. (25)

vr is chosen as follows:

vr = c1er1 + xr + λ1z1, (26)

where λ1 is a positive constant at the disposal of thedesigner and z1 = ∫ t

0 er1(t) dt is the integral of thetracking error. By adding this integral action into (26),the convergence of the tracking error to zero at thesteady state can be enforced, despite the presence ofthe disturbance and model uncertainty in the trackingcontrol system.

Define the er2 as follows:

er2 = vr − vm = c1er1 + xr + λ1z1 − vm

= c1er1 + λ1z1 + er1. (27)

Then

er2 = c1er1 + xr + λ1z1 − vm

= c1er1 + xr + λ1er1 − Amxm2 − Bmum. (28)

From (27), we can obtain

er1 = −c1er1 − λ1z1 + er2. (29)

Substituting (29) into (28) we get

er2 = c1(−c1er1 − λ1z1 + er2) + xr + λ1er1

− Amxm2 − Bmum.

After the above derivation, we analyze the stabilityand asymptotic convergence performance of the pro-posed controller based on the Lyapunov theory.

The Lyapunov function is designed as below:

V1 = λ1

2z2

1+ 1

2e2r1 + 1

2e2r2. (30)

Then

V1 = λ1z1z1 + er1er1 + er2er2

= λ1z1er1 + er1(−c1er1 − λ1z1 + er2)

+ er2[c1(−c1er1 − λ1z1 + er2) + xr


+ λ1er1 − Amxm2 − Bmum

]= er1(−c1er1 + er2) + er2

[c1(−c1er1 − λ1z1

+ er2) + xr + λ1er1 − Amxm2 − Bmum

]. (31)

Designed reference control input

um = 1

Bm

[−c21er1 − c1λ1z1 + c1er2 + er1

+ c2er2 + xr + λ1er1 − Amxm2], (32)

where c2 is a positive constant at the disposal of thedesigner.

Substituting (32) into (31), we get

V1 = −c1e2r1 − c2e

2r2 ≤ 0. (33)

The global uniform boundedness of the tracking er-ror signal er1, the integral action z1, and the velocityerror signal er2 are guaranteed by the definition of theLyapunov function in (30) and the non-positivity ofthe Lyapunov derivative in (33).

For robust controller design, we define the error be-tween system output x1 and reference output xm1 : e =x1 − xm1.

Define the sliding surface

s = e − Ame. (34)

We now define a new Lyapunov function

V2 = 1

2s2. (35)

Equation (36) can now be derived from (34) by

s = e − Ame = (x2 − xm2) − Am(x1 − xm1)

= x2 − xm2 − Amx1 + Amxm1

= (x2 − Ax1) − (xm2 − Amxm1) + x1(A − Am)

= B(x)u + F − Bmum + x1(A − Am),

B−1(x)s = u + B−1(x)F − B−1(x)Bmum

+ B−1(x)x1(A − Am).

(36)

According to previous assumption, we can assume that

βmin ≤ B−1(x) ≤ βmax,

αmin ≤ B−1(x)(A − Am) ≤ αmax,

B−1(x)F ≤ Dmax.

(37)

Let

α = αmax + αmin

2, Δα = αmax − αmin

2,

β = βmax + βmin

2, Δβ = βmax − βmin

2.

(38)

Design controller u

u = −ks + βBmum − αx1 − [Dmax + Δβ|Bmum|

− Δα|x1|]sgn(s). (39)

Substituting (39) into (36) we obtain

B−1(x)s = −ks + βBmum − αx1

− [Dmax + Δβ|Bmum| + Δα|x1|

]sgn(s)

+ B−1(x)F − B−1(x)Bmum

+ B−1(x)x1(A − Am)

= (β − B−1(x)

)Bmum − Δβ|Bmum|sgn(s)

+ [B−1(x)(A − Am) − α

]x1

− Δα|x1|sgn(s) + B−1(x)F

− Dmaxsgn(s) − ks, (40)

because

B−1(x) − β ≤ Δβ,

B−1(x)(A − Am) − α ≤ Δα.(41)

Equation (42) can now be derived from (35) by

B−1(x)V2 = B−1(x)ss

= (β − B−1(x)

)Bmums − Δβ|Bmum||s|

+ [B−1(x)(A − Am) − α

]x1s

− Δα|x1||s|+ B−1(x)F s − Dmax|s| − ks2 ≤ 0.

(42)

Equation (42) shows that the design of control in-put satisfies the approaching sliding condition, andhence ensures that the system will be asymptotical sta-ble. Since the desired position reference signal xr isbounded by the assumption and tracking error er1 ande are also bounded, we know that the actual positionoutput x1 is globally uniformly bounded. The controlsystem diagram is shown in Fig. 6.


Fig. 6 The block diagramof control system

Fig. 7 Position regulationtracking results for differentinlet pressures

Fig. 8 Voltage output forposition regulation tracking


Fig. 9 Tracking referenceposition sinusoidaltrajectory(xr = a sin(2πf t),a = 10,12,15 mm forf = 0.05 Hz)

Fig. 10 Voltage output forxr = a sin(2πf t),a = 10,12,15 mm forf = 0.05 Hz

4 Experimental results

As shown in Fig. 1, the experimental pneumatic sys-tem consists of a piston air motor, proportional valve,ball screw table, linear scale, and DSP controller. Thevalve has a critical frequency of 128 Hz in the spoolstroke. The ball screw table has an anti-backlash ballnut design, and the accuracy of the liner scale 0.5 µmis employed to feed back the ball screw table posi-tion. The mismatched uncertainties arise for the designby using a liquid tank which is mounted on the table

to produce a time-varying payload whose mass variesfrom 10 kg to 5 kg with an inflow rate of 0.2 kg/s.

The proposed control strategy is implemented inthe DSP controller with a 1 ms sampling time. Theoutput voltage is constrained by ±5 V. The referencemodel variable is set to the following specifications:Am = −85, Bm = 88. These values are designed un-der a supply pressure of 3 bar. The pneumatic systemis required to start from x(0) = 0 m to the desired po-sition and the supply pressure is regulated at 5 bar,4 bar, 3 bar at the beginning of the experiment. The


Fig. 11 Tracking error forxr = a sin(2πf t),a = 10,12,15 mm forf = 0.05 Hz

Fig. 12 Tracking referenceposition sinusoidaltrajectory(xr = a sin(2πf t),a = 20,22,25 mm forf = 0.025 Hz)

three lines plotted in each figure represent the resultsunder different pressures.

The desired reference position signals and ex-perimental tracking trajectories for position track-ing are shown in Fig. 7 and the output voltage isshown in Fig. 8. It can be seen in Fig. 7 that thetracking reference signal for the system is with-out large overshoot for each different supply pres-sure. In this case, the steady-state error is less than3 µm and steady state can be enforced regardless ofthe presence of disturbance or modeling uncertainty.

For trajectory tracking, the designed reference po-sition trajectories are two different sinusoid signalsxr = a sin(2πf t): a = 10,12,15 mm for f = 0.05 Hz

and a = 20,22,25 mm for f = 0.025 Hz. Figures 9,10, and 11 (a = 10,12,15 mm, f = 0.05 Hz) showthe results for tracking performance, output voltage,and tracking error, respectively. Figures 12, 13, and14 show the tracking performance results for a =20,22,25 mm, f = 0.025 Hz. It should be notedfrom the tracking error results that the proposed robustbackstepping controller with integral action can signif-


Fig. 13 Voltage output forxr = a sin(2πf t),a = 20,22,25 mm forf = 0.025 Hz

Fig. 14 Tracking error forxr = a sin(2πf t),a = 20,22,25 mm forf = 0.025 Hz

icantly reduce error leading to almost complete con-vergence to the given reference position signal evenwith transient dynamics. With the proposed controller,the piston air motor can obtain enough torque inputto quickly reduce the initial tracking error, while therobust backstepping controller generates a significantamount of control effort at the beginning to bring thepiston air motor up to the desired trajectory. The track-ing error remains small during the whole transition forthe different inlet pressures. The performance is con-sistent for a considerable period of time during opera-

tion. The device is effective for most pneumatic servoapplications in industry.

5 Conclusions

In this paper, we present a non-linear controller devel-oped based on robust model reference backsteppingfor industrial pneumatic motion control applications,which improves the system tracking response. Thesystem model is first described in a normal form. Un-certainties can be observed to enter the system due to


the time-varying payload and force disturbances. Theexperimental results demonstrate the effectiveness ofour design. The robust model reference backsteppingcontroller is shown to achieve superior performance.The controller is proven capable of giving asymptoticconvergence performance of the tracking error and theexperimental results justify the probability of the con-trol strategy.

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A model reference robust multiple-surfaces design for tracking control of radial pneumatic motion systems