Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472 Received for review: 2021-04-26
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-08-04
DOI:10.5545/sv-jme.2021.7236 Original Scientific Paper Accepted for publication: 2021-09-01
*Corr. Author’s Address: Hanoi University of Science and Technology, No. 1, Dai Co Viet str., Hai Ba Trung dist., Hanoi city, Vietnam, bo.tranxuan@hust.edu.vn 458
Nonlinear Control of a Pneumatic Actuator  
Based on a Dynamic Friction Model
Tran, X.B.
Xuan Bo Tran
Hanoi University of Science and Technology, School of Transportation Engineering, Vietnam
This paper presents a new controller for the position of a pneumatic actuator. The controller is designed based on the multiple-surface sliding 
control method in combination with a frictional compensator. The multiple-surface sliding control method is applied to deal with the nonlinear 
characteristics of the pneumatic system, and the frictional compensator is applied to compensate for the friction force in the pneumatic 
actuator. The friction force is estimated based on a dynamic friction model (the LuGre model). Both simulation and experimental studies are 
done to evaluate the new controller. The evaluation results indicate significant improvement in the tracking position error of the new controller 
comparing to the multiple-surface sliding controller without friction compensation and other nonlinear controllers.
Keywords: pneumatic actuator, nonlinear control, friction compensation, dynamic friction model
Highlights
• 	 The new controller with friction compensation is proposed to control the position of a pneumatic cylinder.
• 	 A minimal position peak error of 1 mm in steady-state condition can be obtained by the new controller with the step desired 
positions.
• 	 A minimal position peak error of 2.2 mm in steady-state condition can be obtained by the new controller with the sinusoidal 
desired positions. 
• 	 By using the new controller, the relative position peak error is reduced by 10 %, and the relative root means squared (RMS) 
error is reduced by 6.5 % in comparison to those of the multiple-surface sliding (MSS) controller without friction compensation. 
0  INTRODUCTION
Pneumatic systems are widely used in industrial 
environments because they can provide many 
advantages, including low cost, high power-to-weight 
ratio, cleanness, ease of maintenance and replacement, 
cheap and available supply of air sources [1]. The 
pneumatic systems are also preferred in hot and/or 
humid environments where electric systems cannot be 
applied [2]. However, the compressibility of air, the 
nonlinearity of valves and friction in the pneumatic 
actuator are disadvantages of pneumatic systems. 
They cause the system to have high nonlinearities 
and modelling uncertainties [3] to [5]. Therefore, the 
application of linear control strategies is not suitable 
for achieving fast and highly accurate responses for 
the pneumatic actuators’ position [6] to [8]. Many 
nonlinear and advanced control strategies have 
been employed to overcome the nonlinearities and 
uncertainties of the pneumatic systems. Bobrow and 
Jabbari [9] applied an adaptive control method based 
on the linear dynamics of the system to control the 
position of a pneumatic actuator. An adaptive fuzzy-
proportional-derivative (PD) controller was applied 
by Gao and Feng [10] for a one-degree pneumatic 
actuator. The full-state feedback was used in the 
control law for simultaneous parameter identification 
and tracking control. A combination of an adaptive 
control strategy and neural network was proposed 
by Chen et al. [11] for an electro-pneumatic servo 
system; the neural network was used to compensate 
for constructing a linearized model of the nonlinear 
system, and the robust adaptive controller was used 
to perform the model-matching for the uncertain 
linearized model of the system. Gross and Rattan [12] 
applied multilayer neural networks to compensate 
for the nonlinear nature of the dynamic system in 
conjunction with a proportional-integral-derivative 
(PID) feedback controller. The advantages of the 
above-advanced control strategies are that they allow 
updating system parameters and controller parameters. 
However, the quality of control performances greatly 
depends on the accuracy of the mathematical model of 
the system.
For the sliding mode control method, the first 
positioning application for pneumatic actuators was 
made by Paul et al. [13], Tang and Walker [14], and 
Surgenor and V aughan [15]. This robust control method 
allows handling nonlinearities and compensates for the 
mismatched uncertainties of the mathematical model. 
However, one of the main drawbacks of this kind of 
control strategy for pneumatic systems is the need for 
acceleration feedback. To overcome this difficulty, 
Acarman et al. [16] and Liu et al. [17] used observers 
to estimate the acceleration and Pandian et al. [18] 
proposed the use of differential pressure feedback 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
459 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
instead of acceleration feedback. Recently, Tsai 
and Huang [19] proposed a multiple-surface sliding 
controller (multiple-surface sliding (MSS) controller) 
for a pneumatic servo system. This controller was 
designed based on three sliding surfaces of position, 
velocity, and force, and this controller can exhibit the 
smallest control error among the controllers proposed 
thus far for the pneumatic actuators. However, the 
friction force was omitted in designing the MSS 
controller.
Armstrong-Helouvry [20] has shown that the 
control performances of a mechanical system can be 
significantly improved when the system controller 
is designed with friction compensation. Koch and 
Reichhartinger [21] and Ayalew and Jablokow [22] 
have applied the sliding model control method 
combined with model-based friction compensation 
in the controlling position of hydraulic actuators; 
they have achieved improved control performances. 
However, only simple nonlinear friction models such 
as static or Coulomb friction models were used in 
these studies. Tran and Yanada [23] have shown that 
the static friction models are not enough or useless in 
predicting dynamic friction behaviours in pneumatic 
cylinders, especially in the cases where the cylinders 
operate at low and oscillating velocity conditions. 
Therefore, dynamic friction models must be used to 
achieve highly accurate control performances for the 
pneumatic cylinder position. To date, many dynamic 
friction models have been proposed [24] to [29]. 
Among these, the dynamic LuGre friction model [25] 
is most used. This dynamic friction model can observe 
basic friction characteristics in most mechanical 
systems such as pre-sliding displacement, stick-slip 
motion, varying break-away force, and hysteresis. In 
addition, this model is quite suitable for theoretical 
calculations and is easy to implement. Tran and 
Yanada [23] have also experimentally examined 
the friction characteristics in pneumatic cylinders 
and have shown that the LuGre friction model can 
relatively adequately simulate all the measured 
friction characteristics. However, according to the 
author’s survey, the application of the LuGre friction 
model combined with the MSS control method for 
controlling the position of the pneumatic actuators has 
not been studied.
In this paper, the multiple-surface control method 
combining with friction compensation based on the 
LuGre friction model for controlling the position of a 
pneumatic cylinder is studied. An electro-pneumatic 
servo system with a double-acting pneumatic cylinder 
and two pneumatic proportional flow control valves 
is considered to carry out this work. A general 
mathematical model of the system is built first. The 
control law of the new controller is then designed, and 
the absolute stability of the system is analysed. The 
new controller is evaluated using both simulations 
and experiments. The control performances of the 
new controller are compared with those of the MSS 
controller and other nonlinear controllers. Finally, the 
effectiveness of the new controller is shown. 
The rest of this paper is organized as follows. 
In Section 1, the electro-pneumatic servo system is 
introduced, and its mathematical model is developed. 
Section 2 shows the synthesis of the system controller. 
A stable analysis of the closed-loop system is also 
shown in this section. The simulation evaluations are 
shown in Section 3, and the experimental evaluations 
are shown in Section 4. Finally, the main conclusion is 
drawn in Section 5.
1  SYSTEM MODEL
In this paper, we consider an electro-pneumatic servo 
system, as shown in Fig. 1. In this system, a double-
acting pneumatic cylinder is used to drive a load mass 
M. 
The motion of the cylinder piston is controlled by 
two 3/2 electro-pneumatic proportional flow control 
valves, Valves 1 and 2. Valve 1 is connected to the left 
chamber, and V alve 2 is connected to the right chamber 
of the cylinder. The valves’ operation is controlled 
by the voltage signals u
1
 and u
2
. According to the 
characteristics of the valves used in the experimental 
Fig. 1.  Schematic diagram of the system

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
460 Tran, X.B.
system, as shown later in Section 4.1, the voltage 
values of u
1
 and u
2
 vary from 0 to 5 VDC. If 2.5 ≤ u
i
 ≤ 5 
VDC (i = 1 and 2), the valves operate at the left side 
position, letting air from the compressor flow into the 
cylinder chambers; and if 0 ≤ u
i
 < 2.5 VDC, the valves 
operate at the right-side position, releasing air from 
the cylinder chambers to the atmosphere. Therefore, 
the cylinder piston will extend if 2.5 ≤ u
1
 ≤ 5 VDC and 
0 ≤ u
2
 < 2.5 VDC. Conversely, the cylinder piston will 
extract if 0 ≤ u
1
 < 2.5 VDC and 2.5 ≤ u
2
 ≤ 5 VDC. The 
flow rates through the valves depend on the voltage 
values of u
1
 and u
2
.
To develop a mathematical model of the system, 
the following assumptions are used:
a) The used air is an ideal gas, and its kinetic energy 
is negligible in the cylinder chambers.
b) The leakages of the cylinder are negligible.
c) The temperature variation in cylinder chambers is 
negligible with respect to the supply temperature.
d) The pressure and the temperature in the cylinder 
chambers are homogeneous.
e) The evolution of the gas in each chamber is a 
polytropic process.
f) The supply and ambient pressures are constant.
The mass flow rate  m
i
 (i = 1 and 2) that flows 
into or flows out from the chambers of the pneumatic 
cylinder can be derived in terms of the voltage inputs 
u
i
 of the two valves as follow:
  m
p
k
RT
Hu u
p
k
RT
Hu u
i
ib sV ii
ie iV ii

  
  


1
2
25 25 5
25 02
.,.
., .. ,
5







 (1)
where p
s
 and p
i
 are respectively the supply air pressure 
and the pressure in the cylinder’s chambers, k is the 
specific heat ratio, R is the gas constant, and T is the 
temperature of the supply source. H
V1
 and H
V2
 are 
respectively the valve coefficients. β
ib
 and β
ie 
are the 
pressure coefficients and are given by:

ib
i
s
k
k
i
s
k
k
i
s
k
k
k
p
p
p
p
p
pk
























2
1
1
2
1
1
2
1
,
1 1
1
05 8
2
1
.,
,
p
pk
i
s
k
k


















 (2)

ie
atm
i
k
k
atm
i
k
k
atm
i
k
p
p
p
p
p
pk




















2
1
1
2
1
1
2
1
,
 





















k
k
atm
i
k
k
p
pk
1
1
05 8
2
1
.,
, (3)
where p
atm
 is the atmosphere pressure.
The dynamic relationship between the mass flow 
rates  m
1
,  m
2
 and the pressures p
1
, p
2
 in the cylinder 
chambers can be given by continuous equation as 
follows:
 


p
k
V
RTmp Av
p
k
V
RTmp Av
1
1
11 1
2
2
22 2




,
, (4)
where v is the piston velocity, A
1
 and A
2
 are the piston 
areas. V
1
 and V
2
 are respectively to the volumes in the 
left and right chambers of the cylinder and are given 
as follows:
 
VV Ax
VV ALx
11 01
22 02

 
,
() , (5)
where L is the piston stroke, x is the piston position, 
V
10
 and V
20
 are, respectively, the dead volumes in the 
left and right chambers of the cylinder. The motion 
equation of the cylinder piston according to Newton’s 
second law is given by:
 Ma pA pA F
fr

11 22
, (6)
where M is the load mass of the piston, a is the piston 
acceleration and F
fr
 is the friction force. In [23], Tran 
and Yanada have measured the friction characteristics 
of the pneumatic cylinders and have shown that most 
of the measured characteristics can be simulated by 
the dynamic LuGre friction model with relatively 
good accuracy. Therefore, in this study, the LuGre 
model is used to describe the friction force F
fr 
in the 
pneumatic cylinder. The LuGre friction model is 
described as follows:
  wv
w
qv
v 


0
, (7)
 qv FF F
cl st cl
vv
s
n
() exp,
/
  

 (8)
 Fw wv
fr
 
012
 , (9)
where w is the mean deflection of the elastic bristle 
between two surfaces made in contact, σ
0
 is the 
stiffness of the elastic bristle, q(v) is the Stribeck 
function, F
st
 is the static friction force, F
cl
 is the 
Coulomb friction force, v
s
 is the Stribeck velocity, n 
is the exponent that affects the slope of the Stribeck 
curve, σ
1
 is the micro-viscous friction coefficient, and 
σ
2
 is the viscous friction coefficient.
Setting the valve signals u
1
 and u
2
 according to 
the control law u of the system as follows:

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
461 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
 
uu
uu
1
2
25
25


.,
., (10)
then substituting Eq. (10) into Eq. (1) and along with 
Eq. (4), we obtain:
 p
kA pv
V
kk RT
V
sign up H
sign up H
bV
eV
1
11
11
11
11
1
1
 
 
 
()
()


s
2 2
2
22
22
21
1
1









 

u
p
kA pv
V
kk RT
V
sign up H
sign
bV
,
()
(


s
u up H
u
eV
)
,










22 2
 (11)
By setting xx xx
12 1
= = ;  and x
3
 = p
1
A
1
 – p
2
A
2
,  
Eq. (6) then becomes:
 


xx
x
M
x
F
M
fr
12
23
1


,
, (12)
Taking time derivative of x
3
 along with Eq. (11) 
can be achieved as follows:
  xPtBtu
3
 (,)( ,), xx (13)
where x = [x
1 
x
2 
x
3
] is state-vector and
 Pt
kp x
V
kp x
V
(,) x  
AA
1
2
12
1
2
2
22
2
, (14)
Btkk RT
u
pH
V
pH
V
bV
eV
(,)
sgn( )
x 















1
2
11 1
1
22 22
2


s
A
A
1 1
2
11 21
1
21 2
2

























sgn( ) u
pH
V
pH
V
eV
bs V


A
A
 














. (15)
It can be realized from Eqs. (1) to (15) that the 
dynamics of the electro-pneumatic servo system are 
highly nonlinear due to the effects of compressibility 
of air in the cylinder chambers, the nonlinearity of the 
valves, and friction in the pneumatic cylinder.
2  CONTROL STRATEGY
2.1 Control Design
In this section, we design a controller so that the 
piston position x
1
 can track the desired trajectory x
1d
 
with high accuracy under the effect of the system 
nonlinearities (the compressibility of air in the cylinder 
chambers and the nonlinearity of the valves) and 
under the existence of friction. In [19], Tsai and Huang 
have applied the MSS controller to deal with the 
system nonlinearities and have obtained the smallest 
position error of 5 mm to sinusoidal desired position 
trajectories among the controllers proposed thus far 
for the pneumatic cylinders. However, the friction 
force was considered as a bounded uncertain force 
in the controller design. It is shown by Armstrong-
Helouvry [20] that a controller combining with a 
friction compensator has the potential to improve the 
control performances of a control system. Therefore, 
in this study, we propose a new position controller 
of the cylinder piston by combining the MSS control 
method with a friction observer. The friction observer 
is designed based on the dynamic LuGre friction 
model. The block diagram of the closed-loop system 
using the new controller is shown in Fig. 2. 
To design the new controller, the following 
assumptions of the system parameters are considered: 
i) the mass load is bounded by M
min
 ≤ M ≤ M
max
; 
ii) the piston velocity is bounded by |x
2
| ≤ α
1
; iii) the 
friction force is bounded by |F
fr
| ≤ F
st
; iv) the pressures 
Fig. 2.  Block diagram of the closed-loop system using new controller

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
462 Tran, X.B.
in the cylinder chambers are bounded by p
atm
 ≤ p
i
 ≤ p
s
, 
i = 1 and 2. Basing on these assumptions, the bounds 
for F
fr 
/M and P(x,t) can be identified as follows:
 
F
M
F
M
fr
st

min
, 
2
 (16)
 Pt
kp A
V
kp A
V
ss
(,). x 
1
2
1
10
2
2
1
20
3

 (17)
The bound for B(x,t) can be found as:
 0  BB tB
minm ax
(,), x (18)
where B
min
 and B
max 
are determined from Eqs. (2), (3) 
and (15) as:
Bp kk RT
HA
V
HA
V
Bp
k
VV
atm
max
min
.,
.











05 8
05 8
2
1
11
10
22
20
s
 



















k
k
V
V
kk RT
HA
VA L
HA
VA L
1
11
10 1
22
20 2
, (19)
Let choosing an estimation B

 of the gain B(x,t) 
as the mean of the above bounds:
 BB B

= () ,
minm ax
/ 12
 (20)
then bounds in Eq. (18) can be written in the form:
 


1
Bt
B
(,)
,
x

 (21)
where    BB
maxm in
.
12
Defining three sliding surfaces of position, 
velocity, and applied force as follows:
 
sxx
sxx
sxx
11 1
22 2
33 3



d
d
d
,
,
, (22)
with the desired velocity and applied force are selected 
as follows:
 xx
s
21 1
1
1
dd
  

, (23)
 xM xF
s
dd
fr
32
  



2
2
2
, (24)
where τ
i
 (i = 1 and 2) are positive constants and θ
i 
are 
the boundary layer thicknesses of the surfaces s
i
. F fr

 
is the friction force that is estimated by the observer 
basing on the LuGre model as follows:
 Fw wx fr
 
  
0121
,
.
 (25)
 
.
wx
w
qx
xs
z




  
1
0
1
11


()
, (26)
where τ
z
 is a positive constant. By introducing 


ww w 
 
as the estimating error of the mean 
deflection of the contacting bristle and by combining 
with Eqs. (7) and (26), the time derivative of the error
 w can be obtained as:
 



w
xw
qx
s
z



10
1
1

 . (27)
Taking time derivatives of x
2d
 and x
3d
 along with 
Eq. (12) obtains:
  xx xx
dd d 21
1
1
21
  


, (28)
 
.
    xM xMxx
M
M
x
F
M
dd dd
fr
31
1
1
1
2
2
2
1
1
2
2
3
1
 
















 





F fr

. (29)
Defining
    xM xMxx M
x
M
dd dd 31 1
1
1
1
2
2
2
1
1
2
2
3
 














, as 
known part and 

xM
F
M
F
d
fr
fr
32
1
1
2
2












.
, as 
unknown part of  x
d 3
, then we have:
   xxx
ddd 33 13 2
. (30)
In the ideal case where there is no friction force, 
the bounds of   xx x
dd d 23 13 2
,a nd can be identified as
   xx xx
dd dd 21
1
1
11 2
  
maxm ax max
,


 (31)
    xM xM xx
M
dd dd 31 1
1
1
1
2
2
2
1
1
 

maxm ax maxm ax max
max






 

2
2
12
31








pA pA
M
x
sa tm
d
min
max
,  (32)
  x
d 32
= 0. (33)
Taking the Eqs. (12) and (13) into consideration, 
the time derivatives of three sliding surfaces can be 
derived as
  ss
s
12 1
1
1
 

, (34)
 

s
M
s
M
FF
M
s
fr
fr
23 2
2
2
11 1
 

 

, (35)

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
463 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
  sPtBtu x
d 3
  (,)( ,) . xx
3
 (36)
Eqs. (27) and (31) to (36) will be used later in 
Section 2.2 to analyse the stable properties of the 
closed-loop system. The control law of the system is 
proposed as follows:
 u
B
xP t
s
 






1
3
3
3


3d1
(,). x 

 (37)
where θ
3
 is the boundary thickness of the sliding 
surface s
3
, and τ
3
 is a positive constant. It can be noted 
that adding the friction force estimate F fr

 in 
calculating the desired force x
3d
 in Eq. (24) affects the 
control law u of the system in Eq. (37) through the 
sliding surface s
3
.
2.2  Stability Analysis
This section presents the analysis of the stability 
properties of the closed-loop system basing on the 
Lyapunov direct method [30]. First, let us consider a 
Lyapunov function for the third sliding surface s
3
 as 
Gs
33
2
1
2
= . Taking the time derivative of G
3
 and 
combining it with the Eqs. (30), (36), and (37), we 
obtain:


 Gs
Bt
B
Pt
Bt
B
xx
dd 33 33
11 
















(,)
(,)
(,) x
x
x
12  




Bt
B
s
(,)
.
x



3
3
3
2
 (38)
By selecting


 
33 33
1
11      



 x
d1max
 with γ
3
 > 0  along with Eqs. (31) to (33), then we obtain:
    


Gs
x
s s
d
33
3
3
3
3
3
3
2
3
1
1
1 
 
 


















 


1max
, (39)
if we choose θ
3
 ≤ |s
3
| then 

G
s
33
3
2
3
0   

. This 
means that s
3
 converges asymptotically to its boundary 
layer, i.e., the force x
3
 converges to the desired force 
x
3d
. Similarly, let us consider a Lyapunov function for 
the second sliding surface s
2
 as Gs
22
2
1
2
= . Taking 
the time derivative of G
2
 and combining it with Eq. 
(35), we obtain:



Gs ss
M
s
M
FF
M
s
fr
fr
22 22 32
2
2
11 1
 










. (40)
By assuming that the estimated friction force is 
bounded by | F fr

| ≤ F
st
 and by selecting  
 
 
23 2
11
2  






M
MM
F
st max
minm in
 with γ
2
 > 0, then,

Gs
M
s
ss
F
s
st 22 33
2
2
2
2
2
2
2
1
12 






















min




2 2
. (41)
If we choose 

2
3
3
2

s
s then 

G
s
22
2
2
2
0   

. 
This proves that s
2
 converges to its boundary layer 
asymptotically, i.e., the cylinder velocity x
2
 converges 
to the desired velocity x
2d
. Finally, a Lyapunov 
function for the first sliding surface s
1
 is defined as 
Gs w
z
11
22
1
2
1
2


 . Taking the time derivative of G
1
 
and combining it with Eqs. (27) and (34), we obtain:
 



 Gs s
s xw
qx
ws
z
11 21
1
2
1
10
2
1
1
 

 



. (42)
By selecting τ
1
 = θ
2
 + γ
1
 with γ
1
 > 0 and noting that 
 ws ws
1
2
1
2
1
2
1
2
 , then we obtain:
 


Gss
s
s
qx
z
11 22
1
1
1
1
1
2
20
1
1
2
1
2



























 



 w
2
. (43)
It can be noted in Eq. (8) that the Stribeck 
function qx 
1

 is strictly positive and is bounded 
by 0
1
   Fq xF
cl st
 . So, if we select 

1
2
2
1

s
s , 


1
1
2

 and 

z
qx


2
20
1

 then 

G
1
0 < . Therefore, s
1
 
converges asymptotically to its boundary layer, i.e., 
the piston position x
1
 converges to the desired position 
x
1d
. This result, therefore, verifies the system’s 
stability.
3  SIMULATION EVALUATION
In this section, simulation studies are carried out 
to evaluate the effectiveness of the new controller. 
Simulations were done by MATLAB/Simulink 
software. Both the new controller and the MSS 
controller were simulated with the same desired 
position inputs. The step and sinusoidal desired 
position inputs with different amplitudes and 
frequencies were used. The parameters of the 
system used in simulations are listed in Table 1. The 
maximum velocity of the piston was set to be α
1
 = 1 
m/s according to the cylinder’s specification. The 
static parameters of the LuGre friction model were 
given by F
st
 = 30 N, F
cl
 = 5.6 N, v
s
 = 0.01 m/s, n = 2.5, 
σ
2
 = 25 Ns/m for the positive velocity direction (i.e., 
for extending stroke of the piston) and by F
st 
= 25 N, 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
464 Tran, X.B.
F
cl
 = 5.8 N, v
s
 = 0.015 m/s, n = 1.2, σ
2
 = 25 Ns/m for 
the negative velocity direction (i.e., for extending 
stroke of the piston). The dynamic parameters of 
the LuGre model were given by σ
0
 = 8×10
3
 N/m and 
σ
1
 = 0.1 Ns/m for both directions of the velocity. 
These parameters of the LuGre friction model were 
identified in advance for the pneumatic cylinder in a 
previous study [23]. The parameters of the controllers 
used in simulations were chosen as σ
1
 = 0.002, σ
2
 = 0.1, 
σ
3
 = 0.5, γ
1
 = 0.01, γ
2
 = γ
3
 = 1, α
2 
= 60 and the parameter 
of the friction force observer was chosen as τ
z 
= 5000.
Table 1.  System parameters used in simulation
Parameter [unit] Value Parameter [unit] Value
M [kg] 0.5 H
V1
 [m
2
/V] 5×10
–7
A
1
 [m
2
] 4.9×10
–4
H
V2
 [m
2
/V] 6×10
–7
A
2
 [m
2
] 4.12×10
–4
T [K] 295
L [m] 0.3 R [Nm/(kgK)] 287
V
10
 [m
3
] 9.8×10
–7
p
s
 [bar] 5
V
20
 [m
3
] 8.24×10
–7
p
atm
 [bar] 0.1
k 1.3997
Fig. 3 shows a comparison of the control 
performances obtained by simulations  between the 
new controller and the MSS controller with a step 
desired position input of the piston position x
1d
 = 0.25 
m. The piston is set at an initial position of 0 m. 
Fig. 3a shows that both the controllers exhibit good 
control performances with no overshoot. However, 
it can be observed that the rise time achieved by the 
new controller (0.26 s) is smaller than that of the MSS 
controller (0.5 s). In addition, the maximum tracking 
error in steady-state achieved by the new controller 
(0.5 mm) is also smaller than that of the MSS 
controller (1 mm) (Fig. 3b).
Fig. 4 shows simulation comparisons of the 
control performances between the new controller and 
the MSS controller for four different constant desired 
positions: x
1d 
= 0.01 m (Fig 4a), 0.05 m (Fig. 4b), 0.1 
m (Fig. 4c) and 0.2. m (Fig. 4c). The initial positions 
of the piston are 0 m. Fig. 4 shows that the controller 
can provide shorter rise times and smaller position 
errors for four cases of the desired position. The 
maximum tracking errors obtained under steady-state 
conditions are also 0.5 mm for the new controller and 
1 mm for the MSS controller.
Fig. 5 shows a comparison between the new 
controller and the MSS controller for a sinusoidal 
desired input x
1d
 = 0.15 + 0.1sin(2pft) m at the 
frequency f = 0.1 Hz. The figure shows that the new 
controller can track the desired position faster, i.e., a 
shorter rise time, than the MSS controller. In addition, 
the tracking error achieved in a steady-state condition 
of the new controller is much smaller than that by 
the MSS controller. The maximum tracking error 
in the steady-state condition of the new controller 
is 1.5 mm, whereas the maximum tracking error of 
the MSS controller is 4.8 mm. Other comparisons 
between two controllers with higher frequencies of 
the desired position input x
1d
 = 0.15 + 0.1sin (2pft) 
m are shown in Figs. 6 and 7. The position amplitude 
remains the same as in Fig. 5, but the frequency is 
increased to 0.5 Hz and 1 Hz. As compared with Fig. 
5, although the abilities to track the desired position 
of the two controllers are inferior when the frequency 
is increased, the new controller also provides smaller 
errors than the MSS controller. In Fig. 6 for f = 0.5 
Hz, the maximum error in a steady-state condition is 
2.8 mm for the new controller and is 6.5 mm for the 
MSS controller. In Fig. 7 for f = 1 Hz, the maximum 
error in a steady-state condition is 4.5 mm for the new 
controller and is 7.8 mm for the MSS controller. The 
results obtained in Figs. 3 to 7 verify by simulation 
that the new controller can provide better control 
performances than the MSS controller.
Fig. 3.  Comparing control results of the new controller and the 
MSS controller with a step desired position input x
1d
 = 0.25 m; 
a) piston position, b) control position error (simulation)

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
465 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
Fig. 6.  Control performances of the new controller and of the MSS 
controller with a sinusoidal desired position 
x
1d
 = 0.15+ 0.1sin (2pft) m, f = 0.5 Hz by simulation;  
a) tracking position; b) tracking error.
Fig. 5.  Control performances of the new controller and of the MSS 
controller with a sinusoidal desired position  
x
1d
 = 0.15+ 0.1sin (2pft) m, f = 0.1 Hz by simulation;  
a) tracking position; b) tracking error
Fig. 4.  Control results of the new controller and the MSS controller for different step desired positions;  
a) x
1d
 = 0.01 m, b) x
1d
 = 0.05 m; c) x
1d
 = 0.1 m; d) x
1d
 = 0.2 m (experiment)

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
466 Tran, X.B.
Fig. 7.  Control performances of the new controller and of the MSS 
controller with a sinusoidal desired position  
x
1d
 = 0.15+ 0.1sin (2pft) m, f = 1 Hz by simulation;  
a) tracking position; b) tracking error
4  EXPERIMENTAL EVALUATION
4.1 Experimental System
Fig. 8 shows the experimental system used for the 
experimental evaluation of the new controller. The 
system used a pneumatic double-acting cylinder 
(1). The piston rod was connected to a load mass 
(2) which slides on a guiding bar (3). The load mass 
was set at 0.5 kg. Two proportional pneumatic valves 
(4) were used to control flow rates supplying to the 
cylinder chambers. These valves were controlled 
by current signals ranging from 0 A to 1 A. The 
current signals were supplied by two amplifiers (5). 
The input signals of the valve amplifiers are voltage 
values ranging from 0 VDC to 5 VDC. For the flow 
characteristic of the valves, it is noted that if the 
valves’ inputs vary from 0.5 A to 1 A (corresponding 
to 2.5 VDC to 5 VDC of the valve amplifiers’ inputs), 
the valves will provide air from the supply source 
into the cylinder chambers and if the valves’ inputs 
vary from 0 A to 0.5 A (corresponding to 0 VDC to 
2.5 VDC of the valve amplifiers’ inputs), the valves 
will release air from the cylinder chambers to the 
atmosphere. A position sensor with a measuring 
range from 0 m to 0.3 m (6) was used to measure the 
piston displacement. Two pressures, p
1
 and p
2
, in the 
cylinder chambers were measured by two pressure 
sensors (7). A data acquisition card (8) that includes 
both 12 bits analogue-to-digital converter (ADC) and 
12 bits digital-to-analogue converter (DAC) was used 
to acquire the displacement and the pressure signals 
from the sensors and to send the voltage signals to 
the valve amplifiers. The data acquisition device 
was connected to a personal computer (9). The main 
technical specifications of the devices are listed in 
Table 2. Microsoft Visual C++ software was used to 
communicate between the data acquisition device 
and the computer. Each step of the experimental 
programme was executed in a period of 1.16 ms.
The piston velocity that is used in calculating the 
control law was not measured in this experimental 
system. It was calculated by an approximate 
Fig. 8.  Experimental system

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
467 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
differentiation of the measured piston displacement. 
The noise in the calculated velocity signal was filtered 
using an acausal low-pass filter with a bandwidth of 
30 Hz. The compressed air was supplied to the system 
by a compressor through an air preparation unit, and 
the supply pressure p
s
 was set at 5 bar.
4.2  Experimental Results
In this section, experimental performances of the 
new controller and the MSS controller are compared 
to evaluate the effectiveness of the new controller. 
Similarly to simulation, constant and sinusoidal 
desired position inputs were used. In addition, the 
control performances of the new controller are also 
compared to those of the other nonlinear controllers. 
The values of the controller’s parameters and the 
parameters of the LuGre friction model were given as 
the same as those used in the simulation.
Fig. 9 compares the experimental performances 
between the new controller and the MSS controller 
with a desired step position input of x
1d
 = 0.25 m. 
Similarly to the results obtained in the simulation 
results in Fig. 3, both controllers exhibit accurate 
position tracking with the maximum tracking errors in 
a steady-state condition below 1 mm, but the rise time 
of the piston obtained by the new controller (0.2 s) is 
less than the rise time of the MSS controller (0.3 s). 
It is additionally noted in Fig. 9 that at the beginning, 
the piston remains at its original position for 0.1 s. 
This result may be caused by the compressibility of 
the air in the cylinder chamber and/or by the delay 
characteristic of the pneumatic valves.
Fig. 9.  Comparing control results of the new controller and the 
MSS controller with a desired constant position input  
x
1d
 = 0.25 m; a) piston position,  
b) control position error (experiment)
Table 2.  Main equipment used in the experimental system
Equipment Model Main technical specifications Manufacturer
Compressor PK 1090A
Maximum pressure: 10 bar 
Maximum flow rate: 185 l/min
PUMA
Air preparation unit GC 600-20
Maximum pressure: 10 bar 
Maximum flow rate: 1000 l/min
Filter element: 40 m
AIRTAC
Pneumatic cylinder CM2L25-300
Piston diameter: 0.025 m
Rod diameter: 0.01 m
Piston rod: 0.3 m
SMC
Proportional pneumatic valves VEF3121-1-02
Maximum pressure: 10 bar 
Maximum flow rate: 720 l/min 
Current input: 0 A to 1 A
SMC
Valve amplifier VEA 250
Voltage input: 0 VDC to 5 VDC
Curent output: 0 A to 1 A
Accuracy: ± 1 % F .S
SMC
Position sensor LWH0300
Measurement range: 0 m to 0.3 m
Accuracy: 0.4 % F .S
NOVOTECHNIK
Pressure sensor PSE 540
Measurement range: 0 bar to 10 bar
Accuracy: 2 % F .S
SMC
Data acquisition card (ADC/DAC) USB4711
Analog input channels: 16
Analog output channels: 2
ADVANTECH
Personal computer Dell Vostro 260
CPU: Intel Core I3-2120 3.3 GHz
Memory: 8GB RAM
DELL

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
468 Tran, X.B.
Fig. 10 compares the experimental performances 
between the new controller and the MSS controller at 
four step-desired positions: 0.01 m (Fig. 10a), 0.05 
m (Fig. 10b), 0.1 m (Fig. 10c), and 0.2 m (Fig. 10d). 
Similarly to the simulation results in Fig. 4, Fig. 10 
shows that the new controller provides better control 
performance than the MSS controller with shorter rise 
times and smaller steady-state errors in all cases. The 
maximum position errors in steady-state conditions 
are 0.5 mm for the new controller and 1 mm for the 
MSS controller. It can be additionally noted in Fig. 10 
that the period at which the piston remains at its initial 
position is increased by reducing the desired position. 
This result can be explained by that when the desired 
position is reduced, the initial flow rate supplied to the 
cylinder is small and thus increases the delay time of 
the piston. 
Fig. 11 shows an experimental comparison 
between the new controller and the MSS controller 
with a sinusoidal desired position x
1d
 = 0.15+ 0.1sin 
(2pft) m at a frequency f = 0.1 Hz. Fig. 11 shows 
that the peak position error obtained in a steady-state 
condition is 8.4 mm (corresponding to 4.2 % of the 
maximum cross displacement) for the MSS controller 
and is 4.8 mm (2.4 %) for the new controller. When 
calculating the root means squared error (RMS error) 
in a steady-state condition, the results show that the 
Fig. 10.  Control results of the new controller and the MSS controller for different desired step position inputs;  
a) x
1d
 = 0.01 m, b) x
1d
 = 0.05 m; c) x
1d
 = 0.1 m; d) x
1d
 = 0.2 m (experiment)
Fig. 11.  Control performances of the MSS controller and the new 
controller with a sinusoidal input x
1d
 = 0.15+ 0.1sin (2pft) m, 
f = 0.1 Hz by experiments; a) control position; b) tracking error

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
469 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
RMS error obtained for the MSS controller is 3.7 
mm (1.85 % of the maximum cross displacement) 
while the RMS error obtained for the new controller 
is 2.1 mm (1.05 %). The calculated velocity and the 
estimated friction force by the new controller are 
shown in Fig. 12. When the piston starts moving, the 
friction force is suddenly increased to a very larger 
value of about 30 N in the first haft cycle of the piston 
motion. After that, the friction force reduces and varies 
under a trapezoidal form between the value from -13 
N to 20 N in the steady-state condition. The maximum 
friction force at the positive velocity direction is 
larger than at the negative velocity direction. In Fig. 
11, the large values of the tracking error obtained by 
the MSS controller appear mainly at high velocities, 
and they can be eliminated by the new controller. 
At high velocities, the friction force becomes large 
due to the effect of the viscous friction force. Thus, 
the compensation of friction to the MSS controller 
significantly reduces the tracking error at these 
velocities. The tracking errors of the two controllers 
achieved by experiments are larger than those 
achieved by simulation in Fig. 3. However, like the 
simulation results, the experimental result also shows 
the improvement in the control performances of the 
new controller comparing with the MSS controller.
Fig. 12.  Control performances of the MSS controller and the new 
controller with a sinusoidal input x
1d
 = 0.15+ 0.1sin (2pft) m,  
f = 0.1 Hz by experiments; a) control position; b) tracking error
Fig. 13 compares the percentage rates of the 
peak position error and of the RMS error in steady-
state condition between two controllers at different 
displacement amplitude values (A) of the desired 
position input x
1d 
= 0.15+Asin(2pft) m. The varying 
frequency is fixed at f = 0.1 Hz. The results show 
that both the peak error and the RMS error of both 
controllers increase with decreasing the displacement 
amplitude A. However, these errors are significantly 
reduced when using the new controller. When the 
amplitude A varies from 0.01 m to 0.1 m, the peak 
error can be reduced from 1.8 % to 10 %, and the 
RMS error can be reduced from 0.8 % to 6.5 %. 
The minimum peak error of 2.2 mm can be obtained 
at the amplitude A = 0.01 m. This result of the new 
controller indicates better control performances than 
other nonlinear and robust control methods [9] to [21].
Fig. 13.  Comparison of the percentage rate of peak error and 
RMS error between the new controller and the MSS controller at 
different displacement amplitudes A and fixed frequency 0.1 Hz 
of the desired position input x
1d
 = 0.15+Asin(2πft) m
Fig. 14.  Comparison of the percentage rate of peak error and 
RMS error between the new controller and the MSS controller at 
different frequencies f and fixed displacement amplitude A = 0.1 
m of the desired position input x
1d
 = 0.15+ 0.1sin (2pft) m

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
470 Tran, X.B.
Fig. 14 compares the percentage rates of the 
peak position error and of the RMS error in steady-
state condition between two controllers at different 
frequencies (f) of the desired position input x
1d 
= 
0.15+Asin(2pft) m. The displacement amplitude is 
fixed at A = 0.1 m. The results show that both the peak 
error and the RMS error of both controllers grow with 
increasing the frequency. However, the peak error and 
the RMS error are also significantly reduced when 
using the new controller. When the frequency f varies 
from 0.1 Hz to 1 Hz, the peak error can be reduced 
from 1 % to 3.5 % and the RMS error can be reduced 
from 0.8 % to 2 %. Figs. 15 and 16 show the control 
performances of two controllers at two frequencies 
0.5 Hz and 1 Hz.
5  CONCLUSIONS
In this study, a new controller was proposed for 
improving position control performances of a 
pneumatic cylinder. It was designed by combining the 
multiple-surface sliding control method with a model-
based friction compensator, which was designed 
based on the dynamic LuGre friction model. Both 
simulation and experimental studies were performed 
to evaluate the new controller. The study results show 
that by using the new controller, the relative position 
peak error is reduced 10 %, and the relative RMS error 
is reduced by 6.5 % when comparing those of the 
MSS controller without friction compensation. The 
new controller also provides improved control error 
compared to other nonlinear and advanced controllers 
with a minimal peak error of 2.2 mm. It is noted that 
the parameters of the LuGre model used in the friction 
observer in this study were identified in advance. 
Therefore, a methodology for online parameter 
estimation of the LuGre model when implementing 
the new controller will be considered in the future.
6  ACKNOWLEDGEMENTS
This research was an extension of the work in [31]. 
This research was funded by the Hanoi University 
of Science and Technology (HUST) under Project 
number T2020-PC-014. 
7  NOMENCLATURES
a  piston acceleration, [m/s
2
]
A
i  
piston area (i = 1, 2), [m
2
]
B
max 
maximum value of the function B(x,t)
B
min
  minimum value of the function B(x,t)
B

  
estimated value of the function B(x,t)
F
cl  
coulomb friction force, [N]
Fig. 15.  Control performances of the MSS controller and the new 
controller with a sinusoidal input x
1d
 = 0.15+ 0.1sin (2pft) m, 
f = 0.5 Hz by experiments; a) control position; b) tracking error
Fig. 16.  Control performances of the MSS controller and the new 
controller with a sinusoidal input x
1d
 = 0.15+ 0.1sin (2pft) m,  
f = 1 Hz by experiments; a) control position; b) tracking error

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
471 Nonlinear Control of a Pneumatic Actuator Based on a Dynamic Friction Model 
F
fr
  friction force, [N]
F fr

  estimated friction force, [N]
F
st 
: maximum static friction force, [N]
G
i 
: Lyapunov function (i = 1, 2, 3)
H
Vi 
: valve coefficient (i = 1, 2), [m
2
/V]
k : specific heat ratio
L : piston stroke, [m]
M : load mass, [kg]
M
max 
maximum load mass, [kg]
M
min 
minimum load mass, [kg]
 m
i
  mass flow rate (i = 1, 2), [kg/s]
n  exponent for Stribeck curve
p
atm
  atmosphere pressure, [bar]
p
i
  pressure (i = 1, 2), [bar]
p
s
  supply pressure, [bar]
P(x, t) function of the states’
B(x, t) function of the states’
q(v)  Stribeck function, [N]
R  gas constant, [Nm/(kgK)]
s
i 
 sliding surface (i = 1, 2, 3)
T  Temperature, [K]
u  control law, [V]
u
i 
 valve control input (i = 1, 2), [V]
v  piston velocity, [m/s]
V
i 
 volume of the cylinder chamber (i = 1, 2),  
 [m
3
]
V
i0
  dead volume in the cylinder chamber 
 (i = 1, 2), [m
3
]
v
s 
: Stribeck velocity, [m/s]
x : piston position, [m]
x : state vector
x
1 
: piston position, [m]
x
1d 
: desired position of the piston, [m]
x
2 
: piston velocity, [m/s]
x
2d 
: desired velocity of the piston, [m/s]
 x
2d max
 maximum value of  x
2d
, [m/s
2
]
x
3 
: piston force generated by the pressures, [N]
x
3d 
: desired force, [N] 
 x
3d1
  known part of  x
3d
 x
3d1 max
 maximum value of  x
3d1
 x
3d2
  unknown part of  x
3d
w  mean deflection of bristles, [m]


ww w   estimated mean deflection of bristles, [m]
 w  estimating error of the mean deflection of 
 bristles, [m]
α
1
  maximum cylinder velocity, [m/s]
α
2
  maximum value of F
fr 
/M, [N/kg]
α
3
  maximum value of P(x, t)
β
ib
  pressure coefficient (i = 1, 2)
β
ie
  pressure coefficient (i = 1, 2)
γ
i 
: positive constant (i = 1, 2, 3)
λ : constant
σ
0 
: stiffness of bristles, [N/m]
σ
1
  micro-viscous friction coefficient for bristles,  
 [Ns/m]
σ
2
  viscous friction coefficient, [Ns/m]
τ
i
  positive constant (i = 1, 2, 3)
τ
z
  positive constant
θ
i
  boundary layer thickness of the surfaces s
i 
 
(i = 1, 2, 3)
8  REFERENCES
[1] Beater, P. (2007). Pneumatic Drives: System Design, Modelling 
and Control. Springer, Berlin, DOI:10.1007/978-3-540-69471-
7.
[2] Karponis, D., Koya, Y., Miyazaki, R., Kanno, T., Kawashima, K. 
(2019). Evaluation of a pneumatic surgical robot with dynamic 
force feedback. Journal of Robotic Surgery, vol. 13, p. 413-
421, DOI:10.1007/s11701-018-0878-2.
[3] Tressler, J.M., Clement, T., Kazerooni, H., Lim, M. (2002). 
Dynamic behavior of pneumatic systems for lower extremity 
extenders. Proceedings of IEEE International Conference 
on Robotics and Automation, p. 3248-3253, DOI:10.1109/
ROBOT.2002.1013727.
[4] Rao, Z., Bone, G.M. (2008). Nonlinear modeling and control 
of servo pneumatic actuators. IEEE Transaction on Control 
Systems Technology, vol. 16, no. 3, p. 562-569, DOI:10.1109/
TCST.2007.912127.
[5] Harris, P.G., O’Donnell, G.E., Whelan, T. (2012). Modelling 
and identification of industrial pneumatic drive system. The 
International Journal of Advanced Manufacturing Technology, 
vol. 58, p. 1075-1086, DOI:10.1007/s00170-011-3447-7.
[6] Wang, J., Pu, J., Moore, P. (1999). A practical control strategy 
for servo-pneumatic actuator systems. Control Engineering 
Practice, vol. 7, no. 12, p. 1483-1488, DOI:10.1016/S0967-
0661(99)00115-X.
[7] Chillari, S., Guccione, S., Muscato, G. (2001). An experimental 
comparison between several pneumatic position control 
methods. Proceedings of the IEEE Conference on Decision and 
Control, vol. 2, p. 1168-1173, DOI:10.1109/CDC.2001.981043.
[8] Merola, A., Colacino, D., Cosentino, C., Amato, F. (2018). 
Model-based tracking control design, implementation of 
embedded digital controller and testing of a biomechatronic 
device for robotic rehabilitation. Mechatronics, vol. 52, p. 70-
77, DOI:10.1016/j.mechatronics.2018.04.006.
[9] Bobrow, J.E.; Jabbari, F. (1991). Adaptive pneumatic force 
actuation and position control. ASME Journal of Dynamic 
Systems, Measurement, and Control, vol. 113, no. 2, p. 267-
272, DOI:10.1115/1.2896374.
[10] Gao, X., Feng, Z.J. (2005). Design study of an adaptive 
fuzzy-PD controller for pneumatic servo system. Control 
Engineering Practice, vol. 13, no. 1, p. 55-65, DOI:10.1016/j.
conengprac.2004.01.001.
[11] Chen, Y., Tao, G., Liu, H. (2019). High precision adaptive robust 
neural network control of a servo pneumatic system. Appled 
Sciences., vol. 9, no. 17, p. 1-17, DOI:10.3390/app9173472.
[12] Gross, D.C., Rattan, K.S. (1997). Pneumatic cylinder trajectory 
tracking control using a feedforward multilayer neural network, 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 458-472
472 Tran, X.B.
Proceedings of IEEE Aerospace and Electronics Conference, 
vol. 2, p. 777-784, DOI:10.1109/NAECON.1997.622728.
[13] Paul, A.K., Mishra, J.K.; Radke, M.G. (1994). Reduce order 
sliding mode control for pneumatic actuator. IEEE Transaction 
on Control Systems Technology, vol. 2, no. 3, p. 271-276. 
DOI:10.1109/87.317984.
[14] Tang, J.J., Walker, G.G. (1995). Variable structure control of 
a pneumatic actuator. ASME Journal of Dynamic Systems, 
Measurement, and Control, vol. 117, no. 1, p. 88-92, 
DOI:10.1115/1.2798526.
[15] Surgenor, B.W., Vaughan, N.D. (1997). Continuous sliding 
mode control of a pneumatic actuator. ASME Journal of 
Dynamic Systems, Measurement, and Control, vol. 119, no. 3, 
p. 578-581, DOI:10.1115/1.2801298.
[16] Acarman, T., Hatipoglu, C., Ozguner, U. (2001). A robust 
nonlinear controller design for a pneumatic actuator. 
Proceedings of the IEEE American Control Conference, vol. 6, 
p. 4490-4495, DOI:10.1109/ACC.2001.945686.
[17] Liu, Y.T, Kung, T.T, Chang, K.M., Chen, S.Y. (2013). Observer-
based adaptive sliding mode control for pneumatic servo 
system. Precision Engineering, vol. 37, no. 3, p. 522-530, 
DOI:10.1016/j.precisioneng.2012.12.003.
[18] Pandian, S.R., Hayakawa, Y., Kanazawa, Y., Kamoyama, Y., 
Kawamura, S. (1997). Practical design of a sliding mode 
controller for pneumatic actuators. ASME Journal of Dynamic 
Systems,Measurement, and Control, vol. 119, no. 4, p. 666-
674, DOI:10.1115/1.2802376.
[19] Tsai, Y.C., Huang, A.C. (2008). Multiple-surface sliding 
controller design for pneumatic servo systems. 
Mechatronics, vol. 18, no. 9, p. 506-512, DOI:10.1016/j.
mechatronics.2008.03.006.
[20] Armstrong-Helouvry, B. (1991). Control of Machines with 
Friction, Springer, Boston, DOI:10.1007/978-1-4615-3972-8.
[21] Koch, S., Reichhartinger, M. (2016). Observer-based sliding 
mode control of hydraulic cylinders in the presence of unknown 
load forces, e & i Elektrotechnik und Informationstechnik, vol. 
133, p. 253-260, DOI:10.1007/s00502-016-0418-6.
[22] Ayalew, B., Jablokow, K.W. (2006). Cascaded robust 
nonlinear position tracking control of an electrohydraulic 
actuator: sliding mode and feed forward. Proceedings of the 
ASME International Mechanical Engineering Congress and 
Exposition. Fluid Power Systems and Technology, p. 95-102, 
DOI:10.1115/IMECE2006-13943.
[23] Tran, X.B., Yanada, H. (2013). Dynamic friction behaviors of 
pneumatic cylinders. Intelligent Control and Automation, vol. 
4, no. 2, p. 180-190, DOI:10.4236/ica.2013.42022.
[24] Armstrong-Helouvry, B., Dupont, P., Canudas de Wit, C. (1994). 
A survey of models analysis tools and compensation methods, 
for the control of machines with friction. Automatica, vol. 30, 
no. 7, p. 1083-1138, DOI:10.1016/0005-1098(94)90209-7.
[25] Canudas de Wit, C., Olsson, H., Aström, K.J., Lischinsky, P. 
(1995). A new model for control of systems with friction. IEEE 
Transaction on Automatic Control, vol. 40, no. 3, p. 419-425, 
DOI:10.1109/9.376053.
[26] Lampaert, V., Swevers, J., Al-Bender, F. (2002). Modification 
of the Leuven integrated friction model structure. IEEE 
Transaction on Automatic Control, vol. 47, no. 4, p. 683-687, 
DOI:10.1109/9.995050.
[27] Dupont, P., Hayward, V., Armstrong, B., Altpeter, F. (2002). 
Single state elastoplastic friction models. IEEE Transactions 
on Automatic Control, vol. 47, no. 5, p. 787-792, DOI:10.1109/
TAC.2002.1000274.
[28] Al-Bender, F., Lampaert, V., Swevers, J. (2005). The generalized 
Maxwell-slip model: a novel model for friction simulation and 
compensation. IEEE Transaction on Automatic Control, vol. 50, 
no. 11, p. 1883-1887, DOI:10.1109/TAC.2005.858676.
[29] Tran, X.B., Hafizah, N., Yanada, H. (2012). Modeling of dynamic 
friction behaviors of hydraulic cylinders. Mechatronics, vol. 22, 
no. 1, p. 65-75, DOI:10.1016/j.mechatronics.2011.11.009.
[30] Slotine, J.J.E., Li, W. (1991). Applied Nonlinear Control, 
Prentice Hall, London.
[31] Tran, X.B., Nguyen, V.L., Nguyen, N.C., Pham, D.T., Phan, V.L. 
(2020). Sliding mode control for a pneumatic servo system 
with friction compensation. In: Sattler, K.U., Nguyen, D., Vu, 
N., Tien Long, B., Puta, H. (eds). Advances in Engineering 
Research and Application. ICERA 2019. Lecture Notes 
in Networks and Systems, vol. 104. Springer, Cham, 
DOI:10.1007/978-3-030-37497-6_75.