*Corr. Author’s Address: School of Mechanical and Electrical Engineering, Henan University of Technology, Zhengzhou, 450001, China, wanzhenshuai@haut.edu.cn 771
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780 Received for review: 2022-07-14
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-10-27
DOI:10.5545/sv-jme.2022.284 Original Scientific Paper Accepted for publication: 2022-12-16
Adaptive Super-twisting Sliding Mode Control  
of Hydraulic Servo Actuator with Nonlinear Features  
and Modeling Uncertainties
Wan, Z – Fu, Y . – Yue, L. – Liu, C.
Zhenshuai Wan
1,2*
 – Yu Fu
1,2
 – Longwang Yue
1
 – Chong Liu
1
1 
Henan University of Technology, School of Mechanical and Electrical Engineering, China 
2 
Henan University of Technology, Key Laboratory of Grain Information Processing  
and Control of Ministry of Education, China
This study proposes a novel adaptive super-twisting sliding mode controller (ASTSMC) for hydraulic servo actuator with nonlinear features and 
modeling uncertainties. In the proposed method, an extended state observer (ESO) is utilized to estimate the value of lumped uncertainties. 
The core feature of this paper is the combination of ASTSMC with ESO to compensate disturbance in hydraulic servo actuator. Moreover, the 
proposed ASTSMC does not need to obtain the bound of uncertainties in advance and ensures that the sliding variable and its derivative 
reach to zero in a finite time. In addition, the stability of the closed-loop is proved by Lyapunov theory. Simulation and experiment results 
demonstrate that the proposed ASTSMC can effectively mitigate the lumped uncertainties and obviously improve the tracking performance.
Keywords: hydraulic servo actuator, nonlinear features, modeling uncertainties, super-twisting sliding mode control
Highlights
• 	 The dynamic mathematical model of hydraulic servo actuator is established considering nonlinear features and modeling 
uncertainties.
• 	 The ESO is used to estimate the unmeasured system state and lumped uncertainties.
• 	 The ASTSMC is adopted to compensate the disturbance and further improve the tracking precision.
• 	 The simulation and experiment results validate the effectiveness of the ASTSMC based on ESO.
0  INTRODUCTION
Hydraulic servo actuator is widely employed in 
modern industrial, such as heavy vehicle [1] and 
[2], load simulator [3], hot-pressing equipment [4], 
hydraulic manipulator [5], due to the virtues of small 
size-to-power ratio, high control precision, fast 
response performance and strong bearing capacity [6] 
to [8]. However, the nonlinear features and modeling 
uncertainties of hydraulic servo actuator complicates 
the dynamic model and hinders position tracking 
performance. The nonlinear features are mainly 
caused by pressure-flow characteristic of servo valve 
and nonlinear friction of hydraulic actuator [9] to [12]. 
While the modeling uncertainties are mostly caused 
by time-varying hydraulic parameters, unmodeled 
friction and external disturbance [13] to [15]. Hence, 
the traditional linear control schemes have become 
more and more difficult to satisfy the high precision 
position tracking control requirement of modern 
hydraulic servo actuator. Importantly, it is essential to 
study high performance control strategy for hydraulic 
servo actuator.
In recent years, numerous control schemes have 
been proposed, such as adaptive control [16], robust 
control [17], backstepping control [18], sliding mode 
control (SMC) [19], and intelligent control [20] to 
improve the control performance of hydraulic servo 
system. As an effective control method, SMC can cope 
with uncertainties and achieve asymptotic tracking 
performance [21] to [23]. However, the inevitable 
chattering of SMC caused by discontinuous control 
input is not acceptable for practical systems. To solve 
this problem, the continuous switching function, such 
as continuous saturation function and hyperbolic 
function, is used to replace the discontinuous symbol 
function in conventional SMC. Although the tracking 
error of improved control scheme is bounded, it loses 
the asymptotic tracking performance. The high order 
sliding mode controller can ensure the continuity of 
SMC and obtain the asymptotic tracking performance. 
However, it needs the derivative information of the 
sliding mode variable, which is often unattainable in 
practice, so it is difficult to be realized in engineering 
practice. In hydraulic servo system, only parts of 
states can be measured, and load or disturbance 
cannot be measured directly. Hence, an ESO is used 
to estimate the immeasurable system state variables 
and lumped uncertainties in this paper [24]. The 
proposed ASTSMC can effectively avoid the above 
drawbacks, but the controller gain related to the 
upper bound of modeling uncertainty needs to be 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
772 Wan, Z – Fu, Y. – Yue, L. – Liu, C.
set artificially, which is conservative to some extent. 
The feedforward adaptive control law based on 
model is introduced into the ASTSMC to improve the 
control precision of actuator. Moreover, the proposed 
method does not need to know the exact bound of 
modeling uncertainty, rather designs an adaptive law 
to constantly adjust the controller gain associated 
with the bound. In particular, ASTSMC can make 
the tracking error converge asymptotically to a small 
adjustable range near zero in finite time.
The rest of this paper is organized as follows. 
Section 1 gives system description and dynamical 
model. The ASTSMC design process and theoretical 
result are presented in section 2. Simulation results 
and discussion are depicted in section 3. Section 4 
carries experimental setup and comparative results. 
Finally, the conclusions are summarized in section 5.
1  SYSTEM DESCRIPTION AND DYNAMICAL MODEL
The schematic diagram of the hydraulic servo actuator 
is given in Fig. 1, which mainly includes pump, motor, 
servo valve and hydraulic actuator. 
Fig. 1.  Schematic diagram of the hydraulic servo actuator
The system mainly consists of three subsystems 
which are the servo system, the hydraulic part and 
the mechanical section. The hydraulic part supplies 
energy for overall equipment. The servo system 
provides the hydraulic pressure to the mechanical 
section by controlling the piston’s movement. As an 
executive part, the performance of actuator directly 
influences the precision of hydraulic driven system. 
However, many hydraulic servo actuator models have 
not adequately account for the impact of nonlinear 
features and modeling uncertainties. To improve the 
position control performance of the hydraulic servo 
actuator, the dynamic model considering various 
disturbance and uncertainties is used to replace the 
traditional linear model.
According to the Newton’s second law, the 
dynamic equation of load can be described as
 my=PAB yf t
L
  −− () , (1)
where m is the equivalent load mass, y is the load 
displacement, P
L
 is the load pressure, A is of effective 
piston area of hydraulic cylinder, B is viscous friction 
coefficient, f(t) is the nonlinear features and modeling 
uncertainties.
Ignoring external leakage of hydraulic cylinder, 
the dynamic equation of load pressure can be written 
as
 
V
t
L
4
e
Lt L
P=QA yC Pq t

   () , (2)
where V
t
 is total volume of hydraulic actuator, β
e 
is 
the effective bulk modulus of the hydraulic fluid, Q
L
 
is load flow, C
t
 is the internal leakage coefficient of 
hydraulic cylinder, q(t) is the disturbance.
The load flow of servo valve can be constructed 
as
 Qk uP uP
Lt sL
 sign() , (3)
where k
t
 is the flow gain coefficient, u is the input 
control voltage of the servo valve, P
s
 is the supply 
pressure of the pump, sign(u) is defined as
 sign
if
if
()
,
,
. u
u
u






10
10
 (4)
For the ease of calculation, let the system states 
be defined as
 xx xx yyy
TT
= = [, ,] [,,] .
123
  (5)
Combing Eqs. (1) to (5), the state space model of 
the hydraulic servo actuator is given by
 



xx
xx
xf uPuf xf xd t
L
12
23
31 22 33


 





(, )( )( )( )
, (6)
where

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
773 Adaptive Super-twisting Sliding Mode Control of Hydraulic Servo Actuator with Nonlinear Features and Modeling Uncertainties  
 
ft
Ak
mV
Pu P
ft
mV
AB Cx
ft
et
t
sL
e
t
t
e
1
2
2
2
3
4
4
4
() ()
()
()







sign
m mV
Cm
BV
x
dt
A
mV
qt
C
mV
ft
ft
m
t
t
t
e
e
t
et
t







 
4
44
3


() () ()
()

 












. 
The greatest difficulties in this model is the 
high nonlinearity with respect the control signal and 
unmodeled disturbance for hydraulic servo actuator. 
To cope with this problem, the ASTSMC is utilized 
to drive the load track the desired trajectory as closely 
as possible.
2  CONTROLLER DESIGN
2.1 Extended State Observer
Set the lumped uncertainties of d(t) as an extended 
state x
4
, then Eq. (6) can be rewritten as
 





xx
xx
xf uPuf xf xx
xd t
L
12
23
31 22 33 4
4


 








(, )( )( )
()
. (7)
Note that q(t) and f(t) are bounded, thus d(t) is 
bounded by a known positive constant δ.
Design an ESO of the Eq. (7) as
xx xx
xx xx
xf uP ufx
L
 
 
 
12
1
1
1
23
2
1
1
3
12 2
 

 

   


,     

 










fx xx x
xx x
 

3 3
4
3
1
1
4
4
1
1


, (8)
where xi

 is the estimation states of the ESO, f
i

 is the 
estimations of f
i
, β
i
 is the observe gains to be 
determined.
Define 
ii
i xx 

, the dynamic of error is
 




  
  
AL xxBd F
AL CB dF
w
w
()
,
1
1
 (9)
where 

f
i
 is the estimation error of f
i
, and 
 
AL C 




























0100
0010
0001
0000
1
2
3
4
,,




1 1
0
0
0
0
0
0
1
0
0
0
12 3





























T
w
B
F
fu ff
,,
 
 











.
To ensure the pole of matrix (A – LC) on the 
left half plane, the characteristic polynomial can be 
written as
 
00
4
() () () ss IA LC sw   , (10)
where w
0
 is the bandwidth of the observe.
The parameters of ESO are selected as
  
10 20
2
30
3
40
4
46 44   ww ww ,, ,. (11)
Assumption 1: The unmodeled disturbances are 
bounded and satisfy
 dt Md tM Ft M () ,( ), () , ≤≤ ≤
12 3

 (12)
where M
1
, M
2
, M
3
 are unknown positive constants.
Theorem 1: The estimated error of the ESO can 
be expressed as
 Lim
t
t

 () . 0 (13)
Proof: Define A
1
=A – LC, then
 
 

() exp( )()e xp((( ))
exp((( )) ()
tA tA tF d
At Bdt
t
w
 

 11
0
1
0
0
t t
d

. (14)
Using the property of matrix norms, Eq. (14) can 
be represented as
 



() exp( )( )
exp((( )) ()
exp((( ))
tA t
At Bd td
At F
w
t




1
1
0
1
0
0
t t
d

. (15)
Based on Eq. (10), the eigenvalues of A
1
 are 
λ
1
 = λ
2
 = λ
3
 = – w
0
, then there exists κ > 1 such that for 
all t ≥  0
 
exp( )e xp( )
exp((( )) exp( () )
.
At w
At wt
10
10

 






 
 (16)
Then the Eq. (15) can be represented as

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
774 Wan, Z – Fu, Y. – Yue, L. – Liu, C.
 
 
() exp( )( )
exp( )e xp( ).
tw t
M
w
wt
M
w
wt

     
0
1
0
0
3
0
0
0
11 (17)
Therefore, if the w
0
 goes to infinity, the ε(t) would 
be convergent to zero.
2.2  Adaptive Super-twisting Sliding Mode Controller
The tracking errors are defined as
 
zxx
zxx
zxx
d
d
d
11 1
22 1
33 1










, (18)
where z
1
, z
2
, z
3 
are tracking error of position, velocity 
and acceleration, respectively.
The sliding mode surface is defined as
 sk zk zz 
11 22 3
, (19)
where k
1
 and k
2
 are positive constants.
The derivative of s can be written as
 

sk zk zz
kz kz fu Pu fx fx x
Ld

 
12 23 3
12 23 12 23 31
(, )( )( ). (20)
The control input of super-twisting sliding mode 
controller is designed as
 u
fu P
fx fx x
kz kz s
L
d
u

 

1
1
22 33 1
11 23
12
1
(, )
() ()
/

  
 s sign
sign
()
()
,
s
s
u






















2
0
2
   
 (21)
where u
1
 and u
2
 are feedforward control law and 
robust control law, α and θ are time varying controller 
gains.
The adaptive laws are defined as
 
 
 








1
2
2
/
,
sign s
 (22)
where γ
1
, ω, θ, ν are positive constants.
According to Eqs. (21) and (22), the dynamics of 
s are represented as
 


ss sd t
s
 









12
2
/
() ()
()
.
sign
sign
 (23)
Define state vector    [, ][ || () ,]
/
 
12
12 TT
ss sign 
   [, ][ || () ,]
/
 
12
12 TT
ss sign , then the unmodeled uncertainties 
are expressed as
 dt xt ss xt () (,)( )( ,) ,
/
  
12
1
sign (24)
where ρ(x,t) is a positive function.
Then, the new state equation is defined as
 

    A (25)
where A








1
2
1
0
1



(,) xt
.
A Lyapunov function is represented as
 VV   
0
1
0
2
2
0
2
1
2
1
2 


 () () , (26)
where α
0
 and β
0
 are positive constants. V
0
 is defined 
as follows
 V
T
0
   P , (27)
where P 










42
21
2
, λ is a positive constant.
The derivative of V
0
 is written as
 

V
TTTT
T
0
1
1
2


      
   
PP AP PA ()
||
,

 (28)
where
 
  
  
 
 







 
 
11 12
21 22
2
24 4
2
,( )( ),
()
11
12
=
=
 
   
   

 
4
24 4
2
2
,
() ,.   
21 22
==
When α satisfies
 
 

 
 






()
()
()
()
.
4
1
24
12 1
2
1
22
1
 (29)
We have 
 V
T
0
11
 



 || ||
.     (30)
According to the following inequalities
 



minm ax
/
/
min
/
() ()
()
,
P¾ ¾P¾P ¾
¾
P
22
1
12 2
0
12
12







T
s
V
 (31)
where λ
max
(P) and λ
min
(P) are maximum and minimum 
eigenvalue of matrix P, respectively.
Thus, Eq. (31) can be rewritten as
 

VV
00
12
  
/
, (32)

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
775 Adaptive Super-twisting Sliding Mode Control of Hydraulic Servo Actuator with Nonlinear Features and Modeling Uncertainties  
where    
min
/
max
() /( )
12
PP .
Then, the time derivative can be given as
 




VV
V
 
 
0
1
0
2
0
0
12
1
0
2
0
11
11









() ()
() (
/
) )
,







  
1
1
0
2
2
0
22
 (33)
where     min, ,
12
.
 



VV  
 











0
12
1
0
2
0
1
1
0
2
2
0
11
22
/
() ()
. (34)
Based on Eq. (22), there exists positive constants 
α
0
 and β
0
, which satisfy α  – α
0
 < 0 and β – β
0
 < 0. Then 
Eq. (33) can be represented as
 




 
 
 
  





11 22
11 22
22
22
/, /
/, /
,
sv
sv
 (35)
Assuming ν = 0, we know that s →0  in  finite  time, 
and
 t
Vt
f

2
12
0
/
()
.

 (36)
3  SIMULATION RESULTS
To show the trajectory tracking performance of the 
presented ASTSMC, PID and SMC schemes are 
first utilized for simulation comparison. It should be 
noted that all controller parameters are set through 
a preliminary tuning process. The parameters of 
hydraulic servo actuator are listed in table 1.
1) PID: The gains of PID controller are tunned as  
K
p
 = 120, K
i
 = 10, K
d
 = 0.1 to balance the steady-
state error and transient response performance.
2) SMC: Based on SMC scheme, the control law is 
designed as
 u
fu P
fx fx x
kz kz ks
L
d


 






1
1
22 33 1
11 23 3
(, )
() ()
()
,

sign
 (37)
 where k
1
 = 4 × 10
3
, k
2
 = 2 × 10
3
, k
3
 = 2 × 10
2
.
3)  ASTSMC: The parameters of proposed ASTSMC 
are given as k
1
 = 2 × 10
3
, k
2
 = 6 × 10
2
, k
3
 = 2 × 10
2
 
γ
1
 = 2, ω = 6, θ = 5,  ν = 2. Note that the sign 
function in Eq. (21) is replaced with saturation 
function for meeting the requirement of the 
control input continuous.
Table 1.  The parameters of hydraulic servo actuator
Parameter Value
m 300 kg
V
t
9×10
-5 
m
3
ρ 900 kg/m
3
B 1200 N·s/m
C
t
4×10
-3
β
e
6.9×10
8 
Pa
K
t
7.2×10
-7 
m/V
A 3.14×10
-4 
m
2
To validate the advantages of the proposed 
ASTSMC, a sinusoidal signal x
d
 = 60 sin(20πt) mm 
is used as reference trajectory. The position tracking 
performance and tracking error are given in Figs. 
2 and 3, from which we can see that the proposed 
ASTSMC is better than the SMC and PID. This is 
because the proposed ASTSMC can estimate the 
unknown dynamics by ESO and compensate that 
by adaptive control law. Compared with the PID 
controller, the tracking errors of SMC and ASTSMC 
are substantially reduced by 18.3 % and 48.8 %, 
respectively. Fig. 4 shows the control input of the 
three controllers. It is noted that owing to the adaptive 
mechanism in ASTSMC, its control input is smaller 
than PID and SMC. The observation performance of 
ESO to external interfererence is shown in Fig. 5. It 
is clearly indicated that the ESO can estimate the state 
variavbe and lumped uncertainties accurately.
In order to further authenticate the rationality 
of ASTMC, multi-frequency sinusoidal signal 
x
d
 = 50 sin(10πt) + 40 sin(25πt) + 20 sin(50πt) [mm] 
is selected as reference signal. Also, the simulation 
results are shown in Figs. 6 and 7. It is noted that 
three controllers can track the reference trajectory 
accurately. However, the proposed ASTSMC obtains 
the smallest tracking error than SMC and PID, which 
verifies the superiority of the proposed controller. The 
maximum tracking errors of PID, SMC and ASTSM 
are 20.471 mm, 14.237 mm, 8.690 mm, respectively.
In order to quantitatively compare the tracking 
performance of different controllers, maximum 
absolute value of tracking error M
e
, average tracking 
error μ
e
, and standard deviation of tracking error σ
e
 are 
adopted as performance indices. Table 2 summarizes 
the performance indices of different controllers for 
sinusoidal and multi-frequency sinusoidal motion 
reference signal. It can be found that the proposed 
ASTSMC produces the smallest values among three 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
776 Wan, Z – Fu, Y. – Yue, L. – Liu, C.
controllers. In addition, the performance indices of 
SMC are better than PID. The simulation results 
clearly demonstrate that the proposed ASTSM can 
provide a better control performance for the hydraulic 
servo actuator with unknown dynamics than the others 
under different reference trajectories.
Fig. 2.  Position tracking of sinusoidal motion
Fig. 3.  Tracking error of sinusoidal motion
Fig. 4.  Control law of sinusoidal motion
Fig. 5.  The observation performance of ESO
Fig. 6.  Position tracking of multi-frequency sinusoidal motion
Fig. 7.  Tracking error of multi-frequency sinusoidal motion

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
777 Adaptive Super-twisting Sliding Mode Control of Hydraulic Servo Actuator with Nonlinear Features and Modeling Uncertainties  
Table 2.  Comparison results of performance indices
Signal Controller M
e
μ
e
σ
e
Sinusoidal
PID 6.147 1.682 1.225
SMC 5.023 1.459 1.111
ASTSMC 3.145 0.881 0.595
Multi-frequency 
sinusoidal
PID 20.471 3.909 3.825
SMC 14.237 2.267 2.463
ASTSMC 8.690 1.556 1.435
4  EXPERIMENTAL RESULTS
In this section, a hydraulic servo actuator 
experimental setup is used to demonstrate the 
effectiveness of the proposed control scheme. The 
diagram of experimental setup is shown in Fig 8. The 
host computer offers human-computer interaction 
interface for compiling programs and adjusting 
controller parameters. The Target computer reads the 
feedback signal real time and feeds it back to the host 
computer by TCP/IP protocol. The digital control 
signal is converted to analogue signal by D/A card 
and processed by signal conditioner, and then sent to 
the servo valve to drive the hydraulic servo actuator. 
The position and pressure information are collected by 
position sensor and pressure sensor, respectively. The 
A/D card obtains the sensors information and sends 
them to target computer to form closed-loop control 
system by signal conditioner.
Fig. 8.  The diagram of experimental setup
The load is first commanded to track a low-
speed motion x
d
 = 40sin(10 πt) [mm]. The obtained 
position tracking and tracking error are displayed 
in Figs. 9 and 10. It can be seen that the proposed 
controller ASTSMC delivers smaller tracking error 
than PID and SMC, because they use ESO to estimate 
the lumped uncertainties. In addition, the tracking 
error of all controllers occurs chattering when the 
trajectory is reversed due to the unmodeled dynamic 
characteristic and measurement noise. However, the 
chattering value of the proposed ASTSMC is smaller 
to other controllers, which means that the control 
scheme based ESO and adaptive law is very helpful 
to alleviate the effects from lumped uncertainties in 
hydraulic system.
Fig. 9.  Position tracking of x
d 
= 40sin(10 πt) [mm]
Fig. 10.  Tracking error of x
d 
= 40sin(10 πt) [mm]
To further verify the superiority of the ASTMC, 
a high-speed motion x
d 
= 40sin(15 πt) is performed 
as reference signal. The experimental comparison 
results of three controllers are exhibited in Figs. 11 
and 12. As presented, the proposed ASTSMC attains 
better tracking precision in comparison to the other 
controllers. This is because that the control gains in 
ASTSMC can dynamically be adjusted as unknown 
uncertainties changes.
In this case, a large amplitude motion signal x
d 
= 80sin(15 πt) mm is chose as the reference trajectory 
with the amplitude of 80 mm and the frequency 7.5 
Hz. The corresponding position tracking performance 
and tracking error are presented in Figs. 13 and 14. 
It is noted that the three controllers are all able 
to suppress the nonlinear features and modeling 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
778 Wan, Z – Fu, Y. – Yue, L. – Liu, C.
controllers. The tracking error of ASTSMC is always 
within in 8 mm, showing a good tracking precision.
uncertainties for such a large amplitude tracking test. 
However, the proposed ASTSMC shows excellent 
tracking performance than the other two compared 
Fig. 11.  Position tracking of x
d 
= 40sin(15 πt) [mm]
Fig. 12.  Tracking error of x
d 
= 40sin(15 πt) [mm]
Fig. 13.  Position tracking of x
d 
= 80sin(15 πt) [mm]
Fig. 14.  Tracking error of x
d 
= 80sin(15 πt) [mm]
Fig. 15.  Comparison performance indices of controllers
Fig. 16. Position tracking of x
d 
= 50sin(10 πt)+20sin(40 πt) [mm]

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)12, 771-780
779 Adaptive Super-twisting Sliding Mode Control of Hydraulic Servo Actuator with Nonlinear Features and Modeling Uncertainties  
The performance indices of three controllers for 
different sinusoidal motion are shown in Fig. 15. One 
can find that the values of performance indices with 
the ASTSMC are the smallest among all controllers. 
The maximum errors of above experimental situation 
are 1.6412 mm and 2.1231 mm for amplitude 40 mm, 
respectively. In particular, the maximum of relative 
average error of ASTSMC is within 1.35 %, which 
demonstrates the effectiveness of the proposed control 
scheme.
Fig. 17.  Tracking error of x
d 
= 50sin(10 πt)+20sin(40 πt) [mm]
Fig. 18.  Control law of x
d 
= 50sin(10 πt)+20sin(40 πt) [mm]
Multi-frequency sinusoidal experimental results 
x
d 
= 50sin(10 πt)+20sin(40 πt) are shown in  Figs. 16 
and 17. It can be seen that the presented controller 
ASTSMC can track the reference signal accurately, 
and the tracking error is smaller than SMC and PID. 
Especially, PID controller gives the worst control 
performance. It is evident that the tracking error of 
ASTSMC is within 5 mm, and that of PID, SMC is 
within 22 mm and 15 mmm, respectively, which 
proves the high-accuracy tracking performance of the 
designed control scheme again. The control inputs of 
different controllers are shown in Fig 18. As shown, 
although the input of all controller is bounded, the 
input of ASTSMC is smaller than PID and SMC.
5  CONCLUSIONS
In this paper, a ASTSMC scheme based on ESO 
has been proposed for hydraulic servo actuator 
with nonlinear features and modeling uncertainties. 
First, the dynamical mathematical model containing 
various nonlinear and uncertainties is established. The 
parameter adaptive law is used to update the controller 
gain in real time to avoid the conservativeness caused 
by artificial setting. The obtained control input is 
continuous, which avoids the chatter problem of 
traditional sliding mode controller. The stability 
analysis demonstrates that the tracking error of the 
system converges asymptotically to an arbitrarily 
small range near zero in finite time, and the 
convergence rate and the bounds of steady-state error 
can be adjusted by parameters. Extensive comparative 
simulation and experimental results show that the 
proposed ASTSMC can make the position trajectory 
track the reference command well and satisfy the high 
precision control of the servo system.
6  ACKNOWLEDGEMENTS
This work is supported by the Key Science 
and Technology Program of Henan Province 
(222102220104) and the High-Level Talent 
Foundation of Henan University of Technology 
(2020BS043).
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