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Research Article
Terminal sliding mode tracking control for a class of SISO uncertain
nonlinear systems
Mou Chen a,n, Qing-Xian Wu a, Rong-Xin Cui b
a College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Chinab College of Marine Engineering, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e i n f o
Article history:
Received 11 July 2011Received in revised form
14 September 2012
Accepted 28 September 2012
This paper was recommended for publica-
tion by Dr. Jeff Pieper
Keywords:
Uncertain nonlinear system
Input saturation
Disturbance observer
Terminal sliding mode control
Tracking control
a b s t r a c t
In this paper, the terminal sliding mode tracking control is proposed for the uncertain single-input and
single-output (SISO) nonlinear system with unknown external disturbance. For the unmeasureddisturbance of nonlinear systems, terminal sliding mode disturbance observer is presented.
The developed disturbance observer can guarantee the disturbance approximation error to converge
to zero in the finite time. Based on the output of designed disturbance observer, the terminal sliding
mode tracking control is presented for uncertain SISO nonlinear systems. Subsequently, terminal
sliding mode tracking control is developed using disturbance observer technique for the uncertain SISO
nonlinear system with control singularity and unknown non-symmetric input saturation. The effects of
the control singularity and unknown input saturation are combined with the external disturbance
which is approximated using the disturbance observer. Under the proposed terminal sliding mode
tracking control techniques, the finite time convergence of all closed-loop signals are guaranteed via
Lyapunov analysis. Numerical simulation results are given to illustrate the effectiveness of the
proposed terminal sliding mode tracking control.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
During the past several decades, a lots of robust control
schemes have been developed for the uncertain nonlinear system
[1–4]. Sliding mode control (SMC) has received much attention as
an efficient control technique for handling systems with large
uncertainties, nonlinearities, and bounded external disturbances
[5–10]. The SMC was proposed for a class of nonlinear stochastic
systems with sector nonlinearities and deadzones in [9]. In [10],
the sliding-mode framework was given for multi-input affine
nonlinear systems. The systematic output-sliding control design
methodology was investigated for multivariable nonlinear sys-
tems [11]. The dynamic SMC was proposed for multi-input and
multi-output (MIMO) systems with additive uncertainties in [12].In [13], model-reference tracking SMC was presented for uncer-
tain plants with arbitrary relative degree. Due to the robust
enhanced ability, SMC has been widely used in many practical
control systems to improve the control system performance [14].
In [15], robust SMC was proposed for the inverted-pendulum
system. The second order SMC algorithm was presented for a twin
tank system in [16]. However, the basic SMC cannot guarantee
closed-loop system states converge to the equilibrium in the
finite time.
In order to render the system state reach the equilibrium in
the finite time, terminal sliding mode control (TSMC) has been
proposed and used in various control problems. In [17], TSMC of
MIMO linear systems was studied. Robust TSMC technique was
developed for rigid robotic manipulators in which MIMO term-
inal switching plane variable vector was first defined in [18].
In [19], the fast terminal dynamics was proposed and used in
the design of the sliding-mode control for SISO nonlinear
dynamical systems. The adaptive fuzzy TSMC was proposed for
linear systems with mismatched time-varying uncertainties in
[20]. In [21], TSM tracking control was developed for a class of
nonminimum phase systems with uncertain dynamics. In [19],the fast terminal dynamics was proposed and used in the design
of the sliding-mode control for SISO nonlinear dynamical
systems. Output regulation of nonlinear systems with delay
was studied using TSMC in [22]. In [23], the design of variable
structure control with terminal sliding modes was developed for
multi-input uncertain linear systems. TSMC was proposed for
uncertain dynamic systems with pure-feedback form in [24].
In the most of existing research results, the upper boundary
of external disturbances is directly used to design TSMC.
To explore the information about the characteristic of external
disturbances, the disturbance observer can be used to estimate
the system external disturbance.
Contents lists available at SciVerse ScienceDirect
journal homepage: www .elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.09.009
n Corresponding author.
E-mail address: chenmou@nuaa.edu.cn (M. Chen).
Please cite this article as: Chen M, et al. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISATransactions (2012), http://dx.doi.org/10.1016/j.isatra.2012.09.009
ISA Transactions ] (]]]]) ] ]]–]]]
8/9/2019 Terminal Sliding Mode Tracking Control for a Class of SISO Uncertain Nonlinear Systems
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In recent years, various design technologies of disturbance
observers have been intensively studied [25–28]. In these litera-
tures, the disturbance observers were used to approximate
external disturbance, and robust control were designed based
on the output of disturbance observers. In [25,26], fuzzy distur-
bance observer was proposed and its application was studied
for discrete-time and continuous-time systems. The general
framework was given for nonlinear systems subject to distur-
bances using disturbance observer based control (DOBC) techni-ques [27]. The nonlinear PID predictive control was proposed
based on disturbance observers in [28]. In [29], the nonlinear
disturbance observer was proposed for multivariable minimum-
phase systems with arbitrary relative degrees. The nonlinear
disturbance observer for robotic manipulators was developed in
[30]. In [31], composite disturbance-observer-based control and
TSMC were investigated for uncertain structural systems. In [32],
the nonlinear disturbance observer-based approach was proposed
for longitudinal dynamics of a missile. The disturbance attenua-
tion and rejection problem was investigated for a class of
MIMO nonlinear systems using DOBC framework in [33]. In
[34], composite DOBC and H 1 control were proposed for complex
continuous models. Composite DOBC and TSMC were studied for
nonlinear systems with disturbances in [35]. In [36], sliding mode
synchronization control was proposed for uncertain chaotic
systems using disturbance observer. Robust DOBC was presented
for time delay uncertain systems in [37]. In [38], sliding mode
control was developed for a class of uncertain nonlinear systems
based on disturbance observer. To guarantee the approximation
error of disturbance observer converge to zero in the finite time,
the terminal sliding mode disturbance observer is developed in
this paper.
On the other hand, the various control input constraints will
degrade the control system performance which attract the atten-
tion of many researchers. Adaptive fuzzy output feedback control
was proposed for the uncertain nonlinear systems with unknown
backlash-like hysteresis in [39]. In [40], adaptive fuzzy output
feedback tracking backstepping control was presented for strict-
feedback nonlinear systems with unknown dead zones. The
control input saturation is the most important non-smooth
nonlinearity which should be explicitly considered in the control
design. In [41–45], the analysis and design of control systems
with input saturation constraints have been studied. Robust
adaptive neural network control was proposed for a class of
uncertain MIMO nonlinear systems with input nonlinearities
[41]. In [42], the neural network tracking control was developed
for ocean surface vessels with input saturation. Hover control was
studied for an unmanned aerial vehicle (UAV) with backstepping
design including input saturations in [43]. In [44], robust stability
analysis and fuzzy-scheduling control were developed for non-
linear systems subject to actuator saturation. Globally stable
adaptive control was presented for minimum phase SISO plants
with input saturation [45]. The nonlinear control was proposed toachieve the attitude maneuver of a three-axis stabilized flexible
spacecraft with control input nonlinearity [46]. Furthermore, the
control system design becomes more complicated if the nonlinear
system exists the control singularity. Thus, the robust tracking
control needs to be further developed for the uncertain nonlinear
system with control singularity case and unknown non-
symmetric input saturation. In this paper, the terminal sliding
mode tracking control is developed using disturbance observer
technique for the uncertain nonlinear system with the control
singularity and unknown non-symmetric input saturation.
This work is motivated by the terminal sliding mode tracking
control of nonlinear systems with external disturbance, control
singularity and unknown input saturation. The organization of
the paper is as follows. Section 2 details the problem formulation.
The terminal sliding mode disturbance observer is proposed in
Section 3. Section 4 presents the terminal sliding mode tracking
control based on the disturbance observer. Terminal sliding mode
tracking control is designed for uncertain nonlinear systems with
control singularity and input saturation in Section 5. Simulation
studies are shown in Section 6 to demonstrate the effectiveness of
our proposed approaches, followed by some concluding remarks
in Section 7.
2. Problem formulation
Consider the uncertain SISO nonlinear system described in the
form of
_ x i ¼ xiþ 1, i¼ 1,2, . . . ,n1
_ xn ¼ f ð xÞþ g ð xÞuþd
y ¼ x1 ð1Þwhere x¼ ½ x1, x2, . . . , xnT is the system measurable state, f ð xÞAR
and g ð xÞAR are known nonlinear functions, respectively, uAR is
the system control input, yAR is the system output and dAR is
the external disturbance, respectively.
To proceed the design of terminal sliding mode control for theuncertain SISO nonlinear system (1), the following assumption
and lemma are required:
Lemma 1 (Man and Yu [17]). Assume that there exists the contin-
uous positive definite function V (t ) which satisfies the following
inequality:
_V ðt ÞþaV ðt ÞþlV gðt Þr0, 8t 4t 0 ð2Þthen V (t ) converges to the equilibrium point in finite time t s
t srt 0 þ 1
að1þgÞ ln aV 1gðt 0Þþl
l ð3Þ
where a40, l40 and 0ogo1.
In this paper, the control objective is to design thedisturbance-observer-based terminal sliding mode tracking con-
trol and make the system output follow a given desired output of
the nonlinear system in the presence of input saturation, control
singularity and external disturbance. For the desired system
output yd, the proposed terminal sliding mode control must
ensure that all closed-loop signals are convergent in the finite
time.
Assumption 1 (Tee and Ge [47]). For all t 40, there exists D i40
such that J yðiÞd ðt ÞJrDi, i¼ 1,2, . . . ,n.
3. Design of disturbance observer with finite time
convergence ability
In this section, the design process of the terminal sliding
mode observer will be given. To design a sliding mode distur-
bance observer with finite time convergence property, the follow-
ing auxiliary variable is introduced:
s¼ z xn ð4Þwhere z is given by
_ z ¼ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞþ g ð xÞu ð5Þwhere q0 and p0 are odd positive integers with p0oq0 . Design
parameters k, b and e are positive and b49d9.
The terminal sliding mode disturbance estimate d̂ is given by
^
d ¼ksb signðsÞes p0=q0
9 f ð xÞ9 signðsÞ f ð xÞ ð6Þ
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]]2
Please cite this article as: Chen M, et al. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISATransactions (2012), http://dx.doi.org/10.1016/j.isatra.2012.09.009
8/9/2019 Terminal Sliding Mode Tracking Control for a Class of SISO Uncertain Nonlinear Systems
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Differentiating (4), and considering (1) and (5) yield
_s ¼ _ z _ xn
¼ ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞ f ð xÞd ð7Þ
Theorem 1. Considering the uncertain SISO nonlinear system (1),
the terminal sliding mode disturbance observer is designed according
to (4)–(6). Then, the disturbance approximation error of the proposed
terminal sliding mode disturbance observer is convergent in the finite
time.
Proof. Consider the Lyapunov function candidate
V o ¼ 12s
2 ð8Þ
The time derivative of V o is given by
_V o ¼ s_s ¼ sðksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞ f ð xÞdÞrks
2bs signðsÞesð p0 þq0Þ=q0 þ9s9 9d9
rks2esð p0 þq0Þ=q0
r2kV o2ð p0 þq0Þ=2q0eV ð p0 þq0Þ=2q0o ð9Þ
Invoking (1) and (4)–(6), we obtain~d ¼ d̂d ¼ ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞ f ð xÞd
¼ ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞ f ð xÞ _ xnþ f ð xÞþ g ð xÞu¼ ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞþ g ð xÞu _ xn
¼ _ z _ xn ¼ _s ð10Þ
According to Lemma 1 and (9), we can know that the auxiliary
variable s converges to the equilibrium point in the finite time.
Due to the finite time convergence property of the auxiliary
variable s, we can conclude that the disturbance approximation
error ~d of the proposed disturbance observer is convergent in the
finite time. This concludes the proof. &
Remark 1. As the same as the design of existing sliding mode
disturbance observers [48,49], the proposed terminal sliding
mode disturbance observer still requires that the external dis-
turbance d has a known upper boundary, i.e., b49d9. Comparing
with the existing results, the proposed terminal sliding mode
disturbance observer can guarantee the disturbance estimate
error to converge to zero in the finite time.
4. Terminal sliding mode tracking control design
In this section, we develop the tracking control scheme for the
case where all states are available using terminal sliding mode
control approach. To develop the terminal sliding mode tracking
control, we define
s1 ¼ y yd ð11ÞThe nth time derivative of s1 is given by
sðnÞ1 ¼ yðnÞ yðnÞ
d ¼ _ xn yðnÞ
d ð12Þ
Considering (12), the recursive procedure for TSMC of uncer-
tain nonlinear systems (1) is written as [19]
s2 ¼ _s1 þa1s1 þb1s p1=q1
1
s3 ¼ _s2 þa2s2 þb2s p2=q2
2
^
sn ¼ _sn1 þan1sn1 þbn1s pn1=qn1
n1 þs ð13Þwhere ai40 and bi40. pi and qi, i¼ 1,2, . . . ,nþ1 are positive odd
integers with p ioqi.
For each s i, we have
_si ¼ €si1 þ d
dt ½ai1si1 þbi1s
pi1=qi1
i1
¼ €si1 þai1 _si1 þbi1
d
dt s pi1=qi1
i1 ð14Þ
Thus, the jth-order derivative of s i is
sð jÞi ¼
sð jþ 1Þi1 þ
dð jÞ
dt ð jÞ ½ai
1si
1
þbi
1s
pi1=qi1
i1 ð15
ÞIn accordance with (12)–(15), the derivative of sn is
_sn ¼ sðnÞ1 þ
Xn1
j ¼ 1
a jsðn jÞ j
þXn1
j ¼ 1
b jd
ðn jÞ
dt ðn jÞ s
p j=q j j
þ _s ð16Þ
Considering (1) and (12), we have
_sn ¼ _ xn yðnÞd þ
Xn1
j ¼ 1
a jsðn jÞ j
þXn1
j ¼ 1
b jd
ðn jÞ
dt ðn jÞ s
p j=q j j
þ _s
¼ f ð xÞþ g ð xÞuþd yðnÞd þ
Xn1
j ¼ 1
a jsðn jÞ j
þXn1
j ¼ 1
b jd
ðn jÞ
dt ðn jÞ s
p j=q j j
þ _s ð17Þ
Without loss of generality, we assume that g ð xÞa0 [50] in
this section. The singular case will be considered in the nextsection. Then, the disturbance-observer-based terminal sliding
mode tracking control is designed as
u¼ u0
g ð xÞ
u0 ¼ f ð xÞ yðnÞd þ
Xn1
j ¼ 1
a jsðn jÞ j
þXn1
j ¼ 1
b jd
ðn jÞ
dt ðn jÞ s
p j=q j j
þ d̂þdsnþms pn=qnn
ð18Þwhere d40 and m40.
The above design procedure of the terminal sliding mode
control can be summarized in the following theorem, which
contains the results for disturbance-observer-based terminal
sliding mode tracking control of uncertain nonlinear systems
with external disturbance.
Theorem 2. Considering the uncertain nonlinear system (1) with the
external disturbance and assuming full state information available, the
terminal sliding mode disturbance observer is designed as (4)–(6).
Under the proposed terminal sliding mode tracking control (18), all
signals of the closed-loop system are convergent in the finite time.
Proof. Substituting (18) into (17) yields
_sn ¼dsnms pn=qnn þdd̂þ _s
¼ dsnms pn=qnn ~dþ _s ð19Þ
Invoking (10), we obtain
_sn ¼dsnms p
n
=qnn ð20Þ
Consider the Lyapunov function candidate
V ¼ 12s
2n ð21Þ
Invoking (9) and (20), the time derivative of V is
_V rds2nmsð pn þqnÞ=qn
n
r2dV m2ð pn þqnÞ=2qnV ð pn þqnÞ=2qn ð22Þ
According to Lemma 1 and (22), we can know that all closed-
loop signals converge to the equilibrium point in the finite time.
This concludes the proof. &
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]] 3
Please cite this article as: Chen M, et al. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISATransactions (2012), http://dx.doi.org/10.1016/j.isatra.2012.09.009
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Remark 2. Although the known upper boundary of the external
disturbance is required in the design of disturbance observer, the
boundary of the external disturbance does not directly utilize to
design tracking control scheme. Furthermore, the output of
terminal sliding mode disturbance observer is used to design
terminal sliding mode tracking control (18). Thus, the proposed
disturbance-observer-based terminal sliding mode tracking
control can explore the information about the characteristic of
disturbances. It is worth to point out that sn in (13) has anadditional term s. The reason is that s should be considered in the
sliding mode surface sn to analyze the stability of the whole
closed-loop system.
5. Terminal sliding mode control design for uncertain
nonlinear systems with control singularity and input
saturation
In Section 4, we assume g ð xÞa0 for the uncertain nonlinear
system (1). However, there exists the feasibility of g ð xÞ ¼ 0 at a
moment in the practical system which leads to the control
singularity. On the other hand, input saturation as the most
important non-smooth nonlinearity should be explicitly consid-
ered in the control design. Thus, we propose the terminal slidingmode tracking control for the uncertain nonlinear system (1)
with control singularity case and unknown non-symmetric input
saturation in this section. Considering the unknown input satura-
tion constraints, the control input u is given by
u¼umax if v4umax
v if uminrvrumax
umin if voumin
8><>: ð23Þ
where v is the designed control input command. umin and umax are
the unknown parameters of control input saturation. Here,
uminaumax denotes the non-symmetric input saturation.
For further facilitate analyze the integrated effect of the
control singularity and unknown non-symmetric input satura-
tion, the robust tracking control command is given byv¼ g ð xÞð g 2ð xÞþtÞ1vr ð24Þwhere t40 is a design parameter and vr will be given later.
It is clear that
g 2ð xÞð g 2ð xÞþtÞ1 ¼ 1tð g 2ð xÞþtÞ1 ð25ÞSubstituting (24) into (1), we obtain
_ xn ¼ f ð xÞþ g ð xÞðvþDuÞþd
¼ f ð xÞþvr þdþ g ð xÞDutð g 2ð xÞþtÞ1vr ð26Þwhere Du¼ uv.
Since the upper and the lower limits of the non-symmetric
input saturation are unknown. Hence, Du is unknown. To handle
the unknown Du, we define the compound disturbance as
D¼ dþ g ð xÞDutð g 2ð xÞþeÞ1vr ð27ÞConsidering (27), we obtain
_ xn ¼ f ð xÞþvr þD ð28ÞSimilarly, the recursive procedure for TSMC of uncertain non-
linear systems (1) with the control singularity case and unknown
non-symmetric input saturation is designed as (11)–(17). Invok-
ing (17) and (28), we have
_sn ¼ _ xn yðnÞd þ
Xn1
j ¼ 1
a jsðn jÞ j
þXn1
j ¼ 1
b jd
ðn jÞ
dt ðn jÞ s
p j=q j j
þ _s
¼ f ð xÞþvr þD yðnÞd þXn1
j ¼ 1
a jsðn jÞ j
þXn1
j ¼ 1
b jd
ðn jÞ
dt ðn
j
Þ
s p j=q j j
þ _s ð29Þ
Due to the unknown compound disturbance D, the terminal
sliding mode disturbance observer needs to be developed to
estimate it. Thus, the same auxiliary variable is introduced:
s¼ z xn ð30Þwhere z is given by
_ z ¼ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞþvr ð31ÞBased on (30) and (31), the terminal sliding mode disturbance
estimate D̂ is given by
D̂ ¼ksb signðsÞes p0=q0 9 f ð xÞ9 signðsÞ f ð xÞ ð32Þwhere D̂ is the estimate of compound disturbance D and bZ9D9.
Based on the output of the terminal sliding mode disturbance,
the terminal sliding mode tracking control is designed as
vr ¼ f ð xÞþ yðnÞd Xn1
j ¼ 1
a jsðn jÞ j
Xn1
j ¼ 1
b jd
ðn jÞ
dt ðn jÞ s
p j=q j j
D̂dsnms pn=qnn
ð33Þwhere a i40 , b i40, d40 and m40.
The above design procedure and analysis can be summarized
in the following theorem, which contains the results for the
uncertain nonlinear system (1) with the control singularity caseand unknown non-symmetric input saturation.
Theorem 3. Consider the uncertain SISO nonlinear system (1) with
the external disturbance and suppose full state information is
available. The terminal sliding mode disturbance observer is designed
as (30)–(32). The terminal sliding mode tracking control is decided
by (24) and (33). Under the proposed terminal sliding mode tracking
control, all signals of the closed-loop system are convergent in the
finite time.
Proof. Substituting (33) into (29) yields
_sn ¼dsnms pn=qnn þDD̂þ _s
¼ dsnms pn=qnn ~Dþ _s ð34Þ
Similar with ~d, considering (30)–(32), we have
~D ¼ _s ð35Þ
Thus, (34) can be rewritten as
_sn ¼dsnms pn=qnn ð36Þ
Consider the Lyapunov function candidate
V ¼ 12s
2n ð37Þ
Considering (9) and (36), the time derivative of V is given by
_V rds2nmsð pn þqnÞ=qn
n
r2dV m2ð pn þqnÞ=2qnV ð pn þqnÞ=2qn ð38Þ
According to Lemma 1 and (38), we know that all closed-loop
signals converge to the equilibrium point in the finite time. This
concludes the proof. &
Remark 3. From (27), the integrated effects of control singularity
and unknown non-symmetric input saturation are treated as a
part of the external disturbance which is approximated using the
terminal sliding mode disturbance observer (30)–(32). Although
the uncertain nonlinear system (1) has the feasibility of control
singularity and unknown non-symmetric input saturation,
Lyapunov analysis shows that the trajectory of the closed-loop
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]]4
Please cite this article as: Chen M, et al. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISATransactions (2012), http://dx.doi.org/10.1016/j.isatra.2012.09.009
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system is asymptotically convergent in the finite time under the
proposed disturbance-observer-based terminal sliding mode
tracking control. In general, the design parameter b should be
chosen as a large positive constant to guarantee the design
requirement of the proposed sliding mode disturbance observer.
6. Simulation study
In this section, simulation results are given to illustrate theeffectiveness of the proposed control techniques using two
different examples.
6.1. Simulation example for Van der Pol circuits system
Let us consider the following Van der Pol circuits system [50]
_ x1 ¼ x2
_ x2 ¼2 x1 þ3ð1 x21Þ x2 þuþd
y¼ x1 ð39Þwhere the external disturbance d ¼ 2 sinð0:1pt Þþ3 sinð0:2
ffiffiffiffiffiffiffiffiffiffi t þ1
p Þ.
Following, the extensive simulation to demonstrate the effec-
tiveness of the proposed terminal sliding mode tracking control.To proceed the design of terminal sliding mode disturbance
observer and terminal sliding mode tracking control, the design
parameters are chosen as k¼2500, b¼ 4, e¼ 0:5, p0 ¼ 5, q0 ¼ 9,
a1 ¼ 50, b1 ¼ 0:5, p1 ¼ p2 ¼ 5, q1 ¼q2 ¼ 7, d¼ 60 and m¼ 0:8.
The initial state conditions are arbitrarily chosen as x0 ¼ 0:1,
x2 ¼ 0:2, d̂ ¼ 0:2 and D̂ ¼ 0. The desired trajectory is taken as
yd ¼ 2 sinð0:2t Þ.
6.1.1. Without considering input saturation for Van der Pol circuits
system
In this subsection, we do not consider the control input
saturation. The terminal sliding mode disturbance observer and
the terminal sliding mode tracking control are given by
s¼ z x2
_ z ¼ksb signðsÞes p0=q0 92 x1 þ3ð1 x21Þ x29 signðsÞþu
s1 ¼ x1 yds2 ¼ _s1 þa1s1 þb1s
p1=q1
1 þs
d̂ ¼ksb signðsÞes p0=q0 92 x1 þ3ð1 x21Þ x29 signðsÞ
þ2 x13ð1 x21Þ x2
u¼ 2 x13ð1 x21Þ x2 þ € yda1 _s1b1 _s
p1=q1
1 d̂ds2ms p2=q2
2 ð40Þ
Under the proposed disturbance-observer-based terminal slid-
ing mode tracking control (40), from Figs. 1 and 2, we can observe
that the tracking performance is satisfactory and the tracking
error quickly converge to zero in a short finite time for the
uncertain nonlinear system (39) in the presence of the time-varying external disturbance. To show the good control perfor-
mance of the proposed control terminal sliding mode control, we
compare the control performance under the developed control
approach with the developed disturbance-observer-based sliding
mode control in [38]. The comparison result is presented in Fig. 2
which illustrates the effectiveness of the proposed control scheme
and can obtain better control performance. According to Fig. 3, we
can see that the control input is bounded and convergent.
The disturbance estimate output and the disturbance estimate
error are presented in Figs. 4 and 5, respectively. Based on these
simulation results, we can obtain that the proposed disturbance-
observer-based terminal sliding mode tracking control is valid for
the uncertain nonlinear system with the time-varying external
disturbance.
6.1.2. Considering input saturation for Van der Pol circuits system
To illustrate the effectiveness of the proposed terminal sliding
mode tracking control for the Van der Pol circuits system with
input saturation constraint, the saturation values is given by
umax ¼ 15:0, umin ¼20 in the simulation study. The terminal
0 5 10 15 20 25 30 35 40 45 50
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Time [s]
0 0.2 0.4
0
0.1
0.2
Time [s]
Fig. 1. Output x1 (solid line) follows desired trajectory yd (dashed line) without
input saturation for Van der Pol circuits system.
0 5 10 15 20 25 30 35 40 45 50
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Time [s]
s 1
Tracking error using the approach developed in [38]Tracking error using the approach developed in this paper
Fig. 2. Tacking errors without input saturation for Van der Pol circuits system.
0 5 10 15 20 25 30 35 40 45 50
−300
−250
−200
−150
−100
−50
0
50
100
150
200
Time [s]
u
Fig. 3. Control input without input saturation for Van der Pol circuits system.
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]] 5
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sliding mode disturbance observer and the terminal sliding mode
tracking control are designed as
s¼ z x2
_ z ¼ksb signðsÞes p0=q0 92 x1 þ3ð1 x21Þ x29 signðsÞþvr
s1 ¼ x1 yds2 ¼ _s1 þa1s1 þb1s
p1=q1
1 þs
D̂ ¼
ksb sign
ðsÞes p0=q0
9
2 x1
þ3ð1 x2
1Þ x29 sign
ðsÞþ2 x13ð1 x2
1Þ x2
vr ¼ 2 x13ð1 x21Þ x2 þ € yda1 _s1b1 _s
p1=q1
1 d̂þds2ms p2=q2
2 ð41Þ
The tracking results of the uncertain nonlinear system with
input saturation are shown in Figs. 6 and 7 under the disturbance-
observer-based terminal sliding mode tracking control (40).
Although, there exists the non-symmetric input saturation and
the time-varying external disturbance, the tracking performance
is still satisfactory and the tracking error converges to zero.
The control input is shown in Fig. 8 and the non-symmetric input
saturation of the control input signal is observed. On the other
0 5 10 15 20 25 30 35 40 45 50
0
1
2
3
4
5
6
Time [s]
0 1 2 30.5
1
1.5
2
2.5
3
Time [s]
Fig. 4. The disturbance estimate output d̂ (solid line) and the disturbance d
(dashed line).
0 5 10 15 20 25 30 35 40 45 50
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Time [s]
D i s t u r b a n c e e s t i m a t e e r r o r
Fig. 5. The disturbance estimate error.
0 5 10 15 20 25 30 35 40 45 50
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Time [s]
Fig. 6. Output x1 (solid line) follows desired trajectory yd (dashed line) with input saturation for Van der Pol circuits system.
0 5 10 15 20 25 30 35 40 45 50
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time [s]
s 1
Fig. 7. Tacking errors with input saturation for Van der Pol circuits system.
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]]6
Please cite this article as: Chen M, et al. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISATransactions (2012), http://dx.doi.org/10.1016/j.isatra.2012.09.009
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0 5 10 15 20 25 30 35 40 45 50
−25
−20
−15
−10
−5
0
5
10
15
20
Time [s]
u
Fig. 8. Control input with input saturation for Van der Pol circuits system.
0 5 10 15 20 25 30 35 40 45 50−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time [s]
Fig. 9. Output x1 (solid line) follows desired trajectory yd (dashed line) without
input saturation for duffing forced oscillation systems.
0 5 10 15 20 25 30 35 40 45 50
−200
−150
−100
−50
0
50
100
150
200
Time [s]
u
Fig. 11. Control input without input saturation for duffing forced oscillation
systems.
0 5 10 15 20 25 30 35 40 45 50
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time [s]
s 1
Tracking error using the approach developed in this paper Tracking error using the approach developed in [38]
Fig. 10. Tacking errors without input saturation for duffing forced oscillation
systems.
0 5 10 15 20 25 30 35 40 45 50
−1.5
−1
−0.5
0
0.5
1
1.5
Time [s]
Fig. 12. Output x1 (solid line) follows desired trajectory yd (dashed line) with
input saturation for duffing forced oscillation systems.
0 5 10 15 20 25 30 35 40 45 50
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Time [s]
S 1
Fig. 13. Tacking errors with input saturation for duffing forced oscillation systems.
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]] 7
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hand, the chattering phenomena appear in the input control
signals which are excited by the sliding mode.
6.2. Simulation example of duffing forced oscillation systems
Consider the duffing forced oscillation system as [2]
_ x1 ¼ x2
_ x2 ¼0:1 x2 x31 þ12 cosðt Þþuþd
y¼ x1
ð42
Þwhere the external disturbance d ¼ 1:5 sinð0:2t Þþ2 cosð0:2t þ1Þ.
To demonstrate the effectiveness of the terminal sliding mode
disturbance observer and terminal sliding mode tracking control,
all design parameters are chosen as k¼2500, b ¼ 4, e ¼ 0:5, p0 ¼ 5,
q0 ¼ 9, a1 ¼ 50, b1 ¼ 0:5, p1 ¼ p2 ¼ 5, q1 ¼q2 ¼7, d¼ 60 and
m¼ 0:8. The initial state conditions are arbitrarily chosen as
x0 ¼ 0:2, x2 ¼ 0, d̂ ¼ 0:4 and D̂ ¼ 0. The desired trajectory is
considered as yd ¼ 0:8ðsinðt Þþsinð0:5t ÞÞ.
6.2.1. Without considering input saturation for duffing forced
oscillation systems
In this subsection, the control input saturation is not consid-
ered for the duffing forced oscillation system. The terminal sliding
mode disturbance observer is designed as (4)–(6). In accordancewith (18), the proposed terminal sliding mode tracking control
is designed. Under the proposed disturbance-observer-based
terminal sliding mode tracking control scheme, from Figs. 9
and 10, the satisfactory tracking performance is observed and
the tracking error can quickly converge to zero for the duffing
forced oscillation system (42) in the presence of the time-varying
external disturbance. To show the better control performance of
the proposed terminal sliding mode control, the comparison
control performance under the developed control approach with
the developed disturbance-observer-based sliding mode control
in [38] is shown in Fig. 10. The comparison results show that the
proposed control scheme can obtain better control performance.
From Fig. 11, the control input is presented and the convergence
is noticed. In accordance with these simulation results, we
can conclude that the proposed disturbance-observer-based
terminal sliding mode tracking control is valid for the duffing
forced oscillation system (42) with the time-varying external
disturbance.
6.2.2. Considering input saturation for duffing forced oscillation
systems
In this subsection, the effectiveness of the proposed terminal
sliding mode tracking control will be shown for the duffing forced
oscillation system (42) with input saturation constraint and
unknown external disturbance. In this simulation, the saturation
values are given byumax ¼10:0 and umin ¼9:5. The terminal sliding
mode disturbance observer is designed as (30)–(32). The terminal
sliding mode tracking control is decided by (24) and (33).
The tracking results of the uncertain nonlinear system with input
saturation are shown in Figs. 12 and 13 under the disturbance-
observer-based terminal sliding mode tracking control (40).
Although, there exists the non-symmetric input saturation and the
time-varying external disturbance, the tracking performance is still
satisfactory and the tracking error converges to zero quickly.
The control input is shown in Fig. 14 and the non-symmetric input
saturation of the control input signal is observed.
From these simulation results of two cases, we can obtain that
the developed terminal sliding mode control scheme based on
disturbance observer is valid. And the developed terminal sliding
mode disturbance observer can modify the control performance
of the terminal sliding mode control.
7. Conclusion
In this paper, the disturbance-observer-based terminal sliding
mode tracking control has been proposed for a class of uncertain
nonlinear systems. To improve the ability of the disturbance
attenuation and system performance robustness, the terminal
sliding mode disturbance observer has been developed to approx-
imate the system disturbance in the finite time. Based on the
output of the disturbance observer, the disturbance-observer-
based terminal sliding mode tracking control has been presented
0 5 10 15 20 25 30 35 40 45 50
−10
−8
−6
−4
−2
0
2
4
6
8
10
Time [s]
u
Fig. 14. Control input with input saturation for duffing forced oscillation systems.
M. Chen et al. / ISA Transactions ] (]]]]) ] ]]–]]]8
Please cite this article as: Chen M, et al. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISATransactions (2012), http://dx.doi.org/10.1016/j.isatra.2012.09.009
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for the uncertain nonlinear system with the non-symmetric input
saturation and the time-varying external disturbance. The stabi-
lity of the closed-loop system has been proved using rigorous
Lyapunov analysis. Finally, simulation results have been used to
illustrate the effectiveness of the proposed robust terminal sliding
mode tracking control scheme.
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