terminal sliding mode tracking control for a class of siso uncertain nonlinear systems

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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: [email protected] (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

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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


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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


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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



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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.



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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.



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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.









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.



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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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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





terminal sliding mode tracking control for a class of siso uncertain nonlinear systems

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