1. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014)
February 27 – 28, 2014
Quasi-sliding mode control of chaos in Fractional-
order Duffing system
Kishore Bingi
Electrical Engineering Department
National Institute of Technology Calicut, India
kishore9860@gmail.com
Susy Thomas
Professor and Head of the department
Electrical Engineering Department
National Institute of Technology Calicut, India
susy@nitc.com
Abstract—In this paper a quasi-sliding mode controller for
control of chaos in Fractional-order duffing system is designed.
Here, the designed sliding mode control law is to make the
Fractional-order duffing system globally asymptotically stable
and it also guarantees the system globally asymptotically in the
presence of uncertainties and external disturbances. Finally
numerical results demonstrate the effectiveness of the proposed
controller.
Index Terms—Chaos, Fractional-order duffing system, Quasi-
sliding mode
I. INTRODUCTION
Fractional calculus is three centuries old as conventional
calculus, but not very popular among engineering and sciences.
However, its applications to physics and engineering have just
started in recent decades. The beauty of the subject is the
solution of the Fractional derivative (or) integral. After the
invention of Grunwald-Letnikov derivative, Riemann-Liouville
and Caputo definition the applications are rapidly grown up
because it was found that many of the systems can be elegantly
modeled with the help of Fractional derivative.
Chaotic behavior of dynamic systems can be utilized in
many real-world applications such as engineering, finance,
microbiology, biology, physics, robotics, mathematics,
economics, philosophy, meteorology, computer science, and
civil engineering and so on. From the investigation of
researchers it was found that Functional-order chaotic systems
possess memory and display more sophisticated dynamics
compared to its Integer-order systems.
Recently, the control of chaos in Fractional-order systems
has been one of the most interesting topics, and many
researchers have made great contributions. For example, in [1],
a state feedback control law was proposed for control of chaos
in Fractional-order Chen system. In [2], a control algorithm is
proposed for Fractional-order Liu system to improve the
projective synchronization in the integer order systems. In [3],
an active control methodology for controlling chaotic behavior
of a Fractional-order version of Rossler system was presented.
The main feature of the designed controller is its simplicity for
practical implementation. In [4], the Fractional Routh-Hurwitz
conditions are used to control chaos in Fractional-order
modified autonomous Van der Pol-duffing system to its
equilibrium. In [5], a non-linear state feedback control in ODE
system to Fractional-order systems is studied. In [6], a classical
PID controller is designed for Fractional-order systems with
time delays.
In this paper, the Fractional-order duffing system is
introduced and to control chaos in this system, a Quasi-sliding
mode controller is proposed. The proposed control law makes
the states of the system asymptotically stable. Simulation
results illustrate that the controller can easily eliminate chaos
and stabilize the system on the sliding surface.
This paper is organized as follows. Section II contains the
basic definitions about Fractional Calculus. Section III
describes about Fractional-order duffing system. A Quasi-
sliding mode controller is proposed to control chaos in
Fractional-order duffing system in Section IV. In Section V the
concluding comments are given.
II. BASIC DEFINITIONS
Definition 1: The continuous integral-differential operator
is defined as
 














t
0
α
α
α
α
t
0,dτ
01,
0,
dt
d
D



(1)
Definition 2: The Grunwald-Letnikov derivative definition
of order  is describes as
   jhtf
jh
tfD
j
j
h
t 





 



0
0
1
1
lim)(



(2)
Definition 3: Suppose that the unstable Eigen values of a
focus points are 2,12,12,1  j . The necessary condition to
exhibit double scroll attractor is the Eigen value 2,1 remaining
in the unstable region. The condition for commensurate
derivative is

2. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014)
February 27 – 28, 2014









i
i
aq



tan
2
(3)
III. FRACTIONAL-ORDER DUFFING SYSTEM
Duffing system was introduced by Georg. Duffing with
negative linear stiffness, damping and periodic excitation is
often written in the form
 tFxxxx  cos3
  (4)
The equation (4) is rewritten as a system of first order
autonomous differential equations in the form:
 
       tFtxtxty
dt
dy
ty
dt
dx


cos
3

(5)
From equation (5), the Fractional Duffing system is
obtained by replacing conventional derivatives by fractional
derivatives.
   
         tFtxtxtytyD
tytxD
q
t
q
t


cos
32
1

(6)
1q , 2q are Fractional derivatives and F,,,  are system
parameters.
Here if 21 qq  , then the Fractional-order duffing system is
called commensurate Fractional-order duffing system.
Otherwise we call the system as non-commensurate Fractional-
order duffing system.
The Jacobian matrix of the duffing system (5) is








 2
3
10
x
J
The fixed points (equilibrium) of the Integer-order duffing
system with parameters 1,3.0,15.0,1,1  F
are A  0,0728.1 , B  0,1667.0 and C  0,9061.0 and their
corresponding Eigen values are,
For A we get 9278.0,0778.12,1  ,
For B we get j4122.1075.02,1  , and
For C we get j4122.1075.02,1  .
Here the Eigen value for corresponding equilibrium point A
is saddle points which satisfy the stability condition of chaotic
behavior.
Figure 1 shows the chaotic attractor of Integer-order
duffing system with parameters simulation time stsim 200
and with initial condition )13.0,21.0( .
We choose 98.021 qq Figure 2 shows the chaotic
attractor of commensurate Fractional-order duffing system and
Figure 3 shows the time response of the states of the
commensurate Fractional-order duffing system with
parameters, simulation time 05.0,200  hstsim and with
initial condition )13.0,21.0( .
For 98.0,95.0 21  qq Figure 4 shows the chaotic
attractor of non-commensurate Fractional-order duffing system
and Figure 5 shows the time response of the states of the non-
commensurate Fractional-order duffing system with
parameters, simulation time 05.0,200  hstsim and with
initial condition )13.0,21.0( .
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-1.5
-1
-0.5
0
0.5
1
1.5
X
YFigure 1: Chaotic attractor of Integer-order duffing system with
parameters, simulation time stsim 200 and with initial
condition )13.0,21.0( .
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-1.5
-1
-0.5
0
0.5
1
1.5
X
Y
Figure 2: Chaotic attractor of commensurate Fractional-order duffing
system with parameters, 98.0
21
qq simulation time stsim 200 ,
05.0h and with initial condition )13.0,21.0( .
IV. QUASI SLIDING-MODE CONTROL OF FRACTIONAL-ORDER
DUFFING SYSTEM
The sliding mode control scheme involves: 1) selection of
sliding surface that represents a desirable system dynamic

3. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014)
February 27 – 28, 2014
behavior, 2) finding a switching control law that a sliding mode
exists on every point of the sliding surface.
The control input )(tu is added to the last state equation in
order to control chaos.
0 20 40 60 80 100 120 140 160 180 200
-2
-1
0
1
2
Time
X
0 20 40 60 80 100 120 140 160 180 200
-1
-0.5
0
0.5
1
Time
Y
Figure 3: Time response of the states of the commensurate Fractional-
order duffing system with parameters, 98.0
21
qq simulation
time stsim 200 , 05.0h and with initial condition )13.0,21.0( .
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-1.5
-1
-0.5
0
0.5
1
1.5
X
Y
Figure 4: Chaotic attractor of non-commensurate Fractional-order duffing
system with parameters, 98.02,95.01  qq simulation time stsim 200 ,
05.0h and with initial condition )13.0,21.0( .
Therefore the Fractional-order duffing system can be
described as follows:
   
          )(cos32
1
tutFtxtxtytyD
tytxD
q
t
q
t



(7)
The sliding mode control )(tu in equation (7) has following
structure:
)()()( tututu sweq  (8)
Where )(tueq , the equivalent control and )(tusw , the switching
control of the system.
0 20 40 60 80 100 120 140 160 180 200
-2
-1
0
1
2
Time
X
0 20 40 60 80 100 120 140 160 180 200
-1
-0.5
0
0.5
1
Time
YFigure 5: Time response of the states of the non-commensurate Fractional-
order duffing system with parameters, 98.02,95.01  qq simulation
time stsim 200 , 05.0h and with initial condition )13.0,21.0( .
Let us choose the sliding surface )(ts as
  

t
q
t dttytxtyDts
0
1
)()()()( 2
 (9)
For sliding mode control method, the sliding surface and its
derivative must be zero.
  0)()()()(
0
12
 

t
q
t dttytxtyDts  (10)
0)()()()( 2
 tytxtyDts
q
t  (11)
Therefore
        0)()()(cos
)()()()(
3
2


tytxtutFtxtxty
tytxtyDts
eq
q
t


  )cos()(1)()( 3
tFtxtxtueq   (12)
The switching control )(tusw is chosen in order to satisfy the
sliding condition
)()( ssignKtusw  (13)
Where K is the gain of the controller and )(ssign is the
Signum function.
Therefore, the total control law can be defined as
)()cos()()1)(()( 3
ssignKtFtxtxtu   (14)

4. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014)
February 27 – 28, 2014
Selecting a Lyapunov function 2
)(
2
1
tsV 
Here, the time derivative of the Lyapunov function is given by
      
     
  
0
))()()cos()(1)(
cos()(
)()(cos)(
)()(
3
3
3





sK
txssignKtFtxtx
tFtxtxts
txtutFtxtxts
tstsV




Therefore, it confirms the existence of sliding mode
dynamics and the closed loop system is globally asymptotically
stable.
Consider the system (7) being perturbed by uncertainties
and external disturbance which can be modeled as
   
         
)()(),(
cos
21
32
1
tutdyxd
tFtxtxtytyD
tytxD
q
t
q
t



 (15)
Where ),(1 yxd , the uncertainties in the states and )(2 td ,
the external disturbance are assumed to be bounded i.e.
11 ),( dyxd  and 22 )( dtd  .
For the Lyapunov function 2
2
1
sV 
      
       
0)(
)()(
)(),()cos()(
1)(cos)(
)()(cos)(
21
21
3
3
3






sddK
txssignK
tdyxdtFtx
txtFtxtxts
txtutFtxtxts
ssV




Therefore, it confirms the closed loop system in the
presence of uncertainties and external disturbance with the
sliding mode controller is globally asymptotically stable
when 21 ddK  .
In order to avoid the chattering effect in the control
input )(tu , one of the obvious solutions to make the control
function continuous/smooth is to approximate the
discontinuous signum function by continuous/smooth sigmoid
function.


)(
)(
))((
ts
ts
tssign (16)
Here  is a small positive scalar.
Therefore, the modified control input )(tu can be defined
as




)(
)(
)cos()()1)(()( 3
ts
ts
KtFtxtxtu (17)
For commensurate Fractional-order Duffing system, the
states of the system (7) under the controller (17) and the sliding
surface (10) are illustrated in Figure 6 and with uncertainties in
the states )sin()cos(45.0),(1 yxyxd  and with external
disturbance )sin(5.0)(2 ttd  is illustrated in Figure 7 when
gain of the controller K=1.0 and with initial condition
)13.0,21.0( .
0 50 100
-0.1
0
0.1
0.2
0.3
State X(t)
X
Time
0 50 100
-0.1
0
0.1
0.2
0.3
State Y(t)
Y
Time
0 50 100
-0.5
0
0.5
1
Sliding surface
S
Time
0 50 100
-2
-1
0
1
Controller
U
Time
Figure 6: Time response of the states of the controlled commensurate
Fractional-order duffing system with simulation time s
sim
t 100 .
0 50 100
-0.1
0
0.1
0.2
0.3
State X(t)
X
Time
0 50 100
-0.2
-0.1
0
0.1
0.2
State Y(t)
Y
Time
0 50 100
-0.5
0
0.5
1
Sliding surface
S
Time
0 50 100
-2
-1
0
1
Controller
U
Time
Figure 7: Time response of the states of the controlled commensurate
Fractional-order duffing system in the presence of uncertainties and external
disturbance with simulation time s
sim
t 100 .
For non-commensurate Fractional-order Duffing system,
the states of the system (7) under the controller (17) and the
sliding surface (10) are illustrated in Figure 8 and with
uncertainties in the states )sin()cos(45.0),(1 yxyxd  and
with external disturbance )sin(5.0)(2 ttd  is illustrated in

5. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014)
February 27 – 28, 2014
Figure 9 when gain of the controller K=1.0 and with initial
condition )13.0,21.0( .
From the obtained results, it is clear that the proposed
controller is good at controlling the chaos in Fractional order
duffing system.
0 50 100
-0.1
0
0.1
0.2
0.3
State X(t)
X
Time
0 50 100
-0.1
0
0.1
0.2
0.3
State Y(t)
Y
Time
0 50 100
-0.5
0
0.5
1
Sliding surface
S
Time
0 50 100
-2
-1
0
1
Controller
U
Time
Figure 8: Time response of the states of the controlled non-commensurate
Fractional-order duffing system with simulation time s
sim
t 100 .
0 50 100
-0.1
0
0.1
0.2
0.3
State X(t)
X
Time
0 50 100
-0.2
-0.1
0
0.1
0.2
State Y(t)
Y
Time
0 50 100
-0.5
0
0.5
1
Sliding surface
S
Time
0 50 100
-2
-1
0
1
Controller
U
Time
Figure 9: Time response of the states of the controlled non-commensurate
Fractional-order duffing system in the presence of uncertainties and external
disturbance with simulation time s
sim
t 100 .
V. CONCLUSION
In this paper, According to Lyapunov stability theorem, the
quasi-sliding mode controller is designed to control chaos in
Fractional-order duffing system. Based on the sliding mode
control method the states of the Fractional-order duffing
system have been stabilized. Finally, the numerical results will
demonstrate the effectiveness of the proposed controller.
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