quasi sliding mode control of chaos in fractional order duffing system
TRANSCRIPT
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 [email protected]
Susy Thomas Professor and Head of the department
Electrical Engineering Department National Institute of Technology Calicut, India
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
αt
0,dτ
01,
0,dt
d
D
(1)
Definition 2: The Grunwald-Letnikov derivative definition of order is describes as
jhtfjh
tfDj
j
ht
00
11
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
Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014
i
iaq
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:
tFtxtxtydt
dy
tydt
dx
cos3
(5)
From equation (5), the Fractional Duffing system is obtained by replacing conventional derivatives by fractional derivatives.
tFtxtxtytyD
tytxD
qt
qt
cos32
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
23
10
xJ
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
Y
Figure 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, 98.021qq simulation time stsim 200 ,
05.0h 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
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, 98.021qq simulation
time stsim 200 , 05.0h 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 ,
05.0h and with initial condition )13.0,21.0( .
Therefore the Fractional-order duffing system can be described as follows:
)(cos32
1
tutFtxtxtytyD
tytxD
qt
qt
(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
Y
Figure 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 , 05.0h and with initial condition )13.0,21.0( .
Let us choose the sliding surface )(ts as
tqt dttytxtyDts
0
1)()()()( 2 (9)
For sliding mode control method, the sliding surface and its derivative must be zero.
0)()()()(
0
12
tqt dttytxtyDts (10)
0)()()()( 2 tytxtyDtsqt (11)
Therefore
0)()()(cos
)()()()(
3
2
tytxtutFtxtxty
tytxtyDts
eq
qt
)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)
Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014
Selecting a Lyapunov function 2)(2
1tsV
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
qt
qt
(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
1sV
0)(
)()(
)(),()cos()(
1)(cos)(
)()(cos)(
21
213
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
tstssign (16)
Here is a small positive scalar.
Therefore, the modified control input )(tu can be defined
as
)(
)()cos()()1)(()( 3
ts
tsKtFtxtxtu (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
0.3State X(t)
X
Time
0 50 100-0.1
0
0.1
0.2
0.3State Y(t)
Y
Time
0 50 100-0.5
0
0.5
1Sliding surface
S
Time
0 50 100-2
-1
0
1Controller
U
Time
Figure 6: Time response of the states of the controlled commensurate
Fractional-order duffing system with simulation time ssimt 100 .
0 50 100-0.1
0
0.1
0.2
0.3State X(t)
X
Time
0 50 100-0.2
-0.1
0
0.1
0.2State Y(t)
Y
Time
0 50 100-0.5
0
0.5
1Sliding surface
S
Time
0 50 100-2
-1
0
1Controller
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 ssimt 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
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
0.3State X(t)
X
Time
0 50 100-0.1
0
0.1
0.2
0.3State Y(t)
Y
Time
0 50 100-0.5
0
0.5
1Sliding surface
S
Time
0 50 100-2
-1
0
1Controller
U
Time
Figure 8: Time response of the states of the controlled non-commensurate
Fractional-order duffing system with simulation time ssimt 100 .
0 50 100-0.1
0
0.1
0.2
0.3State X(t)
X
Time
0 50 100-0.2
-0.1
0
0.1
0.2State Y(t)
Y
Time
0 50 100-0.5
0
0.5
1Sliding surface
S
Time
0 50 100-2
-1
0
1Controller
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 ssimt 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.
REFERENCES
[1] Chunguang Li a, Guanrong Chen, “Chaos in the fractional order Chen system and its control”, Chaos, Solutions and Fractals, vol.22, pp. 549–554, 2004.
[2] YS Deng, “Fractional order Liu-system synchronization and its application in multimedia security”, Communications, Circuits and Systems (ICCCAS), 2010 International Conference, pp.769-772, 2010.
[3] Alireza K. Golmankhaneh, Roohiyeh Arefi, Dumitru Baleanu, “The Proposed Modified Liu System with Fractional Order”, Advances in Mathematical Physics, Article ID 186037, 2013.
[4] Matouk, A.E. “Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit” Commun. Nonlinear Science and Numerical Simulation, Vol. 16, pp. 975–986. 2011.
[5] Yamin Wang, Xiaozhou Yin, Yong Liu, “Control Chaos in System with Fractional Order, “Journal of Modern Physics, Vol. 3, pp. 496-501, 2012.
[6] Hitay Ozbay, Catherine Bonnet, Andre Ricardo Fioravanti, “PID controller design for fractional-order systems with time delays, “Systems & Control Letters, Vol. 61 pp. 18–23, 2012.