SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEMS

International Journal of Information Sciences and Techniques (IJIST) Vol.1, No.1, July 2011
20
SLIDING MODE CONTROLLER DESIGN FOR
SYNCHRONIZATION OF SHIMIZU-MORIOKA
CHAOTIC SYSTEMS
Sundarapandian Vaidyanathan1
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
ABSTRACT
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems
(Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the
complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov
stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode
control method is very effective and convenient to achieve global chaos synchronization of the identical
Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the
synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
KEYWORDS
Sliding Mode Control, Chaos Synchronization, Chaotic Systems, Shimizu-Morioka Systems.
1. INTRODUCTION
Chaotic systems are dynamical systems that are highly sensitive to initial conditions. The
sensitive nature of chaotic systems is commonly called as the butterfly effect [1]. Synchronization
of chaotic systems is a phenomenon which may occur when two or more chaotic oscillators are
coupled or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly
effect which causes the exponential divergence of the trajectories of two identical chaotic systems
started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly a
very challenging problem.
In most of the chaos synchronization approaches, the master-slave or drive-response formalism is
used. If a particular chaotic system is called the master or drive system and another chaotic
system is called the slave or response system, then the idea of the synchronization is to use the
output of the master system to control the slave system so that the output of the slave system
tracks the output of the master system asymptotically.
Since the pioneering work by Pecora and Carroll ([2], 1990), chaos synchronization problem has
been studied extensively and intensively in the literature [2-17]. Chaos theory has been applied to
a variety of fields such as physical systems [3], chemical systems [4], ecological systems [5],
secure communications [6-8], etc.
In the last two decades, various schemes have been successfully applied for chaos
synchronization such as PC method [2], OGY method [9], active control method [10-12],
adaptive control method [13-14], time-delay feedback method [15], backstepping design method
[16], sampled-data feedback method [17], etc.

21
In this paper, we derive new results based on the sliding mode control [18-20] for the global
chaos synchronization of identical Shimizu-Morioka systems ([21], Shimizu and Morioka, 1980).
In robust control systems, the sliding mode control method is often adopted due to its inherent
advantages of easy realization, fast response and good transient performance as well as its
insensitivity to parameter uncertainties and external disturbances.
This paper has been organized as follows. In Section 2, we describe the problem statement and
our methodology using sliding mode control (SMC). In Section 3, we discuss the global chaos
synchronization of identical Shimizu-Morioka chaotic systems. In Section 4, we summarize the
main results obtained in this paper.
2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC
In this section, we describe the problem statement for the global chaos synchronization for
identical chaotic systems and our methodology using sliding mode control (SMC).
Consider the chaotic system described by
( )
x Ax f x
= +
& (1)
where n
x∈R is the state of the system, A is the n n
× matrix of the system parameters and
: n n
f →
R R is the nonlinear part of the system.
We consider the system (1) as the master or drive system.
As the slave or response system, we consider the following chaotic system described by the
dynamics
( )
y Ay f y u
= + +
& (2)
where n
y ∈R is the state of the system and m
u ∈R is the controller to be designed.
If we define the synchronization error as
,
e y x
= − (3)
then the error dynamics is obtained as
( , ) ,
e Ae x y u
η
= + +
& (4)
where
( , ) ( ) ( )
x y f y f x
η = − (5)
The objective of the global chaos synchronization problem is to find a controller u such that
lim ( ) 0
t
e t
→∞
= for all (0) .
n
e ∈R

22
To solve this problem, we first define the control u as
( , )
u x y Bv
η
= − + (6)
where B is a constant gain vector selected such that ( , )
A B is controllable.
Substituting (5) into (4), the error dynamics simplifies to
e Ae Bv
= +
& (7)
which is a linear time-invariant control system with single input .
v
Thus, the original global chaos synchronization problem can be replaced by an equivalent
problem of stabilizing the zero solution 0
e = of the system (7) by a suitable choice of the sliding
mode control. In the sliding mode control, we define the variable
1 1 2 2
( ) n n
s e Ce c e c e c e
= = + + +
L (8)
where [ ]
1 2 n
C c c c
= L is a constant vector to be determined.
In the sliding mode control, we constrain the motion of the system (7) to the sliding manifold
defined by
{ }
| ( ) 0
n
S x s e
= ∈ =
R
which is required to be invariant under the flow of the error dynamics (7).
When in sliding manifold ,
S the system (7) satisfies the following conditions:
( ) 0
s e = (9)
which is the defining equation for the manifold S and
( ) 0
s e =
& (10)
which is the necessary condition for the state trajectory ( )
e t of (7) to stay on the sliding manifold
.
S
Using (7) and (8), the equation (10) can be rewritten as
[ ]
( ) 0
s e C Ae Bv
= + =
& (11)
Solving (11) for ,
v we obtain the equivalent control law
1
eq ( ) ( ) ( )
v t CB CA e t
−
= − (12)
where C is chosen such that 0.
CB ≠
Substituting (12) into the error dynamics (7), we obtain the closed-loop dynamics as

23
1
( )
e I B CB C Ae
−
 
= −
 
& (13)
The row vector C is selected such that the system matrix of the controlled dynamics
1
( )
I B CB C A
−
 
−
  is Hurwitz, i.e. it has all eigenvalues with negative real parts. Then the
controlled system (13) is globally asymptotically stable.
To design the sliding mode controller for (7), we apply the constant plus proportional rate
reaching law
sgn( )
s q s k s
= − −
& (14)
where sgn( )
⋅ denotes the sign function and the gains 0,
q > 0
k > are determined such that the
sliding condition is satisfied and sliding motion will occur.
From equations (11) and (14), we can obtain the control ( )
v t as
[ ]
1
( ) ( ) ( ) sgn( )
v t CB C kI A e q s
−
= − + + (15)
which yields
[ ]
[ ]
1
1
( ) ( ) , if ( ) 0
( )
( ) ( ) , if ( ) 0
CB C kI A e q s e
v t
CB C kI A e q s e
−
−
− + + >
=
− + − <



(16)
Theorem 1. The master system (1) and the slave system (2) are globally and asymptotically
synchronized for all initial conditions (0), (0) n
x y R
∈ by the feedback control law
( ) ( , ) ( )
u t x y Bv t
η
= − + (17)
where ( )
v t is defined by (15) and B is a column vector such that ( , )
A B is controllable. Also, the
sliding mode gains ,
k q are positive.
Proof. First, we note that substituting (17) and (15) into the error dynamics (4), we obtain the
closed-loop error dynamics as
[ ]
1
( ) ( ) sgn( )
e Ae B CB C kI A e q s
−
= − + +
& (18)
To prove that the error dynamics (18) is globally asymptotically stable, we consider the candidate
Lyapunov function defined by the equation
2
1
( ) ( )
2
V e s e
= (19)
which is a positive definite function on .
n
R

24
Differentiating V along the trajectories of (18) or the equivalent dynamics (14), we get
2
( ) ( ) ( ) sgn( )
V e s e s e ks q s s
= = − −
& & (20)
which is a negative definite function on .
n
R
This calculation shows that V is a globally defined, positive definite, Lyapunov function for the
error dynamics (18), which has a globally defined, negative definite time derivative .
V
&
Thus, by Lyapunov stability theory [22], it is immediate that the error dynamics (18) is globally
asymptotically stable for all initial conditions (0) .
n
e ∈R
This means that for all initial conditions (0) ,
n
e R
∈ we have
lim ( ) 0
t
e t
→∞
=
Hence, it follows that the master system (1) and the slave system (2) are globally and
asymptotically synchronized for all initial conditions (0), (0) .
n
x y ∈R
This completes the proof.
3. GLOBAL SYNCHRONIZATION OF IDENTICAL SHIMIZU-MORIOKA
CHAOTIC SYSTEMS USING SLIDING MODE CONTROL
3.1 Theoretical Results
In this section, we apply the sliding mode control results derived in Section 2 for the global chaos
synchronization of identical Shimizu-Morioka (SM) chaotic systems ([21], Shimizu and Morioka,
1980).
Thus, the master system is described by the SM dynamics
1 2
2 1 2 1 3
2
3 1 3
x x
x x x x x
x x x
λ
ξ
=
= − −
= −

(21)
where 1 2 3
, ,
x x x are state variables and , , ,
α β γ ω are positive, constant parameters of the system.
The slave system is described by the controlled SM dynamics
1 2 1
2 1 2 1 3 2
2
3 1 3 3
y y u
y y y y y u
y y y u
λ
ξ
= +
= − − +
= − +

(22)
where 1 2 3
, ,
y y y are state variables and 1 2 3
, ,
u u u are the controllers to be designed.
The Shimizu-Morioka systems (21) and (22) are chaotic when 0.75
λ = and 0.45.
ξ =

25
Figure 1 illustrates the double-wing butterfly-shaped attractor of the SM system (21).
Figure 1. State Orbits of the Shimizu-Morioka Chaotic System
The chaos synchronization error is defined by
, ( 1,2,3)
i i i
e y x i
= − = (23)
The error dynamics is easily obtained as
1 2 1
2 1 2 1 3 1 3 2
2 2
3 3 1 1 3
e e u
e e e y y x x u
e e y x u
λ
ξ
= +
= − − + +
= − + − +

(24)
We write the error dynamics (24) in the matrix notation as
( , )
e Ae x y u
η
= + +
(25)
where
0 1 0
1 0 ,
0 0
A λ
ξ
 
 
= −
 
 
−
 
1 3 1 3
2 2
1 1
0
( , )
x y y y x x
y x
η
 
 
= − +
 
 
−
 
and
1
2
3
u
u u
u
 
 
=  
 
 
. (26)
The sliding mode controller design is carried out as detailed in Section 2.
First, we set u as

26
( , )
u x y Bv
η
= − + (27)
where B is chosen such that ( , )
A B is controllable.
We take B as
1
1 .
1
B
 
 
=  
 
 
(28)
In the chaotic case, the parameter values are
0.75
λ = and 0.45.
ξ =
The sliding mode variable is selected as
[ ] 1 2 3
2 1 2 2 2
s Ce e e e e
= = − = + − (29)
which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 5
k = and 0.1.
q =
We note that a large value of k can cause chattering and an appropriate value of q is chosen to
speed up the time taken to reach the sliding manifold as well as to reduce the system chattering.
From Eq. (15), we can obtain ( )
v t as
1 2 3
( ) 11 6.25 9.1 0.1 sgn( )
v t e e e s
= − − + − (30)
Thus, the required sliding mode controller is obtained as
( , )
u x y Bv
η
= − + (31)
where ( , ),
x y B
η and ( )
v t are defined as in the equations (26), (28) and (30).
By Theorem 1, we obtain the following result.
Theorem 2. The identical Shimizu-Morioka chaotic systems (21) and (22) are globally and
asymptotically synchronized for all initial conditions with the sliding mode controller u defined
by (31).
3.2 Numerical Results
In this section For the numerical simulations, the fourth-order Runge-Kutta method with time-
step 6
10
h −
= is used to solve the Shimizu-Morioka chaotic systems (21) and (22) with the sliding
mode controller u given by (31) using MATLAB.
In the chaotic case, the parameter values are

27
0.75
λ = and 0.45.
ξ =
The sliding mode gains are chosen as
5
k = and 0.1.
q =
The initial values of the master system (21) are taken as
1 2 3
(0) 30, (0) 25, (0) 14
x x x
= = =
and the initial values of the slave system (22) are taken as
1 2 3
(0) 4, (0) 18, (0) 20
y y y
= = =
Figure 2 illustrates the complete synchronization of the identical Shimizu-Morioka chaotic
systems (21) and (22).
Figure 2. Synchronization of Identical Shimizu-Morioka Chaotic Systems

28
4. CONCLUSIONS
In this paper, we have deployed sliding mode control (SMC) to achieve global chaos
synchronization for the identical Shimizu-Morioka chaotic systems (1980). Our synchronization
results for the identical Shimizu-Morioka chaotic systems have been proved using Lyapunov
stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding
mode control method is very effective and convenient to achieve global chaos synchronization for
the identical Shimizu-Morioka chaotic systems. Numerical simulations are also shown to
illustrate the effectiveness of the synchronization results derived in this paper using the sliding
mode control.
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[18] Slotine, J.E. Sastry, S.S. (1983) “Tracking control of nonlinear systems using sliding surface
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Author
Dr. V. Sundarapandian is a Professor (Systems Control Engineering) in the R
D Centre at Vel Tech Dr. RR Dr. SR Technical University, Chennai, India. His
current research areas are Linear and Nonlinear Control Systems, Chaos Theory,
Dynamical Systems, Soft Computing, etc. He has published over 150 refereed
international journal publications. He has published over 45 papers in International
Conferences and over 90 papers in National Conferences. He has published two
text-books titled Numerical Linear Algebra and Probability, Statistics and
Queueing Theory with Prentice Hall of India, New Delhi, India. He is a Senior
Member of AIRCC and the Editor-in-Chief of the AIRCC international journals –
International Journal of Instrumentation and Control Systems and International
Journal of Control Theory and Computer Modeling. He is the Associate Editor of
many Control Journals like International Journal of Control Theory and Applications, International
Journal of Computer Information and Sciences, and International Journal of Advances in Science and
Technology. He has delivered many Key Note Lectures on Nonlinear Control Systems, Modern Control
Systems, Chaos and Control, Mathematical Modelling, Scientific Computing with SCILAB/MATLAB, etc.

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