ANALYSIS AND GLOBAL CHAOS CONTROL OF THE HYPERCHAOTIC LI SYSTEM VIA SLIDING CONTROL

International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
DOI:10.5121/ijitca.2013.3108 101
ANALYSIS AND GLOBAL CHAOS CONTROL OF THE
HYPERCHAOTIC LI SYSTEM VIA SLIDING CONTROL
Sundarapandian Vaidyanathan1
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, T.N., INDIA
sundarvtu@gmail.com
ABSTRACT
In this paper, we derive new results for global chaos control of chaotic systems via sliding control. We also
explore the analysis and global chaos control of the hyperchaotic Li system (2005) using sliding control.
The global chaos control results via sliding control have been established using Lyapunov stability theory.
Numerical simulations using MATLAB are shown to validate and depict the effectiveness of the global
chaos control of the hyperchaotic Li system.
KEYWORDS
Sliding Control, Chaos, Hyperchaos, Control, Hyperchaotic Li System.
1. INTRODUCTION
Hyperchaotic systems have received considerable attention from many physical scientists and
engineers. A hyperchaotic dynamical system is defined as a chaotic system having more than one
positive Lyapunov exponent (LE) in its Lyapunov spectrum. The first hyperchaotic system was
discovered by Rössler in 1979 [1].
During the last two decades, hyperchaotic systems have been studied using mathematical control
theory and implemented using electronic oscillators [2-3]. Realization of hyperchaotic systems by
circuit design has been applied in engineering areas such as secure communication
[4-7], synchronization [8-9], encryption [10], etc. Thus, designing hyperchaos and control of
hyperchaotic systems have become important research problems.
The control of chaotic system is to design state feedback control laws that stabilize the chaotic
systems around the unstable equilibrium points. Chaos and control of chaotic dynamical systems
are research problems that have both received rapid attention in the recent decades [11-20].
In this paper, we derive new results based on the sliding mode control [21-23] for the global
control of hyperchaotic Li system ([24], 2005). In robust control theory, the sliding mode control
method is often adopted due to its inherent advantages of easy realization, fast response and good
transient performance and its insensitivity to parameter uncertainties and external disturbances.

102
This paper has been organized as follows. In Section 2, we describe the global control of a chaotic
system using sliding mode control (SMC). In Section 3, we derive results for the global chaos
control of the hyperchaotic Li system (2005). Section 4 contains a summary of the main results.
2. GLOBAL CONTROL OF A CHAOTIC SYSTEM USING SMC
In this paper, we consider a general chaotic system described by
( )
x Ax f x u
= + +
& (1)
In Eq. (1), n
x∈R is the state of the system, A is the n n
× constant matrix of the system
parameters, : n n
f →
R R is the nonlinear part of the system and u is the control input.
The goal of the global chaos control problem is to seek a controller u such that
lim ( ) 0
t
x t
→∞
= for all (0) .
n
x ∈R (2)
To solve this global problem, we adopt the sliding control method.
As a first step, we define the control u as
( )
u f x Bv
= − + (3)
where B is a constant 1
n× matrix selected such that ( , )
A B is completely controllable.
Substituting (3) into (1), the state dynamics becomes
x Ax Bv
= +
& (4)
which is a linear time-invariant control system having single input .
v
Thus, the original global chaos control problem is equivalent to the problem of stabilizing the
zero solution 0
x = of the linear system (4) by means of a suitable choice of the sliding control.
In the following, we use Lyapunov stability theory to solve the equivalent control problem.
In the sliding control, we first define the variable
1 1 2 2
( ) n n
s x Cx c x c x c x
= = + + +
L (5)
where
[ ]
1 2 n
C c c c
= L
is a constant vector to be determined.
In the sliding control, we restrict the motion of the system (4) to the sliding manifold defined by
{ }
| ( ) 0
n
S x s x
= ∈ =
R
which is required to be invariant under the flow of the dynamics (4).

103
When in sliding manifold ,
S the system (4) satisfies the following conditions:
( ) 0
s x = (6)
which is the defining equation for the manifold S and
( ) 0
s x =
& (7)
which is the necessary condition for the state trajectory ( )
x t of (4) to stay on the sliding manifold
.
S
Using (4) and (5), the equation (7) can be rewritten as
[ ]
( ) 0
s x C Ax Bv
= + =
& (8)
Solving (8) for ,
v we obtain the equivalent control law
1
eq ( ) ( ) ( )
v t CB CA x t
−
= − (9)
where C is chosen such that 0.
CB ≠
Substitution of (9) into the state dynamics (4) yields the closed-loop dynamics as
1
( )
x I B CB C Ax
−
 
= −
 
& (10)
The row vector C is chosen in such a way that the system matrix of the controlled dynamics
1
( )
I B CB C A
−
 
−
  is Hurwitz, i.e. it has all eigenvalues in the open left-half of the complex
plane. Then the controlled system (10) is globally asymptotically stable.
To design the sliding mode controller for (4), we apply the constant plus proportional rate
reaching law
sgn( )
s q s k s
= − −
& (11)
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 (8) and (11), we can obtain the control ( )
v t as
[ ]
1
( ) ( ) ( ) sgn( )
v t CB C kI A x q s
−
= − + + (12)
which yields
[ ]
[ ]
1
1
( ) ( ) , if ( ) 0
( )
( ) ( ) , if ( ) 0
CB C kI A x q s x
v t
CB C kI A x q s x
−
−
− + + >
=
− + − <



(13)

104
Theorem 1. The global control problem for the chaotic system (1) is solved by applying the
feedback control law
( ) ( ) ( )
u t f x Bv t
= − + (14)
where ( )
v t is defined by (12) and B is a column vector such that ( , )
A B is controllable.
Proof. First, we note that substituting (14) and (12) into the chaotic dynamics (1), we arrive at
the closed-loop error dynamics
[ ]
1
( ) ( ) sgn( )
x Ax B CB C kI A x q s
−
= − + +
& (15)
To prove that the closed-loop state dynamics (15) is globally asymptotically stable, we consider
the candidate Lyapunov function defined by the equation
2
1
( ) ( )
2
V x s x
= (16)
which is a positive definite function on .
n
R
Differentiating V along the trajectories of (15) or the equivalent dynamics (11), we get
2
( ) ( ) ( ) sgn( )
V e s x s x ks q s s
= = − −
& & (17)
which is a negative definite function on .
n
R
Thus, we have shown that V is a globally defined, positive definite, Lyapunov function for the
state dynamics (15) and that V
& is a globally defined, negative definite function.
Hence, by Lyapunov stability theory [25], it follows that the state dynamics (15) is globally
asymptotically stable.
This completes the proof.
3. GLOBAL CHAOS CONTROL OF THE HYPERCHAOTIC LI SYSTEM VIA
SLIDING CONTROL
3.1 Theoretical Results
In this section, we apply the sliding control results derived in Section 2 for the global chaos
control of the hyperchaotic Li system (2005).
Thus, we consider the controlled hyperchaotic Li dynamics

105
1 2 1 4 1
2 1 1 3 2 2
3 3 1 2 3
4 2 3 4 4
( )
x a x x x u
x dx x x cx u
x bx x x u
x x x rx u
= − + +
= − + +
= − + +
= + +

(18)
where 1 2 3 4
, , ,
x x x x are state variables, , , , ,
a b c d r are positive, constant parameters of the system
and 1 2 3 4
, , ,
u u u u are the controls to be designed.
We write the state dynamics (18) in the matrix notation as
( )
x Ax f x u
= + +
(19)
Where
0 1
0 0
,
0 0 0
0 0 0
a a
d c
A
b
r
−
 
 
 
=
 
−
 
 
1 3
1 2
2 3
0
( )
x x
f x
x x
x x
 
 
−
 
=
 
 
 
and
1
2
3
4
u
u
u
u
u
 
 
 
=
 
 
 
. (20)
The 4-D system exhibits hyperchaotic behaviour when the parameter values are taken as
35, 3, 12, 7, 0.6
a b c d r
= = = = =
(see Figure 1).
Figure 1. Phase Portrait of the Hyperchaotic Li System

106
Using the results of Section 2, we build a sliding controller for the global chaos control of the
hyperchaotic Li system (18).
First, we define the controlu as
( )
u f x Bv
= − + (21)
where B is carefully selected such that ( , )
A B is completely controllable. We take B as
1
1
.
1
1
B
 
 
 
=
 
 
 
(22)
The sliding mode variable is selected as
[ ] 1 2 3 4
4 1 1 3 4 3
s Cx x x x x x
= = − = + + − (23)
which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 5
k = and 0.1.
q =
From Eq. (12), we can obtain ( )
v t as
1 2 3 4
( ) 37.67 52.33 0.67 4.27 0.03sgn( )
v t x x x x s
= − − + − (24)
Thus, the required sliding mode controller is obtained as
( )
u f x Bv
= − + (25)
By Theorem 1, we obtain the following result.
Theorem 2. The hyperchaotic Li system (18) is globally asymptotically stabilized for all initial
conditions with the sliding mode controller u defined by (25).
3.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 8
10
h −
= is
used to solve the controlled hyperchaotic Li system (18) with the sliding mode controller u given
by (25) using MATLAB.
In the hyperchaotic case, the parameter values are given by
35, 3, 12, 7, 0.6
a b c d r
= = = = =
The sliding mode gains are chosen as
5
k = and 0.1.
q =

107
The initial values of the hyperchaotic Li system (18) are taken as
1 2 3 4
(0) 20, (0) 15, (0) 8, (0) 12
x x
x x
= = − = =
Figure 2 illustrates the chaos control of the hyperchaotic Li system (18).
Figure 2. Chaos Control of the Hyperchaotic Li System
4. CONCLUSIONS
In this paper, we have derived new results for the global chaos control of chaotic systems via
sliding control. We have also explored the analysis and global chaos control of the hyperchaotic
Li system (2005). Our global chaos control results have been established using Lyapunov stability
theory. Numerical simulations using MATLAB are also shown to illustrate the effectiveness of
the SMC-based control results derived for the hyperchaotic Li system.
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Author
Dr. V. Sundarapandian earned his Doctor of Science degree in Electrical and Systems
Engineering from Washington University, St. Louis, USA in May 1996. He is Professor
and Dean at the R D Centre at Vel Tech Dr. RR Dr. SR Technical University,
Chennai, Tamil Nadu, India. He has published over 290 papers in refereed international
journals. He has published over 180 papers in National and International Conferences.
He is the Editor-in-Chief of the AIRCC Journals - International Journal of
Instrumentation and Control Systems, International Journal of Control Systems and
Computer Modelling, International Journal of Information Technology, Control and
Automation, International Journal of Chaos, Control, Modelling and Simulation and International Journal
of Information Technology, Modeling and Computing. His research interests are Linear and Nonlinear
Control Systems, Chaos Theory and Control, Soft Computing, Optimal Control, Operations Research,
Mathematical Modelling and Scientific Computing. He has delivered several Keynote lectures on
nonlinear control systems, chaos theory, mathematical modelling and scientific computing using
MALTAB and SCILAB.

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