ADAPTIVE CONTROL AND SYNCHRONIZATION OF SPROTT-I SYSTEM WITH UNKNOWN PARAMETERS

International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI), Vol.1, No.1, August 2012
29
ADAPTIVE CONTROL AND SYNCHRONIZATION OF
SPROTT-I SYSTEM WITH UNKNOWN PARAMETERS
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 derives new results for the adaptive control and synchronization design of the Sprott-I chaotic
system (1994), when the system parameters are unknown. First, we build an adaptive controller to
stabilize the Sprott-I chaotic system to its unstable equilibrium at the origin. Then we build an adaptive
synchronizer to achieve global chaos synchronization of the identical Sprott-I chaotic systems with
unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the
Sprott-I chaotic system have been established using adaptive control theory and Lyapunov stability
theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control
and synchronization schemes derived in this paper for the Sprott-I chaotic system.
KEYWORDS
Adaptive Control, Chaos, Chaotic Systems, Synchronization, Sprott-I System.
1. INTRODUCTION
Chaotic systems are nonlinear dynamical systems which are extremely sensitive to changes in
initial conditions and also exhibit random-like behaviour in its deterministic motion.
Experimentally, chaos was first discovered by Lorenz ([1], 1963) while he was simulating
weather models. A chaotic system simpler than the Lorenz system was proposed by Rössler
([2], 1976). The theoretical equations of the Rössler system were later found to be useful in
modelling equilibrium in chemical reactions.
The control of chaotic systems is to design state feedback control laws that stabilizes the chaotic
systems around the unstable equilibrium points. Active control technique is used when the
system parameters are known and adaptive control technique is used when the system
parameters are unknown [3-4].
Synchronization of chaotic systems is a phenomenon that may occur when two or more chaotic
attractors are coupled or when a hyperchaotic attractor drives another hyperchaotic attractor. In
the last two decades, there has been significant interest in the literature on the synchronization
of chaotic and hyperchaotic systems [5-16].
In 1990, Pecora and Carroll [5] introduced a method to synchronize two identical chaotic
systems and showed that it was possible for some chaotic systems to be completely
synchronized. From then on, chaos synchronization has been widely explored in a variety of
fields including physical systems [6], chemical systems [7], ecological systems [8], secure
communications [9-10], etc.
The pioneering work by Pecora and Carroll (1990) has been followed by a variety of impressive
approaches in the literature such as the sampled-data feedback method [11], OGY method [12],

30
time-delay feedback method [13], backstepping method [14], active control method [15-20],
adaptive control method [21-25], sliding mode control method [26-28], etc.
This paper is organized as follows. In Section 2, we give a description of the Sprott-I chaotic
system (Sprott, [29], 1994). In Section 3, we derive results for the adaptive control of Sprott-I
chaotic system with unknown parameters. In Section 4, we derive results for the adaptive
synchronization of the identical Sprott-I chaotic systems with unknown parameters. Section 5
contains a summary of the main results derived in this paper.
2. SYSTEM DESCRIPTION
The Sprott-I system ([29], 1994) is described by the 3D dynamics
1 2
2 1 3
2
3 1 2 3
x ax
x x bx
x cx x x
= −
= +
= + −
&
&
&
(1)
where 1 2 3
, ,
x x x are the state variables of the system and , ,
a b c are constant, positive parameters
of the system.
The system (1) is chaotic when the parameter values are taken as
0.2, 1
a b
= = and 1
c = (2)
Figure 1 describes the strange attractor of the Sprott-I chaotic system (1).
Figure 1. The Strange Attractor of the Sprott-I Chaotic System

31
When the parameter values are taken as in (2) for the Sprott-I chaotic system (1), the system
linearization matrix at the equilibrium point 0 (0,0,0)
E = is given by
0 0.2 0
1 0 1
1 0 1
A
−
 
 
=  
 
−
 
which has the eigenvalues
1 2
1.1345, 0.0672 0.5900 i
λ λ
= − = + and 3 0.0672 0.5900 i
λ = −
Since 2
λ and 3
λ are unstable eigenvalues of ,
A it follows from Lyapunov stability theory [30]
that the Sprott-I system (1) is unstable at the equilibrium point 0 (0,0,0).
E =
3. ADAPTIVE CONTROL OF THE SPROTT-I CHAOTIC SYSTEM
3.1 Theoretical Results
In this section, we design adaptive control law for globally stabilizing the Sprott-I system
(1994), when the parameter values are unknown.
Thus, we consider the controlled Sprott-I system, which is described by the 3D dynamics
1 2 1
2 1 3 2
2
3 1 2 3 3
x ax u
x x bx u
x cx x x u
= − +
= + +
= + − +
&
&
&
(3)
where 1 2
,
u u and 3
u are feedback controllers to be designed using the states 1 2 3
, ,
x x x and
estimates ˆ
ˆ ˆ
, ,
a b c of the unknown parameters , ,
a b c of the system.
In order to ensure that the controlled system (3) globally converges to the origin asymptotically,
we consider the following adaptive control functions
1 2 1 1
2 1 3 2 2
2
3 1 2 3 3 3
ˆ
ˆ
ˆ
u ax k x
u x bx k x
u cx x x k x
= −
= − − −
= − − + −
(4)
where ˆ
ˆ,
a b and ĉ are estimates of the parameters ,
a b and ,
c respectively, and ,( 1,2,3)
i
k i =
are positive constants.

32
Substituting the control law (4) into the controlled Sprott-I dynamics (3), we obtain
1 2 1 1
2 3 2 2
3 1 3 3
ˆ
( )
ˆ
( )
ˆ
( )
x a a x k x
x b b x k x
x c c x k x
= − − −
= − −
= − − −
&
&
&
(5)
Let us now define the parameter errors as
ˆ
ˆ,
a b
e a a e b b
= − = − and ˆ
c
e c c
= − (6)
Using (6), the closed-loop dynamics (5) can be written compactly as
1 2 1 1
2 3 2 2
3 1 3 3
a
b
c
x e x k x
x e x k x
x e x k x
= − −
= −
= − −
&
&
&
(7)
For the derivation of the update law for adjusting the parameter estimates ˆ
ˆ,
a b and ˆ,
c the
Lyapunov approach is used.
Consider the quadratic Lyapunov function
( )
2 2 2 2 2 2
1 2 3 1 2 3
1
( , , , , , )
2
a b c a b c
V x x x e e e x x x e e e
= + + + + + (8)
which is a positive definite function on 6
.
R
Note also that
ˆ
ˆ ˆ
, ,
a b c
e a e b e c
= − = − = −
&
& &
& & & (9)
Differentiating V along the trajectories of (7) and using (9), we obtain
( ) ( ) ( )
2 2 2
1 1 2 2 3 3 1 2 2 3 1 3
ˆ
ˆ ˆ
a b c
V k x k x k x e x x a e x x b e x x c
= − − − + − − + − + −
&
& &
& (10)
In view of Eq. (10), the estimated parameters are updated by the following law:
1 2 4
2 3 5
1 3 6
ˆ
ˆ
ˆ
a
b
c
a x x k e
b x x k e
c x x k e
= − +
= +
= +
&
&
&
(11)
where 4 5
,
k k and 6
k are positive constants.
Next, we prove the following result.

33
Theorem 1. The controlled Sprott-I system (1) with unknown parameters is globally and
exponentially stabilized for all initial conditions 3
(0)
x R
∈ by the adaptive control law (4),
where the update law for the parameters is given by (11) and , ( 1, ,6)
i
k i = K are positive
constants.
Proof. Substituting (11) into (10), we get
2 2 2 2 2 2
1 1 2 2 3 3 4 5 6
a b c
V k x k x k x k e k e k e
= − − − − − −
& (12)
which is a negative definite function on 6
.
R
Thus, by Lyapunov stability theory [30], it is immediate that the controlled Sprott-I system (7)
is globally exponentially stable and also that the parameter estimation errors , ,
a b c
e e e
exponentially converge to zero with time.
This completes the proof.
3.2 Numerical Results
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the chaotic
system (3) with the adaptive control law (4) and the parameter update law (11).
The parameters of the Sprott-I system (3) are selected as
0.2, 1
a b
= = and 1.
c =
For the adaptive and update laws, we take
5, ( 1,2, ,6).
i
k i
= = K
Suppose that the initial values of the estimated parameters are
ˆ
ˆ ˆ
(0) 4, (0) 12, (0) 9
a b c
= = =
The initial state of the controlled Sprott-I system (3) is taken as
1 2 3
(0) 9, (0) 14, (0) 10
x x x
= = − =
When the adaptive control law (4) and the parameter update law (11) are used, the controlled
modified Sprott-I system converges to the equilibrium 0 (0,0,0)
E = exponentially as shown in
Figure 2.
The time-history of the parameter estimates is shown in Figure 3.
The time-history of the parameter estimation errors is shown in Figure 4.

34
Figure 2. Time Responses of the Controlled Sprott-I System
Figure 3. Time-History of the Parameter Estimates ˆ
ˆ ˆ
( ), ( ), ( )
a t b t c t

35
Figure 4. Time-History of the Parameter Estimates , ,
a b c
e e e
4. ADAPTIVE SYNCHRONIZATION OF IDENTICAL SPROTT-I CHAOTIC
SYSTEMS
4.1 Theoretical Results
In this section, we discuss the adaptive synchronization of identical Sprott-I chaotic systems
(1994) with unknown parameters.
As the master system, we consider the Sprott-I dynamics described by
1 2
2 1 3
2
3 1 2 3
x ax
x x bx
x cx x x
= −
= +
= + −

(13)
where , ( 1,2,3)
i
x i = are the state variables and , ,
a b c are unknown system parameters.
As the slave system, we consider the controlled Sprott-I system described by
1 2 1
2 1 3 2
2
3 1 2 3 3
y ay u
y y by u
y cy y y u
= − +
= + +
= + − +

(14)
where , ( 1,2,3)
i
y i = are the state variables and , ( 1,2,3)
i
u i = are adaptive controllers to be
designed.

36
The synchronization error is defined by
1 1 1
2 2 2
3 3 3
e y x
e y x
e y x
= −
= −
= −
(15)
Then the error dynamics is obtained as
1 2 1
2 1 3 2
2 2
3 1 3 2 2 3
e ae u
e e be u
e ce e y x u
= − +
= + +
= − + − +

(16)
Let us now define the adaptive control functions 1 2 3
( ), ( ), ( )
u t u t u t as
1 2 1 1
2 1 3 2 2
2 2
3 1 3 2 2 3 3
ˆ
ˆ
ˆ
u ae k e
u e be k e
u ce e y x k e
= − −
= − − −
= − + − + −
(17)
where ˆ
ˆ,
a b and ĉ are estimates of the parameters ,
a b and ,
c respectively, and ,( 1,2,3)
i
k i = are
positive constants.
Substituting the control law (17) into (16), we obtain the error dynamics as
1 2 1 1
2 3 2 2
3 1 3 3
ˆ
( )
ˆ
( )
ˆ
( )
e a a e k e
e b b e k e
e c c e k e
= − − −
= − −
= − −

(18)
Let us now define the parameter errors as
ˆ
ˆ ˆ
, ,
a b c
e a a e b b e c c
= − = − = − (19)
Substituting (19) into (18), the error dynamics simplifies to
1 2 1 1
2 3 2 2
3 1 3 3
a
b
c
e e e k e
e e e k e
e e e k e
= − −
= −
= −

(20)
Consider the quadratic Lyapunov function
( )
2 2 2 2 2 2
1 2 3 1 2 3
1
( , , , , , )
2
a b c a b c
V e e e e e e e e e e e e
= + + + + + (21)
which is a positive definite function on 6
.
R

37
Note also that
ˆ
ˆ ˆ
, ,
a b c
e a e b e c
= − = − = −

(22)
Differentiating V along the trajectories of (20) and using (22), we obtain
( ) ( ) ( )
2 2 2
1 1 2 2 3 3 1 2 2 3 1 3
ˆ
ˆ ˆ
a b c
V k e k e k e e e e a e e e b e e e c
= − − − + − − + − + −

(23)
In view of Eq. (23), the estimated parameters are updated by the following law:
1 2 4
2 3 5
1 3 6
ˆ
ˆ
ˆ
a
b
c
a e e k e
b e e k e
c e e k e
= − +
= +
= +

(24)
where 4 5
,
k k and 6
k are positive constants.
Theorem 2. The identical Sprott-I systems (13) and (14) with unknown parameters are globally
and exponentially synchronized for all initial conditions by the adaptive control law (17), where
the update law for parameters is given by (24) and ,( 1, ,6)
i
k i = K are positive constants.
Proof. Substituting (24) into (23), we get
2 2 2 2 2 2
1 1 2 2 3 3 4 5 6
a b c
V k e k e k e k e k e k e
= − − − − − −
(25)
From (25), we find that V
is a negative definite function on 6
.
R
Thus, by Lyapunov stability theory [30], it is immediate that the synchronization error and the
parameter error decay to zero exponentially with time for all initial conditions.
4.2 Numerical Results
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the two
systems of differential equations (13) and (14) with the adaptive control law (17) and the
parameter update law (24).
We take the parameter values as in the chaotic case, viz.
0.2, 1, 1
a b c
= = =
We take the positive constants , ( 1, ,8)
i
k i = K as
5
i
k = for 1,2, ,6.
i = K
Suppose that the initial values of the estimated parameters are
ˆ
ˆ ˆ
(0) 6, (0) 12, (0) 8
a b c
= = =

38
We take the initial values of the master system (13) as
1 2 3
(0) 12, (0) 15, (0) 4
x x x
= = − =
We take the initial values of the slave system (14) as
1 2 3
(0) 23, (0) 10, (0) 7
y y y
= = = −
Figure 5 shows the adaptive chaos synchronization of the identical Sprott-I systems.
Figure 6 shows the time-history of the parameter estimates ˆ
ˆ ˆ
( ), ( ), ( ).
a t b t c t From this figure, it
is clear that the parameter estimates converge to the chosen values of , ,
a b c respectively.
Figure 7 shows the time-history of the parameter estimation errors , , .
a b c
e e e
Figure 5. Adaptive Synchronization of the Sprott-I Systems

39
Figure 6. Time-History of the Parameter Estimates ˆ
ˆ ˆ
( ), ( ), ( )
a t b t c t
Figure 7. Time-History of the Parameter Estimation Error , ,
a b c
e e e

40
5. CONCLUSIONS
In this paper, we applied adaptive control theory for the stabilization and synchronization of the
Sprott-I system (1994) with unknown system parameters. First, we designed adaptive control
laws to stabilize the Sprott-I system to its unstable equilibrium point at the origin based on the
adaptive control theory and Lyapunov stability theory. Then we derived adaptive
synchronization scheme and update law for the estimation of system parameters for the identical
Sprott-I systems with unknown parameters. Our synchronization schemes were established
using Lyapunov stability theory. Since the Lyapunov exponents are not required for these
calculations, the proposed adaptive control method is very effective and convenient to achieve
chaos control and synchronization of the Sprott-I chaotic system. Numerical simulations are
shown to demonstrate the effectiveness of the proposed adaptive stabilization and
synchronization schemes.
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41
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Author
Dr. V. Sundarapandian obtained his Doctor of Science degree in Electrical and
Systems Engineering from Washington University, St. Louis, USA in May 1996.
He is a Professor at the R D Centre at Vel Tech Dr. RR Dr. SR Technical
University, Chennai, Tamil Nadu, India. He has published over 260 refereed
international publications. He has published over 170 papers in National and
International Conferences. He is the Editor-in-Chief of International Journal of
Instrumentation and Control Systems, International Journal of Control Systems
and Computer Modelling, and International Journal of Information Technology,
Control and Automation. 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 Key Note Lectures on
Control Systems, Chaos Theory and Control, Scientific Computing using MATLAB/SCILAB, etc.

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