Synchronization of the Hyperchaotic Lu System Using Feedback Linearizatio Controllers

1. Mohamed Zribi Nejib Smaoui
Haitham Salim
Department of Electrical Engineering,
Kuwait University,
1

2.  Overview of Chaos
 What is synchronization?
 Motivation of the work
 Models of the master and the slave systems
 Feedback Linearization controller (three inputs)
 Feedback Linearization controller (two inputs)
 Secure communication using chaos
synchronization
 Conclusion
2

3.  Chaos Theory
 Describes the behavior of certain dynamical
systems that are highly sensitive to initial conditions
 Future dynamics are fully defined by their initial
conditions
 Even a slight difference in the initial conditions
between two identical chaotic systems results in a
big difference in the trajectories after some time
3

4.  Edward Lorenz’s discovery
 Weather prediction simulation in 1961
 He wanted to see a sequence of data again
 He started the simulation in the middle of its course
 a value like 0.506127 was printed as 0.506
 The result was completely different from the
weather calculated before
4

5.  Chaotic behavior observed in the laboratory:
 Electrical Circuits
 Lasers
 Oscillating chemical reactions
 Fluid dynamics
 Mechanical devices
 Chaotic behavior observed in nature:
 Molecular vibrations
 Weather and climate
 Solar System
5

6.  The Hypechaotic Lu System
( )
36, 3,
20
0.35 1.3
x a y x w
y x z cy
z x y bz
w x z rw
a b
c
r
= − +
= − +
= −
= +
= =
=
− < ≤
&
&
&
&
6

7. Synchronization
 The process of maintaining one operation in step
with another
 Started in the 17th
century with the discovery of
Huygens that two very weakly coupled pendulum
clocks become synchronized in phase
7

8. The problem of synchronizing two chaotic
systems is equivalent to a control problem.
8

9. Model of the Master System:
( )= − +
= − +
= −
= +
&
&
&
&
m m m m
m m m m
m m m m
m m m m
x a y x w
y x z cy
z x y bz
w x z rw
1
3
2
Model of the Slave System:
( )= − +
= − + +
= − +
= + +
&
&
&
&
s s s s
s s s s
s s s s
s s s s
x a y x w
y x z cy u
z x y bz u
w x z rw u
9

10. 10
Model of the Master System:
( )= − +
= − +
= −
= +
&
&
&
&
m m m m
m m m m
m m m m
m m m m
x a y x w
y x z cy
z x y bz
w x z rw
1
3
2
Model of the Slave System:
( )= − +
= − + +
= − +
= + +
&
&
&
&
s s s s
s s s s
s s s s
s s s s
x a y x w
y x z cy u
z x y bz u
w x z rw u

11. 1
2
3
4
1 2 1 4
2 3 1 1 1
3 2 1 3 3
4 2 1 3 2
Define,
The errors equations are such that:
( )
= −
= −
= −
= −
= − +
= − − + +
= + − +
= + + +
&
&
&
&
s m
s m
s m
s m
s m
s m
s m
e x x
e y y
e z z
e w w
e a e e e
e x e z e ce u
e x e y e be u
e x e z e re u
11

12. 1 1 1 2 1 2
2 1 3 1 4 2 4
3 1
when applied to the error system guarantee
the asymptotic convergence of the
The Feedback linearization control laws when
errors
m=3:
when
to
m
s m
m
u ae z e ce e
u e x e z e re e
u y e
γ
γ
= − + − −
= − − − − −
= −
zero as t .→ ∞
12

13. 13

14. 14

15. 15

16. 16

17. 1
2 3 3 4 2 1 3
1
3 1 2 1 2
1 1 1 2 1 2 4
2
The feedback linearization control laws with n=2:
(f ( )) if |x |
if |x |
(f ( )) if |x
s s m sx
s m s
s
e x e y e be
u
x e z e ce e
e ae ae e
u
α α ε
γ ε
α α
−
+ + + − >
= 
+ − − ≤
− + + − + +
= 2 3 3 4 2 1 3
3 1 4 2 4
when applied to
|
(f ( ))
if |x |
when the error system guarantee
the asymptotic convergence of the errors
to zero as t
s
a
s mx
s m s
e x e y e be
x e z e re e
ε
α α
γ ε
>

+ + + + −

− − − − ≤
.→ ∞
17

18. 1 3 1 2 2 1 4
3 1 4
2 2 3 1 2
1 2 1 4
2 1 3
1 2 3 4 1 2
where,
f ( ) ( )
+
f ( ) ( )
+( ) ( )
-b( )
, , , , , are positive scalars
s m
s m
s s s s m s
m m m m
s m
a x e z e ce a ae ae e
x e z e re
ay ax w e x e z e ce x
x z cy e ae ae e y
x e y e be
α α α α γ γ
= − − + − − +
+ +
= − + + − − +
− + + − +
+ −
is a small scalarε 18

19. 19

20. 20

21. 21

22. 22

23.  At the transmitter side, consider a hyperchaotic Lu
system which we will denote as the master system.
 Assume that the message to be sent is(t) is a binary
signal consisting of a sequence of zeros and ones.
 Add the information message is(t) to the first state of the
master system xm(t) to obtain the combined signal s(t).
Send the signals s(t), ym(t), zm(t) and wm(t) using a
public channel.
23

24.  The transmitted signals s(t), ym(t), zm(t) and
wm(t) will be corrupted with noise due to the
added noise of the public channel and
therefore they will be denoted by
24
( ), ( ), ( ), ( )m m ms t y t z t w t

25.  At the receiver side, we will use a hyperchaotic Lu system
which is referred to as the salve system
 We will then use the proposed control schemes to
synchronize the master and the slave systems. Note that
the errors between the states of the master and the slave
systems are now such that:
25
1
2
3
4
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
s
s m
s m
s m
e t x t s t
e t y t y t
e t z t z t
e t w t w t
= −
= −
= −
= −

26.  To recover the information message, we subtract the first state of the slave
system xs(t) from the received signal and then obtain the information
signal corrupted with some additive noise due to the public channel.
 The recovered noisy information signal can be filtered so that the exact
information signal is recovered. We will denote the filter's function by F(.)
and the received message can be recovered as follows:
where is(t) is the sent information signal, is the noisy received
signal and ir (t) is the actual information signal after passing through the
filter.
26
( )s t
( ) ( ( )) ( ( ) ( ))r r si t F i t F s t x t= = −
( )ri t

27.
 Note that every time the message signal's amplitude is zero,
it takes a small time for the controller to synchronize the
master and the slave systems. Therefore, the filter is
designed such that it will check the values of
every approximately 85% to 95% of each time period T in
order to make sure that the synchronization is obtained
whenever a signal is(t) of amplitude zero is sent. Otherwise,
the master and the slave systems are not synchronized and
the output of the filter is a signal whose amplitude is one.
27
( ) ( )ss t x t−

28. Noise
28
Hyperchaotic System
Hyperchaotic System

29. Noise
29
Hyperchaotic System
Hyperchaotic System

30. 30

31. 31

32. 32

33. 33

34. 34
 The synchronization of two hyperchaotic Lu systems is
investigated in this paper.
 Two different cases are studied. The first case is when the
number of inputs to the slave system is three. The second case
which is investigated is the more realistic case when the
number of inputs to the slave system is 2.

35. 35
 For each of the cases, a feedback linearization controller is
proposed. Both controllers ensure the asymptotic
convergence of the errors between the states of the master
and the slave hyperchaotic Lu systems to zero as time tends
to infinity.
 The simulation results clearly show that the proposed control
schemes are able to synchronize the master and the slave
hyperchaotic Lu systems when the two systems start from
different initial conditions.

36. 36
 The proposed control schemes are used for
secure communication purposes. The
transmitted message is a binary signal
consisting of a sequence of zeros and ones.
The simulation results indicate that the
proposed synchronization controllers are able
to recover the transmitted signal even in the
presence of zero mean Gaussian noise.