An Exponential Observer Design for a Class of Chaotic Systems with Exponential Nonlinearity

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 5 Issue 1, November-December 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
@ IJTSRD | Unique Paper ID – IJTSRD38233 | Volume – 5 | Issue – 1 | November-December 2020 Page 1550
An Exponential Observer Design for a Class of
Chaotic Systems with Exponential Nonlinearity
Yeong-Jeu Sun
Professor, Department of Electrical Engineering, I-Shou University, Kaohsiung, Taiwan
ABSTRACT
In this paper, a class of generalized chaotic systems with exponential
nonlinearity is studied and the state observation problem of such systems is
explored. Using differential inequality with time domain analysis, a practical
state observer for such generalized chaotic systems is constructed to ensure
the global exponential stability of the resulting error system. Besides, the
guaranteed exponential decay rate can be correctly estimated.Finally,several
numerical simulations are giventodemonstratethevalidity,effectiveness,and
correctness of the obtained result.
KEYWORDS: Generalized chaotic systems with exponential nonlinearity, state
observer, exponential decay rate, Ten-ring chaotic system
How to cite this paper: Yeong-Jeu Sun
"An Exponential Observer Design for a
Class of Chaotic Systems withExponential
Nonlinearity"
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of Trend in Scientific
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ISSN: 2456-6470,
Volume-5 | Issue-1,
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1. INTRODUCTION
In the past few years, various types of chaotic systems have
been widely studied; see, for example, [1-4] and the
references therein. The investigation of chaotic systems not
only allows us to understand the chaotic characteristics, but
we can use the research results in various chaos
applications; such as image processing and chaotic secure
communication. Because the state variables of a chaotic
system are highly sensitive to initial values and the output
signal is unpredictable, the state variables of a chaotic
system are always more difficult to estimate than those of a
non-chaotic system.
Due to the excessive number of state variables or the lack of
measurement equipment, the state variablesofreal physical
systems are often difficult to estimate; see, for example, [5-
10]. For chaotic systems, designing a suitable state observer
has always been one of the goals pursued by researchers
engaged in nonlinear systems.
In this paper, the state observer for a class of generalized
chaotic systems with exponential nonlinearity is explored
and studied. Based on the differential inequality and time-
domain approach, a state observer of such generalized
chaotic systems will be developed to guarantee the global
exponential stability of the resulting error system. In
addition, the guaranteed exponential decay rate can be
accurately estimated. Finally, some numerical exampleswill
be provided to illustrate the effectiveness of the obtained
results.
2. PROBLEM FORMULATION AND MAIN RESULTS
In this paper, we consider the following generalized chaotic
system with exponential nonlinearity
( ) ( ) ( )
t
x
a
t
x
a
t
x 2
2
1
1
1 +
=
& (1a)
( ) ( ) ( ) ( )
( )
t
x
t
x
t
x
f
t
x 3
2
1
1
2 ,
,
=
& (1b)
( ) ( ) ( )
,
2
1
3
3
3
t
x
e
t
x
a
t
x +
−
=
& (1c)
( ) ( ) ( ),
2 2
1 t
x
t
x
t
y +
= (1d)
where ( ) ( ) ( ) ( )
[ ] 3
3
2
1
: ℜ
∈
=
T
t
x
t
x
t
x
t
x is the state vector,
( ) ℜ
∈
t
y is the system output, 1
f is a smooth function, and
3
2
1 ,
, a
a
a are the parameters of the system (1), with 0
3 >
a
and 1
2
2 a
a > . In addition, we assume that the signal of ( )
t
x1
is bounded.
Remark 1: It is emphasizedthatthefamousTen-ringchaotic
system [4] is a special case of the system (1).
It is a well-known fact that since states are not always
available for direct measurement, states must be estimated.
The objective of this paper is to search a suitable state
observer for the nonlinear system (1) such that the global
exponential stability of the resulting error systems can be
guaranteed. In whatfollows, x denotestheEuclidean norm
of the column vector x and a denotestheabsolutevalueof a
real number a.
IJTSRD38233

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Before presenting the main result, let us introduce a
definition which will be used in the main theorem.
Definition 1. The system (1) is exponentially state
reconstructible if there exist a state estimator
( ) ( ) ( )
( )
t
y
t
z
h
t
z
E ,
=
& and positivenumbers k and α suchthat
( ) ( ) ( ) ( ) 0
,
exp
: ≥
∀
−
≤
−
= t
t
k
t
z
t
x
t
e α ,
where ( )
t
z expresses the reconstructed state of the system
(1). In this case, the positive number α is called the
exponential decay rate.
Now we present the main result.
Theorem 1. The system (1) is exponentially state
reconstructible. Besides, a suitable stateobserverisgiven by
( ) ( ) ( ) ( ),
2 2
1
2
1
1 t
y
a
t
z
a
a
t
z +
−
=
& (2a)
( ) ( ) ( ),
2 1
2 t
z
t
y
t
z −
= (2b)
( ) ( ) ( )
,
2
1
3
3
3
t
z
e
t
z
a
t
z +
−
=
& (2c)
with the guaranteed exponential decay rate






−
=
2
,
2
min
: 3
1
2
a
a
a
α .
Proof. Define ( )
t
x
M 1
≥ , from (1), (2) with
( ) ( ) ( ) { },
3
,
2
,
1
,
: ∈
∀
−
= i
t
z
t
x
t
e i
i
i (3)
it can be readily obtained that
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
t
y
a
t
z
a
a
t
x
a
t
x
a
t
z
t
x
t
e
2
1
2
1
2
2
1
1
1
1
1
2 −
−
−
+
=
−
= &
&
&
( ) ( ) ( )
[ ]
( ) ( ) ( )
( ) ( ) ( )
[ ]
t
z
t
x
a
a
t
y
a
t
z
a
a
t
x
t
y
a
t
x
a
1
1
1
2
2
1
2
1
1
2
1
1
2
2
2
−
−
−
=
−
−
−
−
+
=
( ) ( ) .
0
,
2 1
1
2 ≥
∀
−
−
= t
t
e
a
a
It results that
( ) ( ) ( )
[ ] .
0
,
2
exp
0 1
2
1
1 ≥
∀
−
−
= t
t
a
a
e
t
e (4)
Moreover, form (1)-(4), we have
( ) ( ) ( )
( ) ( )
[ ] ( ) ( )
[ ]
t
z
t
y
t
x
t
y
t
z
t
x
t
e
1
1
2
2
2
2
2 −
−
−
=
−
=
( ) ( )
[ ]
t
z
t
x 1
1
2 −
−
=
( )
t
e1
2
−
=
( ) ( )
[ ] .
0
,
2
exp
0
2 1
2
1 ≥
∀
−
−
−
= t
t
a
a
e (5)
Define ( )
t
x
M 1
≥ and form (1)-(5), it yields
( ) ( ) ( )
t
z
t
x
t
e 3
3
3
&
&
& −
=
( ) ( )
( ) ( )
t
z
t
x
e
t
z
a
e
t
x
a
2
1
2
1
3
3
3
3 −
+
+
−
=
( ) ( ) ( )
t
z
t
x
e
e
t
e
a
2
1
2
1
3
3 −
+
−
=
( ) ( ) ( ) ( ) ( ) ( )
[ ]
t
z
t
x
e
e
t
e
t
e
a
t
e
t
e
2
1
2
1
3
2
3
3
3
3 2
2 −
+
−
=
⇒ &
This implies that
( )
[ ] ( ) ( ) ( ) ( )
[ ]
t
z
t
x
e
e
t
e
t
e
a
dt
t
e
d 2
1
2
1
3
2
3
3
2
3
2
2 −
+
−
=
( ) ( )
( ) ( )
[ ]2
3
2
3
3
2
3
3
2
1
2
1
1
2
t
z
t
x
e
e
a
t
e
a
t
e
a
−
+
+
−
≤
( ) ( ) ( )
[ ]2
3
2
3
3
2
1
2
1
1 t
z
t
x
e
e
a
t
e
a −
+
−
=
( )
[ ] ( ) ( )
[ ]
t
z
t
x
t
a
t
a
e
e
a
e
dt
t
e
e
d 2
1
2
1
3
3
3
2
3
−
≤
⇒
( ) ( )
( ) ( )
[ ]
( ) ( )
[ ]
( )
( ) ( ) ( )
∫
∫
∫
−
−
+
⋅
≤
−
≤
−
≤
−
⇒
t
t
a
a
M
t
M
t
t
z
t
x
t
a
t
a
dt
e
e
e
M
a
e
dt
t
z
t
x
e
a
dt
e
e
a
e
e
t
e
e
0
2
1
1
3
0
2
1
2
1
3
0 3
2
3
2
3
1
2
2
2
2
1
2
1
3
3
0
0
2
1
0
( )
( ) ( )
( )
[ ]
t
a
a
M
e
a
a
e
e
M
a
e
1
2
2
2
1
2
1
1
3
1
2
0
0
2
−
−
−
⋅
−
⋅
+
⋅
=
( ) ( )
( )
( ) ( )
( )
[ ]
( )
( ) ( )
( )] t
a
M
t
a
a
a
t
a
M
t
a
e
e
a
a
e
e
M
a
e
e
e
a
a
e
e
M
a
e
e
e
t
e
3
2
3
1
2
3
2
3
0
2
0
0
2
2
0
0
2
0
2
3
1
2
1
1
3
2
1
2
1
1
3
2
3
2
3
−
+
−
−
−
−
+




−
⋅
+
⋅
≤
−
⋅
−
⋅
+
⋅
+
≤
⇒
( ) ( )
( ) ( )




−
⋅
+
⋅
≤
⇒
1
2
1
1
3
3
2
0
0
2
2
a
a
e
e
M
a
e
t
e
M
( )] .
0
,
0 2
2
1
2
3
3
≥
∀
⋅
+
−
t
e
e
t
a
(6)
Consequently, by (4)-(6), we conclude that
( ) ( ) ( ) ( ) ( )
,
0
,
exp
2
3
2
2
2
1
≥
∀
⋅
≤
+
+
=
t
t
k
t
e
t
e
t
e
t
e α
with ( ) ( ) ( )
( ) ( )
1
2
1
1
3
2
3
2
1
2
0
0
2
0
0
5
2
a
a
e
e
M
a
e
e
e
k
M
−
⋅
+
⋅
+
= . This
completes the proof. □
3. NUMERICAL SIMULATIONS
Example 1: Consider the following Ten-ring chaotic system
[4]:
( ) ( ) ( ),
20
20 2
1
1 t
x
t
x
t
x +
=
& (7a)
( ) ( ) ( ) ( ) ( ),
28
5
.
0 3
1
2
1
2 t
x
t
x
t
x
t
x
t
x −
−
=
& (7b)
( ) ( ) ( )
,
2
2
1
3
3
t
x
e
t
x
t
x +
−
=
& (7c)
( ) ( ) ( ),
2 2
1 t
x
t
x
t
y +
= (7d)

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Comparison of (7) with (1), one has
2
,
20
,
20 3
2
1 =
=
= a
a
a , and
( ) ( ) ( )
( )
( ) ( ) ( ) ( ).
28
5
.
0
,
,
3
1
2
1
3
2
1
1
t
x
t
x
t
x
t
x
t
x
t
x
t
x
f
−
−
=
By Theorem 1, we conclude that the system (7) is
exponentially state reconstructible by the state observer
( ) ( ) ( ),
20
20 1
1 t
y
t
z
t
z +
−
=
& (8a)
( ) ( ) ( ),
2 1
2 t
z
t
y
t
z −
= (8b)
( ) ( ) ( )
,
2
2
1
3
3
t
z
e
t
z
t
z +
−
=
& (8c)
with the guaranteed exponential decay rate 1
:=
α . The
typical state trajectories of the systems (7) and (8) are
depicted in Figure 1 and Figure 2, respectively. Besides, the
time response of error states between the systems (7) and
(8) is shown in Figure 3.
4. CONCLUSION
In this paper, a class of generalized chaotic system with
exponential nonlinearity has been studied and the state
observation problem of such systems has been explored.
Using differential inequality with time domain analysis, a
practical state observer forsuchgeneralizedchaoticsystems
has been built to ensure the global exponential stability of
the resulting error system. Moreover, the guaranteed
exponential decay rate can be correctly calculated. Finally,
several numerical simulations have been offered to
demonstrate the validity and effectiveness of the obtained
result.
ACKNOWLEDGEMENT
The author thanks the Ministry of Science and Technology of
Republic of China forsupportingthisworkundergrantMOST
109-2221-E-214-014. Besides, the authorisgrateful toChair
Professor Jer-Guang Hsieh for the useful comments.
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0 2 4 6 8 10 12 14 16 18 20
-100
0
100
200
300
400
500
600
t (sec)
x1(t);
x2(t);
x3(t)
x1: the Blue Curve
x2: the Green Curve
x3: the Red Curve
Figure 1: Typical state trajectories of the system (7).
0 2 4 6 8 10 12 14 16 18 20
-100
0
100
200
300
400
500
600
t (sec)
z1(t);
z2(t);
z3(t)
z1: the Blue Curve
z2: the Green Curve
z3: the Red Curve
Figure 2: Typical state trajectories of the system (8).
0 1 2 3 4 5 6 7 8 9 10
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
t (sec)
e1(t);
e2(t);
e3(t)
e1: the Blue Curve
e2: the Green Curve
e3: the Red Curve
Figure 3: The time response of error states.

An Exponential Observer Design for a Class of Chaotic Systems with Exponential Nonlinearity

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