![International Journal of Trend in
International Open Access Journal
ISSN No: 2456
@ IJTSRD | Available Online @ www.ijtsrd.com
Exponential State Observer Design
Uncertain Chaotic
Professor, Department of Electrical Engineering,
ABSTRACT
In this paper, a class of uncertain chaotic and non
chaotic systems is firstly introduced and the state
observation problem of such systems is explored.
Based on the time-domain approach with integra
differential equalities, an exponential state observer
for a class of uncertain nonlinear systems is
established to guarantee the global exponential
stability of the resulting error system. Besides, the
guaranteed exponential convergence rate can be
calculated correctly. Finally, numerical simulations
are presented to exhibit the feasibility and
effectiveness of the obtained results.
Key Words: Chaotic system, state observer design,
uncertain systems, exponential convergence rate
1. INTRODUCTION
In recent years, various chaotic systems
widely explored by scholars; see, for example, [1
and the references therein. Frequently, chaotic signals
are often the main cause of system instability and
violent oscillations. Moreover, chaos often occurs in
various engineering systems and applied physics
instance, ecological systems, secure communicati
and system identification. Based on practical
considerations, not all state variables of most systems
can be measured. Furthermore, the design of the state
estimator is an important work when the sensor fails.
Undoubtedly, the state observer design of
with both chaos and uncertainties tends to be more
difficult than those without chaos and uncertainties.
For the foregoing reasons, the observer design of
uncertain chaotic systems is really significant
essential.
In this paper, the observability problem for
uncertain nonlinear chaotic or non-chaotic
investigated. By using the time-domain approach with
International Journal of Trend in Scientific Research and Development (IJTSRD)
International Open Access Journal | www.ijtsrd.com
ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov
www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018
Exponential State Observer Design for a Class
Uncertain Chaotic and Non-Chaotic Systems
Yeong-Jeu Sun
f Electrical Engineering, I-Shou University, Kaohsiung, Taiwan
In this paper, a class of uncertain chaotic and non-
chaotic systems is firstly introduced and the state
observation problem of such systems is explored.
domain approach with integral and
differential equalities, an exponential state observer
for a class of uncertain nonlinear systems is
established to guarantee the global exponential
stability of the resulting error system. Besides, the
guaranteed exponential convergence rate can be
calculated correctly. Finally, numerical simulations
are presented to exhibit the feasibility and
Chaotic system, state observer design,
uncertain systems, exponential convergence rate
chaotic systems have been
; see, for example, [1-8]
Frequently, chaotic signals
are often the main cause of system instability and
Moreover, chaos often occurs in
various engineering systems and applied physics; for
instance, ecological systems, secure communication,
Based on practical
considerations, not all state variables of most systems
Furthermore, the design of the state
estimator is an important work when the sensor fails.
design of systems
with both chaos and uncertainties tends to be more
without chaos and uncertainties.
the foregoing reasons, the observer design of
significant and
lity problem for a class of
chaotic systems is
domain approach with
integral and differential equalities
observer for a class of uncertain
will be provided to ensure the global exponential
stability of the resulting error system. In
guaranteed exponential convergence rate can be
precisely calculated. Finally, some numerical
simulations will be given to demonstrate the
effectiveness of the main result.
This paper is organized as follows. The problem
formulation and main results
2. Several numerical simulations are
3 to illustrate the main result. Finally, conclusion
remarks are drawn in Section
paper, n
ℜ denotes the n-dimensional
xxx T
⋅=: denotes the Euclidean norm of the
vector x, and a denotes the
number a.
2. PROBLEM FORMULATION AND MAI
RESULTS
In this paper, we explore the following
nonlinear systems:
( ) ( ) ( ) ( )( ),,, 32111 txtxtxftx ∆=&
( ) ( ) ( ) ( )( ),, 31222 txtxftaxtx +=&
( ) ( )( ) ( )( ),14333 txftxftx +=&
( ) ( ) ,0,3 ≥∀= ttbxty
( ) ( ) ( )[ ] [T
xxxxxx 2010321 000 =
where ( ) ( ) ( ) ( )[ ]321: ∈=
T
txtxtxtx
( ) ℜ∈ty is the system output,
initial value, 1f∆ is uncertain function,
indicate the parameters of the system
0≠b . Besides, in order to ensure the existence and
uniqueness of the solution, we assume that
Research and Development (IJTSRD)
www.ijtsrd.com
1 | Nov – Dec 2018
Dec 2018 Page: 1158
Class of
Chaotic Systems
Shou University, Kaohsiung, Taiwan
equalities, a new state
uncertain nonlinear systems
ensure the global exponential
stability of the resulting error system. In addition, the
guaranteed exponential convergence rate can be
precisely calculated. Finally, some numerical
simulations will be given to demonstrate the
ult.
This paper is organized as follows. The problem
are presented in Section
simulations are given in Section
to illustrate the main result. Finally, conclusion
remarks are drawn in Section 4. Throughout this
dimensional real space,
the Euclidean norm of the column
he absolute value of a real
ROBLEM FORMULATION AND MAIN
the following uncertain
(1a)
(1b)
(1c)
(1d)
]T
x30 , (1e)
3
ℜ∈ is the state vector,
is the system output, [ ]T
xxx 302010 is the
is uncertain function, and ℜ∈ba,
parameters of the systems, with 0<a and
n order to ensure the existence and
uniqueness of the solution, we assume that all the](https://image.slidesharecdn.com/159exponentialstateobserverdesignforaclassofuncertainchaoticandnon-chaoticsystems-190405092640/85/Exponential-State-Observer-Design-for-a-Class-of-Uncertain-Chaotic-and-Non-Chaotic-Systems-1-320.jpg)
![International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456
@ IJTSRD | Available Online @ www.ijtsrd.com
functions of 1f∆ and () { }4,3,2, ∈∀⋅ ifi are smooth
the inverse function of 4f exists. The chaotic sprott E
system is a special case of systems (1) with
2
12 xf = , 13 =f , 14 4xf −= , and 1−=a . It is a well
fact that since states are not always available for direct
measurement, particularly in the event of sensor
failures, states must be estimated. The
paper is to search a novel state observer
uncertain nonlinear systems (1) such that
exponential stability of the resulting error systems can
be guaranteed.
Before presenting the main result, the state
reconstructibility is provided as follows.
Definition 1
The uncertain nonlinear systems (1) are
state reconstructible if there exist a state observer
( ) 0,, =yzzf & and positive numbers κ and
( ) ( ) ( ) ( ) 0,exp: ≥∀−≤−= tttztxte ακ ,
where ( )tz represents the reconstructed state of
systems (1). In this case, the positive number
called the exponential convergence rate.
Now, we are in a position to present the main results
for the exponential state observer of uncertain
(1).
Theorem 1.
The uncertain systems (1) are exponentially state
reconstructible. Moreover, a suitable state observer is
given by
( ) ( ) ( ) ,
11
3
1
41
−= −
ty
b
fty
b
ftz & (2a)
( ) ( ) ( ) ( )( ),, 31222 tztzftaztz +=& (2b)
( ) ( ) 0,
1
3 ≥∀= tty
b
tz . (2c)
In this case, the guaranteed exponentia
rate is given by a−=:α .
Proof. For brevity, let us define the observer error
( ) ( ) ( ) { }3,2,1,: ∈∀−= itztxte iii and 0≥t . (3)
From (1d) and (2c), one has
( ) ( ) ( )
( ) ( )ty
b
ty
b
tztxte
11
333
−=
−=
.0,0 ≥∀= t (4)
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456
www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018
are smooth and
chaotic sprott E
(1) with 321 xxf =∆ ,
. It is a well-known
fact that since states are not always available for direct
vent of sensor
he aim of this
state observer for the
(1) such that the global
the resulting error systems can
nting the main result, the state
reconstructibility is provided as follows.
are exponentially
state reconstructible if there exist a state observer
and α such that
the reconstructed state of
(1). In this case, the positive number α is
called the exponential convergence rate.
a position to present the main results
uncertain systems
exponentially state
, a suitable state observer is
the guaranteed exponential convergence
observer error
From (1c), it can be readily obtained that
( )( ) ( ) ( )( )txftxtxf 33314 −= & .
It results that
( ) ( ) ( )( )( )
( ) ( )
−=
−=
−
−
ty
b
fty
b
f
txftxftx
11
3
1
4
333
1
41
&
&
Thus, one has
( ) ( ) ( )
( ) ( )
( ) ( )
−−
−=
−=
−
−
ty
b
fty
b
f
ty
b
fty
b
f
tztxte
11
11
3
1
4
3
1
4
111
&
&
,0,0 ≥∀= t
in view of (5) and (2a). In addition
(4), and (6), it is easy to see that
( ) ( ) ( )
( ) ( ) ( )( )[ ]
( ) ( ) ( )( )[ ]
( ) ( ) ( )( )[ ]
( ) ( ) ( )( )[ ]
( ) ( )[ ]tztxa
txtxftaz
txtxftax
tztzftaz
txtxftax
tztxte
22
3122
3122
3122
3122
222
,
,
,
,
−=
+−
+=
+−
+=
−= &&&
( ) .0,2 ≥∀= ttae
It follows that
( ) ( ) .0,022 ≥∀=− ttaete&
Multiplying with ( )at−exp yields
( ) ( ) ( ) ( ) 0expexp 22 =−⋅−−⋅ attaeatte&
Hence, it can be readily obtained tha
( ) ( )[ ] .0,0
exp2
≥∀=
−⋅
t
dt
atted
Integrating the bounds from 0
( ) ( )[ ] ,00
exp
00
2
==
−⋅
∫∫ dtdt
dt
atted
tt
This implies that
( ) ( ) ( ) 0,exp022 ≥∀⋅= tatete .
Thus, from (4), (6), and (7), we have
( ) ( ) ( ) ( )
( ) ( ) .0,exp02
2
3
2
2
2
1
≥∀⋅=
++=
tate
tetetete
Consquently, we conclude that t
suitable state observer with the guaranted exponential
convergence rate a−=α . This completes the proof.
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Dec 2018 Page: 1159
From (1c), it can be readily obtained that
(5)
(6)
In addition, from (1b), (1c),
is easy to see that
yields
.0,0 ≥∀ t
Hence, it can be readily obtained that
and t , it results
0≥∀ t .
(7)
, we have
conclude that the system (2) is a
the guaranted exponential
. This completes the proof. □](https://image.slidesharecdn.com/159exponentialstateobserverdesignforaclassofuncertainchaoticandnon-chaoticsystems-190405092640/85/Exponential-State-Observer-Design-for-a-Class-of-Uncertain-Chaotic-and-Non-Chaotic-Systems-2-320.jpg)
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Chaotic Systems
- 1. International Journal of Trend in International Open Access Journal ISSN No: 2456 @ IJTSRD | Available Online @ www.ijtsrd.com Exponential State Observer Design Uncertain Chaotic Professor, Department of Electrical Engineering, ABSTRACT In this paper, a class of uncertain chaotic and non chaotic systems is firstly introduced and the state observation problem of such systems is explored. Based on the time-domain approach with integra differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and effectiveness of the obtained results. Key Words: Chaotic system, state observer design, uncertain systems, exponential convergence rate 1. INTRODUCTION In recent years, various chaotic systems widely explored by scholars; see, for example, [1 and the references therein. Frequently, chaotic signals are often the main cause of system instability and violent oscillations. Moreover, chaos often occurs in various engineering systems and applied physics instance, ecological systems, secure communicati and system identification. Based on practical considerations, not all state variables of most systems can be measured. Furthermore, the design of the state estimator is an important work when the sensor fails. Undoubtedly, the state observer design of with both chaos and uncertainties tends to be more difficult than those without chaos and uncertainties. For the foregoing reasons, the observer design of uncertain chaotic systems is really significant essential. In this paper, the observability problem for uncertain nonlinear chaotic or non-chaotic investigated. By using the time-domain approach with International Journal of Trend in Scientific Research and Development (IJTSRD) International Open Access Journal | www.ijtsrd.com ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Exponential State Observer Design for a Class Uncertain Chaotic and Non-Chaotic Systems Yeong-Jeu Sun f Electrical Engineering, I-Shou University, Kaohsiung, Taiwan In this paper, a class of uncertain chaotic and non- chaotic systems is firstly introduced and the state observation problem of such systems is explored. domain approach with integral and differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and Chaotic system, state observer design, uncertain systems, exponential convergence rate chaotic systems have been ; see, for example, [1-8] Frequently, chaotic signals are often the main cause of system instability and Moreover, chaos often occurs in various engineering systems and applied physics; for instance, ecological systems, secure communication, Based on practical considerations, not all state variables of most systems Furthermore, the design of the state estimator is an important work when the sensor fails. design of systems with both chaos and uncertainties tends to be more without chaos and uncertainties. the foregoing reasons, the observer design of significant and lity problem for a class of chaotic systems is domain approach with integral and differential equalities observer for a class of uncertain will be provided to ensure the global exponential stability of the resulting error system. In guaranteed exponential convergence rate can be precisely calculated. Finally, some numerical simulations will be given to demonstrate the effectiveness of the main result. This paper is organized as follows. The problem formulation and main results 2. Several numerical simulations are 3 to illustrate the main result. Finally, conclusion remarks are drawn in Section paper, n ℜ denotes the n-dimensional xxx T ⋅=: denotes the Euclidean norm of the vector x, and a denotes the number a. 2. PROBLEM FORMULATION AND MAI RESULTS In this paper, we explore the following nonlinear systems: ( ) ( ) ( ) ( )( ),,, 32111 txtxtxftx ∆=& ( ) ( ) ( ) ( )( ),, 31222 txtxftaxtx +=& ( ) ( )( ) ( )( ),14333 txftxftx +=& ( ) ( ) ,0,3 ≥∀= ttbxty ( ) ( ) ( )[ ] [T xxxxxx 2010321 000 = where ( ) ( ) ( ) ( )[ ]321: ∈= T txtxtxtx ( ) ℜ∈ty is the system output, initial value, 1f∆ is uncertain function, indicate the parameters of the system 0≠b . Besides, in order to ensure the existence and uniqueness of the solution, we assume that Research and Development (IJTSRD) www.ijtsrd.com 1 | Nov – Dec 2018 Dec 2018 Page: 1158 Class of Chaotic Systems Shou University, Kaohsiung, Taiwan equalities, a new state uncertain nonlinear systems ensure the global exponential stability of the resulting error system. In addition, the guaranteed exponential convergence rate can be precisely calculated. Finally, some numerical simulations will be given to demonstrate the ult. This paper is organized as follows. The problem are presented in Section simulations are given in Section to illustrate the main result. Finally, conclusion remarks are drawn in Section 4. Throughout this dimensional real space, the Euclidean norm of the column he absolute value of a real ROBLEM FORMULATION AND MAIN the following uncertain (1a) (1b) (1c) (1d) ]T x30 , (1e) 3 ℜ∈ is the state vector, is the system output, [ ]T xxx 302010 is the is uncertain function, and ℜ∈ba, parameters of the systems, with 0<a and n order to ensure the existence and uniqueness of the solution, we assume that all the
- 2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 @ IJTSRD | Available Online @ www.ijtsrd.com functions of 1f∆ and () { }4,3,2, ∈∀⋅ ifi are smooth the inverse function of 4f exists. The chaotic sprott E system is a special case of systems (1) with 2 12 xf = , 13 =f , 14 4xf −= , and 1−=a . It is a well fact that since states are not always available for direct measurement, particularly in the event of sensor failures, states must be estimated. The paper is to search a novel state observer uncertain nonlinear systems (1) such that exponential stability of the resulting error systems can be guaranteed. Before presenting the main result, the state reconstructibility is provided as follows. Definition 1 The uncertain nonlinear systems (1) are state reconstructible if there exist a state observer ( ) 0,, =yzzf & and positive numbers κ and ( ) ( ) ( ) ( ) 0,exp: ≥∀−≤−= tttztxte ακ , where ( )tz represents the reconstructed state of systems (1). In this case, the positive number called the exponential convergence rate. Now, we are in a position to present the main results for the exponential state observer of uncertain (1). Theorem 1. The uncertain systems (1) are exponentially state reconstructible. Moreover, a suitable state observer is given by ( ) ( ) ( ) , 11 3 1 41 −= − ty b fty b ftz & (2a) ( ) ( ) ( ) ( )( ),, 31222 tztzftaztz +=& (2b) ( ) ( ) 0, 1 3 ≥∀= tty b tz . (2c) In this case, the guaranteed exponentia rate is given by a−=:α . Proof. For brevity, let us define the observer error ( ) ( ) ( ) { }3,2,1,: ∈∀−= itztxte iii and 0≥t . (3) From (1d) and (2c), one has ( ) ( ) ( ) ( ) ( )ty b ty b tztxte 11 333 −= −= .0,0 ≥∀= t (4) International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 are smooth and chaotic sprott E (1) with 321 xxf =∆ , . It is a well-known fact that since states are not always available for direct vent of sensor he aim of this state observer for the (1) such that the global the resulting error systems can nting the main result, the state reconstructibility is provided as follows. are exponentially state reconstructible if there exist a state observer and α such that the reconstructed state of (1). In this case, the positive number α is called the exponential convergence rate. a position to present the main results uncertain systems exponentially state , a suitable state observer is the guaranteed exponential convergence observer error From (1c), it can be readily obtained that ( )( ) ( ) ( )( )txftxtxf 33314 −= & . It results that ( ) ( ) ( )( )( ) ( ) ( ) −= −= − − ty b fty b f txftxftx 11 3 1 4 333 1 41 & & Thus, one has ( ) ( ) ( ) ( ) ( ) ( ) ( ) −− −= −= − − ty b fty b f ty b fty b f tztxte 11 11 3 1 4 3 1 4 111 & & ,0,0 ≥∀= t in view of (5) and (2a). In addition (4), and (6), it is easy to see that ( ) ( ) ( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )[ ] ( ) ( )[ ]tztxa txtxftaz txtxftax tztzftaz txtxftax tztxte 22 3122 3122 3122 3122 222 , , , , −= +− += +− += −= &&& ( ) .0,2 ≥∀= ttae It follows that ( ) ( ) .0,022 ≥∀=− ttaete& Multiplying with ( )at−exp yields ( ) ( ) ( ) ( ) 0expexp 22 =−⋅−−⋅ attaeatte& Hence, it can be readily obtained tha ( ) ( )[ ] .0,0 exp2 ≥∀= −⋅ t dt atted Integrating the bounds from 0 ( ) ( )[ ] ,00 exp 00 2 == −⋅ ∫∫ dtdt dt atted tt This implies that ( ) ( ) ( ) 0,exp022 ≥∀⋅= tatete . Thus, from (4), (6), and (7), we have ( ) ( ) ( ) ( ) ( ) ( ) .0,exp02 2 3 2 2 2 1 ≥∀⋅= ++= tate tetetete Consquently, we conclude that t suitable state observer with the guaranted exponential convergence rate a−=α . This completes the proof. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Dec 2018 Page: 1159 From (1c), it can be readily obtained that (5) (6) In addition, from (1b), (1c), is easy to see that yields .0,0 ≥∀ t Hence, it can be readily obtained that and t , it results 0≥∀ t . (7) , we have conclude that the system (2) is a the guaranted exponential . This completes the proof. □
- 3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 @ IJTSRD | Available Online @ www.ijtsrd.com 3. NUMERICAL SIMULATIONS Consider the uncertain nonlinear system 321 xxcf ⋅∆=∆ , 2 12 xf = , 13 =f , (8a) 14 4xf −= , 1−=a , 2=b , 11 ≤∆≤− c . (8b) By Theorem 1, we conclude that the systems (1) with (8) is exponentially state reconstructible by the state observer ( ) ( ) , 4 1 8 1 1 + − = tytz & (9a) ( ) ( ) ( ),2 122 tztztz +−=& (9b) ( ) ( ) .0, 2 1 3 ≥∀= ttytz (9c) The typical state trajectory of the uncertain (1) with (8) is depicted in Figure 1. Furthermore time response of error states is depicted in Fig From the foregoing simulations results, it is seen that the uncertain systems (1) with (8) are state reconstructible by the state observer o the guaranted exponential convergence rate 4. CONCLUSION In this paper, a class of uncertain chaotic and non chaotic systems has been introduced observation problem of such system studied. Based on the time-domain approach with integral and differential equalities, a observer for a class of uncertain nonlinear been constructed to ensure the global exponential stability of the resulting error system. guaranteed exponential convergence rate precisely calculated. Finally, numerical simulations have been presented to exhibit the effectiveness feasibility of the obtained results. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106-2221 MOST 106-2813-C-214-025-E, and MOST E-214-030. Besides, the author is grateful to Professor Jer-Guang Hsieh for the useful REFERENCES 1. S. Xiao and Y. Zhao, “A large class of chaotic sensing matrices for compressed sensing Processing, vol. 149, pp. 193-203, 201 2. R. Zhang, D. Zeng, S. Zhong, K. Shi, “New approach on designing stochastic sampled data controller for exponential synchronization of International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 systems (1) with , we conclude that the uncertain is exponentially state uncertain systems Furthermore, the is depicted in Figure 2. From the foregoing simulations results, it is seen that are exponentially state reconstructible by the state observer of (9), with the guaranted exponential convergence rate 1=α . chaotic and non- and the state observation problem of such systems has been domain approach with , a novel state uncertain nonlinear systems has the global exponential the resulting error system. Moreover, the exponential convergence rate can be Finally, numerical simulations the effectiveness and Ministry of Science and of Republic of China for supporting this 2221-E-214-007, , and MOST 107-2221- grateful to Chair for the useful comments. A large class of chaotic sensing matrices for compressed sensing,” Signal , 2018. Shi, and J. Cui, New approach on designing stochastic sampled- ntial synchronization of chaotic Lur’e systems,” Hybrid Systems, vol. 29, pp. 3. J. Kim and H. Ju, “Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding system Chaos, Solitons & Fractals 2018. 4. F.H. Hsiao, “Chaotic synchronization cryptosystems combined with RSA encryption algorithm,” Fuzzy Sets and Systems 109-137, 2018. 5. J. Park and P. Park, “ ∞H control for synchronization of chaotic Lur systems with time delays,” Institute, vol. 355, pp. 8005 6. A.K. Shikha, “Chaotic analysis and combination combination synchronization of a novel hyperchaotic system without any equilibria Chinese Journal of Physics 2018. 7. R. Zhang, D. Zeng, S. Zhong, K “New approach on designing stochastic sampled data controller for exponential synchronization of chaotic Lur’e systems,” Hybrid Systems, vol. 29, pp. 8. J. Kim and H. Ju, “Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems Chaos, Solitons & Fractals, vol. 2018. 9. D. Astolfi, L. Marconi, L “Low-power peaking-free high Automatica, vol. 98, pp. 169 10. S. Xiao, Y. Zhang, and triggered network-based state observer design of positive systems,” Information Sciences pp. 30-43, 2018. 11. D. Bernal, “State observers in the design of eigenstructures for enhanced sensitivity,” Mechanical Systems and Signal Processing 110, pp. 122-129, 2018. 12. H. Hammouri, F.S. Ahmed “Observer design based on immersion technics and canonical form,” Systems & Control Letters vol. 114, pp. 19-26, 2018. 13. S. Li, H. Wang, A. Aitouche, and N. Christov “Sliding mode observer design for fault and International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Dec 2018 Page: 1160 ,” Nonlinear Analysis: , pp. 303-321, 2018. Hausdorff dimension of the Yorke pairs for some chaotic dynamical coupled expanding systems,” Chaos, Solitons & Fractals, vol. 109, pp. 246-251, Chaotic synchronization cryptosystems combined with RSA encryption Fuzzy Sets and Systems, vol. 342, pp. sampled-state feedback control for synchronization of chaotic Lur’e ,” Journal of the Franklin 8005-8026, 2018. Chaotic analysis and combination- combination synchronization of a novel aotic system without any equilibria,” Chinese Journal of Physics, vol. 56, pp. 238-251, Zhong, K. Shi, and J. 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- 4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 @ IJTSRD | Available Online @ www.ijtsrd.com disturbance estimation using Takagi European Journal of Control, vol. 44 2018. Figure 1: Typical state trajectory Figure 0 -8 -6 -4 -2 0 2 4 6 8 x1(t);x2(t);x3(t) 0 -0.5 0 0.5 1 1.5 2 e1(t);e2(t);e3(t) e1=e2=0 e3 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 ing Takagi-Sugeno,” 44, pp. 114-122, 14. A. Sassi, H.S. Ali, M. Zasadzinski, and K. Abderrahim, “Adaptive observer design for a class of descriptor nonlinear systems,” European Journal of Control, vol. 44, pp. 90 trajectory es of the uncertain nonlinear systems (1) ure 2: The time response of error states. 50 100 150 t (sec) x1: the Blue Curve x2: the Green Curve x3: the Red Curve 5 10 15 t (sec) International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Dec 2018 Page: 1161 A. Sassi, H.S. Ali, M. Zasadzinski, and K. Abderrahim, “Adaptive observer design for a class of descriptor nonlinear systems,” European Journal of Control, vol. 44, pp. 90-102, 2018. (1) with (8).