873

Sensors and Materials, Vol. 31, No. 3 (2019) 873–881
MYU Tokyo
S & M 1821

Simplified State Observer of Hyperchaotic Lü Attractor
with One Voltage-sensed Signal
Chi-Hsin Yang, Hui-Cun Shen, Kun-Chieh Wang,* and Hong-Yi Chen
Zhaoqing University, Zhaoqing City, Guangdong Province 526061, P.R. China(Received November 23, 2018; accepted January 18, 2019)

Keywords: simplified state observer, voltage measurement, hyperchaotic Lü attractor, exponential
stability

The main goal of this study is to develop a simplified state observer for a four-dimensional
hyperchaotic Lü attractor implemented by a circuit system. The elements of the state vector canbe fully reconstructed by feeding the proposed state observer with only one continuously sensed
voltage measurement. By utilizing the time-domain approach, the estimated state errors areproven to satisfy global and exponential stability, and the exponential convergence rate can be
specified via the values of system parameters. Numerical studies are also presented to illustrate
the effectiveness of the proposed scheme.

1.

Introduction

Since the introduction of a method of synchronizing two identical chaotic systems withdifferent initial conditions in the pioneering work of Pecora and Carroll,(1) chaos control andsynchronization have become a very important topic in nonlinear science. Owing to the huge
range of potential applications, chaos control and chaos synchronization(2–9) have received
a great deal of attention among researchers in many fields, such as physical(10) and chemicalsystems,(11,12) power converters,(13) ecological and biological systems,(14,15) financial systems,(16)and secure communication.(17,18)
In applications, it is either inefficient or impractical to measure all the elements of the statevector under practical hardware limitations. Moreover, a state observer can be applied tocompute the reconstruction and take the place of sensor signals in the event of sensor failures.
For these reasons, state observer design is an essential and meaningful task for chaos control
and synchronization. Numerous studies(19–21) on the state observer design of many chaotic
systems have been reported. However, the system configurations of state observers in theprevious works are sometimes complicated and almost all the state variables of the measured
drive system must be supplied.
In a reasonable and practical way, the configuration of the state observer should be
simplified and the number of measured state inputs in the chaotic system should be small.
In the literature, simple state observers with the state of one drive system as the input were*

Corresponding author: e-mail: kcwang0104@qq.com
https://doi.org/10.18494/SAM.2019.2224
ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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Sensors and Materials, Vol. 31, No. 2 (2019)

investigated by Sun(22,23) for three-dimensional generalized Lorenz and Chen chaotic systems.
However, such a simple state observer with the feeding of one state variable of the drive system
has not yet been addressed for more complex and higher-dimensional systems.
Motivated by the aforementioned studies, a simplified state observer design for the fourdimensional (4D) hyperchaotic Lü attractor implemented by a circuit system(24) is developed
utilizing the time-domain approach. The state vector can be fully reconstructed by the proposed
state observer with one continuously sensed voltage signal of the 4D chaotic circuit system. The
estimated state errors are proven to guarantee global and exponential stability. Furthermore,
the exponential convergence rates can also be specified accurately via the parameters of sucha system. Simulations are performed to show the validity of the main results. The rest of thispaper is organized as follows. In Sect. 2, a problem description is given and the main results
are derived. In Sect. 3, simulations to demonstrate the effectiveness of the main results arereported. Finally, conclusions are given in Sect. 4.

2.

Problem Description and Main Results

A 4D hyperchaotic Lü attractor introducing state feedback based on the Lü system(25,26) wasconstructed in Ref. 24. In this system with the output y (t ) including the time series of the firststate variable, x1 (t ), has the form x1 = a ( x2 − x1 ) + x4

x2 cx2 − x1 x3
=−bx3 + x1 x2
 x3 =
= x4 dx4 + x1 x3
y (t ) = x1 (t )(1)Twhere x (t ) = [ x1 (t ) x2 (t ) x3 (t ) x4 (t ) ] and a, b, c, and d are the state vector and systemparameters, respectively. It has three quadratic nonlinearities and thus has more complex
dynamical behaviors than normal chaotic systems. Chen et al.(24) proved that the dynamical
behaviors of the system in Eq. (1) switch among a periodic orbit, a chaotic state, and a
hyperchaotic state with increasing parameter d. When the system parameters are a = 36, b =3, and c = 20, it was shown that the system in Eq. (1) has a periodic orbit for −1.03 ≤ d ≤ −0.46,
a chaotic attractor for −0.46 < d ≤ −0.35, and hyperchaotic attractors for −0.35 < d ≤ 1.30. The
hyperchaotic Lü attractor with the system parameters a = 36, b = 3, c = 20, and d = 0.5, and theinitial conditions ( x1 (0), x2 (0), x3 (0), x4 (0)) = (0.1, 0.1, 25, 0.1) is depicted in Fig. 1.
The circuit design in Ref. 24 to realize the 4D hyperchaotic Lü system is shown in Fig. 2.
The output voltages, xi , i = 1, , 4, in the illustrated circuit are equivalent to the state variablesof the 4D system. It was concluded that the occurrence of chaotic or hyperchaotic behavior
depended on the position of switch K and the values of the resistors in the circuit. From theviewpoint of practical usage, it is easy to continuously sense one of the output voltages, forSensors and Materials, Vol. 31, No. 3 (2019)

Fig. 1.

875

4D hyperchaotic Lü attractor with a = 36, b = 3, c = 20, and d = 0.5.

Fig. 2.

Circuit implementation of 4D hyperchaotic Lü system.(24)

example, the first one x1, and record the measurements as the time series using a suitable
voltage sensing system.
In the following, a simplified state observer design utilizing the sensed time signals of thefirst voltage x1 of the circuit system in Eq. (1) is developed. Before proceeding to the main
results, a definition that will be applied in the main theorem is presented. For the design of asimplified state observer of the system in Eq. (1), an equivalent system representation of thesystem in Eq. (1) will also be introduced in a lemma.
Definition. The states of system (1) can be globally and exponentially reconstructed ifTthe state obser ver xˆ (t ) = f ( x (t ), y (t )) exists, where xˆ (t ) = [ xˆ (t ) x (t ) x (t ) x (t ) ]
1

T

2

3

4

xˆ (t ) = [ xˆ1 (t ) x2 (t ) x3 (t ) x4 (t ) ] expresses the reconstructed state vector of system (1) and876

Sensors and Materials, Vol. 31, No. 2 (2019)

positive constants A and B such thate (=t)

x (t ) − xˆ (t ) ≤ A exp(− Bt ), ∀t ≥ 0

for any initial condition xˆ (0) in a set D4 ⊆ R 4 . In this case, the positive constant B is named theguaranteed exponential convergent rate.
Lemma. The system in Eq. (1) can be transformed into the following equivalent systemrepresentation:c − adc−d−σ (t ) +( x1 (t ) + ax1 (t ))σ (t ) =
a −1
a −1

1
x =
( x1 (t ) + ax1 (t ) − σ (t )) 2 (t )
a −1


a
−1
t)
( x1 (t ) + ax1 (t )) +σ (t )
 x4 (=a −1
a −1
 x3 (t ) =−bx3 (t ) + x1 (t ) x2 (t )(2)where the auxiliary state variable σ (t ) is defined asσ=(t ) x2 (t ) + x4 (t ) .

(3)

Proof. The first equation in the system (1) can be expressed asax2 + x4 = x1 + ax1 .

(4)Then, x2 (t ) and x4 (t ) can be solved from Eqs. (3) and (4) and expressed as1( x1 (t ) + ax1 (t ) − σ (t ))
a −1
−1
a
x4 (=
t)
( x1 (t ) + ax1 (t )) +σ (t ) .
a −1
a −1
x2=
(t )

(5)

Substituting the second and fourth equations in the system in Eq. (1) into the time derivative of
the auxiliary state variable defined in Eq. (3) yieldsσ (t ) = x2 (t ) + x4 (t ) = cx2 (t ) + dx4 (t ).

(6)

The first-order differential equation of the auxiliary state variable σ (t ) is obtained bysubstituting Eq. (5) into Eq. (6).

Sensors and Materials, Vol. 31, No. 3 (2019)c − ada −1877

σ (t ) =σ (t ) +−c−d( x1 (t ) + ax1 (t ))
a −1

(7)

This completes the proof of Lemma.
Theorem. The states of system (1) can be globally and exponentially reconstructed in the caseof d < c a. A suitable state observer is given by xˆ1 (t ) = y (t )1 xˆ2=(t )( y (t ) + ay (t ) − σˆ (t ))

a −1
,

−1
a
 xˆ4 (=t)
( y (t ) + ay (t )) +σˆ (t )
a −1
a −1

 ˆ
 x3 (t ) = −b xˆ3 (t ) + xˆ1 (t ) xˆ2 (t )(8)where σˆ (t ) is the reconstructed auxiliary state variable and can be evaluated using the following
equation:σˆ (t ) = −kσˆ (t ) +

=
k
with

c−d
( y (t ) + ay (t )) ,
a −1

(9)

c − ad> 0.
a −1

Proof. Consider the state errors between the state variables of systems in Eqs. (1) and (8)defined by e=i (t ) xi (t ) − xˆi (t ), ∀i ∈ {1, 2,3, 4}. It can be readily found thate1 (t ) = x1 (t ) − xˆ1 (t ) = 0, ∀t ≥ 0 .

(10)

To obtain e2 (t ) = 0 and e4 (t ) = 0, two linear equations containing e2 (t ) and e4 (t ) should be
proposed.
ae2 (t ) + e4 (t )= a ( x2 (t ) − xˆ2 (t )) + ( x4 (t ) − x4 (t )) 1
= ax2 (t ) + x4 (t ) − a ( y (t ) + ay (t ) − σˆ (t )) 
 a −1a −1σˆ (t ) −
( y (t ) + ay (t )) +a −1
 a −1= x1 (t ) + ax1 (t ) − ( y (t ) + ay (t )) = 0
		(11)

878

Sensors and Materials, Vol. 31, No. 2 (2019)eσ=(t ) σ (t ) − σˆ (t )c − adc−dc−dσ (t ) +( x1 (t ) + ax1 (t )) − kσˆ (t ) +( y (t ) + ay (t ))
=
−
a −1
a −1
a −1
= −keσ (t )t ) e2 (t ) + e4 (=t ) eσ (0) exp(−kt ), ∀t ≥ 0⇒ eσ (==k
The stability of eσ (t ) is guaranteed by the fact that
concluded thatc − ad> 0, that is, d < c a . It can bea −1e2 (t ) + e4 (t ) = eσ (0) exp(− kt ) = 0, ∀t ≥ tsettling > 0(12)where tsettling > 0 is the settling time for eσ (t ) approaching zero. Thus, e2 (t ) = 0 and e4 (t ) = 0
are obtained by solving Eqs. (11) and (12). Consequently, the stability of the estimated error
e3 (t ) is derived ase=
3 (t ) x3 (t ) − xˆ3 (t )=−bx3 (t ) + x1 (t ) x2 (t ) + bxˆ3 (t ) − xˆ1 (t ) xˆ2 (t ))=−b[ x3 (t ) − xˆ3 (t )] + x1 (t )[ x2 (t ) − xˆ2 (t )]=−be3 (t ), ∀t ≥ tsettling > 0.

This implies thatd[e3 (t ) exp(bt )] = 0dt(t ) e3 (0) exp(−bt )
⇒ e3=

(13)

⇒
=
e3 (t ) e3 (0) exp(−bt ), ∀t ≥ tsettling > 0.
Equations (10) to (13) yielde (t ) =e12 (t ) + e22 (t ) + e32 (t ) + e42 (t ) ≤ e3 (0) exp(−bt ), ∀t ≥ 0 .

According to Definition, the states of the system in Eq. (1) can be globally and exponentiallyreconstructed by the state observer in Eq. (8) associated with Eq. (9). This completes the proof.

3.

Numerical Studies

a 36,=b 3,=
c 20, andConsider the 4D hyperchaotic Lü attractor in Eq. (1) with=d ∈ (−1.03,1.3]. Assume that the time signals of the first voltage x1 of the circuit system areSensors and Materials, Vol. 31, No. 3 (2019)879

produced by performing a suitably practical measurement. From Theorem, it is concluded thatthe system in Eq. (1) can be globally and exponentially reconstructed by the following simplified
5 / 9. (1) can be globally and exponentiallystate observer in the case of −1.03 ≤ d < c a =reconstructed by the following simplified state observer:
 xˆ1 (t ) = y (t ) xˆ2 (t ) = 1 ( y (t ) + 36 y (t ) − σˆ (t ))35, xˆ4 (t ) = −1 ( y (t ) + 36 y (t )) + 36 σˆ (t ))
)
35
35

 ˆ
ˆ
ˆ
ˆ
x
(
t
)
x
(
t
)
+
x
(
t
)
2
1
 x3 (t ) = −3 3

(14)

In Eq. (14), σˆ (t ) is the reconstructed auxiliary state variable and can be evaluated asσˆ (t ) = −20 − 36d
20 − dσˆ (t ) +( y (t ) + 36 y (t )) .
35
35

(15)

20 − 36d
and B2 = 3 for the35
reconstructed auxiliary state variable σˆ (t ) and the reconstructed state variable xˆ3 (t ) ,respectively. In the following, simulations on two cases of 4D hyperchaotic Lü attractors are
performed to illustrate the efficiency of the proposed simple state observer.
The numerical simulations are carried out using MATLAB/SIMULINK software. The
fourth-order Runge–Kutta method with a time step of 0.0001 is used to solve the system. The
initial conditions of the 4D hyperchaotic Lü attractor (1) and proposed simplified state observer(0) y=(0) 10, x2 (0) = 10, x3 (0) = 10, x4 (0) = 10, and σˆ (0) = 0, xˆ3 (0) = 0,
are selected to be x1=
respectively. From Eqs. (1) and (8), the following initial values are evaluated: y (0) = 10,xˆ2 (0) = 10.57, xˆ4 (0) = −10.57. The time responses of the estimated state errors between the

The guaranteed exponential convergence rates are B1 =Fig. 3.
−0.3.

Time responses of estimated errors for d =Fig. 4. Time responses of the system in Eq. (1) andstate observer in Eq. (14) associated with Eq. (15) ford = −0.3.

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Sensors and Materials, Vol. 31, No. 2 (2019)

Fig. 5.

Time responses of estimated errors for d = 0.3.

Fig. 6. Time responses of the system in Eq. (1) andstate observer in Eq. (14) associated with Eq. (15) for
d = 0.3.

simplified state observer (14) associated with Eq. (15) and the system in Eq. (1) for the case ofd = −0.3 are depicted in Fig. 3. In this case, the guaranteed exponential convergence rate is
B1 = 0.88. Time responses of the system in Eq. (1) and the simplified state observer in Eq. (14)associated with Eq. (15) are shown in Fig. 4. It is shown that the reconstructed states of Eq. (14)associated with Eq. (15) are readily convergent to the states of the system in Eq. (1). In the caseof d = 0.3, the guaranteed exponential convergence rate is B1 = 0.263. The time responses of thereconstructed errors and state responses are illustrated in Figs. 5 and 6, respectively. From theabove simulation results, it is shown that the system in Eq. (1) can be globally and exponentiallyreconstructed using the proposed simplified state observer in Eq. (14) associated with Eq. (15).

4.

Conclusions

In this study, the 4D hyperchaotic Lü attractor implemented by the circuit system proposedby Chen et al.(24) is considered and the state reconstruction problem of this system is studied.
Utilizing the time domain approach, a simplified state observer applied to one continuously
measured voltage signal of the 4D hyperchaotic circuit system is designed, and state errors areproven to satisfy global and exponential stability. The guaranteed exponential convergence
rate can be estimated from the system parameters provided that d < c a is satisfied. Finally,
simulations are performed to demonstrate the effectiveness of the proposed scheme.

References
1
2
3
4
5
6
7

L. M. Pecora and T. L. Carroll: Phys. Rev. Lett. 64 (1990) 821. https://doi.org/10.1103/PhysRevLett.64.821
A. Loría: IEEE Trans. Circuits Syst. II 56 (2009) 674. 10.1109/TCSII.2009.2025624
C. C. Yang: Appl. Math. Comput. 216 (2010) 1316. https://doi.org/10.1016/j.amc.2010.02.026
C. C. Yang: Nonlinear Dyn. 63 (2011) 447. https://doi.org/10.1007/s11071-010-9814-3
C. C. Yang: J. Franklin Inst. 349 (2012) 349. https://doi.org/10.1016/j.jfranklin.2011.11.013
C. C. Yang: Appl. Math. Comput. 219 (2013) 5152. https://dx.doi.org/10.1016/j.amc.2012.11.003
C. C. Yang: Arabian J. Sci. Eng. 39 (2014) 2413. https://10.1007/s13369-013-0768-x

Sensors and Materials, Vol. 31, No. 3 (2019)

8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

881

R. Luo and Y. Zeng: Nonlinear Dyn. 80 (2015) 989. https://doi.org/10.1007/s11071-015-1923-6
J. Yu, J. Le, and D. Liu: Optik 130 (2017) 1053. https://doi.org/10.1016/j.ijleo.2016.11.108
K. S. Sudheer and M. Sabir: Phys. Lett. A 373 (2009) 1847. https://doi.org/10.1016/j.physleta.2009.03.052
T. P. Chang: Sens. Mater. 30 (2018) 1911. https://dx.doi.org/10.18494/SAM.2018.2012
C. J. Xu and Y. Wu: Appl. Math. Modell. 39 (2015) 2295. https://doi.org/10.1016/j.apm.2014.10.030
V. Avrutin, E. Mosekilde, Z. T. Zhusubaliyev, and L. Gardini: Chaos 25 (2015) 043114. https://dx.doi.
org/10.1063/1.4918299
S. K. Dana, P. K. Roy, and J. Kurths: Complex Dynamics in Physiological Systems: From Heart to Brain (Springer,
New York, 2009) p. 127. https://dx.doi.org 10.1007/978-1-4020-9143-8
B. Blasius and L. Stone: Int. J. Bifurcation Chaos 10 (2000) 2361. https://doi.org/10.1142/S0218127400001511
C. Lin: J. Control Sci. Eng. 2018 (2018) 4976380. https://doi.org/10.1155/2018/4976380
R. Martfnez-Guerra and Y. Wen: Int. J. Bifurcation Chaos 18 (2008) 235. https://doi.org/10.1142/
S0218127408020264
W. D. Chang, S. P. Shih, and C. Y. Chen: J. Control Sci. Eng. 2015 (2015) 471913. https://dx.doi.
org/10.1155/2015/471913
C. Hua and X. Guan: Phys. Lett. A 334 (2005) 382. https://doi.org/10.1016/j.physleta.2004.11.050
R. Martfnez-Guerra, W. Yu, and E. Cisneros-Saldana: Chaos Solitons Fractals 36 (2008) 1141. https://doi.
org/10.1016/j.chaos.2006.07.039
J. Zheng and B. Cui: Chaos Solitons Fractals 112 (2018) 141. https://doi.org/10.1016/j.chaos.2018.04.038
Y. J. Sun: Chaos Solitons Fractals 39 (2009) 1641. https://doi.org/10.1016/j.chaos.2007.06.043
Y. J. Sun: Phys. Lett. A 374 (2010) 933. https://doi.org/10.1016/j.physleta.2009.12.019
A. Chen, J. Lu, and S. Yu: Physica A 364 (2006) 103. https://doi.org/10.1016/j.physa.2005.09.039
J. Lu and G. Chen: Int. J. Bifurcation Chaos 12 (2002) 659. https://doi.org/10.1142/S0218127402004620
J. Lu, G. Chen, and S. Zhang: Int. J. Bif urcation Chaos 12 (2002) 1001. htt ps://doi.org/10.1142/
S0218127402004851