Investigation on Synchronization of Two Identical Class of Chaotic Systems Using Back-Stepping Control Technique in the Presence of Time Delays

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Investigation on Synchronization of Two Identical Class of Chaotic
Systems Using Back-stepping Control Technique in the Presence of
Time Delays
Riddhi Mohan Bora1, Bharat Bhushan Sharma2, Bhabani Shankar Dey3and Indra Narayan Kar4
Abstract— Time delay analysis is very crucial when it comes
to the synchronization of chaotic systems. All practical dy-
namical systems are described by differential equations, hence
the presence of time delay in dynamics becomes a pivotal
part of the system analysis. So, in this article, an attempt
has been made to study the synchronization problem of two
identical same-ordered time-delayed chaotic systems, in strict-
feedback form as described in (2) and (3) in master and
slave configuration. Unlike other conventional nonlinear control
techniques, the back-stepping control design strategy is used
to serve the purpose of synchronization. The main advantage
of using the back-stepping control technique is that it utilizes
only a single scalar controller to attain synchronization among
different chaotic systems. A generalized control function u∈Ris
designed for the same. For verification of the theoretical results,
the 3D time-delayed Genesio-Tesi chaotic system is considered
both a master and slave system. Simulation results are provided
to support the proposition.
I. INTRODUCTION
Over the past three decades, chaos theory and the synchro-
nization of chaos systems have emerged as one of the most
intriguing areas of study due to their wide range of potential
applications in fields as diverse as biomedical, chemical,
communication, electrical, mechanical, and many others [1]–
[5]. Chaotic systems’ dynamics are analogous to random
processes. However, it is controlled by exact mathematical
differential equations that are extremely reliant on the initial
conditions. Controlling and stabilizing chaotic systems, as
well as synchronizing two or more chaotic systems, is a
challenging topic of study, hence many authors have offered
different methods to do so. Non-linear feedback control [6],
active control [7], adaptive control [8], sliding mode control
[9], observer design [10], and LMI-based control [11], etc.
are all examples of effective control methods.
The number of controllers required in the majority of
the above-mentioned control strategies is the fundamental
disadvantage in various applications [6]–[11]. System design
frequently becomes uneconomical as the number of con-
trollers increases. Therefore, reducing the number of con-
trollers is crucial throughout the design and manufacturing
process. To address this problem, back-stepping control is
*This is a collaborative work with the Indian Institute of Technology
Delhi and the National Institute of Technology Hamirpur.
1,3,4are with the Control and Automation Group of the Electrical Engi-
neering Department, Indian Institute of Technology Delhi, Hauz Khas, New
Delhi, 110016, India eez228208@iitd.ac.in
2is with the department of Electrical Engineering, National Institute
of Technology Hamirpur, Hamirpur, Himachal Pradesh, 177005, India
bhushan@nith.ac.in
often used in nonlinear controller design. Since its first use
by Miroslav Krstic et al. in [12], it has had extensive usage
in the regulation and synchronization of nonlinear systems
of integer order. Lyapunov’s theory is the basis for the
methodical approach used here. Each step in the process
results in a revised control rule for the subsequent phase, and
the state variables are managed independently as subsystem
controllers. Each step’s controller is developed with the help
of the appropriate Lyapunov function (a function of the states
of the drive and response systems), guaranteeing the stability
of its subsystem.
However, the classic back-stepping approach has the con-
straint that it can only be used for achieving synchronization
of those systems whose state equations are in the strict-
feedback form and in which both the drive and response
systems have the same order. This restriction prevents it
from being used to achieve synchronization of any other
systems. In spite of the fact that the back-stepping approach
is subject to the restrictions described above, the authors of
the works [13], [14] have made an attempt to demonstrate
the widespread applicability of the back-stepping technique
for systems that have what is known as a non-strict-feedback
form. In addition, a significant number of research articles
[15]–[19] have been published on the topic of employing
the back-stepping control strategy to synchronize two chaotic
and hyperchaotic systems of the same order. It has previously
been proved that two chaotic systems of the same order that
belong to the class of triangular feedback may synchronize
with one another [17], [20], [21]. In his article [22], Shihua
Chen showed the application of the back-stepping approach
to developing a highly effective generalized process for
constructing a scalar controller to achieve synchronization
between two same-ordered broad classes of chaotic systems
in the broad Strict Feedback form. The literature survey
implies that using back-stepping control very few works have
been done on the synchronization of chaotic or hyperchaotic
systems in the presence of time delays in the system. Though
some of the noteworthy works in the similar line are as
follows [23]–[26]. Great efforts can be seen in the work
[27], where authors investigated the synchronization and
stabilization problem of time-delayed chaotic systems using
the backstepping control design technique. In their work, par-
ticularly strict feedback systems have not been considered for
master and slave systems. But our work specifically focuses
on systems that are in generalized triangular feedback form
(as described in (1)) which is a special class of strict feedback
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systems. Also, our proposed generalized control function has
a different structure from the one described in [27]. To gain
further insights, one can consult the comprehensive research
conducted in [28].
Motivated by the research gaps found in the literature
survey, the following important contributions are enlisted
here.
•The back-stepping technique is utilized to propose a
constructive solution for achieving time-delayed syn-
chronization between a drive system and a response
system that belongs to a specific class of strict-feedback
form.
•A control function with a generalized structure has been
developed using the back-stepping control technique
to achieve synchronization between two systems of a
particular class (as described in (2) and (3)) in the
presence of time delays.
The structure of this paper is as follows: Section I provides
a brief historical account of the evolution of chaotic systems
and back-stepping techniques, along with a review of the
literature. This section also discusses the research gaps
and motivations based on the literature review. Section II
presents the proposed class of chaotic systems in strict-
feedback form, along with the mathematical formulation. The
problem formulation is described in section III, followed by
the proposed methodology and a systematic proof to derive
the generalized control function, which is presented as a
theorem. In section IV, an example is given to illustrate
the application of the proposed theoretical procedure, with
proper derivations. The numerical simulation and results are
discussed in section V, with a detailed analysis of all the
figures. Finally, section VI concludes the paper with closing
remarks.
II. GEN ERALIZED STRU CTU RE OF THE CHAOT IC
SYS TEM S
The control and synchronization of chaotic dynamical
systems are often investigated by employing low-dimensional
systems as benchmark examples to verify and validate con-
trol theoretic methods and algorithms. Our focus lies on a
generalized category of chaotic systems that can be observed
in existing literature, encompassing a range of benchmark
chaotic or hyper-chaotic systems like the Genesio-Tesi sys-
tem, Arneodo system, Jerk System, Hyper-chaotic jerk sys-
tem, Duffing oscillator, Van-Der-Pol oscillator, among others.
The dynamics for such a class of systems can be expressed
as:
˙x1=x2
˙x2=x3
.
.
.
˙xn−1=xn
˙xn=−
n
∑
i=1
Kixi+fn(x1,x2,...,xn) + ˆ
f(t)
(1)
where, vector X= [x1,x2,...,xn]Trepresents state vari-
ables of the system i.e. X∈ℜn; with nas order of the
system. fn(x1,x2,...,xn)is the nonlinear function present
in the nth state dynamics. The ˆ
f(t)is a continuous time-
dependent function. Ki’s, are constants associated with the
linear terms in the last state equation of the system where i
varies from 1 to n.
Without loss of generality, this class of systems can be
called Generalized Triangular Feedback (GTF) form.
III. PRO BLE M FORM ULATI ON
For the analysis purpose, let us consider two identical
systems, namely, master and slave in the following structure:
˙xi=xi+1,i=1,2,...,(n−1)
˙xn=−
n
∑
i=1
Kixi(t−τi) + fn(x1,x2,...,xn)(2)
where, Xis the state vector of the master system of the
appropriate dimension. τiare the time delays associated with
the linear parts of the last equation. Here, for the simplicity
of analysis, constant time delay has been considered.
Likewise, let us consider a controlled slave system that
is required to synchronize with the system (2) in a similar
manner.
˙yi=yi+1,i=1,2,...,(n−1)
˙yn=−
n
∑
i=1
Qiyi(t−τ1i) + gn(y1,y2,...,yn) + u(3)
where, Y, and Qiare defined similarly as the master system.
τ1iare the time delays (constant delay) associated with
the linear parts of the last equation. Our goal is to design
the control function u∈Rusing the back-stepping control
technique.
The error dynamics can be formulated as follows:
E(t) = Y(t)−X(t) = Y−X= [e1(t),e2(t),...,en(t)]T
Theorem 1: Using the control function given in (4), we can
obtain the global synchronization between the system (3) and
(2), i.e., the error dynamics E(t)will approach zero as time
(t)→∞.
u=χn(x1,x2,...,xn)−Θn(y1,y2,...,yn)(4)
where;
χn(x1,x2,...,xn) = χn−2(x1,x2,...,xn−1) + χn−1(x1,x2,...,xn)
+
n−1
∑
k=1
∂
∂xk
[χn−1(x1,x2,...,xn)]xk+1
+fn(x1,x2,...,xn)−
n
∑
k=1
Kixi(t−τi)
and
Θn(y1,y2,...,yn) = Θn−2(y1,y2,...,yn−1) + Θn−1(y1,y2,...,yn)
+
n−1
∑
k=1
∂
∂yk
[Θn−1(y1,y2,...,yn)]yk+1
+gn(y1,y2,...,yn)−
n
∑
k=1
Qiyi(t−τ1i)
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Proof: Let us first define an auxiliary variable p1=y1−x1.
Step 1: Introducing an auxiliary variable for the systems
delineated in (2) and (3);
p1=y1−x1=Θ0(y1)−χ0(x1)
where,
Θ0(y1) = y1and χ0(x1) = x1(5)
Next, select the first Lyapunov function as:
V1(p1) = 1
2p2
1
upon taking the time derivative, one can express the result
as ˙
V1=p1˙p1
˙
V1=−p2
1+p1[˙p1+p1]
where new variable p2is defined as p2=p1+˙p1
p2= [y1−x1] + [ ˙y1−˙x1]
Using relevant dynamic equations from (3) and (2) in the
above expression, we can obtain;
p2= [y1−x1] + [y2−x2]
⇒p2=Θ1(y1,y2)−χ1(x1,x2)
where,
χ1(x1,x2) = (x1+x2)and Θ1(y1,y2) = (y1+y2)(6)
Step 2: Choosing the second Lyapunov function to be:
V2(p1,p2) = V1(p1) + 1
2p2
2
the time derivative of the above function can be defined as
˙
V2=−p2
1+p1p2+p2˙p2
⇒˙
V2=−p2
1−p2
2+p2[˙p2+p2+p1]
Again using the dynamics of the master and slave system
from (2) and (3), we get:
˙
V2=−p2
1−p2
2+p2[p1+p2+d
dt [Θ1(y1,y2)−χ1(x1,x2)]]
⇒˙
V2=−p2
1−p2
2+p2[p1+p2+d
dt Θ1(y1,y2)−d
dt χ1(x1,x2)]
⇒˙
V2=−p2
1−p2
2+p2[p1+p2+∂Θ1(y1,y2)
∂y1
dy1
dt
+∂Θ1(y1,y2)
∂y2
dy2
dt −[∂ χ1(x1,x2)
∂x1
dx1
dt +∂ χ1(x1,x2)
∂x2
dx2
dt ]]
To simplify the above equation, let us assume another
auxiliary variable p3as
p3=p1+p2+∂Θ1(y1,y2)
∂y1
dy1
dt +∂Θ1(y1,y2)
∂y2
dy2
dt
−[∂ χ1(x1,x2)
∂x1
dx1
dt +∂ χ1(x1,x2)
∂x2
dx2
dt ]
Using the dynamics from (2) and (3), we get:
p3= [y1−x1] + [Θ1(y1,y2)−χ1(x1,x2)] + ∂Θ1(y1,y2)
∂y1
y2
+∂Θ1(y1,y2)
∂y2
y3−[∂ χ1(x1,x2)
∂x1
x2+∂ χ1(x1,x2)
∂x2
x3]
It can be further expressed in general form as:
p3= [y1+Θ1(y1,y2) + ∂Θ1(y1,y2)
∂y1
y2+∂Θ1(y1,y2)
∂y2
y3]
−[x1+χ1(x1,x2) + ∂ χ1(x1,x2)
∂x1
x2+∂ χ1(x1,x2)
∂x2
x3]
⇒p3=Θ2(y1,y2,y3)−χ2(x1,x2,x3)
where,
Θ2(y1,y2,y3) = y1+Θ1(y1,y2) + y2+y3
χ2(x1,x2,x3) = x1+χ1(x1,x2) + x2+x3
(7)
So we can write now:
˙
V2=−p2
1−p2
2+p2p3
Further, we can generalize the partial Lyapunov function for
ith step (ivaries from 3 to (n−1)), by writing
Vi=Vi−1+1
2p2
i(8)
Its time derivative along the solution of (2) and (3) will be:
dVi
dt =−
i−1
∑
k=1
p2
k+pi[pi−1+d
dt pi]
⇒dVi
dt =−
i
∑
k=1
p2
k+pi[pi−1+pi+d
dt pi]
⇒dVi
dt =−
i
∑
k=1
p2
k+pi[pi−1+pi
+
i
∑
k=1
∂
∂yk
[Θi−1(y1,y2,...,yi)]yk+1
−
i
∑
k=1
∂
∂xk
[χi−1(x1,x2,...,xi)]xk+1]
So we can write that;
dVi
dt =−
i
∑
k=1
p2
k+pipi+1(9)
The last step: The Lyapunov function may now be repre-
sented as follows, having reached its nth and final stage:
V=Vn−1+1
2p2
n
⇒dV
dt =−
n−1
∑
k=1
p2
k+pn[pn−1+d
dt pn]
⇒dV
dt =−
n
∑
k=1
p2
k+pn[pn−1+pn+d
dt pn]
(10)
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Now, the mathematical expressions of pn−1,pnand d
dt pncan
be written as follows:
pn=Θn−1(y1,y2,...,yn)−χn−1(x1,x2,...,xn)
pn−1=Θn−2(y1,y2,...,yn−1)−χn−2(x1,x2,...,xn−1)
d
dt pn=d
dt [Θn−1(y1,y2, .., yn)−χn−1(x1,x2, .., xn)]
⇒d
dt pn=d
dt Θn−1(y1,y2, .., yn)−d
dt χn−1(x1,x2, .., xn)
⇒d
dt pn=
n−1
∑
k=1
∂
∂yk
Θn−1(y1,y2, .., yn)d
dt yk
+∂
∂yn
Θn−1(y1,y2, .., yn)d
dt yn
−
n
∑
k=1
∂
∂xk
χn−1(x1,x2, .., xn)d
dt xk
⇒d
dt pn=
n−1
∑
k=1
∂
∂yk
Θn−1(y1,y2, .., yn)yk+1
+∂
∂yn
Θn−1(y1,y2, .., yn)
[−
n
∑
i=1
Qiyi(t−τ1i) + gn(y1,y2,...,yn) + u]
−
n
∑
k=1
∂
∂xk
χn−1(x1,x2, .., xn)xk+1
(11)
Next, by substituting all the given values of pn−1,pnand
d
dt pnin (10), we get:
dV
dt =−
n
∑
k=1
p2
k+pn[Θn(y1,y2,...,yn)−χn(x1,x2,...,xn)
+∂
∂yn
Θn−1(y1,y2,...,yn)u]
We can now substitute the value:
∂
∂yn
Θn−1(y1,y2,...,yn) = 1
This statement holds true for all instances falling under
the category of the Generalized Triangular Form (GTF) as
outlined in section II (1).
The derivative of the nth Lyapunov function over time may
thus be expressed as:
dV
dt =−
n
∑
k=1
p2
k+pn[Θn(y1,y2,...,yn)
−χn(x1,x2,...,xn) + u]
(12)
where;
χn(x1,x2,...,xn) = χn−2(x1,x2,...,xn−1) + χn−1(x1,x2,...,xn)
+
n−1
∑
k=1
∂
∂xk
[χn−1(x1,x2,...,xn)]xk+1
+fn(x1,x2,...,xn)−
n
∑
k=1
Kixi(t−τi)
and
Θn(y1,y2,...,yn) = Θn−2(y1,y2,...,yn−1) + Θn−1(y1,y2,...,yn)
+
n−1
∑
k=1
∂
∂yk
[Θn−1(y1,y2,...,yn)]yk+1
+gn(y1,y2,...,yn)−
n
∑
k=1
Qiyi(t−τ1i)
Now, after this, if we select the control function as:
u=χn(x1,x2,...,xn)−Θn(y1,y2,...,yn)(13)
the time derivative of the Lyapunov function in (12) will
become dV
dt =−
n
∑
k=1
p2
k=−2V
Solving this simple differential equation, one can get:
V=V0(exp(−2t))
So, it is evident that when t→∞for any arbitrary initial
condition V0, the Vexponentially converges to zero. It further
ensures that the p1,p2,. .., pnare also stabilized, and so
the errors (e1,e2,e3,...,en) are converging. Finally, we can
conclude that with the help of the proposed scalar controller,
synchronization is achieved between the time-delayed master
and slave systems with the general description given in (2)
and (3).
IV. APP LIC ATI ON EXA MPL E
Let us take a time-delayed 3D Genesio-Tesi system as the
Master system, whose mathematical description can be given
as:
˙x1=x2
˙x2=x3
˙x3=−cx1(t−τ1)−bx2(t−τ2)−ax3(t−τ3) + x2
1
(14)
τ1,τ2, and τ3are considered to be the time delays associated
with the system dynamics.
Likewise, in the same fashion, the controlled slave system
is considered as follows:
˙y1=y2
˙y2=y3
˙y3=−cy1(t−τ11)−by2(t−τ12)−ay3(t−τ13) + y2
1+u
(15)
τ11,τ12 and τ13 are considered to be the time delays asso-
ciated with the system dynamics. Using the back-stepping
technique the system given in (15) will get synchronized
with (14).
The application of Theorem 1 enables us to formulate
the following controller u∈ℜ, which ensures the global
synchronization between the master system and the slave
system illustrated above.
u=−2(y3−x3)−3(y2−x2)−2(y1−x1)−(y2
1−x2
1)
+c[y1(t−τ11)−x1(t−τ1)] + b[y2(t−τ12)−x2(t−τ2)]
+a[y3(t−τ13)−x3(t−τ3)]
(16)
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Note: The proof involves extensive calculations and the steps
are exactly similar to the ones explained in the proof of
Theorem 1. Refer to Appendix A in [29] for detailed proof.
V. NU MER ICA L SIMULATIO NS AND RESULTS
The simulation parameters are chosen as a=1.2,b=
2.92,c=6, in such a way that the systems remain in chaotic
behavior. To solve the delay differential equations, we use
DDE23 solvers in MATLAB 2013a software. The history
matrix is chosen as [2,−1,3,0,1,−5]and the constant delays
are taken to be τi=0.02 seconds and τ1i=0.02 seconds,
where ivaries from 1 to 3 in our example.
Fig. 1. States of the Master and the Slave systems vs Time when controller
uis activated.
0 10 20 30 40 50 60 70
−25
−20
−15
−10
−5
0
5
10
15
Time (sec)
e1,e2,e3 Errors
e1
e2
e3
Fig. 2. Synchronizing Errors [e1,e2,e3]vs Time when controller uis
activated.
Fig. 1, shows the synchronized states of the master and
slave systems while the control function uis activated at
time (t) = 20 seconds. Simultaneously, Fig. 2 showcased the
error dynamics [e1,e2,e3]Twhich are approaching zero as the
time →∞, under the effect of controller u. Fig. 3 displays
the 3D phase portraits of the master and slave systems,
respectively. When the controller uis not activated, then the
error dynamics are not tending to zero value, which is visible
in Fig. 4. The corresponding un-synchronized state dynamics
of both the master and slave systems are drawn in Fig. 5.
−5
0
5
−5
0
5
−15
−10
−5
0
5
10
15
x3 and y3
−5
0
5
−5
0
5
−5
0
5
10
15
x2
x1
x3
−5
0
5
−5
0
5
−20
0
20
y3
3D Phase Portrait of the Slave Sytem
y2
Overlapping of 3D Phase Portraits of
Master and Slave System
3D Phase Portrait of the Master Sytem
y1
x1 and y1
x2 and y2
Fig. 3. 3D Phase Portraits of Slave and Master systems, respectively, when,
controller uis activated.
0 10 20 30 40 50 60 70
−25
−20
−15
−10
−5
0
5
10
15
20
Time (sec)
e1,e2,e3 Errors
e1
e2
e3
Fig. 4. Synchronizing Errors [e1,e2,e3]vs Time when u=0.
0 10 20 30 40 50 60 70
−10
0
10
x1 and y1 state
Time (sec)
x1
y1
0 10 20 30 40 50 60 70
−10
0
10
x2 and y2 state
Time (sec)
x2
y2
0 10 20 30 40 50 60 70
−20
0
20
x3 and y3 state
Time (sec)
x3
y3
Fig. 5. States of the Master and the Slave systems vs Time when u=0.
VI. CON CLUSIONS AND FUTURE SC OP E S
A generalized methodology is derived to synchronize two
special classes of chaotic or hyperchaotic systems in the
strict-feedback form given in (2) and (3) respectively. The
generic scalar control functions uis derived to support the
results. One benchmark example; such as the 3D time-
delayed Genesio-Tesi system is considered both a master
and a slave system to prove the usefulness of the process
as an application to our proposed methodology. With the
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availability of time-delayed information on the states of
the systems, the controller can easily be implemented. The
proposed controller can also be used for the synchronization
of two non-identical systems which have similar structures
as given in (1).
As future scopes, some of the points are highlighted as
follows: the practical validation of this proposed technique
is yet to explore. Being an inherent property of any dy-
namic system, a time delay study like this finds so many
empirical applications. One can design controllers when the
time delays are considered to be in the nonlinear state-
dependent functions in the system dynamics. Considering
state-dependent time delays and time-varying delays the
same problem can be studied for more insights (see [30],
[31]).
ACK NO WL EDG EME NT
The authors appreciate the reviewers’ careful consideration
of the article and their thoughtful comments.
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