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IIUM Engineering Journal, Vol. 17, No. 1, 2016 Pourmehdi et al.
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CHAOS CONTROL AND SYNCHRONIZATION
USING SYNERGETIC CONTROLLER WITH
FRACTIONAL AND LINEAR EXTENDED
MANIFOLD
MORTEZA POURMEHDI, ABOLFAZL RANJBAR NOEI*, AND JALIL SADATI
Department of Electrical & Computer Engineering,
Babol University of Technology, Babol, Iran.
morteza.pourmehdi1366@gmail.com; *a.ranjbar@nit.ac.ir; and j.sadati@nit.ac.ir
(Received: 02 Jun. 2015; Accepted: 20 Sept. 2015; Published on-line: 30 Apr. 2016)
ABSTRACT: In this manuscript, for the first time, a fractional-order manifold in a
synergetic approach using a fractional order controller is introduced. Furtheremore, in the
synergetic theory a macro variable is expended into a linear combination of state variables.
An aim is to increase the convergence rate as well as time response of the whole closed
loop system. Quality of the proposed controller is investigated to control and synchronize
a nonlinear chaotic Coullet system in comparison with an integer order manifold
synergetic controller. The stability of the proposed controller is proven using the Lyapunov
method. In this regard stabilizing control effort is yielded. Simulation result confirm
convergence of states towards zero. This is achieved through a control effort with fewer
oscillations and lower amplitude of signls which confirm feasibility of the control effort
in practice.
ABSTRAK: Manuskrip ini memperkenalkan, buat kali pertama, manifold pecahan
pesanan dalam pendekatan sinergi yang menggunakan pengawal pecahan pesanan. Dalam
teori sinergi pembolehubah makro diperluaskan menjadi satu kombinasi linear
pembolehubah keadaan. Tujuannya adalah untuk meningkatkan kadar penumpuan serta
masa tindak balas keseluruhan sistem lingkaran tertutup. Kualiti pengawal yang
dicadangkan dikaji dari segi mengawal dan menyelaras sistem Coullet tak linear
berbanding dengan pengawal sinergi pesanan integer manifold. Kestabilan pengawal yang
dicadangkan terbukti dengan penggunaan kaedah Lyapunov. Keadaan ini menghasilkan
kawalan yang stabil. Simulasi menghasilkan penumpuan keadaan ke arah sifar. Ini dicapai
melalui usaha kawalan yang mempunyai kurang pengayunan dan amplitud isyarat lebih
rendah dan ini mengesahkan bahawa usaha kawalan dalam amalan boleh dilaksanakan.
KEYWORDS: Synergetic control theory; Fractional order system; Synchronization;
Nonlinear chaotic Coullet system; Chaos control
1. INTRODUCTION
Synergetic control theory is primarily reported by Russian researchers [1]. In essence,
a synergetic controller is of an analytical topic with high flexibility in defining dynamical
manifold. Fractional order controllers are shown providing quality and robust performance
in the presence of uncertainties and disturbances for nonlinear systems [2]. During the
design procedure, undesired dynamics can be eliminated by introducing dynamical
constraints, defined in a manifold. In this regard specifications of desired performance can
be applied to the system according to this manifold. Elimination of undesired dynamics and
reduction of the system’s order are some issues of desired control specifications. The aim
of the synergetic controller is to force states of the system to reach and remain on a manifold
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through macro variables. These variables are selected according to the control goals. When
system reaches to the desired manifold, its behavior is usually controlled by a first order
differential equation. Innovatively a combination of fractional controller and the synergetic
controller is used to improve the tuning process and the convergence of states.
Although synergetic controller is recently proposed, several applications are
successfully reported in different fields of engineering. This controller is successfully
applied in nonlinear power electronic and industrial processes. Nonlinear power system
stabilizers [3],power converters for pulse current charging [4], DC-DC boost converters [5]
and control of chaotic oscillation in power systems [6] are such of these applications.
Fractional calculus by over 300 years’ history is a generalization of integer order
derivative and integral into a real order one. In recent decades, applications of fractional
calculus in modeling and control are widely reported. Several works such as [7-16] deals
with fractional calculus. Fractional order dynamics are involved in deferent systems such as
viscoelasticity materials, electrochemical processes, electrical machines and etc. It is shown
in many fields of science and engineering that the fractional operators can provide more
accurate modeling[17,18]. During modeling of physical systems which exhibit hereditary,
diffusion and viscoelasticity properties [19-22], heat transfer [23], [24], [25], [12] and
fractional PID [26] controllers can be mentioned as such examples of applying fractional
order controllers in different field. Recently [27] incorporates a combination of synergetic
controller with fuzzy theory.
In this manuscript, fractional derivative in the Caputo sense [15] will be utilized. The
current research proposes an idea to combine both synergetic controller and fractional
operators to control a fractional order chaotic system. This proposed approach shows
improving time response and the convergence rate.
The rest of the paper is organized as follows:
In section 2, basics of synergetic theory are introduced. Section 3 briefly describes the
Caputo fractional calculus. Synergetic controller with fractional order and together with
combined linear extension of the manifold is proposed for the first time in section 4. The
stability of the control system is analyzed in this section. Section 5 investigates with quality
of the proposed method through a chaos control of a fractional order system.
Synchronization is made possible in section 6 using the proposed fractional-order synergetic
structure. Finally, section 7 closes the work by a conclusion.
2. BASICS OF SYNERGETIC THEORY
Synergetic method generates a control effort through defining a macro variable
introducing required dynamic as a manifold. Assume that the system is represented in the
following nonlinear state space equations:
(1) ̇ = ( , , )
Where denotes vector of the state variable, u refers to the vector of the control input,
defines the time variable and finally ( , , ) indicates a nonlinear relation between states
and input. Synergetic design uses macro variable ψ( ) in either linear or nonlinear function
of the state variable such as:
(2) = ( )
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Appropriate control law drives states towards and stays on manifold of ( ) = 0. The
designer may choose macro variable according to the control specifications and
requirements [28]. Macro variables are usually considered as a liner combination of the state
variables. Number of macro variables may be defined as much as the control input channels
(multi input). A dynamical equation of manifold may be defined as follows [2]:
(3) ̇ + = 0, > 0
where denotes the design parameter. This eventually adjusts rate of the convergence of
the states in the manifold (2). Equation (3) immediately yields:
(4) ( ) = , ≥ 0
The convergence of ( ) towards the manifold ( ) = 0 is guaranteed with any bounded
initial condition where here the rate of convergence will be tuned by appropriate choice of
the design parameter . Roll of parameter can be found using the following chain rule of
Eqn.(4):
(5) ̇ = = ̇
Substituting equation Eqn.(5) into Eqn.(3), involving the system Eqn.(1) achieves:
(6) T (x, u, t) + = 0
Finally, an appropriate control law will be achieved using Eqn.(6) considering other
specifications. Each manifold imposes a new constraint to the system which decreases the
order of the system.
3. THE CAPUTO FRACTIONAL DERIVATIVE
In fractional calculus several methods of Grünwald–Letnikov, Riemann–Liouville and
Caputo are commonly used. In this manuscript, the Caputo fractional derivative [15] is used.
Benefits of the Caputo technique are studied in [29-32]. This is because initial conditions in
the Caputo are of an integer order which is physically realizable. In the Riemann–Liouville
definition the initial condition is of a fractional order that is less realizable in practice as
well as increases the complexity [15]. The Caputo fractional derivative is defined as
follows:
(7) ( ) =
1
Γ( − )
( )( )
( − )
where Γ(. ) is the Euler Gamma function. For − 1 < < , initial condition of the
fractional order differential equation is the same as the integer order one [15]. A unified
formula for fractional order integral of ( ) i.e. ( ), with ∈ (0,1) in sense of the
Caputo in the time interval [0,t] is defined as follows [15]:
(8) ( ) =
1
Γ( )
( − ) ( )
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Likewise ( ), − 1 < < is a Caputo fractional order derivative of as defines
as follows [15]:
(9) ( ( )) = ( ) −
( )( )
!
( − )
For = 0, = 0 and 0 ≤ < 1 [15] It is shown that he following relation is yielded:
(10) ( ( )) = ( ) − (0)
Furthermore, the following relations are also held for the Caputo method [15]:
(11) ( ( )) = ( ), ( = 0,1,2, … ; − 1 < < )
Meanwhile:
(12)
( ( )) = ( ( )) = ( )
( )(0) = 0 , ∈ ℕ , = , + 1, … ,
( = 0,1,2, … ; − 1 < < )
In the following, the above fractional calculus is combined with the synergetic control
theory to obtain appropriate control law.
4. METHODOLOGY: ANALYTICAL INVESTIGATION OF THE
STABILITY OF FRACTIONAL-ORDER MANIFOLD
SYNERGETIC CONTROLLER
In this section, the synergetic controller in Eqn.(2) to Eqn.(6) gains a fractional
manifold. The combination gives advantages of both the synergetic theory and the fractional
calculus. The latter provides more flexibility and degree of freedom to design the fractional
controller. When a higher integer order is needed a small size fractional controller provides
better results [17, 33-35].The idea is also developed to consider a linear combination of state
variables. In this regard, a fractional order form of the synergetic manifold of Eqn.(3) is
proposed as follows:
(13) ( ) + ( ) = 0, > 0
where is a real number in the interval ∈ (0, 1). In the following, performance of the
proposed controller is compared with conventional synergetic controller of integer order
manifold when fractional order chaotic Coullet system [36] is controlled. Chaotic systems
are found very sensitive to initial conditions. This means two identical systems but with a
minor deviation in their initial conditions may produce a completely different result. Hence,
it is difficult predict the behavior of these systems. The term Chaos arises from a
deterministic dynamic with nearly stochastic (almost unpredictable) behavior. The chaos
effect and control is a more challenging topic. Therefore the following section addressess
control of chaotic Coullet dynamic.
4.1 Stabilizing the Chaos
The Coullet system is used in a synchronization approach, which is as follows:
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(14)
=
=
= + + +
Parameters of the Coullet system are chosen as a = 0.8, b = -1.1, c = -0.45, d = -1 and =
0.95 together with the state initial conditions: x (0) = −0.8, x (0) = 1.2 and x (0) = 0.2.
State response of the system (4.2) to initial conditions is simulated in 50 seconds where
depicted in Fig. 1 for − . Fourth plot in Fig. 1 represents a phase portrait of the system.
Fig. 1: States (x1-x3) responses of together with a phase portrait (fourth).
As can be seen from the first three graphs in Fig. 1, state x1-x3 are of chaotic (neither
periodic nor converging). The fourth plot also confirms states fail to converge to any point.
To deeply verify the chaos, an FFT of x1 is assessed and plotted in Fig. 2, using power GUI
facility in the MATLABTM simulink.
As can be seen from Fig. 2, all of the frequency components occur in the frequency
spectrum. This is in confirmation that the signal is not pure oscillatory. This is main
characteristics of chaos (similar to frequency characteristics of noise) where all frequencies
can be seen in the chaos [5-6, 37-38]. Although this is for x1, similar results are achieved
for x2 and x3.
In order to control the chaos, a control effort ( )is applied to the third equation of
Eqn.(4.2):
(15)
=
=
= + + + +
0 20 40 60
-1
-0.5
0
0.5
1
1.5
Time (s)
A
m
pl
itu
de
X1 State
0 10 20 30 40 50
-1
-0.5
0
0.5
1
1.5
Time (s)
A
m
pl
itu
de
X2 State
0 20 40 60
-1
-0.5
0
0.5
1
Time (s)
A
m
pl
itu
de
X3 State
-2
0
2
-2
0
2
-1
0
1
x1x2
x3
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Fig. 2: Time (the above) and frequency (bottom) response of state x1 of the
Coullet system.
The control takes places using the synergetic technique. The duty is to provide zero
convergence of the state variables. The macro variable is defined as follows [2]:
(16) = + +
In order to make more degree of freedom, arbitrary gains of , and are innovatively
added to the macro variable in Eqn. (16). This generalization expands the macro as in the
following:
(17) = + +
It will be shown that the proposed macro in Eqn. (17) increases the performance. The
required control effort can be obtained when the conventional manifold in Eqn. (16) i.e. ψ =
x + x + x = 0 is substituted into Eqn. (13), which is as follows:
(18)
= −
1 ( ) + ( ) + ( + + + )
+ ( ) + ( ) + ( )
Likewise, anew control law is derived for the proposed manifold in Eqn. (17) using the
following fractional dynamical equation of macro variables:
(19) ( ) + ( ) = 0, > 0
Substitution of Eqn. (19) into Eqn. (13) yields:
(20) ( ) + ( ) + ( ) + ( ) + ( ) + ( ) = 0
0 10 20 30 40 50
-0.5
0
0.5
1
Selected signal: 50 cycles. FFT window (in red): 1 cycles
Time (s)
0 10 20 30 40 50
0
10
20
30
40
50
60
Frequency (Hz)
Fundamental (1Hz) = 0.3419 , THD= 78.96%
M
ag
(
%
o
f F
un
da
m
en
ta
l)
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where ( ) is -order derivative of state variable in the Caputo sense as:
(21) ( ) = = /
By substituting dynamical equation Eqn. (15) into Eqn. (20), the following equation is
obtained:
(22)
( ) + ( ) + ( + + + + )
+ ( ) + ( ) + ( ) = 0
The control input u is finally extracted form (22) which is as follows
(23)
= −
1 ( ) + ( ) + ( + + + )
+ ( ) + ( ) + ( )
The stability of the proposed controller will be analyzed through the following theorem:
Theorem: Consider the Coullet fractional order system in Eqn. (15). If the control law in
Eqn. (23) is applied, system states asymptotically converge to zero.
Proof: The Lyapunov function is candidate as in the following:
(24)
=
1
2
(ψ( )) ≥ 0
The conventional derivative yields:
(25)
V̇ = (ψ( ))( )ψ( ) = ψ( )ψ( )
A fractional form of the macro-variable i.e. ψ( ), is achieved as follows:
(26)
( ) = ( ) + ( ) + ( )
= ( ) + ( ) + ( + + + + )
Replacement the control effort ( ) from Eqn. (23) into Eqn. (26) provides:
(27)
( ) = ( ) + ( ) + ( + + +
−
1 ( ) + ( ) + ( + + + )
+ ( ) + ( ) + ( )
= −
1 ( ) + ( ) + ( ) = −
1 ( )
Substituting ψ( ) = − ψ( )from Eqn. (27) into Eqn. (25) achieves:
IIUM Engineering Journal, Vol. 17, No. 1, 2016 Pourmehdi et al.
122
(28) V̇ = −
1 ( ) ( ) = −
1
( ( ) ) ≤ 0
Inequality Eqn. (28) guarantees strictly decreasing nature of the Lyapunov function in Eqn.
(22). This confirms that the control input Eqn. (21) is a stabilizing control effort for system
Eqn. (15).
5. RESULTS: APPLICATION OF THE PROPOSED SYNERGETIC
CONTROLLER TO CONTROL A CHAOS
A simulation is carried out for the fractional order Coullet system gaining both the
manifold in Eqn. (16) [2] and the proposed manifold in Eqn. (17). Time responses of three
states are shown in Fig. 3, Fig. 4 and Fig. 5 when initial conditions are assumed as (0) =
−0.8, (0) = 1.2, (0) = 0.2 and T = 0.1 whilst gains are chosen = 3, = 5 and
= 1 using Eqn. (3) to Eqn. (6).
From Fig. 3 it can be seen that by considering extra gains into Eqn. (16) the macro variable
in Eqn. (17) causes reduction in the overshoot. Meanwhile better performance with respect
to the conventional macro variable in Eqn. (17) is also achieved. However, errors in two
methods approach zero. The time behavior of the second state variable in Fig. 4 is similar to
the first state variable as in Fig. 3. In contrast to those two state variables, the third state
variable in Fig. 5 provides more negative peak with respect to that of the conventional macro
in Eqn. (16). However, the proposed macro in Eqn. (17) provides a faster convergence rate
(with the price of more negative peak). This necessitates to tune those gains in an optimal
way rather than simple selection of = 3, = 5 and = 1. Fortunately, this probable
draw back requires no more attention to provide the control effort which is depicted in Fig.
6 which makes the lack of negative peak negligible.
Fig. 3: Time response of state variable
x in fractional order Coullet system with
conventional macro Eqn. (16) as =
+ + = 0 and the proposed
macro Eqn. (17) as = + +
= 0.
Fig. 4: Time response of state variable x in
fractional order Coullet system with
conventional macro Eqn. (16) as = +
+ = 0 and the proposed macro Eqn.
(17) as = + + = 0.
0 5 10 15
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time(s)
x 1
The proposed method
method in [2]
0 5 10 15
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time(s)
x 2
The proposed method
method in [2]
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123
Fig. 5: Time response of state
variable x in fractional order
Coullet system with conventional
macro Eqn. (16) as = + +
= 0 and the proposed macro
Eqn. (17) as = + +
= 0.
Fig. 6: The required control input for
using the conventional macro in Eqn.
(16) as = + + = 0 and
the proposed macro Eqn. (17) as =
+ + = 0.
6. SYNERGETIC SYNCHRONIZATION OF FRACTIONAL-ORDER
COULLET SYSTEM
Basically, chaos synchronization means forcing two systems work in a same way in
a master-slave structure. The designed nonlinear control system obtains signals from the
master to control the slave dynamics. Block diagram of the master and slave
synchronization using the synergetic controller is shown in Fig. 7.
Fig. 7: Block diagram of the master and slave synchronization using the synergetic
controller.
The goal is to synchronize a fractional-order Coullet system assuming the master as
in Eqn. (29) to be followed by the slave as in Eqn. (5.2) using the proposed synergetic
controller Eqn. (23).
(29)
=
=
= + + +
Supposing (0) = −0.8, (0) = 1.2 and (0) = 0.2 where the makes the system
chaotic. The slave dynamics is similarly shown by:
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Time(s)
x 3
The proposed method
method in [2]
0 5 10 15
-6
-5
-4
-3
-2
-1
0
1
Time(s)
co
nt
ro
l e
ff
or
t
The proposed method
method in [2]
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124
(30)
=
=
= + + + +
Initial conditions of the slave are assumed different as (0) = 1, (0) =
−1.2 and (0) = −0.4. The error is defined as discrepancy of the corresponding states, i.e.
= − for i = 1,2,3. Deducing the master dynamic from the slave leads to:
(31)
=
=
= + + + +
The same procedure as in Eqn. (19) to Eqn. (23) using the proposed linear gains leads to
generate the following control effort:
(32)
= −
1 ( ) + ( ) + ( + + + )
+ ( ) + ( ) + ( )
A proper selection of control parameters ki, i.e. k1 = 3, k2 = 5.75 and k3 = 1 provides lim
→
= 0. Outcomes of the synchronization are shown in Fig. 108, Fig. 9 and Fig. 10. The proof
is similar to that of in Eqn. (24) to Eqn. (28). This immediately means that output of the
slave asymptotically follows the master state i.e. = . This confirms that the
synchronization between master and slave is made possible.
Fig. 8: Synchronization of states and
together with the = − .
Fig. 9: Synchronization of states and
together with the = − .
Fig. 10: Synchronization of states and together with the = − .
0 5 10 15 20
-4
-2
0
2
Time(s)
x 1
,y
1
x
1
y
1
0 5 10 15 20
-4
-2
0
2
Time(s)
e 1
=x
1-
y 1
0 5 10 15 20
-2
0
2
Time(s)
x 2
,y
2
x
2
y
2
0 5 10 15 20
0
1
2
3
Time(s)
e 2
=x
2-
y 2
0 5 10 15 20
-2
-1
0
1
Time(s)
x3
,y
3
x3
y3
0 5 10 15 20
-2
-1
0
Time(s)
E
rr
or
=
x3
-y
3
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7. DISCUSSION AND CONCLUSION
In this paper, a new controller is proposed by considering a fractional derivative in
linear weighted combination of state variables in the synergetic controller. The degree of
freedom is also increased by adding control coefficients , , and to the designed
synergetic manifold. By tuning parameters , and frequency of oscillations is
decreased when convergence is finally achieved. The stability of the proposed configuration
is proven using the Lyapunov theory. The achievement confirms performance of the
proposed structure. The parameters , and are determined by a trial and error
process, while they can be determined using intelligent algorithm such as PSO (Particle
swarm optimization) or other algorithms to obtain better results.
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