ArticlePDF Available

Abstract and Figures

We present sufficient conditions for the preservation of stability of fractional-order systems, and then we use this result to preserve the synchronization, in a master-slave scheme, of fractional-order systems. The systems treated herein are autonomous fractional differential linear and nonlinear systems with commensurate orders lying between 0 and 2, where the nonlinear ones can be described as a linear part plus a nonlinear part. These results are based on stability properties for equilibria of fractional-order autonomous systems and some similar properties for the preservation of stability in integer order systems. Some simulation examples are presented only to show the effectiveness of the analytic result.
Content may be subject to copyright.
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 928930, 16 pages
doi:10.1155/2012/928930
Research Article
Preservation of Stability and Synchronization of
a Class of Fractional-Order Systems
Armando Fabi
´
an Lugo-Pe
˜
naloza, Jos
´
e Job Flores-Godoy,
and Guillermo Fern
´
andez-Anaya
Departamento de F
´
ısica y Matem
´
aticas, Universidad Iberoamericana, Prol. Paseo de la Reforma 880,
Lomas de Santa Fe, 01210 M
´
exico, DF, Mexico
Correspondence should be addressed to
Armando Fabi
´
an Lugo-Pe
˜
naloza, armando.lugo@correo.uia.mx
Received 18 April 2012; Revised 11 August 2012; Accepted 12 August 2012
Academic Editor: Ricardo Femat
Copyright q 2012 Armando Fabi
´
an Lugo-Pe
˜
naloza et al. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
We present sucient conditions for the preservation of stability of fractional-order systems, and
then we use this result to preserve the synchronization, in a master-slave scheme, of fractional-
order systems. The systems treated herein are autonomous f ractional dierential linear and
nonlinear systems with commensurate orders lying between 0 and 2, where the nonlinear ones can
be described as a linear part plus a nonlinear part. These results are based on stability properties for
equilibria of fractional-order autonomous systems and some similar properties for the preservation
of stability in integer order systems. Some simulation examples are presented only to show the
eectiveness of the analytic result.
1. Introduction
The applications of fractional calculus to science and engineering have been growing in the
last few years 1; this is due in part to the properties of these operators. The applications,
specifically, that involve fractional-order chaotic systems or their synchronization had been
one of the principal subjects of investigation; some of these works are 26. There is also
several works concerning chaotic systems or complex networks of integer order or their
synchronization, for example, 710. There are many dierent works on the synchronization
of fractional autonomous systems that can be described as a linear plus a nonlinear part 11
14, in such works several schemes are proposed to ensure that the error dynamics satisfies
the conditions from the celebrated theorem for autonomous commensurate dierential
systems with fractional order between 0 and 1 by 15; this means that the error dynamics
must hold a linear relation in order to achieve the synchronization. There is also a scheme
proposed in 16, based on 17, where the dynamical system of the synchronization error
2 Mathematical Problems in Engineering
can be nonlinear, which when viewed from the analytical point can be important because it
does not restrict the error dynamics to be only linear.
There is some other interesting theme, the preservation of stability and synchroniza-
tion, which is the main issue in this work. This problem can be stated as follows: if we have an
original autonomous nonlinear system that can be described as a linear plus a nonlinear part
whose origin is stable, we want to investigate some kinds of modifications that can occur to
the fractional order, the linear part, and the nonlinear part in such a way that the origin of the
modified system is also stable. This subject is important because such modifications can be
interpreted as perturbations on the system. Note that the modification of the linear part of the
vector field associated with the fractional dierential equation modifies some local properties
of the vector field at the point of equilibrium, in particular local stability. In 18, the authors
developed two results for the preservation of stability of integer-order nonlinear systems; one
of such results gives conditions for the preservation of stability between systems of dierent
orders of the state vector but does not give direct insight on the transformations, and the
other result gives more insight but in return is a little more restrictive because as part of the
hypothesis it asks for diagonalizability of the linear part of the system. In 19, the authors
have reached conditions for the preservation of stability for integer-order systems in the
presence of nonlinear modifications to the Jacobian matrix; such modifications can be applied
on the characteristic polynomial or in form of a nonlinear polynomial matrix evaluation.
The main objective of this work is to state under which conditions a certain family of
transformations applied to the fractional order, the linear part, and t he nonlinear part of an
autonomous fractional dierential system with commensurate order will preserve stability
of the origin. It is important to point out that this analytical result is of relevance for its
relation with robustness not for the use of an advanced controller in the stabilization or the
synchronization. As far as the authors know, this problem has not been addressed for the case
of fractional-order systems.
In Section 2, we present the definitions and some results on the stability of autonomous
commensurate fractional-order systems. In Section 3, the main results are stated in form
of propositions and corollaries. Based on these propositions, in Section 4 , we present a
methodology to illustrate how these results can be used and this is complemented by the
application of this methodology in two examples of simulation presented in Section 5. Finally
in Section 6, we present the obtained conclusions.
2. Preliminary Results
There are several definitions of a fractional derivative of order α R
2022. We will
use the Caputo fractional operator because the meaning of the initial conditions for systems
described using this operator is the same as for integer-order systems.
Definition 2.1 Caputo fractional derivative. The Caputo fractional derivative of order α R
of a function x is defined as see 20
x
α
t
0
D
α
t
x
1
Γ
m α
t
t
0
d
m
x
τ
m
t τ
mα1
,
2.1
where m 1 α<m, d
m
xτ/dτ
m
is the mth derivative of x in the usual sense, m N,andΓ
is the gamma function. Throughout the paper, we use indistinctly x
α
x
α
t,x xt.
Mathematical Problems in Engineering 3
We recall some previous results on the stability of autonomous commensurate
fractional-order systems that are related to our study.
2.1. Autonomous Commensurate Fractional-Order Linear Systems Stability
Given an autonomous fractional-order system with state space representation
x
α
Ax Bu,
y Cx,
2.2
where A R
n×n
, B R
n×m
, C R
p×n
, the state vector x R
n
, the input vector u R
m
,andthe
output vector y R
p
.
Definition 2.2 see 15. The fractional-order autonomous system 2.2
x
α
Ax, with x
0
x
0
,
2.3
is said to be
i stable if and only if forall >0 δ δ> 0, such that given x
0
then
xt <for all t 0;
ii asymptotically stable if and only if it is stable and lim
t →∞
xt 0.
Firstly, we will introduce some results on fractional-order systems stability. First for
0 <α<1, we have the celebrated Theorem 15 that gives us necessary and sucient
conditions for the asymptotic stability of the origin of a type of autonomous linear fractional-
order systems; such conditions involve the argument of the eigenvalues of the system matrix.
Theorem 2.3. The autonomous system
x
α
Ax, with x
t
0
x
0
, 0 <α<1,
2.4
is asymptotically stable if and only if | argspecA| π/2,wherespecA is the set of all
the eigenvalues of A. Also, the state vector x decays towards 0 and meets the following condition:
x <Nt
α
, t>0, α>0.
And for 1 <α<2, we have a similar result 23.
Theorem 2.4. The autonomous fractional dierential system
x
α
Ax, t > t
0
,
2.5
with initial conditions x
k
t
0
x
k
k 0, 1, with the Caputo derivative and where x R
n
, A
R
n×n
is asymptotically stable if and only if | argspecA| π/2. In this case, the components
of the state decay towards 0 like t
α1
. Moreover, the system 2.5 is stable if and only if either it is
4 Mathematical Problems in Engineering
asymptotically stable, or those eigenvalues which satisfy | argspecA| απ/2 have the same
algebraic and geometric multiplicities.
2.2. Commensurate Fractional-Order Nonlinear Systems Stability
Given a commensurate fractional-order system with the Caputo fractional operator
x
α
f
t, x
2.6
with initial condition xt
0
x
0
, α 0, 1, f : t
0
, × Ω R
n
is piecewise continuous in
t and locally Lipschitz in x on t
0
, × Ω,andΩ R
n
is a domain that contains the origin
x 0.
The equilibrium point of 2.6 is defined as follows 24.
Definition 2.5. The constant x
e
is an equilibrium point of the fractional-order system 2.6 if
and only if ft, x
e
0.
Without loss of generality, let the equilibrium point be x x
e
0. In this definition,
we are considering that the result of the derivative of a constant is zero because we are using
only the Caputo fractional operator.
Definition 2.6 the Lyapunov stability. The equilibrium point x 0 of the system 2.6 is said
to be
1 stable, if for all >0 δ>0 such that if x
0
then x <,for all t 0.
Otherwise the equilibrium point is called unstable;
2 asymptotically stable, if it is stable and in addition the following equality holds:
lim
t →∞
x
0.
2.7
As a starting point for the construction of our own results, we can use the following
result for the stability of the origin of commensurate fractional-order systems with 0 <α<1
17.
Theorem 2.7. Consider the n-dimensional nonlinear fractional-order dynamic system
x
α
Ax g
x
,
2.8
with a constant linear regular matrix A, a nonlinear function gx of the states x, and 0 <α<1.If
1 the zero solution of x
α
Ax is asymptotically stable and αρA > 1;
2 g00 and lim
x→0
gx/x0,whereρA is the spectral radius of A,
then x 0, 0 t
0
t is a stable solution of the system 2.8.
Mathematical Problems in Engineering 5
The following result is valid for the asymptotic stability of systems with 1 <α<2.
Consider the n-dimensional nonlinear fractional-order dynamic system with the Caputo
derivative
x
α
Ax g
t, x
,t>t
0
,
2.9
under the initial conditions
x
αk
t
tt
0
x
k1
k 1, 2
,
2.10
where x R, matrix A R
n×n
,and1<α<2, gt, x : t
0
, × R
n
R
n
is a continuous
function in which gt, 00; moreover, gt, x holds the Lipschitz condition with respect to
x.
Theorem 2.8. If the matrix A such that | argspecA|
/
0, | argspecA| π/2, α1/A <
2, and suppose that the function gt, x satisfies uniformly
lim
x →∞
g
t, x
x
0,t
t
0
,
,
2.11
then the zero solution of 2.9 is asymptotically stable.
The proof of this theorem for the Caputo derivative follows from the proof of Theorem
3.3 in 23 and the application of Lemma 2.7 in 23 and Gronwall-Bellman inequality.
3. Preservation of Stability
So once given all these stability results, we need to give a definition for the preservation of
stability in fractional-order systems in order to be in the possibility to state the conditions in
form of a proposition.
Definition 3.1. Given an asymptotically stable autonomous commensurate fractional-order
linear system of the kind
x
α
Ax,
3.1
where A R
n×n
, x R
n
,0 <α<2andA PJ
A
P
1
. If one has a transformation ψ :
R
× R
n×n
R
× R
n×n
, namely, ψα, Aαβ, MA, such that the new system
x
αβ
MAx, with 0 1
3.2
is also asymptotically stable, where MA PJ
M
J
A
P
1
, M R
n×n
, for some matrix M
PJ
M
P
1
, where J
M
and J
A
are Jordan matrices, then one says that ψ is an asymptotically
stability preserving transformation for commensurate fractional-order autonomous linear
systems.
6 Mathematical Problems in Engineering
We should notice that for the matrices M and A we are using the same matrix P .Itis
also worth to mention that, given the Jordan matrix J
A
that corresponds to the matrix A,the
Jordan matrix J
M
, that represents the modifications, must have the same order and type of
Jordan blocks that J
A
. The reason behind this fact is that in several applications we will need
that MA R
n×n
. But it should also be noticed that another canonical form, instead of Jordan
form, could result more convenient in the construction of state feedback controllers.
Now in a similar way we state the definition for this concept in commensurate
fractional-order nonlinear systems.
Definition 3.2. Given a commensurate fractional-order nonlinear system of the kind
x
α
Ax g
t, x
,
3.3
where A R
n×n
, A PJ
A
P
1
, x R
n
,0 <α<2, g : t
0
, × Ω R
n
is piecewise
continuous in t and locally Lipschitz in x on t
0
, × Ω,andΩ R
n
is a domain that contains
the origin and the origin itself is a stable solution of the system. If one has a transformation
Ψ : R
×R
n×n
×C
k
R
n
,R
n
R
×R
n×n
×C
k
R
n
,R
n
, namely, Ψα, A, g· αβ, MA, cg·,
in such a way that in the new system
x
αβ
MAx cg
t, x
,
3.4
the origin is also a stable solution, where c R, MA PJ
M
J
A
P
1
, M R
n×n
, for some matrix
M PJ
M
P
1
, where J
M
and J
A
are Jordan matrices, then one calls to that transformation a
stability preserving transformation for commensurate fractional-order nonlinear systems.
Remark 3.3. In Definition 3.2 , for the case where 0 <α<1, the nonlinear part is considered
as autonomous, that is, for the system 3.3, we have x
α
Ax gx, and for the modified
system 3.4, we have x
αβ
MAx cgx.
Now based on the Theorems 2.3, 2.4, 2.7,and2.8, and the results from 18 for the
preservation of stability for integer-order systems, the following criterion for the preservation
of stability in autonomous commensurate fractional-order systems can be stated as follows.
Proposition 3.4. Consider an autonomous commensurate fractional-order nonlinear system of the
form
x
α
Ax g
x
3.5
with x R
n
, A R
n×n
, g : D R
n
R
n
is a continuous function, D is a neighborhood of the origin
for 0 <α<1.LetA R
n×n
with the argument of its kth e igenvalue denoted by θ
k
argλ
k
A.
Given a transformation Ψα, A, g·αβ, MA, cg· such that the new system is
x
αβ
MAx cg
x
,
3.6
Mathematical Problems in Engineering 7
where c R, M R
n×n
, 0 1, φ
k
argλ
k
M is the argument of the kth eigenvalue of M,
A PJ
A
P
1
, M PJ
M
P
1
.Alsoletφ
a
k
θ
k
απ/2, φ
b
k
θ
k
απ/2, φ
max
max
k
{φ
a
k
},
φ
min
min
k
{φ
b
k
},if
φ
min
k
max
3.7
for each k 1, 2,...,n, and if the system x
α
Ax is asymptotically stable, αρA > 1, g0
0, lim
x→0
gx/x 0, and ρMA ρA, then one claims that such transformation is a
stability preserving transformation for fractional-order systems of the kind of 3.5.
Proof . Summarizing the initial hypothesis, the original system 3.5 holds the conditions from
Theorem 2.7, so we have |θ
k
| π/2 for k 1, 2,...,n.
By the hypothesis ρMA ρA, ρA > 1, and 0 1, we have that αβρMA > 1,
and we have asked for gx to hold g00 and lim
x→0
gx/x 0. As a result we need
the asymptotic stability of the system x
αβ
MAx to hold the conditions of Theorem 2.7 for
the new system 3.6.
By the properties of the complex numbers, and based on the fact that J
M
and J
A
are
Jordan matrices with the same structure and that MA PJ
M
J
A
P
1
, in order to assure that
the system x
αβ
MAx is asymptotically stable, we need for | argspecMA| βπ/2 to
hold, so first we want for
φ
k
θ
k
π
2
,k
{
1, 2,...,n
}
,
3.8
to hold. The last part of the hypothesis states that the inequality 3.7holds. From the
right part of 3.7, we know that given that each φ
k
is greater than φ
max
, we have that
φ
k
> θ
k
απ/2. And similarly from the left part we have that θ
k
απ/2
k
for
any φ
k
. T hen these two parts together give us precisely that 3.8 holds, and taking from
the hypothesis that 0 1, we have that αβ α and therefore |φ
k
θ
k
| βπ/2,
and thus the modified system holds all the conditions for the linear part from Theorem 3.
From the demonstration of Theorem 3 given in 17, we can observe that cgx also holds the
corresponding conditions; therefore we claim that Ψ is a stability preserving transformation
for the fractional-order autonomous systems of the form of 3.5.
Now we have a similar result for systems with fractional orders lying between 1 and
2.
Proposition 3.5. Consider a partially autonomous commensurate fractional-order nonlinear system
of the form
x
α
Ax g
t, x
3.9
with 1 <α<2, x R
n
, A R
n×n
, g : t
0
, × D R
n
R
n
is a continuous function, D is a
neighborhood of the origin, and gx holds the Lipschitz condition with respect to x.
Let A R
n×n
with the argument of its kth eigenvalue denoted by θ
k
argλ
k
A.Givena
transformation Ψα, A, gt, · αβ, MA, cgt, · such that the new system is
x
αβ
MAx cg
t, x
,
3.10
8 Mathematical Problems in Engineering
where c R, 1 < β < 2, M R
n×n
, φ
k
argλ
k
M is the argument of the kth eigenvalue of
M, A PJ
A
P
1
, M PJ
M
P
1
.Alsoletφ
a
k
θ
k
απ/2, φ
b
k
θ
k
απ/2, φ
max
max
k
{φ
a
k
},
φ
min
min
k
{φ
b
k
},if
φ
min
k
max
3.11
for each k 1, 2,...,n, and if | argspecA|
/
0, |arg(spec(A))| π/2, α 1/A < 2,
β 1, β<α2 1/MA, lim
x 0
gt, x/x 0 is uniformly satisfied for t t
0
, ,
MA≥A, and | argspecM|
/
0, then one claims that such transformation is a stability
preserving transformation for fractional-order systems of the kind of 3.9.
Proof. The first part of the hypothesis is that the original system 3.9 holds the conditions of
Theorem 2.8; therefore, we have that |θ
k
| π/2,fork 1, 2,...,n, and that its origin is an
asymptotically stable solution.
Another part of the hypothesis is that gt, x holds the conditions from Theorem 2.8,
and from the proof of Theorem 2.8, we observe that such conditions also hold for cgt, x.
We have also asked for MA≥A, α 1/A< 2, and β<α2 1/MA to
hold, and thus we have that αβ 1/MA < 2. As a result, we only need | argspecMA| >
αβπ/2 to hold the conditions from Theorem 2.8 for the new system 3.10.
By very similar arguments as in the proof of Proposition 3.4, we have that 3.8 holds
and we have asked for β 1 to hold; therefore conditions from Theorem 2.8
are satisfied and
we claim that Ψ is a stability preserving transformation for the fractional-order autonomous
systems of the form of 3.5.
The following corollaries are a direct consequence of Proposition 1 and Theorem 3.3 in
23.
Corollary 3.6. Consider a linear autonomous commensurate fractional-order system of the form
x
α
Ax, with 0 <α<2,
3.12
that holds the conditions from Theorem 2.4 or Theorem 2.3 accordingly, if one has a transformation
Ψα, Mαβ, MA such that the new system is
x
αβ
MAx,
3.13
with all the variables and matrices as defined before, and if the inequality
φ
min
k
max
3.14
holds, with φ
k
, φ
min
, and φ
max
as defined before, then one claims that Ψ is an asymptotic stability
preserving transformation for the autonomous commensurate fractional-order linear systems of the
form of 3.12.
Remark 3.7. If we take the case of α 1 for Proposition 3.4,orProposition3.5 we will have
similar conditions for the preservation of stability of the origin for an integer-order nonlinear
Mathematical Problems in Engineering 9
system ˙x Ax gx. And if we take the case of α 1 for Corollary 3.6, we will have
the conditions f or the preservation of asymptotic stability of an integer-order linear system
˙x Ax, without the modification over the order, of course.
Remark 3.8. Notice that the conditions from 18, 19 are dierent from the ones that we
obtain choosing α 1inProposition3.4 and cannot be obtained as a particular case of
Proposition 3.4.
4. Preservation of Synchronization
Given the conditions from Proposition 3.4, we want to illustrate how to use these criteria to
ensure the preservation of stabilization and synchronization.
4.1. Preservation of Stabilization of Autonomous Commensurate
Fractional-Order Nonlinear Systems
First for the stabilization of a system, we know from previous works that for an autonomous
fractional commensurate order system of the form x
α
Ax gxu, where gx holds,
the conditions from Theorem 2.7, and with A PJ
A
P
1
, we can choose a control u K
1
x,
K
1
R
n×n
,withK
1
R
n×n
in such a way that for the system x
α
AK
1
xgx, the origin
is a stable solution. But in this particular case we want that A K
1
PJ
A
J
K
1
P
1
; therefore,
we need to construct K
1
as K
1
PJ
K
1
P
1
; the Jordan form J
K
1
also has the restriction that its
Jordan blocks must be of the same order and type as the ones of J
A
.
Now for the modified system x
αβ
MAx cgxu, we will use a similar control
i.e., with the same K
1
defined as u MK
1
x, where M : PJ
M
P
1
holds the conditions
from 1 and all the other matrices defined as before, in such a way that for the system x
αβ
MA MK
1
x cgx, the origin is also a stable solution.
4.2. Preservation of Complete Practical Synchronization of Autonomous
Commensurate Fractional-Order Nonlinear Systems
First we need to describe the synchronization scheme. Let us consider two fractional-order
systems as the master and the slave system, respectively,
x
α
M
A
M
x
M
g
x
M
,x
α
S
A
S
x
S
g
x
S
w,
y
M
h
M
x
M
,y
S
h
S
x
S
,
4.1
where x
M
R
n
is the state vector of the master system, y
M
: R
p
is the output of the master
system, x
S
R
n
is the state vector of the slave system, w R
n
is the control input, and
y
S
R
p
is the output of the slave system.
In this synchronization scheme, the master system represents the target dynamics,
while the slave system represents the system to be controlled.
Let us consider that all the outputs are available, only to illustrate the eectiveness
of the method by showing more states on the graphs of the synchronization error in the
10 Mathematical Problems in Engineering
examples, of course the order of the output can be less than the order of the state vector. This
consideration will lead us to the synchronization error:
e x
M
x
S
, 4.2
we must find a function w such that et is bounded in a subset that contains the origin,
because the result presented in 17 only gives conditions for the stability of the origin,
not for asymptotic stability. Because of this fact, this approach is called complete practical
synchronization.
We specifically choose the control
w g
x
M
g
x
S
g
x
M
x
S
A
M
x
M
A
S
x
S
A
S
x
M
x
S
K
2
x
M
x
S
, 4.3
where K
2
R
n×n
, in such a way that the error dynamics is given by
e
α
x
α
M
x
α
S
A
S
K
2
e g
e
.
4.4
As before, given that A
S
P
A
S
J
A
S
P
1
A
S
, we will have A
S
K
2
P
A
S
J
A
S
J
K
2
P
1
A
S
,
and therefore we need to construct K
2
as K
2
P
A
S
J
K
2
P
1
A
S
in such a way that the origin of
the dynamic system of the error is stable. Again the Jordan form J
K
has the restriction that its
Jordan blocks must be of the same order and type as the ones of J
A
S
.
Then, we want to illustrate what kind of transformations can be applied to the master
and slave systems in such a way that the same K
2
still stabilizes the origin of the modified
synchronization error system. And to do this we define the modification matrices M
M
:
P
A
M
J
M
P
1
A
M
and M
S
: P
A
S
J
M
P
1
A
S
that hold the conditions from Proposition 3.4. With these
modifications applied to each of the systems in the following way:
x
α
M
M
M
A
M
x
M
g
x
M
x
α
S
M
S
A
S
x
S
g
x
S
w,
4.5
with the synchronization error defined as in 4.2, and the control defined as
w g
x
M
g
x
S
g
e
M
M
A
M
x
M
M
S
A
S
x
S
M
S
A
S
x
S
M
S
K
2
e 4.6
the only change is that instead of the term K
2
e now is M
S
K
2
e, one has that the autonomous
commensurate fractional order dynamical system of the error is
e
α
M
S
A
S
K
2
e g
e
M
S
A
S
M
S
K
2
e g
e
.
4.7
Given that one has constructed the matrix M in such a way that it holds the conditions from
Proposition 3.4 it is straightforward to prove that the origin of the new dynamic system of the
error is also stable. Note t hat the modification of the linear part of the vector field associated
to the fractional dierential equation, modifies the manifold of synchronization, but not the
stability.
Mathematical Problems in Engineering 11
5. Examples
5.1. Preservation of Stabilization for Fractional-Order Lorenz Systems
Let us take the Lorenz system with commensurate fractional-order α 0.97 that can be
written as 5:
x
α
Ax g
x
u
σσ 0
ρ 10
00
β
x
0
x
1
x
3
x
1
x
2
K
1
x 5.1
with x x
1
x
2
x
3
T
, σ 10, ρ 28, β 8/3, initial conditions x09 514
T
.This
system is chaotic as it is claimed in 25.
The objective is to stabilize the system and then apply a modification that holds the
conditions of Proposition 3.4, to illustrate the validity of the analytical results.
In order to do this the next, we choose u K
1
x.With
J
K
1
17 0 0
0150
004
,K
1
5.1552 9.2338 0
25.8545 3.1552 0
004
, 5.2
the eigenvalues of the new matrix A K
1
are λ
1
≈−5.827, λ
2
≈−3.172, and λ
3
≈−6.666, with
this and the fact that gx holds the conditions from Theorem 2.7 as it has been demonstrated
in 17, we know that the origin of the controlled system is a stable solution.
Now we are interested in verifying what will happen if we propose a modification
such as αβ 0.95, β 0.97938, c 0.8, and
J
M
11 0 0
09 0
0010.4
,M
10.2595 0.5771 0
1.6159 9.7403 0
0010.4
, 5.3
it can be easily verified that M holds the conditions of Proposition 3.4. The eigenvalues of
the modified system MA K
1
are λ
1
≈−64.105, λ
2
≈−28.550, and λ
3
≈−69.333. Given that
this eigenvalues hold the conditions of Theorem 2.7 and that 0.8gx also holds the rest of the
conditions, the origin of the modified controlled system is also a stable solution.
In Figures 1 and 4, the simulation step and time were 0.004 and 50 s, respectively. In
the first 25 s, the system was the original one u 0, and for the last 25 s the control u was
activated, for the unmodified and modified systems Figure 2.
5.2. Preservation of Complete Practical Synchronization for
Fractional-Order Chen Systems
In this example specifically, we will make the synchronization of two fractional-order Chen
systems with identical parameters and orders but dierent initial conditions, and because of
these consideration, we will have that A
M
A
S
, and therefore we drop the indices for the
matrices A and M.
12 Mathematical Problems in Engineering
40
20
0
20
40
60
x
1
,x
2
,x
3
05
Time (s)
1510 20 25 30 35 40 45 50
x
1
x
2
x
3
a The unmodified controlled system x
α
A Kx gx
400
200
0
200
400
600
800
x
1
,x
2
,x
3
0 5 10 15
Time (s)
20 25 30 35 40 45 50
x
1
x
2
x
3
b The modified controlled system x
α
MA Kx gx
Figure 1: Graphs of the states versus time of the controlled systems.
10
0
10
20
30
40
50
x
3
20
10
0
10
20 40
20
0
20
40
x
1
x
2
a The system x
α
A Kx gx
0
100
200
300
400
500
600
700
x
3
400
400
200
200
0
0
200
200
400
400
x
1
x
2
b The system x
α
MA Kx gx
Figure 2: Phase plane of the unmodified and modified systems.
We use the structure of the Chen system with fractional-order α as presented in 3.
For the slave system, we have
x
α
S
Ax
S
g
x
S
aa0
d ad 0
00b
x
S
0
x
S
1
x
S
3
x
S
1
x
S
2
w 5.4
Mathematical Problems in Engineering 13
40
20
0
20
40
05
10 15 20
25 30
35
40 45
50
Time (s)
e
1
,e
2
,e
3
e
1
e
2
e
3
a Synchronization error of the systems without modification
40
20
0
20
40
05
10 15 20 25 30 35 40 45 50
Time (s)
e
1
,e
2
,e
3
e
1
e
2
e
3
b Synchronization error of the modified systems
Figure 3: Errors graph with the control law applied at t 25 s.
with w as defined in 4.6, x
S
x
S
1
x
S
2
x
S
3
T
, a 35, b 3, d 28, and α 0.975, initial
conditions x
S
03010
T
.
Now, for the master system, we take
x
α
M
Ax
M
g
x
M
aa0
d ad 0
00b
x
M
0
x
M
1
x
M
3
x
M
1
x
M
2
5.5
with x
M
x
M
1
x
M
2
x
M
3
T
, a 35, b 3, d 28, α 0.975, and initial conditions x
M
0
9 514
T
.
Both master and slave systems are chaotic, as far as the conditions from 25. For the
control law 4.3, we have
J
K
2
26 0 0
0280
004
,K
2
30.1130 34.5700 0
6.9140 32.1130 0
004
, 5.6
and for the modification we take c 0.9, αβ 0.96, β 0.9846, and
J
M
1.075 0 0
00.955 0
001.2
,M
1.0841 0.0768 0
0.0154 0.9459 0
001.2
. 5.7
14 Mathematical Problems in Engineering
5
10
15
20
25
30
35
40
45
40
20
0
20
40
40
20
0
20
40
Master
Slave
x
M
3
,x
S
3
x
M
1
,x
S
1
x
M
2
,x
S
2
a The unmodified synchronization
Master
Slave
5
10
15
20
25
30
35
40
45
40
20
0
20
40
30
20
10
0
10
20
30
x
M
3
,x
S
3
x
M
2
,x
S
2
x
M
1
,x
S
1
b The modified synchronization
Figure 4: Phase plane of the unmodified and modified synchronizations; the master systems are in blue,
and the slave systems in black; the control law is applied at t 50 s.
With K
2
as proposed, it is assured that the synchronization error system without
modification holds the conditions of Theorem 2.7. It can be easily verified that M and 0.9gx
hold the conditions from Proposition 3.4, in such a way that we can assure that this is an
example of preservation of synchronization.
Again the simulation step and time were 0.0028 and 50 s, respectively. In the first 25 s
the system was the original one w 0, and for the last 25 s, the control w was activated, for
the unmodified and modified systems.
In Figure 3, we can observe how the application of the control law, with the same K
2
,
stabilizes the origin of the unmodified and the modified synchronization error systems.
In Figure 4, we can observe the way the slave system follows to the master system for
the unmodified and the modified systems with the same K
2
. All the simulations were made
using the algorithms presented in 26.
From several simulations, we have observed that under large variations in the
parameters these transformations do not preserve chaos. This limits the possible variations
in the transformations because, as it is well known, the chaos in dynamical systems is very
sensitive to variations in the parameters.
6. Conclusions
As far as the authors know, this is the first time that the preservation of autonomous
commensurate fractional order systems stability is made considering transformations that
aect the fractional order, the linear part, and the nonlinear part of the vector field of the
dierential equation.
Furthermore, we have explained how these results can be used to ensure the
preservation of stabilization and the preservation of synchronization of autonomous
commensurate fractional-order systems and, through the presented examples, we have also
showed the eectiveness of the results.
Mathematical Problems in Engineering 15
It is also worth to mention that there are some other results on the stability of
fractional-order autonomous systems 25, 27, 28that can be used in a similar way to
Proposition 3.4 to state the conditions for the preservation of asymptotic stability of the
solutions, for 0 <α<1.
References
1 R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ,
USA, 2000.
2 J. L
¨
u and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,”
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 775–
858, 2006.
3 C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons and
Fractals, vol. 22, no. 3, pp. 549–554, 2004.
4 C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order R
¨
ossler equations,” Physica A, vol.
341, no. 1–4, pp. 55–61, 2004.
5 I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review
Letters, vol. 91, no. 3, Article ID 034101, 4 pages, 2003.
6 T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua’s system,” IEEE
Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995.
7 J. L
¨
u, S. Yu, H. Leung, and G. Chen, “Experimental verification of multidirectional multiscroll chaotic
attractors,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 1, pp. 149–165, 2006.
8 J. L
¨
u and G. Chen, “A time-varying complex dynamical network model and its controlled
synchronization criteria,” IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 841–846, 2005.
9 J. Zhou, J.-A. Lu, and J. L
¨
u, “Adaptive synchronization of an uncertain complex dynamical network,”
IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 652–656, 2006.
10 J. L
¨
u, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical
networks,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 4, pp. 787–796, 2004.
11 Y. Tang and J.-A. Fang, “Synchronization of N-coupled fractional-order chaotic systems with ring
connection,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 401–412,
2010.
12 G. Peng, Y. Jiang, and F. Chen, “Generalized projective synchronization of fractional order chaotic
systems,” Physica A, vol. 387, no. 14, pp. 3738–3746, 2008.
13
X. Y. Wang and J. M. Song, “Synchronization of the fractional order hyperchaos Lorenz systems with
activation feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no.
8, pp. 3351–3357, 2009.
14 J. G. Lu, “Chaotic dynamics of the fractional-order L
¨
u system and its synchronization,” Physics Letters,
Section A, vol. 354, no. 4, pp. 305–311, 2006.
15 D. Matignon, “Stability results for fractional dierential equations with applications to control
processing,” in Proceedings of the Computational Engineering in System Application, pp. 963–968, 1996.
16 R. Mart
´
ınez-Mart
´
ınez, H. Jard
´
on-Kojakhmetov, J. A. Le
´
on, and G. Fern
´
andez-Anaya, Estabilizaci
´
on de
Redes Complejas Fraccionarias de Sistemas de Lorenz y Sistemas de Chen, AMCA, 2009.
17 X.-J. Wen, Z.-M. Wu, and J.-G. Lu, “Stability analysis of a class of nonlinear fractional-order systems,”
IEEE Transsactions on Circuits and Systems-II, vol. 55, pp. 1178–1182, 2008.
18 G. Fern
´
andez-Anaya, J. J. Flores-Godoy, R. F emat, and J. J.
´
Alvarez-Ram
´
ırez, “Preservation of stability
and synchronization in nonlinear systems,” Physics Letters, Section A, vol. 371, no. 3, pp. 205–212, 2007.
19 D. Becker-Bessudo, G. Fernandez-Anaya, and J. J. Flores-Godoy, “Preserving synchronization under
matrix product modifications,” Physica A, vol. 387, no. 26, pp. 6631–6645, 2008.
20 I. Podlubny, Fractional Dierential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 9th
edition, 1999.
21 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Dierential Equations,
John Wiley & Sons, New York, NY, USA, 1993.
22 I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional
dierentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.
23
F. Zhang and C. Li, “Stability analysis of fractional dierential systems with order lying in 1, 2,”
Advances in Dierence Equations, vol. 2011, Article ID 213485, 17 pages, 2011.
16 Mathematical Problems in Engineering
24 Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leer stability of fractional order nonlinear dynamic
systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009.
25 M. S. Tavazoei and M. Haeri, “Chaotic attractors in incommensurate fractional order systems,” Physica
D. Nonlinear Phenomena, vol. 237, no. 20, pp. 2628–2637, 2008.
26 I. Petras, Fractional Order Chaotic Systems, 2010, http://www.mathworks.com/matlabcentral/
fileexchange/27336-fractional-order-chaotic-systems.
27 W. Deng, C. Li, and J. L
¨
u, “Stability analysis of linear fractional dierential system with multiple time
delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007.
28 M. S. Tavazoei and M. Haeri, “A necessary condition for double scroll attractor existence in fractional-
order systems,” Physics Letters, Section A, vol. 367, no. 1-2, pp. 102–113, 2007.
... has been the subject of investigation in [18,30,42]. Asymptotic stability results have been established under the following conditions: ...
Article
Full-text available
Abstract We extend the (integer-order) Halanay inequality with distributed delay to the fractional-order case between one and two. The main feature is the passage from integer order to noninteger order between one and two. This case of order between one and two is more delicate than the case between zero and one because of several difficulties explained in this paper. These difficulties are encountered, in fact, in general differential equations. Here we show that solutions decay to zero as a power function in case the delay kernel satisfies a general (integral) condition. We provide a large class of admissible functions fulfilling this condition. The even more complicated nonlinear case is also addressed, and we obtain a local stability result of power type. Finally, we give an application to a problem arising in neural network theory and an explicit example.
Article
Full-text available
Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this paper, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network.
Article
Full-text available
The stability of n-dimensional linear fractional differential systems with commensurate order 1<α<2 and the corresponding perturbed systems are investigated. By using the Laplace transform, the asymptotic expansion of the Mittag-Leffler function, and the Gronwall inequality, some conditions on stability and asymptotic stability are given.
Article
In this paper, we propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann-Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion.
Article
In this paper, based on the idea of a nonlinear observer, a new method is proposed and applied to “generalized projective synchronization” for a class of fractional order chaotic systems via a transmitted signal. This synchronization approach is theoretically and numerically studied. By using the stability theory of linear fractional order systems, suitable conditions for achieving synchronization are given. Numerical simulations coincide with the theoretical analysis.
Article
In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The lowest order for chaos to be able to appear in this system is found to be 0.1. Master–slave synchronization of chaotic fractional-order Ikeda delay systems with linear coupling is also studied.
Article
In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under modification through matrix multiplication. The objective is to show that under a defined multiplicative group, hyperbolic critical points are preserved along the stable and unstable manifolds. The properties of this multiplicative group were determined through the use of simultaneous Jordan decomposition. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of different modified systems.
In this paper, the synchronization of N-coupled fractional-order chaotic systems with ring connection is firstly investigated in detail. Based on stability criteria of fractional-order system, the synchronization of N-coupled fractional-order chaotic systems with unidirectional coupling and bidirectional coupling is achieved. Moreover, some appropriate comparisons are made to contrast to some of existing results. Finally, some numerical examples are provided to illustrate and verify the effectiveness of the proposed schemes.