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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 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 f ractional 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.

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 2–6. There is also

several works concerning chaotic systems or complex networks of integer order or their

synchronization, for example, 7–10. There are many diﬀerent 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 diﬀerential

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 diﬀerential 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 diﬀerent

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 diﬀerential 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

20–22. 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

dτ

m

t − τ

m−α−1

dτ,

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

<δthen

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 suﬃcient

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| >απ/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 diﬀerential 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| >απ/2. In this case, the components

of the state decay towards 0 like t

−α−1

. 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

<δthen 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 g00 and lim

x→0

gx/x0,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

α−k

t

tt

0

x

k−1

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, 00; moreover, gt, x holds the Lipschitz condition with respect to

x.

Theorem 2.8. If the matrix A such that | argspecA|

/

0, | argspecA| >απ/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, Aαβ, MA, such that the new system

x

αβ

MAx, with 0 <β≤ 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, Ψα, 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 Ψα, 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 <β≤ 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

>φ

k

>φ

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

| >απ/2 for k 1, 2,...,n.

By the hypothesis ρMA ≥ ρA, ρA > 1, and 0 <β≤ 1, we have that αβρMA > 1,

and we have asked for gx to hold g00 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| >αβπ/2 to

hold, so ﬁrst we want for

φ

k

θ

k

>α

π

2

,k∈

{

1, 2,...,n

}

,

3.8

to hold. The last part of the hypothesis states that the inequality 3.7 holds. 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 >φ

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 <β≤ 1, we have that αβ ≤ α and therefore |φ

k

θ

k

| >αβπ/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 Ψα, 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/α < β < 2/α, 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

>φ

k

>φ

max

3.11

for each k 1, 2,...,n, and if | argspecA|

/

0, |arg(spec(A))| >απ/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

| >απ/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

Ψα, Mαβ, 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

>φ

k

>φ

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 diﬀerent 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 gxu, 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

A−K

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 cgxu, 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 eﬀectiveness

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 diﬀerential 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 diﬀerent 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

00−b

⎤

⎦

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

03010

T

.

Now, for the master system, we take

x

α

M

Ax

M

g

x

M

⎡

⎣

−aa0

d − ad 0

00−b

⎤

⎦

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

aﬀect the fractional order, the linear part, and the nonlinear part of the vector ﬁeld of the

diﬀerential 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 eﬀectiveness 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, 28 that 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.

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