Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 785861, 14 pages
http://dx.doi.org/10.1155/2015/785861

Research Article

Chaotic Tolerant Synchronization Analysis with Propagation Delay and Actuator Faults

Zhang Qunli

Department of Mathematics, Heze University, Heze, Shandong 274015, China

Received 20 June 2015; Accepted 24 August 2015

Academic Editor: Laura Gardini

Copyright © 2015 Zhang Qunli. 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.

Abstract.

The criteria for tolerant synchronization with a constant propagation delay and actuator faults are presented by using matrix analysis techniques. A new algorithm, which constructs the extended error systems in order to make the conservation of the stability lower, is proposed. Based on proper Lyapunov-Krasovskii functional, the novel delay-dependent fault tolerant synchronization analyses are derived. Finally, numerical examples show the effectiveness of the proposed method.

1. Introduction

The realization of OGY (Ott-Grebogi-Yorker) chaos-control method [1] and PC (Pecora and Carroll) synchronization method [2] has been attracting researchers’ attention since the 1990s. Chaos synchronization [3–13] is of great practical significance and has aroused great interest in recent years.

They all focused on the design of synchronization under normal operating conditions in the above works described. But, in practical chaotic secure communication systems, sensors, actuators, and inner components may inevitably fail, which can lead to sharp performance decline of chaotic secure. For the reason, fault tolerant synchronization and control [14–20] of chaotic systems have been the hot topic of intensive researches recently.

More recent works studied the fault tolerant synchronization and control, but they were limited to construct Lyapunov-Krasovskii functional

which separates and , where is the synchronization error and is the error of fault function. Motivating the limitation and the extended transformation [21, 22], the effective new method, which does the extended transformation for the error system, to consider Lyapunov-Krasovskii functional

which do not separate and , for the chaotic fault tolerant synchronization with a constant propagation delay and actuator faults, makes the conservation descent. Finally, numerical examples are given to verify the above method.

Notations. In this paper, , and denote, respectively, the real number, the real -vectors, and the real matrices. The superscript “” stands for the transpose of a matrix. The symbol  , where and are symmetric matrices, means that - is positive definite (positive semidefinite). is the identity matrix of appropriate dimensions. “” denotes the matrix entries implied by symmetry.

2. Preliminaries and Systems Description

Consider a chaotic master system with the actuator faults item in the following form:

where is the measurable state vector. is the output vector, and , and are proper dimension constant matrices. , are known continuous nonlinear functions. is the delay.

Assumption 1. There exist the matrices , and the nonlinear functions , and satisfy

for all
The slave system linked with the chaotic master system (3) is described by

where is the measurable state vector and is a constant propagation delay.
Let , , and , and then

Lemma 2 (see [23, 24]). For any constant matrix , , scalar , and vector function such that the integrations concerned are well defined, and then

Lemma 3 (see [25]). Let ,  , and be real matrices of appropriate dimensions, with satisfying . Then, the following inequalities are equivalent:

(1);(2)there exists a scalar such that .

Lemma 4 (see [26–28]). Let , and be real matrices of appropriate dimensions, with satisfying . Then, one has the following:

(1)for any scalar ,

(2)for any matrix and scalar , such that ,

3. Fault Tolerant Synchronization Analysis

3.1. Fault Tolerant Synchronization Analysis When and Are Derivable on

Let , and we have

Suppose , and then

where

From the assumption, we have

So, we can get from assumption that

in which the proper dimension matrices , , are arbitrary.

That is,

where

is any constant. Consider

where

Imitating the above inference, we obtain from inequality (5) the following:

where

We choose Lyapunov-Krasovskii functional

Differentiating with respect to , the following result is yielded:

From Lemma 2, we have

where

From model (12), we have

where

From formulas (18), (20), (24), and (26), we get

where .

Based on the above derivation, we have the following result.

Theorem 5. The fault tolerant synchronization (3) and (6) is achieved if there exist constants , , , and , the positive definite matrices , , and , and the matrices , , , , such that the matrix .

Remark 6. After the extended transformation , system (7) is turned into system (12). Based on the Lyapunov functional in [15–21], we take Lyapunov functional

and we get matrix . The conservation of stability of error system (7) can be decreased by choosing the matrices , , and the constant .

Corollary 7. When in system (3), result similar to Theorem 5 can be obtained.

When , inequality (18) is transformed to

where

is any constant. The proper dimension matrices , , are arbitrary.

We choose Lyapunov functional

and differentiating with respect to and using Lemma 2 yield

where

From model (12), we have

where

Based on the above derivation, we have the following result.

Corollary 8. The fault tolerant synchronization (3) and (5) is achieved if there exist constants , the positive definite matrices , , and the matrices , , , ,  , , such that matrix .

3.2. Fault Tolerant Synchronization Analysis When and Are Derivable on

Let and ,   be proper dimension matrices, with satisfying , where are proper dimension matrices and is identity matrix. Then,

Suppose ; we have

According to , , we get

where . Consider

where

From the assumption, we have

where

is any constant. The proper dimension matrices , , are arbitrary

where

We choose Lyapunov function

where , and it is easy to know that .

The derivation of on is

From Lemma 2, we have

where

From model (3), we have

where

where the proper dimension matrices is arbitrary.

From formulas (42), (44), (48), and (50), we get

where

where

From Lemma 3, we obtain that is equivalent to

Based on the above derivation, we have the following result.

Theorem 9. The fault tolerant synchronization (3) and (6) is achieved if there exist constants , , the positive definite matrices , , and the matrices , , , , , , such that matrix , where

Remark 10. After extending system (37) to singular system (40) by the transformation , taking proper matrices , , and the constant can decrease the conservation of stability of error system (37) by constructing the Lyapunov functional

When , inequality (26) is transformed to

where

is any constant. The proper dimension matrices , , are arbitrary.
We choose Lyapunov function

where , and it is easy to know that , and differentiating with respect to and using Lemma 2 yield

where

From model (12), we have

where

where the proper dimension matrices is arbitrary.

Based on the above derivation, we have the following result.

Corollary 11. The fault tolerant synchronization (3) and (5) is achieved if there exist constants , , the positive definite matrices , , and the matrices , , , , , , such that matrix , where

4. Numerical Examples

Example 1. Consider a typical delayed Hopfield neural networks [29–32] with two neurons

When and are derivable on , we take

Simulation results are shown in Figure 1.
When and are derivable on , we take

Simulation results are shown in Figure 2.

Figure 1: The state responses of and when and are derivable on .

Figure 2: The state responses of and when and are derivable on .

Example 2. Consider chaotic Lü system [31, 32]; that is,

where . We choose

When and are derivable on , we take

Simulation results are shown in Figure 3.
When and are derivable on , we take

Simulation results are shown in Figure 4.

Figure 3: The state responses of and when and are derivable on .

Figure 4: The state responses of and when and are derivable on .

5. Conclusions

In this paper, taking into account the constant propagation delay and actuator faults and the extended transformation of the error systems, the problem of the chaotic fault tolerant synchronization has been addressed. The conservation of stability of the error systems is lower than that of other existing literatures.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11071141), the Natural Science Foundation of Shandong Province of China (ZR2011AL018, ZR2011AQ008), a Project of Shandong Province Higher Educational Science and Technology Program (J13LI02, J11LA06), and the Research Fund Project of Heze University (XY10KZ01).

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