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 functionalwhich 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 functionalwhich 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 satisfyfor all
The slave system linked with the chaotic master system (3) is described bywhere 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 haveSuppose , and thenwhereFrom the assumption, we haveSo, we can get from assumption thatin which the proper dimension matrices , , are arbitrary.
That is,whereis any constant. ConsiderwhereImitating the above inference, we obtain from inequality (5) the following:whereWe choose Lyapunov-Krasovskii functionalDifferentiating with respect to , the following result is yielded:From Lemma 2, we havewhereFrom model (12), we havewhereFrom formulas (18), (20), (24), and (26), we getwhere .
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 functionaland 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 towhereis any constant. The proper dimension matrices , , are arbitrary.
We choose Lyapunov functionaland differentiating with respect to and using Lemma 2 yieldwhereFrom model (12), we havewhereBased 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 haveAccording to , , we getwhere . ConsiderwhereFrom the assumption, we havewhereis any constant. The proper dimension matrices , , are arbitrarywhereWe choose Lyapunov functionwhere , and it is easy to know that .
The derivation of on isFrom Lemma 2, we havewhereFrom model (3), we havewherewhere the proper dimension matrices is arbitrary.
From formulas (42), (44), (48), and (50), we getwherewhere
From Lemma 3, we obtain that is equivalent toBased 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 functionalWhen , inequality (26) is transformed towhereis any constant. The proper dimension matrices , , are arbitrary.
We choose Lyapunov functionwhere , and it is easy to know that , and differentiating with respect to and using Lemma 2 yieldwhereFrom model (12), we havewherewhere 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 neuronsWhen and are derivable on , we takeSimulation results are shown in Figure 1.
When and are derivable on , we takeSimulation results are shown in Figure 2.
Example 2. Consider chaotic Lü system [31, 32]; that is,where . We chooseWhen and are derivable on , we takeSimulation results are shown in Figure 3.
When and are derivable on , we takeSimulation results are shown in Figure 4.
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).