Research Article | Open Access
Chaotic Tolerant Synchronization Analysis with Propagation Delay and Actuator Faults
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.
The realization of OGY (Ott-Grebogi-Yorker) chaos-control method  and PC (Pecora and Carroll) synchronization method  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 3 (see ). Let , , and be real matrices of appropriate dimensions, with satisfying . Then, the following inequalities are equivalent:(1);(2)there exists a 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.
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 .
When , inequality (18) is transformed towhereis any constant. The proper dimension matrices , , are arbitrary.
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 .
From Lemma 3, we obtain that is equivalent toBased on the above derivation, we have the following result.
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 havewhere