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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 378376, 8 pages
Finite-Time Synchronization of Singular Hybrid Coupled Networks
1Research Center for Complex Systems and Network Sciences, and Department of Mathematics, Southeast University, Nanjing 210096, China
2Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Received 20 December 2012; Accepted 11 February 2013
Academic Editor: Jong Hae Kim
Copyright © 2013 Cong Zheng and Jinde Cao. 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.
This paper investigates finite-time synchronization of the singular hybrid coupled networks. The singular systems studied in this paper are assumed to be regular and impulse-free. Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled networks under a state feedback controller by using finite-time stability theory. A numerical example is finally exploited to show the effectiveness of the obtained results.
In recent years, singular systems, also known as descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems, are attracting more and more attentions from many fields of scientific research because they can better describe a larger class of dynamic systems than the regular ones. Many results of regular systems have been extended to the area about singular systems such as [1–21]. For example, stability (robust stability or quadratic stability) and stabilization for singular systems have been studied via LMI approach in [2–8]; robust control (or , control and robust dissipative filtering) for singular systems has been discussed in [9–16]; synchronization (or state estimation) for singular complex networks has been considered in [17–21].
Synchronization is an interesting and important characteristic in the coupled networks. There are a lot of results in regular coupled networks. Recently, some authors study synchronization of the singular systems such as [17–21] and the references therein. In , Xiong et al. introduced the singular hybrid coupled systems to describe complex network with a special class of constrains. They gave a sufficient condition for global synchronization of singular hybrid coupled system with time-varying nonlinear perturbation based on Lyapunov stability theory. Synchronization issues are studied for singular systems with delays by using Linear Matrix Inequality (LMI) approach . Koo et al. considered synchronization of singular complex dynamical network with time-varying delays . Li et al. in  investigated synchronization and state estimation for singular complex dynamical networks with time-varying delays. Li et al. in  investigated robust control of synchronization for uncertain singular complex delayed networks with stochastic switched coupling.
Finite-time synchronization or finite-time control is interesting topic for its practical application. There are some results on finite-time stability [22–26], finite-time synchronization [27–33], finite-time consensus or agreement [34–37], and finite-time observers . However, these results are obtained for regular systems. Up to now, to the best of our knowledge, few authors studied finite-time synchronization of singular hybrid coupled systems whose structures are more complex than those in [27–33]. Considering the important role of synchronization of complex networks, the finite-time synchronization of singular hybrid coupled networks is worth studying.
Motivated by the previous discussions, in this paper, we investigate finite-time synchronization of singular hybrid complex systems. Some sufficient conditions for it are obtained by the state feedback controller based on the finite-time stability theory. Finally, a numerical example is exploited to illustrate the effectiveness of the obtained result.
The rest of this paper is organized as follows. In Section 2, a singular hybrid coupled system is given, and some preliminaries are briefly outlined. In Section 3, some sufficient criteria are derived for the finite-time synchronization of the proposed singular system by the feedback controller. In Section 4, an example is provided to show the effectiveness of the obtained results. Some conclusions are finally drawn in Section 5.
2. Model Formulation and Some Preliminaries
Consider a singular hybrid coupled system as follows: where represents the state vector of the th node, are constant matrices, and may be singular. Without loss of generality, we will assume that . is a vector-value function. The constant denotes the coupling strength, and is inner-coupling matrix between nodes. describes the linear coupling configuration of the network, which satisfies
Definition 2. The singular system (1) is said to be synchronized in the finite time, if for a suitable designed feedback controller, there exists a constant (which depends on the initial vector value ), such that and for , .
Assumption 3. Assume that the singular system (1) is connected in the sense that there are no isolated clusters; that is, the matrix is an irreducible matrix.
With Assumption 3, we obtain that zero is an eigenvalue of with multiplicity , and all the other eigenvalues of are strictly negative, which are denoted by . At the same time, since is a symmetric matrix, there exists a unitary matrix such that with and .
Let be a function to which all are expected to synchronize in the finite time. That is, the synchronization state is . Suppose that satisfies the equation . Let . We can obtain the following singular error system:
Assumption 4. Assume that there exist nonnegative constants such that
Assumption 5. There exist matrices such that where , .
Lemma 6 (see ). Suppose that the function is differentiable (the derivative of at is in fact its right derivative) and , , , where , are two constants. Then, for any given , satisfies the following inequality: with given by .
Lemma 7 (Jensen’s Inequality). If are positive numbers and , then
3. Main Results
In this section, we consider the finite-time synchronization of the singular coupled network (1) under the appropriate controllers. In order to control the states of all nodes to the synchronization state in finite time, we apply some simple controllers , to system (1). Then, the controlled system can be written as Then, we have where , .
With Assumption 5, it follows from the proof of Theorem 1 in  and Lemma 2.2 in  that the pair is regular and impulse-free; that is, there exist nonsingular matrices satisfying that where , . So, system (13) is equivalent to where , , and . And , , , , , and .
In order to achieve our aim, we design the following controllers: where is a tunable constant, and the real number satisfies . So, we obtain . That is, .
Remark 8. From (17), the controllers are dependent not only on the coupled matrix , but also on the singular matrix . And from the shape of controllers, we only use the states of slow subsystems (15) in controllers , but we do not consider the states of fast subsystems (16). It is very special. It is interesting for our future research to design more general controller which makes the singular hybrid coupled networks synchronize in finite time.
Theorem 9. Suppose that Assumptions 3, 4, and 5 hold. Under the controllers (17), the singular system (1) is synchronized in a finite time , where , is the initial condition of , and and are defined as (25).
Proof. Consider the following Lyapunov function:
The derivative of along the trajectory of system (13) is
By using Assumption 4, one can get the following inequality:
where and .
Define , where , , , and . Using (7) and (8) (see ), one can obtain that and ; then,
Substituting (8), (21), and (23) into (20) and letting , while , , one has
Let From (22), we get By the use of (24)–(26) and Lemma 7, we can obtain that
From Lemma 6, we have that the solutions of system (15) are globally asymptotically stable with respect to in the finite time ; that is, where .
In the following, we show that are globally asymptotically stable with respect to in the finite time . From (16), one has Similar to the proof of Lemma 2.2 in  and the proof of Theorem 1 in , let , which implies that . One has Then, that is,
With Assumption 4, there must exist nonsingular matrices satisfying the equalities , , where , . Moreover, nonsingular matrices can be suitably chosen to satisfy , for . Therefore, one can obtain , and for from (28) and (32), . Consequently, , and for , . The proof is completed.
If , system (1) is a general nonsingular coupled network. By using the controllers similar to in (17), we can derive the finite-time synchronization of system (1). For simplicity, let . Then, we have the following.
Remark 11. Since the conditions in Assumption 5 are not strict LMIs problems, they cannot be solved directly by the LMI Matlab Toolbox. According to Lemma 1 in , Lemma 1 in , and Remark 3 in , if matrix has the decomposition as where , , and with for , then Assumption 5 can be transformed into a strict LMIs problem.
Corollary 12. Suppose that Assumptions 3 and 4 hold, and matrix has the decomposition as (34) in Remark 11. By the controllers (17), the singular hybrid coupled network (1) can be synchronized in the finite time in the sense of Definition 2, if there exist matrices , , and , , such that where , , and .
Suppose that we choose the average state of all node states as synchronized state; that is, . We have similar results. Before giving these results, we need some assumptions as follows:
Assumption 2′. Assume that there exist nonnegative constants such that
Assumption 3′. There exist matrices such that where , and.
Theorem 13. Suppose that Assumptions 3, 2′, and 3′ hold. By the controllers (17), the singular hybrid coupled network (1) can be synchronized to the average state of all node states in the finite time in the sense of Definition 2.
Corollary 14. Suppose that Assumptions 3 and 2′ hold, and matrix has the decomposition as (34) in Remark 11. By the controllers (17), if there exist matrices , , and , , such that where , , , , and , the singular hybrid coupled network (1) can be synchronized to the average state of all node states in the finite time.
Remark 15. In this paper, we study finite-time synchronization of the singular hybrid coupled networks when the singular systems studied in this paper are assumed to be regular and impulse-free. However, it may be more complicated when we do not assume in advance that the systems are regular and impulse free. Synchronization or finite-time synchronization of singular coupled systems is worth discussing without the assumption that the considered systems are regular and impulsive free.
4. An Illustrative Example
In this section, a numerical example will be given to verify the theoretical results obtained earlier.
Example 16. Consider the following singular hybrid coupled network which is similar to one given in :
where , , , , , , and
Since is symmetric matrix and its six eigenvalues are and, there exists a unitary matrix
such that and .
Choose , , and satisfying Assumptions 3, 4, and 5. Under the controllers defined in Theorem 9, the singular hybrid system (39) can be synchronized in the finite time according to Theorem 9 if . If the controller gain , . Corresponding simulation results are shown in Figures 1 and 2 with initial conditions .
In this paper, we discuss finite-time synchronization of the singular hybrid coupled networks with the assumption that the considered singular systems are regular and impulsive-free. Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled networks under a state feedback controller by finite-time stability theory. A numerical example is finally exploited to show the effectiveness of the obtained results. It will be an interesting topic for the future researches to extend new methods to study synchronization, robust control, pinning control, and finite-time synchronization of singular hybrid coupled networks without the assumption that the considered singular systems are regular and impulsive-free.
This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 61272530 and 11072059 and the Jiangsu Provincial Natural Science Foundation of China under Grant no. BK2012741.
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