Finite-Time Synchronization of Singular Hybrid Coupled Networks

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.


Introduction
In recent years, singular systems, also known as descriptor systems, generalized state-space systems, differentialalgebraic 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 .For example, stability (robust stability or quadratic stability) and stabilization for singular systems have been studied via LMI approach in [2][3][4][5][6][7][8]; robust control (or  2 ,  ∞ control and robust dissipative filtering) for singular systems has been discussed in [9][10][11][12][13][14][15][16]; synchronization (or state estimation) for singular complex networks has been considered in [17][18][19][20][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][18][19][20][21] and the references therein.In [17], 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 [18].Koo et al. considered synchronization of singular complex dynamical network with time-varying delays [19].Li et al. in [20] investigated synchronization and state estimation for singular complex dynamical networks with time-varying delays.Li et al. in [21] investigated robust  ∞ control of synchronization for uncertain singular complex delayed networks with stochastic switched coupling.
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 finitetime 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.

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   () ∈ R  ,  = 1, 2, . . ., , to system (1).Then, the controlled system can be written as Then, we have where V  =   ,  = ( 1 ,  2 , . . .,   ).
With Assumption 5, it follows from the proof of Theorem 1 in [2] and Lemma 2.2 in [3] that the pair (,  +   Γ) is regular and impulse-free; that is, there exist nonsingular matrices   ,   ∈ R × satisfying that where   ∈ R × ,  = 1, 2, . . ., .So, system ( 13) is equivalent to where ) .In order to achieve our aim, we design the following controllers: where ) . ( > 0 is a tunable constant, and the real number  satisfies 0 <  < 1.So, we obtain 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  1  of slow subsystems (15) in controllers V  , but we do not consider the states  2  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.
If rank() = , system (1) is a general nonsingular coupled network.By using the controllers   similar to V  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 [17], Lemma 1 in [9], and Remark 3 in [18], if matrix  has the decomposition as where  = ( where   > (4/)( − 1) ‖   ‖,  = ∑  =1   , 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   ∈ R × ,   ≥ 0, and   ∈ R (−)× ,  = 1, 2, . . ., , such that where   > (4/)( − 1) ‖   ‖,   = ( 1     1  +  2   ),  = 2, . . ., ,  = ∑  =1   , and   = ∑  =1, ̸ =    , 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.

An Illustrative Example
In this section, a numerical example will be given to verify the theoretical results obtained earlier.

Conclusions
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 impulsivefree.Some sufficient conditions are derived to ensure finitetime 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.