Finite-Time Synchronization of Singular Hybrid Coupled Networks

Abstract

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.

1. Introduction

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 𝐻₂, 𝐻_∞ 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 [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 𝐻_∞ 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 [38]. 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: 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗(𝑡)Γ𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 represents the state vector of the 𝑖th node, 𝐴,𝐸∈ℝ^𝑛×𝑛 are constant matrices, and 𝐸 may be singular. Without loss of generality, we will assume that 0<rank(𝐸)=𝑟<𝑛. 𝑓(𝑥_𝑖(𝑡),𝑡) is a vector-value function. The constant 𝑐>0 denotes the coupling strength, and Γ=diag(𝛾₁,𝛾₂,…,𝛾_𝑛)∈ℝ^𝑛×𝑛 is inner-coupling matrix between nodes. 𝐵=(𝑏_𝑖𝑗)_𝑁×𝑁 describes the linear coupling configuration of the network, which satisfies 𝑏_𝑖𝑗=𝑏_𝑗𝑖𝑏,for𝑖≠𝑗,_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑏_𝑖𝑗,𝑖=1,2,…,𝑁. (2)

Remark 1. If rank(𝐸)=𝑛, then system (1) is a general nonsingular coupled network. We will also give a sufficient condition of the finite-time synchronization for this circumstance. See Corollary 10.

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 𝑡^∗>0 (which depends on the initial vector value 𝑥(0)=(𝑥^𝑇₁(0),𝑥^𝑇₂(0),…,𝑥^𝑇_𝑁(0))^𝑇), such that lim_{𝑡→𝑡∗}‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖=0 and ‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≡0 for 𝑡>𝑡^∗, 𝑖,𝑗=1,2,…,𝑁.

Assumption 3. Assume that the singular system (1) is connected in the sense that there are no isolated clusters; that is, the matrix 𝐵 is an irreducible matrix.

With Assumption 3, we obtain that zero is an eigenvalue of 𝐵 with multiplicity 1, and all the other eigenvalues of 𝐵 are strictly negative, which are denoted by 0=𝜆₁>𝜆₂≥⋯≥𝜆_𝑁. At the same time, since 𝐵 is a symmetric matrix, there exists a unitary matrix 𝑊=(𝑊₁,𝑊₂,…,𝑊_𝑁)∈ℝ^𝑛×𝑛 such that 𝐵=𝑊Λ𝑊^𝑇 with 𝑊𝑊^𝑇=𝐼 and Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑁).

Let 𝑠(𝑡) be a function to which all 𝑥_𝑖(𝑡) are expected to synchronize in the finite time. That is, the synchronization state is 𝑠(𝑡). Suppose that 𝑠(𝑡) satisfies the equation 𝐸𝑠(̇𝑡)=𝐴𝑠(𝑡)+𝑓(𝑠(𝑡),𝑡). Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑁. We can obtain the following singular error system: 𝐸𝑒_𝑖̇(𝑡)=𝐴𝑒_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡−𝑓(𝑠(𝑡),𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗(𝑡)Γ𝑒_𝑗(𝑡),𝑖=1,2,…,𝑁. (3)

Let 𝑒(𝑡)=(𝑒₁(𝑡),𝑒₂(𝑡),…,𝑒_𝑁(𝑡)), 𝑦(𝑡)=𝑒(𝑡)𝑊; then system (3) can be written as 𝐸𝑒̇(𝑡)=𝐴𝑒(𝑡)+𝐹(𝑒(𝑡),𝑡)+𝑐Γ𝑒(𝑡)𝐵^𝑇,𝐸𝑦̇(𝑡)=𝐴𝑦(𝑡)+𝐹(𝑒(𝑡),𝑡)𝑊+𝑐Γ𝑦(𝑡)Λ, (4) where 𝐹(𝑒(𝑡),𝑡) = (𝑓(𝑥₁(𝑡),𝑡)−𝑓(𝑠(𝑡),𝑡),𝑓(𝑥₂(𝑡),𝑡)−𝑓(𝑠(𝑡),𝑡),…,𝑓(𝑥_𝑁(𝑡),𝑡)−𝑓(𝑠(𝑡),𝑡)), 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑁(𝑡)),and 𝑦_𝑖(𝑡)=𝑒(𝑡)𝑊_𝑖∈ℝ^𝑛, 𝑖=1,2,…,𝑁. Then, system (4) can be written as 𝐸𝑦̇_𝑖(𝑡)=𝐴𝑦_𝑖(𝑡)+𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑐𝜆_𝑖Γ𝑦_𝑖(𝑡)=𝐴+𝑐𝜆_𝑖Γ𝑦_𝑖(𝑡)+𝐹(𝑒(𝑡),𝑡)𝑊_𝑖. (5)

Therefore, the finite-time synchronization problem of system (1) is equivalent to the finite-time stabilization of system (5) at the origin under the suitable controllers 𝑢_𝑖, 𝑖=1,2,…,𝑁.

Assumption 4. Assume that there exist nonnegative constants 𝐿_𝑖 such that 𝑓𝑥_𝑖(𝑡),𝑡−𝑓(𝑠(𝑡),𝑡)≤𝐿_𝑖𝑥_𝑖(𝑡)−𝑠(𝑡),𝑖=1,2,…,𝑁. (6)

Assumption 5. There exist matrices 𝑃_𝑖 such that 𝐸^𝑇𝑃_𝑖=𝑃^𝑇_𝑖𝐴𝐸≥0,𝑖=1,2,…,𝑁,_𝑇𝑃₁+𝑃^𝑇₁𝐴<0,(𝐴+𝑐𝜆_𝑖Γ)^𝑇𝑃_𝑖+𝑃^𝑇_𝑖𝐴+𝑐𝜆_𝑖Γ≤−𝜂_𝑖𝐼,𝑖=2,…,𝑁, (7)(8) where 𝜂_𝑖>2𝐿(𝑁−1)‖𝑃_𝑖‖, 𝐿=∑^𝑁_𝑖=1𝐿_𝑖.

Lemma 6 (see [26]). Suppose that the function 𝑉(𝑡)∶[𝑡₀,∞)→[0,∞) is differentiable (the derivative of 𝑉(𝑡) at 𝑡₀ is in fact its right derivative) and 𝑉̇(𝑡)≤−𝐾(𝑉(𝑡))^𝛼, ∀𝑡≥0, 𝑉(𝑡₀)≥0, where 𝐾>0, 0<𝛼<1 are two constants. Then, for any given 𝑡₀, 𝑉(𝑡) satisfies the following inequality: 𝑉^1−𝛼(𝑡)≤𝑉^1−𝛼𝑡₀−𝐾(1−𝛼)𝑡−𝑡₀,𝑡₀≤𝑡≤𝑡^∗,𝑉(𝑡)≡0,∀𝑡>𝑡^∗, (9) with 𝑡^∗ given by 𝑡^∗=𝑡₀+𝑉^1−𝛼(𝑡₀)/𝐾(1−𝛼).

Lemma 7 (Jensen’s Inequality). If 𝑎₁,𝑎₂,…,𝑎_𝑛 are positive numbers and 0<𝑟<𝑝, then ^𝑛_𝑖=1𝑎^𝑝_𝑖^(1/𝑝)≤^𝑛_𝑖=1𝑎^𝑟_𝑖^(1/𝑟). (10)

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 𝑠(𝑡) in finite time, we apply some simple controllers 𝑢_𝑖(𝑡)∈ℝ^𝑛,𝑖=1,2,…,𝑁, to system (1). Then, the controlled system can be written as 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑢_𝑖,𝑖=1,2,…,𝑁. (11) Then, we have 𝐸𝑒_𝑖̇(𝑡)=𝐴𝑒_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡−𝑓(𝑠(𝑡),𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑢_𝑖,𝐸𝑦̇_𝑖(𝑡)=𝐴+𝑐𝜆_𝑖Γ𝑦_𝑖(𝑡)+𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑣_𝑖, (12)(13) where 𝑣_𝑖=𝑢𝑊_𝑖, 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑁).

With Assumption 5, it follows from the proof of Theorem  1 in [2] and Lemma  2.2 in [3] that the pair (𝐸,𝐴+𝑐𝜆_𝑖Γ) is regular and impulse-free; that is, there exist nonsingular matrices 𝑀_𝑖,𝑄_𝑖∈ℝ^𝑛×𝑛 satisfying that 𝑀_𝑖𝐸𝑄_𝑖=diag𝐼_𝑟,𝑀,0_𝑖𝐴+𝑐𝜆_𝑖Γ𝑄_𝑖=diag𝐴_𝑖,𝐼_𝑛−𝑟, (14) where 𝐴_𝑖∈ℝ^𝑟×𝑟, 𝑖=1,2,…,𝑁. So, system (13) is equivalent to 𝑦̇¹_𝑖(𝑡)=𝐴_𝑖𝑦¹_𝑖(𝑡)+𝑀¹_𝑖𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑀¹_𝑖𝑣_𝑖,0=𝑦²_𝑖(𝑡)+𝑀²_𝑖𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑀²_𝑖𝑣_𝑖, (15)(16) where 𝑄_𝑖⁻¹𝑦_𝑖(𝑡)=^𝑦1𝑖_𝑦^(𝑡)_2𝑖(𝑡), 𝑦¹_𝑖(𝑡)∈ℝ^𝑟, and 𝑦²_𝑖(𝑡)∈ℝ^𝑛−𝑟. And 𝑀_𝑖=^𝑀1𝑖_𝑀2𝑖, 𝑀¹_𝑖∈ℝ^𝑟×𝑛, 𝑀²_𝑖∈ℝ^{(𝑛−𝑟)×𝑛}, 𝑄_𝑖=(^𝑄1_𝑖^𝑄2_𝑖), 𝑄¹_𝑖∈ℝ^𝑛×𝑟, and 𝑄²_𝑖∈ℝ^{𝑛×(𝑛−𝑟)}.

In order to achieve our aim, we design the following controllers: 𝑣_𝑖=−𝑘𝑀_𝑖⁻¹sign𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖^𝛽, (17) where 𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖=𝑀_𝑖𝐸𝑦_𝑖(𝑡)=𝑀_𝑖𝐸𝑄_𝑖𝑄_𝑖⁻¹𝑦_𝑖(𝑡)=𝐼_𝑟0𝑦00𝑦^1𝑖2_𝑖=𝑦¹_𝑖0,𝑀_𝑖𝐸𝑦_𝑖(𝑡)^𝛽=𝑦¹_𝑖1(𝑡)^𝛽,…,𝑦¹_𝑖𝑟(𝑡)^𝛽,0,…,0_𝑛−𝑟^𝑇,sign𝑀_𝑖𝐸𝑦_𝑖(𝑡)=diagsign𝑦¹_𝑖1(𝑡),sign𝑦¹_𝑖2(𝑡)𝑦,…,sign¹_𝑖𝑟(𝑡),0,…,0_𝑛−𝑟. (18)𝑘>0 is a tunable constant, and the real number 𝛽 satisfies 0<𝛽<1. So, we obtain 𝑢=(𝑣₁,𝑣₂,…,𝑣_𝑁)𝑊⁻¹=𝑣(𝜉₁,𝜉₂,…,𝜉_𝑁). That is, 𝑢_𝑖=𝑣𝜉_𝑖.

Remark 8. From (17), the controllers 𝑢_𝑖 are dependent not only on the coupled matrix 𝐵, but also on the singular matrix 𝐸. And from the shape of controllers, we only use the states 𝑦¹_𝑖 of slow subsystems (15) in controllers 𝑣_𝑖, but we do not consider the states 𝑦²_𝑖 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 𝑡^∗=𝑡₀+(𝑉^(1−𝛽)/2(𝑡₀)/𝑎𝑘𝑏^{−(1+𝛽)/2}(1−𝛽)), where 𝑉(𝑡₀)=∑^𝑁_𝑖=1𝑦^𝑇_𝑖(𝑡₀)𝐸^𝑇𝑃_𝑖𝑦_𝑖(𝑡₀)=∑^𝑁_𝑖=1(𝑒(𝑡₀)𝑊_𝑖)^𝑇𝐸^𝑇𝑃_𝑖(𝑒(𝑡₀)𝑊_𝑖), 𝑒(𝑡₀) is the initial condition of 𝑒(𝑡), and 𝑎 and 𝑏 are defined as (25).

Proof. Consider the following Lyapunov function: 𝑉(𝑡)=^𝑁_𝑖=1𝑦^𝑇_𝑖(𝑡)𝐸^𝑇𝑃_𝑖𝑦_𝑖(𝑡). (19) The derivative of 𝑉(𝑡) along the trajectory of system (13) is 𝑉̇(𝑡)=^𝑁_𝑖=1𝑦^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝐴+𝑐𝜆_𝑖Γ𝑦_𝑖(𝑡)+𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑣_𝑖+𝐴+𝑐𝜆_𝑖Γ𝑦_𝑖(𝑡)+𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑣_𝑖^𝑇𝑃_𝑖𝑦_𝑖(𝑡=_𝑁𝑖=1)𝑦^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝐴+𝑐𝜆_𝑖Γ+𝐴+𝑐𝜆_𝑖Γ^𝑇𝑃_𝑖𝑦_𝑖(𝑡)+2𝑦^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝐹(𝑒(𝑡),𝑡)𝑊_𝑖−2𝑘𝑦^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝑀_𝑖⁻¹sign𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖^𝛽. (20) By using Assumption 4, one can get the following inequality: 𝐹(𝑒(𝑡),𝑡)𝑊_𝑖=^𝑁_𝑘=1𝑓𝑥_𝑘(𝑡),𝑡−𝑓(𝑠(𝑡),𝑡)𝑊_𝑖𝑘≤^𝑁_𝑘=1𝐿_𝑘𝑒_𝑘(𝑡=_𝑁𝑘=1)𝐿_𝑘𝑦(𝑡)𝜉_𝑘≤^𝑁_𝑘=1𝐿_𝑘𝑦(𝑡)=𝐿𝑦(𝑡)≤𝐿^𝑁_𝑘=1𝑦_𝑘(𝑡), (21) where 𝑊_𝑖=(𝑊_𝑖1,𝑊_𝑖2,…,𝑊_𝑖𝑛)^𝑇 and (𝜉₁,𝜉₂,…,𝜉_𝑁)=𝑊⁻¹=𝑊^𝑇.
Define 𝑀_𝑖^−𝑇𝑃_𝑖𝑄_𝑖=^{𝑃1𝑖𝑃2𝑖}_{𝑃3𝑖𝑃4𝑖}, where 𝑃¹_𝑖∈ℝ^𝑟×𝑟, 𝑃²_𝑖∈ℝ^{𝑟×(𝑛−𝑟)}, 𝑃³_𝑖∈ℝ^{(𝑛−𝑟)×𝑟}, and 𝑃⁴_𝑖∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)}. Using (7) and (8) (see [2]), one can obtain that 𝑃¹_𝑖=(𝑃¹_𝑖)^𝑇>0 and 𝑃²_𝑖=0; then,𝑉(𝑡)=^𝑁_𝑖=1𝑦^𝑇_𝑖(𝑡)𝐸^𝑇𝑃_𝑖𝑦_𝑖(𝑡)=^𝑁_𝑖=1𝑦¹_𝑖(𝑡)^𝑇𝑃¹_𝑖𝑦¹_𝑖(𝑡),−2𝑘𝑦^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝑀_𝑖⁻¹sign𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖𝑀_𝑖𝐸𝑒(𝑡)𝑊_𝑖^𝛽=−2𝑘𝑦^𝑇_𝑖(𝑡)𝑃^𝑇_𝑖𝑀_𝑖⁻¹sign𝑦¹_𝑖1(𝑡)𝑦¹_𝑖1(𝑡)^𝛽𝑦,…,sign¹_𝑖𝑟(𝑡)𝑦¹_𝑖𝑟(𝑡)^𝛽,0,…,0=−2𝑘𝑦^𝑇_𝑖(𝑡)𝑄_𝑖^−𝑇𝑃¹_𝑖0𝑃³_𝑖𝑃⁴_𝑖^𝑇𝑀_𝑖𝑀_𝑖⁻¹×diagsign𝑦¹_𝑖(𝑡)𝑦,0^1𝑖(𝑡)^𝛽0=−2𝑘𝑦_𝑖^1𝑇(𝑡)𝑃_𝑖^1𝑇sign𝑦¹_𝑖(𝑡)𝑦¹_𝑖(𝑡)^𝛽≤−2𝑘𝜆_min𝑃_𝑖^1𝑇𝑦¹_𝑖(𝑡)^1+𝛽. (22)(23)
Substituting (8), (21), and (23) into (20) and letting 𝜂₁=0, while 𝜂_𝑖≥2𝐿(𝑁−1)‖𝑃_𝑖‖, 𝑖=2,…,𝑁, one has 𝑉̇(𝑡)≤^𝑁_𝑖=1−𝜂_𝑖𝑦_𝑖(𝑡)²+2𝐿𝑃_𝑖^𝑁_𝑗=1𝑦_𝑗(𝑡)𝑦_𝑖(𝑡)−2𝑘𝜆_min𝑃_𝑖^1𝑇𝑦¹_𝑖(𝑡)^1+𝛽≤^𝑁_𝑖=1−2𝑘𝜆_min𝑃_𝑖^1𝑇𝑦¹_𝑖(𝑡)^1+𝛽. (24)
Let 𝑎≜min_𝑖𝜆_min𝑃_𝑖^1𝑇,𝑏≜max_𝑖𝜆_max𝑃_𝑖^1𝑇. (25) From (22), we get 𝑎^𝑁_𝑖=1𝑦¹_𝑖(𝑡)²≤𝑉(𝑡)≤𝑏^𝑁_𝑖=1𝑦¹_𝑖(𝑡)². (26) By the use of (24)–(26) and Lemma 7, we can obtain that 𝑉̇(𝑡)≤−2𝑎𝑘^𝑁_𝑖=1𝑦¹_𝑖(𝑡)^1+𝛽≤−2𝑎𝑘^𝑁_𝑖=1𝑦_𝑖(𝑡)^2(1+𝛽)/21≤−2𝑎𝑘𝑏𝑉(𝑡)^(1+𝛽)/2=−2𝑎𝑘𝑏^{−(1+𝛽)/2}(𝑉(𝑡))^(1+𝛽)/2. (27)
From Lemma 6, we have that the solutions 𝑦¹_𝑖(𝑡) of system (15) are globally asymptotically stable with respect to 𝑦¹_𝑖(𝑡)=0 in the finite time 𝑡^∗; that is, lim_{𝑡→𝑡∗}𝑦¹_𝑖(𝑡)=0,𝑦¹_𝑖(𝑡)=0for𝑡≥𝑡^∗, (28) where 𝑡^∗=𝑡₀+(𝑉^(1−𝛽)/2(𝑡₀)/𝑎𝑘𝑏^{−(1+𝛽)/2}(1−𝛽)).
In the following, we show that 𝑦²_𝑖(𝑡) are globally asymptotically stable with respect to 𝑦²_𝑖(𝑡)=0 in the finite time 𝑡^∗. From (16), one has 𝑦²_𝑖(𝑡)≤𝑀²_𝑖𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑀²_𝑖𝑣_𝑖. (29) Similar to the proof of Lemma  2.2 in [3] and the proof of Theorem  1 in [17], let 𝑀²_𝑖(𝑀²_𝑖)^𝑇=𝐼_𝑛−𝑟, which implies that ‖𝑀²_𝑖‖=1. One has 𝑦²_𝑖(𝑡)≤𝐹(𝑒(𝑡),𝑡)𝑊_𝑖+𝑢_𝑖𝑊_𝑖≤𝐿^𝑁_𝑗=1𝑦_𝑗(𝑡)+−𝑘𝑀_𝑖⁻¹sign𝑀_𝑖𝐸𝑦_𝑖(𝑡)×𝑀_𝑖𝐸𝑦_𝑖(𝑡)^𝛽≤𝐿^𝑁_𝑗=1𝑄_𝑗𝑦¹_𝑗(𝑡)+𝑦²_𝑗(𝑡)+𝑘𝑀_𝑖⁻¹×𝑦¹_𝑖(𝑡)^𝛽. (30) Then, ^𝑁_𝑖=1𝑦²_𝑖(𝑡)≤^𝑁_𝑖=1𝐿^𝑁_𝑗=1𝑄_𝑗𝑦¹_𝑗(𝑡)+𝑦²_𝑗(𝑡𝑀)+𝑘_𝑖⁻¹𝑦¹_𝑖(𝑡)^𝛽; (31) that is, 1−𝑁𝐿𝑄^𝑁_𝑖=1𝑖𝑦²_𝑖(𝑡)≤^𝑁_𝑖=1𝑁𝐿𝑄_𝑖𝑦¹_𝑗(𝑡𝑀)+𝑘_𝑖⁻¹𝑦¹_𝑖(𝑡)^𝛽. (32)
With Assumption 4, there must exist nonsingular matrices 𝑀_𝑖,𝑄_𝑖∈ℝ^𝑛×𝑛 satisfying the equalities 𝑀_𝑖𝐸𝑄_𝑖=diag{𝐼_𝑟,0}, 𝑀_𝑖(𝐴+𝑐𝜆_𝑖Γ)𝑄_𝑖=diag{𝐴_𝑖,𝐼_𝑛−𝑟}, where 𝐴_𝑖∈ℝ^𝑟×𝑟, 𝑖=1,2,…,𝑁. Moreover, nonsingular matrices 𝑄_𝑖 can be suitably chosen to satisfy 1−𝑁𝐿‖𝑄_𝑖‖>0, for ∀𝑖∈{1,2,…,𝑁}. Therefore, one can obtain lim_{𝑡→𝑡∗}‖𝑦²_𝑖(𝑡)‖=0, and ‖𝑦²_𝑖(𝑡)‖=0 for 𝑡≥𝑡^∗ from (28) and (32), 𝑖=1,2,…,𝑁. Consequently, lim_{𝑡→𝑡∗}‖𝑒_𝑖(𝑡)‖=0, and ‖𝑒_𝑖(𝑡)‖=0 for 𝑡≥𝑡^∗, 𝑖=1,2,…,𝑁. The proof is completed.

If rank(𝐸)=𝑛, system (1) is a general nonsingular coupled network. By using the controllers 𝑢_𝑖 similar to 𝑣_𝑖 in (17), we can derive the finite-time synchronization of system (1). For simplicity, let 𝐸=𝐼_𝑛. Then, we have the following.

Corollary 10. When 𝐸=𝐼_𝑛, under Assumptions 3 and 4, let the controllers 𝑢_𝑖(𝑡) be as follows: 𝑢_𝑖(𝑡)=−𝑘sign𝑒_𝑖(𝑡)𝑒_𝑖(𝑡)^𝛽,𝑖=1,2,…,𝑁; (33) system (1) is synchronized in a finite time.

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 𝐸 has the decomposition as 𝐼𝐸=𝑈^𝑟0Ξ00^𝑟00𝐼_𝑛−𝑟𝑉^𝑇, (34) where 𝑈=(𝑈₁,𝑈₂), 𝑉=(𝑉₁,𝑉₂), and Ξ_𝑟=diag{𝜚₁,…,𝜚_𝑟} with 𝜚_𝑖>0 for 𝑖=1,2,…,𝑟, then Assumption 5 can be transformed into a strict LMIs problem.

Corollary 12. Suppose that Assumptions 3 and 4 hold, and matrix 𝐸 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 𝑇_𝑖∈ℝ^𝑟×𝑟, 𝑇_𝑖≥0, and 𝑆_𝑖∈ℝ^{(𝑛−𝑟)×𝑟}, 𝑖=1,2,…,𝑁, such that 𝐴^𝑇𝑈₁𝑇₁𝑈^𝑇₁𝐸+𝑈₂𝑆₁+𝑈₁𝑇₁𝑈^𝑇₁𝐸+𝑈₂𝑆₁^𝑇𝐴<0,𝐴+𝑐𝜆_𝑖Γ^𝑇𝑈₁𝑇_𝑖𝑈^𝑇₁𝐸+𝑈₂𝑆_𝑖+𝑈₁𝑇_𝑖𝑈^𝑇₁𝐸+𝑈₂𝑆_𝑖^𝑇𝐴+𝑐𝜆_𝑖Γ≤−𝜂_𝑖𝐼, (35) where 𝜂_𝑖>2𝐿(𝑁−1)‖𝑃_𝑖‖, 𝑃_𝑖=(𝑈₁𝑇_𝑖𝑈^𝑇₁𝐸+𝑈₂𝑆_𝑖),𝑖=2,…,𝑁, and 𝐿=∑^𝑁_𝑖=1𝐿_𝑖.

Suppose that we choose the average state of all node states as synchronized state; that is, 𝑠(𝑡)=(1/𝑁)∑^𝑁_𝑘=1𝑥_𝑘(𝑡). We have similar results. Before giving these results, we need some assumptions as follows:

Assumption  2′. Assume that there exist nonnegative constants 𝐿_𝑖𝑗 such that 𝑓𝑥_𝑖(𝑡),𝑡−𝑓𝑥_𝑗(𝑡),𝑡≤𝐿_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡),𝑖,𝑗=1,2,…,𝑁. (36)

Assumption  3′. There exist matrices 𝑃_𝑖 such that 𝐸^𝑇𝑃_𝑖=𝑃^𝑇_𝑖𝐴𝐸≥0,𝑖=1,2,…,𝑁,_𝑇𝑃₁+𝑃^𝑇₁𝐴<0,𝐴+𝑐𝜆_𝑖Γ^𝑇𝑃_𝑖+𝑃^𝑇_𝑖𝐴+𝑐𝜆_𝑖Γ≤−𝜍_𝑖𝐼,𝑖=2,…,𝑁, (37) where 𝜍_𝑖>(4𝐿/𝑁)(𝑁−1)‖𝑃_𝑖‖,𝐿=∑^𝑁_𝑖=1𝐿_𝑖, and𝐿_𝑖=∑^𝑁_{𝑘=1,𝑘≠𝑖}𝐿_𝑖𝑘.

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 𝐸 has the decomposition as (34) in Remark 11. By the controllers (17), if there exist matrices 𝑇_𝑖∈ℝ^𝑟×𝑟, 𝑇_𝑖≥0, and 𝑆_𝑖∈ℝ^{(𝑛−𝑟)×𝑟}, 𝑖=1,2,…,𝑁, such that 𝐴^𝑇𝑈₁𝑇₁𝑈^𝑇₁𝐸+𝑈₂𝑆₁+𝑈₁𝑇₁𝑈^𝑇₁𝐸+𝑈₂𝑆₁^𝑇𝐴<0,𝐴+𝑐𝜆_𝑖Γ^𝑇𝑈₁𝑇_𝑖𝑈^𝑇₁𝐸+𝑈₂𝑆_𝑖+𝑈₁𝑇_𝑖𝑈^𝑇₁𝐸+𝑈₂𝑆_𝑖^𝑇𝐴+𝑐𝜆_𝑖Γ≤−𝜂_𝑖𝐼, (38) where 𝜂_𝑖>(4𝐿/𝑁)(𝑁−1)‖𝑃_𝑖‖, 𝑃_𝑖=(𝑈₁𝑇_𝑖𝑈^𝑇₁𝐸+𝑈₂𝑆_𝑖), 𝑖=2,…,𝑁, 𝐿=∑^𝑁_𝑖=1𝐿_𝑖, and 𝐿_𝑖=∑^𝑁_{𝑘=1,𝑘≠𝑖}𝐿_𝑖𝑘, 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 [18]: 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡+𝑐⁶_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑢_𝑖,𝑖=1,2,…,6, (39) where 𝑥_𝑖(𝑡)=(𝑥¹_𝑖(𝑡),𝑥²_𝑖(𝑡))^𝑇, 𝑓(𝑥_𝑖(𝑡),𝑡)=((1/15)tanh(𝑥¹_𝑖(𝑡)),tanh(𝑥²_𝑖(𝑡)))^𝑇, 𝑠(𝑡)=(0,0)^𝑇, 𝐿_𝑖=1/15, 𝐿=2/5, 𝑐=1, and ,.𝐸=8000,𝐴=−1011−10,Γ=1001𝐵=−5111111−4111011−4101111−4101101−4110101−3 (40) Since 𝐵 is symmetric matrix and its six eigenvalues are 𝜆₁=0,𝜆₂=−3,𝜆₃=−4,𝜆₄=−5,𝜆₅=−6,and𝜆₆=−6, there exists a unitary matrix 𝑊=𝑊₁,…,𝑊₆=−1√61000−√6−2√6−1√61√610−√21√60−1√610−√210−√61√6−1√61√601√21√60−1√601√210−√61√6−1√6−2√6100√60 (41) such that 𝐵=𝑊Λ𝑊^𝑇 and Λ=diag(0,−3,−4,−5,−6,−6).
Choose 𝑀_𝑖=1110−𝜆_𝑖01,𝑄_𝑖=1801810−𝜆_𝑖1𝜆_𝑖,𝑃−10_𝑖=,1001 (42)𝜂_𝑖>4, 𝑖=1,2,…,6, and  𝛽=1/2 satisfying Assumptions 3, 4, and 5. Under the controllers 𝑢_𝑖 defined in Theorem 9, the singular hybrid system (39) can be synchronized in the finite time 𝑡^∗=7.1858 according to Theorem 9 if 𝑘=1. If the controller gain 𝑘=5, 𝑡^∗=1.4372. Corresponding simulation results are shown in Figures 1 and 2 with initial conditions 𝑒(0)=^1320.72.50.3_{0.5−10.8−21.5−0.5}.

Figure 1 

Error variable 𝑒1𝑖(𝑡)(𝑖=1,2,…,6) of system (39) with 𝑘=1.

Figure 2 

Error variable 𝑒2𝑖(𝑡)(𝑖=1,2,…,6) of system (39) with 𝑘=1.

5. 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 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.

Acknowledgments

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.