Synchronization and Control of Linearly Coupled Singular Systems

Abstract

The synchronization and control problem of linearly coupled singular systems is investigated. The uncoupled dynamical behavior at each node is general and can be chaotic or, otherwise the coupling matrix is not assumed to be symmetrical. Some sufficient conditions for globally exponential synchronization are derived based on Lyapunov stability theory. These criteria, which are in terms of linear matrix inequality (LMI), indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold. The controllers are designed for state feedback control and pinning control, respectively. Finally, a numerical example is provided to illustrate the effectiveness of the proposed conditions.

1. Introduction

Many natural and synthetic systems, such as neural systems, social systems, WWW, food webs, and electrical power grids, can all be described by complex networks. For decades, complex networks have been focused on by scientists from various fields, for instance, sociology, biology, mathematics, and physics.

Linear coupled ordinary differential equations (LCODEs) are a large class of dynamical systems with continuous time and state, as well as discrete space, for describing coupled oscillators [1]. The LCODE model is widely used to describe the models in nature and engineering. Dhamala et al. [2] study spike-burst neural activity and the transitions to a synchronized state using a model of linearly coupled bursting neurons. Lloyd and Jansen [3] present an extension of an analysis technique for an ecologically motivated spatial meta population model. Perez-Munuzuri et al. [4] study the dynamics of linearly coupled Chua circuits with application to image processing and many other cases. Many complicated dynamical behaviors of coupled chaotic oscillators have been widely studied [5–10]. Among them, the synchronization of LCODEs has been an active area for decades [11].

Synchronization is very universal in daily life. In 1665, Huygens found the synchronous wing of two pendulums [12]. In 1680, Kempfer found fireflies’ synchronized flashing. In 2000, Néda et al. [13] elaborate the mechanism of synchronized clapping in concert halls. There are countless heart cells synchronously oscillating in our hearts. Some researches indicate that thousands of Routers may synchronously send messages on the internet which will trigger network congestion. Now, the synchronization of several systems plays an important role in nuclear magnetic resonance spectrometers, signal generators, laser equipments, superconducting materials, and communication systems. The synchronization technique for LCODEs is applied in many fields. The autoprinciples for parallel image processing are presented in [4, 14, 15]. Lu and Chen [16] apply synchronization technique to recognize an image with strong robustness in recognition. The different architectures of the coupled chaotic system for communication are presented in [16–19] and many other papers. Hoppensteadt and Izhikevich [20] propose an architecture of coupled neural networks to store and retrieve complex oscillatory patterns as synchronization states. Therefore, study of the synchronization of the LCODEs is an important and necessary step for both understanding the dynamics of complex networks and practical design of coupled oscillators.

Recently, synchronization of LCODEs has attracted much attention from researchers in different fields. Wu and Chua [21] define a distance between the state of the coupled system and the synchronization manifold and propose a methodology for discussing global convergence for the complete regular coupling configuration. Nijmeijer and Mareels [22] view the problem of synchronization as a special case of the observer design problem and provide a reasonable comprehensive framework for synchronization issues. In [8, 23], the local stability of the synchronization manifold is studied via a linearization method named the “master stability function” method. Wang and Chen [8, 9] study synchronization of randomly coupled networks such as small-world lattices and scale-free networks. Lu and Chen [24] investigate the stability of the synchronization manifold on the basis of geometrical analysis of the synchronization manifold in which the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles. Liu et al. [25] show that the 𝐿₂ norm of the synchronization error vector presents a suitable measure of both swiftness and vibration of network synchronization and design an LQR controller to drive the network onto some homogeneous stationary states and minimize the 𝐿₂ norm of the output of the linearized network. Lu and Ho [26] distill a quantity convenient to calculate even for large-scale networks from the coupling matrix to characterize the synchronizability of the corresponding dynamical networks and obtain some criteria to guarantee the globally exponential synchronization. Liu et al. [27] propose a synchronization strategy of adjusting adaptively a node’s coupling strength based on the node’s local generalized synchronization information. Fan et al. [28] propose a new scheme for synchronization between two or more complex networks using scalar signals under pinning control. Guan et al. [29] introduce the concept of control topology and design an impulsive controller to achieve the exponential synchronization of complex networks with system delay and multiple coupling delays.

It is worthwhile to mention that we usually have to consider some algebraic constraints of complex networks in modeling the real-world problems. For instance, communication resources are always limited and required to be allocated to different levels of privileged users. Hence, special constraints are needed in the resource allocation process. Constructing a complex networks model with a set of constraints is necessary and indispensable. For this purpose, Xiong et al. [30] introduce linearly coupled singular systems to describe complex networks with a special class of constraints. Singular systems are very complicated since one does not know whether the solution exists, not to mention the stability or synchronization of the solution for the system regularity and impulse elimination. In fact, singular systems give a more general description of physical systems than the normal one, and many studies have extended concepts and results from the normal systems theory into the realm of singular systems [26–44]. In [30], the synchronization problem of linearly coupled singular systems with the coupling matrix assumed to be symmetrical and irreducible is investigated, and a sufficient condition for globally asymptotic synchronization is derived based on the Lyapunov stability theory.

In this paper, we further investigate the synchronization problem of linearly coupled singular systems. The model considered here is general. At each node, the uncoupled system defined by 𝐸𝑥̇=𝐴𝑥+𝑓(𝑥,𝑡) can have various dynamical behaviors. As for the coupling matrix, we do not assume symmetry. We find out that even though the synchronization manifold can be stable, the individual state may be unstable. We construct a reference state making use of the left eigenvector corresponding to eigenvalue zero of the coupling matrix and obtain criteria for globally exponential synchronization based on the Lyapunov stability theory. We design the controllers for state feedback control and pinning control. The conditions presented in the paper are based on strict linear matrix inequalities (LMIs) and can be solved directly by LMI toolbox of Matlab.

The remainder of the paper is organized as follows. In Section 2, the criteria for synchronization of the linearly coupled singular systems are obtained. In Section 3, We investigate the state feedback control and pinning control problem for synchronization of the linearly coupled singular systems. We show a simulation example to demonstrate the effectiveness of our results in Section 4. This paper is concluded in Section 5.

Notation. Lip(𝐿) denotes the class of Lipschitz continuous function with Lipschitz constant 𝐿. 𝐼_𝑛 stands for the 𝑛×𝑛 identity matrix. The superscript 𝑇 and * represent transpose and conjugate transpose, respectively. For a square matrix 𝐴, if 𝐴 has only real eigenvalues, then 𝜆_𝑀(𝐴) denote the maximal eigenvalues of 𝐴. For real symmetric matrices 𝐴 and 𝐵, 𝐴>𝐵(𝐴≥𝐵) means that 𝐴−𝐵 is positive definite (or positive semidefinite). The matrices, if not explicitly stated, are assumed to have compatible dimensions.

2. Synchronization of Linearly Coupled Singular Systems

Consider the following linearly coupled network: 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁, (1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈𝑅^𝑛 is state variable of node 𝑖, 𝑖=1,2,…,𝑁, and 𝑓∶𝑅^𝑛×𝑅→𝑅^𝑛 is a time-varying nonlinear function. 𝐸,𝐴∈𝑅^𝑛×𝑛 are constant matrices; 0<rank(𝐸)=𝑟<𝑛. The scalar 𝑐>0 is the coupling strength. 𝑏_𝑖𝑗≥0 denotes the coupling coefficient from node 𝑗 to node 𝑖, for 𝑖,𝑗=1,…,𝑁, 𝑖≠𝑗. 𝑏_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝑏_𝑖𝑗, 𝑖=1,2,…,𝑁. Γ=diag(𝛾₁,…,𝛾_𝑛) denotes the inner connection at each node with 𝛾_𝑖≥0, 𝑖=1,…,𝑛, which means that two nodes are connected by their 𝑖th component if 𝛾_𝑖>0.

Definition 1 (see [24]). The coupled system (1) is globally exponentially synchronized if there exist 𝛼>0,𝑡₀>0, and 𝑀>0, such that for each solution 𝑥_𝑖(𝑡), 𝑖=1,2,…,𝑁, ‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≤𝑀𝑒^−𝛼𝑡 holds for all 𝑡>𝑡₀,𝑖,𝑗=1,2,…,𝑁.

The objective of this section is to find the conditions ensuring the coupled system (1) globally exponentially synchronized.

Denote that 𝐵=(𝑏_𝑖𝑗)_𝑁×𝑁. The following lemma characterizes the right and left eigenvectors of 𝐵 corresponding to eigenvalue 0, which play key roles in the investigation of the synchronization of the coupled system (1).

Lemma 2 (see [24]). If 𝐵 is irreducible, then the following are valid.(1)B has an eigenvalue 0 with multiplicity 1 and the right eigenvector [1,1,…,1]^𝑇.(2)If 𝜆 is an eigenvalue of 𝐵 and 𝜆≠0, then Re(𝜆)<0.(3)Suppose that 𝜉=[𝜉₁,𝜉₂,…,𝜉_𝑁]^𝑇∈𝑅^𝑁 (without loss of generality, assume ∑^𝑁_𝑖=1𝜉_𝑖=1) is the left eigenvector of 𝐵 corresponding to eigenvalue 0; then, 𝜉_𝑖>0 holds for all 𝑖=1,2,…,𝑁.

In the following discussion, we always assume that 𝐵 is irreducible and 𝑓(𝑥,𝑡)∈Lip(𝐿).

Let 𝑥(𝑡)=∑^𝑁_𝑖=1𝜉_𝑖𝑥_𝑖(𝑡), 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥(𝑡), 𝑓_𝑖(𝑥,𝑡)=𝑓(𝑥_𝑖,𝑡)−∑^𝑁_𝑘=1𝜉_𝑘𝑓(𝑥_𝑘(𝑡),𝑡), 𝑖=1,2,…,𝑁. Noticing that ∑^𝑁_𝑗=1𝑏_𝑖𝑗=0, 𝑖=1,2,…,𝑁 and ∑^𝑁_𝑖=1𝜉_𝑖𝑏_𝑖𝑗=0, 𝑗=1,2,…,𝑁, we have 𝐸𝑒_𝑖̇(𝑡)=𝐴𝑒_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑒_𝑗(𝑡)+𝑓_𝑖(𝑥,𝑡),𝑖=1,2,…,𝑁. (2) Denote that 𝑒(𝑡)=[𝑒₁(𝑡),…,𝑒_𝑁(𝑡)], and 𝐹(𝑡)=[𝑓₁(𝑥,𝑡),…,𝑓_𝑁(𝑥,𝑡)]; then, 𝐸𝑒̇(𝑡)=𝐴𝑒(𝑡)+𝑐Γ𝑒(𝑡)𝐵^𝑇+𝐹(𝑥,𝑡). (3)

Let 𝐵^𝑇=𝑊𝐽𝑊⁻¹ be the Jordan decomposition of 𝐵^𝑇, where 𝐽=diag(𝐽₁,𝐽₂,…,𝐽_𝑙),   𝐽_𝑗 is 𝜎_𝑗-order Jordan block, 𝑗=1,2,…,𝑙, and ∑^𝑙_𝑗=1𝜎_𝑗=𝑁. Denote the diagonal element of 𝐽_𝑗 by 𝜆_𝑗, 𝑗=1,2,…,𝑙. Without loss of generality, we assume that 𝜆₁=0, and then 𝐽₁=0 is a scalar.

Define 𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑁(𝑡)=𝑒(𝑡)𝑊, (4) and denote that 𝑊=[𝑊₁,𝑊₂,…,𝑊_𝑁]=(𝑤_𝑖𝑗)_𝑁×𝑁; then, 𝑦_𝑖(𝑡)=𝑒(𝑡)𝑊_𝑖,  𝑖=1,2,…,𝑁.

Denote that 𝜌₀=0 and 𝜌_𝑗=∑^𝑗_𝑘=1𝜎_𝑘, 𝑗=1,2,…,𝑙; then, we can rewrite the coupled system (3) as 𝐸𝑦̇_𝑖(𝑡)=𝐴+𝑐𝜆_𝑘Γ𝑦_𝑖(𝑡)+𝐹(𝑥,𝑡)𝑊_𝑖,𝑖=𝜌_𝑘−1𝐸𝑦̇+1,𝑘=1,2,…,𝑙,_𝑖(𝑡)=𝐴+𝑐𝜆_𝑘Γ𝑦_𝑖(𝑡)+𝑐Γ𝑦_𝑖−1(𝑡)+𝐹(𝑥,𝑡)𝑊_𝑖,𝜌_𝑘−1+1<𝑖<𝜌_𝑘,𝑘=2,…,𝑙. (5)

For 𝑗=1,2,…,𝑙, define 𝑧_𝑗=𝑦_{𝜌𝑗−1+1}𝑦_{𝜌𝑗−1+2}⋮𝑦_𝜌𝑗,𝐴_𝑗=𝐴+𝑐𝜆_𝑗Γ𝑐Γ𝐴+𝑐𝜆_𝑗Γ⋱⋱𝑐Γ𝐴+𝑐𝜆_𝑗Γ,𝐸_𝑗=𝐸𝐸⋱𝐸_{(𝜎𝑗𝑛)×(𝜎𝑗𝑛)},𝐹_𝑗()=𝐹()𝑊𝑥,𝑡𝑥,𝑡_{𝜌𝑗−1+1}𝐹(𝑥,𝑡)𝑊_{𝜌𝑗−1+2}⋮𝐹(𝑥,𝑡)𝑊_𝜌𝑗, (6) and then the coupled system (5) can be rewritten as 𝐸_𝑗𝑧̇_𝑗(𝑡)=𝐴_𝑗𝑧_𝑗(𝑡)+𝐹_𝑗(𝑥,𝑡),𝑗=1,2,…,𝑙. (7)

Let 𝑧(𝑡)=[𝑧^𝑇₁(𝑡),𝑧^𝑇₂(𝑡),…,𝑧^𝑇_𝑙(𝑡)]^𝑇; then, with the previous decompositions, synchronization of the coupled system (1) is equivalent to 𝑧(𝑡)→0. In the following, instead of investigating 𝑥(𝑡), we investigate dynamical behaviors of 𝑧(𝑡) directly.

For the sake of convenient expression, we give the following definition.

Definition 3 (see [45, 46]). The pair (𝐸,𝐴) is said to be regular if det(𝑠𝐸−𝐴) is not identical zero. The pair (𝐸,𝐴) is said to be impulse-free if deg(det(𝑠𝐸−𝐴))=rank(𝐸).

Suppose that 𝑊⁻¹=[𝑊^∼₁,…,𝑊^∼_𝑁]=(𝑤^∼_𝑖𝑗)_𝑁×𝑁. For 𝑗=2,…,𝑙, 𝑘=1,2,…,𝑁, 𝑠=1,2,…,𝑁, define 𝛼_𝑗=^𝜎𝑗_𝑑=1^𝑁_𝑖=1𝑤_{𝑖,𝜌𝑗−1+𝑑}²,𝛽_𝑘𝑗=^𝜎𝑗_𝑑=1𝑤_{𝑘,𝜌𝑗−1+𝑑}^𝑁_𝑖=1𝑤_{𝑖,𝜌𝑗−1+𝑑},𝛾_𝑘𝑗=𝛼_𝑗𝜉_𝑘+𝛽_𝑘𝑗,𝜙_𝑘𝑠=^𝑙𝑁_𝑗=2𝑖=1𝛾^∼_𝑖𝑗𝑤^∗∼_𝑘𝑖𝑤_𝑠𝑖, (8) and denote 𝜙𝜙=²²𝜙₂₃⋯𝜙_2𝑁𝜙₃₂𝜙₃₃⋯𝜙_3𝑁𝜙⋮⋮⋱⋮_𝑁2𝜙_𝑁3⋯𝜙_𝑁𝑁,Φ=𝜙⊗𝐼_𝑛. (9)

In the following, we give the main conclusion of this section.

Theorem 4. If there exists a scalar 𝜀>0 and matrices 𝑃_𝑗∈𝑅^{(𝜎𝑗𝑛)×(𝜎𝑗𝑛)}, 𝑗=2,3,…,𝑙, satisfying 𝐸^𝑇_𝑗𝑃_𝑗=𝑃^𝑇_𝑗𝐸_𝑗≥0,𝑗=2,3,…,𝑙,𝑃+2𝐿²Φ+𝜀𝐼_{(𝑁−1)𝑛}<0, (10)(11) where 𝑃=diag(𝑃^𝑇₂𝑃₂+𝐴^∗₂𝑃₂+𝑃^𝑇₂𝐴₂,…,𝑃^𝑇_𝑙𝑃_𝑙+𝐴^∗_𝑙𝑃_𝑙+𝑃^𝑇_𝑙𝐴_𝑙), then the coupled system (1) is globally exponentially synchronized.

Proof. First, note that 𝜆₁=0 and corresponding left eigenvector 𝜉 satisfies that ∑^𝑁_𝑖=1𝜉_𝑖=1. This corresponds to synchronization on the manifold 𝑧₁(𝑡)=0.
Next, we consider the manifolds 𝑧₂(𝑡)=0,𝑧₃(𝑡)=0,…,𝑧_𝑙(𝑡)=0.
Choose the Lyapunov function candidate as follows: 𝑉(𝑡)=12^𝑙_𝑗=2𝑧^∗_𝑗(𝑡)𝑃^𝑇_𝑗𝐸_𝑗𝑧_𝑗(𝑡), (12) and then the derivative of 𝑉(𝑡) along the trajectory of (7) yields 𝑉̇(𝑡)=12^𝑙_𝑗=2𝑧^∗_𝑗𝐴^∗_𝑗𝑃_𝑗+𝑃^𝑇_𝑗𝐴_𝑗𝑧_𝑗+𝐹^∗_𝑗(𝑥,𝑡)𝑃_𝑗𝑧_𝑗+𝑧^∗_𝑗𝑃^𝑇_𝑗𝐹_𝑗()≤1𝑥,𝑡2^𝑙_𝑗=2𝑧^∗_𝑗𝑃^𝑇_𝑗𝑃_𝑗+𝐴^∗_𝑗𝑃_𝑗+𝑃^𝑇_𝑗𝐴_𝑗𝑧_𝑗+𝐹^∗_𝑗(𝑥,𝑡)𝐹_𝑗(𝑥,𝑡). (13)
Since 𝑒_𝑖(𝑡)=𝑦(𝑡)𝑊^∼_𝑖,𝑖=1,2,…,𝑁, 𝑓(𝑥,𝑡)∈Lip(𝐿), and𝐹_𝑗(𝑥,𝑡)=^𝑁_𝑘∑₌₁𝑤_{𝑘,𝜌𝑗−1+1}𝑓𝑥_𝑘,𝑡−𝑓(𝑥,𝑡)+∑^𝑁_𝑣=1𝜉_𝑣𝑓(𝑥,𝑡)−𝑓𝑥_𝑣,𝑡_𝑁𝑘∑₌₁𝑤_{𝑘,𝜌𝑗−1+2}𝑓𝑥_𝑘,𝑡−𝑓(𝑥,𝑡)+∑^𝑁_𝑣=1𝜉_𝑣𝑓(𝑥,𝑡)−𝑓𝑥_𝑣⋮,𝑡_𝑁𝑘∑₌₁𝑤_{𝑘,𝜌𝑗}𝑓𝑥_𝑘,𝑡−𝑓(𝑥,𝑡)+∑^𝑁_𝑣=1𝜉_𝑣𝑓(𝑥,𝑡)−𝑓𝑥_𝑣,𝑡, (14)we have𝐹^∗_𝑗(𝑥,𝑡)𝐹_𝑗(𝑥,𝑡)=^𝜎𝑗𝑁_𝑑=1^𝑁_𝑘=1𝑠=1𝑤^∗_{𝑘,𝜌𝑗−1+𝑑}𝑤_{𝑠,𝜌𝑗−1+𝑑}×𝑓^∗𝑥_𝑘,𝑡−𝑓^∗()×𝑓𝑥𝑥,𝑡_𝑠,𝑡−𝑓(𝑥,𝑡)+2^𝑁_𝑣=1𝜉_𝑣𝑓^∗𝑥_𝑘,𝑡−𝑓^∗()×𝑓𝑥𝑥,𝑡_𝑣()+,𝑡−𝑓𝑥,𝑡_{𝑁𝑁𝑣=1𝑖=1}𝜉_𝑣𝜉_𝑖𝑓^∗𝑥_𝑣,𝑡−𝑓^∗()×𝑓𝑥𝑥,𝑡_𝑖,𝑡−𝑓(𝑥,𝑡)≤2^𝜎𝑗_𝑑=1^𝑁_𝑠=1𝑤_{𝑠,𝜌𝑗−1+𝑑}×^𝑁_𝑘=1𝑤_{𝑘,𝜌𝑗−1+𝑑}𝑓𝑥_𝑘,𝑡−𝑓(𝑥,𝑡)²+2^𝜎𝑗_𝑑=1^𝑁_𝑖=1𝑤_{𝑖,𝜌𝑗−1+𝑑}²×^𝑁_𝑣=1𝜉_𝑣𝑓𝑥_𝑣,𝑡−𝑓(𝑥,𝑡)²≤2𝐿^2𝑁_𝑘=1𝛾_𝑘𝑗𝑒^∗_𝑘𝑒_𝑘=2𝐿^2𝑁_𝑘=1𝛾^∼_𝑘𝑗𝑊^∗_𝑘𝑦^∗(𝑡)𝑦(𝑡)𝑊^∼_𝑘=2𝐿^2𝑁𝑁_𝑣=2𝑠=2^𝑁_𝑘=1𝛾^∼_𝑘𝑗𝑤^∗∼_𝑣𝑘𝑤_𝑠𝑘𝑦^∗_𝑣𝑦_𝑠,𝑉̇(𝑡)≤12^𝑙_𝑗=2𝑧^∗_𝑗𝑃^𝑇_𝑗𝑃_𝑗+𝐴^∗_𝑗𝑃_𝑗+𝑃^𝑇_𝑗𝐴_𝑗𝑧_𝑗+2𝐿^2𝑁𝑁_𝑘=2𝑠=2𝜙_𝑘𝑠𝑦^∗_𝑘𝑦_𝑠; (15) that is, 𝑉̇(𝑡)≤12𝑧^∗(𝑡)𝑃+2𝐿²Φ𝑧(𝑡), (16) where 𝑧(𝑡)=[𝑧^𝑇₂(𝑡),…,𝑧^𝑇_𝑙(𝑡)]^𝑇.
According to (11), one has 𝑉̇(𝑡)<−12𝜀^𝑁_𝑗=2𝑧^∗_𝑗𝑧_𝑖𝜀≤−𝜅𝑉(𝑡), (17) where 𝜅=max{𝜆_𝑀(𝑃^𝑇_𝑗𝐸_𝑗),𝑗=2,3,…,𝑙}.
Denote that 𝑊=^∼𝑤^∼₂₁𝑤₂₂⋯𝑤^∼_∼2𝑁𝑤^∼₃₁𝑤₃₂⋯𝑤^∼_3𝑁⋮⋮⋱⋮_∼𝑤^∼_𝑁1𝑤_𝑁2⋯𝑤^∼_𝑁𝑁, (18)𝛾_𝑖=∑^𝑙_𝑗=2𝛾_𝑖𝑗, 𝑖=1,2,…,𝑁, and Γ=diag(𝛾₁,…,𝛾_𝑁); then, 𝜙=(𝑊^𝑇)^∗Γ𝑊^𝑇, which implies that 𝜙 and Φ are both positive definite matrices.
It follows from (10) and (11) that the pairs (𝐸_𝑗,𝐴_𝑗), 𝑗=2,…,𝑙, are all regular and impulse-free [45]; that is, there exist nonsingular matrices 𝑀_𝑗,𝑄_𝑗∈𝐶^{(𝜎𝑗𝑛)×(𝜎𝑗𝑛)} satisfying that 𝑀_𝑗𝐸_𝑗𝑄_𝑗=diag𝐼_𝜎𝑗𝑟,𝑀,0_𝑗𝐴_𝑗𝑄_𝑗=diag𝐴^∼_𝑗,𝐼_{𝜎𝑗(𝑛−𝑟)}, (19) for 𝑗=2,3,…,𝑁.
By (10) and (11), we can also obtain that (𝑀_𝑗⁻¹)^∗𝑃_𝑗𝑄_𝑗=^𝑃𝑗10_{𝑃𝑗3𝑃𝑗4}, where 𝑃_𝑗1∈𝑅^{(𝜎𝑗𝑟)×(𝜎𝑗𝑟)} and 𝑃_𝑗1>0, 𝑗=2,3,…,𝑁 [36].
Let 𝑄_𝑗⁻¹𝑧_𝑗=[𝑧^𝑇_𝑗1,𝑧^𝑇_𝑗2]^𝑇 and 𝑀_𝑗𝐹_𝑗(𝑥,𝑡)=[(𝐹_𝑗1(𝑥,𝑡))^𝑇,(𝐹_𝑗1(𝑥,𝑡))^𝑇]^𝑇, where 𝑧_𝑗1∈𝐶^𝜎𝑗𝑟 and 𝐹_𝑗1(𝑥,𝑡)∈𝐶^𝜎𝑗𝑟, 𝑗=1,2,…,𝑙, and then the system (7) can be rewritten as 𝑧̇_𝑗1(𝑡)=𝐴^∼_𝑗𝑧_𝑗1(𝑡)+𝐹_𝑗1(),𝑧𝑥,𝑡_𝑗2(𝑡)=−𝐹_𝑗2(),𝑉(𝑡)=1𝑥,𝑡𝑗=1,2,…,𝑙,2^𝑙_𝑗=2𝑧^∗_𝑗1𝑃_𝑗1𝑧_𝑗1. (20)(21) Since the matrix 𝑃_𝑗1 is positive definite, 𝑧_𝑗1(𝑡), 𝑖=2,3,…,𝑙, are globally exponentially stable. In the following, we show that 𝑧_𝑗2(𝑡), 𝑗=2,3,…,𝑙, are also globally exponentially stable; thus, the coupled system (1) is globally exponentially synchronized.
Denote that 𝑀_𝑗=[𝑀^𝑇_𝑗1𝑀^𝑇_𝑗2]^𝑇, where 𝑀_𝑗1∈𝐶^{(𝜎𝑗𝑟)×(𝜎𝑗𝑛)}, 𝑗=1,2,…,𝑙. Without loss of generality, we assume that ‖𝑀_𝑗2‖=1 [40].
From (20), we have 𝑧_𝑗2(𝑡)²≤𝑀_𝑗2²𝐹_𝑗(𝑥,𝑡)²=𝐹^∗_𝑗(𝑥,𝑡)𝐹_𝑗(𝑥,𝑡)≤2𝐿^2𝑁𝑁_𝑣=2𝑠=2𝜂_𝑗𝑣𝑠𝑦^∗_𝑣𝑦_𝑠, (22) where 𝜂_𝑗𝑣𝑠=∑^𝑁_𝑘=1𝛾^∼_𝑘𝑗𝑤^∗∼_𝑣𝑘𝑤_𝑠𝑘, 𝑗=2,3,…,𝑙, 𝑣=1,2,…,𝑁, 𝑠=1,2,…,𝑁.
For 𝑗=2,3,…,𝑙, define Ψ_𝑗=𝜂_𝑗22𝐼_𝑛𝜂_𝑗23𝐼_𝑛⋯𝜂_𝑗2𝑁𝐼_𝑛𝜂_𝑗32𝐼_𝑛𝜂_𝑗33𝐼_𝑛⋯𝜂_𝑗3𝑁𝐼_𝑛𝜂⋮⋮⋱⋮_𝑗𝑁2𝐼_𝑛𝜂_𝑗𝑁3𝐼_𝑛⋯𝜂_𝑗𝑁𝑁𝐼_𝑛, (23) and let 𝜆_Ψ𝑗 stand for the maximal eigenvalue of the matrix Ψ_𝑗; then, 𝑧_𝑗2(𝑡)²≤2𝐿²𝜆_Ψ𝑗𝜆_𝑄×^𝑙_𝑘=2𝑧_𝑘1(𝑡)²+𝑧_𝑘2(𝑡)²,𝑗=2,3,…,𝑙,_𝑙𝑗=2𝑧_𝑗2(𝑡)²≤2𝐿²𝜆_𝑄𝜆_Ψ^𝑙_𝑘=2𝑧_𝑘1(𝑡)²+𝑧_𝑘2(𝑡)², (24)(25)where 𝜆_𝑄 is the maximal eigenvalue of the matrix diag(𝑄^∗₂𝑄₂,𝑄^∗₃𝑄₃,…,𝑄^∗_𝑙𝑄_𝑙) and 𝜆_Ψ=∑^𝑙_𝑗=2𝜆_Ψ𝑗.
From (25), we have 1−2𝐿²𝜆_𝑄𝜆_Ψ^𝑙_𝑗=2𝑧_𝑗2(𝑡)²≤2𝐿²𝜆_𝑄𝜆_Ψ^𝑙_𝑘=2𝑧_𝑘1(𝑡)². (26) By choosing suitably 𝑄₂,𝑄₃,…,𝑄_𝑙 such that 1−2𝐿²𝜆_𝑄𝜆_Ψ>0, we have ^𝑙_𝑗=2𝑧_𝑗2(𝑡)²≤2𝐿²𝜆_𝑄𝜆_Ψ1−2𝐿²𝜆_𝑄𝜆_Ψ^𝑙_𝑘=2𝑧_𝑘1(𝑡)²; (27) consequently, 𝑧_𝑗2(𝑡), 𝑗=2,3,…,𝑁, are globally exponentially stable.

The following lemma is the combination of Theorem 4 and Lemma 2 of [33].

Lemma 5 (see [33]). Suppose that 𝐸,𝐴∈𝑅^𝑛×𝑛, 𝐵∈𝑅^𝑛×𝑞, 𝐶∈𝑅^𝑝×𝑛, 𝛾>0 is a scalar, 0<rank(𝐸)=𝑟<𝑛,𝐸=𝐸_𝐿𝐸^𝑇_𝑅, 𝐸_𝐿,𝐸_𝑅∈𝑅^𝑛×𝑟 are of full column rank, 𝑈,𝑉∈𝑅^{𝑛×(𝑛−𝑟)} are of full column rank and composed of bases of Null(𝐸) and Null(𝐸^𝑇), respectively. There exists 𝑃∈𝑅^𝑛×𝑛 satisfying 𝑃^𝑇𝐴+𝐴^𝑇𝑃+𝐵𝐵^𝑇+𝑃^𝑇𝐶^𝑇𝐶𝑃𝛾²<0,𝐸^𝑇𝑃=𝑃^𝑇𝐸≥0, (28) if and only if there exist a symmetric matrix 𝑌∈𝑅^𝑛×𝑛 and a matrix 𝑆∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} satisfying Φ₁Φ₂Φ^𝑇₂−𝛾²𝐼<0,𝐸^𝑇_𝐿𝑌𝐸_𝐿>0, (29) where Φ₁=𝐴^𝑇𝑆₁+𝑆^𝑇₁𝐴+𝐵𝐵^𝑇, Φ₂=𝑆^𝑇₁𝐶^𝑇, and 𝑆₁=𝑌𝐸+𝑉𝑆𝑈^𝑇.

Let 𝜆_Φ be the maximal eigenvalue of the matrix Φ, 𝐸_𝑗=𝐸_𝑗𝐿𝐸^𝑇_𝑗𝑅, where 𝐸_𝑗𝐿, 𝐸_𝑗𝑅∈𝑅^{(𝜎𝑗𝑛)×(𝜎𝑗𝑟)} are of full column rank, and 𝑈_𝑗, 𝑉_𝑗∈𝑅^{(𝜎𝑗𝑛)×(𝜎𝑗(𝑛−𝑟))} are of full column rank and composed of bases of Null(𝐸_𝑗) and Null(𝐸^𝑇_𝑗), respectively, 𝑗=2,3,…,𝑙.

Based on Theorem 4 and Lemma 5, we have the following conclusion.

Theorem 6. If there exist a scalar 𝜀>0, symmetric matrices 𝑌_𝑗∈𝑅^{(𝜎𝑗𝑛)×(𝜎𝑗𝑛)}, and matrices 𝑆_𝑗∈𝑅^{(𝜎𝑗(𝑛−𝑟))×(𝜎𝑗(𝑛−𝑟))} satisfying 2𝐿²𝜆_Φ+𝜀𝐼_𝜎𝑗𝑛−𝐴^∗_𝑗𝐴_𝑗𝐴^∗_𝑗+Λ^𝑇_𝑗𝐴_𝑗+Λ_𝑗−𝐼_𝜎𝑗𝑛<0,𝐸^𝑇_𝑗𝐿𝑌_𝑗𝐸_𝑗𝐿>0, (30) where Λ_𝑗=𝑌_𝑗𝐸_𝑗+𝑉_𝑗𝑆_𝑗𝑈^𝑇_𝑗, 𝑗=2,3,…,𝑙, then the coupled system (1) is globally exponentially synchronized.

When 𝐵 is a real symmetric matrix, 𝐽=diag(𝜆₁,𝜆₂,…,𝜆_𝑁) and 𝑊 is an orthogonal matrix, where 0=𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁 are the eigenvalues of 𝐵. The coupled system (3) can be rewritten as 𝐸𝑦̇_𝑖(𝑡)=𝐴+𝑐𝜆_𝑖Γ𝑦_𝑖(𝑡)+𝐹(𝑥,𝑡)𝑊_𝑖,𝑖=1,2,…,𝑁. (31)

For 𝑗=2,…,𝑁,  𝑘=1,2,…,𝑁, and 𝑣=1,2,…,𝑁, define 𝛼_𝑠𝑗=^𝑁_𝑖=1𝑤_𝑖𝑗²,𝛽_𝑠𝑘𝑗=𝑤_𝑘𝑗^𝑁_𝑖=1𝑤_𝑖𝑗,𝛾_𝑠𝑘𝑗=𝛼_𝑗𝜉_𝑘+𝛽_𝑘𝑗,𝜙_𝑠𝑘𝑣=^𝑁𝑁_𝑗=2𝑖=1𝛾^∼_𝑠𝑖𝑗𝑤^∗∼_𝑘𝑖𝑤_𝑣𝑖,𝜙_𝑠=𝜙_𝑠22𝜙_𝑠23⋯𝜙_𝑠2𝑁𝜙_𝑠32𝜙_𝑠33⋯𝜙_𝑠3𝑁𝜙⋮⋮⋱⋮_𝑠𝑁2𝜙_𝑠𝑁3⋯𝜙_𝑠𝑁𝑁,Φ_𝑠=𝜙_𝑠⊗𝐼_𝑛, (32) and denote that 𝜆_Φ𝑠 is the maximal eigenvalue of Φ_𝑠, 𝐸=𝐸_𝐿𝐸^𝑇_𝑅, 𝐸_𝐿,𝐸_𝑅∈𝑅^𝑛×𝑟 are of full column rank, 𝑈,𝑉∈𝑅^{𝑛×(𝑛−𝑟)} are of full column rank and composed of bases of Null(𝐸) and Null(𝐸^𝑇), respectively.

Similar to the proof of Theorem 6, we have the following conclusion.

Theorem 7. Suppose that 𝐵 is a real symmetric matrix. If there exist a scalar 𝜀>0, a symmetric matrix 𝑌∈𝑅^𝑛×𝑛 and a matrix 𝑆∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} satisfying 𝐸^𝑇_𝐿𝑌𝐸_𝐿>0,2𝐿²𝜆_Φ𝑠+𝜀𝐼_𝑛−𝐴+𝑐𝜆_𝑖Γ^𝑇𝐴+𝑐𝜆_𝑖Γ𝐴+𝑐𝜆_𝑖Γ^𝑇+Λ^𝑇𝐴+𝑐𝜆_𝑖Γ+Λ−𝐼_𝑛<0, (33)(34) where Λ=𝑌𝐸+𝑉𝑆𝑈^𝑇, 𝑖=2,3,…,𝑁, and then the coupled system (1) is globally exponentially synchronized.

Remark 8. By Schur complement, the inequality (34) is equivalent to 2𝐿²𝜆_Φ𝑠+𝜀𝐼_𝑛+Λ^𝑇Λ+𝐴^𝑇Λ+Λ^𝑇𝐴+𝑐𝜆_𝑖ΓΛ+Λ^𝑇Γ<0. (35) When ΓΛ+Λ^𝑇Γ>0, if there exists a scalar 𝑐₁>0 satisfying 2𝐿²𝜆_Φ𝑠+𝜀𝐼_𝑛+Λ^𝑇Λ+𝐴^𝑇Λ+Λ^𝑇𝐴+𝑐₁𝜆₂ΓΛ+Λ^𝑇Γ<0, (36) then inequality (34) is right for all 𝑐∈[𝑐₁,∞). When ΓΛ+Λ^𝑇Γ<0, if there exists a scalar 𝑐₂>0 satisfying 2𝐿²𝜆_Φ𝑠+𝜀𝐼_𝑛+Λ^𝑇Λ+𝐴^𝑇Λ+Λ^𝑇𝐴+𝑐₂𝜆_𝑁ΓΛ+Λ^𝑇Γ<0, (37) then inequality (34) is right for all 𝑐∈[0,𝑐₂].

3. Control of Linearly Coupled Singular Systems

In this section, we investigate the state feedback control and pinning control problems of linearly coupled singular systems.

3.1. State Feedback Control

Suppose that the linearly coupled network with a state feedback controller is 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝐷𝑢_𝑖(𝑡),𝑢_𝑖(𝑡)=−𝐾𝑥_𝑖(𝑡),𝑖=1,2,…,𝑁, (38) where 𝐷∈𝑅^𝑛×𝑚 is constant matrix, and 𝐾∈𝑅^𝑚×𝑛 is feedback gain matrix.

We continue to use the symbols in Section 2, and let 𝐷_𝑗=diag(𝐷,𝐷,…,𝐷), 𝐾_𝑗=diag(𝐾,𝐾,…,𝐾), 𝐷_𝑗∈𝑅^{(𝜎𝑗𝑛)×(𝜎𝑗𝑚)}, 𝐾_𝑗∈𝑅^{(𝜎𝑗𝑚)×(𝜎𝑗𝑛)}, 𝑗=1,2,…,𝑙; then, the closed loop coupled network (38) can be rewritten as 𝐸_𝑗𝑧̇_𝑗(𝑡)=𝐴_𝑗−𝐷_𝑗𝐾_𝑗𝑧_𝑗(𝑡)+𝐹_𝑗(𝑥(𝑡),𝑡). (39)

Follow the proof of Theorem 6, we have the following conclusion.

Theorem 9. If there exist a scalar 𝜀>0, symmetric matrices 𝑌_𝑗∈𝑅^{(𝜎𝑗𝑛)×(𝜎𝑗𝑛)}, matrices 𝑆_𝑗∈𝑅^{(𝜎𝑗(𝑛−𝑟))×(𝜎𝑗(𝑛−𝑟))}, and matrix 𝐾∈𝑅^𝑚×𝑛 satisfying 𝐸^𝑇_𝑗𝐿𝑌_𝑗𝐸_𝑗𝐿>0,2𝐿²𝜆_Φ+𝜀𝐼_𝜎𝑗𝑛−𝐴^∗_𝑗𝐴_𝑗+𝐴^∗_𝑗𝐷_𝑗𝐾_𝑗+𝐾^𝑇_𝑗𝐷_𝑗𝐴_𝑗𝐴^∗_𝑗−𝐾^𝑇_𝑗𝐷^𝑇_𝑗+Λ^𝑇_𝑗𝐴_𝑗−𝐷_𝑗𝐾_𝑗+Λ_𝑗−𝐼_𝜎𝑗𝑛<0, (40) where Λ_𝑗=𝑌_𝑗𝐸_𝑗+𝑉_𝑗𝑆_𝑗𝑈^𝑇_𝑗, 𝑗=1,2,…,𝑙, then the closed loop network (39) is globally exponentially synchronized with the state feedback 𝑢_𝑖(𝑡)=−𝐾𝑥_𝑖(𝑡), 𝑖=1,2,…,𝑁.

3.2. Pinning Control

Suppose that the linearly coupled network with a single controller is 𝐸𝑥̇₁(𝑡)=𝐴𝑥₁(𝑡)+𝑓𝑥₁(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_1𝑗Γ𝑥_𝑗(𝑡)−𝐾^∼𝑒₁(𝑡),𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡),𝑡+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=2,3,…,𝑁, (41) where 𝐾^∼∈𝑅^𝑛×𝑛 is the feedback gain matrix.

Denote that 𝐸=diag(𝐸₂,𝐸₃,…,𝐸_𝑙), 𝐴=diag(𝐴₂,𝐴₃,…,𝐴_𝑙), 𝐹(𝑥,𝑡)=[𝐹^𝑇₂(𝑥,𝑡), 𝐹^𝑇₃(𝑥,𝑡),…,𝐹^𝑇_𝑙(𝑥,𝑡)]^𝑇, 𝑧=[𝑦^𝑇₂,𝑦^𝑇₃,…,𝑦^𝑇_𝑙]^𝑇, Γ_𝑖=𝜉₁(∑^𝑁_𝑘=1𝑤_𝑘𝑖)𝐾^∼−𝑤_1𝑖𝐼_𝑛, 𝑖=2,3,…,𝑁, and Γ^∼=[Γ^𝑇₂,Γ^𝑇₃,…,Γ^𝑇_𝑁]^𝑇[𝑤^∼₂₁𝐼_𝑛,𝑤^∼₃₁𝐼_𝑛,…,𝑤^∼_𝑁1𝐼_𝑛]; then, the coupled network (41) can be rewritten as 𝐸𝑧̇=𝐴+Γ^∼𝑧+𝐹(𝑥,𝑡). (42)

Denote that 𝛼^∼=^𝑁_𝑗=2^𝑁_𝑠=1𝑤_𝑠𝑗²,_∼𝛽_𝑘=^𝑁_𝑗=2𝑤_𝑘𝑗^𝑁_𝑠=1𝑤_𝑠𝑗,𝛾^∼_𝑘=𝛼^∼𝜉_𝑘+𝛽^∼_𝑘,𝑘=1,2,…,𝑁,_∼𝜙_𝑣𝑠=^𝑁∼_𝑘=1𝛾𝑤_𝑘^∼∗∼_𝑣𝑘𝑤_𝑠𝑘,𝑣,𝑠=2,3,…,𝑁,_∼𝜙=^∼𝜙^∼₂₂𝜙₂₃⋯𝜙^∼_∼2𝑁𝜙^∼₃₂𝜙₃₃⋯𝜙^∼_3𝑁⋮⋮⋱⋮_∼𝜙^∼_𝑁2𝜙_𝑁3⋯𝜙^∼_𝑁𝑁,Φ^∼=𝜙^∼⊗𝐼_𝑛, (43) and let 𝐸=𝐸_𝐿𝐸^𝑇_𝑅, where 𝐸_𝐿,𝐸_𝑅∈𝑅^{((𝑁−1)𝑛)×((𝑁−1)𝑟)} are of full column rank, and 𝑈,𝑉∈𝑅^{((𝑁−1)𝑛)×((𝑁−1)(𝑛−𝑟))} are of full column rank and composed of bases of Null(𝐸) and Null(𝐸^𝑇); then, we have the following conclusion.

Theorem 10. If there exist a scalar 𝜀>0, symmetric matrices 𝑌∈𝑅^{((𝑁−1)𝑛)×((𝑁−1)𝑛)}, matrices 𝑆∈𝑅^{((𝑁−1)(𝑛−𝑟))×((𝑁−1)(𝑛−𝑟))}, and matrix 𝐾^∼∈𝑅^𝑛×𝑛 satisfying 𝐸^𝑇_𝐿𝑌𝐸_𝐿>0,2𝐿Φ^2∼+𝜀𝐼_{(𝑁−1)𝑛}−𝐴^∗𝐴−𝐴Γ^∗∼−Γ^∼∗𝐴−𝑊_𝑘𝐴^∗+Γ^∗+Λ^𝑇𝐴+Γ+Λ−𝐼_{(𝑁−1)𝑛}<0, (44) where Λ=𝑌𝐸+𝑉𝑆𝑈^𝑇 and 𝑊_𝑘=[𝑤^∼₂₁𝐼_𝑛,𝑤^∼₃₁𝐼_𝑛,…,𝑤^∼_𝑁1𝐼_𝑛]^∗∑^𝑁_𝑗=2(|𝑤_1𝑗|²𝐼_𝑛−𝜉₁𝑤_1𝑗(∑^𝑁_𝑘=1𝑤^∗_𝑘𝑗)𝐾^∼∗−𝜉₁𝑤^∗_1𝑗(∑^𝑁_𝑘=1𝑤_𝑘𝑗)𝐾^∼)[𝑤^∼₂₁𝐼_𝑛,𝑤^∼₃₁𝐼_𝑛,…,𝑤^∼_𝑁1𝐼_𝑛]; then, the coupled network (42) is globally exponentially synchronized.

4. A Simulation Example

In this section, we give a numerical example to demonstrate the effectiveness of our results.

Consider linearly coupled singular systems (1), where𝑥_𝑖(𝑡)=𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡)^𝑇,𝑓𝑥_𝑖(𝑡)=1,𝑡𝑥15tanh_𝑖1(𝑡),tanh𝑥_𝑖2(𝑡)^𝑇,𝑖=1,2,…,6, (45)1𝐿=15,𝑐=1,Γ=𝐼₂,,.𝐸=8000,𝐴=0−3−4−4𝐵=−2100101−5111101−4210011−5212110−5121101−5 (46)

The Jordan canonical form of 𝐵 is 𝐽=0000000−2.5325000000−5.7338+1.0405𝑖000000−5.7338−1.0405𝑖100000−6100000−6, (47) and the left eigenvector of 𝐵 corresponding to eigenvalue zero is 𝜉=[0.3508,0.1667,0.1298,0.0853,0.1809,0.0866]^𝑇. (48)

Let 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡),𝑥₅(𝑡),𝑥₆(𝑡)𝜉,𝑖=1,2,…,6, (49) and the simulation results of original system are shown in Figure 1.

(a)

(b)

Figure 1 

Curves of error variable 𝑒𝑖(𝑡)(𝑖=1,2,…,6) of system (1).

From Theorem 9, the feedback gain matrix of state feedback control is 𝐾=0.3291−3.4872−2.5034−1.8884. (50) The simulation results of state feedback control are shown in Figure 2.

(a)

(b)

Figure 2 

Curves of error variable 𝑒𝑖(𝑡)(𝑖=1,2,…,6) of system (38).

From Theorem 10, the feedback gain matrix of pinning control is 𝐾^∼=3.4992−2.7554−1.00901.4619. (51) The simulation results of pinning control are shown in Figure 3.

(a)

(b)

Figure 3 

Curves of error variable 𝑒𝑖(𝑡)(𝑖=1,2,…,6) of system (41).

In Figure 1, the error variables are divergent, which indicates that the original system is not synchronized. In Figures 2 and 3, the error variables converge to zero, which indicates that both the state feedback control and the pinning control can synchronize the network. However, the effects of state feedback control and pinning control have no obvious difference.

5. Conclusion

In this paper, the synchronization and control problem of linearly coupled singular systems has been investigated. The dynamical behaviors at each node and the coupling configuration are general and the coupling matrix is not assumed symmetrical. A reference state was proposed making use of the left eigenvector associated with eigenvalue zero of the coupling matrix, and some criteria for globally exponential synchronization were obtained based on the Lyapunov stability theory. The controllers are designed for state feedback control and pinning control to achieve globally exponential synchronization.

The significance of the paper, we think, can be summarized as follows. (1) The present paper applies a special reference state to linearly coupled singular systems and investigates the synchronization and control problem of these systems. Such work is rare in the existing literature to our knowledge. (2) The conditions adopted in this paper are described by strict linear matrix inequalities and can be solved directly by LMI toolbox of Matlab.

However, there are many problems unsolved about the synchronization and control problems of linearly coupled singular systems. For example, in this paper, only the linear coupling case was treated, and the nonlinear coupling case has not been involved; for the dynamical behaviors at each node, we only considered the regular singular systems, and the nonregular singular system case needs to be further studied.

Acknowledgments

This work was supported by NCET and partially by the NSFC under Grant 10531030 and 90818020 and Research Project of Zhejiang Province Education Department under Grant Y201018827.