# Synchronization and Control of Linearly Coupled Singular Systems

Authors: Fang Qingxiang; Peng Jigen; Cao Feilong
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 byEx˙=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 Section2, 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. In 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 λM(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:(1)Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bijΓxj(t),i=1,2,…,N,
where xi(t)=[xi1(t),xi2(t),…,xin(t)]T∈Rn is state variable of node i, i=1,2,…,N, and f:Rn×R→Rn is a time-varying nonlinear function. E,A∈Rn×n are constant matrices; 0<rank(E)=r<n. The scalar c>0 is the coupling strength. bij≥0 denotes the coupling coefficient from node j to node i, for i,j=1,…,N,  i≠j. bii=-∑j=1,j≠iN‍bij,  i=1,2,…,N. Γ=diag(γ1,…,γn) denotes the inner connection at each node with γi≥0,  i=1,…,n, which means that two nodes are connected by their ith component if γi>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 xi(t), i=1,2,…,N, ∥xi(t)-xj(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 thatB=(bij)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]).
IfB 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)
Ifλ is an eigenvalue of B and λ≠0, then Re(λ)<0.(3)
Suppose thatξ=[ξ1,ξ2,…,ξN]T∈RN (without loss of generality, assume ∑i=1N‍ξi=1) is the left eigenvector of B corresponding to eigenvalue 0; then, ξi>0 holds for all i=1,2,…,N.In the following discussion, we always assume thatB is irreducible and f(x,t)∈Lip(L).Letx^(t)=∑i=1N‍ξixi(t), ei(t)=xi(t)-x^(t), fi(x,t)=f(xi,t)-∑k=1N‍ξkf(xk(t),t), i=1,2,…,N. Noticing that ∑j=1N‍bij=0, i=1,2,…,N and ∑i=1N‍ξibij=0, j=1,2,…,N, we have
(2)Ee˙i(t)=Aei(t)+c∑j=1N‍bijΓej(t)+fi(x,t),i=1,2,…,N.
Denote that e(t)=[e1(t),…,eN(t)], and F(t)=[f1(x,t),…,fN(x,t)]; then,
(3)Ee˙(t)=Ae(t)+cΓe(t)BT+F(x,t).LetBT=WJW-1 be the Jordan decomposition of BT, where J=diag(J1,J2,…,Jl),   Jj is σj-order Jordan block, j=1,2,…,l, and ∑j=1l‍σj=N. Denote the diagonal element of Jj by λj, j=1,2,…,l. Without loss of generality, we assume that λ1=0, and then J1=0 is a scalar.Define(4)y(t)=[y1(t),y2(t),…,yN(t)]=e(t)W,
and denote that W=[W1,W2,…,WN]=(wij)N×N; then, yi(t)=e(t)Wi,  i=1,2,…,N.Denote thatρ0=0 and ρj=∑k=1j‍σk, j=1,2,…,l; then, we can rewrite the coupled system (3) as
(5)Ey˙i(t)=(A+cλkΓ)yi(t)+F(x,t)Wi,Ey˙i(t)=cci=ρk-1+1,k=1,2,…,l,Ey˙i(t)=(A+cλkΓ)yi(t)+cΓyi-1(t)+F(x,t)Wi,Ey˙i(t)=(A+cλkΓ)ρk-1+1<i<ρk,k=2,…,l.Forj=1,2,…,l, define
(6)zj=[yρj-1+1yρj-1+2⋮yρj],Aj=[A+cλjΓcΓA+cλjΓ⋱⋱cΓA+cλjΓ],Ej=[EE⋱E](σjn)×(σjn),Fj(x,t)=[F(x,t)Wρj-1+1F(x,t)Wρj-1+2⋮F(x,t)Wρj],
and then the coupled system (5) can be rewritten as
(7)Ejz˙j(t)=Ajzj(t)+Fj(x,t),j=1,2,…,l.Letz(t)=[z1T(t),z2T(t),…,zlT(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 thatW-1=[W~1,…,W~N]=(w~ij)N×N. For j=2,…,l,  k=1,2,…,N, s=1,2,…,N, define
(8)αj=∑d=1σj(∑i=1N‍|wi,ρj-1+d|)2‍,βkj=∑d=1σj‍(|wk,ρj-1+d|∑i=1N‍|wi,ρj-1+d|),γkj=αjξk+βkj,ϕks=∑j=2l‍∑i=1N‍γijw~ki*w~si,
and denote
(9)ϕ=[ϕ22ϕ23⋯ϕ2Nϕ32ϕ33⋯ϕ3N⋮⋮⋱⋮ϕN2ϕN3⋯ϕNN],Φ=ϕ⊗In.In the following, we give the main conclusion of this section.Theorem 4.
If there exists a scalarε>0 and matrices Pj∈R(σjn)×(σjn),  j=2,3,…,l, satisfying
(10)EjTPj=PjTEj≥0,j=2,3,…,l,(11)P^+2L2Φ+εI(N-1)n<0,
where P^=diag(P2TP2+A2*P2+P2TA2,…,PlTPl+Al*Pl+PlTAl), then the coupled system (1) is globally exponentially synchronized.Proof.
First, note thatλ1=0 and corresponding left eigenvector ξ satisfies that ∑i=1N‍ξi=1. This corresponds to synchronization on the manifold z1(t)=0.
Next, we consider the manifoldsz2(t)=0,z3(t)=0,…,zl(t)=0.
Choose the Lyapunov function candidate as follows:(12)V(t)=12∑j=2l‍zj*(t)PjTEjzj(t),
and then the derivative of V(t) along the trajectory of (7) yields
(13)V˙(t)=12∑j=2l‍(zj*(Aj*Pj+PjTAj)zj+Fj*(x,t)Pjzj+zj*PjTFj(x,t))≤12∑j=2l‍(zj*(PjTPj+Aj*Pj+PjTAj)zj+Fj*(x,t)Fj(x,t)).
Sinceei(t)=y(t)W~i,i=1,2,…,N, f(x,t)∈Lip(L), and(14)Fj(x,t)=[∑k=1N‍wk,ρj-1+1[(f(xk,t)-f(x^,t))+∑v=1N‍ξv(f(x^,t)-f(xv,t))]∑k=1N‍wk,ρj-1+2[(f(xk,t)-f(x^,t))+∑v=1N‍ξv(f(x^,t)-f(xv,t))]⋮∑k=1N‍wk,ρj[(f(xk,t)-f(x^,t))+∑v=1N‍ξv(f(x^,t)-f(xv,t))]],we have(15)Fj*(x,t)Fj(x,t)=∑d=1σj‍∑k=1N‍∑s=1N‍wk,ρj-1+d*ws,ρj-1+d×[∑v=1N‍(f*(xk,t)-f*(x^,t))×(f(xs,t)-f(x^,t))+2∑v=1N‍ξv(f*(xk,t)-f*(x^,t))×(f(xv,t)-f(x^,t))+∑v=1N‍∑i=1N‍ξvξi(f*(xv,t)-f*(x^,t))×(f(xi,t)-f(x^,t))∑i=1N‍]≤2∑d=1σj‍(∑s=1N‍|ws,ρj-1+d|)×∑k=1N‍|wk,ρj-1+d|∥f(xk,t)-f(x^,t)∥2+2(∑d=1σj‍(∑i=1N‍|wi,ρj-1+d|)2)×∑v=1N‍ξv∥f(xv,t)-f(x^,t)∥2≤2L2∑k=1N‍γkjek*ek=2L2∑k=1N‍γkjW~k*y*(t)y(t)W~k=2L2∑v=2N‍∑s=2N‍(∑k=1N‍γkjw~vk*w~sk)yv*ys,V˙(t)≤12(∑j=2l‍∑s=2N‍zj*(PjTPj+Aj*Pj+PjTAj)zj+2L2∑k=2N‍∑s=2N‍ϕksyk*ys);
that is,
(16)V˙(t)≤12z^*(t)(P^+2L2Φ)z^(t),
where z^(t)=[z2T(t),…,zlT(t)]T.
According to (11), one has
(17)V˙(t)<-12ε∑j=2Nzj*zi≤-εκV(t),
where κ=max{λM(PjTEj),j=2,3,…,l}.
Denote that(18)W^=[w~21w~22⋯w~2Nw~31w~32⋯w~3N⋮⋮⋱⋮w~N1w~N2⋯w~NN],γ^i=∑j=2l‍γij, i=1,2,…,N, and Γ^=diag(γ^1,…,γ^N); then, ϕ=(W^T)*Γ^W^T, which implies that ϕ and Φ are both positive definite matrices.
It follows from (10) and (11) that the pairs (Ej,Aj), j=2,…,l, are all regular and impulse-free [45]; that is, there exist nonsingular matrices Mj,Qj∈C(σjn)×(σjn) satisfying that
(19)MjEjQj=diag(Iσjr,0),MjAjQj=diag(A~j,Iσj(n-r)),
for j=2,3,…,N.
By (10) and (11), we can also obtain that (Mj-1)*PjQj=[Pj10Pj3Pj4], where Pj1∈R(σjr)×(σjr) and Pj1>0, j=2,3,…,N [36].
LetQj-1zj=[zj1T,zj2T]T and MjFj(x,t)=[(Fj1(x,t))T,(Fj1(x,t))T]T, where zj1∈Cσjr and Fj1(x,t)∈Cσjr, j=1,2,…,l, and then the system (7) can be rewritten as
(20)z˙j1(t)=A~jzj1(t)+Fj1(x,t),zj2(t)=-Fj2(x,t),j=1,2,…,l,(21)V(t)=12∑j=2l‍zj1*Pj1zj1.
Since the matrix Pj1 is positive definite, zj1(t),  i=2,3,…,l, are globally exponentially stable. In the following, we show that zj2(t), j=2,3,…,l, are also globally exponentially stable; thus, the coupled system (1) is globally exponentially synchronized.
Denote thatMj=[Mj1TMj2T]T, where Mj1∈C(σjr)×(σjn),  j=1,2,…,l. Without loss of generality, we assume that ∥Mj2∥=1 [40].
From (20), we have
(22)∥zj2(t)∥2≤∥Mj2∥2∥Fj(x,t)∥2=Fj*(x,t)Fj(x,t)≤2L2∑v=2N‍∑s=2N‍ηjvsyv*ys,
where ηjvs=∑k=1N‍γkjw~vk*w~sk, j=2,3,…,l, v=1,2,…,N,  s=1,2,…,N.
Forj=2,3,…,l, define
(23)Ψj=[ηj22Inηj23In⋯ηj2NInηj32Inηj33In⋯ηj3NIn⋮⋮⋱⋮ηjN2InηjN3In⋯ηjNNIn],
and let λΨj stand for the maximal eigenvalue of the matrix Ψj; then,
(24)∥zj2(t)∥2≤2L2λΨjλQ×∑k=2l‍(∥zk1(t)∥2+∥zk2(t)∥2),j=2,3,…,l,(25)∑j=2l‍∥zj2(t)∥2≤2L2λQλΨ∑k=2l‍(∥zk1(t)∥2+∥zk2(t)∥2),where λQ is the maximal eigenvalue of the matrix diag(Q2*Q2,Q3*Q3,…,Ql*Ql) and λΨ=∑j=2l‍λΨj.
From (25), we have
(26)(1-2L2λQλΨ)∑j=2l‍∥zj2(t)∥2≤2L2λQλΨ∑k=2l‍∥zk1(t)∥2.
By choosing suitably Q2,Q3,…,Ql such that 1-2L2λQλΨ>0, we have
(27)∑j=2l‍∥zj2(t)∥2≤2L2λQλΨ1-2L2λQλΨ∑k=2l‍∥zk1(t)∥2;
consequently, zj2(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 thatE,A∈Rn×n,  B∈Rn×q,  C∈Rp×n, γ>0 is a scalar, 0<rank(E)=r<n,E=ELERT, EL,ER∈Rn×r are of full column rank, U,V∈Rn×(n-r) are of full column rank and composed of bases of Null(E) and Null(ET), respectively. There exists P∈Rn×n satisfying
(28)PTA+ATP+BBT+PTCTCPγ2<0,ETP=PTE≥0,
if and only if there exist a symmetric matrix Y∈Rn×n and a matrix S∈R(n-r)×(n-r) satisfying
(29)[Φ1Φ2Φ2T-γ2I]<0,ELTYEL>0,
where Φ1=ATS1+S1TA+BBT,  Φ2=S1TCT, and S1=YE+VSUT.LetλΦ be the maximal eigenvalue of the matrix Φ, Ej=EjLEjRT, where EjL, EjR∈R(σjn)×(σjr) are of full column rank, and Uj,  Vj∈R(σjn)×(σj(n-r)) are of full column rank and composed of bases of Null(Ej) and Null(EjT), respectively, j=2,3,…,l.Based on Theorem4 and Lemma 5, we have the following conclusion.Theorem 6.
If there exist a scalarε>0, symmetric matrices Yj∈R(σjn)×(σjn), and matrices Sj∈R(σj(n-r))×(σj(n-r)) satisfying
(30)[(2L2λΦ+ε)Iσjn-Aj*AjAj*+ΛjTAj+Λj-Iσjn]<0,EjLTYjEjL>0,
where Λj=YjEj+VjSjUjT,  j=2,3,…,l, then the coupled system (1) is globally exponentially synchronized.WhenB is a real symmetric matrix, J=diag(λ1,λ2,…,λN) and W is an orthogonal matrix, where 0=λ1≥λ2≥⋯≥λN are the eigenvalues of B. The coupled system (3) can be rewritten as
(31)Ey˙i(t)=(A+cλiΓ)yi(t)+F(x,t)Wi,i=1,2,…,N.Forj=2,…,N,  k=1,2,…,N, and v=1,2,…,N, define
(32)αsj=(∑i=1N‍|wij|)2,βskj=|wkj|∑i=1N‍|wij|,γskj=αjξk+βkj,ϕskv=∑j=2N‍∑i=1N‍γsijw~ki*w~vi,ϕs=[ϕs22ϕs23⋯ϕs2Nϕs32ϕs33⋯ϕs3N⋮⋮⋱⋮ϕsN2ϕsN3⋯ϕsNN],Φs=ϕs⊗In,
and denote that λΦs is the maximal eigenvalue of Φs, E=ELERT,  EL,ER∈Rn×r are of full column rank, U,V∈Rn×(n-r) are of full column rank and composed of bases of Null(E) and Null(ET), respectively.Similar to the proof of Theorem6, we have the following conclusion.Theorem 7.
Suppose thatB is a real symmetric matrix. If there exist a scalar ε>0, a symmetric matrix Y∈Rn×n and a matrix S∈R(n-r)×(n-r) satisfying
(33)ELTYEL>0,(34)[(2L2λΦs+ε)In-(A+cλiΓ)T(A+cλiΓ)(A+cλiΓ)T+ΛTA+cλiΓ+Λ-In]<0,
where Λ=YE+VSUT, 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
(35)(2L2λΦs+ε)In+ΛTΛ+ATΛ+ΛTA+cλi(ΓΛ+ΛTΓ)<0.
When ΓΛ+ΛTΓ>0, if there exists a scalar c1>0 satisfying
(36)(2L2λΦs+ε)In+ΛTΛ+ATΛ+ΛTA+c1λ2(ΓΛ+ΛTΓ)<0,
then inequality (34) is right for all c∈[c1,∞). When ΓΛ+ΛTΓ<0, if there exists a scalar c2>0 satisfying
(37)(2L2λΦs+ε)In+ΛTΛ+ATΛ+ΛTA+c2λN(ΓΛ+ΛTΓ)<0,
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(38)Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bijΓxj(t)+Dui(t),ui(t)=-Kxi(t),i=1,2,…,N,
where D∈Rn×m is constant matrix, and K∈Rm×n is feedback gain matrix.We continue to use the symbols in Section2, and let Dj=diag(D,D,…,D), Kj=diag(K,K,…,K),  Dj∈R(σjn)×(σjm), Kj∈R(σjm)×(σjn), j=1,2,…,l; then, the closed loop coupled network (38) can be rewritten as
(39)Ejz˙j(t)=(Aj-DjKj)zj(t)+Fj(x(t),t).Follow the proof of Theorem6, we have the following conclusion.Theorem 9.
If there exist a scalarε>0, symmetric matrices Yj∈R(σjn)×(σjn), matrices Sj∈R(σj(n-r))×(σj(n-r)), and matrix K∈Rm×n satisfying
(40)EjLTYjEjL>0,[(2L2λΦ+ε)Iσjn-Aj*Aj+Aj*DjKj+KjTDjAjAj*-KjTDjT+ΛjTAj-DjKj+Λj-Iσjn]<0,
where Λj=YjEj+VjSjUjT,  j=1,2,…,l, then the closed loop network (39) is globally exponentially synchronized with the state feedback ui(t)=-Kxi(t), i=1,2,…,N.


### 3.2. Pinning Control

Suppose that the linearly coupled network with a single controller is(41)Ex˙1(t)=Ax1(t)+f(x1(t),t)+c∑j=1N‍b1jΓxj(t)-K~e1(t),Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bijΓxj(t),i=2,3,…,N,
where K~∈Rn×n is the feedback gain matrix.Denote thatE^=diag(E2,E3,…,El), A^=diag(A2,A3,…,Al), F^(x,t)=[F2T(x,t), F3T(x,t),…,FlT(x,t)]T, z^=[y2T,y3T,…,ylT]T, Γi=ξ1(∑k=1N‍wki)K~-w1iIn, i=2,3,…,N, and Γ~=[Γ2T,Γ3T,…,ΓNT]T[w~21In,w~31In,…,w~N1In]; then, the coupled network (41) can be rewritten as
(42)E^z^˙=(A^+Γ~)z^+F^(x,t).Denote that(43)α~=∑j=2N‍(∑s=1N‍|wsj|)2,β~k=∑j=2N‍|wkj|∑s=1N‍|wsj|,γ~k=α~ξk+β~k,k=1,2,…,N,ϕ~vs=∑k=1N‍γ~kw~vk*w~sk,v,s=2,3,…,N,ϕ~=[ϕ~22ϕ~23⋯ϕ~2Nϕ~32ϕ~33⋯ϕ~3N⋮⋮⋱⋮ϕ~N2ϕ~N3⋯ϕ~NN],Φ~=ϕ~⊗In,
and let E^=E^LE^RT, 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 scalarε>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~∈Rn×n satisfying
(44)E^LTY^E^L>0,[2L2Φ~+εI(N-1)n-A^*A^-A^*Γ~-Γ~*A^-WkA^*+Γ^*+Λ^TA^+Γ^+Λ^-I(N-1)n]<0,
where Λ^=Y^E^+V^S^U^T and Wk=[w~21In,w~31In,…,w~N1In]*∑j=2N‍(|w1j|2In-ξ1w1j(∑k=1N‍wkj*)K~*-ξ1w1j*(∑k=1N‍wkj)K~)[w~21In,w~31In,…,w~N1In]; then, the coupled network (42) is globally exponentially synchronized.


## 3.1. State Feedback Control

Suppose that the linearly coupled network with a state feedback controller is(38)Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bijΓxj(t)+Dui(t),ui(t)=-Kxi(t),i=1,2,…,N,
where D∈Rn×m is constant matrix, and K∈Rm×n is feedback gain matrix.We continue to use the symbols in Section2, and let Dj=diag(D,D,…,D), Kj=diag(K,K,…,K),  Dj∈R(σjn)×(σjm), Kj∈R(σjm)×(σjn), j=1,2,…,l; then, the closed loop coupled network (38) can be rewritten as
(39)Ejz˙j(t)=(Aj-DjKj)zj(t)+Fj(x(t),t).Follow the proof of Theorem6, we have the following conclusion.Theorem 9.
If there exist a scalarε>0, symmetric matrices Yj∈R(σjn)×(σjn), matrices Sj∈R(σj(n-r))×(σj(n-r)), and matrix K∈Rm×n satisfying
(40)EjLTYjEjL>0,[(2L2λΦ+ε)Iσjn-Aj*Aj+Aj*DjKj+KjTDjAjAj*-KjTDjT+ΛjTAj-DjKj+Λj-Iσjn]<0,
where Λj=YjEj+VjSjUjT,  j=1,2,…,l, then the closed loop network (39) is globally exponentially synchronized with the state feedback ui(t)=-Kxi(t), i=1,2,…,N.


## 3.2. Pinning Control

Suppose that the linearly coupled network with a single controller is(41)Ex˙1(t)=Ax1(t)+f(x1(t),t)+c∑j=1N‍b1jΓxj(t)-K~e1(t),Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bijΓxj(t),i=2,3,…,N,
where K~∈Rn×n is the feedback gain matrix.Denote thatE^=diag(E2,E3,…,El), A^=diag(A2,A3,…,Al), F^(x,t)=[F2T(x,t), F3T(x,t),…,FlT(x,t)]T, z^=[y2T,y3T,…,ylT]T, Γi=ξ1(∑k=1N‍wki)K~-w1iIn, i=2,3,…,N, and Γ~=[Γ2T,Γ3T,…,ΓNT]T[w~21In,w~31In,…,w~N1In]; then, the coupled network (41) can be rewritten as
(42)E^z^˙=(A^+Γ~)z^+F^(x,t).Denote that(43)α~=∑j=2N‍(∑s=1N‍|wsj|)2,β~k=∑j=2N‍|wkj|∑s=1N‍|wsj|,γ~k=α~ξk+β~k,k=1,2,…,N,ϕ~vs=∑k=1N‍γ~kw~vk*w~sk,v,s=2,3,…,N,ϕ~=[ϕ~22ϕ~23⋯ϕ~2Nϕ~32ϕ~33⋯ϕ~3N⋮⋮⋱⋮ϕ~N2ϕ~N3⋯ϕ~NN],Φ~=ϕ~⊗In,
and let E^=E^LE^RT, 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 scalarε>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~∈Rn×n satisfying
(44)E^LTY^E^L>0,[2L2Φ~+εI(N-1)n-A^*A^-A^*Γ~-Γ~*A^-WkA^*+Γ^*+Λ^TA^+Γ^+Λ^-I(N-1)n]<0,
where Λ^=Y^E^+V^S^U^T and Wk=[w~21In,w~31In,…,w~N1In]*∑j=2N‍(|w1j|2In-ξ1w1j(∑k=1N‍wkj*)K~*-ξ1w1j*(∑k=1N‍wkj)K~)[w~21In,w~31In,…,w~N1In]; 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), where(45)xi(t)=[xi1(t),xi2(t)]T,f(xi(t),t)=115[tanh(xi1(t)),tanh(xi2(t))]T,i=1,2,…,6,(46)L=115,c=1,Γ=I2,E=[8000],A=[0-3-4-4],B=[-2100101-5111101-4210011-5212110-5121101-5].The Jordan canonical form ofB is
(47)J=[0000000-2.5325000000-5.7338+1.0405i000000-5.7338-1.0405i100000-6100000-6],
and the left eigenvector of B corresponding to eigenvalue zero is
(48)ξ=[0.3508,0.1667,0.1298,0.0853,0.1809,0.0866]T.Let(49)ei(t)=xi(t)-[x1(t),x2(t),x3(t),x4(t),x5(t),x6(t)]ξ,iii=1,2,…,6,
and the simulation results of original system are shown in Figure 1.Curves of error variableei(t)(i=1,2,…,6) of system (1).
(a)
(b)From Theorem9, the feedback gain matrix of state feedback control is
(50)K=[0.3291-3.4872-2.5034-1.8884].
The simulation results of state feedback control are shown in Figure 2.Curves of error variableei(t)(i=1,2,…,6) of system (38).
(a)
(b)From Theorem10, the feedback gain matrix of pinning control is
(51)K~=[3.4992-2.7554-1.00901.4619].
The simulation results of pinning control are shown in Figure 3.Curves of error variableei(t)(i=1,2,…,6) of system (41).
(a)
(b)In Figure1, 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.



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*Source: 230741-2013-04-03.xml*