# H

Authors: Bowen Du; Dianfu Ma
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper investigates the synchronization of complex dynamical networks with coupling delays and external disturbances by applying local feedback injections to a small fraction of nodes in the whole network. Based onH∞ control theory, some delay-independent and -dependent synchronization criteria with a prescribed H∞ disturbances attenuation index are derived for such controlled networks in terms of linear matrix inequalities (LMIs), which guarantee that by placing a small number of feedback controllers on some nodes, the whole network can be pinned to reach network synchronization. A simulation example is included to validate the theoretical results.


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## Body

## 1. Introduction

Large real communication networked systems have become a hot research topic for a rather long time [1, 2]. Typical examples include the Internet, which is an enormous network of many routers connected by physical or wireless links with information packets flowing on them, and traffic and transportation network [3–5]. Recently, dynamical processes of the complex dynamical networks such as synchronization have been extensively investigated [6–21]. The synchronization discussed here is a kind of typical collective behaviors and basic motions in nature [22]. However, in the case where the whole network cannot synchronize by itself, some controllers may be designed and applied to force the network to synchronize the configuration and states. What is more, the real-world traffic networks normally have a large number of nodes; therefore, it is usually difficult to control a network by adding the controllers to all nodes.In manipulating various networks, pinning control is a simple and cost-effective technique for control, stabilization, and synchronization [7–12]. Wang and Chen [7] introduced a uniform model of complex dynamical networks by considering dynamical elements of a network as nodes and exploited a pinning control technique for scale-free chaotic dynamical networks, where local feedback injections were applied to a small portion of nodes so as to control the entire network. Reference [8] provided a clear explanation on why significantly less local controllers were needed by the specifically selective pinning scheme, which pinned the most highly connected nodes in a scale-free network, than that required by the randomly pinning scheme, and why there was no significant difference between the two schemes for controlling random-graph networks. In [9, 10], the idea of pinning control was again used to stabilize complex dynamical networks with nonlinear couplings onto some homogenous states. Several adaptive synchronization criteria were given in [11] by using Lyapunov stability theory and pinning control method. Reference [12] further solved some fundamental problems on how the local controllers on the pinned nodes affect the global network synchronization. The common feature of the work in [7–12] is that there are no coupling delays in the network. Networks with coupling delays have also received a great deal of attention. The time delays are usually caused by finite speed of information processing and communication, and their existence make dynamical behaviors of the networks much more complicated [13]. Pinning control of general complex dynamical networks with time-delay was given in [14, 15].Moreover, in real physical systems, some noises or external disturbances always exist that may cause instability and poor performance and thereby destroying the synchronization performance. TheH∞ control theory has been seen as an effective tool to reduce the effect of the noises or disturbances in chaos synchronization [23–25]. Reference [23] proposed the H∞ control concept to reduce the effect of the disturbance on the available output to within a prescribed level. References [24, 25] proposed two dynamic feedback approaches for H∞ synchronization of chaotic systems with and without time-delay. Therefore, how to reduce the effect of the noises or disturbances in synchronization of complex dynamical networks should also be paid attention to. New H∞ synchronization and state estimation problems were proposed for an array of coupled discrete time-varying stochastic complex networks over a finite horizon [17]. However, to the best of our knowledge, the pinning synchronization of complex networks with coupling delays and external disturbances has not yet been established, which motivates the present study.In this paper, we aim to deal with the pinning synchronization for complex dynamical networks with coupling delays and external disturbances byH∞ control theory. By linearizing the controlled network on the synchronization state, the synchronization problem can be viewed as a normal H∞ control problem. Then by some necessary model transforms, both delay-independent and -dependent synchronization conditions with a given H∞ disturbances attenuation index γ are derived in terms of linear matrix inequalities (LMIs), which guarantee that by placing a small number of feedback controllers on some nodes, the whole network can be pinned to reach network synchronization. Finally, simulation results show that the network reaches the desired synchronization performance under the pinning control when there exist coupling delays and external disturbances.Throughout this paper,In denotes the n×n identity matrix; ℝn and ℝn×m denote the n-dimensional and the n×m-dimensional Euclidean spaces, respectively; X<0 represents that X is the symmetric negative matrix; given a matrix Y, σ-(Y) represents its largest singular value; the superscripts T and -1 stand for matrix transposition and matrix inverse; in symmetric block matrices, * is used as an ellipsis for terms induced by symmetry; the notation ⊗ denotes the Kronecker product; the space of square-integrable vector functions over [0,∞) is denoted by L2[0,∞), and for w(t)∈L2[0,∞), its normalized energy is defined by ∥w(t)∥2=(∫0∞‍wT(t)w(t)dt)1/2.


## 2. Model Description and Preliminaries

Consider the following complex dynamical network model with a coupling delay(1)x˙i(t)=f(xi(t))+c∑j=1N‍aijΓxj(t-d)x˙i(t)=+Bwi(t)+ui(t),zi(t)=Cxi(t),i=1,2,…,N,
where f:ℝn→ℝn is a continuously differentiable function, xi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝn are the state variables of node i, and ui(t)∈ℝn is the control input. wi(t)∈ℝp is the external disturbance input that belongs to L2[0,∞), and zi(t)∈ℝq is the output vector. d≥0 is the time delay (we assume that all delays are the same in the network), the constant c>0 is coupling strength, Γ=(γij)∈ℝn×n is a inner-coupling matrix, if some pairs (i,j), 1≤i, j≤n with γij≠0, then it means that the two coupled nodes are linked through their ith and jth state variables, respectively. A=(aij)∈ℝN×N is the outer-coupling matrix of the network, in which aij is defined as follows: if there is a connection between node i and node j(j≠i), then aij=aji=1; otherwise, aij=aji=0(j≠i), and the diagonal elements of matrix A are defined by
(2)aii=-∑j=1,j≠iN‍aij=-∑j=1,j≠iN‍aji,i=1,2,…,N.
Suppose that network (1) is connected in the sense that there are no isolated clusters, that is, A is an irreducible matrix. B and C are known real matrices with appropriate dimensions.Before stating the main results of this paper, some preliminaries need to be given for convenient analysis.Lemma 1.
SupposeA is a real symmetric and irreducible matrix, in which aij≥0(j≠i) and aii=-∑j=1,j≠iN‍aij, nonzero matrix D=diag(d1,d2,…,dN) satisfies di≥0(1≤i≤N). Let B1=A-D, then(i)
all the eigenvalues ofB1 are less than 0;(ii)
there exists an orthogonal matrixU∈ℝN×N such that UTBU=diag(λ1,λ2,…,λN), where λ1,λ2,…,λN are the eigenvalues of B1.The proof of Lemma1 is omitted here since it can be deduced using linear algebra theory such as in [26].Lemma 2 (see [19]).
Assume thata and b are vectors, then for any positive-definite matrix X, the following inequality holds:
(3)-2aTb≤infX>0{aTXa+bTX-1b}.Lemma 3 (see [27]).
The LMI(4)[Q(x)S(x)ST(x)R(x)]>0,
where Q(x)=QT(x), R(x)=RT(x), and S(x) depend affinely on x, is equivalent to R(x)>0, Q(x)-S(x)R-1(x)ST(x)>0.


## 3.H∞-Based Pinning Synchronization of Complex Dynamical Networks with a Coupling Time Delay

In this section, we present the synchronization of a network (1) to an isolate node, which is assumed as
(5)s˙(t)=f(s(t)),z-(t)=Cs(t),
in which s(t) can be an equilibrium point, a periodic orbit, and even a chaotic orbit in the phase space. To achieve the above goal, we apply the pinning control strategy on a small fraction δ(0<δ≤1) of the nodes in network (1). Suppose that nodes i1,i2,…,ik are selected to be under pinning control, and ik+1,il+2,…,iN are the unselected nodes, where k=[δN] stands for the smaller but nearest integer to the real number δN. Certainly, k is not less than 0 for pinning control. Then, the controlled network can be described as
(6)x˙il(t)=f(xil(t))+c∑j=1N‍ailjΓxj(t-d)x˙il(t)=+Bwil(t)+uil,l=1,2,…,k,x˙il(t)=f(xil(t))+c∑j=1N‍ailjΓxj(t-d)x˙il(t)=+Bwil(t),l=k+1,k+2,…,N.
For simplicity, we use the local linear negative feedback control law as follows:
(7)uil=-cdilΓ(xil(t-d)-s(t-d)),l=1,2,…,k,
where the feedback gain dil>0.Combining (6)-(7) and letting dil=0, l=k+1, k+2,…,N, we have
(8)x˙i(t)=f(xi(t))+c∑j=1N‍aijΓxj(t-d)x˙i(t)=-cdiΓ(xi(t-d)-s(t-d))+Bwi(t),zi(t)=Cxi(t),i=1,2,…,N.
Denote
(9)ei(t)=xi(t)-s(t),i=1,2,…,N.
The error dynamical system is then described by
(10)e˙i(t)=f(ei(t)+s(t))-f(s(t))e˙i(t)=+c∑j=1N‍aijΓej(t-d)-cdiΓei(t-d)+Bwi(t),z-i(t)=Cei(t),i=1,2,…,N.
Linearizing the system (10) on the state s(t) yields the following error dynamical system:
(11)e˙i(t)=J(t)ei(t)+c∑j=1N‍aijΓej(t-d)e˙i(t)=-cdiΓei(t-d)+Bwi(t),z-i(t)=Cei(t),i=1,2,…,N,
where J(t)∈ℝn×n is the Jacobian of f on s(t). Obviously, if all the errors ei(t)(i=1,2,…,N) uniformly asymptotically tend to zero, then the network (1) realizes synchronization.Define the following matrices:(12)D=diag(d1,d2,…,dN)∈ℝN×N,e(t)=(e1T(t),…,eNT(t))T∈ℝnN,w(t)=(w1T(t),…,wNT(t))T∈ℝpN,z(t)=(z-1T(t),…,z-NT(t))T∈ℝqN.
By using the Kronecker product, the error dynamical system (13) can be rewritten in the following matrix form:
(13)e˙(t)=(IN⊗J(t))e(t)+(cB1⊗Γ)e˙(t)=×e(t-d)+(IN⊗B)w(t),z(t)=(IN⊗C)e(t),
where B1=A-D. For the system (13), the attenuating ability of its stability performance against external disturbances can be quantitatively measured by the H∞ norm of the transfer function matrix Tzw(s) from the external disturbance w(t) to the controlled output z(t), which is defined by [28]:
(14)∥Tzw(s)∥∞=supv∈Rσ-(Tzw(jv))∥Tzw(s)∥∞=sup0≠w(t)∈L2[0,∞)∥z(t)∥2∥w(t)∥2.To the reach desired synchronization performance against external disturbances, we need to design the local linear negative feedback control lawuil, l=1,2,…,k in (7) such that the following are achieved:(i)
the error dynamical system (13) is asymptotically stable with w(t)=0;(ii)
∥Tzw(s)∥∞<γ holds for a prescribed H∞ index γ>0, or equivalently the system (13), satisfying the following dissipation inequality:
(15)∫0∞‍∥z(t)∥2dt<γ2∫0∞‍∥w(t)∥2dt,∀w∈L2[0,∞).In summary, the pinning synchronization control problem of complex dynamical networks with external disturbances is transformed into the above H∞ control problem.It is easy to see that the matrixB1 satisfies Lemma 1; let λN≤⋯≤λ2≤λ1<0 be the eigenvalues of matrix B1; thus there exists an orthogonal matrix U∈ℝN×N such that
(16)UTB1U=Λ=diag{λ1,λ2,…,λN}.
Perform the orthogonal transform
(17)e^(t)=(UT⊗In)e(t),w^(t)=(UT⊗Ip)w(t),z^(t)=(UT⊗Iq)z(t).
Then from (13), we have
(18)e^˙(t)=(IN⊗J(t))e^(t)+(cΛ⊗Γ)e^(t-d)e^˙(t)=+(IN⊗B)w^(t)z^(t)=(IN⊗C)e^(t).
Further, it can be easily proved that Tz^w^(s)=(UT⊗In)Tzw(s)(U⊗In), which leads to ∥Tzw(s)∥∞=∥Tz^w^(s)∥∞ according to the definition of H∞ norm defined in (14).We can rewrite (17) into the following equations:
(19)e^˙i(t)=J(t)e^i(t)+cλiΓe^i(t-d)+Bw^i(t),z^i(t)=Ce^i(t),i=1,2,…,N.
By the definition of H∞ norm given in (14), if ∥Tz^iw^i(s)∥∞<γ holds for all i=1,2,…,N, then ∥Tz^w^(s)∥∞<γ follows. To summarize, the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ, if the N equations (18) are all asymptotically stable and satisfy the H∞ index γ.


### 3.1. Delay-Independent Condition

Theorem 4.
For a given indexγ>0, if there exist two symmetric positive-definite matrices P,Q∈ℝn×n, such that
(20)[J(t)TP+PJ(t)cλNPΓPBCT*-Q00**-γI0***-γI]>0
is satisfied. Then, the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ holds for all d∈[0,∞), which implies that network synchronization is reached asymptotically with H∞ disturbance attenuation index γ.Proof.
First, we study the stability of the system (18) without external disturbances, that is, w^i(t)=0. Define a Lyapunov-Krasovskii function
(21)V(e^i(t))=e^i(t)TPe^i(t)+∫t-dt‍e^i(θ)TQe^i(θ)dθ
with positive definite matrices P,Q∈ℝn×n. The time derivative of V(e^i(t)) is
(22)V˙(e^i(t))=e^i(t)T[JT(t)P+PJ(t)+Q]e^i(t)V˙(e^i(t))=+2e^i(t-d)TcλiΓTPe^i(t)V˙(e^i(t))=-e^i(t-d)TQe^i(t-d).
From Lemma 2, we have
(23)2e^i(t-d)TcλiΓTPe^i(t)≤e^i(t-d)TQe^i(t-d)2e^i(t-d)TcλiΓTPe^i(t)≤+c2λi2e^i(t)TPΓQ-1ΓTPe^i(t).
So, we can obtain
(24)V˙(e^i(t))=e^i(t)T[JT(t)P+PJ(t)+QV˙(e^i(t))=e^i(t)T+c2λi2PΓQ-1ΓTP]e^i(t)V˙(e^i(t))≤e^i(t)T[JT(t)P+PJ(t)+QV˙(e^i(t))=e^i(t)T+c2λN2PΓQ-1ΓTP]e^i(t).
By Lemma 3, inequality (19) implies
(25)JT(t)P+PJ(t)+Q+c2λN2PΓQ-1ΓTP<0.
Thus, the system (18) is asymptotically stable when w^i(t)=0.
Subsequently, we discuss the performance of system (18) with nonzero disturbance w^i(t). Consider the cost function
(26)J(w^i)=∫0∞‍(γ-1z^i(t)Tz^i(t)-γw^i(t)Tw^i(t))dt.
Under the assumption that the initial state is zero-valued, we have
(27)J(w^i)=∫0∞‍(γ-1z^i(t)Tz^i(t)-γw^i(t)Tw^i(t))J(w^i)=+V˙(e^i(t))dt-V(e^i(∞))J(w^i)≤∫0∞‍(γ-1z^i(t)Tz^i(t)-γw^i(t)Tw^i(t))J(w^i)=+V˙(e^i(t))dtJ(w^i)≤∫0∞‍ξT(t)Θξ(t)dt,
where
(28)Θ=[JT(t)P+PJ(t)+Q+c2λN2PΓQ-1ΓTP+γ-1CTCPB*-γI],ξ(t)=[x(t)w(t)].
By Lemma 3, inequality (19) is equivalent to Θ<0; thus, we have
(29)∫0∞‍z^i(t)Tz^i(t)dt≤γ2∫0∞‍w^i(t)Tw^i(t)dt
for N equations in the system (18), which completes the proof.


### 3.2.  Delay-Dependent Condition

Lemma 5 (see [29]).
Consider the following time-delay system:(30)x˙(t)=Ax(t)+Adx(t-d)+Bw(t),z^(t)=Cx(t),
where d∈[0,d*] is the constant time delay. Given a scalar γ>0, the system is asymptotically stable and with H∞ performance index γ, if there exist symmetric positive-definite matrices P1,S,R=[R1R2R2TR3] and matrices P2,P3,W=[W1W2W3W4] such that(31)Ξ=[Ξ11Ξ12P2TBd*(W1T+P1)d*(W3T+P2T)-W3TAdCT*Ξ22P3TBd*W2Td*(W4T+P3T)-W4TAd0**-γ2I0000***-d*R1-d*R200****-d*R3-S0******-I]<0holds, where
(32)Ξ11=(A+Ad)TP2+P2T(A+Ad)Ξ11=+(W3TAd+AdTW3)+S,Ξ12=P1-P2T+(A+Ad)TP2+AdTW4Ξ22=-P3-P3T+d*AdTR3Ad.Theorem 6.
For a given indexγ>0, the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ for all d∈[0,d*]; that is, network synchronization is achieved asymptotically with H∞ index γ under the pinning control (7) if there exist symmetric positive-definite matrices P1,S,R=[R1R2R2TR3] and matrices P2,P3,W=[W1W2W3W4] such that(33)Φ=[ϕ11ϕ12P2TBd*(W1T+P1)d*(W3T+P2T)-cλiW3TΓCT*ϕ22P3TBd*W2Td*(W4T+P3T)-cλiW4TΓ0**-γ2I0000***-d*R1-d*R200****-d*R3-S0******-I]<0
is satisfied fori=1,2,…,N, where
(34)ϕ11=(J(t)+cλiΓ)TP2+P2T(J(t)+cλiΓ)ϕ11=+(cλiW3TΓ+cλiΓTW3)+S,ϕ12=P1-P2T+(J(t)+cλiΓ)TP2+cλiΓTW4ϕ22=-P3-P3T+d*c2λi2ΓTR3Γ.Proof.
Due to Lemma5, all N equations in the system (18) are asymptotically stable and satisfy the H∞ performance index γ, which implies that the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ for all d∈[0,d*].Remark 7.
Due to the convex property of LMIs, actually only two LMIs associated with the largest and the smallest eigenvaluesλ1 and λN need to be verified. This helps to significantly reduce the computational burden caused by the number of nodes in network, especially when the number N is very large.Remark 8.
According to Theorems4 and 6, if the network is given, the synchronization conditions are determined by J(t), Γ, B, C, the eigenvalues of matrix B1, the coupling strength c, and the disturbance attenuation index γ. Hence, the stabilization of such networks using feedback control is determined by the dynamics of each uncoupled node, the inner-coupling matrix, the coupling matrix, the disturbance input matrix, the output matrix, and the feedback gain matrix of the network. Generally, the number of controllers is preferred to be very small with the entire network size N. According to Lemma 1, B1 is symmetric and negative even if D has only one nonzero element. So in such case, appropriate c, γ, and D may make Lemma 1 holds. It is concluded that such a complex dynamical network can be pinned to its equilibrium by using only one controller.Remark 9.
In fact, the multiagent systems are the special cases of complex networks, so if we choose all the agents to be pinned, the method developed in this paper can be extended to the multi-agent systems in [30] and the corresponding delay-independent and -dependent consensus criteria can be derived.


## 3.1. Delay-Independent Condition

Theorem 4.
For a given indexγ>0, if there exist two symmetric positive-definite matrices P,Q∈ℝn×n, such that
(20)[J(t)TP+PJ(t)cλNPΓPBCT*-Q00**-γI0***-γI]>0
is satisfied. Then, the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ holds for all d∈[0,∞), which implies that network synchronization is reached asymptotically with H∞ disturbance attenuation index γ.Proof.
First, we study the stability of the system (18) without external disturbances, that is, w^i(t)=0. Define a Lyapunov-Krasovskii function
(21)V(e^i(t))=e^i(t)TPe^i(t)+∫t-dt‍e^i(θ)TQe^i(θ)dθ
with positive definite matrices P,Q∈ℝn×n. The time derivative of V(e^i(t)) is
(22)V˙(e^i(t))=e^i(t)T[JT(t)P+PJ(t)+Q]e^i(t)V˙(e^i(t))=+2e^i(t-d)TcλiΓTPe^i(t)V˙(e^i(t))=-e^i(t-d)TQe^i(t-d).
From Lemma 2, we have
(23)2e^i(t-d)TcλiΓTPe^i(t)≤e^i(t-d)TQe^i(t-d)2e^i(t-d)TcλiΓTPe^i(t)≤+c2λi2e^i(t)TPΓQ-1ΓTPe^i(t).
So, we can obtain
(24)V˙(e^i(t))=e^i(t)T[JT(t)P+PJ(t)+QV˙(e^i(t))=e^i(t)T+c2λi2PΓQ-1ΓTP]e^i(t)V˙(e^i(t))≤e^i(t)T[JT(t)P+PJ(t)+QV˙(e^i(t))=e^i(t)T+c2λN2PΓQ-1ΓTP]e^i(t).
By Lemma 3, inequality (19) implies
(25)JT(t)P+PJ(t)+Q+c2λN2PΓQ-1ΓTP<0.
Thus, the system (18) is asymptotically stable when w^i(t)=0.
Subsequently, we discuss the performance of system (18) with nonzero disturbance w^i(t). Consider the cost function
(26)J(w^i)=∫0∞‍(γ-1z^i(t)Tz^i(t)-γw^i(t)Tw^i(t))dt.
Under the assumption that the initial state is zero-valued, we have
(27)J(w^i)=∫0∞‍(γ-1z^i(t)Tz^i(t)-γw^i(t)Tw^i(t))J(w^i)=+V˙(e^i(t))dt-V(e^i(∞))J(w^i)≤∫0∞‍(γ-1z^i(t)Tz^i(t)-γw^i(t)Tw^i(t))J(w^i)=+V˙(e^i(t))dtJ(w^i)≤∫0∞‍ξT(t)Θξ(t)dt,
where
(28)Θ=[JT(t)P+PJ(t)+Q+c2λN2PΓQ-1ΓTP+γ-1CTCPB*-γI],ξ(t)=[x(t)w(t)].
By Lemma 3, inequality (19) is equivalent to Θ<0; thus, we have
(29)∫0∞‍z^i(t)Tz^i(t)dt≤γ2∫0∞‍w^i(t)Tw^i(t)dt
for N equations in the system (18), which completes the proof.


## 3.2.  Delay-Dependent Condition

Lemma 5 (see [29]).
Consider the following time-delay system:(30)x˙(t)=Ax(t)+Adx(t-d)+Bw(t),z^(t)=Cx(t),
where d∈[0,d*] is the constant time delay. Given a scalar γ>0, the system is asymptotically stable and with H∞ performance index γ, if there exist symmetric positive-definite matrices P1,S,R=[R1R2R2TR3] and matrices P2,P3,W=[W1W2W3W4] such that(31)Ξ=[Ξ11Ξ12P2TBd*(W1T+P1)d*(W3T+P2T)-W3TAdCT*Ξ22P3TBd*W2Td*(W4T+P3T)-W4TAd0**-γ2I0000***-d*R1-d*R200****-d*R3-S0******-I]<0holds, where
(32)Ξ11=(A+Ad)TP2+P2T(A+Ad)Ξ11=+(W3TAd+AdTW3)+S,Ξ12=P1-P2T+(A+Ad)TP2+AdTW4Ξ22=-P3-P3T+d*AdTR3Ad.Theorem 6.
For a given indexγ>0, the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ for all d∈[0,d*]; that is, network synchronization is achieved asymptotically with H∞ index γ under the pinning control (7) if there exist symmetric positive-definite matrices P1,S,R=[R1R2R2TR3] and matrices P2,P3,W=[W1W2W3W4] such that(33)Φ=[ϕ11ϕ12P2TBd*(W1T+P1)d*(W3T+P2T)-cλiW3TΓCT*ϕ22P3TBd*W2Td*(W4T+P3T)-cλiW4TΓ0**-γ2I0000***-d*R1-d*R200****-d*R3-S0******-I]<0
is satisfied fori=1,2,…,N, where
(34)ϕ11=(J(t)+cλiΓ)TP2+P2T(J(t)+cλiΓ)ϕ11=+(cλiW3TΓ+cλiΓTW3)+S,ϕ12=P1-P2T+(J(t)+cλiΓ)TP2+cλiΓTW4ϕ22=-P3-P3T+d*c2λi2ΓTR3Γ.Proof.
Due to Lemma5, all N equations in the system (18) are asymptotically stable and satisfy the H∞ performance index γ, which implies that the error dynamical system (13) is asymptotically stable and ∥Tzw(s)∥∞<γ for all d∈[0,d*].Remark 7.
Due to the convex property of LMIs, actually only two LMIs associated with the largest and the smallest eigenvaluesλ1 and λN need to be verified. This helps to significantly reduce the computational burden caused by the number of nodes in network, especially when the number N is very large.Remark 8.
According to Theorems4 and 6, if the network is given, the synchronization conditions are determined by J(t), Γ, B, C, the eigenvalues of matrix B1, the coupling strength c, and the disturbance attenuation index γ. Hence, the stabilization of such networks using feedback control is determined by the dynamics of each uncoupled node, the inner-coupling matrix, the coupling matrix, the disturbance input matrix, the output matrix, and the feedback gain matrix of the network. Generally, the number of controllers is preferred to be very small with the entire network size N. According to Lemma 1, B1 is symmetric and negative even if D has only one nonzero element. So in such case, appropriate c, γ, and D may make Lemma 1 holds. It is concluded that such a complex dynamical network can be pinned to its equilibrium by using only one controller.Remark 9.
In fact, the multiagent systems are the special cases of complex networks, so if we choose all the agents to be pinned, the method developed in this paper can be extended to the multi-agent systems in [30] and the corresponding delay-independent and -dependent consensus criteria can be derived.


## 4. Simulation Results

In this section, we give an example to demonstrate the effectiveness of the proposed method.We assume that the controlled network (1) consists of 10 identical Lorenz systems [31], which is described by
(35)x˙i(t)=f(xi(t))+c∑j=110‍aijΓxj(t-d)x˙i(t)=+Bwi(t)+ui(t),zi(t)=Cxi(t),i=1,2,…,10,
where the inner-coupling matrix is Γ=diag(1,1,1), the coupling matrix is
(36)A=[-41001110001-30000011000-10000100000-20101001000-20001010010-20000100000-30110111000-30001001010-41000000101-2],B=[011], and C=[100]. In order to clearly reflect the effect of external disturbances to the synchronization performance, the disturbance is assumed to be wi(t)=7sin(0.2t). The node dynamics are then given by
(37)x˙i1=σ(xi2-xi1),x˙i2=rxi1-xi2-xi1xi3,x˙i3=xi1xi2-bxi3,
where σ is called Prandtl number and assumed as σ>1, r>1. There are three equilibriums which are the origin
(38)p=[b(r-1)b(r-1)r-1]T,q=[-b(r-1)-b(r-1)r-1]T.
The Lorenz system is symmetrical with respect to the xi3 axis. Denote r*=σ(σ+b+3)/(σ-b-1). It is known that p and q are unstable and chaos occurs if r>r* Let σ=10, r=28, and b=8/3, which means that p and q are unstable equilibria, and chaos occurs at the same time. A typical behavior of a Lorenz chaos system is the butterfly effect shown in Figure 1. Now the objective here is to stabilize this network (30) onto the unstable equilibrium p with H∞ disturbance attenuation index γ by applying the local linear feedback pinning control (8).Figure 1
Lorenz chaos.Consider the network with a bounded coupling delay0≤d≤d*=0.02, and the coupling strength is chosen as c=3. Take the initial condition as xi(s)=0, s∈[-0.02,0]. Three nodes are pinned and the feedback gains are designed as d1=d7=d9=2; then a minimum value of γ=0.82 is obtained by applying Theorem 6 with
(39)P1=[0.7590-0.06970.0334-0.06970.5061-0.82530.0334-0.82534.0335],P2=[0.79020.41870.9993-0.46811.0807-1.30120.0072-0.33121.3426],P3=[1.2139×104-7.4842×1031.8941×1041.6926×1041.3199×104-1.6672×104-1.6925×104-1.3198×1041.6672×104],S=[2.8504×1052.6204×1052.6492×1052.6204×1051.3184×106-2.0552×1062.6492×105-2.0552×1067.4047×106],R3=[3.9131×108-3.9789×107-4.0882×107-3.9789×1073.8412×108-2.5207×107-4.0882×107-2.5207×1074.1503×108].
Figure 2 shows the control results when d=0.015. Figure 3 shows the control results when we select 5 nodes as the controlled nodes. It can be seen that if we select more pinned nodes, it will cost less time to reach stable state.Figure 2
Control results of the network whenc=3, d1=d7=d9=3, and d=0.015.Figure 3
Control results of the network whenc=3, d1=d2=d4=d7=d9=3, and d=0.015.


## 5. Conclusions

This paper addressed the pinning synchronization problem for a group of complex dynamical networks with coupling delays and external disturbances, by transforming it into a normalH∞ control problem. Specifically, delay-independent and delay-dependent conditions in terms of LMI were both derived to ensure the synchronization of networks with a prescribed H∞ disturbance attenuation index. It deserves pointing out that the obtained results can be extended to complex dynamical network with coupling heterogeneous delays or time-varying delays by the same method introduced in this paper.



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*Source: 275205-2013-05-08.xml*