# Enhancement of the Quality and Robustness in Synchronization of Nonlinear Lur'e Dynamical Networks

Authors: Shiyun Xu; Yong Tang; Huadong Sun; Ziguan Zhou; Ying Yang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

In order to improve the synchronous reliability and dependability of complex dynamical networks, methods need to be proposed to enhance the quality and robustness of the synchronization scheme. The
present study focuses on the robust fault detection issue within the synchronization for a class of nonlinear
dynamical networks composed by identical Lur'e systems. Sufficient conditions in terms of linear
matrix inequalities (LMIs) are established to guarantee global robustℋ−/ℋ∞ synchronization of the
network. Under such a synchronization scheme, the error dynamical system is globally asymptotically
stable, the effect of external disturbances is suppressed, and at the same time, the network is sensitive to
possible faults based on a mixed ℋ−/ℋ∞ performance. The fault sensitivity ℋ− index, moreover, can
be optimized via a convex optimization algorithm. The effectiveness and applicability of the analytical
results are demonstrated through a network example composed by the Chua's circuit, and it shows that
the quality and robustness of synchronization has been greatly enhanced.


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## Body

## 1. Introduction

In daily life, many physical systems can be characterized by various complex network models whose nodes are the elements of the network and the edges represent the interactions among them [1]. Treated as typical versions of large-scale systems, the notion of complex dynamical networks has drawn more and more attentions in recent years [2, 3]. One of the interesting and significant phenomena in complex dynamical networks is the synchronization of all dynamical nodes, which is a kind of typical collective behaviors and basic motions in nature [4–9]. Aiming at deriving global synchronization conditions, attempts have been made to consider the synchronization for a special class of networks composed of nonlinear Lur’e systems [10–12]. The main reason is that, in various fields of theory and engineering applications, vast amounts of nonlinear systems can be represented as the Lur’e type, including the Chua’s circuit [13], the Goodwin model [14], and the swarm model [15]. Primary methods of dealing with such problems, among others, are developed under the framework of absolute stability theory [16].In order to improve the synchronous reliability and dependability, methods have been proposed to enhance the quality and robustness of the synchronization scheme. Due to the instability and poor performance that caused by noise or disturbances, it is reasonable to take the noise phenomenon into account during the synchronization process of complex dynamical networks [17, 18]. On the other hand, research in fault diagnosis has been gaining increasing consideration worldwide in the past decades [19–23]. One of the key issues related to fault detection is concerned with its robustness. Large amounts of the relevant jobs have been done for the linear systems in order to examine the robust fault detection (RFD) problem (see [22, 23] and the references therein). In a recent work, we have investigated the robust fault sensitive synchronization of nonlinear Lur’e systems coupled in a master-slave fashion [24]. Similarly, in complex dynamical networks, since it is inevitable for faults to happen within each of the single node, a fault-free synchronization process cannot always be guaranteed. Even though, there is a few work concentrating upon the RFD problem of large-scale nonlinear systems, and hardly there is any previous work that brought the notion “fault” into physical aspects such as synchronization of nonlinear dynamical networks.Based on these considerations, this present study considers the fault detection and disturbance rejection problem within robust synchronization for a class of dynamical networks. The network model is composed by identical nodes with each node being a perturbed nonlinear Lur’e system, while at the same time, subject to possible faults. The main challenge in evaluating the synchronization scheme is to distinguish failures from other disturbances, and accordingly, theℋ-/ℋ∞ paradigm is introduced [25]. For the purpose of description, the robustness objectives during synchronization are considered in virtue of the ℋ∞ norm, while the fault sensitivity specifications are expressed by utilizing the formulation of ℋ- index. In this manner, the closed-loop error system is asymptotically stable with the ℋ∞-norm from the disturbance input to controlled output reduced to a prescribed level, and at the same time, with the ℋ- performance index maximized. By transforming the synchronization problem of dynamical networks into absolute stability problem of corresponding error systems as well as applying Lur’e system method in control theory [16], sufficient conditions to the global robust ℋ-/ℋ∞ synchronization within nonlinear Lur’e networks are developed in terms of sets of linear matrix inequalities (LMI) [26]. Furthermore, the derived high-dimensional LMI condition is simplified into three groups of lower-dimensional LMIs, which are easier to handle. It should be pointed out that no linearization technique is involved through derivation of all the synchronization criteria.The rest of the paper is organized as follows. Section2 proposes the model to be examined in this study, and gives the mathematical formulations of the global robust ℋ-/ℋ∞ synchronization problem to be solved. In Section 3, the global robust ℋ∞ synchronization scheme of the networks is firstly studied, based on which the criteria on ℋ-/ℋ∞ synchronization are then proposed in virtue of the LMI technique. Moreover, performance analysis of the network is also discussed in this part. The dynamical network composed by ten identical Chua’s circuits is adopted as a numerical example in Section 4, and Section 5 closes the paper.


## 2. Notations and Preliminaries

The notations used in this study are fairly standard.ℝn×n is the set of n×n real matrices. For a matrix A, AT denotes its transpose. He is the Hermit operator with HeA=A+AT. If A is a real symmetric negative definite matrix, it is shown by A<0. diag(·) implies a diagonal or block-diagonal matrix. A⊗B indicates the Kronecker product of an n×m matrix A and a p×q matrix B, that is,
(2.1)A⊗B=(a11B⋯a1mB⋮⋱⋮an1B⋯anmB).
If not explicitly stated, matrices are assumed to have compatible dimensions, and the terms replaced by * of a matrix refer to the terms in a symmetric position that do not need to be written out.


### 2.1.  Basic Knowledge on Lur’e Systems

The basic model of nonlinear Lur’e systems subject to input noise and possible faults considered in this paper is described by(2.2)ẋ=Ax+Bφ(y)+Bdd0+Bff0,y=Cx,z=Hx+Df,
where x∈ℝn is the state vector and z∈ℝm represents the measurement output vector. d0∈ℝp is an unknown input vector (including disturbance, uninterested fault as well as some norm-bounded unstructured model uncertainty) belonging to L2[0,+∞), while f0∈ℝq denotes the process, sensor, or actuator fault vector to be detected and isolated. Depending on specific situations under consideration, f0 and d0 can be modeled as different types of signals. The matrices A,B,C,D,Bd,Bf and H, are known constant matrices with appropriate dimensions. Nonlinearity φ:ℝm×ℝ+→ℝm is continuous and locally Lipschitz in the first argument with φ(0)=0,y=(y1T,…,ymT)T and φ(y)=(φ1T(y1),…,φmT(ym))T, where the functions φl(yl),l=1,2…,m are assumed to satisfy the following inequalities:(2.3)0≤φl(yl)yl≤θlyl2,l=1,2…,m,
where θl∈ℝ,l=1,2…,m. Denoting Θ0=diag(θ1,θ2,…,θm), it is obvious to get(2.4)φT(y)(φ(y)-Θ0y)≤0,
and the nonlinearity φ(y) is said to be in the sector [0,Θ0] if it satisfies (2.4).Definition 2.1.
Nonlinear system (2.2) is said to be absolutely stable with respect to the sector [0,Θ0], if the equilibrium point x=0 is globally asymptotically stable for the nonlinearity φ(y) satisfying (2.4).In order to characterize the influence of the disturbance and fault input, several definitions are introduced.Definition 2.2.
Consider the following transfer functiond0↦z of system (2.2):
(2.5)Kzd0(s)≜H(sI-A)-1Bd.
Then its ℋ∞ norm is defined as ∥Kzd0∥∞=supd0∈𝕃2σ¯[Kzd0(jω)]=supd0∈𝕃2(∥Kzd0d0∥2/∥d0∥2), where σ¯ represents the maximal singular value.Definition 2.3.
For system (2.2), the transfer function from the input f0 to output z is given as:
(2.6)Kzf0(s)≜H(sI-A)-1Bf+D,
whose ℋ- index is defined by ∥Kzf0(s)∥-[0,ω¯]≜infω∈[0,ω¯]σ̲[Kzf0(jω)] =inff0∈𝕃2(∥Kzf0f0∥2/∥f0∥2), where σ̲ stands for the minimum singular value and ω¯ denotes the frequency band [0,ω¯].Remark 2.4.
Theℋ- index defined has been widely adopted to measure the sensitivity of residual to fault in the frequency domain. A system is said to possess a better level of RFD, if the ℋ∞ norm of its transfer function from the disturbances to the performance variable is small; meanwhile, the ℋ- index of the transfer function from fault to the output variable is large [19]. Various kinds of ℋ-/ℋ∞ performance criteria have been proposed to determine the RFD issue [21], and the performance is mostly adopted as a trade-off between robustness and sensitivity. In this study, for the sake of simplicity, we will consider the case of maximizing the fault sensitivity ∥Krf(s)∥- with disturbance attenuation ∥Krd(s)∥∞ being a prescribed constant.


### 2.2. Dynamical Networks Composed of Lur’e Nodes

Consider a class of complex dynamical network model with each node being a general Lur’e system (2.2) shown as follows:(2.7)ẋi=Axi+Bφ(yi)+∑j=1NgijΓzj+Bdd0+Bff0,yi=Cxi,zi=Hxi+Df0,i=1,2,…,N,
where xi∈ℝnandzi∈ℝm are the state and measurement output of the ith node, respectively. d0 and f0 are defined as in system (2.2), which are supposed to be the same with respect to each node. The inner coupling matrix Γ=(τij)n×n denotes the coupling pattern between two nodes. G=(gij)N×N is the outer coupling matrix, standing for the coupling configuration of the network. If there is a connection between node i and node j(i≠j), then gij=gji=1; otherwise, gij=gji=0(i≠j). The row sums of G are zero, that is, ∑j=1,j≠iNgij=-gii,i=1,2,…,N. Let φi=(φi1T,…,φimT)T∈Rm and φ(yi)=(φ1T(yi1)⋯φmT(yim))T∈Rm with the following properties:(2.8)0≤φl(yil)yil≤θ1lyil2,0≤φl′(yil)≤θ1l,i=1,2,…,N,l=1,2,…,m.
Denote Θ1=diag(θ11,…,θ1m) then the nonlinear function φ(y) belongs to the sector [0,Θ1].Lemma 2.5 (Wu [27]).
The eigenvalues of an irreducible matrixG0=(G0ij)∈RN×N with ∑j=1,j≠iNG0ij=-G0ii,i=1,2,…,N satisfy the following.(i)
0 is an eigenvalue ofG0 associated with the eigenvector (1,1,…,1)T.(ii)
IfG0ij≥0 for all 1≤i,j≤N,i≠j, then the real parts of all eigenvalues of G0 are less than or equal to 0 and all possible eigenvalues with zero part are 0. In fact, 0 is its eigenvalue of multiplicity 1.Assume that the network (2.7) has no isolate clusters; namely, the network is connected. Under this circumstance, the coupling matrix G is symmetric and irreducible; hence it satisfies all the properties given in Lemma 2.5. Besides, suppose that the coupling matrix G has q distinct different eigenvalues λ1,…,λq; then there exists a nonsingular matrix U with UTU=IN such that UTGU=Λ, where Λ is in the following form:(2.9)Λ=diag⁡(λ1,λ2,…,λ2︸m2,λ3,…,λ3︸m3,…,λq,…,λq︸mq).
Here, λ1=0 is the maximum eigenvalue of multiply 1 and λi is the eigenvalue of multiply mi,i=2,3,…,q satisfying m2+⋯+mq=N-1 and λ2>λ3>⋯>λq.Definition 2.6.
Whend0=f0=0, the dynamical network (2.7) is said to achieve global (asymptotical) synchronization if
(2.10)lim⁡t→∞‖xi-xs‖2=0,i=1,2,…,N,
where ∥·∥2 means the Euclidean norm. xs∈Rn is a solution of an isolate node given by
(2.11)ẋs=Axs+Bφ(ys),ys=Cxs,zs=Hxs,
which can be an equilibrium point, a periodic orbit, or even a nonperiodic orbit.From the properties of the internal coupling matrixG, the following condition holds:(2.12)ẋs=Axs+Bφ(Cxs)+∑j=1NgijΓHxs.
Define error signals ei=xi-xs and residual signals ri=zi-zs for i=1,2,…,N. By subtracting (2.12) from (2.7), one arrives at the dynamics of synchronization residual error:(2.13)ėi=Aei+Bϕ(Cei;xs)+∑j=1NgijΓHej+Bdd0+Bff0+∑j=1NgijΓDf0,ri=Hei+Df0,i=1,2,…,N,
where ϕ(Cei;xs)=φ(Cei+Cxs)-φ(Cxs). Let ϕ(·)=(ϕ1T(·),…,ϕmT(·))T; it is not difficult to derive from (2.8) that for ciTe≠0, the nonlinear functions ϕl(clTei;xs), l=1,2,…,m, satisfy the following sector restrictions:(2.14)0≤ϕl(clTei;xs)clTei=φ(clTei+clTxs)-φ(clTxs)clTei≤θ1l,i=1,2,…,N,l=1,2,…,m,
which leads to(2.15)ϕl(clTei;xs)(ϕl(clTei;xs)-θ1lclTei)≤0,i=1,2,…,N,l=1,2,…,m,
and thus ϕ(Cei;xs) also belongs to the sector [0,Θ1].Remark 2.7.
Based on the basic knowledge of synchronization, the residual error dynamics must be asymptotically stable in order for the whole process to work. Note that the dynamics of the residual error signalr depends not only on f0,d0, and ϕ(y) but also on the states of each isolated node xi. In consequence, this study aims at ensuring the residual error dynamical system to be sensitive to possible faults in the regard of ℋ- index, but the error dynamics also remain robustly asymptotically stable to external disturbance in the ℋ∞ sense. Under such circumstances, the dynamical network composed of Lur’e nodes is said to achieve global synchronization with a guaranteed ℋ-/ℋ∞ performance.Reformulating system (2.13) in virtue of the Kronecker product [28] as(2.16)ė=(IN⊗A+G⊗ΓH)e+(IN⊗B)Φ((IN⊗C)e;Xs)+(IN⊗Bd)d+(IN⊗Bf+G⊗ΓD)f≜A¯e+B¯Φ(C¯e;Xs)+B¯dd+B¯ff,r=(IN⊗H)e+(IN⊗D)f≜H¯e+D¯f,
where(2.17)e=(e1e2⋮eN)∈RNn,d=(d0d0⋮d0)∈RNp,f=(f0f0⋮f0)∈RNq,Φ(C¯e;Xs)=(ϕ(Ce1;xs)ϕ(Ce2;xs)⋮ϕ(CeN;xs))∈RNm,Xs=(xsxs⋮xs)∈RNn,
with Θ=IN⊗Θ1∈RNm>0 and Φ(C¯e;Xs) belonging to the sector [0,Θ]. accordingly, the residual error dynamical system (2.16) can be treated as an Nn-dimensional nonlinear Lur’e system, and the ℋ-/ℋ∞ synchronization of the nonlinear dynamical networks (2.7) can be transformed into the performance analysis and stabilization problem of the corresponding residual error dynamics (2.16).For system (2.16), the transfer function d↦r is given by Krd(s)≜H¯(sI-A¯)-1B¯d, while Krf(s)≜H¯(sI-A¯)-1B¯f+D¯ denotes the transfer function f↦r. Specifically speaking, the main objective of this present study is to determine under what condition the residual error dynamics (2.16) could be asymptotically stable and, at the same time, satisfy the following conditions:(2.18)‖Krf(s)‖->β,‖Krd(s)‖∞<γ,
where γ is a prescribed positive constant, and β is a constant to be optimized. By applying the well-known Parseval theorem to the frequency-domain expressions (2.18), where the ratios ℋ∞ norm and ℋ- index are presented in Definitions 2.2 and 2.3, respectively, we arrive at the equivalent statements as follows:(2.19)J1=∫0∞[r(t)Tr(t)-γ2dT(t)d(t)]dt<0,(2.20)J2=∫0∞[r(t)Tr(t)-β2fT(t)f(t)]dt>0.Accordingly, the definition of robustℋ-/ℋ∞ synchronization is derived as follows.Definition 2.8.
The dynamical networks composed of nonlinear Lur’e nodes in (2.7) are said to achieve global robust ℋ-/ℋ∞ synchronization with disturbance attenuation γ and fault sensitivity β over the frequency range [0,ω¯] (where ω¯ could be both finite and infinite), if with zero disturbance and zero fault, the synchronization residual error signal (2.16) is asymptotically stable, while with zero initial condition and given constants γ>0,β>0, conditions (2.19)-(2.20) hold.


## 2.1.  Basic Knowledge on Lur’e Systems

The basic model of nonlinear Lur’e systems subject to input noise and possible faults considered in this paper is described by(2.2)ẋ=Ax+Bφ(y)+Bdd0+Bff0,y=Cx,z=Hx+Df,
where x∈ℝn is the state vector and z∈ℝm represents the measurement output vector. d0∈ℝp is an unknown input vector (including disturbance, uninterested fault as well as some norm-bounded unstructured model uncertainty) belonging to L2[0,+∞), while f0∈ℝq denotes the process, sensor, or actuator fault vector to be detected and isolated. Depending on specific situations under consideration, f0 and d0 can be modeled as different types of signals. The matrices A,B,C,D,Bd,Bf and H, are known constant matrices with appropriate dimensions. Nonlinearity φ:ℝm×ℝ+→ℝm is continuous and locally Lipschitz in the first argument with φ(0)=0,y=(y1T,…,ymT)T and φ(y)=(φ1T(y1),…,φmT(ym))T, where the functions φl(yl),l=1,2…,m are assumed to satisfy the following inequalities:(2.3)0≤φl(yl)yl≤θlyl2,l=1,2…,m,
where θl∈ℝ,l=1,2…,m. Denoting Θ0=diag(θ1,θ2,…,θm), it is obvious to get(2.4)φT(y)(φ(y)-Θ0y)≤0,
and the nonlinearity φ(y) is said to be in the sector [0,Θ0] if it satisfies (2.4).Definition 2.1.
Nonlinear system (2.2) is said to be absolutely stable with respect to the sector [0,Θ0], if the equilibrium point x=0 is globally asymptotically stable for the nonlinearity φ(y) satisfying (2.4).In order to characterize the influence of the disturbance and fault input, several definitions are introduced.Definition 2.2.
Consider the following transfer functiond0↦z of system (2.2):
(2.5)Kzd0(s)≜H(sI-A)-1Bd.
Then its ℋ∞ norm is defined as ∥Kzd0∥∞=supd0∈𝕃2σ¯[Kzd0(jω)]=supd0∈𝕃2(∥Kzd0d0∥2/∥d0∥2), where σ¯ represents the maximal singular value.Definition 2.3.
For system (2.2), the transfer function from the input f0 to output z is given as:
(2.6)Kzf0(s)≜H(sI-A)-1Bf+D,
whose ℋ- index is defined by ∥Kzf0(s)∥-[0,ω¯]≜infω∈[0,ω¯]σ̲[Kzf0(jω)] =inff0∈𝕃2(∥Kzf0f0∥2/∥f0∥2), where σ̲ stands for the minimum singular value and ω¯ denotes the frequency band [0,ω¯].Remark 2.4.
Theℋ- index defined has been widely adopted to measure the sensitivity of residual to fault in the frequency domain. A system is said to possess a better level of RFD, if the ℋ∞ norm of its transfer function from the disturbances to the performance variable is small; meanwhile, the ℋ- index of the transfer function from fault to the output variable is large [19]. Various kinds of ℋ-/ℋ∞ performance criteria have been proposed to determine the RFD issue [21], and the performance is mostly adopted as a trade-off between robustness and sensitivity. In this study, for the sake of simplicity, we will consider the case of maximizing the fault sensitivity ∥Krf(s)∥- with disturbance attenuation ∥Krd(s)∥∞ being a prescribed constant.


## 2.2. Dynamical Networks Composed of Lur’e Nodes

Consider a class of complex dynamical network model with each node being a general Lur’e system (2.2) shown as follows:(2.7)ẋi=Axi+Bφ(yi)+∑j=1NgijΓzj+Bdd0+Bff0,yi=Cxi,zi=Hxi+Df0,i=1,2,…,N,
where xi∈ℝnandzi∈ℝm are the state and measurement output of the ith node, respectively. d0 and f0 are defined as in system (2.2), which are supposed to be the same with respect to each node. The inner coupling matrix Γ=(τij)n×n denotes the coupling pattern between two nodes. G=(gij)N×N is the outer coupling matrix, standing for the coupling configuration of the network. If there is a connection between node i and node j(i≠j), then gij=gji=1; otherwise, gij=gji=0(i≠j). The row sums of G are zero, that is, ∑j=1,j≠iNgij=-gii,i=1,2,…,N. Let φi=(φi1T,…,φimT)T∈Rm and φ(yi)=(φ1T(yi1)⋯φmT(yim))T∈Rm with the following properties:(2.8)0≤φl(yil)yil≤θ1lyil2,0≤φl′(yil)≤θ1l,i=1,2,…,N,l=1,2,…,m.
Denote Θ1=diag(θ11,…,θ1m) then the nonlinear function φ(y) belongs to the sector [0,Θ1].Lemma 2.5 (Wu [27]).
The eigenvalues of an irreducible matrixG0=(G0ij)∈RN×N with ∑j=1,j≠iNG0ij=-G0ii,i=1,2,…,N satisfy the following.(i)
0 is an eigenvalue ofG0 associated with the eigenvector (1,1,…,1)T.(ii)
IfG0ij≥0 for all 1≤i,j≤N,i≠j, then the real parts of all eigenvalues of G0 are less than or equal to 0 and all possible eigenvalues with zero part are 0. In fact, 0 is its eigenvalue of multiplicity 1.Assume that the network (2.7) has no isolate clusters; namely, the network is connected. Under this circumstance, the coupling matrix G is symmetric and irreducible; hence it satisfies all the properties given in Lemma 2.5. Besides, suppose that the coupling matrix G has q distinct different eigenvalues λ1,…,λq; then there exists a nonsingular matrix U with UTU=IN such that UTGU=Λ, where Λ is in the following form:(2.9)Λ=diag⁡(λ1,λ2,…,λ2︸m2,λ3,…,λ3︸m3,…,λq,…,λq︸mq).
Here, λ1=0 is the maximum eigenvalue of multiply 1 and λi is the eigenvalue of multiply mi,i=2,3,…,q satisfying m2+⋯+mq=N-1 and λ2>λ3>⋯>λq.Definition 2.6.
Whend0=f0=0, the dynamical network (2.7) is said to achieve global (asymptotical) synchronization if
(2.10)lim⁡t→∞‖xi-xs‖2=0,i=1,2,…,N,
where ∥·∥2 means the Euclidean norm. xs∈Rn is a solution of an isolate node given by
(2.11)ẋs=Axs+Bφ(ys),ys=Cxs,zs=Hxs,
which can be an equilibrium point, a periodic orbit, or even a nonperiodic orbit.From the properties of the internal coupling matrixG, the following condition holds:(2.12)ẋs=Axs+Bφ(Cxs)+∑j=1NgijΓHxs.
Define error signals ei=xi-xs and residual signals ri=zi-zs for i=1,2,…,N. By subtracting (2.12) from (2.7), one arrives at the dynamics of synchronization residual error:(2.13)ėi=Aei+Bϕ(Cei;xs)+∑j=1NgijΓHej+Bdd0+Bff0+∑j=1NgijΓDf0,ri=Hei+Df0,i=1,2,…,N,
where ϕ(Cei;xs)=φ(Cei+Cxs)-φ(Cxs). Let ϕ(·)=(ϕ1T(·),…,ϕmT(·))T; it is not difficult to derive from (2.8) that for ciTe≠0, the nonlinear functions ϕl(clTei;xs), l=1,2,…,m, satisfy the following sector restrictions:(2.14)0≤ϕl(clTei;xs)clTei=φ(clTei+clTxs)-φ(clTxs)clTei≤θ1l,i=1,2,…,N,l=1,2,…,m,
which leads to(2.15)ϕl(clTei;xs)(ϕl(clTei;xs)-θ1lclTei)≤0,i=1,2,…,N,l=1,2,…,m,
and thus ϕ(Cei;xs) also belongs to the sector [0,Θ1].Remark 2.7.
Based on the basic knowledge of synchronization, the residual error dynamics must be asymptotically stable in order for the whole process to work. Note that the dynamics of the residual error signalr depends not only on f0,d0, and ϕ(y) but also on the states of each isolated node xi. In consequence, this study aims at ensuring the residual error dynamical system to be sensitive to possible faults in the regard of ℋ- index, but the error dynamics also remain robustly asymptotically stable to external disturbance in the ℋ∞ sense. Under such circumstances, the dynamical network composed of Lur’e nodes is said to achieve global synchronization with a guaranteed ℋ-/ℋ∞ performance.Reformulating system (2.13) in virtue of the Kronecker product [28] as(2.16)ė=(IN⊗A+G⊗ΓH)e+(IN⊗B)Φ((IN⊗C)e;Xs)+(IN⊗Bd)d+(IN⊗Bf+G⊗ΓD)f≜A¯e+B¯Φ(C¯e;Xs)+B¯dd+B¯ff,r=(IN⊗H)e+(IN⊗D)f≜H¯e+D¯f,
where(2.17)e=(e1e2⋮eN)∈RNn,d=(d0d0⋮d0)∈RNp,f=(f0f0⋮f0)∈RNq,Φ(C¯e;Xs)=(ϕ(Ce1;xs)ϕ(Ce2;xs)⋮ϕ(CeN;xs))∈RNm,Xs=(xsxs⋮xs)∈RNn,
with Θ=IN⊗Θ1∈RNm>0 and Φ(C¯e;Xs) belonging to the sector [0,Θ]. accordingly, the residual error dynamical system (2.16) can be treated as an Nn-dimensional nonlinear Lur’e system, and the ℋ-/ℋ∞ synchronization of the nonlinear dynamical networks (2.7) can be transformed into the performance analysis and stabilization problem of the corresponding residual error dynamics (2.16).For system (2.16), the transfer function d↦r is given by Krd(s)≜H¯(sI-A¯)-1B¯d, while Krf(s)≜H¯(sI-A¯)-1B¯f+D¯ denotes the transfer function f↦r. Specifically speaking, the main objective of this present study is to determine under what condition the residual error dynamics (2.16) could be asymptotically stable and, at the same time, satisfy the following conditions:(2.18)‖Krf(s)‖->β,‖Krd(s)‖∞<γ,
where γ is a prescribed positive constant, and β is a constant to be optimized. By applying the well-known Parseval theorem to the frequency-domain expressions (2.18), where the ratios ℋ∞ norm and ℋ- index are presented in Definitions 2.2 and 2.3, respectively, we arrive at the equivalent statements as follows:(2.19)J1=∫0∞[r(t)Tr(t)-γ2dT(t)d(t)]dt<0,(2.20)J2=∫0∞[r(t)Tr(t)-β2fT(t)f(t)]dt>0.Accordingly, the definition of robustℋ-/ℋ∞ synchronization is derived as follows.Definition 2.8.
The dynamical networks composed of nonlinear Lur’e nodes in (2.7) are said to achieve global robust ℋ-/ℋ∞ synchronization with disturbance attenuation γ and fault sensitivity β over the frequency range [0,ω¯] (where ω¯ could be both finite and infinite), if with zero disturbance and zero fault, the synchronization residual error signal (2.16) is asymptotically stable, while with zero initial condition and given constants γ>0,β>0, conditions (2.19)-(2.20) hold.


## 3. Main Results

The intention of this part is to investigate the fault sensitivity as well as disturbance rejection ability of the complex dynamical network (2.7). In order to quantify these two performance indices, one borrows the concept of ℋ- index and ℋ∞-norm defined in the previous section.


### 3.1. Globalℋ∞ Synchronization of Nonlinear Lur’e Networks

In this subsection, we first consider the case that there is no fault existed in the network by extending previous results onℋ∞ synchronization between two identical Lur’e systems to that of nonlinear Lur’e dynamical networks. Accordingly, the network model is described by(3.1)ẋi=Axi+Bφ(yi)+∑j=1NGijΓzj+Bdd0,yi=Cxi,zi=Hxi,i=1,2,…,N,
and the corresponding error dynamics in form of Kronecker product is expressed as(3.2)ė=A¯e+B¯Φ(C¯e;Xs)+B¯dd,r=H¯e.
The disturbance rejection problem within the synchronization of nonlinear dynamical network (3.1) is summarized in the following definition.Definition 3.1.
Given constant scalarγ>0, the dynamical network (3.1) is said to achieve global robust ℋ∞ synchronization, if system (3.2) is globally asymptotically stable with zero disturbance, and meanwhile, the performance index (2.19) is satisfied with zero initial conditions.The robustℋ∞ synchronization can be determined in virtue of the following criterion.Theorem 3.2.
Suppose thatγ>0 is a prescribed constant. For a given scalar α, if there exist positive-definite matrices P=PT>0, diagonal matrices Δ1=diag(δ1,…,δm)>0, Π1=diag(π1,…,πm)>0, and Ω1=diag(ω1,…,ωm)>0, and matrices Q¯1 and Q¯2 such that the following LMI
(3.3)Ξ1=(-He(Q¯1A¯)+H¯TH¯C¯TΘΔ-Q¯1B¯P¯+Q¯1-A¯TQ¯2TC¯TΘΩ-Q¯1B¯d*-HeΔ-B¯TQ¯2T00**HeQ¯2C¯TΠ-Q¯2B¯d**-HeΩ0****-γ2I)<0
holds, where Δ=IN⊗Δ1,Π=IN⊗Π1,andΩ=IN⊗Ω1, then the dynamical network (3.1) achieves global robust ℋ∞ synchronization with disturbance attenuation γ.Proof.
See AppendixA.Remark 3.3.
Theorem3.2 has provided a sufficient condition for the global robust ℋ∞ synchronization of nonlinear Lur’e networks by introducing slack matrices Q¯1 and Q¯2 into LMI (3.3). It is thus expected that Theorem 3.2 will be less conservative than some existing results due to the increasing freedom of these slack variables [29]. With the derived ℋ∞ synchronization conditions on Lur’e networks, the fault detection issue will then be examined in the next subsection. However, if the number of nodes is large, condition (3.3) would become a high-dimensional LMI, which is rather tedious to verify. To this end, both of these criteria will be further simplified to the test of three groups of lower-dimensional LMIs.


### 3.2. Fault Detection within Globalℋ∞ Synchronization

The RFD within a synchronization configuration can be treated as a multiple objective design task; that is, the design objective is not only being as sensitive as possible to faults such that early detection of faults is possible, but on the other hand, the sensitivity of possible faults is maximized, also suppressing the effect of disturbances and modeling errors on the synchronization error and subsequently on the residual, in order to prevent the synchronization process from being destroyed. Next theorem gives an LMI formulation for global robustℋ-/ℋ∞ synchronization.Theorem 3.4.
Suppose thatγ>0,β>0 are prescribed constant scalars. For a given constant α, if there exist a positive-definite matrix P=PT>0, diagonal matrices Δ1=diag(δ11,…,δ1m)>0,Π1=diag(π11,…,π1m)>0,and Ω1=diag(ω11,…,ω1m)>0, and matrices Q¯1andQ¯2 such that LMIs (3.3) as well as
(3.4)Ξ2=(-He(Q¯1A¯)-H¯TH¯C¯TΘΔ-Q1B¯P¯+Q¯1-A¯TQ¯2TC¯TΘΩ-H¯D¯-Q¯1B¯f*-HeΔ-B¯TQ¯2T00**HeQ¯2C¯TΠ-Q¯2B¯f**-HeΩ0****β2I-D¯TD¯)<0
hold, then the dynamical network in (2.7) achieves global robust ℋ-/ℋ∞ synchronization with disturbance attenuation γ and fault sensitivity β.Proof.
See AppendixB.Theorem 3.5.
Suppose thatα,β>0 and γ>0 are given scalars. If there exist matrices Wi>0,Vi, and diagonal matrices Δ1>0,Π1>0,and Ω1>0 such that the following conditions for i=1,2 and q hold:
(3.5)(Υ11+HTHCTΘ1Δ1-ViBΥ31CTΘ1Ω1-ViBd*-HeΔ1-αBTViT00**HeαViCTΠ1-αViBd**-HeΩ10****-γ2I)<0,(3.6)(Υ11-HTHCTΘ1Δ1-ViBΥ31CTΘ1Ω1-HD-ViBf-λiViΓD*-HeΔ1-αBTViT00**HeαViCTΠ1-αViBf-λiαViΓD**-HeΩ10****β2I-DTD)<0,
where Υ11=-He(ViA+λiViΓH) and Υ31=Wi+Vi-αATViT-λiαHTΓTViT, then the conditions given in Theorem 3.4 are ensured.Proof.
See AppendixC.Corollary 3.6.
For a constantα, let β>0 and γ>0 be prescribed constant scalars. The dynamical network (2.7) is said to achieve global robust ℋ-/ℋ∞ synchronization with disturbance attenuation γ and fault sensitivity β, if there exist matrices Wi>0,ViTi, and diagonal matrices Δ1>0,Ω1>,Π1>0 such that the LMI conditions (3.5)-(3.6) hold for i=1,2 and q (corresponding to the largest, second largest, and smallest eigenvalues, resp.).Remark 3.7.
If the number of nodesN is large, the ℋ-/ℋ∞ synchronization criterion of the dynamical network would become a group of LMIs with rather high dimensions. In order to tackle this problem, the synchronization of the nN×nN-dimensional network has been disposed in a lower n-dimensional space through verifying three groups of n-dimensional LMIs in Corollary 3.6, and the derived conditions are quite convenient to use.As an immediate consequence, we arrive at the simplified criterion for global robustℋ∞ synchronization of nonlinear dynamical network (3.1) summarized as in the following corollary.Corollary 3.8.
For a constantα, let β>0 and γ>0 be prescribed constant scalars. If there exist matrices Wi>0, ViTi, and diagonal matrices Δ1>0,Ω1>,Π1>0 such that the LMIs (3.5) for i=1,2 and q are feasible, then the dynamical network (2.7) is said to achieve global robust ℋ∞ synchronization.


### 3.3.ℋ-/ℋ∞ Performance Analysis

It comes from Corollary3.6 that the ℋ-/ℋ∞ synchronization within a dynamical network can be cast into that of three sets of independent systems whose dimensions are the same as that of each isolate node. Namely, if the following systems(3.7)ėλi=(A+λiΓH)eλi+Bϕ(Ceλi)+(Bf+λiΓD)f0+Bdd0,rλi=Heλi+Df0,
satisfy (3.5)-(3.6) for i=1,2andq, then the conditions given in Definition 2.8 will be guaranteed. Suppose the transfer function of system (3.7) from d0↦rλi and f0↦rλi for i=1,2,…,N as Krdi and Krfi, respectively. Then denote(3.8)Krdλ=diag⁡(Krd1,…,KrdN),Krfλ=diag⁡(Krf1,…,KrfN),
where KrdλandKrfλ are in the following form:(3.9)Krdλ=(IN⊗H)(sI-IN⊗A-Λ⊗ΓH)-1(IN⊗Bd),Krfλ=(IN⊗H)(sI-IN⊗A-Λ⊗ΓH)-1(IN⊗Bf-Λ⊗ΓD)+(IN⊗D).On the other hand, consider the following system:(3.10)ėλ=(IN⊗A+Λ⊗ΓH)eλ+(IN⊗B)Φ((IN⊗C)e;Xs)+(IN⊗Bd)d+(IN⊗Bf+Λ⊗ΓD)frλ=(IN⊗H)eλ+(IN⊗D)f,
where eλ=(eλ1T,…,eλNT)Tandrλ=(rλ1T,…,rλNT)T. It can be found that the transfer functions from d↦rλ and f↦rλ of system (3.10) are just those defined in (3.8). Moreover, by carrying out unitary transformation, Krdλ is similar to Krd, and so do Krfλ and Krf. Recall the definition of the ℋ∞ norm and ℋ- index previously stated in Definitions 2.2 and 2.3; then we arrive at the following relationships between the ℋ∞ norms of Krd and Krdi as well as the ℋ- indexes of Krf and Krfi for i=1,2,…,N:(3.11)‖Krd‖∞=‖Krdλ‖∞=max⁡i=1,…,N‖Krdi‖∞,‖Krf‖-=‖Krfλ‖-=min⁡i=1,…,N‖Krfi‖-.Conditions (3.11) show that the ℋ∞ norm of the transfer function from d↦r in (2.16) equals to the maximum of those of the N systems (3.7), whilst the corresponding ℋ- index is the minimum value within those of (3.7). Accordingly, the RFD of the network (2.7) can be cast into those of (3.7); thus we have the following corollary.Corollary 3.9.
For a given scalarγ>0, the performance indexes of the dynamical network (2.7) satisfy ∥Krd∥∞<γ,and∥Krf∥->β, if maxi=1,…,N∥Krdi∥∞<γ and mini=1,…,N∥Krfi∥->β hold in the decoupled systems (3.7) for i=1,2,…,N.The following corollary presents a method of deriving the maximum value of fault sensitivity and, at the same time, suppresses the external disturbance to a prescribed level for the global robustℋ-/ℋ∞ synchronization of network (2.7).Corollary 3.10.
The nonlinear dynamical networks (2.7) are said to achieve global synchronization with guaranteed ℋ∞ performance γ and the maximum fault detection sensitivity β0=ρ, where ρ is the global minimum of the following generalized eigenvalue minimization problem with respect to matrices Wi>0,Vi for i={1,2,q} as well as diagonal matrices Δ1>0,Π1>0, and Ω1>0:
(3.12)min⁡-ρ(Υ11-HTHCTΘ1Δ1-ViBΥ13-HD-ViBf*-HeΔ1Π1C-αBTViT0**HeαVi-αViBf***ρI-DTD)<0,
as well as the LMI condition (3.5) holds. Here, Υ11 and Υ13 are described in Theorem 3.5 with constant scalars α and γ>0 prescribed.


## 3.1. Globalℋ∞ Synchronization of Nonlinear Lur’e Networks

In this subsection, we first consider the case that there is no fault existed in the network by extending previous results onℋ∞ synchronization between two identical Lur’e systems to that of nonlinear Lur’e dynamical networks. Accordingly, the network model is described by(3.1)ẋi=Axi+Bφ(yi)+∑j=1NGijΓzj+Bdd0,yi=Cxi,zi=Hxi,i=1,2,…,N,
and the corresponding error dynamics in form of Kronecker product is expressed as(3.2)ė=A¯e+B¯Φ(C¯e;Xs)+B¯dd,r=H¯e.
The disturbance rejection problem within the synchronization of nonlinear dynamical network (3.1) is summarized in the following definition.Definition 3.1.
Given constant scalarγ>0, the dynamical network (3.1) is said to achieve global robust ℋ∞ synchronization, if system (3.2) is globally asymptotically stable with zero disturbance, and meanwhile, the performance index (2.19) is satisfied with zero initial conditions.The robustℋ∞ synchronization can be determined in virtue of the following criterion.Theorem 3.2.
Suppose thatγ>0 is a prescribed constant. For a given scalar α, if there exist positive-definite matrices P=PT>0, diagonal matrices Δ1=diag(δ1,…,δm)>0, Π1=diag(π1,…,πm)>0, and Ω1=diag(ω1,…,ωm)>0, and matrices Q¯1 and Q¯2 such that the following LMI
(3.3)Ξ1=(-He(Q¯1A¯)+H¯TH¯C¯TΘΔ-Q¯1B¯P¯+Q¯1-A¯TQ¯2TC¯TΘΩ-Q¯1B¯d*-HeΔ-B¯TQ¯2T00**HeQ¯2C¯TΠ-Q¯2B¯d**-HeΩ0****-γ2I)<0
holds, where Δ=IN⊗Δ1,Π=IN⊗Π1,andΩ=IN⊗Ω1, then the dynamical network (3.1) achieves global robust ℋ∞ synchronization with disturbance attenuation γ.Proof.
See AppendixA.Remark 3.3.
Theorem3.2 has provided a sufficient condition for the global robust ℋ∞ synchronization of nonlinear Lur’e networks by introducing slack matrices Q¯1 and Q¯2 into LMI (3.3). It is thus expected that Theorem 3.2 will be less conservative than some existing results due to the increasing freedom of these slack variables [29]. With the derived ℋ∞ synchronization conditions on Lur’e networks, the fault detection issue will then be examined in the next subsection. However, if the number of nodes is large, condition (3.3) would become a high-dimensional LMI, which is rather tedious to verify. To this end, both of these criteria will be further simplified to the test of three groups of lower-dimensional LMIs.


## 3.2. Fault Detection within Globalℋ∞ Synchronization

The RFD within a synchronization configuration can be treated as a multiple objective design task; that is, the design objective is not only being as sensitive as possible to faults such that early detection of faults is possible, but on the other hand, the sensitivity of possible faults is maximized, also suppressing the effect of disturbances and modeling errors on the synchronization error and subsequently on the residual, in order to prevent the synchronization process from being destroyed. Next theorem gives an LMI formulation for global robustℋ-/ℋ∞ synchronization.Theorem 3.4.
Suppose thatγ>0,β>0 are prescribed constant scalars. For a given constant α, if there exist a positive-definite matrix P=PT>0, diagonal matrices Δ1=diag(δ11,…,δ1m)>0,Π1=diag(π11,…,π1m)>0,and Ω1=diag(ω11,…,ω1m)>0, and matrices Q¯1andQ¯2 such that LMIs (3.3) as well as
(3.4)Ξ2=(-He(Q¯1A¯)-H¯TH¯C¯TΘΔ-Q1B¯P¯+Q¯1-A¯TQ¯2TC¯TΘΩ-H¯D¯-Q¯1B¯f*-HeΔ-B¯TQ¯2T00**HeQ¯2C¯TΠ-Q¯2B¯f**-HeΩ0****β2I-D¯TD¯)<0
hold, then the dynamical network in (2.7) achieves global robust ℋ-/ℋ∞ synchronization with disturbance attenuation γ and fault sensitivity β.Proof.
See AppendixB.Theorem 3.5.
Suppose thatα,β>0 and γ>0 are given scalars. If there exist matrices Wi>0,Vi, and diagonal matrices Δ1>0,Π1>0,and Ω1>0 such that the following conditions for i=1,2 and q hold:
(3.5)(Υ11+HTHCTΘ1Δ1-ViBΥ31CTΘ1Ω1-ViBd*-HeΔ1-αBTViT00**HeαViCTΠ1-αViBd**-HeΩ10****-γ2I)<0,(3.6)(Υ11-HTHCTΘ1Δ1-ViBΥ31CTΘ1Ω1-HD-ViBf-λiViΓD*-HeΔ1-αBTViT00**HeαViCTΠ1-αViBf-λiαViΓD**-HeΩ10****β2I-DTD)<0,
where Υ11=-He(ViA+λiViΓH) and Υ31=Wi+Vi-αATViT-λiαHTΓTViT, then the conditions given in Theorem 3.4 are ensured.Proof.
See AppendixC.Corollary 3.6.
For a constantα, let β>0 and γ>0 be prescribed constant scalars. The dynamical network (2.7) is said to achieve global robust ℋ-/ℋ∞ synchronization with disturbance attenuation γ and fault sensitivity β, if there exist matrices Wi>0,ViTi, and diagonal matrices Δ1>0,Ω1>,Π1>0 such that the LMI conditions (3.5)-(3.6) hold for i=1,2 and q (corresponding to the largest, second largest, and smallest eigenvalues, resp.).Remark 3.7.
If the number of nodesN is large, the ℋ-/ℋ∞ synchronization criterion of the dynamical network would become a group of LMIs with rather high dimensions. In order to tackle this problem, the synchronization of the nN×nN-dimensional network has been disposed in a lower n-dimensional space through verifying three groups of n-dimensional LMIs in Corollary 3.6, and the derived conditions are quite convenient to use.As an immediate consequence, we arrive at the simplified criterion for global robustℋ∞ synchronization of nonlinear dynamical network (3.1) summarized as in the following corollary.Corollary 3.8.
For a constantα, let β>0 and γ>0 be prescribed constant scalars. If there exist matrices Wi>0, ViTi, and diagonal matrices Δ1>0,Ω1>,Π1>0 such that the LMIs (3.5) for i=1,2 and q are feasible, then the dynamical network (2.7) is said to achieve global robust ℋ∞ synchronization.


## 3.3.ℋ-/ℋ∞ Performance Analysis

It comes from Corollary3.6 that the ℋ-/ℋ∞ synchronization within a dynamical network can be cast into that of three sets of independent systems whose dimensions are the same as that of each isolate node. Namely, if the following systems(3.7)ėλi=(A+λiΓH)eλi+Bϕ(Ceλi)+(Bf+λiΓD)f0+Bdd0,rλi=Heλi+Df0,
satisfy (3.5)-(3.6) for i=1,2andq, then the conditions given in Definition 2.8 will be guaranteed. Suppose the transfer function of system (3.7) from d0↦rλi and f0↦rλi for i=1,2,…,N as Krdi and Krfi, respectively. Then denote(3.8)Krdλ=diag⁡(Krd1,…,KrdN),Krfλ=diag⁡(Krf1,…,KrfN),
where KrdλandKrfλ are in the following form:(3.9)Krdλ=(IN⊗H)(sI-IN⊗A-Λ⊗ΓH)-1(IN⊗Bd),Krfλ=(IN⊗H)(sI-IN⊗A-Λ⊗ΓH)-1(IN⊗Bf-Λ⊗ΓD)+(IN⊗D).On the other hand, consider the following system:(3.10)ėλ=(IN⊗A+Λ⊗ΓH)eλ+(IN⊗B)Φ((IN⊗C)e;Xs)+(IN⊗Bd)d+(IN⊗Bf+Λ⊗ΓD)frλ=(IN⊗H)eλ+(IN⊗D)f,
where eλ=(eλ1T,…,eλNT)Tandrλ=(rλ1T,…,rλNT)T. It can be found that the transfer functions from d↦rλ and f↦rλ of system (3.10) are just those defined in (3.8). Moreover, by carrying out unitary transformation, Krdλ is similar to Krd, and so do Krfλ and Krf. Recall the definition of the ℋ∞ norm and ℋ- index previously stated in Definitions 2.2 and 2.3; then we arrive at the following relationships between the ℋ∞ norms of Krd and Krdi as well as the ℋ- indexes of Krf and Krfi for i=1,2,…,N:(3.11)‖Krd‖∞=‖Krdλ‖∞=max⁡i=1,…,N‖Krdi‖∞,‖Krf‖-=‖Krfλ‖-=min⁡i=1,…,N‖Krfi‖-.Conditions (3.11) show that the ℋ∞ norm of the transfer function from d↦r in (2.16) equals to the maximum of those of the N systems (3.7), whilst the corresponding ℋ- index is the minimum value within those of (3.7). Accordingly, the RFD of the network (2.7) can be cast into those of (3.7); thus we have the following corollary.Corollary 3.9.
For a given scalarγ>0, the performance indexes of the dynamical network (2.7) satisfy ∥Krd∥∞<γ,and∥Krf∥->β, if maxi=1,…,N∥Krdi∥∞<γ and mini=1,…,N∥Krfi∥->β hold in the decoupled systems (3.7) for i=1,2,…,N.The following corollary presents a method of deriving the maximum value of fault sensitivity and, at the same time, suppresses the external disturbance to a prescribed level for the global robustℋ-/ℋ∞ synchronization of network (2.7).Corollary 3.10.
The nonlinear dynamical networks (2.7) are said to achieve global synchronization with guaranteed ℋ∞ performance γ and the maximum fault detection sensitivity β0=ρ, where ρ is the global minimum of the following generalized eigenvalue minimization problem with respect to matrices Wi>0,Vi for i={1,2,q} as well as diagonal matrices Δ1>0,Π1>0, and Ω1>0:
(3.12)min⁡-ρ(Υ11-HTHCTΘ1Δ1-ViBΥ13-HD-ViBf*-HeΔ1Π1C-αBTViT0**HeαVi-αViBf***ρI-DTD)<0,
as well as the LMI condition (3.5) holds. Here, Υ11 and Υ13 are described in Theorem 3.5 with constant scalars α and γ>0 prescribed.


## 4. Numerical Examples

A lower-dimensional dynamical network model is concerned in this part so as to demonstrate the applicability and effectiveness of the approaches proposed in the previous sections. Throughout our numerical simulations, each node of the network is supposed to be a concrete Chua’s circuit, which is frequently observed in various fields of theory and engineering applications [30].In the first stage, it will be shown that how the results derived in Section3.1 can be used to guarantee the global robust ℋ∞ synchronization of the dynamical network (2.7). Let us take a group of ten dimensionless state equations of Chua’s oscillators, for example, where one of the node system is shown as system Sa in Figure 1, a=1,2,…,10:(4.1)(v̇a1v̇a2i̇a3)=(1C1(va2-va1R-g(va1))+∑j=110GajR1C1Hvj11C2(va1-va2R+ia3+iad)-1L(va2+R0ia3)).
Here, R0 and R are linear resistors. The voltages across the capacitors C1 and C2 are denoted by va1 and va2,ia3 is the current through the inductances L, and iad is an external disturbance current that system 𝒮a subjects to. The nonlinear characteristic g(v1) represents the current through the nonlinear resistor NR, which is a piecewise-linear function expressed as(4.2)g(va1)=M1va1+12(M0-M1)[|va1+1|-|va1-1|],
and it satisfies min{M0,M1}≤g′(v1)≤min{M0,M1}.Figure 1
Coupling of any of the two identical Chua’s circuits with disturbance signal,a=1,2,…,N, and b=1,2,…,N.Suppose that each node of the dynamical network developed by (2.7) is a circuit in the form of (4.1). The possible coupling between two arbitrary Chua’s circuits, as shown in Figure 1, indicates that there is a connection from Sb to Sa but none from Sa to Sb, where the element F plays the role of unicommunication. Depending on different values of the controller gain, the resistor R1 can be adjusted. It is straightforward to reformulate system (4.1) into the Lur’e form as(4.3)ẋi=Axi+Bφ(Cxi)+∑j=110gijΓHxj+Bddi,
where(4.4)x=(v1v2i3),A=(-p(M0+1)p01-110-q-s),B=(-p(M1-M0)00),C=(100),Bd=(01C20),di=iad,
and the nonlinear function φ(Cx)=(1/2)(|x1+1|-|x1-1|) satisfies the sector condition on [0,1]. Furthermore, suppose the output equation to be as(4.5)zi=Hxi,
with parameter matrix H=(100). Choose system parameters as R=C2=1, p=1/RC1=5.5,q=1/L=7.3,s=R0/L=4,M0=-1/7,M1=2/7. In the following, R1=0.3 Ω is taken. The network topology is assumed as star-like with ten nodes; thus G has the eigenvalues as follows:(4.6)λ1=0,λ2=⋯=λ9=-1,λ10=-10.
Picking α=3, and prescribing disturbance attenuation γ=0.9, we arrive at the feasible solutions given in Appendix B by solving the LMIs (3.5), which, according to Corollary 3.8, implies that the dynamical network composed of Chua’s circuits has achieved the global robust ℋ∞ synchronization.Simulation results also confirm the effectiveness of the design. Figure2 depicts the time response of synchronization error of the nominal dynamical network without disturbance signal d(t), and it shows that the synchronization error converges to zero exponentially. Herein, initial values are taken arbitrarily.The time responses of synchronization errors of the nominal dynamical network.
(a)(b)(c)To observe theℋ∞ performance with disturbance attenuation, assume the unknown input noise disturbance di to be as(4.7)di(t)=0.5sin⁡(2t),t≥0,i=1,2,…,10.
Accordingly, the time response of the output residual error of Lur’e dynamical network with the above disturbance signals and zero initial conditions are shown in Figure 3.Figure 3
The fault-free residual responsesri with di(t)=0.5sin(2t), i=1,2,…,10.In what follows, let us consider the global robustℋ-/ℋ∞ synchronization of the dynamical network (4.4) in the presence of fault signal f. For the purpose of illustration, the process fault is supposed to be a faulty current flowing in the same direction as im3 along with the leftmost branch of each of the circuits, which will be simulated as two different types. Accordingly, it leads to(4.8)ẋi=Axi+Bφ(Cxi)+∑j=110gijΓHxj+Bddi+Bff,zi=Hxi+Df,i=1,2,…,10
with Bf=(01/C2-R0/L)T and D=1.Remainingγ=0.9 and picking the fault sensitivity β=0.6, we arrive at solution of the LMI (3.5)-(3.6) with α=3 presented in Appendix C, which on its turn ensures that the network (4.4) has achieved global robust ℋ-/ℋ∞ synchronization in the presence of possible faults and external disturbances.As for the corresponding simulation results, first let the process fault be a pulse of unit amplitude occurred from5s to 10s (and is zero otherwise). The generated residual signals ri(t), i=1,2,…,10 are depicted in Figure 4(a), from which one observes that the effect of the disturbance input di(t) on the residual error signal ri(t), i=1,2,…,10 has been greatly reduced, and the residuals have rather large amplitudes so that the synchronization process remains sensitive to fault. With the same disturbance di(t), then redo the simulation with a different:(4.9)f2(t)={0.4(t-5),10≤t<20,0,elsewhere,
and the results are plotted in Figure 4(b).The residual responsesri(t) with di(t)=0.5sin(2t) and (a) f(t)=f1(t); (b) f(t)=f2(t), i=1,2,…,10.
(a)(b)By solving the generalized eigenvalue problem corresponding to the minimization problem given in Corollary3.6, we get estimates of the maximum values of fault sensitivity as β1m=0.7961, β2m=⋯=β9m=0.8548, and β10m=0.9524., which also guarantees in terms of Corollary 3.6 that Lur’e dynamical networks achieve ℋ-/ℋ∞ synchronization with β<β0m, where β0m=mini=1,…,10{βim}=0.7961.


## 5. Conclusion and Future Work

Aiming at enhancing the reliability and robustness of synchronization, the global robustℋ-/ℋ∞ synchronization scheme has been introduced into a class of nonlinear dynamical networks in the existence of possible faults and external disturbances. The criterion on synchronization was developed in virtue of the LMI technique such that each of the node systems of the network is robustly synchronized as well as sensitive to faults according to a mixed ℋ-/ℋ∞ performance. Since both of the external disturbance and system fault are, respectively, considered, such synchronization scheme proposed here may be more practical than the synchronization in the previous literature. Moreover, the fault sensitivity H- index could be optimized via a convex optimization algorithm. In order to demonstrate the effectiveness and applicability of the derived results, a low-dimensional dynamical network with each node being a Chua’s circuit has been adopted as an example.As for future work, it will be interesting to study the synchronization of complex networks with different disturbance from various sources. Also, it is possible to extend the present results to stochastic complex networks [31–34].



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