# Non-fragile $$H_{\infty}$$ state estimation for discrete-time complex networks with randomly occurring time-varying delays and channel fadings

Non-fragile $$H_{\infty}$$ state estimation for discrete-time complex networks with randomly... Abstract In this article, the non-fragile state estimation problem is investigated for a class of discrete time-delay nonlinear complex networks with both randomly occurring gain variations (ROGVs) and channel fadings. Two sequences of random variables obeying the Bernoulli distribution are employed to describe the phenomena of randomly occurring time-varying delays and ROGVs. Moreover, the phenomenon of channel fadings occurs in a random way and the fading probability is allowed to be uncertain but within a given interval. Through stochastic analysis and Lyapunov functional approach, sufficient conditions are derived for the existence of the desired estimator that guarantees both the exponential mean-square stability and the prescribed $$H_\infty$$ performance of the estimation error dynamics. The explicit expression of such estimators is also characterized by resorting to the semidefinite programming technique. Finally, a simulation example is provided to show the usefulness and effectiveness of the proposed state estimation scheme. 1. Introduction Complex networks are made up of a lot of highly interconnected dynamical units and therefore display very complicated dynamics. Many phenomena in nature can be modelled as complex networks, for example, the brain structure as a network of neurons, the social interaction as a network of people and the Internet as a network of routers or domains. The complex connections of these networks can be represented in terms of nodes, edges and coupling strengths. As is well known, random graphs are able to describe the large-scale networks with no explicit design principles, and therefore the early study on complex networks has been the territory of graph theory following the seminal work in Erdös & Rényi (1960). Since 1990, due to the introduction of scale-free networks (Barabási & Albert, 1999) and small-world networks (Watts & Strogatz, 1998), the dynamical behaviors of complex networks have attracted an ever-increasing research interest from a variety of communities such as mathematics, physics, technology and life sciences. As a result, a great deal of dynamic analysis issues have been extensively investigated for complex networks such as stability and synchronization (see Albert & Barabási, 2002; Boccaletti et al., 2006; Xiang & Chen, 2007; Karimi & Gao, 2010; Liu et al., 2016; Wang et al., 2016; Zhang et al., 2017b, and the references therein). Because of its clear practical insight, the state estimation problem has long been a fundamental issue for scientists and engineers in the past few decades (Xie et al., 1991; Brown & Hwang, 1992; Shi et al., 1999; Mahmoud, 2004; Shen et al., 2011; Liu et al., 2016; Wen et al., 2016). So far, different kinds of methodologies have been developed for the state estimation issues, among which Kalman filter and $$H_{\infty}$$ filter are two of the most popular ones. Kalman filter is an algorithm that uses a series of measurements observed over time and produces estimates of unknown variables. Traditional Kalman filter is based on accurate mathematical model that requires the assumption that the exogenous noise is a strict Gauss process or a Gauss noise sequence. When a priori statistical information on the external noise signals is unknown, the Kalman filter cannot be applied. In order to conquer this problem, the $$H_{\infty}$$ filter is employed in which the external noise signal is assumed to be energy bounded and the main objective is to minimize the $$H_{\infty}$$ norm from the process noise to the estimation error. The state estimation problem is particularly important for complex networks simply because of the large scale of the network as well as the unaffordable cost in directly acquiring the network states. Recently, the $$H_{\infty}$$ state estimation problems for complex networks have gained much research interest, see e.g., Ding et al. (2012), Shen et al. (2011) and Liu et al. (2008). In Liu et al. (2008), a synchronization problem has been investigated for an array of coupled complex discrete-time networks with both the discrete and distributed time delays, then a state estimator is designed to estimate the network states through available output measurements. The synchronization and state estimation problems have been studied for a class of coupled discrete time-varying stochastic complex networks over a finite horizon in Shen et al. (2011). Traditionally, most available state estimator design approaches rely on the implicit assumption that the designed estimator can be accurately implemented. This assumption, however, is not always true in reality as the state estimators do have a certain degree of imprecisions when it comes to implementation. The estimator gain might be subject to inevitable fluctuations because of finite resolution measuring instruments, round-off errors in numerical computations, random failures and repairs of components, and the need to provide practicing engineers with safe-tuning margins, see Keel & Bhattacharyya (1997), Hu et al. (2012), Chang & Yang (2012), Lien & Yu (2007) and Yang & Che (2008). As such, the estimator should be designed to be insensitive or non-fragile against the gain variations. On the other hand, due to unpredictable changes of network conditions, the variations of the estimator gains may appear in a probabilistic way with certain types and intensity. In this case, the phenomenon of randomly occurring gain variations (ROGVs) should be taken into careful consideration in the course of estimator design. Note that, very recently, a non-fragile controller with ROGVs has been studied in Fang & Park (2013) to address the synchronization problem of neural networks, and a non-fragile $$H_\infty$$ controller has been dealt with in Li et al. (2015) for a class of discrete-time systems subject to ROGVs and infinite-distributed delays. On another active research front, due to the ever-increasing popularity of communication networks (Sheng et al., 2017; Zou et al., 2017), network-induced phenomena (e.g., packet dropouts Sahebsara et al. 2007; Wang et al. 2013; Ding et al. 2017b; Luo et al. 2017, communication delays Fang & Park 2013; Luo et al. 2017 and signal quantization Liu et al. 2016; Ding et al. 2017a) have been well studied for state estimation and control problems of networked systems. However, the issue of channel fadings has received relatively less attention in spite of its practical significance in wireless mobile communications. Generally speaking, the primary cause for channel fadings is the multipath propagation and the shadowing from obstacles, when the electromagnetic waves do not directly reach the sensor due to obstacles that block the line of sight path. In practice, it is known that wireless channels are sensitive to fading effects Elia (2005), which constitute as one of the most dominant features in wireless communication links. If not dealt with adequately, the phenomenon of channel fadings would inevitably deteriorate the performance of the state estimators in case of fading measurements. Up to now, some pioneering works have been reported in the literature concerning networked control systems with fading channels (see e.g., Elia, 2005; Mostofi & Murray, 2009; Garone et al., 2012; Quevedo et al., 2012; Xiao et al., 2012; Zhang et al., 2014, and the references therein). Nevertheless, the corresponding non-fragile $$H_{\infty}$$ state estimation problem for complex networks had gain very little research attention, not to mention the case when ROGVs, channel fadings and random delays are simultaneously present. Summarizing the discussion made so far, (1) note that the corresponding researches for the discrete-time complex networks been gaining increasing research attention because complex networks could be potentially applied in many real-world systems; (2) the phenomenon of ROGVs is often unavoidable and should be taken into account when designing estimators and (3) in wireless communications, channel fadings often happen in a random fashion, and the original fading model needs further improvement. Motivated by the above discussion, a seemingly natural idea is to examine how to design the state estimators for discrete time-delay nonlinear complex networks subject to ROGVs in the presence of measurement transmission over a fading channel. To the best of the authors’ knowledge, such a research problem has not been fully investigated yet and the main purpose of this paper is therefore to shorten such a gap by addressing the non-fragile $$H_{\infty}$$ state estimation problem for discrete time-delay nonlinear complex networks with ROGVs and channel fadings. The main contributions of this paper can be highlighted as follows: (1) A comprehensive complex network model is established that caters for randomly occurring time-varying delays and channel fadings, and a unified framework is put forward for the estimator design ensuring the error dynamics of the state estimation to be exponentially mean-square stable with guaranteed $$H_{\infty}$$ performance constraint. (2) A novel model is proposed to account for channel fadings, where the coefficient of channel fadings has the probability density function on a given interval. The rest of this article is organized as follows: In Section 2, the model for the problem under consideration is presented and some assumptions are made on ROGVs and channel coefficients. Sufficient conditions for the exponentially stability and $$H_\infty$$-performance index of the error dynamics of the state estimation are obtained in Section 3. An illustrative example is provided in Section 4 to demonstrate the effectiveness of the main results, and Section 5 concludes the article with some discussion on future research directions. Notation: The notation used in the article is fairly standard. The superscript ‘$$T$$’ stands for matrix transposition, $${\mathbb{R}}^n$$ denotes the $$n$$-dimensional Euclidean space and $${\mathbb{R}}^{m\times n}$$ is the set of all real matrices of dimension $$m\times n$$. $$I$$ and $$0$$ represent the identity matrix and zero matrix, respectively. The notation $$P>0$$ means that $$P$$ is real symmetric and positive definite. $$l_2([0,+\infty),{\mathbb{R}}^n)$$ is the space of square summable sequences. The notation $$\|A\|$$ refers to the norm of a matrix $$A$$ defined by $$\|A\| =\sqrt{{\rm trace}(A^TA)}$$. In symmetric block matrices or complex matrix expressions, we use an asterisk ($$\ast$$) to represent a term that is induced by symmetry, and diag$$\{\ldots \}$$ stands for a block-diagonal matrix. The symbol $$\otimes$$ denotes the Kronecker product. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation and preliminaries The system under consideration is shown in Fig. 1, in which the local measurement is transmitted to the remote estimator via fading channels. We will discuss each part one by one in the sequential section. Fig. 1. View largeDownload slide Structure of the estimation system. Fig. 1. View largeDownload slide Structure of the estimation system. Consider the following discrete-time complex network with randomly occurring time-varying delays consisting of $$N$$ coupled nodes of the form: \begin{eqnarray}\label{seq:1} \left\{ \begin{aligned} x_i(k+1)=&\;f(x_i(k))+\sum_{m=1}^{h}\beta_m(k)g(x_i(k-\tau_m(k)))+\sum_{j=1}^{N}w_{ij}{\it{\Gamma}} x_j(k)+B_iv(k)+h_i(x_i(k))\omega(k)\\ z_i(k)=&\;E_ix_i(k)\\ x_i(s) =&\phi_i(s), \forall s \in {\mathcal{S}}\triangleq\{-\textrm{max}\{\overline{d}_1,\cdots,\overline{d}_m\},\cdots,-1,0\}, i=1,2,\cdots,N \end{aligned}\right. \end{eqnarray} (1) where $$x_i(k)\in {\mathbb{R}}^n$$ and $$z_i(k)\in {\mathbb{R}}^r$$ are, respectively, the state vector and the output of the $$i$$-th node. $$f(\cdot)$$ and $$g(\cdot)$$ are nonlinear vector valued functions and $$h(\cdot)$$ is the noise intensity function, satisfying certain conditions to be given later. $$v(k)$$ represents the exogenous disturbance input belonging to $$l_2([0,\infty), {\mathbb{R}})$$ and $$\omega(k)$$ is an one-dimensional zero-mean Gaussian white noise sequence on a probability space $$({\it{\Omega}},{\mathcal{F}},\{{\mathcal{F}}_t\}_{t\geq0}, \textrm{Prob})$$ with $${\mathbb{E}}\{\omega^2(k)\} = 1$$. $$B_i$$ and $$E_i$$ are known constant matrices with appropriate dimensions. $${\it{\Gamma}}=\textrm{diag}\{\gamma_1,\gamma_2,\cdots,\gamma_n\}\geq0$$ is an inner-coupling matrix linking the $$j$$th state variable if $$\gamma_j\neq0$$, and $$W=(w_{ij})\in {\mathbb{R}}^{N\times N}$$ is the coupled configuration matrix of the network with $$w_{ij}\geq0 (i\neq j)$$ but not all zero. As usual, the coupling configuration matrix $$W =(w_{ij})$$ is symmetric (i.e., $$W = W^T$$). The positive integers $$\tau_m(k) (m=1,2,\cdots, h)$$ denote the time-varying delays satisfying \begin{eqnarray}\label{seq:2} 0<\underline{d}_m\leq\tau_m(k)\leq \overline{d}_m, \end{eqnarray} (2) where $$\underline{d}_m$$ and $$\overline{d}_m$$ are known constant positive integers, representing the lower and upper bounds on the communication delay, respectively. To characterize the phenomena of stochastic interval time-varying delays, we introduce the stochastic variables $$\beta_m(k)\in {\mathbb{R}}$$ that are mutually independent Bernoulli-distributed white sequences. A natural assumption on $$\beta_m(k)$$ can be made as follows: \begin{eqnarray*} \left. \begin{aligned} &\textrm{Prob}\{\beta_m(k)=1\}={\mathbb{E}}\{\beta_m(k)\}=\bar\beta_m,\\ &\textrm{Prob}\{\beta_m(k)=0\}=1-\bar\beta_m. \end{aligned} \right. \end{eqnarray*} The nonlinear vector-valued functions $$f(\cdot), g(\cdot)$$ and the noise intensity function $$h(\cdot)$$ are continuous and satisfy \begin{eqnarray}\label{seq:3} &[f(x)-f(y)-U_1(x-y)]^T[f(x)-f(y)-U_2(x-y)]\leq0,\\ \end{eqnarray} (3) \begin{eqnarray} &[g(x)-g(y)-V_1(x-y)]^T[g(x)-g(y)-V_2(x-y)]\leq0,\label{seq:4}\\ \end{eqnarray} (4) \begin{eqnarray} &\|h_i(x)-h_j(y)\|^2\leq\|L(x-y)\|^2, \,\,\, \forall x,y\in{\mathbb{R}}^n, \label{seq:5} \end{eqnarray} (5) where $$U_1, U_2, V_1, V_2$$ and $$L$$ are known constant matrices (see Chu, 2001; Liang et al., 2014). For notation simplicity, we let \begin{eqnarray*} x(k)&=&\big[x^T_1(k) x^T_2(k) \cdots x^T_N(k)\big]^T,\\ B&=&\big[B^T_1 B^T_2 \cdots B^T_N\big]^T,\\ F(x(k))&=&\big[f^T(x_1(k)) f^T(x_2(k)) \cdots f^T(x_N(k))\big]^T,\\ G(x(k))&=&\big[g^T(x_1(k)) g^T(x_2(k)) \cdots g^T(x_N(k))\big]^T,\\ H(x(k))&=&\big[h_1^T(x_1(k)) h_2^T(x_2(k)) \cdots h_N^T(x_N(k))\big]^T. \end{eqnarray*} By utilizing the Kronecker product, we can rewrite the complex network (1) as the following compact form: \begin{eqnarray}\label{seq:6} x(k+1)=F(x(k))+\sum_{m=1}^{h}\beta_m(k)G(x(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})x(k)+Bv(k)+H(x(k))\omega(k). \end{eqnarray} (6) The fading measurement output $$y_i(k) \in {\mathbb{R}}^m$$ from the $$i$$-th node is of the form, \begin{eqnarray}\label{seq:7} y_i(k) = \lambda_i(k)C_ix_i(k), i= 1, 2,\cdots,N, \end{eqnarray} (7) where $$C_i\in {\mathbb{R}}^{m\times n}$$ is a known constant matrix. The random variable $$\lambda_i(k)\in {\mathbb{R}}$$, which accounts for the phenomena of channel fadings, has the probability density function on the interval $$[\underline{\lambda}_i, \overline{\lambda}_i]$$ with mathematical expectation $$\mu_i$$ and variance $$\sigma^2_i$$, where $$0<\underline{\lambda}_i <\mu_i-\sigma_i<\mu_i+\sigma_i<\overline{\lambda}_i<1$$. In order to estimate the states of the complex network (6), we take the phenomena of ROGVs into account and construct the following state estimator: \begin{eqnarray}\label{seq:8} \left\{ \begin{aligned} \hat{x}(k+1)=&F(\hat{x}(k))+\sum_{m=1}^{h}\beta_m(k)G(\hat{x}(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})\hat{x}(k)\\ &+(K+\alpha(k){\it{\Delta}} K(k))[y(k)-C\hat{x}(k)]\\ \hat{z}(k)=&E\hat{x}(k) \end{aligned}\right. \end{eqnarray} (8) with \begin{eqnarray*} \hat{x}(k)&=&\big[\hat{x}^T_1(k) \hat{x}^T_2(k) \cdots \hat{x}^T_N(k)\big]^T, \hat{z}(k)=\big[\hat{z}^T_1(k) \hat{z}^T_2(k) \cdots \hat{z}^T_N(k)\big]^T,\\ y(k)&=&\big[y^T_1(k) y^T_2(k) \cdots y^T_N(k)\big]^T, C=\textrm{diag}\{C_1,C_2,\cdots,C_N\},\\ F(\hat x(k))&=&\big[f^T(\hat x_1(k)) f^T(\hat x_2(k)) \cdots f^T(\hat x_N(k))\big]^T,\\ G(\hat x(k))&=&\big[g^T(\hat x_1(k)) g^T(\hat x_2(k)) \cdots g^T(\hat x_N(k))\big]^T,\\ E&=&\textrm{diag}\{E_1,E_2,\cdots,E_N\}, K=\textrm{diag}\{K_1,K_2,\cdots,K_N\},\\ {\it{\Delta}} K(k)&=&\textrm{diag}\{{\it{\Delta}} K_1(k),{\it{\Delta}} K_2(k),\cdots,{\it{\Delta}} K_N(k)\},\\ \alpha(k)&=&\textrm{diag}\{\alpha_1(k)I,\alpha_2(k)I,\cdots,\alpha_N(k)I\}, \end{eqnarray*} where $$\hat x_i(k)\in {\mathbb{R}}^n$$ is the estimate of the state $$x_i(k)$$, $$\hat z_i(k)\in {\mathbb{R}}^r$$ is the estimate of the output $$z_i (k)$$, and $$K_i\in {\mathbb{R}}^{n\times m}$$ is the estimator gain matrix to be determined. $$\alpha_i(k)\in {\mathbb{R}} (i=1,2,\cdots,N)$$ are mutually independent Bernoulli-distributed white sequences. A natural assumption on $$\alpha_i(k)$$ can be made as follows: \begin{eqnarray*} \left. \begin{aligned} &\textrm{Prob}\{\alpha_i(k)=1\}={\mathbb{E}}\{\alpha_i(k)\}=\bar\alpha_i,\\ &\textrm{Prob}\{\alpha_i(k)=0\}=1-\bar\alpha_i. \end{aligned} \right. \end{eqnarray*} Moreover, for each $$k$$, $$\alpha_i(k)$$ is independent of $$\beta_m(k)$$ and $$\lambda_{i}(k)$$. The uncertain perturbation matrix $${\it{\Delta}} K_i(k)\in {\mathbb{R}}^{n\times m}$$ is defined as follows: \begin{eqnarray}\label{seq:9} {\it{\Delta}} K_{i}(k)=J_{i}{\it{\Delta}}(k)M, \quad i\in \{1,2,\cdots,N\}, \end{eqnarray} (9) where $$J_i$$ and $$M$$ are known constant matrices with appropriate dimensions, and $${\it{\Delta}}(k)$$ is unknown matrix function satisfying \begin{eqnarray}\label{seq:10} {\it{\Delta}}^T(k){\it{\Delta}}(k)\leq I. \end{eqnarray} (10) Remark 1 The state estimation problem has been investigated in Ding et al. (2012) for an array of discrete time-delay nonlinear complex networks with randomly occurring sensor saturations and randomly varying sensor delays. In the real networked world, the parameter gain variations, which cannot be ignored, may be subject to random changes in environmental circumstances, for instance, network-induced random failures and repairs of components, sudden environmental disturbances, etc. It is worth pointing out that the models proposed in (7) and (8) provide a novel framework to account for the phenomenon of both ROGVs and channel fadings, which have not been taken into account in Ding et al. (2012). The stochastic variable $$\alpha_i(k)$$ is used to describe the random nature of estimator gain variations. In Srikanth et al. (2000), it has been proposed that analysis of a probabilistic system is to determine the probability density function over an interval. The stochastic variable $$\lambda_i(k)$$, whose probability density function is on a predetermined interval, characterizes the phenomenon of the probabilistic channel fadings. Such a description is more suitable for reflecting parameter variations of the estimator in a random fashion, particularly in the transmission of wireless communication over fading channels. In addition, the application on interval statistical characteristics have been received enough attention, see Srikanth et al. (2000) and Wei et al. (2013). Remark 2 Recently, the control and filtering issues with channel fadings have been investigated for traditional networked systems where the statistical characteristics are certain, see Zhang et al. (2014); Zhang et al. (2016) and the reference therein. In practice, the statistics of channel fadings could be obtained via offline statistical tests. From the viewpoint of statistics, any identification is not absolutely accurate, and should drop into an interval with given degree of confidence. Therefore, the description on channel fadings used in this article is more reasonable and the obtained results are more general and reliable. In addition, ROGVs and channel fadings with time-varying statistical characteristics on channel coefficients have been discussed in Zhang et al. (2016) for T-S fuzzy systems. In comparison with the research in Zhang et al. (2016), the main differences in this article are shown in the following two aspects: (1) different from the time-varying statistical characteristics, a predetermined interval is utilized to describe the channel fadings, which results into some interval-dependent analysis and design conditions and (2) the design is nontrivial and is with challenges coming from complex dynamical behavior, heterogeneous measurements, ROGVs as well as channel fadings with interval statistics. Let $$e(k)=\big[e^T_1(k) e^T_2(k) \cdots e^T_N(k)\big]^T$$ with $$e_i(k)=x_i(k)-\hat{x}_i(k)$$ being the state estimator error. Denote \begin{eqnarray*} \hat{F}(e(k))&=&\big[\hat{f}^T(e_1(k)) \hat{f}^T(e_2(k)) \cdots \hat{f}^T(e_N(k))\big]^T\\ &:=&F(x(k))-F(\hat{x}(k)),\\ \hat{G}(e(k))&=&\big[\hat{g}^T(e_1(k)) \hat{g}^T(e_2(k)) \cdots \hat{g}^T(e_N(k))\big]^T\\ &:=&G(x(k))-G(\hat{x}(k)),\\ \lambda(k)&:=&\textrm{diag}\{\lambda_1(k)I,\lambda_2(k)I,\cdots,\lambda_N(k)I\}. \end{eqnarray*} Then, from (6) and (8), we obtain the following system governing the estimate error dynamics: \begin{eqnarray}\label{seq:11} \left\{ \begin{aligned} x(k+1)=&F(x(k))+\sum_{m=1}^{h}(\hat\beta_m(k)+\bar\beta_m)G(x(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})x(k)+Bv(k)+H(x(k))\omega(k)\\ e(k+1)=&\hat{F}(e(k))+\sum_{m=1}^{h}(\hat\beta_m(k)+\bar\beta_m)\hat{G}(e(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})e(k)+Bv(k)+H(x(k))\omega(k)\\ &-(K+(\hat\alpha(k)+\bar\alpha){\it{\Delta}} K(k))(\hat\lambda(k)+\mu-I_N)C x(k)-(K+(\hat\alpha(k)+\bar\alpha){\it{\Delta}} K(k))Ce(k) \end{aligned} \right.\nonumber\\ \end{eqnarray} (11) where \begin{eqnarray*} \hat\beta_m(k)&=&\beta_m(k)-\bar\beta_m, \hat\alpha_i(k)=\alpha_i(k)-\bar\alpha_i, \hat\lambda_i(k)=\lambda_i(k)-\mu_i,\\ \hat\alpha(k)&=&\textrm{diag}\{\hat\alpha_1(k)I,\hat\alpha_2(k)I,\cdots,\hat\alpha_N(k)I\}, \bar\alpha=\textrm{diag}\{\bar\alpha_1I,\bar\alpha_2I,\cdots,\bar\alpha_NI\},\\ \hat\lambda(k)&=&\textrm{diag}\{\hat\lambda_1(k)I,\hat\lambda_2(k)I,\cdots,\hat\lambda_N(k)I\}, \mu=\textrm{diag}\{\mu_1I,\mu_2I,\cdots,\mu_NI\}. \end{eqnarray*} It is clear that \begin{eqnarray*} &&{\mathbb{E}}\{\hat\beta_m(k)\}=0,\quad {\mathbb{E}}\{\hat\beta_m^2(k)\}\triangleq\hat\beta_m^\ast=\bar\beta_m(1-\bar\beta_m),\\ &&{\mathbb{E}}\{\hat\alpha_i(k)\}=0,\quad {\mathbb{E}}\{\hat\alpha_i^2(k)\}\triangleq\hat\alpha_i^\ast=\bar\alpha_i(1-\bar\alpha_i),\\ &&{\mathbb{E}}\{\hat\lambda_i(k)\}=0,\quad {\mathbb{E}}\{\hat\lambda_i^2(k)\}=\sigma^2_i. \end{eqnarray*} Next, by setting $$\eta(k) = [x^T(k) e^T(k)]^T$$ and defining the output error $$\tilde{z}(k)=z(k)-\hat z(k)$$, we have the following augmented system: \begin{eqnarray}\label{seq:12} \left\{ \begin{aligned} \eta(k+1)=&{\mathcal{W}}(k)\eta(k)+{\mathcal{F}}(k)+\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))+\sum_{m=1}^{h}\hat\beta_m(k){\mathcal{G}}(k-\tau_m(k))\\ &+[I_2\otimes\hat\lambda(k)]{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha(k)\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes(\mu\hat\alpha(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\\ &+[I_2\otimes(\bar\alpha\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes\hat\alpha(k)]{\mathcal{K}}_3(k)\eta(k)+{\mathcal{B}}v(k)+{\mathcal{H}}(k)\omega(k)\\ \tilde{z}(k)=&{\mathcal{E}}\eta(k)\\ \eta(s)=&\big[\phi^T_1(s) \phi^T_2(s) \cdots \phi^T_N(s) \phi^T_1(s) \phi^T_2(s) \cdots \phi^T_N(s)\big]^T, s \in\{-\textrm{max}\{\overline{d}_1,\cdots,\overline{d}_m\},\cdots,0\} \end{aligned} \right.\nonumber\\ \end{eqnarray} (12) where \begin{eqnarray*} {\mathcal{F}}(k)&=&[F^T(x(k)) \hat{F}^T(e(k))]^T,\\ {\mathcal{G}}(k)&=&[G^T(x(k)) \hat{G}^T(e(k))]^T,\\ {\mathcal{H}}(k)&=&[H^T(x(k)) H^T(x(k))]^T,\\ {\mathcal{B}}&=&[B^T B^T]^T, {\mathcal{E}}=[0 E], {\mathcal{S}}=[I_N 0],\\ {\mathcal{W}}(k)&=&{\mathcal{W}}+{\it{\Delta}}{\mathcal{W}}(k),\\ {\mathcal{W}}&=&\left[ \begin{array}{cc} (W\otimes{\it{\Gamma}}) & 0 \\ -(\mu-I_N)KC & (W\otimes{\it{\Gamma}})-KC\\ \end{array} \right],\\ {\it{\Delta}}{\mathcal{W}}(k)&=&\left[ \begin{array}{cc} 0 & 0 \\ -\bar\alpha(\mu-I_N){\it{\Delta}} K(k) C & -\bar\alpha{\it{\Delta}} K(k)C\\ \end{array} \right],\\ {\mathcal{K}}_1&=&\left[ \begin{array}{cc} 0 \\ -KC \\ \end{array} \right], {\mathcal{K}}_2(k)=\left[ \begin{array}{cc} 0 \\ -{\it{\Delta}} K(k)C & \\ \end{array} \right], \\ {\mathcal{K}}_3(k)&=&\left[ \begin{array}{cc} 0 & 0 \\ {\it{\Delta}} K(k)C & -{\it{\Delta}} K(k)C\\ \end{array} \right].\\ \end{eqnarray*} Definition 1 The augmented system (12) with $$v(t)=0$$ is said to be exponentially mean-square stable if there exist two scalars $$\nu>0$$ and $$0<\kappa<1$$ such that $$\label{seq:13} {\mathbb{E}}\{\|\eta(k)\|^{2}\} \leq \nu \kappa^{k}\sup_{s\in{\mathcal{S}}}{\mathbb{E}}\{\|\phi (s)\|^2\},\quad \forall k\geq 0.$$ (13) We are now ready to describe the non-fragile $$H_{\infty}$$ state estimation problem for the complex network (1). Specifically, we aim to design the state estimator (8), i.e., look for parameter matrix $$K_i$$ such that the following two requirements are simultaneously satisfied: (i) the augmented system (12) with $$v(t)=0$$ is exponentially mean-square stable. (ii) under the zero-initial condition, the output error $$\tilde z(t)$$ satisfies the $$H_{\infty}$$ performance constraint: $$\label{seq:14} \frac{1}{N}\sum_{k=0}^{\infty}{\mathbb{E}}\{\|\tilde z(k)\|^{2}\} \leq\gamma^2\sum_{k=0}^{\infty} \| v (k)\|^2,$$ (14) for all nonzero $$v(t)$$, where $$\gamma>0$$ is a given disturbance attenuation level. Remark 3 The $$H_{\infty}$$ performance index $$\gamma>0$$ is used to quantify the attenuation level of the estimation error dynamics against exogenous disturbances. Based on (12) and (14), it is easy to know that the value of $${\mathbb{E}}\{\|\tilde z(k)\|^{2}\}$$ would become larger if the number of the nodes increases. In theory, the $$H_\infty$$ disturbance attenuation level for the overall network should account for the average disturbance rejection performance which is insensitive to the quantity changes of the nodes in the estimator design. In order to achieve this purpose, the scalar of $$1/N$$ is employed to accommodate the average index over the complex network so that the scalar reflects the practical significance of the $$H_\infty$$ disturbance attenuation level. 3. Main results In this section, let us deal with the state estimation problem in the mean square for the complex network (6). First, we introduce several lemmas to be used in the sequel. Lemma 1 (Schur complement) Given constant matrices $${\it{\Sigma}}_1$$, $${\it{\Sigma}}_2$$, $${\it{\Sigma}}_3$$, where $${\it{\Sigma}}_1={\it{\Sigma}}^T_1$$ and $$0<{\it{\Sigma}}_2={\it{\Sigma}}^T_2$$. Then $${\it{\Sigma}}_1+{\it{\Sigma}}^T_3{\it{\Sigma}}^{-1}_2{\it{\Sigma}}_3<0$$ if and only if $$\label{seq:15} \left[ \begin{array}{cc} {\it{\Sigma}}_1 & {\it{\Sigma}}^T_3 \\ {\it{\Sigma}}_3 & -{\it{\Sigma}}_2 \\ \end{array} \right]<0\quad \textrm{or} \quad \left[ \begin{array}{cc} -{\it{\Sigma}}_2 & {\it{\Sigma}}_3 \\ {\it{\Sigma}}^T_3 & {\it{\Sigma}}_1 \\ \end{array} \right]<0.$$ (15) Lemma 2 (S-procedure) Let $$L=L^T$$, $$H, M$$ and $$N$$ be real matrices of appropriate dimensions with $$M$$ satisfying $$MM^T\leq I$$. Then, $$L+HMN+N^TM^TH^T<0$$, if and only if there exists a positive scalar $$\varepsilon>0$$ such that $$L+\varepsilon^{-1}HH^T+\varepsilon N^TN<0$$ or equivalently $$\label{seq:16} \left[ \begin{array}{ccc} L & H & \varepsilon N^T \\ H^T & -\varepsilon I & 0 \\ \varepsilon N& 0 & -\varepsilon I \\ \end{array} \right]<0.$$ (16) Now, we have the following analysis result that serves as a theoretical basis for the subsequent design problem. Theorem 1 Let the estimator parameters $$K_i (i=1,2,\cdots,N)$$ and a prescribed $$H_{\infty}$$ performance $$\gamma > 0$$ be given. Then, the zero solution of the augmented system (12) (with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) = 0$$) is exponentially mean-square stable if there exist positive definite matrices $$P_1>0, P_2>0, Q_{1m}>0$$ and $$Q_{2m}>0 (m=1,2,\cdots, h)$$, and positive scalars $$\delta_1, \delta_2, \delta_3$$ such that the following matrix inequalities hold: \begin{eqnarray}\label{seq:th1a} \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &{\it{\Sigma}}^T_1 \\ \ast & -{\mathcal{P}}_1\\ \end{array} \right]<0, \end{eqnarray} (17) where \begin{eqnarray*} P&=&\textrm{diag}\{I_{N}\otimes P_1,I_{N}\otimes P_2\}, {\mathcal{P}}_1=I_2\otimes P, \\ Q_m&=&\textrm{diag}\{I_{N}\otimes Q_{1m},I_{N}\otimes Q_{2m}\},\\ \check{U}_1&=&I_{2N}\otimes[(U^T_1U_2+U^T_2U_1)/2], \\ \check{U}_2&=&I_{2N}\otimes[(U^T_1+U^T_2)/2],\\ \check{V}_1&=&I_{2N}\otimes[(V^T_1V_2+V^T_2V_1)/2], \\ \check{V}_2&=&I_{2N}\otimes[(V^T_1+V^T_2)/2], \check{L}=I_{N}\otimes L,\\ {\it{\Phi}}_1&=&\left[ \begin{array}{ccccc} {\it{\Phi}}_{11} & {\it{\Phi}}_{12}& {\it{\Phi}}_{13}& 0& 0\\ \ast &{\it{\Phi}}_{22} & 0& 0& 0\\ \ast & \ast & {\it{\Phi}}_{33} & 0& 0\\ \ast & \ast & \ast & {\it{\Phi}}_{44}& 0\\ \ast & \ast & \ast & 0& {\it{\Phi}}_{55}\\ \end{array} \right],\\ {\it{\Phi}}_{11}&=&-P-\delta_1\check U_1-\delta_2\check V_1+\delta_3{\mathcal{S}}^T\check L^T\check L{\mathcal{S}}+\frac{1}{N}{\mathcal{E}}^T{\mathcal{E}},\\ {\it{\Phi}}_{12}&=&\delta_1\check U_2, {\it{\Phi}}_{13}=[\delta_2\check V_2 \underbrace{0 \cdots 0}_h], {\it{\Phi}}_{22}=-\delta_1I_{2N},\\ {\it{\Phi}}_{33}&=&\textrm{diag}\{\sum^h_{m=1}(1+\overline{d}_m-\underline{d}_m)Q_m-\delta_2I_{2N}, \hat\beta^{\ast}_1P- Q_1, \cdots, \hat\beta^{\ast}_hP- Q_h\},\\ {\it{\Phi}}_{44}&=&P-\frac{1}{2}\delta_3I_{2N}, {\it{\Phi}}_{55}=-\gamma^2I,\\ {\it{\Sigma}}_1&=&\left[ \begin{array}{ccccc} P{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0& P{\mathcal{B}}\\ {\it{\Lambda}}_1 P{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ \end{array} \right],\\ {\it{\Sigma}}_{13}&=&[0 \bar\beta_1P \bar\beta_2P \cdots \bar\beta_hP],\\ {\it{\Lambda}}_1&=&\textrm{diag}\{\sigma_1I,\cdots,\sigma_NI,\sigma_1I,\cdots,\sigma_NI\}. \end{eqnarray*} Proof. First, in order to show that the augmented system (12) (with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) = 0$$) is exponentially stable, we choose the following Lyapunov functional: $$\label{seq:18} V(k)=\sum_{i=1}^3V_i(k),$$ (18) where \begin{eqnarray*} V_1(k)&=& \eta^T(k)P\eta(k),\\ V_2(k)&=& \sum_{m=1}^h\sum_{i=k-\tau_m(k)}^{k-1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i),\\ V_3(k)&=& \sum_{m=1}^h\sum_{n=k-\overline{d}_m+1}^{k-\underline{d}_m}\sum_{i=n}^{k-1}{\mathcal{G}}^T(i)Q_m {\mathcal{G}}(i).\\ \end{eqnarray*} Calculating the difference of $$V_1(k)$$ along the trajectory of (12) we have \begin{eqnarray*}\label{seq:19a} {\it{\Delta}} V_1(k) &=&V_1(k+1)-V_1(k)\\ &=&\big({\mathcal{W}}(k)\eta(k)+{\mathcal{F}}(k)+\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))+\sum_{m=1}^{h}\hat\beta_m(k){\mathcal{G}}(k-\tau_m(k))\\ &&+[I_2\otimes\hat\lambda(k)]{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha(k)\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes(\mu\hat\alpha(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\\ &&+[I_2\otimes(\bar\alpha\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes\hat\alpha(k)]{\mathcal{K}}_3(k)\eta(k)+{\mathcal{B}}v(k)+{\mathcal{H}}(k)\omega(k)\big)^TP\\ &&\times \big({\mathcal{W}}(k)\eta(k)+{\mathcal{F}}(k)+\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))+\sum_{m=1}^{h}\hat\beta_m(k){\mathcal{G}}(k-\tau_m(k))\\ &&+[I_2\otimes\hat\lambda(k)]{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha(k)\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes(\mu\hat\alpha(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\\ &&+[I_2\otimes(\bar\alpha\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes\hat\alpha(k)]{\mathcal{K}}_3(k)\eta(k)+{\mathcal{B}}v(k)+{\mathcal{H}}(k)\omega(k)\big)-\eta^T(k)P\eta(k) \end{eqnarray*} Then, takeing the mathematical expectation, we obtain \begin{eqnarray}\label{seq:19} &&{\mathbb{E}}\{{\it{\Delta}} V_1(k)\}\notag\\ &&\quad={\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}(k)\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\left(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\right)^TP\left(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\right)\notag\\ &&\qquad+2\left(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\right)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k)+2[I_2\otimes\bar\alpha\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes(\hat\alpha^\ast\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k) +[I_2\otimes(\mu^T\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+2[I_2\otimes(\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_3(k)\eta(k) +[I_2\otimes(\bar\alpha^T\bar\alpha\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes\hat\alpha^\ast]\eta^T(k){\mathcal{K}}^T_3(k)P{\mathcal{K}}_3(k)\eta(k) +v(k)^T{\mathcal{B}}^TP{\mathcal{B}}v(k)+{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\Big\}, \end{eqnarray} (19) where $$\sigma^2=\textrm{diag}\{\sigma^2_1I,\sigma^2_2I,\cdots,\sigma^2_NI\}.$$ Also, it can be obtained that \begin{eqnarray*} \begin{array}{l} \begin{split} {\mathbb{E}}\Big\{{\it{\Delta}} V_2\Big\} =& {\mathbb{E}}\Big\{V_2(k+1)-V_2(k)\Big\}\\ =& {\mathbb{E}}\left\{\sum_{m=1}^h\left[\sum_{i=k+1-\tau_m(k+1)}^{k}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)-\sum_{i=k-\tau_m(k)}^{k-1} {\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\right]\right\}\\ =& {\mathbb{E}}\Bigg\{\sum_{m=1}^h\Bigg[{\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k)+\sum_{i=k+1-\tau_m(k+1)}^{k-1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\\ &-\sum_{i=k+1-\tau_m(k)}^{k-1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)-{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Bigg]\Bigg\}\\ \leq& {\mathbb{E}}\Bigg\{\sum^h_{m=1}\Bigg[{\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k)-{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\\ &+\sum^{k-\underline{d}_m}_{i=k-\overline{d}_m+1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\Bigg]\Bigg\}, \end{split}\\ \end{array} \end{eqnarray*} \begin{eqnarray}\label{seq:20} \begin{array}{l} \begin{split} {\mathbb{E}}\Big\{{\it{\Delta}} V_3\Big\} =& {\mathbb{E}}\Big\{V_3(k+1)-V_3(k)\Big\}\\ =& {\mathbb{E}}\Bigg\{\sum_{m=1}^h\Bigg[\sum_{n=k-\overline{d}_m+2}^{k-\underline{d}_m+1}\sum_{i=n}^{k}{\mathcal{G}}^T(i)Q_m {\mathcal{G}}(i)-\sum_{n=k-\overline{d}_m+1}^{k-\underline{d}_m}\sum_{i=n}^{k-1}{\mathcal{G}}^T(i))Q_m {\mathcal{G}}(i)\Bigg]\Bigg\}\\ =& {\mathbb{E}}\Bigg\{\sum_{m=1}^h\Bigg[\sum_{n=k-\overline{d}_m+2}^{k-\underline{d}_m+1}\sum_{i=n}^{k}{\mathcal{G}}^T(i)Q_m {\mathcal{G}}(i)-\sum_{n=k-\overline{d}_m+1}^{k-\underline{d}_m}\sum_{i=n}^{k}{\mathcal{G}}^T(i))Q_m {\mathcal{G}}(i)\\ &+(\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k)\Bigg]\Bigg\}\\ \leq& {\mathbb{E}}\Bigg\{\sum^h_{m=1}\Big[(\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -\sum^{k-\underline{d}_m}_{i=k-\overline{d}_m+1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\Bigg]\Bigg\}.\\ \end{split}\\ \end{array} \end{eqnarray} (20) When $${\it{\Delta}} K_{i}(k)=0$$ and $$v(k) =0$$, we can obtain from (19) and (20) that \begin{eqnarray}\label{seq:21} {\mathbb{E}}\{{\it{\Delta}} V(k)\} &\leq&{\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)\notag\\ &&+2\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&+[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k) +{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\notag\\ &&+\sum^h_{m=1}\Bigg[(1+\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Bigg]\Bigg\}\notag\\ &\leq& {\mathbb{E}}\Bigg\{\xi^T(k)\bar{\it{\Phi}}\xi(k)+\xi^T(k)\bar{\it{\Sigma}}^T_1{\mathcal{P}}_1\bar{\it{\Sigma}}_1\xi(k)\Bigg\}, \end{eqnarray} (21) where \begin{eqnarray*} \xi(k)&=&[\eta(k)^T {\mathcal{F}}^T(k) {\mathcal{G}}^T_m(k) {\mathcal{H}}^T(k)]^T,\\ {\mathcal{G}}_m(k)&=&[{\mathcal{G}}^T(k) {\mathcal{G}}^T(k-\tau_1(k)) {\mathcal{G}}^T(k-\tau_2(k)) \cdots {\mathcal{G}}^T(k-\tau_h(k))]^T,\\ \bar{\it{\Phi}}&=&\left[ \begin{array}{cccc} -P & 0 & 0 & 0\\ \ast & 0 & 0 & 0\\ \ast & \ast & \bar{\it{\Phi}}_{33}& 0 \\ \ast & \ast & \ast & P\\ \end{array} \right],\bar{\it{\Sigma}}_1=\left[ \begin{array}{ccccc} P{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0\\ {\it{\Lambda}}_1 P{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0\\ \end{array} \right]\!,\\\\ \bar {\it{\Phi}}_{33}&=&\textrm{diag}\left\{ \sum^h_{m=1}(1+\overline{d}_m-\underline{d}_m)Q_m, \hat\beta^{\ast}_1P- Q_1, \cdots, \hat\beta^{\ast}_hP- Q_h\right\}. \end{eqnarray*} Subsequently, we rewrite (3) and (4) as $$\label{seq:22} \begin{array}{rcl} &&\left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}} (k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check U_1 & - \check U2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}}(k) \\ \end{array} \right]\leq0,\\ &&\left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check V1 & - \check V2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]\leq0.\\ \end{array}$$ (22) Similarly, (5) can be rewritten as $$\label{seq:23} \begin{array}{rcl} &&\left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} -{\mathcal{S}}^T\check L^T\check L{\mathcal{S}} & 0 \\ \ast & \frac{1}{2}I_{2N} \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]\leq0.\\ \end{array}$$ (23) For the augmented system (13) with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) =0$$, it follows from (21–23) that $$\label{seq:24} \begin{array}{rcl} {\mathbb{E}}\{{\it{\Delta}} V(k)\} &\leq& {\mathbb{E}}\left\{\xi^T(k)\bar{\it{\Phi}}\xi(k)\vphantom{\left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}} (k) \\ \end{array} \right]^T}\right.\\ &&-\delta_1\left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}} (k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check U_1 & -\check U2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}}(k) \\ \end{array} \right]\\ &&-\delta_2\left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check V1 & -\check V2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]\\ &&\left.-\delta_3\left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} -{\mathcal{S}}^T\check L^T\check L{\mathcal{S}} & 0 \\ \ast & \frac{1}{2}I_{2N} \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]\right\}\\ &=& {\mathbb{E}}\Big\{\xi^T(k){\it{\Phi}}^\ast\xi(k)+\xi^T(k)\bar{\it{\Sigma}}^T_1{\mathcal{P}}_1\bar{\it{\Sigma}}_1\xi(k)\Big\},\\ \end{array}$$ (24) where \begin{eqnarray*} {\it{\Phi}}^\ast&=&\left[ \begin{array}{cccc} {\it{\Phi}}^\ast_{11} & {\it{\Phi}}_{23}& {\it{\Phi}}_{13}&0\\ \ast &{\it{\Phi}}_{22} & 0 &0\\ \ast & \ast & {\it{\Phi}}_{33} &0\\ \ast & \ast & \ast &{\it{\Phi}}_{44}\\ \end{array} \right],\\ {\it{\Phi}}^\ast_{11}&=&-P-\delta_1\check U_1-\delta_2\check V_1+\delta_3{\mathcal{S}}^T\check L^T\check L{\mathcal{S}}. \end{eqnarray*} By applying Lemma 1, we know that $${\mathbb{E}}\{{\it{\Delta}} V (k)\} < 0$$ if and only if (17) is true. Furthermore, along the same way of the proof of Theorem 1 in Ding et al. (2012), it can be concluded that the augmented system (12) with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) = 0$$ is exponentially mean-square stable. Next, it remains to show that, under zero initial condition, the output error $$\tilde z(k)$$ satisfies the $$H_{\infty}$$ performance constraint (14). For this purpose, we introduce the following index: $$\label{seq:25} J(n)={\mathbb{E}}\left\{\sum_{k=0}^n \big[\frac{1}{N}\tilde z^T(k)\tilde z(k)-\gamma^2v^T(k) v(k)\big]\right\},$$ (25) where $$n$$ is a nonnegative integer. Obviously, our aim is to show $$J(n) < 0$$ under the zero-initial condition. Set $$\tilde\xi(k)=[\xi^T(k) v^T(k)]^T$$. Along the trajectory of the augmented system (12) with $${\it{\Delta}} K_i(k) = 0$$, it follows from (24) and (25) that \label{seq:26} \begin{aligned} J(n)=& {\mathbb{E}}\sum_{k=0}^{n}\Bigg\{\frac{1}{N}\tilde z^T(k)\tilde z(k) -\gamma^2 v^T(k) v(k)+{\it{\Delta}} V(k)\Bigg\}-{\mathbb{E}}V(n+1)\\ \leq& {\mathbb{E}}\sum_{k=0}^{n}\Bigg\{\frac{1}{N}\tilde z^T(k)\tilde z(k) -\gamma^2 v^T(k) v(k)+{\it{\Delta}} V(k) \Bigg\}\\ \leq& \sum_{k=0}^{n}{\mathbb{E}}\Bigg\{\tilde\xi^T(k){\it{\Phi}}_1\tilde\xi(k)+\tilde\xi^T(k){\it{\Sigma}}^T_1{\mathcal{P}}_1{\it{\Sigma}}_1\tilde\xi(k)\Bigg\}<0.\\ \end{aligned} (26) Letting $$n\rightarrow+\infty$$, it follows from the above inequality that \begin{equation*} \frac{1}{N}\sum_{k=0}^{\infty}{\mathbb{E}}\{\|\tilde z(k)\|^{2}\} \leq\gamma^2\sum_{k=0}^{\infty} \| v (k)\|^2. \end{equation*} and the proof is now complete. □ Up to now, the analysis problem of the estimator performance has been solved. Next, let us consider the $$H_{\infty}$$ estimator design problem for the complex network (1). The following result can be easily accessible from Theorem 1, and the proof is therefore omitted. Theorem 2 Let the disturbance attenuation level $$\gamma > 0$$ be given. For the discrete complex networks (1), the augmented system (12) is exponentially mean-square stable and satisfies the $$H_{\infty}$$ performance constraint (14) for all nonzero $$v(k)$$ if there exist positive definite matrices $$P_1>0, P_2>0, Q_{1m}>0, Q_{2m}>0 (m=1,2,\cdots, h)$$ and $$X_i>0 (i=1,2,\cdots, N)$$, and positive scalars $$\delta_1, \delta_2, \delta_3$$ satisfying \begin{eqnarray}\label{seq:th2a} \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &{\it{\Sigma}}^T_2 \\ \ast & -{\mathcal{P}}_1\\ \end{array} \right]<0, \end{eqnarray} (27) where \begin{equation*} \begin{array}{rcl} {\it{\Sigma}}_2&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0& P{\mathcal{B}}\\ {\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ \end{array} \right], \tilde{\mathcal{K}}_1=\left[ \begin{array}{cc} 0 \\ -XC \\ \end{array} \right]\!, \\[12pt] \tilde{\mathcal{W}}&=&\left[ \begin{array}{cc} (I_N\otimes P_1)(W\otimes{\it{\Gamma}}) & 0 \\ -(\mu-I_N)XC & (I_N\otimes P_2)(W\otimes{\it{\Gamma}})-XC\\ \end{array} \right]\!,\\[9pt] X&=&\textrm{diag}\{X_{1},X_{2},\cdots,X_{N}\}.\\ \end{array} \end{equation*} and other parameters are defined as in Theorem 1. Moreover, if the above inequality is feasible, the desired state estimator gains can be determined by $$\label{seq:28} K_i=P^{-1}_2X_i.$$ (28) Finally, the previous results obtained for the nominal state estimation error dynamics will be extended to non-fragile system with ROGVs described in (12). Theorem 3 Consider the non-fragile state estimation problem concerning error dynamics (12) and let $$\gamma>0$$ be a given scalar. The augmented system (12) is exponentially mean-square stable satisfies the $$H_{\infty}$$ performance constraint for all nonzero $$v(k)$$ if there exist matrices $$P>0$$, $$R_l>0$$$$(l=1,2, \cdots, L)$$, $$Q_m>0 (m=1, \cdots, h)$$ and $$X_j$$, positive scalars $$\delta_1, \delta_2, \delta_3$$ and $$\varepsilon$$ satisfying the following linear matrix inequalities (LMIs) $$\label{seq:th3a} \left[ \begin{array}{ccc} {\mathcal{L}} & {\mathcal{H}} & \varepsilon {\mathcal{N}}^T\\ {\mathcal{H}}^T &-\varepsilon I & 0 \\ \varepsilon {\mathcal{N}} &0 & -\varepsilon I \\ \end{array} \right]<0,$$ (29) where \begin{eqnarray*} {\mathcal{L}}&=& \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &{\it{\Sigma}}^T_3 \\ \ast & -{\mathcal{P}}_2\\ \end{array} \right], {\mathcal{J}}=\textrm{diag}\{P_2J_1,P_2J_2,\cdots,P_2J_N\},\\ {\mathcal{H}}&=&[0 0 0 0 0 {\mathcal{R}}^T_1 0 {\mathcal{R}}^T_2{\it{\Lambda}}^T_2 {\mathcal{R}}^T_2{\it{\Lambda}}^T_3 {\mathcal{R}}^T_2{\it{\Lambda}}^T_4 {\mathcal{R}}^T_3{\it{\Lambda}}^T_5]^T,\\ {\mathcal{N}}&=&[{\mathcal{M}} \underbrace{0 \cdots 0}_{10}], {\mathcal{P}}_2=I_6\otimes P, \bar{\mathcal{M}}=I_N\otimes M, {\mathcal{M}}=\textrm{diag}\{\bar{\mathcal{M}}C,\bar{\mathcal{M}}C\},\\ {\mathcal{R}}_1&=&\left[ \begin{array}{cc} 0 & 0 \\ -\bar\alpha(\mu-I_N){\mathcal{J}} & -\bar\alpha{\mathcal{J}}\\ \end{array} \right] , {\mathcal{R}}_2=\left[ \begin{array}{cc} 0 & 0 \\ -{\mathcal{J}} & 0\\ \end{array} \right] , {\mathcal{R}}_3=\left[ \begin{array}{cc} 0 & 0 \\ {\mathcal{J}} & -{\mathcal{J}}\\ \end{array} \right] ,\\ {\it{\Sigma}}_3&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0 & P{\mathcal{B}}\\ \sqrt{2}{\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ \end{array} \right],\\ {\it{\Sigma}}_{13}&=&[0 \bar\beta_1P \bar\beta_2P \cdots \bar\beta_hP],\\ {\it{\Lambda}}_2&=&\textrm{diag}\{\sqrt{2}\sigma_1\bar\alpha_1I,\cdots,\sqrt{2}\sigma_N\bar\alpha_NI,\sqrt{2}\sigma_1\bar\alpha_1I,\cdots,\sqrt{2}\sigma_N\bar\alpha_NI\},\\ {\it{\Lambda}}_3&=&\textrm{diag}\{\sigma_1\sqrt{\hat\alpha^\ast_1}I,\cdots,\sigma_N\sqrt{\hat\alpha^\ast_N}I,\sigma_1\sqrt{\hat\alpha^\ast_1}I,\cdots,\sigma_N\sqrt{\hat\alpha^\ast_1}I\},\\ {\it{\Lambda}}_4&=&\textrm{diag}\{\mu_1\sqrt{2\hat\alpha^\ast_1}I,\cdots,\mu_N\sqrt{2\hat\alpha^\ast_N}I,\mu_1\sqrt{2\hat\alpha^\ast_1}I,\cdots,\mu_N\sqrt{2\hat\alpha^\ast_N}I\},\\ {\it{\Lambda}}_5&=&\textrm{diag}\{\sqrt{2\hat\alpha^\ast_1}I,\cdots,\sqrt{2\hat\alpha^\ast_N}I,\sqrt{2\hat\alpha^\ast_1}I,\cdots,\sqrt{2\hat\alpha^\ast_N}I\},\\ \end{eqnarray*} and other parameters are defined as in Theorem 1. Proof. When $${\it{\Delta}} K_{i}(k)\neq0$$, from (19), (20) and the elementary inequality $$2a^Tb\geq a^T a + b^T b$$, we can obtain \begin{eqnarray}\label{seq:30} &&{\mathbb{E}}\{{\it{\Delta}} V(k)\}\notag\\ &&\quad\leq{\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)\notag\\ &&\qquad+2\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k)+2[I_2\otimes\bar\alpha\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes(\hat\alpha^\ast\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k) +[I_2\otimes(\mu^T\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+2[I_2\otimes(\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_3(k)\eta(k) +[I_2\otimes(\bar\alpha^T\bar\alpha\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes\hat\alpha^\ast]\eta^T(k){\mathcal{K}}^T_3(k)P{\mathcal{K}}_3(k)\eta(k) +v(k)^T{\mathcal{B}}^TP{\mathcal{B}}v(k)+{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\notag\\ &&\qquad+\sum^h_{m=1}\Big[(1+\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Big]\Big\}\notag\\ &&\quad\leq{\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)\notag\\ &&\qquad+2\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha^\ast\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad +2[I_2\otimes(\mu^T\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k) +2[I_2\otimes(\bar\alpha^T\bar\alpha\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+2[I_2\otimes\hat\alpha^\ast]\eta^T(k){\mathcal{K}}^T_3(k)P{\mathcal{K}}_3(k)\eta(k) +v(k)^T{\mathcal{B}}^TP{\mathcal{B}}v(k)+{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\notag\\ &&\qquad+\sum^h_{m=1}\Big[(1+\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Big]\Big\}. \end{eqnarray} (30) Along the trajectory of the augmented system (12) with $${\it{\Delta}} K_i(k)\neq0$$, taking (24) and (30) into consideration, we have that \label{seq:31} \begin{aligned} J(n)=& {\mathbb{E}}\sum_{k=0}^{n}\Big\{\frac{1}{N}\tilde z^T(k)\tilde z(k) -\gamma^2 v^T(k) v(k)+{\it{\Delta}} V(k)\Big\}-{\mathbb{E}}V(n+1)\\ \leq& \sum_{k=0}^{n}{\mathbb{E}}\Big\{\tilde\xi^T(k){\it{\Phi}}_1\tilde\xi(k)+\tilde\xi^T(k)\hat{\it{\Sigma}}^T_3{\mathcal{P}}_2\hat{\it{\Sigma}}_3\tilde\xi(k)\Big\}\leq0.\\ \end{aligned} (31) By using Lemma 1, it follows from (31) that \begin{eqnarray}\label{seq:32} \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &\hat{\it{\Sigma}}^T_3 \\ \ast & -{\mathcal{P}}_2\\ \end{array} \right]<0, \end{eqnarray} (32) where \begin{equation*} \begin{array}{rcl} \hat{\it{\Sigma}}_3&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}}+P{\it{\Delta}}{\mathcal{W}}(k) & P & {\it{\Sigma}}_{13}& 0 & P{\mathcal{B}}\\ \sqrt{2}{\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_2P {\mathcal{K}}_2(k){\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_3P {\mathcal{K}}_2(k){\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_4P {\mathcal{K}}_2(k){\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_5P {\mathcal{K}}_3(k)&0 & 0 & 0& 0\\ \end{array} \right],\\ \end{array} \end{equation*} and other parameters are defined as in Theorem 1. Next, according to the parameters defined in this theorem, we can rewrite $$\hat{\it{\Sigma}}_3$$ as follow: \begin{equation*} \begin{array}{rcl} \hat{\it{\Sigma}}_3&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}}+{\mathcal{R}}_1\bar{\it{\Delta}}(k){\mathcal{M}} & P & {\it{\Sigma}}_{13}& 0 & P{\mathcal{B}}\\ \sqrt{2}{\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_2 {\mathcal{R}}_2\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_3 {\mathcal{R}}_2\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_4 {\mathcal{R}}_2\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_5 {\mathcal{R}}_3\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ \end{array} \right],\\ \bar{\it{\Delta}}(k)&=&I_{2N}\otimes{\it{\Delta}}(k).\\ \end{array} \end{equation*} Then, we can change (32) into the following compact form: \begin{eqnarray} && {\mathcal{L}}+ {\mathcal{H}}\bar{\it{\Delta}}(k) {\mathcal{N}}+ {\mathcal{N}}^T\bar{\it{\Delta}}^T(k){\mathcal{H}}^T<0,\quad \label{seq:33} \end{eqnarray} (33) and the corresponding parameters have been defined in this theorem. According to Lemma 2, we can easily obtain (29), and the details are thus omitted. □ Remark 4 In this article, the main result established in Theorem 3 contains all the information about the system parameters, the $$H_{\infty}$$ performance index, the occurring probability of the randomly occurring time-delays, ROGVs and the statistical information of channel coefficients. The main novelty is twofold: (1) a comprehensive complex network model is established that caters for randomly occurring time-varying delays and channel fadings and (2) a non-fragile estimator model is proposed to account for ROGVs. The feasibility of the addressed non-fragile state estimator design problem can be readily checked by solving the LMIs, which can be conducted readily by utilizing Matlab Toolbox (YALMIP 3.0 and SeDuMi 1.1) in a straightforward approach. In the next section, an illustrative example will be provided to show the usefulness and effectiveness of the proposed design method. 4. Illustrative example Consider a discrete time-delayed complex network (1) with three nodes. The coupling configuration matrix is assumed to be $$W = (w_{i j} )_{N\times N}$$, with \begin{eqnarray*} W=\left[ \begin{array}{ccc} -0.6& 0.6& 0\\ 0.3& -0.8& 0.5\\ 0& 0.5& -0.5\\ \end{array} \right],\\ \end{eqnarray*} and the inner-coupling matrix is given as $${\it{\Gamma}}=\rm{diag}\{0.1, 0.1\}$$. The disturbance matrices and the output matrix are \begin{eqnarray*} &&B_1=\left[\begin{array}{@{}cc@{}} 0.21 \\ 0.18\\ \end{array} \right], B_2=\left[ \begin{array}{@{}cc@{}} 0.53\\ 0.32\\ \end{array} \right], B_3=\left[ \begin{array}{@{}cc@{}} 0.25\\ -0.1\\ \end{array} \right],\\ &&C_{1}=\left[ \begin{array}{@{}cc@{}} 0.6 & 0.3\\ -0.1 & 0.4\\ \end{array} \right], C_{2}=\left[ \begin{array}{@{}cc@{}} 0.5 & 0.1\\ 0.3 & 0.4\\ \end{array} \right], C_{3}=\left[ \begin{array}{@{}cc@{}} 0.3 & 0.6\\ 0.05 & 0.4\\ \end{array} \right],\\ &&E_1=[\begin{array}{@{}cc@{}} 0.50 & 0.45\\ \end{array}], E_2=[\begin{array}{@{}cc@{}} -0.35 & 0.55\\ \end{array}], E_3=[\begin{array}{@{}cc@{}} 0.85 & 0.65\\ \end{array}], \\ &&v(k)=0.4*\textrm{rand}(1)/(1 + 0.05k). \end{eqnarray*} The uncertain perturbation matrices in non-fragile state estimator are given as follows: \begin{eqnarray*} &&J_{1}=[ \begin{array}{@{}cc@{}} 0.06 & 0.1\\ \end{array}]^T, J_{2}=[ \begin{array}{@{}cc@{}} 0.05 & -0.06\\ \end{array}]^T, J_{3}=[ \begin{array}{@{}cc@{}} 0.09 & -0.03\\ \end{array}]^T,\\ &&M=[\begin{array}{@{}cc@{}} 0.01 & 0.02\\ \end{array}], {\it{\Delta}}(k)=0.04*\cos(k). \end{eqnarray*} The nonlinear vector-valued functions $$f (x_i (k))$$ and $$g(x_i (k))$$ are chosen as \begin{eqnarray*} &&f (x_i (k))=\left[ \begin{array}{@{}cc@{}} -0.45x_{i1}(k)+0.225x_{i2}(k)+\tanh(0.225x_{i1}(k)) \\ 0.45x_{i2}(k)-\tanh(0.15x_{i2}(k))\\ \end{array}\right], \\ &&g (x_i (k))=\left[ \begin{array}{@{}cc@{}} 0.02x_{i1}(k)+0.06x_{i2}(k) \\ -0.03x_{i1}(k)+0.02x_{i2}(k) +\tanh(0.01x_{i2}(k))\\ \end{array}\right]. \end{eqnarray*} Then, it is easy to see that the constraint (3) and (4) can be met with \begin{eqnarray*} &&U_1=\left[ \begin{array}{@{}cc@{}} -0.45 & 0225\\ 0 &0.30\\ \end{array}\right], U_2=\left[ \begin{array}{@{}cc@{}} -0.225 & 0225\\ 0 &0.45\\ \end{array}\right], \\ &&V_1=\left[ \begin{array}{@{}cc@{}} 0.02 & 0.06\\ -0.03 &0.02\\ \end{array}\right], V_2=\left[ \begin{array}{@{}cc@{}} -0.02 & 0.06\\ -0.02 &0.02\\ \end{array}\right]. \end{eqnarray*} The noise intensity function is simplified to $$h_i(x_i(k))=Lx_i(k)$$ with $$L=\left[ \begin{array}{@{}cc@{}} 0.27& -0.351\\ -0.135& 0.405\\ \end{array} \right].$$ In this example, the channel coefficients’ mathematical expectations are $$\mu_1=0.92, \mu_2=0.89$$ and $$\mu_3=0.71$$, the channel coefficients’ variances are $$\sigma_1^2=0.013, \sigma_2^2=0.02$$ and $$\sigma_3^2=0.022$$. Assume that the time-varying communication delays $$\tau_1(k)$$ and $$\tau_2(k)$$ are random variables whose elements are respectively uniformly distributed in the intervals $$[1, 3]$$ and $$[4, 6]$$, and the probabilities are taken as $$\bar\alpha_1=0.34, \bar\alpha_2=0.22, \bar\alpha_3=0.15,$$ and $$\bar\beta_1=0.15, \bar\beta_2=0.1$$. Our aim is to design a state estimator in the form of (8) such that the estimation error dynamics (11) is exponentially mean-square stable with a guaranteed $$H_\infty$$ norm bound $$\gamma = 0.985$$. By using the MATLAB (with YALMIP 3.0 and SeDuMi 1.1), we solve LMI (29) and obtain a set of feasible solutions as follows: \begin{eqnarray*} P_{1}&&=\left[ \begin{array}{cc} 0.5805 & -0.3976\\ -0.3976 & 1.0885\\ \end{array} \right], P_{2}=\left[ \begin{array}{cc} 0.6146 & -0.1445\\ -0.1445 & 0.8545\\ \end{array} \right], \\ Q_{11}&&=\left[ \begin{array}{cc} 0.8554 & -0.1122\\ -0.1122 & 1.1479\\ \end{array} \right], Q_{12}=\left[ \begin{array}{cc} 0.7793 & -0.0667\\ -0.0667 & 0.9875\\ \end{array} \right], \\ Q_{21}&&=\left[ \begin{array}{cc} 1.0904 & 0.0150\\ 0.0150 & 1.2075\\ \end{array} \right], Q_{22}=\left[ \begin{array}{cc} 1.0312 & 0.0088\\ 0.0088 & 1.1479\\ \end{array} \right], \\ X_1 &&=\left[ \begin{array}{cc} -0.2853 & 0.3743\\ 0.1821 & 0.24621\\ \end{array} \right], X_2 =\left[ \begin{array}{cc} -0.4833 & 0.2157\\ -0.1102 & 0.4251\\ \end{array} \right], \\ X_3 &&=\left[ \begin{array}{cc} -0.6828 & 1.1660\\ 0.1089 & 0.2790\\ \end{array} \right],\\ \delta_1&&=3.9482, \delta_2=8.5298, \delta_3=2.7120. \end{eqnarray*} Then, according to (28), we can obtain the following estimator parameters matrices: $$K_{1}=\left[ \begin{array}{cc} -0.4312 & 0.7047 \\ 0.1401 & 0.4073\\ \end{array} \right], K_{2}=\left[ \begin{array}{cc} -0.8504 & 0.4873\\ -0.2728 & 0.5799\\ \end{array} \right], K_{3}=\left[ \begin{array}{cc} -1.1258 & 2.0556\\ -0.0630& 0.6742\\ \end{array} \right].$$ Simulation results are shown in Figs 2 and 3, where Fig. 2 gives the simulation results of output signal $$z_2(t)$$ and its estimation $$\hat z_2(t)$$. Simulation results of the output estimation errors $$\tilde z_i(k) (i=1,2,3)$$ are shown in Fig. 3. All the simulation results have confirmed that the designed $$H_\infty$$ state estimation performs well. Fig. 2. View largeDownload slide Response of the outputs $$z_2(k)$$ and its estimation. Fig. 2. View largeDownload slide Response of the outputs $$z_2(k)$$ and its estimation. Fig. 3. View largeDownload slide Response of the output estimation errors. Fig. 3. View largeDownload slide Response of the output estimation errors. 5. Conclusions In this article, we have investigated the non-fragile $$H_\infty$$ state estimation problem for a class of discrete time-delay nonlinear complex networks with ROGVs and channel fadings. In order to take ROGVs into account, a novel estimator model has been proposed by using a set of Bernoulli-distributed white sequences with known conditional probabilities. By using the Lyapunov stability theory and stochastic analysis, sufficient conditions for the exponential mean-square stability of the state estimation error dynamic have been obtained and, at the same time, the prescribed $$H_\infty$$ disturbance rejection attenuation level has been guaranteed. Then, the explicit expression of the desired gain parameters has been derived. Finally, the effectiveness of the proposed estimator design is shown by a numerical simulation example. Further research topics include the extension of our results to more general complex networks and also to the $$H_\infty$$ control problem for complex networks with an improved fading model. 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# Non-fragile $$H_{\infty}$$ state estimation for discrete-time complex networks with randomly occurring time-varying delays and channel fadings

, Volume Advance Article – Oct 30, 2017
23 pages

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Publisher
Oxford University Press
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dnx043
Publisher site
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### Abstract

Abstract In this article, the non-fragile state estimation problem is investigated for a class of discrete time-delay nonlinear complex networks with both randomly occurring gain variations (ROGVs) and channel fadings. Two sequences of random variables obeying the Bernoulli distribution are employed to describe the phenomena of randomly occurring time-varying delays and ROGVs. Moreover, the phenomenon of channel fadings occurs in a random way and the fading probability is allowed to be uncertain but within a given interval. Through stochastic analysis and Lyapunov functional approach, sufficient conditions are derived for the existence of the desired estimator that guarantees both the exponential mean-square stability and the prescribed $$H_\infty$$ performance of the estimation error dynamics. The explicit expression of such estimators is also characterized by resorting to the semidefinite programming technique. Finally, a simulation example is provided to show the usefulness and effectiveness of the proposed state estimation scheme. 1. Introduction Complex networks are made up of a lot of highly interconnected dynamical units and therefore display very complicated dynamics. Many phenomena in nature can be modelled as complex networks, for example, the brain structure as a network of neurons, the social interaction as a network of people and the Internet as a network of routers or domains. The complex connections of these networks can be represented in terms of nodes, edges and coupling strengths. As is well known, random graphs are able to describe the large-scale networks with no explicit design principles, and therefore the early study on complex networks has been the territory of graph theory following the seminal work in Erdös & Rényi (1960). Since 1990, due to the introduction of scale-free networks (Barabási & Albert, 1999) and small-world networks (Watts & Strogatz, 1998), the dynamical behaviors of complex networks have attracted an ever-increasing research interest from a variety of communities such as mathematics, physics, technology and life sciences. As a result, a great deal of dynamic analysis issues have been extensively investigated for complex networks such as stability and synchronization (see Albert & Barabási, 2002; Boccaletti et al., 2006; Xiang & Chen, 2007; Karimi & Gao, 2010; Liu et al., 2016; Wang et al., 2016; Zhang et al., 2017b, and the references therein). Because of its clear practical insight, the state estimation problem has long been a fundamental issue for scientists and engineers in the past few decades (Xie et al., 1991; Brown & Hwang, 1992; Shi et al., 1999; Mahmoud, 2004; Shen et al., 2011; Liu et al., 2016; Wen et al., 2016). So far, different kinds of methodologies have been developed for the state estimation issues, among which Kalman filter and $$H_{\infty}$$ filter are two of the most popular ones. Kalman filter is an algorithm that uses a series of measurements observed over time and produces estimates of unknown variables. Traditional Kalman filter is based on accurate mathematical model that requires the assumption that the exogenous noise is a strict Gauss process or a Gauss noise sequence. When a priori statistical information on the external noise signals is unknown, the Kalman filter cannot be applied. In order to conquer this problem, the $$H_{\infty}$$ filter is employed in which the external noise signal is assumed to be energy bounded and the main objective is to minimize the $$H_{\infty}$$ norm from the process noise to the estimation error. The state estimation problem is particularly important for complex networks simply because of the large scale of the network as well as the unaffordable cost in directly acquiring the network states. Recently, the $$H_{\infty}$$ state estimation problems for complex networks have gained much research interest, see e.g., Ding et al. (2012), Shen et al. (2011) and Liu et al. (2008). In Liu et al. (2008), a synchronization problem has been investigated for an array of coupled complex discrete-time networks with both the discrete and distributed time delays, then a state estimator is designed to estimate the network states through available output measurements. The synchronization and state estimation problems have been studied for a class of coupled discrete time-varying stochastic complex networks over a finite horizon in Shen et al. (2011). Traditionally, most available state estimator design approaches rely on the implicit assumption that the designed estimator can be accurately implemented. This assumption, however, is not always true in reality as the state estimators do have a certain degree of imprecisions when it comes to implementation. The estimator gain might be subject to inevitable fluctuations because of finite resolution measuring instruments, round-off errors in numerical computations, random failures and repairs of components, and the need to provide practicing engineers with safe-tuning margins, see Keel & Bhattacharyya (1997), Hu et al. (2012), Chang & Yang (2012), Lien & Yu (2007) and Yang & Che (2008). As such, the estimator should be designed to be insensitive or non-fragile against the gain variations. On the other hand, due to unpredictable changes of network conditions, the variations of the estimator gains may appear in a probabilistic way with certain types and intensity. In this case, the phenomenon of randomly occurring gain variations (ROGVs) should be taken into careful consideration in the course of estimator design. Note that, very recently, a non-fragile controller with ROGVs has been studied in Fang & Park (2013) to address the synchronization problem of neural networks, and a non-fragile $$H_\infty$$ controller has been dealt with in Li et al. (2015) for a class of discrete-time systems subject to ROGVs and infinite-distributed delays. On another active research front, due to the ever-increasing popularity of communication networks (Sheng et al., 2017; Zou et al., 2017), network-induced phenomena (e.g., packet dropouts Sahebsara et al. 2007; Wang et al. 2013; Ding et al. 2017b; Luo et al. 2017, communication delays Fang & Park 2013; Luo et al. 2017 and signal quantization Liu et al. 2016; Ding et al. 2017a) have been well studied for state estimation and control problems of networked systems. However, the issue of channel fadings has received relatively less attention in spite of its practical significance in wireless mobile communications. Generally speaking, the primary cause for channel fadings is the multipath propagation and the shadowing from obstacles, when the electromagnetic waves do not directly reach the sensor due to obstacles that block the line of sight path. In practice, it is known that wireless channels are sensitive to fading effects Elia (2005), which constitute as one of the most dominant features in wireless communication links. If not dealt with adequately, the phenomenon of channel fadings would inevitably deteriorate the performance of the state estimators in case of fading measurements. Up to now, some pioneering works have been reported in the literature concerning networked control systems with fading channels (see e.g., Elia, 2005; Mostofi & Murray, 2009; Garone et al., 2012; Quevedo et al., 2012; Xiao et al., 2012; Zhang et al., 2014, and the references therein). Nevertheless, the corresponding non-fragile $$H_{\infty}$$ state estimation problem for complex networks had gain very little research attention, not to mention the case when ROGVs, channel fadings and random delays are simultaneously present. Summarizing the discussion made so far, (1) note that the corresponding researches for the discrete-time complex networks been gaining increasing research attention because complex networks could be potentially applied in many real-world systems; (2) the phenomenon of ROGVs is often unavoidable and should be taken into account when designing estimators and (3) in wireless communications, channel fadings often happen in a random fashion, and the original fading model needs further improvement. Motivated by the above discussion, a seemingly natural idea is to examine how to design the state estimators for discrete time-delay nonlinear complex networks subject to ROGVs in the presence of measurement transmission over a fading channel. To the best of the authors’ knowledge, such a research problem has not been fully investigated yet and the main purpose of this paper is therefore to shorten such a gap by addressing the non-fragile $$H_{\infty}$$ state estimation problem for discrete time-delay nonlinear complex networks with ROGVs and channel fadings. The main contributions of this paper can be highlighted as follows: (1) A comprehensive complex network model is established that caters for randomly occurring time-varying delays and channel fadings, and a unified framework is put forward for the estimator design ensuring the error dynamics of the state estimation to be exponentially mean-square stable with guaranteed $$H_{\infty}$$ performance constraint. (2) A novel model is proposed to account for channel fadings, where the coefficient of channel fadings has the probability density function on a given interval. The rest of this article is organized as follows: In Section 2, the model for the problem under consideration is presented and some assumptions are made on ROGVs and channel coefficients. Sufficient conditions for the exponentially stability and $$H_\infty$$-performance index of the error dynamics of the state estimation are obtained in Section 3. An illustrative example is provided in Section 4 to demonstrate the effectiveness of the main results, and Section 5 concludes the article with some discussion on future research directions. Notation: The notation used in the article is fairly standard. The superscript ‘$$T$$’ stands for matrix transposition, $${\mathbb{R}}^n$$ denotes the $$n$$-dimensional Euclidean space and $${\mathbb{R}}^{m\times n}$$ is the set of all real matrices of dimension $$m\times n$$. $$I$$ and $$0$$ represent the identity matrix and zero matrix, respectively. The notation $$P>0$$ means that $$P$$ is real symmetric and positive definite. $$l_2([0,+\infty),{\mathbb{R}}^n)$$ is the space of square summable sequences. The notation $$\|A\|$$ refers to the norm of a matrix $$A$$ defined by $$\|A\| =\sqrt{{\rm trace}(A^TA)}$$. In symmetric block matrices or complex matrix expressions, we use an asterisk ($$\ast$$) to represent a term that is induced by symmetry, and diag$$\{\ldots \}$$ stands for a block-diagonal matrix. The symbol $$\otimes$$ denotes the Kronecker product. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation and preliminaries The system under consideration is shown in Fig. 1, in which the local measurement is transmitted to the remote estimator via fading channels. We will discuss each part one by one in the sequential section. Fig. 1. View largeDownload slide Structure of the estimation system. Fig. 1. View largeDownload slide Structure of the estimation system. Consider the following discrete-time complex network with randomly occurring time-varying delays consisting of $$N$$ coupled nodes of the form: \begin{eqnarray}\label{seq:1} \left\{ \begin{aligned} x_i(k+1)=&\;f(x_i(k))+\sum_{m=1}^{h}\beta_m(k)g(x_i(k-\tau_m(k)))+\sum_{j=1}^{N}w_{ij}{\it{\Gamma}} x_j(k)+B_iv(k)+h_i(x_i(k))\omega(k)\\ z_i(k)=&\;E_ix_i(k)\\ x_i(s) =&\phi_i(s), \forall s \in {\mathcal{S}}\triangleq\{-\textrm{max}\{\overline{d}_1,\cdots,\overline{d}_m\},\cdots,-1,0\}, i=1,2,\cdots,N \end{aligned}\right. \end{eqnarray} (1) where $$x_i(k)\in {\mathbb{R}}^n$$ and $$z_i(k)\in {\mathbb{R}}^r$$ are, respectively, the state vector and the output of the $$i$$-th node. $$f(\cdot)$$ and $$g(\cdot)$$ are nonlinear vector valued functions and $$h(\cdot)$$ is the noise intensity function, satisfying certain conditions to be given later. $$v(k)$$ represents the exogenous disturbance input belonging to $$l_2([0,\infty), {\mathbb{R}})$$ and $$\omega(k)$$ is an one-dimensional zero-mean Gaussian white noise sequence on a probability space $$({\it{\Omega}},{\mathcal{F}},\{{\mathcal{F}}_t\}_{t\geq0}, \textrm{Prob})$$ with $${\mathbb{E}}\{\omega^2(k)\} = 1$$. $$B_i$$ and $$E_i$$ are known constant matrices with appropriate dimensions. $${\it{\Gamma}}=\textrm{diag}\{\gamma_1,\gamma_2,\cdots,\gamma_n\}\geq0$$ is an inner-coupling matrix linking the $$j$$th state variable if $$\gamma_j\neq0$$, and $$W=(w_{ij})\in {\mathbb{R}}^{N\times N}$$ is the coupled configuration matrix of the network with $$w_{ij}\geq0 (i\neq j)$$ but not all zero. As usual, the coupling configuration matrix $$W =(w_{ij})$$ is symmetric (i.e., $$W = W^T$$). The positive integers $$\tau_m(k) (m=1,2,\cdots, h)$$ denote the time-varying delays satisfying \begin{eqnarray}\label{seq:2} 0<\underline{d}_m\leq\tau_m(k)\leq \overline{d}_m, \end{eqnarray} (2) where $$\underline{d}_m$$ and $$\overline{d}_m$$ are known constant positive integers, representing the lower and upper bounds on the communication delay, respectively. To characterize the phenomena of stochastic interval time-varying delays, we introduce the stochastic variables $$\beta_m(k)\in {\mathbb{R}}$$ that are mutually independent Bernoulli-distributed white sequences. A natural assumption on $$\beta_m(k)$$ can be made as follows: \begin{eqnarray*} \left. \begin{aligned} &\textrm{Prob}\{\beta_m(k)=1\}={\mathbb{E}}\{\beta_m(k)\}=\bar\beta_m,\\ &\textrm{Prob}\{\beta_m(k)=0\}=1-\bar\beta_m. \end{aligned} \right. \end{eqnarray*} The nonlinear vector-valued functions $$f(\cdot), g(\cdot)$$ and the noise intensity function $$h(\cdot)$$ are continuous and satisfy \begin{eqnarray}\label{seq:3} &[f(x)-f(y)-U_1(x-y)]^T[f(x)-f(y)-U_2(x-y)]\leq0,\\ \end{eqnarray} (3) \begin{eqnarray} &[g(x)-g(y)-V_1(x-y)]^T[g(x)-g(y)-V_2(x-y)]\leq0,\label{seq:4}\\ \end{eqnarray} (4) \begin{eqnarray} &\|h_i(x)-h_j(y)\|^2\leq\|L(x-y)\|^2, \,\,\, \forall x,y\in{\mathbb{R}}^n, \label{seq:5} \end{eqnarray} (5) where $$U_1, U_2, V_1, V_2$$ and $$L$$ are known constant matrices (see Chu, 2001; Liang et al., 2014). For notation simplicity, we let \begin{eqnarray*} x(k)&=&\big[x^T_1(k) x^T_2(k) \cdots x^T_N(k)\big]^T,\\ B&=&\big[B^T_1 B^T_2 \cdots B^T_N\big]^T,\\ F(x(k))&=&\big[f^T(x_1(k)) f^T(x_2(k)) \cdots f^T(x_N(k))\big]^T,\\ G(x(k))&=&\big[g^T(x_1(k)) g^T(x_2(k)) \cdots g^T(x_N(k))\big]^T,\\ H(x(k))&=&\big[h_1^T(x_1(k)) h_2^T(x_2(k)) \cdots h_N^T(x_N(k))\big]^T. \end{eqnarray*} By utilizing the Kronecker product, we can rewrite the complex network (1) as the following compact form: \begin{eqnarray}\label{seq:6} x(k+1)=F(x(k))+\sum_{m=1}^{h}\beta_m(k)G(x(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})x(k)+Bv(k)+H(x(k))\omega(k). \end{eqnarray} (6) The fading measurement output $$y_i(k) \in {\mathbb{R}}^m$$ from the $$i$$-th node is of the form, \begin{eqnarray}\label{seq:7} y_i(k) = \lambda_i(k)C_ix_i(k), i= 1, 2,\cdots,N, \end{eqnarray} (7) where $$C_i\in {\mathbb{R}}^{m\times n}$$ is a known constant matrix. The random variable $$\lambda_i(k)\in {\mathbb{R}}$$, which accounts for the phenomena of channel fadings, has the probability density function on the interval $$[\underline{\lambda}_i, \overline{\lambda}_i]$$ with mathematical expectation $$\mu_i$$ and variance $$\sigma^2_i$$, where $$0<\underline{\lambda}_i <\mu_i-\sigma_i<\mu_i+\sigma_i<\overline{\lambda}_i<1$$. In order to estimate the states of the complex network (6), we take the phenomena of ROGVs into account and construct the following state estimator: \begin{eqnarray}\label{seq:8} \left\{ \begin{aligned} \hat{x}(k+1)=&F(\hat{x}(k))+\sum_{m=1}^{h}\beta_m(k)G(\hat{x}(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})\hat{x}(k)\\ &+(K+\alpha(k){\it{\Delta}} K(k))[y(k)-C\hat{x}(k)]\\ \hat{z}(k)=&E\hat{x}(k) \end{aligned}\right. \end{eqnarray} (8) with \begin{eqnarray*} \hat{x}(k)&=&\big[\hat{x}^T_1(k) \hat{x}^T_2(k) \cdots \hat{x}^T_N(k)\big]^T, \hat{z}(k)=\big[\hat{z}^T_1(k) \hat{z}^T_2(k) \cdots \hat{z}^T_N(k)\big]^T,\\ y(k)&=&\big[y^T_1(k) y^T_2(k) \cdots y^T_N(k)\big]^T, C=\textrm{diag}\{C_1,C_2,\cdots,C_N\},\\ F(\hat x(k))&=&\big[f^T(\hat x_1(k)) f^T(\hat x_2(k)) \cdots f^T(\hat x_N(k))\big]^T,\\ G(\hat x(k))&=&\big[g^T(\hat x_1(k)) g^T(\hat x_2(k)) \cdots g^T(\hat x_N(k))\big]^T,\\ E&=&\textrm{diag}\{E_1,E_2,\cdots,E_N\}, K=\textrm{diag}\{K_1,K_2,\cdots,K_N\},\\ {\it{\Delta}} K(k)&=&\textrm{diag}\{{\it{\Delta}} K_1(k),{\it{\Delta}} K_2(k),\cdots,{\it{\Delta}} K_N(k)\},\\ \alpha(k)&=&\textrm{diag}\{\alpha_1(k)I,\alpha_2(k)I,\cdots,\alpha_N(k)I\}, \end{eqnarray*} where $$\hat x_i(k)\in {\mathbb{R}}^n$$ is the estimate of the state $$x_i(k)$$, $$\hat z_i(k)\in {\mathbb{R}}^r$$ is the estimate of the output $$z_i (k)$$, and $$K_i\in {\mathbb{R}}^{n\times m}$$ is the estimator gain matrix to be determined. $$\alpha_i(k)\in {\mathbb{R}} (i=1,2,\cdots,N)$$ are mutually independent Bernoulli-distributed white sequences. A natural assumption on $$\alpha_i(k)$$ can be made as follows: \begin{eqnarray*} \left. \begin{aligned} &\textrm{Prob}\{\alpha_i(k)=1\}={\mathbb{E}}\{\alpha_i(k)\}=\bar\alpha_i,\\ &\textrm{Prob}\{\alpha_i(k)=0\}=1-\bar\alpha_i. \end{aligned} \right. \end{eqnarray*} Moreover, for each $$k$$, $$\alpha_i(k)$$ is independent of $$\beta_m(k)$$ and $$\lambda_{i}(k)$$. The uncertain perturbation matrix $${\it{\Delta}} K_i(k)\in {\mathbb{R}}^{n\times m}$$ is defined as follows: \begin{eqnarray}\label{seq:9} {\it{\Delta}} K_{i}(k)=J_{i}{\it{\Delta}}(k)M, \quad i\in \{1,2,\cdots,N\}, \end{eqnarray} (9) where $$J_i$$ and $$M$$ are known constant matrices with appropriate dimensions, and $${\it{\Delta}}(k)$$ is unknown matrix function satisfying \begin{eqnarray}\label{seq:10} {\it{\Delta}}^T(k){\it{\Delta}}(k)\leq I. \end{eqnarray} (10) Remark 1 The state estimation problem has been investigated in Ding et al. (2012) for an array of discrete time-delay nonlinear complex networks with randomly occurring sensor saturations and randomly varying sensor delays. In the real networked world, the parameter gain variations, which cannot be ignored, may be subject to random changes in environmental circumstances, for instance, network-induced random failures and repairs of components, sudden environmental disturbances, etc. It is worth pointing out that the models proposed in (7) and (8) provide a novel framework to account for the phenomenon of both ROGVs and channel fadings, which have not been taken into account in Ding et al. (2012). The stochastic variable $$\alpha_i(k)$$ is used to describe the random nature of estimator gain variations. In Srikanth et al. (2000), it has been proposed that analysis of a probabilistic system is to determine the probability density function over an interval. The stochastic variable $$\lambda_i(k)$$, whose probability density function is on a predetermined interval, characterizes the phenomenon of the probabilistic channel fadings. Such a description is more suitable for reflecting parameter variations of the estimator in a random fashion, particularly in the transmission of wireless communication over fading channels. In addition, the application on interval statistical characteristics have been received enough attention, see Srikanth et al. (2000) and Wei et al. (2013). Remark 2 Recently, the control and filtering issues with channel fadings have been investigated for traditional networked systems where the statistical characteristics are certain, see Zhang et al. (2014); Zhang et al. (2016) and the reference therein. In practice, the statistics of channel fadings could be obtained via offline statistical tests. From the viewpoint of statistics, any identification is not absolutely accurate, and should drop into an interval with given degree of confidence. Therefore, the description on channel fadings used in this article is more reasonable and the obtained results are more general and reliable. In addition, ROGVs and channel fadings with time-varying statistical characteristics on channel coefficients have been discussed in Zhang et al. (2016) for T-S fuzzy systems. In comparison with the research in Zhang et al. (2016), the main differences in this article are shown in the following two aspects: (1) different from the time-varying statistical characteristics, a predetermined interval is utilized to describe the channel fadings, which results into some interval-dependent analysis and design conditions and (2) the design is nontrivial and is with challenges coming from complex dynamical behavior, heterogeneous measurements, ROGVs as well as channel fadings with interval statistics. Let $$e(k)=\big[e^T_1(k) e^T_2(k) \cdots e^T_N(k)\big]^T$$ with $$e_i(k)=x_i(k)-\hat{x}_i(k)$$ being the state estimator error. Denote \begin{eqnarray*} \hat{F}(e(k))&=&\big[\hat{f}^T(e_1(k)) \hat{f}^T(e_2(k)) \cdots \hat{f}^T(e_N(k))\big]^T\\ &:=&F(x(k))-F(\hat{x}(k)),\\ \hat{G}(e(k))&=&\big[\hat{g}^T(e_1(k)) \hat{g}^T(e_2(k)) \cdots \hat{g}^T(e_N(k))\big]^T\\ &:=&G(x(k))-G(\hat{x}(k)),\\ \lambda(k)&:=&\textrm{diag}\{\lambda_1(k)I,\lambda_2(k)I,\cdots,\lambda_N(k)I\}. \end{eqnarray*} Then, from (6) and (8), we obtain the following system governing the estimate error dynamics: \begin{eqnarray}\label{seq:11} \left\{ \begin{aligned} x(k+1)=&F(x(k))+\sum_{m=1}^{h}(\hat\beta_m(k)+\bar\beta_m)G(x(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})x(k)+Bv(k)+H(x(k))\omega(k)\\ e(k+1)=&\hat{F}(e(k))+\sum_{m=1}^{h}(\hat\beta_m(k)+\bar\beta_m)\hat{G}(e(k-\tau_m(k)))+(W\otimes{\it{\Gamma}})e(k)+Bv(k)+H(x(k))\omega(k)\\ &-(K+(\hat\alpha(k)+\bar\alpha){\it{\Delta}} K(k))(\hat\lambda(k)+\mu-I_N)C x(k)-(K+(\hat\alpha(k)+\bar\alpha){\it{\Delta}} K(k))Ce(k) \end{aligned} \right.\nonumber\\ \end{eqnarray} (11) where \begin{eqnarray*} \hat\beta_m(k)&=&\beta_m(k)-\bar\beta_m, \hat\alpha_i(k)=\alpha_i(k)-\bar\alpha_i, \hat\lambda_i(k)=\lambda_i(k)-\mu_i,\\ \hat\alpha(k)&=&\textrm{diag}\{\hat\alpha_1(k)I,\hat\alpha_2(k)I,\cdots,\hat\alpha_N(k)I\}, \bar\alpha=\textrm{diag}\{\bar\alpha_1I,\bar\alpha_2I,\cdots,\bar\alpha_NI\},\\ \hat\lambda(k)&=&\textrm{diag}\{\hat\lambda_1(k)I,\hat\lambda_2(k)I,\cdots,\hat\lambda_N(k)I\}, \mu=\textrm{diag}\{\mu_1I,\mu_2I,\cdots,\mu_NI\}. \end{eqnarray*} It is clear that \begin{eqnarray*} &&{\mathbb{E}}\{\hat\beta_m(k)\}=0,\quad {\mathbb{E}}\{\hat\beta_m^2(k)\}\triangleq\hat\beta_m^\ast=\bar\beta_m(1-\bar\beta_m),\\ &&{\mathbb{E}}\{\hat\alpha_i(k)\}=0,\quad {\mathbb{E}}\{\hat\alpha_i^2(k)\}\triangleq\hat\alpha_i^\ast=\bar\alpha_i(1-\bar\alpha_i),\\ &&{\mathbb{E}}\{\hat\lambda_i(k)\}=0,\quad {\mathbb{E}}\{\hat\lambda_i^2(k)\}=\sigma^2_i. \end{eqnarray*} Next, by setting $$\eta(k) = [x^T(k) e^T(k)]^T$$ and defining the output error $$\tilde{z}(k)=z(k)-\hat z(k)$$, we have the following augmented system: \begin{eqnarray}\label{seq:12} \left\{ \begin{aligned} \eta(k+1)=&{\mathcal{W}}(k)\eta(k)+{\mathcal{F}}(k)+\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))+\sum_{m=1}^{h}\hat\beta_m(k){\mathcal{G}}(k-\tau_m(k))\\ &+[I_2\otimes\hat\lambda(k)]{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha(k)\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes(\mu\hat\alpha(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\\ &+[I_2\otimes(\bar\alpha\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes\hat\alpha(k)]{\mathcal{K}}_3(k)\eta(k)+{\mathcal{B}}v(k)+{\mathcal{H}}(k)\omega(k)\\ \tilde{z}(k)=&{\mathcal{E}}\eta(k)\\ \eta(s)=&\big[\phi^T_1(s) \phi^T_2(s) \cdots \phi^T_N(s) \phi^T_1(s) \phi^T_2(s) \cdots \phi^T_N(s)\big]^T, s \in\{-\textrm{max}\{\overline{d}_1,\cdots,\overline{d}_m\},\cdots,0\} \end{aligned} \right.\nonumber\\ \end{eqnarray} (12) where \begin{eqnarray*} {\mathcal{F}}(k)&=&[F^T(x(k)) \hat{F}^T(e(k))]^T,\\ {\mathcal{G}}(k)&=&[G^T(x(k)) \hat{G}^T(e(k))]^T,\\ {\mathcal{H}}(k)&=&[H^T(x(k)) H^T(x(k))]^T,\\ {\mathcal{B}}&=&[B^T B^T]^T, {\mathcal{E}}=[0 E], {\mathcal{S}}=[I_N 0],\\ {\mathcal{W}}(k)&=&{\mathcal{W}}+{\it{\Delta}}{\mathcal{W}}(k),\\ {\mathcal{W}}&=&\left[ \begin{array}{cc} (W\otimes{\it{\Gamma}}) & 0 \\ -(\mu-I_N)KC & (W\otimes{\it{\Gamma}})-KC\\ \end{array} \right],\\ {\it{\Delta}}{\mathcal{W}}(k)&=&\left[ \begin{array}{cc} 0 & 0 \\ -\bar\alpha(\mu-I_N){\it{\Delta}} K(k) C & -\bar\alpha{\it{\Delta}} K(k)C\\ \end{array} \right],\\ {\mathcal{K}}_1&=&\left[ \begin{array}{cc} 0 \\ -KC \\ \end{array} \right], {\mathcal{K}}_2(k)=\left[ \begin{array}{cc} 0 \\ -{\it{\Delta}} K(k)C & \\ \end{array} \right], \\ {\mathcal{K}}_3(k)&=&\left[ \begin{array}{cc} 0 & 0 \\ {\it{\Delta}} K(k)C & -{\it{\Delta}} K(k)C\\ \end{array} \right].\\ \end{eqnarray*} Definition 1 The augmented system (12) with $$v(t)=0$$ is said to be exponentially mean-square stable if there exist two scalars $$\nu>0$$ and $$0<\kappa<1$$ such that $$\label{seq:13} {\mathbb{E}}\{\|\eta(k)\|^{2}\} \leq \nu \kappa^{k}\sup_{s\in{\mathcal{S}}}{\mathbb{E}}\{\|\phi (s)\|^2\},\quad \forall k\geq 0.$$ (13) We are now ready to describe the non-fragile $$H_{\infty}$$ state estimation problem for the complex network (1). Specifically, we aim to design the state estimator (8), i.e., look for parameter matrix $$K_i$$ such that the following two requirements are simultaneously satisfied: (i) the augmented system (12) with $$v(t)=0$$ is exponentially mean-square stable. (ii) under the zero-initial condition, the output error $$\tilde z(t)$$ satisfies the $$H_{\infty}$$ performance constraint: $$\label{seq:14} \frac{1}{N}\sum_{k=0}^{\infty}{\mathbb{E}}\{\|\tilde z(k)\|^{2}\} \leq\gamma^2\sum_{k=0}^{\infty} \| v (k)\|^2,$$ (14) for all nonzero $$v(t)$$, where $$\gamma>0$$ is a given disturbance attenuation level. Remark 3 The $$H_{\infty}$$ performance index $$\gamma>0$$ is used to quantify the attenuation level of the estimation error dynamics against exogenous disturbances. Based on (12) and (14), it is easy to know that the value of $${\mathbb{E}}\{\|\tilde z(k)\|^{2}\}$$ would become larger if the number of the nodes increases. In theory, the $$H_\infty$$ disturbance attenuation level for the overall network should account for the average disturbance rejection performance which is insensitive to the quantity changes of the nodes in the estimator design. In order to achieve this purpose, the scalar of $$1/N$$ is employed to accommodate the average index over the complex network so that the scalar reflects the practical significance of the $$H_\infty$$ disturbance attenuation level. 3. Main results In this section, let us deal with the state estimation problem in the mean square for the complex network (6). First, we introduce several lemmas to be used in the sequel. Lemma 1 (Schur complement) Given constant matrices $${\it{\Sigma}}_1$$, $${\it{\Sigma}}_2$$, $${\it{\Sigma}}_3$$, where $${\it{\Sigma}}_1={\it{\Sigma}}^T_1$$ and $$0<{\it{\Sigma}}_2={\it{\Sigma}}^T_2$$. Then $${\it{\Sigma}}_1+{\it{\Sigma}}^T_3{\it{\Sigma}}^{-1}_2{\it{\Sigma}}_3<0$$ if and only if $$\label{seq:15} \left[ \begin{array}{cc} {\it{\Sigma}}_1 & {\it{\Sigma}}^T_3 \\ {\it{\Sigma}}_3 & -{\it{\Sigma}}_2 \\ \end{array} \right]<0\quad \textrm{or} \quad \left[ \begin{array}{cc} -{\it{\Sigma}}_2 & {\it{\Sigma}}_3 \\ {\it{\Sigma}}^T_3 & {\it{\Sigma}}_1 \\ \end{array} \right]<0.$$ (15) Lemma 2 (S-procedure) Let $$L=L^T$$, $$H, M$$ and $$N$$ be real matrices of appropriate dimensions with $$M$$ satisfying $$MM^T\leq I$$. Then, $$L+HMN+N^TM^TH^T<0$$, if and only if there exists a positive scalar $$\varepsilon>0$$ such that $$L+\varepsilon^{-1}HH^T+\varepsilon N^TN<0$$ or equivalently $$\label{seq:16} \left[ \begin{array}{ccc} L & H & \varepsilon N^T \\ H^T & -\varepsilon I & 0 \\ \varepsilon N& 0 & -\varepsilon I \\ \end{array} \right]<0.$$ (16) Now, we have the following analysis result that serves as a theoretical basis for the subsequent design problem. Theorem 1 Let the estimator parameters $$K_i (i=1,2,\cdots,N)$$ and a prescribed $$H_{\infty}$$ performance $$\gamma > 0$$ be given. Then, the zero solution of the augmented system (12) (with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) = 0$$) is exponentially mean-square stable if there exist positive definite matrices $$P_1>0, P_2>0, Q_{1m}>0$$ and $$Q_{2m}>0 (m=1,2,\cdots, h)$$, and positive scalars $$\delta_1, \delta_2, \delta_3$$ such that the following matrix inequalities hold: \begin{eqnarray}\label{seq:th1a} \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &{\it{\Sigma}}^T_1 \\ \ast & -{\mathcal{P}}_1\\ \end{array} \right]<0, \end{eqnarray} (17) where \begin{eqnarray*} P&=&\textrm{diag}\{I_{N}\otimes P_1,I_{N}\otimes P_2\}, {\mathcal{P}}_1=I_2\otimes P, \\ Q_m&=&\textrm{diag}\{I_{N}\otimes Q_{1m},I_{N}\otimes Q_{2m}\},\\ \check{U}_1&=&I_{2N}\otimes[(U^T_1U_2+U^T_2U_1)/2], \\ \check{U}_2&=&I_{2N}\otimes[(U^T_1+U^T_2)/2],\\ \check{V}_1&=&I_{2N}\otimes[(V^T_1V_2+V^T_2V_1)/2], \\ \check{V}_2&=&I_{2N}\otimes[(V^T_1+V^T_2)/2], \check{L}=I_{N}\otimes L,\\ {\it{\Phi}}_1&=&\left[ \begin{array}{ccccc} {\it{\Phi}}_{11} & {\it{\Phi}}_{12}& {\it{\Phi}}_{13}& 0& 0\\ \ast &{\it{\Phi}}_{22} & 0& 0& 0\\ \ast & \ast & {\it{\Phi}}_{33} & 0& 0\\ \ast & \ast & \ast & {\it{\Phi}}_{44}& 0\\ \ast & \ast & \ast & 0& {\it{\Phi}}_{55}\\ \end{array} \right],\\ {\it{\Phi}}_{11}&=&-P-\delta_1\check U_1-\delta_2\check V_1+\delta_3{\mathcal{S}}^T\check L^T\check L{\mathcal{S}}+\frac{1}{N}{\mathcal{E}}^T{\mathcal{E}},\\ {\it{\Phi}}_{12}&=&\delta_1\check U_2, {\it{\Phi}}_{13}=[\delta_2\check V_2 \underbrace{0 \cdots 0}_h], {\it{\Phi}}_{22}=-\delta_1I_{2N},\\ {\it{\Phi}}_{33}&=&\textrm{diag}\{\sum^h_{m=1}(1+\overline{d}_m-\underline{d}_m)Q_m-\delta_2I_{2N}, \hat\beta^{\ast}_1P- Q_1, \cdots, \hat\beta^{\ast}_hP- Q_h\},\\ {\it{\Phi}}_{44}&=&P-\frac{1}{2}\delta_3I_{2N}, {\it{\Phi}}_{55}=-\gamma^2I,\\ {\it{\Sigma}}_1&=&\left[ \begin{array}{ccccc} P{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0& P{\mathcal{B}}\\ {\it{\Lambda}}_1 P{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ \end{array} \right],\\ {\it{\Sigma}}_{13}&=&[0 \bar\beta_1P \bar\beta_2P \cdots \bar\beta_hP],\\ {\it{\Lambda}}_1&=&\textrm{diag}\{\sigma_1I,\cdots,\sigma_NI,\sigma_1I,\cdots,\sigma_NI\}. \end{eqnarray*} Proof. First, in order to show that the augmented system (12) (with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) = 0$$) is exponentially stable, we choose the following Lyapunov functional: $$\label{seq:18} V(k)=\sum_{i=1}^3V_i(k),$$ (18) where \begin{eqnarray*} V_1(k)&=& \eta^T(k)P\eta(k),\\ V_2(k)&=& \sum_{m=1}^h\sum_{i=k-\tau_m(k)}^{k-1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i),\\ V_3(k)&=& \sum_{m=1}^h\sum_{n=k-\overline{d}_m+1}^{k-\underline{d}_m}\sum_{i=n}^{k-1}{\mathcal{G}}^T(i)Q_m {\mathcal{G}}(i).\\ \end{eqnarray*} Calculating the difference of $$V_1(k)$$ along the trajectory of (12) we have \begin{eqnarray*}\label{seq:19a} {\it{\Delta}} V_1(k) &=&V_1(k+1)-V_1(k)\\ &=&\big({\mathcal{W}}(k)\eta(k)+{\mathcal{F}}(k)+\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))+\sum_{m=1}^{h}\hat\beta_m(k){\mathcal{G}}(k-\tau_m(k))\\ &&+[I_2\otimes\hat\lambda(k)]{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha(k)\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes(\mu\hat\alpha(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\\ &&+[I_2\otimes(\bar\alpha\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes\hat\alpha(k)]{\mathcal{K}}_3(k)\eta(k)+{\mathcal{B}}v(k)+{\mathcal{H}}(k)\omega(k)\big)^TP\\ &&\times \big({\mathcal{W}}(k)\eta(k)+{\mathcal{F}}(k)+\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))+\sum_{m=1}^{h}\hat\beta_m(k){\mathcal{G}}(k-\tau_m(k))\\ &&+[I_2\otimes\hat\lambda(k)]{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha(k)\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes(\mu\hat\alpha(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\\ &&+[I_2\otimes(\bar\alpha\hat\lambda(k))]{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)+[I_2\otimes\hat\alpha(k)]{\mathcal{K}}_3(k)\eta(k)+{\mathcal{B}}v(k)+{\mathcal{H}}(k)\omega(k)\big)-\eta^T(k)P\eta(k) \end{eqnarray*} Then, takeing the mathematical expectation, we obtain \begin{eqnarray}\label{seq:19} &&{\mathbb{E}}\{{\it{\Delta}} V_1(k)\}\notag\\ &&\quad={\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}(k)\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\left(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\right)^TP\left(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\right)\notag\\ &&\qquad+2\left(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\right)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k)+2[I_2\otimes\bar\alpha\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes(\hat\alpha^\ast\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k) +[I_2\otimes(\mu^T\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+2[I_2\otimes(\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_3(k)\eta(k) +[I_2\otimes(\bar\alpha^T\bar\alpha\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes\hat\alpha^\ast]\eta^T(k){\mathcal{K}}^T_3(k)P{\mathcal{K}}_3(k)\eta(k) +v(k)^T{\mathcal{B}}^TP{\mathcal{B}}v(k)+{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\Big\}, \end{eqnarray} (19) where $$\sigma^2=\textrm{diag}\{\sigma^2_1I,\sigma^2_2I,\cdots,\sigma^2_NI\}.$$ Also, it can be obtained that \begin{eqnarray*} \begin{array}{l} \begin{split} {\mathbb{E}}\Big\{{\it{\Delta}} V_2\Big\} =& {\mathbb{E}}\Big\{V_2(k+1)-V_2(k)\Big\}\\ =& {\mathbb{E}}\left\{\sum_{m=1}^h\left[\sum_{i=k+1-\tau_m(k+1)}^{k}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)-\sum_{i=k-\tau_m(k)}^{k-1} {\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\right]\right\}\\ =& {\mathbb{E}}\Bigg\{\sum_{m=1}^h\Bigg[{\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k)+\sum_{i=k+1-\tau_m(k+1)}^{k-1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\\ &-\sum_{i=k+1-\tau_m(k)}^{k-1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)-{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Bigg]\Bigg\}\\ \leq& {\mathbb{E}}\Bigg\{\sum^h_{m=1}\Bigg[{\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k)-{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\\ &+\sum^{k-\underline{d}_m}_{i=k-\overline{d}_m+1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\Bigg]\Bigg\}, \end{split}\\ \end{array} \end{eqnarray*} \begin{eqnarray}\label{seq:20} \begin{array}{l} \begin{split} {\mathbb{E}}\Big\{{\it{\Delta}} V_3\Big\} =& {\mathbb{E}}\Big\{V_3(k+1)-V_3(k)\Big\}\\ =& {\mathbb{E}}\Bigg\{\sum_{m=1}^h\Bigg[\sum_{n=k-\overline{d}_m+2}^{k-\underline{d}_m+1}\sum_{i=n}^{k}{\mathcal{G}}^T(i)Q_m {\mathcal{G}}(i)-\sum_{n=k-\overline{d}_m+1}^{k-\underline{d}_m}\sum_{i=n}^{k-1}{\mathcal{G}}^T(i))Q_m {\mathcal{G}}(i)\Bigg]\Bigg\}\\ =& {\mathbb{E}}\Bigg\{\sum_{m=1}^h\Bigg[\sum_{n=k-\overline{d}_m+2}^{k-\underline{d}_m+1}\sum_{i=n}^{k}{\mathcal{G}}^T(i)Q_m {\mathcal{G}}(i)-\sum_{n=k-\overline{d}_m+1}^{k-\underline{d}_m}\sum_{i=n}^{k}{\mathcal{G}}^T(i))Q_m {\mathcal{G}}(i)\\ &+(\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k)\Bigg]\Bigg\}\\ \leq& {\mathbb{E}}\Bigg\{\sum^h_{m=1}\Big[(\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -\sum^{k-\underline{d}_m}_{i=k-\overline{d}_m+1}{\mathcal{G}}^T(i)Q_m{\mathcal{G}}(i)\Bigg]\Bigg\}.\\ \end{split}\\ \end{array} \end{eqnarray} (20) When $${\it{\Delta}} K_{i}(k)=0$$ and $$v(k) =0$$, we can obtain from (19) and (20) that \begin{eqnarray}\label{seq:21} {\mathbb{E}}\{{\it{\Delta}} V(k)\} &\leq&{\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)\notag\\ &&+2\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&+[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k) +{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\notag\\ &&+\sum^h_{m=1}\Bigg[(1+\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Bigg]\Bigg\}\notag\\ &\leq& {\mathbb{E}}\Bigg\{\xi^T(k)\bar{\it{\Phi}}\xi(k)+\xi^T(k)\bar{\it{\Sigma}}^T_1{\mathcal{P}}_1\bar{\it{\Sigma}}_1\xi(k)\Bigg\}, \end{eqnarray} (21) where \begin{eqnarray*} \xi(k)&=&[\eta(k)^T {\mathcal{F}}^T(k) {\mathcal{G}}^T_m(k) {\mathcal{H}}^T(k)]^T,\\ {\mathcal{G}}_m(k)&=&[{\mathcal{G}}^T(k) {\mathcal{G}}^T(k-\tau_1(k)) {\mathcal{G}}^T(k-\tau_2(k)) \cdots {\mathcal{G}}^T(k-\tau_h(k))]^T,\\ \bar{\it{\Phi}}&=&\left[ \begin{array}{cccc} -P & 0 & 0 & 0\\ \ast & 0 & 0 & 0\\ \ast & \ast & \bar{\it{\Phi}}_{33}& 0 \\ \ast & \ast & \ast & P\\ \end{array} \right],\bar{\it{\Sigma}}_1=\left[ \begin{array}{ccccc} P{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0\\ {\it{\Lambda}}_1 P{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0\\ \end{array} \right]\!,\\\\ \bar {\it{\Phi}}_{33}&=&\textrm{diag}\left\{ \sum^h_{m=1}(1+\overline{d}_m-\underline{d}_m)Q_m, \hat\beta^{\ast}_1P- Q_1, \cdots, \hat\beta^{\ast}_hP- Q_h\right\}. \end{eqnarray*} Subsequently, we rewrite (3) and (4) as $$\label{seq:22} \begin{array}{rcl} &&\left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}} (k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check U_1 & - \check U2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}}(k) \\ \end{array} \right]\leq0,\\ &&\left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check V1 & - \check V2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]\leq0.\\ \end{array}$$ (22) Similarly, (5) can be rewritten as $$\label{seq:23} \begin{array}{rcl} &&\left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} -{\mathcal{S}}^T\check L^T\check L{\mathcal{S}} & 0 \\ \ast & \frac{1}{2}I_{2N} \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]\leq0.\\ \end{array}$$ (23) For the augmented system (13) with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) =0$$, it follows from (21–23) that $$\label{seq:24} \begin{array}{rcl} {\mathbb{E}}\{{\it{\Delta}} V(k)\} &\leq& {\mathbb{E}}\left\{\xi^T(k)\bar{\it{\Phi}}\xi(k)\vphantom{\left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}} (k) \\ \end{array} \right]^T}\right.\\ &&-\delta_1\left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}} (k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check U_1 & -\check U2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{F}}(k) \\ \end{array} \right]\\ &&-\delta_2\left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} \check V1 & -\check V2 \\ \ast & I \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{G}}(k) \\ \end{array} \right]\\ &&\left.-\delta_3\left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]^T \left[ \begin{array}{cc} -{\mathcal{S}}^T\check L^T\check L{\mathcal{S}} & 0 \\ \ast & \frac{1}{2}I_{2N} \\ \end{array} \right] \left[ \begin{array}{c} \eta(k) \\ {\mathcal{H}}(k) \\ \end{array} \right]\right\}\\ &=& {\mathbb{E}}\Big\{\xi^T(k){\it{\Phi}}^\ast\xi(k)+\xi^T(k)\bar{\it{\Sigma}}^T_1{\mathcal{P}}_1\bar{\it{\Sigma}}_1\xi(k)\Big\},\\ \end{array}$$ (24) where \begin{eqnarray*} {\it{\Phi}}^\ast&=&\left[ \begin{array}{cccc} {\it{\Phi}}^\ast_{11} & {\it{\Phi}}_{23}& {\it{\Phi}}_{13}&0\\ \ast &{\it{\Phi}}_{22} & 0 &0\\ \ast & \ast & {\it{\Phi}}_{33} &0\\ \ast & \ast & \ast &{\it{\Phi}}_{44}\\ \end{array} \right],\\ {\it{\Phi}}^\ast_{11}&=&-P-\delta_1\check U_1-\delta_2\check V_1+\delta_3{\mathcal{S}}^T\check L^T\check L{\mathcal{S}}. \end{eqnarray*} By applying Lemma 1, we know that $${\mathbb{E}}\{{\it{\Delta}} V (k)\} < 0$$ if and only if (17) is true. Furthermore, along the same way of the proof of Theorem 1 in Ding et al. (2012), it can be concluded that the augmented system (12) with $${\it{\Delta}} K_i(k) = 0$$ and $$v(k) = 0$$ is exponentially mean-square stable. Next, it remains to show that, under zero initial condition, the output error $$\tilde z(k)$$ satisfies the $$H_{\infty}$$ performance constraint (14). For this purpose, we introduce the following index: $$\label{seq:25} J(n)={\mathbb{E}}\left\{\sum_{k=0}^n \big[\frac{1}{N}\tilde z^T(k)\tilde z(k)-\gamma^2v^T(k) v(k)\big]\right\},$$ (25) where $$n$$ is a nonnegative integer. Obviously, our aim is to show $$J(n) < 0$$ under the zero-initial condition. Set $$\tilde\xi(k)=[\xi^T(k) v^T(k)]^T$$. Along the trajectory of the augmented system (12) with $${\it{\Delta}} K_i(k) = 0$$, it follows from (24) and (25) that \label{seq:26} \begin{aligned} J(n)=& {\mathbb{E}}\sum_{k=0}^{n}\Bigg\{\frac{1}{N}\tilde z^T(k)\tilde z(k) -\gamma^2 v^T(k) v(k)+{\it{\Delta}} V(k)\Bigg\}-{\mathbb{E}}V(n+1)\\ \leq& {\mathbb{E}}\sum_{k=0}^{n}\Bigg\{\frac{1}{N}\tilde z^T(k)\tilde z(k) -\gamma^2 v^T(k) v(k)+{\it{\Delta}} V(k) \Bigg\}\\ \leq& \sum_{k=0}^{n}{\mathbb{E}}\Bigg\{\tilde\xi^T(k){\it{\Phi}}_1\tilde\xi(k)+\tilde\xi^T(k){\it{\Sigma}}^T_1{\mathcal{P}}_1{\it{\Sigma}}_1\tilde\xi(k)\Bigg\}<0.\\ \end{aligned} (26) Letting $$n\rightarrow+\infty$$, it follows from the above inequality that \begin{equation*} \frac{1}{N}\sum_{k=0}^{\infty}{\mathbb{E}}\{\|\tilde z(k)\|^{2}\} \leq\gamma^2\sum_{k=0}^{\infty} \| v (k)\|^2. \end{equation*} and the proof is now complete. □ Up to now, the analysis problem of the estimator performance has been solved. Next, let us consider the $$H_{\infty}$$ estimator design problem for the complex network (1). The following result can be easily accessible from Theorem 1, and the proof is therefore omitted. Theorem 2 Let the disturbance attenuation level $$\gamma > 0$$ be given. For the discrete complex networks (1), the augmented system (12) is exponentially mean-square stable and satisfies the $$H_{\infty}$$ performance constraint (14) for all nonzero $$v(k)$$ if there exist positive definite matrices $$P_1>0, P_2>0, Q_{1m}>0, Q_{2m}>0 (m=1,2,\cdots, h)$$ and $$X_i>0 (i=1,2,\cdots, N)$$, and positive scalars $$\delta_1, \delta_2, \delta_3$$ satisfying \begin{eqnarray}\label{seq:th2a} \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &{\it{\Sigma}}^T_2 \\ \ast & -{\mathcal{P}}_1\\ \end{array} \right]<0, \end{eqnarray} (27) where \begin{equation*} \begin{array}{rcl} {\it{\Sigma}}_2&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0& P{\mathcal{B}}\\ {\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ \end{array} \right], \tilde{\mathcal{K}}_1=\left[ \begin{array}{cc} 0 \\ -XC \\ \end{array} \right]\!, \\[12pt] \tilde{\mathcal{W}}&=&\left[ \begin{array}{cc} (I_N\otimes P_1)(W\otimes{\it{\Gamma}}) & 0 \\ -(\mu-I_N)XC & (I_N\otimes P_2)(W\otimes{\it{\Gamma}})-XC\\ \end{array} \right]\!,\\[9pt] X&=&\textrm{diag}\{X_{1},X_{2},\cdots,X_{N}\}.\\ \end{array} \end{equation*} and other parameters are defined as in Theorem 1. Moreover, if the above inequality is feasible, the desired state estimator gains can be determined by $$\label{seq:28} K_i=P^{-1}_2X_i.$$ (28) Finally, the previous results obtained for the nominal state estimation error dynamics will be extended to non-fragile system with ROGVs described in (12). Theorem 3 Consider the non-fragile state estimation problem concerning error dynamics (12) and let $$\gamma>0$$ be a given scalar. The augmented system (12) is exponentially mean-square stable satisfies the $$H_{\infty}$$ performance constraint for all nonzero $$v(k)$$ if there exist matrices $$P>0$$, $$R_l>0$$$$(l=1,2, \cdots, L)$$, $$Q_m>0 (m=1, \cdots, h)$$ and $$X_j$$, positive scalars $$\delta_1, \delta_2, \delta_3$$ and $$\varepsilon$$ satisfying the following linear matrix inequalities (LMIs) $$\label{seq:th3a} \left[ \begin{array}{ccc} {\mathcal{L}} & {\mathcal{H}} & \varepsilon {\mathcal{N}}^T\\ {\mathcal{H}}^T &-\varepsilon I & 0 \\ \varepsilon {\mathcal{N}} &0 & -\varepsilon I \\ \end{array} \right]<0,$$ (29) where \begin{eqnarray*} {\mathcal{L}}&=& \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &{\it{\Sigma}}^T_3 \\ \ast & -{\mathcal{P}}_2\\ \end{array} \right], {\mathcal{J}}=\textrm{diag}\{P_2J_1,P_2J_2,\cdots,P_2J_N\},\\ {\mathcal{H}}&=&[0 0 0 0 0 {\mathcal{R}}^T_1 0 {\mathcal{R}}^T_2{\it{\Lambda}}^T_2 {\mathcal{R}}^T_2{\it{\Lambda}}^T_3 {\mathcal{R}}^T_2{\it{\Lambda}}^T_4 {\mathcal{R}}^T_3{\it{\Lambda}}^T_5]^T,\\ {\mathcal{N}}&=&[{\mathcal{M}} \underbrace{0 \cdots 0}_{10}], {\mathcal{P}}_2=I_6\otimes P, \bar{\mathcal{M}}=I_N\otimes M, {\mathcal{M}}=\textrm{diag}\{\bar{\mathcal{M}}C,\bar{\mathcal{M}}C\},\\ {\mathcal{R}}_1&=&\left[ \begin{array}{cc} 0 & 0 \\ -\bar\alpha(\mu-I_N){\mathcal{J}} & -\bar\alpha{\mathcal{J}}\\ \end{array} \right] , {\mathcal{R}}_2=\left[ \begin{array}{cc} 0 & 0 \\ -{\mathcal{J}} & 0\\ \end{array} \right] , {\mathcal{R}}_3=\left[ \begin{array}{cc} 0 & 0 \\ {\mathcal{J}} & -{\mathcal{J}}\\ \end{array} \right] ,\\ {\it{\Sigma}}_3&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}} & P & {\it{\Sigma}}_{13}& 0 & P{\mathcal{B}}\\ \sqrt{2}{\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ 0 &0 & 0 & 0& 0\\ \end{array} \right],\\ {\it{\Sigma}}_{13}&=&[0 \bar\beta_1P \bar\beta_2P \cdots \bar\beta_hP],\\ {\it{\Lambda}}_2&=&\textrm{diag}\{\sqrt{2}\sigma_1\bar\alpha_1I,\cdots,\sqrt{2}\sigma_N\bar\alpha_NI,\sqrt{2}\sigma_1\bar\alpha_1I,\cdots,\sqrt{2}\sigma_N\bar\alpha_NI\},\\ {\it{\Lambda}}_3&=&\textrm{diag}\{\sigma_1\sqrt{\hat\alpha^\ast_1}I,\cdots,\sigma_N\sqrt{\hat\alpha^\ast_N}I,\sigma_1\sqrt{\hat\alpha^\ast_1}I,\cdots,\sigma_N\sqrt{\hat\alpha^\ast_1}I\},\\ {\it{\Lambda}}_4&=&\textrm{diag}\{\mu_1\sqrt{2\hat\alpha^\ast_1}I,\cdots,\mu_N\sqrt{2\hat\alpha^\ast_N}I,\mu_1\sqrt{2\hat\alpha^\ast_1}I,\cdots,\mu_N\sqrt{2\hat\alpha^\ast_N}I\},\\ {\it{\Lambda}}_5&=&\textrm{diag}\{\sqrt{2\hat\alpha^\ast_1}I,\cdots,\sqrt{2\hat\alpha^\ast_N}I,\sqrt{2\hat\alpha^\ast_1}I,\cdots,\sqrt{2\hat\alpha^\ast_N}I\},\\ \end{eqnarray*} and other parameters are defined as in Theorem 1. Proof. When $${\it{\Delta}} K_{i}(k)\neq0$$, from (19), (20) and the elementary inequality $$2a^Tb\geq a^T a + b^T b$$, we can obtain \begin{eqnarray}\label{seq:30} &&{\mathbb{E}}\{{\it{\Delta}} V(k)\}\notag\\ &&\quad\leq{\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)\notag\\ &&\qquad+2\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k)+2[I_2\otimes\bar\alpha\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes(\hat\alpha^\ast\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k) +[I_2\otimes(\mu^T\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+2[I_2\otimes(\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_3(k)\eta(k) +[I_2\otimes(\bar\alpha^T\bar\alpha\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+[I_2\otimes\hat\alpha^\ast]\eta^T(k){\mathcal{K}}^T_3(k)P{\mathcal{K}}_3(k)\eta(k) +v(k)^T{\mathcal{B}}^TP{\mathcal{B}}v(k)+{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\notag\\ &&\qquad+\sum^h_{m=1}\Big[(1+\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Big]\Big\}\notag\\ &&\quad\leq{\mathbb{E}}\Big\{\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{W}}\eta(k)+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{F}}(k)+2\eta^T(k){\mathcal{W}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2\eta^T(k){\mathcal{W}}^T(k)P{\mathcal{B}}v(k) +{\mathcal{F}}^T(k)P{\mathcal{F}}(k)+2{\mathcal{F}}^T(k)P\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2{\mathcal{F}}^T(k)P{\mathcal{B}}v(k) +\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)\notag\\ &&\qquad+2\bigg(\sum_{m=1}^{h}\bar\beta_m{\mathcal{G}}(k-\tau_m(k))\bigg)^TP{\mathcal{B}}v(k) +\sum_{m=1}^{h}\hat\beta^{\ast}_m{\mathcal{G}}^T(k-\tau_m(k))P{\mathcal{G}}(k-\tau_m(k))\notag\\ &&\qquad+2[I_2\otimes\sigma^2]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_1P{\mathcal{K}}_1{\mathcal{S}}\eta(k)+[I_2\otimes(\hat\alpha^\ast\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad +2[I_2\otimes(\mu^T\mu\hat\alpha^\ast)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k) +2[I_2\otimes(\bar\alpha^T\bar\alpha\sigma^2)]\eta^T(k){\mathcal{S}}^T{\mathcal{K}}^T_2(k)P{\mathcal{K}}_2(k){\mathcal{S}}\eta(k)\notag\\ &&\qquad+2[I_2\otimes\hat\alpha^\ast]\eta^T(k){\mathcal{K}}^T_3(k)P{\mathcal{K}}_3(k)\eta(k) +v(k)^T{\mathcal{B}}^TP{\mathcal{B}}v(k)+{\mathcal{H}}^T(k)P{\mathcal{H}}(k)-\eta^T(k)P\eta(k)\notag\\ &&\qquad+\sum^h_{m=1}\Big[(1+\overline{d}_m-\underline{d}_m){\mathcal{G}}^T(k)Q_m{\mathcal{G}}(k) -{\mathcal{G}}^T(k-\tau_m(k))Q_m{\mathcal{G}}(k-\tau_m(k))\Big]\Big\}. \end{eqnarray} (30) Along the trajectory of the augmented system (12) with $${\it{\Delta}} K_i(k)\neq0$$, taking (24) and (30) into consideration, we have that \label{seq:31} \begin{aligned} J(n)=& {\mathbb{E}}\sum_{k=0}^{n}\Big\{\frac{1}{N}\tilde z^T(k)\tilde z(k) -\gamma^2 v^T(k) v(k)+{\it{\Delta}} V(k)\Big\}-{\mathbb{E}}V(n+1)\\ \leq& \sum_{k=0}^{n}{\mathbb{E}}\Big\{\tilde\xi^T(k){\it{\Phi}}_1\tilde\xi(k)+\tilde\xi^T(k)\hat{\it{\Sigma}}^T_3{\mathcal{P}}_2\hat{\it{\Sigma}}_3\tilde\xi(k)\Big\}\leq0.\\ \end{aligned} (31) By using Lemma 1, it follows from (31) that \begin{eqnarray}\label{seq:32} \left[ \begin{array}{ccccc} {\it{\Phi}}_1 &\hat{\it{\Sigma}}^T_3 \\ \ast & -{\mathcal{P}}_2\\ \end{array} \right]<0, \end{eqnarray} (32) where \begin{equation*} \begin{array}{rcl} \hat{\it{\Sigma}}_3&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}}+P{\it{\Delta}}{\mathcal{W}}(k) & P & {\it{\Sigma}}_{13}& 0 & P{\mathcal{B}}\\ \sqrt{2}{\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_2P {\mathcal{K}}_2(k){\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_3P {\mathcal{K}}_2(k){\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_4P {\mathcal{K}}_2(k){\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_5P {\mathcal{K}}_3(k)&0 & 0 & 0& 0\\ \end{array} \right],\\ \end{array} \end{equation*} and other parameters are defined as in Theorem 1. Next, according to the parameters defined in this theorem, we can rewrite $$\hat{\it{\Sigma}}_3$$ as follow: \begin{equation*} \begin{array}{rcl} \hat{\it{\Sigma}}_3&=&\left[ \begin{array}{ccccc} \tilde{\mathcal{W}}+{\mathcal{R}}_1\bar{\it{\Delta}}(k){\mathcal{M}} & P & {\it{\Sigma}}_{13}& 0 & P{\mathcal{B}}\\ \sqrt{2}{\it{\Lambda}}_1 \tilde{\mathcal{K}}_1{\mathcal{S}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_2 {\mathcal{R}}_2\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_3 {\mathcal{R}}_2\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_4 {\mathcal{R}}_2\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ {\it{\Lambda}}_5 {\mathcal{R}}_3\bar{\it{\Delta}}(k){\mathcal{M}} &0 & 0 & 0& 0\\ \end{array} \right],\\ \bar{\it{\Delta}}(k)&=&I_{2N}\otimes{\it{\Delta}}(k).\\ \end{array} \end{equation*} Then, we can change (32) into the following compact form: \begin{eqnarray} && {\mathcal{L}}+ {\mathcal{H}}\bar{\it{\Delta}}(k) {\mathcal{N}}+ {\mathcal{N}}^T\bar{\it{\Delta}}^T(k){\mathcal{H}}^T<0,\quad \label{seq:33} \end{eqnarray} (33) and the corresponding parameters have been defined in this theorem. According to Lemma 2, we can easily obtain (29), and the details are thus omitted. □ Remark 4 In this article, the main result established in Theorem 3 contains all the information about the system parameters, the $$H_{\infty}$$ performance index, the occurring probability of the randomly occurring time-delays, ROGVs and the statistical information of channel coefficients. The main novelty is twofold: (1) a comprehensive complex network model is established that caters for randomly occurring time-varying delays and channel fadings and (2) a non-fragile estimator model is proposed to account for ROGVs. The feasibility of the addressed non-fragile state estimator design problem can be readily checked by solving the LMIs, which can be conducted readily by utilizing Matlab Toolbox (YALMIP 3.0 and SeDuMi 1.1) in a straightforward approach. In the next section, an illustrative example will be provided to show the usefulness and effectiveness of the proposed design method. 4. Illustrative example Consider a discrete time-delayed complex network (1) with three nodes. The coupling configuration matrix is assumed to be $$W = (w_{i j} )_{N\times N}$$, with \begin{eqnarray*} W=\left[ \begin{array}{ccc} -0.6& 0.6& 0\\ 0.3& -0.8& 0.5\\ 0& 0.5& -0.5\\ \end{array} \right],\\ \end{eqnarray*} and the inner-coupling matrix is given as $${\it{\Gamma}}=\rm{diag}\{0.1, 0.1\}$$. The disturbance matrices and the output matrix are \begin{eqnarray*} &&B_1=\left[\begin{array}{@{}cc@{}} 0.21 \\ 0.18\\ \end{array} \right], B_2=\left[ \begin{array}{@{}cc@{}} 0.53\\ 0.32\\ \end{array} \right], B_3=\left[ \begin{array}{@{}cc@{}} 0.25\\ -0.1\\ \end{array} \right],\\ &&C_{1}=\left[ \begin{array}{@{}cc@{}} 0.6 & 0.3\\ -0.1 & 0.4\\ \end{array} \right], C_{2}=\left[ \begin{array}{@{}cc@{}} 0.5 & 0.1\\ 0.3 & 0.4\\ \end{array} \right], C_{3}=\left[ \begin{array}{@{}cc@{}} 0.3 & 0.6\\ 0.05 & 0.4\\ \end{array} \right],\\ &&E_1=[\begin{array}{@{}cc@{}} 0.50 & 0.45\\ \end{array}], E_2=[\begin{array}{@{}cc@{}} -0.35 & 0.55\\ \end{array}], E_3=[\begin{array}{@{}cc@{}} 0.85 & 0.65\\ \end{array}], \\ &&v(k)=0.4*\textrm{rand}(1)/(1 + 0.05k). \end{eqnarray*} The uncertain perturbation matrices in non-fragile state estimator are given as follows: \begin{eqnarray*} &&J_{1}=[ \begin{array}{@{}cc@{}} 0.06 & 0.1\\ \end{array}]^T, J_{2}=[ \begin{array}{@{}cc@{}} 0.05 & -0.06\\ \end{array}]^T, J_{3}=[ \begin{array}{@{}cc@{}} 0.09 & -0.03\\ \end{array}]^T,\\ &&M=[\begin{array}{@{}cc@{}} 0.01 & 0.02\\ \end{array}], {\it{\Delta}}(k)=0.04*\cos(k). \end{eqnarray*} The nonlinear vector-valued functions $$f (x_i (k))$$ and $$g(x_i (k))$$ are chosen as \begin{eqnarray*} &&f (x_i (k))=\left[ \begin{array}{@{}cc@{}} -0.45x_{i1}(k)+0.225x_{i2}(k)+\tanh(0.225x_{i1}(k)) \\ 0.45x_{i2}(k)-\tanh(0.15x_{i2}(k))\\ \end{array}\right], \\ &&g (x_i (k))=\left[ \begin{array}{@{}cc@{}} 0.02x_{i1}(k)+0.06x_{i2}(k) \\ -0.03x_{i1}(k)+0.02x_{i2}(k) +\tanh(0.01x_{i2}(k))\\ \end{array}\right]. \end{eqnarray*} Then, it is easy to see that the constraint (3) and (4) can be met with \begin{eqnarray*} &&U_1=\left[ \begin{array}{@{}cc@{}} -0.45 & 0225\\ 0 &0.30\\ \end{array}\right], U_2=\left[ \begin{array}{@{}cc@{}} -0.225 & 0225\\ 0 &0.45\\ \end{array}\right], \\ &&V_1=\left[ \begin{array}{@{}cc@{}} 0.02 & 0.06\\ -0.03 &0.02\\ \end{array}\right], V_2=\left[ \begin{array}{@{}cc@{}} -0.02 & 0.06\\ -0.02 &0.02\\ \end{array}\right]. \end{eqnarray*} The noise intensity function is simplified to $$h_i(x_i(k))=Lx_i(k)$$ with $$L=\left[ \begin{array}{@{}cc@{}} 0.27& -0.351\\ -0.135& 0.405\\ \end{array} \right].$$ In this example, the channel coefficients’ mathematical expectations are $$\mu_1=0.92, \mu_2=0.89$$ and $$\mu_3=0.71$$, the channel coefficients’ variances are $$\sigma_1^2=0.013, \sigma_2^2=0.02$$ and $$\sigma_3^2=0.022$$. Assume that the time-varying communication delays $$\tau_1(k)$$ and $$\tau_2(k)$$ are random variables whose elements are respectively uniformly distributed in the intervals $$[1, 3]$$ and $$[4, 6]$$, and the probabilities are taken as $$\bar\alpha_1=0.34, \bar\alpha_2=0.22, \bar\alpha_3=0.15,$$ and $$\bar\beta_1=0.15, \bar\beta_2=0.1$$. Our aim is to design a state estimator in the form of (8) such that the estimation error dynamics (11) is exponentially mean-square stable with a guaranteed $$H_\infty$$ norm bound $$\gamma = 0.985$$. By using the MATLAB (with YALMIP 3.0 and SeDuMi 1.1), we solve LMI (29) and obtain a set of feasible solutions as follows: \begin{eqnarray*} P_{1}&&=\left[ \begin{array}{cc} 0.5805 & -0.3976\\ -0.3976 & 1.0885\\ \end{array} \right], P_{2}=\left[ \begin{array}{cc} 0.6146 & -0.1445\\ -0.1445 & 0.8545\\ \end{array} \right], \\ Q_{11}&&=\left[ \begin{array}{cc} 0.8554 & -0.1122\\ -0.1122 & 1.1479\\ \end{array} \right], Q_{12}=\left[ \begin{array}{cc} 0.7793 & -0.0667\\ -0.0667 & 0.9875\\ \end{array} \right], \\ Q_{21}&&=\left[ \begin{array}{cc} 1.0904 & 0.0150\\ 0.0150 & 1.2075\\ \end{array} \right], Q_{22}=\left[ \begin{array}{cc} 1.0312 & 0.0088\\ 0.0088 & 1.1479\\ \end{array} \right], \\ X_1 &&=\left[ \begin{array}{cc} -0.2853 & 0.3743\\ 0.1821 & 0.24621\\ \end{array} \right], X_2 =\left[ \begin{array}{cc} -0.4833 & 0.2157\\ -0.1102 & 0.4251\\ \end{array} \right], \\ X_3 &&=\left[ \begin{array}{cc} -0.6828 & 1.1660\\ 0.1089 & 0.2790\\ \end{array} \right],\\ \delta_1&&=3.9482, \delta_2=8.5298, \delta_3=2.7120. \end{eqnarray*} Then, according to (28), we can obtain the following estimator parameters matrices: $$K_{1}=\left[ \begin{array}{cc} -0.4312 & 0.7047 \\ 0.1401 & 0.4073\\ \end{array} \right], K_{2}=\left[ \begin{array}{cc} -0.8504 & 0.4873\\ -0.2728 & 0.5799\\ \end{array} \right], K_{3}=\left[ \begin{array}{cc} -1.1258 & 2.0556\\ -0.0630& 0.6742\\ \end{array} \right].$$ Simulation results are shown in Figs 2 and 3, where Fig. 2 gives the simulation results of output signal $$z_2(t)$$ and its estimation $$\hat z_2(t)$$. Simulation results of the output estimation errors $$\tilde z_i(k) (i=1,2,3)$$ are shown in Fig. 3. All the simulation results have confirmed that the designed $$H_\infty$$ state estimation performs well. Fig. 2. View largeDownload slide Response of the outputs $$z_2(k)$$ and its estimation. Fig. 2. View largeDownload slide Response of the outputs $$z_2(k)$$ and its estimation. Fig. 3. View largeDownload slide Response of the output estimation errors. Fig. 3. View largeDownload slide Response of the output estimation errors. 5. Conclusions In this article, we have investigated the non-fragile $$H_\infty$$ state estimation problem for a class of discrete time-delay nonlinear complex networks with ROGVs and channel fadings. In order to take ROGVs into account, a novel estimator model has been proposed by using a set of Bernoulli-distributed white sequences with known conditional probabilities. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Oct 30, 2017

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