Delay-dependent H∞ filtering for complex dynamical networks with time-varying delays in nonlinear function and network couplings

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Delay-dependent H1 filtering for complex dynamical networks with time-varying delays in nonlinear function and network couplings V.M. Revathi, P. Balasubramaniam, K. Ratnavelu

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S0165-1684(15)00216-9 http://dx.doi.org/10.1016/j.sigpro.2015.06.017 SIGPRO5840

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Received date: 19 January 2015 Revised date: 11 May 2015 Accepted date: 24 June 2015 Cite this article as: V.M. Revathi, P. Balasubramaniam, K. Ratnavelu, Delaydependent H1 filtering for complex dynamical networks with time-varying delays in nonlinear function and network couplings, Signal Processing, http://dx. doi.org/10.1016/j.sigpro.2015.06.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Delay-dependent H∞ filtering for complex dynamical networks with time-varying delays in nonlinear function and network couplings V.M. Revathi †, P. Balasubramaniam1 †, K. Ratnavelu ‡ †Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamilnadu, India. ‡Institute of Mathematical science, Faculty of sciences, University of Malaya - 50603, Kuala Lumpur, Malaysia.

Abstract This paper investigates the H∞ filtering problem for a class of continuous-time complex dynamical networks with time-varying delays in nonlinear function and network couplings. The aim of the addressed problem is to design a H∞ filter against the exogenous disturbances, such that the filtering error system of complex dynamical networks is asymptotically stable and guarantees the desired H∞ performance attenuation level. Based on the Lyapunov stability theory, suitable Lyapunov-Krasovskii functional is constructed in terms of Kronecker product, furthermore, a new delay-dependent sufficient stability conditions are derived in terms of linear matrix inequalities by using reciprocal convex combination approach to obtain less conservative results. Finally, numerical examples are exploited to demonstrate the effectiveness of the proposed theoretical results. Keywords: Complex dynamical networks, Coupling delay, Kronecker product, Reciprocally convex approach, H∞ filtering, Linear matrix inequality. 1. Introduction The study of complex networks handle with the network whose structure is irregular, complex and networks with highly interconnected nodes, which evolves dynamically with respect to time [1]. Nowadays, extensive research work is focused on complex dynamical networks (CDNs) due to its wider applications in airport networks [2], computer networks 1

Corresponding author: [email protected] (P. Balasubramaniam)

Preprint submitted to Signal Processing

June 27, 2015

2 [3], biological networks [4], communication networks [5], etc. These systems exhibit complicated dynamics which are represented by a set of interconnected nodes, edges and coupling strength. Therefore, researchers have paid great attention to analyze the dynamical behavior of CDNs. The techniques of synchronization and state estimation for CDNs can be found in [6]-[9]. For singular CDNs both synchronization and state estimation problem have been investigated in [10]. Recently, authors in [11] have derived the sufficient conditions for adaptive synchronization of CDNs with non-delay and variable delay couplings via pinning control to achieve minimum number of pinning nodes. Robust exponential stabilization and adaptive synchronization for uncertain CDNs with time-delay in the coupling nodes have been discussed in [12, 13]. Synchronization happening between two or more coupled networks is known as ”outer synchronization”. Generalized outer synchronization between two uncertain networks with or without time delay has been proposed in [14]. In real complex network systems, such as in the progress of brain nervous activity, time delay occurs during the information transmission between nerve cells because of the limited speed of signal transmission as well as in the network traffic congestion systems. Thus, presence of time delays (coupling delays) in CDNs is unavoidable. It leads often as a source of instability and poor performance of system behaviors, for instance, see [15]-[18]. The dynamical behavior of CDNs with non-identical nodes are much more complicated than with identical nodes. To deal with different dimensional nodes, authors in [19] have studied the synchronization criteria for time-delayed coupling CDNs with different dimensional nodes by using the decentralized dynamic compensation controllers. Nowadays, various control methods such as adaptive control, sample data control, impulsive control, H∞ control have been explored to realize the synchronization phenomena of CDNs, for details see [11, 13, 19] and references therein. The sample data controller has been applied in [20, 21] to synchronize the CDNs with Markovian jump parameters by considering the time-varying delays in the network. Impulsive control technology introduced in [22, 23] is an efficient method to deal with dynamical systems which cannot ensure continuous disturbance. Very recently, the problem of impulsive control and synchronization of CDNs have

3 been investigated in [24]. By using single impulsive controller, sufficient conditions have been obtained in [25] for pinning synchronization of delayed undirected CDNs. Observer based controller design for complex systems has been an interesting topic in control theory. In practical, an non-fragile observer based robust controller for fractional order CDNs has been designed in [26] by using indirect Lyapunov approach. By introducing fractional-order Lyapunov stability theorem, pinning controller for synchronization of the directed and undirected complex networks have been reported in [27]. It is important to note that, the external disturbances are ubiquitous in nature as well as in complex networks. In this case, the notion of H∞ theory has been determined to reduce the effect of exogenous disturbances and to quantify them within the prescribed level [28, 29]. Based on decentralized observer, the problem of robust H∞ observer-based control for synchronization of a class of CDNs has been investigated in [30]. The novel concept of H∞ synchronization and state estimation for CDNs with mixed delays have been introduced in [31, 32] to quantify against the exogenous disturbance of the complex networks. The problem of filtering or state estimation has been widely applied in the fields of signal processing, image processing and control applications. Among various filtering methods, Kalman filtering deals with minimizing the variance of the state estimation error for a given measurement noise. However, in most practical applications, the statistical assumptions on the external noise signals cannot be known exactly. To overcome this limitation, H∞ filtering technique has been introduced to deal with unavoidable parameter shifts and external disturbances [33, 34]. Nowadays, the H∞ filtering technology has been applied recently to various dynamical timedelay systems such as networked control systems [23, 35], neutral systems [36], singular systems [37], T-S fuzzy systems [38, 39] and so on. Meantime, the delay-dependent filtering analysis for neural networks has been investigated in [40, 41] by employing some inequality techniques to reduce the conservatism. In the literature discussions above, most of the results have been concerned with synchronization and state estimation of CDNs. However, CDNs has its wider applications in science and engineering. Up to now, only limited works have been done with respect to H∞ filter

4 design for CDNs. Recently, the robust H∞ filtering design has been investigated in [42] for a class of discrete-time complex networks which has stochastic packet dropouts and time delays combined with disturbance inputs. The lack of research analysis is probably due to difficulty in designing suitable filter parameters. In order to shorten such a gap, in this paper, suitable full-order filter is designed for continuous-time CDNs with time-varying delays. On the other hand, it has been well known that there exist various approaches to reduce the conservatism of delay-dependent stability results for time-delay systems. The reduction of conservatism means that to increase the feasibility region. In this paper, we have utilized the reciprocal convex combination approach [43] to derive the sufficient conditions for the problem of delay-dependent H∞ filtering for CDNs. To the best of authors knowledge, H∞ filtering analysis for CDNs by using reciprocal convex approach has not been investigated yet. Motivated by the above discussions, this paper is aimed to study the problem of H∞ filtering for continuous-time CDNs with time-varying delays in the coupling nodes. By constructing suitable Lyapunov-Krasovskii functional in terms of Kronecker product, sufficient stability conditions are derived in terms of linear matrix inequalities (LMIs) to guarantee the existence of the designed H∞ filters. The main contributions of this paper lie in the following aspects: (i) For the first time, suitable full-order H∞ filters are designed for each node for continuous-time CDNs with time-varying delays in nonlinear function and network couplings. (ii) The properties of Kronecker product is employed to derive the stability conditions in a more compact form. (iii) In order to reduce the possible conservatism of the result, reciprocal convex combination approach is utilized. (iv) To illustrate the applicability of the proposed results, Barabasi-Albert (BA) scale-free network model is considered. The rest of this paper is organized as follows. A class of continuous-time delayed CDNs consisting of N coupled nodes is presented and some necessary definition and lemmas are provided in Section 2. In Section 3, sufficient conditions are obtained in the form of LMIs and then the H∞ filters are designed. Numerical simulations are given in Section 4, to verify the effectiveness of the designed H∞ filters. Finally, in Section 5 conclusions and future research

5 interests are given. Notation: Throughout this paper, Rn denotes the n dimensional Euclidean space and Rm×n is the space of m × n real matrices. L2 [0, ∞) represents the space of square integrable vector functions over [0, ∞). The Kronecker product of matrices Q ∈ Rm×n and R ∈ Rp×q is a matrix in Rmp×nq and denoted as Q ⊗ R. We use diag{· · ·} as a block-diagonal matrix. A > 0 (< 0) means A is a symmetric positive (negative) definite matrix, A−1 denotes the inverse of matrix A. AT denotes the transpose of matrix A, I is the identity matrix with compatible dimension. 2. Problem formulation and preliminaries Consider the following delayed CDNs consisting of N coupled nodes of the form: ⎧ ⎪ ⎪ ⎪ x˙ i (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ yi (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ zi (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩xi (t)

= Axi (t) + B1 f (xi (t)) + B2 f (xi (t − τ (t))) + +

N

j=1 gij Γ2 xj (t

N

j=1 wij Γ1 xj (t)

− τ (t)) + D1i νi (t),

= Ci xi (t) + D2i ωi (t),

(1)

= Exi (t), = φi (t), ∀t ∈ [−τ, 0], i = 1, 2 . . . N,

where xi (t) ∈ Rn is the state vector of the ith node at time t; νi (t) and ωi (t) are the disturbance inputs which belong to L2 [0, ∞); yi (t) ∈ Rm is the measured output of the ith node; zi (t) ∈ Rq is the signal to be estimated; φi (t) is a compatible vector-valued initial function defined on [−τ, 0]; A, B1 , B2 , E are known constant matrices with appropriate dimensions; D1i ∈ Rn×n , Ci ∈ Rm×n and D2i ∈ Rm×n are some constant matrices; Γ1 and Γ2 are matrices describing the inner-coupling between the subsystems at time t and t − τ (t) respectively; W = [wij ]N ×N and G = [gij ]N ×N are the outer-coupling configuration matrices representing respectively the coupling strength and the topological structure of complex networks; τ (t) is the time-varying

6 delay satisfies the following inequality 0 ≤ τ (t) ≤ τ,

τ˙ (t) ≤ μ < ∞.

(2)

Assumption 1 The nonlinear function f (·) : Rn → Rn is assumed to be continuous and satisfies f (0) = 0. Further the following sector-bounded condition holds [f (x) − f (y) − F1 (x − y)]T [f (x) − f (y) − F2 (x − y)] ≤ 0,

(3)

where F1 and F2 are known real constant matrices with appropriate dimensions. The following full-order filter is designed to estimate the signal zi (t): ⎧ ⎪ ⎪ ⎪ ˆi (t) + BF i yi (t), x ˆ˙ i (t) = AF i x ⎪ ⎪ ⎪ ⎨ ˆi (t), zˆi (t) = E x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x ˆi (t) = φˆi (t), ∀t ∈ [−τ, 0],

(4)

where x ˆi (t) is the filter state vector and AF i , BF i are appropriately dimensioned filter matrices to be designed. With the matrix Kronecker product, the systems (1) and (4) can be rewritten respectively in the following compact form: ⎧ ⎪ ⎪ ⎪ x(t) ˙ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x(t)

2 f (x(t − τ (t))) + (W ⊗ Γ1 )x(t)  1 f (x(t)) + B = Ax(t) +B +(G ⊗ Γ2 )x(t − τ (t)) + D1 ν(t), = Cx(t) + D2 ω(t),  = Ex(t), = φ(t),

∀t ∈ [−τ, 0],

(5)

7 and ⎧ ⎪ ⎪ ⎪ x ˆ˙ (t) ⎪ ⎪ ⎪ ⎨ zˆ(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x ˆ(t)

ˆ(t) + BF y(t), = AF x x =E ˆ(t),

(6)

ˆ = φ(t), ∀t ∈ [−τ, 0],

where 2 = I ⊗ B2 , E  = I ⊗ E,  = I ⊗ A, B 1 = I ⊗ B1 , B A C = diag{C1 , C2 , · · · , CN }, D1 = diag{D11 , D12 , · · · , D1N }, D2 = diag{D21 , D22 , · · · , D2N }, x(t) = [xT1 (t) xT2 (t) · · · xTN (t)]T , T ˆT2 (t) · · · x ˆTN (t)]T , y(t) = [y1T (t) y2T (t) · · · yN (t)]T , x ˆ(t) = [ˆ xT1 (t) x

AF

= diag{AF 1 , AF 2 , · · · , AF N }, BF = diag{BF 1 , BF 2 , · · · , BF N },

T T (t)]T , ω(t) = [ω1T (t) ω2T (t) · · · ωN (t)]T , ν(t) = [ν1T (t) ν2T (t) · · · νN

f (x(t)) = [f T (x1 (t)) f T (x2 (t)) · · · f T (xN (t))]T . ˆT (t)]T and z¯(t) = z(t) − zˆ(t), the augmented By introducing new vector e(t) = [xT (t) x filtering error system is obtained as follows: ⎧ ⎪ ⎪ ⎨e(t) ˙

¯1 f (Ke(t)) + B ¯2 f (Ke(t − τ (t))) + L ¯ 1ω ¯ (t), = A¯1 e(t) + A¯2 Ke(t − τ (t)) + B

⎪ ⎪ ⎩z¯(t)

¯ 2 e(t), =L

¯ e(t) = φ(t), ∀t ∈ [−τ, 0],

(7)

¯ where φ(t) = [φT (t) φˆT (t)]T and ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ ⎤    B1 ⎥ B2 ⎥ ⎢ A + W ⊗ Γ1 0 ⎥ ⎢ G ⊗ Γ2 ⎥ ¯1 = ⎢ ¯2 = ⎢ = ⎣ ⎦, B ⎦, B ⎦, ⎣ ⎣ ⎦ , A¯2 = ⎣ AF 0 0 0 BF C ⎡ ⎤ T      0 ⎢ D1 ⎥ ¯ T T   = ⎣ ¯ (t) = ν (t) ω (t) . ⎦ , L2 = E − E , K = I 0 , ω 0 BF D2 ⎡

A¯1

¯1 L

8 Remark 1 Many phenomena in nature can be modeled as a networks of interconnected dynamical nodes. Examples of such networks include brain structures, social networks, global economic markets, internet, airport networks and so on. These networks have more than one link between two nodes and each of these links have its own property. Airport networks [2] are a kind of complex systems that display significant fluctuations. Such kind of systems can be modeled as a class of CDNs with exogenous disturbance inputs. Definition 2.1 The filtering error system (7) is said to be asymptotically stable with an H∞ performance γ, if it is asymptotically stable with ω ¯ (t) = 0 and, under zero initial condition, there exists γ > 0, such that

1 N

z¯(t) 2 ≤ γ ω ¯ (t) 2 , for any non-zero ω ¯ (t) ∈ L2 [0, ∞).

Lemma 2.1 (Park etal. [43]) Let f1 , f2 , . . . , fN : Rm → R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies min 

{αi |αi >0,

i

αi =1}

  1  fi (t) = fi (t) + max gi,j (t) gi,j(t) αi i

i

i=j

subject to 

⎤

⎡

 ⎢ fi (t) gi,j (t) ⎥ gi,j : Rm → R, gj,i (t)  gi,j (t), ⎣ ⎦≥0 . gi,j (t) fj (t)

Lemma 2.2 (Gu [44]) For any constant matrix M ∈ Rn×n , M = M T > 0, scalar h > 0, such that the following integrations are well defined, then  −h

t

t−h

T

x (s)M x(s)ds ≤ −



t

t−h

  x (s)ds M T

t

t−h

 x(s)ds .

Lemma 2.3 (Li [9]) By the definition of the Kronecker product, the following properties hold. (αA) ⊗ B = A ⊗ (αB), (A + B) ⊗ C = A ⊗ C + B ⊗ C,

9 (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD), (A ⊗ B)T = AT ⊗ B T . Remark 2 It should be noticed that, the H∞ filtering problem for a class of continuous-time CDNs is investigated for the first time in this work and the proposed LMI conditions are given in the form of Kronecker product in a more compact form which can easily solved by using Matlab LMI toolbox. 3. Main Results In this section, the stability analysis problem for the filtering error system (7) with ω ¯ (t) = 0 is investigated by using Lyapunov functional method combining with LMI techniques and ¯ (t) is then the H∞ performance analysis for the filtering error system (7) with nonzero ω derived. Theorem 3.1 For given scalars τ, μ and positive scalars λ1 , λ2 , the filtering error system (7) is asymptotically stable with an H∞ performance γ, if there exist positive definite matrices Ul (l = 1, 2 . . . , 5) and any matrix W1 with appropriate dimension such that the following LMIs hold ⎡

⎡

⎢ U4 ⎣ T W 1

⎢ Φ Ω ⎢ ⎢ ∗ Ω ¯ 66 ⎢ ⎣ ∗ ∗

⎤  W1 ⎥ ⎦ ≥ 0, U4 ⎤ 1 NΥ ⎥ ⎥ 0 ⎥ ⎥ < 0, ⎦ −I

(8)

(9)

10 where ⎡

Φ11 Φ12 Φ13 Φ14 Φ15

⎤

⎥ ⎢ ⎥ ⎢ ⎢ ∗ Φ22 Φ23 Φ24 Φ25 ⎥ ⎥ ⎢ T  ⎥ ⎢ T T T T , Φ = ⎢ ∗ Φ33 0 0 ⎥ ⎥ , Ω = Ω16 Ω26 0 Ω46 Ω56 ⎢ ∗ ⎥ ⎢ ⎥ ⎢ ⎢ ∗ ∗ ∗ Φ44 Φ45 ⎥ ⎦ ⎣ ∗ ∗ ∗ ∗ Φ55  T ¯ 66 = τ 2 L ¯ 1 − γ 2 I, ¯ T K T U4 K L ¯2 0 0 0 0 ,Ω Υ = L 1 with Φ11 = U1 A¯1 + A¯T1 U1 + K T U2 K + K T U3 K + τ 2 A¯T1 K T U4 K A¯1 − K T U4 K  1 , −λ1 K T F1 K, Φ12 = U1 A¯2 + τ 2 A¯T1 K T U4 K A¯2 + K T U4 − K T W

1 , Φ13 = K T W

 2 ¯T T ¯1 + λ1 K T F ¯ ¯1 + τ 2 A¯T1 K T U4 K B ¯ Φ14 = U1 B 2 , Φ15 = U1 B2 + τ A1 K U4 K B2 ,  T + W 1 − U4 − λ2 F  Φ22 = −(1 − μ)U2 + τ 2 A¯T2 K T U4 K A¯2 − U4 + W 1 , Φ23 = U4 − W1 , 1  ¯1 , Φ25 = τ 2 A¯T2 K T U4 K B ¯2 + λ2 F Φ24 = τ 2 A¯T2 K T U4 K B 2 , Φ33 = −U3 − U4 , ¯1 − 2λ1 I, Φ45 = τ 2 B ¯2 Φ55 = τ 2 B ¯2 − 2λ2 I, ¯ T K T U4 K B ¯ T K T U4 K B ¯ T K T U4 K B Φ44 = τ 2 B 1 1 2 ¯ 1 , Ω26 = τ 2 A¯T2 K T U4 K L ¯ 1 , Ω46 = τ 2 B ¯ 1, ¯ 1 + τ 2 A¯T1 K T U4 K L ¯1T K T U4 K L Ω16 = U1 L  ¯1 W    ¯2T K T U4 K L 1 = I ⊗ W1 , F Ω56 = τ 2 B 1 = I ⊗ F1 , F 2 = I ⊗ F2 , F1 = F1T F2 + F2T F1 , F2 = F1T + F2T . Proof Construct the Lyapunov-Krasovskii functional candidate for system (7) as

V (t, e(t)) =

4  n=1

Vn (t, e(t)),

(10)

11 where V1 (t, e(t)) = eT (t)U1 e(t),  t eT (s)K T U2 Ke(s)ds, V2 (t, e(t)) = t−τ (t) t T

 V3 (t, e(t)) = V4 (t, e(t)) = τ

e (s)K T U3 Ke(s)ds,

t−τ  0

−τ



t

t+θ

e˙ T (s)K T U4 K e(s)dsdθ, ˙

with U1 = diag{I ⊗ P1 , I ⊗ P2 }, U2 = I ⊗ P3 , U3 = I ⊗ P4 , U4 = I ⊗ P5 . Taking the time derivative of V (t, e(t)) along the trajectory of the system (7), we obtain  ¯1 f (Ke(t)) + B ¯2 f (Ke(t − τ (t))) V˙ 1 (t, e(t)) = 2eT (t)U1 A¯1 e(t) + A¯2 Ke(t − τ (t)) + B ¯ 1ω ¯ (t)], +L

(11)

V˙ 2 (t, e(t)) = eT (t)K T U2 Ke(t) − (1 − τ˙ (t))eT (t − τ (t))K T U2 Ke(t − τ (t)) ≤ eT (t)K T U2 Ke(t) − (1 − μ)eT (t − τ (t))K T U2 Ke(t − τ (t)), V˙ 3 (t, e(t)) = eT (t)K T U3 Ke(t) − eT (t − τ )K T U3 Ke(t − τ ),  t ˙ −τ e˙ T (s)K T U4 K e(s)ds ˙ V˙ 4 (t, e(t)) = τ 2 e˙ T (t)K T U4 K e(t)  −τ

(12) (13)

t−τ (t)

t−τ (t)

t−τ

e˙ T (s)K T U4 K e(s)ds ˙

 τ  T e (t)K T − eT (t − τ (t))K T U4 τ (t)     τ eT (t − τ (t))K T − eT (t − τ )K T U4 × Ke(t) − Ke(t − τ (t)) − τ − τ (t)   × Ke(t − τ (t)) − Ke(t − τ ) , (14) ⎡ ⎤T ⎢ Ke(t) − Ke(t − τ (t)) ⎥ ˙ −⎣ ≤ τ 2 e˙ T (t)K T U4 K e(t) ⎦ Ke(t − τ (t)) − Ke(t − τ ) ⎡ ⎤ ⎡ ⎤  ⎢ U4 W1 ⎥ ⎢ Ke(t) − Ke(t − τ (t)) ⎥ ×⎣ (15) ⎦ ⎣ ⎦,  T U4 W Ke(t − τ (t)) − Ke(t − τ ) 1 ˙ − ≤ τ 2 e˙ T (t)K T U4 K e(t)

12 where the inequality in (14) has been derived from Lemma 2.2. Further the inequality (15) can be reduced from Lemma 2.1 as ⎡ ⎢ −⎣

where α =

τ (t) τ ,

β=

 −



β α [Ke(t)

− Ke(t − τ (t))]

α β [Ke(t

⎤T ⎡ ⎥ ⎦

⎢ U4 ⎣ T W

⎤  W1 ⎥ ⎦ U4 ⎤

− τ (t)) − Ke(t − τ )] 1  β α [Ke(t) − Ke(t − τ (t))] ⎥ ⎢ ×⎣  ⎦ ≤ 0, − αβ [Ke(t − τ (t)) − Ke(t − τ )] ⎡

τ −τ (t) . τ

  Note that when τ (t) = τ or τ = 0, one can obtain eT (t − τ (t))K T − eT (t − τ )K T = 0 or   eT (t)K T − eT (t − τ (t))K T = 0, respectively. So, the relation (15) still holds. From Assumption 1, for any positive scalars λ1 , λ2 , the following inequalities hold, ⎡ ⎢ −λ1 ⎣

⎤ ⎡ ⎤   ⎢ F 1 −F 2 ⎥ ⎢ Ke(t) ⎥ ⎦ ⎣ ⎦ ≥ 0, ⎣ ∗ 2I f (Ke(t)) ⎤ ⎡ ⎤  −F 2 ⎥ ⎢ Ke(t − τ (t)) ⎥ ⎦ ⎣ ⎦ ≥ 0. 2I f (Ke(t − τ (t)))

⎤T ⎡ Ke(t)

f (Ke(t)) ⎡ ⎤T ⎡ ⎢ Ke(t − τ (t)) ⎥ ⎢ −λ2 ⎣ ⎦ ⎣ f (Ke(t − τ (t)))

⎥ ⎦  F1 ∗

(16)

(17)

Considering (10)-(13) and (15)-(17), we obtain ⎡ ⎢ V˙ (t, e(t)) ≤ V˙ (t, e(t)) − λ1 ⎣

⎤ ⎡ ⎤   ⎢ F 1 −F 2 ⎥ ⎢ Ke(t) ⎥ ⎦ ⎣ ⎦ ⎣ ∗ 2I f (Ke(t)) ⎤ ⎡ ⎤   F 1 −F 2 ⎥ ⎢ Ke(t − τ (t)) ⎥ ⎦ ⎣ ⎦ ∗ 2I f (Ke(t − τ (t)))

⎤T ⎡ Ke(t)

⎥ ⎦

f (Ke(t)) ⎡ ⎤T ⎡ ⎢ Ke(t − τ (t)) ⎥ ⎢ −λ2 ⎣ ⎦ ⎣ f (Ke(t − τ (t))) ⎡ ⎤ ⎢ Φ Ω ⎥ ≤ ξ T (t) ⎣ ⎦ ξ(t), ∗ Ω66

(18)

13 where ¯ 1, ¯ T K T U4 K L ¯ T (t)], Ω66 = τ 2 L ξ T (t) = [η T (t) ω 1 η T (t) = [eT (t) eT (t − τ (t))K T eT (t − τ )K T f T (Ke(t)) f T (Ke(t − τ (t)))]. Now, we prove that for the case of ω ¯ (t) = 0, the filtering error system (7) is asymptotically stable. If ω ¯ (t) = 0, from (18), we obtain V˙ (t, e(t)) ≤ η T (t)Φη(t).

(19)

Hence, one can conclude from (9) that Φ < 0, which implies V˙ (t, e(t)) < 0. Thus, the filtering error system (7) with ω ¯ (t) = 0 is asymptotically stable. ¯ (t), we define For H∞ performance analysis for filtering error system (7) with non-zero ω the following performance index  JT

=

0

T

 1 z¯T (t)¯ z (t) − γ 2 ω ¯ T (t)¯ ω (t) dt. N

(20)

The performance index (20) can be rewritten as  JT

=

0

 ≤

0

 ≤

T

T

T

 T  1 T 2 T z¯ (t)¯ z (t) − γ ω ¯ (t)¯ ω (t) + V˙ (t, e(t)) dt − V˙ (t, e(t))dt N 0   ξ T (t)Θξ(t) dt − V (T, e(T )) 

0

⎤

⎡ ⎢ Φ+ where Θ = ⎣

 ξ T (t)Θξ(t) dt

ΥΥT ∗

Ω ⎥ ⎦. ¯ 66 Ω

By using Schur complement [45], (9) is equivalent to Θ < 0 and consequently, if (8)-(9) hold,

14 then JT < 0, ∀T > 0. That is 

∞

0

which means that

1 N

 1 T z (t) − γ 2 ω ¯ T (t)¯ ω (t) dt < 0, z¯ (t)¯ N

z¯(t) 2 ≤ γ ω ¯ (t) 2 for all nonzero ω ¯ (t). Therefore, the filtering error

system (7) is asymptotically stable with the H∞ performance γ. This completes the proof. Remark 3 Lemma 2.1 and Lemma 2.2 are applied to the corresponding terms in the Lyapunov-Krasovskii functional V4 (t, e(t)) to achieve less conservative results with fewer decision variables. In addition, a more generalized sector-like condition is assumed to well describe the nonlinear functions in the network. The following theorem is derived to design the parameters of the desired filters defined in (4) by using the sufficient conditions established in Theorem 3.1. Theorem 3.2 Given scalars τ, μ and positive scalars λ1 , λ2 , the filtering error system (7) is asymptotically stable with an H∞ performance γ, if there exist positive definite matrices ¯F i (i = 1, 2 . . . , N ), and any matrix W1 with appropriate Pl (l = 1, 2 . . . , 5), matrices A¯F i , B dimension such that the following LMIs hold ⎤

⎡

I ⊗ W1 ⎥ ⎦ ≥ 0, (I ⊗ W1 )T I ⊗ P5 ⎤ 1 ¯ (1) (2) ¯ ¯ ¯ Ω Φ Ω NΥ ⎥ ⎥ ¯ (3) 0 0 ⎥ ∗ Ω ⎥ ⎥ < 0, ⎥ 0 ⎥ ∗ ∗ −γ 2 I ⎦ ∗ ∗ ∗ −I

⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

I ⊗ P5

(21)

(22)

15 where ⎤ ¯ ¯ ¯ ¯ Φ13 Φ14 Φ15 Φ16 ⎥ ⎥ 0 0 0 0 ⎥ ⎥ ⎥ ⎥ ¯ ¯ ¯ ¯ Φ33 Φ34 Φ35 Φ36 ⎥ ⎥, ⎥ ¯ ∗ Φ44 0 0 ⎥ ⎥ ⎥ ¯ 55 Φ ¯ 56 ⎥ ∗ ∗ Φ ⎥ ⎦ ¯ 66 ∗ ∗ ∗ Φ T  T T T T T (2) (1) (1) (1) (2) ¯ = 0 Ω ¯ ¯ ¯ ¯ ,Ω , Ω 0 Ω 0 0 0 0 Ω 13 15 16 12  T ¯ (3) = τ 2 D1T (I ⊗ P5 )D1 − γ 2 I, ¯ = ,Ω Υ I ⊗ E −(I ⊗ E) 0 0 0 0 ⎡

¯ ¯ ⎢ Φ11 Φ12 ⎢ ⎢ ∗ Φ ¯ 22 ⎢ ⎢ ⎢ ⎢ ∗ ∗ ¯ Φ = ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎣ ∗ ∗  T ¯ (1) = ¯ (1) Ω 0 Ω 11

with ¯ 11 = (I ⊗ P1 A) + (I ⊗ P1 A)T + (W ⊗ P1 Γ1 ) + (W ⊗ P1 Γ1 )T + τ 2 (I ⊗ AT P5 A) Φ +τ 2 (W ⊗ AT P5 Γ1 ) + τ 2 (W ⊗ AT P5 Γ1 )T + τ 2 (W T W ⊗ ΓT1 P5 Γ1 ) ¯ 12 = C T B ¯FT , −λ1 (I ⊗ F1 ) + (I ⊗ P3 ) + (I ⊗ P4 ) − (I ⊗ P5 ), Φ ¯ 13 = (G ⊗ P1 Γ2 ) + τ 2 (G ⊗ AT P5 Γ2 ) + τ 2 (W T G ⊗ ΓT P5 Γ2 ) + (I ⊗ P5 ) − (I ⊗ W1 ), Φ 1 ¯ 15 = (I ⊗ P1 B1 ) + τ 2 (I ⊗ AT P5 B1 ) + τ 2 (W T ⊗ ΓT1 P5 B1 ) + λ1 (I ⊗ F2 ), ¯ 14 = I ⊗ W1 , Φ Φ ¯ 22 = A¯F + A¯TF , ¯ 16 = (I ⊗ P1 B2 ) + τ 2 (I ⊗ AT P5 B2 ) + τ 2 (W T ⊗ ΓT1 P5 B2 ), Φ Φ ¯ 33 = −(1 − μ)(I ⊗ P2 ) + τ 2 (GT G ⊗ ΓT P5 Γ2 ) − 2(I ⊗ P5 ) + (I ⊗ W1 )T + (I ⊗ W1 ) Φ 2 −λ2 (I ⊗ F1 ), ¯ 35 = τ 2 (GT ⊗ ΓT P5 B1 ), ¯ 34 = (I ⊗ P5 ) − (I ⊗ W1 ), Φ Φ 2 ¯ 44 = −(I ⊗ P4 ) − (I ⊗ P5 ), ¯ 36 = τ 2 (GT ⊗ ΓT2 P5 B2 ) + λ2 (I ⊗ F2 ), Φ Φ ¯ 56 = τ 2 (I ⊗ B1T P5 B2 ), Φ ¯ 66 = τ 2 (I ⊗ B2T P5 B2 ) − 2λ2 I, ¯ 55 = τ 2 (I ⊗ B1T P5 B1 ) − 2λ1 I, Φ Φ ¯ (1) = τ 2 (GT ⊗ ΓT P5 )D1 , ¯ (1) = (I ⊗ P1 )D1 + τ 2 (I ⊗ AT P5 )D1 + τ 2 (W T ⊗ ΓT P5 )D1 , Ω Ω 1 2 11 13 ¯ (1) = τ 2 (I ⊗ B2T P5 )D1 , Ω ¯ (2) = B ¯F D2 . ¯ (1) = τ 2 (I ⊗ B1T P5 )D1 , Ω Ω 15 16 12

16 Moreover, the parameters of the desired filters are given as AF = (I ⊗ P2 )−1 A¯F ,

¯F , BF = (I ⊗ P2 )−1 B

(23)

¯F = diag{B ¯F 1 , B ¯F 2 , . . . , B ¯F N }. where A¯F = diag{A¯F 1 , A¯F 2 , . . . , A¯F N }, B 

Proof Substituting the values of Ui s(i = 1, 2 . . . 5), (7) into (8) and (9), we obtain (21) and ¯F . Hence, if (22), in which the parameters are defined as (I ⊗ P2 )AF = A¯F , (I ⊗ P2 )BF = B (21)-(22) hold, the filter gain matrices are given by (23). This completes the proof. Remark 4 To develop less conservative delay-dependent stability results, there exist various advanced methods for time-delay systems. For example, the input-output method has been employed in [38] to tackle the time-delays in the sensor networks. In [40] slack matrix method provides less conservative results due to the increased freedom of slack variables. Further, to estimate the upper bound on the derivative of Lyapunov-Krasovskii functional, free-weighting matrix method has been employed in [41, 46, 47] to circumvent the utilization of bounding inequalities for cross products terms. Recently, greater efforts have been made to further reduce the conservatism issue on time-delay systems. In this aspect, the reciprocal convex combination approach has been widely used in the literature [35, 39] to directly relax the integral term of quadratic quantities into the quadratic term of integral quantities together with Jensen’s inequality. The reciprocal convex combination lemma provides not only less conservative results but also it includes only less number of decision variables when compared to free-weighting matrix method. We extend the proposed main results by using free-weighting matrix method as employed in [41] to compare the conservativeness of reciprocal convex approach with free-weighting matrix method. Based on this, the corresponding H∞ performance analysis result for the filtering error system (7) has been obtained through the following corollary. Corollary 3.1 Given scalars τ, μ and positive scalars λ1 , λ2 , the filtering error system (7) is asymptotically stable with an H∞ performance γ, if there exist positive definite matrices ¯F i (i = 1, 2 . . . , N ) with Pl (l = 1, 2 . . . , 5) and matrices Mm , Nm , (m = 1, 2, . . . , 5), A¯F i , B

17 appropriate dimensions such that the following LMI holds ⎡

¯ (2) ¯T ¯T ¯ ¯ (1) Ω M N ⎢ Φ Ω ⎢ ⎢ ∗ Ω ¯ (3) 0 0 0 ⎢ ⎢ ⎢ ∗ −γ 2 I ⎢ ∗ 0 0 ⎢ ⎢ ⎢ ∗ 0 ∗ ∗ −(I ⊗ P5 ) ⎢ ⎢ ⎢ ∗ ∗ ∗ ∗ −(I ⊗ P5 ) ⎢ ⎣ ∗ ∗ ∗ ∗ ∗

1 ¯ NΥ

0 0 0 0 −I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(24)

where ⎡

¯ ¯ ¯ ¯ ¯ ⎢ Φ11 Φ12 Φ13 Φ14 Φ15 ⎢ ⎢ ∗ Φ ¯ 22 0 0 0 ⎢ ⎢ ⎢ ¯ 34 Φ ¯ 35 ¯ 33 Φ ∗ Φ ⎢ ∗ ¯ Φ=⎢ ⎢ ¯ 44 Φ ¯ 45 ⎢ ∗ ∗ ∗ Φ ⎢ ⎢ ⎢ ∗ ¯ 55 ∗ ∗ ∗ Φ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗  T T T ¯ (1) = Ω ¯ (1) 0 Ω ¯ (1) ¯ (1) 0 Ω Ω 11 13 15  T ¯ M = (I ⊗ M1 )T 0 (I ⊗ M2 )T ¯T = N



⎤ ¯ 16 Φ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ¯ 36 ⎥ Φ ⎥ ⎥, ⎥ ¯ Φ46 ⎥ ⎥ ⎥ ¯ 56 ⎥ Φ ⎥ ⎦ ¯ 66 Φ  T T T (2) (1) ¯ ¯ (2) ¯ ,Ω = 0 Ω Ω 16 12 (I ⊗ M3 )T

(I ⊗ M4 )T

T

0 0 0 0 T , (I ⊗ M5 )T

,

T (I ⊗ N1 )T

0 (I ⊗ N2 )T

, (I ⊗ N3 )T (I ⊗ N4 )T (I ⊗ N5 )T  T ¯ (3) = τ D1T (I ⊗ P5 )D1 − γ 2 I ¯ = I ⊗ E −(I ⊗ E) 0 0 0 0 ,Ω Υ

with ¯ 11 = (I ⊗ P1 A) + (I ⊗ P1 A)T + (W ⊗ P1 Γ1 ) + (W ⊗ P1 Γ1 )T + τ (I ⊗ AT P5 A) Φ + τ (W ⊗ AT P5 Γ1 ) + τ (W ⊗ AT P5 Γ1 )T + τ (W T W ⊗ ΓT1 P5 Γ1 ) − λ1 (I ⊗ F1 ) + (I ⊗ P3 ) ¯ 12 = C T B ¯ 13 = (G ⊗ P1 Γ2 ) + τ (G ⊗ AT P5 Γ2 ) ¯FT , Φ + (I ⊗ M1 ) + (I ⊗ M1T ), Φ

18 ¯ 14 = (I ⊗ M T ) − (I ⊗ N1 ), + τ (W T G ⊗ ΓT1 P5 Γ2 ) + (I ⊗ M2T ) − (I ⊗ M1 ) + (I ⊗ N1 ), Φ 3 ¯ 15 = (I ⊗ P1 B1 ) + τ (I ⊗ AT P5 B1 ) + τ (W T ⊗ ΓT1 P5 B1 ) + λ1 (I ⊗ F2 ) + (I ⊗ M4T ), Φ ¯ 22 = A¯F + A¯TF , ¯ 16 = (I ⊗ P1 B2 ) + τ (I ⊗ AT P5 B2 ) + τ (W T ⊗ ΓT1 P5 B2 ) + (I ⊗ M5T ), Φ Φ ¯ 33 = −(1 − μ)(I ⊗ P2 ) + τ (GT G ⊗ ΓT P5 Γ2 ) − (I ⊗ M2 ) − (I ⊗ M T ) + (I ⊗ N2 ) + (I ⊗ N T ) Φ 2 2 2 ¯ 34 = −(I ⊗ M3T ) + (I ⊗ N3T ) − (I ⊗ N2 ), Φ ¯ 35 = τ (GT ⊗ ΓT2 P5 B1 ) − (I ⊗ M4T ) − λ2 (I ⊗ F1 ), Φ ¯ 36 = τ (GT ⊗ ΓT P5 B2 ) + λ2 (I ⊗ F2 ) − (I ⊗ M T ) + (I ⊗ N T ), + (I ⊗ N4T ), Φ 2 5 5 ¯ 45 = −(I ⊗ N4T ) Φ ¯ 46 = −(I ⊗ N5T ), ¯ 44 = −(I ⊗ N3 ) − (I ⊗ N3T ), Φ Φ ¯ 56 = τ (I ⊗ B1T P5 B2 ), Φ ¯ 66 = τ (I ⊗ B2T P5 B2 ) − 2λ2 I, ¯ 55 = τ (I ⊗ B1T P5 B1 ) − 2λ1 I, Φ Φ ¯ (1) = τ (GT ⊗ ΓT P5 )D1 , ¯ (1) = (I ⊗ P1 )D1 + τ (I ⊗ AT P5 )D1 + τ (W T ⊗ ΓT P5 )D1 , Ω Ω 1 2 11 13 ¯ (1) = τ (I ⊗ B2T P5 )D1 , Ω ¯ (2) = B ¯F D2 . ¯ (1) = τ (I ⊗ B1T P5 )D1 , Ω Ω 15 16 12 The parameters of the desired filters are defined in (23). Proof Choose the Lyapunov-Krasovskii functional for the system (7) as V (t, et , i) = V1 (t, et , i) + V2 (t, et , i) + V¯3 (t, et , i), where V1 (t, et , i), V2 (t, et , i) are defined in (10) and V¯3 (t, et , i) =

0 t −τ

t+θ

e˙ T (s)K T U4 K e(s)dsdθ. ˙

Then, by applying the free-weighting matrix method in the derivative of Lyapunov-Krasovskii functional V (t, et , i) and following the similar way of method for proving Theorem 3.1 and Theorem 3.2, one can obtain the LMI (24) in Corollary 3.1. Thus, the filtering error system (7) is asymptotically stable withH∞ performance γ. This completes the proof. 4. Numerical Examples To illustrate the validity of the proposed H∞ filtering design, three numerical examples with simulation results are provided for the delayed CDNs (1). Example 4.1 Consider the delayed CDNs system (1) with N = 3 and its parameters are

19 defined as ⎡

⎤

⎡

⎤

⎡

⎤

⎡

⎤

0 ⎥ ⎢ 0.5 0.8 ⎥ ⎢ 0.4 0.1 ⎥ ⎢ 1 0 ⎥ ⎢ −0.2 A = ⎣ ⎦ , B2 = ⎣ ⎦ , Γ1 = ⎣ ⎦, ⎦ , B1 = ⎣ 0.6 0.5 0.1 0.1 0 1 0 −0.4 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ 0.8 0 ⎥ ⎢ 0.3 0.2 ⎥ ⎢ 0.2 0.1 ⎥ ⎢ 0.4 0.3 ⎥ Γ2 = ⎣ ⎦ , D11 = ⎣ ⎦ , D12 = ⎣ ⎦ , D13 = ⎣ ⎦, 0 0.8 0.1 0.1 0.5 0.2 0.1 0.1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ 0.6 0.2 ⎥ ⎢ 0.9 0.4 ⎥ ⎢ 0.3 0.4 ⎥ ⎢ 0.4 0.5 ⎥ C1 = ⎣ ⎦ , C2 = ⎣ ⎦ , C3 = ⎣ ⎦ , D21 = ⎣ ⎦, 0.6 0.1 0.3 0.8 0.6 0.3 0.6 0.3 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ 0.3 0.5 ⎥ ⎢ 0.6 0.4 ⎥ ⎢ 0.1 0.1 ⎥ D22 = ⎣ ⎦ , D23 = ⎣ ⎦, E = ⎣ ⎦. 0.1 0.3 0.3 0.2 0.3 0.1 The coupling matrices and the nonlinear function f (·) are given by ⎡

⎤

⎤

⎡

0.3 ⎥ ⎢ −0.6 0.3 ⎢ −0.2 0.2 ⎢ ⎥ ⎢ ⎥ , G = ⎢ 0.1 −0.2 W = ⎢ 0.3 −0.6 0.3 ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ 0.1 0.1 0.3 0.3 −0.6 ⎡ 2 ⎢ 0.1xi1 (t)cos (xi1 (t)) − 0.05(xi1 (t) − xi2 (t)) f (xi (t)) = ⎣ −0.05xi1 (t) − 0.1xi2 (t)

0

⎥ ⎥ 0.1 ⎥ ⎥, ⎦ −0.2 ⎤ ⎥ ⎦.

According to (3), the relevant parameter matrices can be chosen as ⎡

⎤

⎡

⎤

⎢ 0.05 0.05 ⎥ ⎢ −0.05 0.05 ⎥ F1 = ⎣ ⎦ , F2 = ⎣ ⎦. 0.05 0.05 −0.15 −0.25 For given γ = 0.6, τ = 0.7 and μ = 0.35, solving the LMIs in Theorem 3.2 using Matlab LMI toolbox, the desired filter matrices and the positive scalars λ1 , λ2 are obtained as ⎡

⎤

⎡

⎤

⎢ −0.5036 −0.2360 ⎥ ⎢ −0.5035 −0.0457 ⎥ AF 1 = ⎣ ⎦ , AF 2 = ⎣ ⎦, 0.2327 −0.5009 0.0425 −0.5008

20 ⎡

⎤

⎡

⎤

⎢ −0.5035 −0.2065 ⎥ ⎢ 0.0068 0.0030 ⎥ AF 3 = ⎣ ⎦ , BF 1 = ⎣ ⎦, 0.2032 −0.5009 0.0059 0.0029 ⎡ ⎤ ⎡ ⎤ ⎢ 0.0057 0.0023 ⎥ ⎢ 0.0046 0.0033 ⎥ BF 2 = ⎣ ⎦ , BF 3 = ⎣ ⎦, 0.0027 0.0018 0.0083 0.0031 λ1 = 43.6673, λ2 = 13.1806. ˆ1 (0) = [0.2 −0.4]T , x2 (0) = [−0.2 0.4]T , By choosing the initial conditions x1 (0) = [−0.4 0.4]T , x ˆ3 (0) = [0.5 − 0.7]T and the disturbance inputs x ˆ2 (0) = [0.6 − 0.5]T , x3 (0) = [−0.5 0.4]T , x νi (t) = sin(t)e−2t , ωi (t) = sin(t)e−4t , the simulation results for Example 4.1 are shown in ˆi (t) and Figure 2 Figures 1 and 2. Figure 1 shows the state trajectories xi (t), its estimates x shows the estimation errors ei (t) for (i = 1, 2, 3). Further, for different values of γ and μ, the maximum allowable upper bound (MAUB) values of τ are presented in Table 1 and Table 3, respectively. Example 4.2 Consider the delayed CDNs system (1) with N = 5 and its parameters are defined as ⎡

⎤

⎢ 0.2 D14 = ⎣ 0.7 ⎡ ⎢ 0.4 C5 = ⎣ 0.5

⎡

⎤

0.3 ⎥ ⎢ 0.6 0.2 ⎦ , D15 = ⎣ 0.3 0.4 0.1 ⎤ ⎡ 0.2 ⎥ ⎢ 0.3 0.2 ⎦ , D24 = ⎣ 0.3 0.4 0.5

⎡

⎤

⎥ ⎢ 0.5 ⎦ , C4 = ⎣ 0.6 ⎤ ⎡ ⎥ ⎢ 0.6 ⎦ , D25 = ⎣ 0.3

0.3 ⎥ ⎦, 0.1 ⎤ 0.4 ⎥ ⎦. 0.2

The coupling matrices are given by ⎡

W

−0.8

⎢ ⎢ ⎢ 0.3 ⎢ ⎢ = ⎢ ⎢ 0.3 ⎢ ⎢ ⎢ 0.1 ⎣ 0.1

0.3

0.3

0.1

0.1

⎤

⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −0.8 0.3 0.1 0.1 ⎥ ⎢ ⎥ ⎢ ⎥ 0.1 −0.8 0.3 0.1 ⎥ , G = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ 0.3 0.1 −0.8 0.3 ⎥ ⎦ ⎣ 0.3 0.1 0.3 −0.8

−0.5 0.2 0.2 0.2 0.2

0.2

0.1

0.1

0.1

⎤

⎥ ⎥ −0.5 0.1 0.1 0.1 ⎥ ⎥ ⎥ 0.1 −0.5 0.1 0.1 ⎥ ⎥, ⎥ ⎥ 0.1 0.1 −0.5 0.1 ⎥ ⎦ 0.1 0.1 0.1 −0.5

21 and the rest of the parameters are same as defined in Example 4.1. For given γ = 0.7, τ = 0.4 and μ = 0.2, solving the LMIs in Theorem 3.2 using Matlab LMI toolbox, the desired filter matrices and the positive scalars are obtained as ⎡ ⎢ −0.5010 −0.0634 AF 1 = ⎣ 0.0626 −0.5002 ⎡ ⎢ −0.5010 −0.0470 AF 3 = ⎣ 0.0462 −0.5002 ⎡ ⎢ −0.5010 −0.0833 AF 5 = ⎣ 0.0825 −0.5002 ⎡ ⎤ ⎢ 0.0015 0.0004 ⎥ BF 2 = ⎣ ⎦, 0.0008 0.0006 ⎡ ⎤ ⎢ 0.0019 0.0009 ⎥ BF 4 = ⎣ ⎦, 0.0018 0.0007

⎤

⎡

⎥ ⎢ −0.5010 −0.0346 ⎦ , AF 2 = ⎣ 0.0338 −0.5002 ⎤ ⎡ ⎥ ⎢ −0.5010 −0.1118 ⎦ , AF 4 = ⎣ 0.1110 −0.5002 ⎤ ⎡ ⎤ ⎥ ⎢ 0.0018 0.0007 ⎥ ⎦ , BF 1 = ⎣ ⎦, 0.0016 0.0007 ⎡ ⎤ ⎢ 0.0012 0.0012 ⎥ BF 3 = ⎣ ⎦, 0.0023 0.0005 ⎡ ⎤ ⎢ 0.0033 −0.0025 ⎥ BF 5 = ⎣ ⎦, 0.0010 0.0034

⎤ ⎥ ⎦, ⎤ ⎥ ⎦,

λ1 = 27.4815, λ2 = 10.5378. ˆ1 (0) = [0.5 −0.1]T , x2 (0) = [−0.8 0.5]T , By choosing the initial conditions x1 (0) = [−0.5 0.1]T x ˆ3 (0) = [0.4 − 0.3]T , x4 (0) = [−0.8 0.5]T , x ˆ4 (0) = x ˆ2 (0) = [0.8 − 0.5]T , x3 (0) = [−0.4 0.3]T x ˆ5 (0) = [0.7 − 0.4]T and the disturbance inputs νi (t) = [0.6 − 0.2]T , x5 (0) = [−0.7 0.4]T , x sin(t)e−6t , ωi (t) = sin(t)e−8t , the simulation results for Example 4.2 are shown in Figures 3 ˆi (t) and Figure 4 shows the and 4. Figure 3 shows the state trajectories xi (t), its estimates x estimation errors ei (t)) for (i = 1, . . . , 5). Further, for different values of γ and μ, the MAUB values of τ are given in Table 2 and Table 4, respectively. Remark 5 It can be analyzed from Table 1 and Table 2 that by increasing the values of γ the MAUB of τ also increases. By comparing the MAUB τ values, it is noticed the MAUB τ values obtained for Example 4.1 are greater than the MAUB τ values obtained for Example 4.2. It shows the effect of time-delays with respect to the performance value γ. In addition,

22 by increasing the nodes, the computational time gets increased and also LMIs are feasible only for smaller time-delay values. Further from Table 3 and Table 4 it can be seen that by increasing the values of μ and for the unknown μ, the MAUB value of τ decreases. It shows the effect of considering the time-varying delays in the networks. Remark 6 From the MAUB values of τ listed in Table 3, one can observe that the maximum value of τ obtained from Theorem 3.2 via reciprocal convex approach is 0.7385 whereas, the maximum value of τ obtained from Corollary 3.1 via free-weighting matrix method is 0.6061. This shows that the proposed delay-dependent stability conditions provide less conservative results. Similarly, from Table 4, it can be analyzed that reciprocal convex approach is less conservative than the free-weighting matrix method. Example 4.3 Many real-world complex networks can be successfully modeled as scale-free networks [48, 49]. In the sequel, we consider a scale-free network consist of five dynamical nodes and the coupling topology are randomly generated by a BA scale-free model from Fig.5, where each node is a continuous-time dynamical system (1) and its parameters are defined as ⎡

⎤

⎡

⎤

⎡

⎤

⎡

⎤

⎢ 0.25 0.75 ⎥ ⎢ 0.14 0.15 ⎥ ⎢ 0.12 0.12 ⎥ ⎢ −0.9 0.26 ⎥ A = ⎣ ⎦ , B2 = ⎣ ⎦ , C1 = ⎣ ⎦, ⎦ , B1 = ⎣ 0.35 0.25 0.45 0.23 0.16 0.11 0.31 −0.8 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ 0.19 0.14 ⎥ ⎢ 0.13 0.14 ⎥ ⎢ 0.55 0.3 ⎥ ⎢ 0.14 0.2 ⎥ C2 = ⎣ ⎦ , C3 = ⎣ ⎦ , C4 = ⎣ ⎦ , C5 = ⎣ ⎦, 0.13 0.18 0.16 0.13 0.6 0.1 0.25 0.43 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 ⎥ ⎢ 0.03 ⎢ 0.13 0.2 ⎥ ⎢ 0.2 0.1 ⎥ Γ1 = Γ2 = ⎣ ⎦ , D11 = ⎣ ⎦ , D12 = ⎣ ⎦, 0 0.03 0.11 0.1 0.25 0.2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ 0.4 0.35 ⎥ ⎢ 0.25 0.3 ⎥ ⎢ 0.6 0.25 ⎥ ⎢ 0.4 0.55 ⎥ D13 = ⎣ ⎦ , D14 = ⎣ ⎦ , D15 = ⎣ ⎦ , D21 = ⎣ ⎦, 0.1 0.1 0.27 0.3 0.45 0.1 0.6 0.3 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.35 0.5 0.6 0.4 0.35 0.2 0.6 0.45 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ D22 = ⎣ ⎦ , D23 = ⎣ ⎦ , D24 = ⎣ ⎦ , D25 = ⎣ ⎦. 0.1 0.35 0.13 0.25 0.4 0.45 0.3 0.25

23 The coupling matrices are given by ⎡

−2

1

⎢ ⎢ ⎢ 1 ⎢ ⎢ W = G = ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣ 1

0

0

1

⎤

⎥ ⎥ −3 1 0 1 ⎥ ⎥ ⎥ 1 −2 1 0 ⎥ ⎥, ⎥ ⎥ 0 1 −2 1 ⎥ ⎦ 1 0 1 −3

and the nonlinear function are assumed to be same as defined in Example 4.1. For given γ = 0.7, τ = 0.4 and μ = 0.6, solving the LMIs in Theorem 3.2 using Matlab LMI toolbox, the desired filter matrices and the positive scalars are obtained as ⎡

⎤

⎡

⎤

⎢ −0.5080 3.3823 ⎥ ⎢ −0.5080 2.5119 ⎥ AF 1 = ⎣ ⎦ , AF 2 = ⎣ ⎦, −3.3740 −0.5027 −2.5036 −0.5027 ⎡ ⎤ ⎡ ⎤ ⎢ −0.5082 3.4571 ⎥ ⎢ −0.5080 1.2384 ⎥ AF 3 = ⎣ ⎦ , AF 4 = ⎣ ⎦, −3.4490 −0.5028 −1.2301 −0.5027 ⎡ ⎤ ⎡ ⎤ ⎢ −0.5080 5.1832 ⎥ ⎢ 0.0084 −0.0045 ⎥ AF 5 = ⎣ ⎦ , BF 1 = ⎣ ⎦, −5.1750 −0.5028 0.0116 −0.0074 ⎡ ⎤ ⎡ ⎤ ⎢ 0.0134 −0.0140 ⎥ ⎢ 0.0095 −0.0043 ⎥ BF 2 = ⎣ ⎦ , BF 3 = ⎣ ⎦, 0.0009 0.0083 0.0138 −0.0137 ⎡ ⎤ ⎡ ⎤ ⎢ −0.0167 0.0121 ⎥ ⎢ 0.0972 −0.1780 ⎥ BF 4 = ⎣ ⎦ , BF 5 = ⎣ ⎦, 0.0297 −0.0180 −0.0362 0.0842 λ1 = 8.5372, λ2 = 4.2713. ˆ1 (0) = [0.5 −0.5]T , x2 (0) = [−0.2 0.5]T , By choosing the initial conditions x1 (0) = [−0.5 0.1]T x ˆ3 (0) = [0.4 − 0.6]T , x4 (0) = [−0.9 0.5]T , x ˆ4 (0) = x ˆ2 (0) = [0.8 − 0.2]T , x3 (0) = [−0.6 0.3]T x ˆ5 (0) = [0.7 − 1.1]T and the disturbance inputs as νi (t) = [0.6 − 0.9]T , x5 (0) = [−1.1 0.4]T , x sin(t)e−2t , ωi (t) = sin(t)e−6t , the simulation results for Example 4.3 are shown in Figures 6

24 ˆi (t) and Figure 7 shows the and 7. Figure 6 shows the state trajectories xi (t), its estimates x estimation errors ei (t) for (i = 1, . . . , 5). Remark 7 The numerical complexity of the derived results is closely related to number of decision variables and the dimension of the LMIs involved. It can be calculated that the number of variables involved in Theorem 3.2 via reciprocal convex approach is 12 [5n(n + 1)] + n2 (1 + 2N ) + 2, where n represents the size of the system matrices and N represents the number of nodes. In order to compare the computational complexity with other existing methods, the number of variables involved via free-weighting matrix method is calculated as 1 2 [4n(n

+ 1)] + n2 (10 + 2N ) + 2, which leads to increase the number of decision variables.

Remark 8 To illustrate the advantage of employing reciprocal convex approach with respect to computation complexity, Table 5 lists the number of decision variables involved for various values of n and N . It can be observed that for n = 2, N = 3, the number of variables by reciprocal convex approach is 45 and by free-weighting matrix method it is 78, which increases the computation time while increasing the number of decision variables. Also, it can be analyzed that when the size of n gets increases, the number of variables is drastically increasing via free-weighting matrix method and the computation time becomes heavier due to the increase in the dimension of the LMIs. Hence, reciprocal convex combination approach provides less conservative result with much less number of decision variables. 5. Conclusions In this paper, a H∞ filtering design is proposed for a class of continuous-time CDNs. The system under consideration includes time-varying delays in the coupling nodes and in the nonlinear function. The novel asymptotic stability conditions have been derived for the filtering error system which satisfies the minimum H∞ performance criteria by utilizing Lyapunov-Krasovskii functional, reciprocal convex combination lemma and the Kronecker product techniques. Then, the desired filter has been established. Finally, numerical examples have been demonstrated to show the effectiveness of the proposed theoretical results. It is well known that in actual systems, complex networks sometimes has finite modes switching

25 from one to another with certain transition rate governed by Markov processes. Hence, in future we will concentrate to extend the proposed main results for filtering analysis of CDNs with Markovian jump parameters and stochastic packet dropouts to develop new H∞ filter design for CDNs. Acknowledgements The authors would like to express their sincere gratitude to the Editor-in-Chief, Associate Editor and Anonymous Reviewers for their valuable and encouraging comments and fruitful suggestions to improve the quality of the manuscript. This work was supported by UGC-BSR - Research fellowship in Mathematical Sciences-2012-13, Govt. of India, New Delhi and HIR project No. UM.C/625/1/HIR/MOHE/13, University of Malaya, Kuala Lumpur, Malaysia. [1] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424 (2006) 175-308. [2] H.T. Zhang, T. Yu, J.P. Sang, X.W. Zou, Dynamic fluctuation model of complex networks with weight scaling behavior and its application to airport networks, Phys. A 393 (2014) 590-599. [3] R. Albert, H. Jeong, A.L. Barabasi, Diameter of the world-wide web, Nature 401 (1999) 130-131. [4] R. Pastor-Satorras, E. Smith, R.V. Sole, Evolving protein interaction networks through gene duplication, J. Theor. Biol. 222 (2003) 199-210. [5] K. Liu, E. Fridman, Networked-based stabilization via discontinuous Lyapunov functionals, Int. J. Robust Nonlinear Control 22 (2012) 420-436. [6] D. Yue, H. Li, Synchronization stability of continuous/discrete complex dynamical networks with interval time-varying delays, Neurocomputing 73 (2010) 809-819.

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27 [17] P. Balasubramaniam, V. Vembarasan, R. Rakkiyappan, Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays, Expert Syst. Appl. 39 (2012) 472-481. [18] N.S. Mkondweni, R. Tzoneva, LabNS2-Co-simulation, Co-emulation, and real-time control toolkit for investigation of networked induced time delays and packet loss in networked control systems, Int. J. Innov. Comput.,Inf. Control 10 (2014) 1-18. [19] L. Zhang, Y. Wang, Q. Wang, S. Zhang, Synchronization for time-delayed coupling complex dynamic networks with different dimensional nodes via decentralized dynamic compensation controllers, Asian J. Control 17 (2015) 1-11. [20] R. Rakkiyappan, N. Sakthivel, S. Lakshmanan, Exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and modedependent probabilistic time-varying delays, Chin. Phys. B. 23 (2014) 020205. [21] X. Liu, H. Xi, Synchronization of neutral complex dynamical networks with Markovian switching based on sampled-data controller, Neurocomputing 139 (2014) 163-179. [22] Z.H. Guan, Z.W. Liu, G. Feng, Y.W. Wang, Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control, IEEE Trans. Circuits Syst. I Reg Papers 57 (2010) 2182-2195. [23] K. Mathiyalagan, J.H. Park, R. Sakthivel, S. Marshal Anthoni, Robust mixed H∞ and passive filtering for networked Markov jump systems with impulses, Signal Process. 101 (2014) 162-173. [24] C. Xie, Y. Xu, D. Tong, Synchronization of time-varying delayed complex networks via impulsive control, Optik-Int. J. Light Electron Opt.125 (2014) 3781-3787. [25] J. Zhou, Q. Wu, L. Xiang, Pinning complex delayed dynamical networks by a single impulsive controller, IEEE Trans. Circuits Syst. I Reg Papers 58 (2011) 2882-2893.

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30 [47] R. Sakthivel, R. Raja, S. Marshal Anthoni, Linear matrix inequality approach to stochastic stability of uncertain delayed BAM neural networks, IMA J. Appl. Math. 78 (2013) 1156-1178. [48] L. Jarina Banu, P. Balasubramaniam, Synchronisation of discrete-time complex networks with randomly occurring uncertainties, nonlinearities and time-delays, Int. J. Syst. Sci. 45 (2014) 1427-1450. [49] A.L. Barabasi, R. Albert, H. Jeong, Mean-Field Theory for Scale-Free Random Networks, Physica 272 (1999) 173-187. Table 1: MAUB τ for different values of γ for Example 4.1.

γ

0.6

0.7

0.8

0.9

1

1.1

τ

0.7385

2.1184

3.0830

3.6093

3.8806

4.0371

0.6 0.4

Amplitude

0.2 0

x1(t) −0.2

xˆ1 (t) x2(t)

−0.4

xˆ2 (t) x3(t)

−0.6 −0.8 0

xˆ3 (t) 5

10

15

20

25

Time (t)

Figure 1: The state trajectories xi (t) and its estimates x ˆi (t) (i = 1, 2, 3) for Example 4.1.

31

0.025 0.02

e1(t)

0.015

e2(t)

0.01

e3(t)

e(t)

0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025 0

5

10

15

20

25

Time (t)

Figure 2: The estimation error responses ei (t) (i = 1, 2, 3) for Example 4.1.

Table 2: MAUB τ for different values of γ for Example 4.2.

γ

0.7

0.8

0.9

1

1.1

1.2

τ

0.4589

0.8692

1.1902

1.4659

1.6740

1.8187

Table 3: MAUB τ for different values of μ with γ=0.6 for Example 4.1.

Methods

0.35

0.5

1

10

100

unknown μ

Theorem 3.1

0.7385

0.7380

0.7380

0.7380

0.7380

0.7380

Corollary 3.1

0.6061

0.5921

0.5921

0.5921

0.5921

0.5921

Table 4: MAUB τ for different values of μ with γ=0.7 for Example 4.2.

Methods

0.2

0.5

1

10

100

unknown μ

Theorem 3.1

0.4589

0.4589

0.4589

0.4589

0.4589

0.4589

Corollary 3.1

0.2085

0.2078

0.2078

0.2078

0.2078

0.2078

32

1

Amplitude

0.5 x1(t) xˆ1(t) x2(t) xˆ2(t) x3(t) xˆ3(t) x4(t) xˆ4(t) x5(t) xˆ5(t)

0

−0.5

−1

−1.5 0

5

10

15

20

25

30

35

Time (t)

Figure 3: The state trajectories xi (t) and its estimates x ˆi (t) (i = 1, . . . , 5) for Example 4.2.

Table 5: Number of variables involved in LMIs for various values of n and N

(n,N) (2,3) (2,5) (3,3)

Reciprocal convex approach 45 61 95

Free-weighting matrix approach 78 94 170

33

0.01 0

e1(t)

−0.01

e2(t) e3(t)

e(t)

−0.02

e4(t) −0.03

e5(t)

−0.04 −0.05 −0.06 0

5

10

15

20

25

30

35

Time (t)

Figure 4: The estimation error responses ei (t) (i = 1, . . . , 5) for Example 4.2.

Figure 5: The structure of BA scale-free complex networks with five nodes

34

1

0.5

Amplitude

0

x1 (t) xˆ1 (t) x2 (t) xˆ2 (t) x3 (t) xˆ3 (t) x4 (t) xˆ4 (t) x5 (t) xˆ5 (t)

−0.5

−1

−1.5 0

5

10

15

20

25

30

35

Time (t)

Figure 6: The state trajectories xi (t) and its estimates x ˆi (t) (i = 1, . . . , 5) for Example 4.3.

1.4

e1 (t)

1.2

e2 (t) e3 (t)

1

e4 (t) e5 (t)

0.8

e(t)

0.6

0.4

0.2

0

−0.2

−0.4 0

5

10

15

20

25

30

Time (t)

Figure 7: The estimation error responses ei (t) (i = 1, . . . , 5) for Example 4.3.

35