Synchronization of discrete-time complex dynamical networks with interval time-varying delays via non-fragile controller with randomly occurring perturbation

Author's Accepted Manuscript

Synchronization of discrete-time complex dynamical networks with interval time-varying delays via non-fragile controller with randomly occurring perturbation M.J. Park, O.M. Kwon, Ju.H. Park, S.M. Lee, E.J. Cha

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S0016-0032(14)00222-1 http://dx.doi.org/10.1016/j.jfranklin.2014.07.020 FI2089

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Cite this article as: M.J. Park, O.M. Kwon, Ju.H. Park, S.M. Lee, E.J. Cha, Synchronization of discrete-time complex dynamical networks with interval timevarying delays via non-fragile controller with randomly occurring perturbation, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2014.07.020 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.

Synchronization of discrete-time complex dynamical networks with interval time-varying delays via non-fragile controller with randomly occurring perturbation M.J. Park† , O.M. Kwon†1 , Ju H. Park‡ , S.M. Lee§ , E.J. Cha¶ †

School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Cheongju 361-763, Republic of Korea

‡

Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 280 Daehak-ro, Gyeongsan 712-749, Republic of Korea. Email: [email protected] §

School of Electronic Engineering, Daegu University, Gyeongsan 712-714, Republic of Korea

¶

Department of Biomedical Engineering, School of Medicine, Chungbuk National University, 52 Naesudong-ro, Cheongju 361-763, Republic of Korea

Abstract This paper addresses synchronization problem for discrete-time complex dynamical networks with interval time-varying delays. In order to achieve the synchronization, a feedback controller subjected to randomly occurring perturbations will be considered. The randomly occurring perturbations are assumed to belong to the Binomial sequence. By constructing a suitable Lyapunov-Krasovskii functional, and utilizing reciprocally convex approach and Finsler’s lemma, the synchronization criteria for the networks are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. The networks are represented by the use of Kronecker product technique. The effectiveness of the proposed methods will be verified via numerical examples.

Keywords: Synchronization; discrete-time complex dynamical networks; time-delay; non-fragile controller.

1

Corresponding author. Email address: [email protected]; Tel.:+82-43-261-2422; Fax:+82-43-263-2419.

1

1

Introduction

Complex dynamic networks (CDNs) are a set of interconnected nodes with specific dynamics. During a few decades, CDNs have received considerable attentions due to their extensive applications in many fields such as the Internet, the World Wide Web, social networks, electrical power grids, global economic markets, and so on. Also, many models were proposed to describe various complex networks, small-world network and scale-free network, etc [1-4]. Before handling these systems, since modern systems use information between each system in networks, these days, we need to pay keen attention to the four following aspects: • When each system in network is unstable, the control design is required to stabilize the system dynamics of CDNs. During the information exchange between each system in networks, there exists perturbations in controller. Also, in implementation of many practical systems such as aircraft and electric circuits, there exist occasionally stochastic perturbations. Thus, the perturbations have influence on the random occurrence of the controller uncertainties. • It is well known that a time-delay often causes undesirable dynamic behaviors such as performance degradation and instability of various systems. Therefore, the study on various problems for CDNs with time-delay has been widely investigated in [5-13]. • Synchronization between interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems [14-18]. Thus, the problem of synchronization of complex networks with time-delay has been a challenging issue in [5-13]. • Most systems use microprocessor or microcontrollers, which are called digital computer, with the necessary input/output hardware to implement the systems. A little more to say, the fundamental character of the digital computer is that it takes compute answers at discrete steps [19]. Therefore, discrete-time modeling for CDNs with time-delay and designing non-fragile controller with randomly occurring perturbation for the concerned CDNs play important roles in many fields of science and engineering applications. Unfortunately, to the best of authors’ knowledge, the problem of synchronization for discrete-time CDNs with time-delay and the randomly occurring perturbation in the controller has not been investigated yet. Very recently, in the work 2

of [20], the non-fragile synchronization controller problem for neural networks with time-delay and randomly occurring controller gain fluctuation is studied. But, in existing works [21-23], the non-fragile controllers are designed without randomly occurring perturbations. Motivated by this mentioned above, in this paper, a synchronization problem will be studied for discrete-time CDNs with interval time-varying delays and randomly occurring perturbation in the controller. This information is one of randomly occurring perturbation with Binomial sequence. In addition, to illustrate Binomial sequence, let X be a random variable that takes on only two possible numerical values, X(Ω) = {0, 1}, where Ω represents the universal set consisting the collection of all objects of interest in a particular context [24]. Multiple independent Bernoulli random variables can be combined to construct more sophisticated random variables. Suppose X is the sum of w independent and identically distributed Bernoulli random variables. Then X is called a binomial random variable with parameters w, number of trials, and p, probability of success for each trial. Put simply, the first and simplest random variable is the Bernoulli random variable used in [20]. Thus, the Binomial sequence is a generalization of the Bernoulli sequence. Also, since delay-dependent analysis make use of the information on the size of time delay, delay-dependent analysis has been paid more attention than delay-independent one [25]. That is, the former is generally less conservative than the latter. Therefore, a great number of results on delay-dependent stability condition for time-delay systems have been reported in the literature [26-31]. To drive these conditions, by construction of a suitable augmented Lyapunov-Krasovskii function, which fractionize the delay interval into two subsections, and utilization of reciprocally convex approach [31] with some added decision variables, a synchronization condition with interval time-varying delays without controller is proposed in Theorem 1. And a existence condition of controller gain for synchronization with interval time-varying delays is introduced in Theorem 2. Based on the results of Theorems 1 and 2, a existence condition of controller with randomly occurring perturbations is derived in Theorem 3 with the LMI framework, which can be solved efficiently by use of standard convex optimization algorithms such as interior-point methods [32]. Moreover, the discrete-time CDNs are represented by use of Kronecker product technique. Finally, two numerical examples are included to show the effectiveness of the proposed methods. Notation: The notations used throughout this paper are fairly standard. Rn is the n-dimensional

3

Euclidean space, and Rm×n denotes the set of all m × n real matrices. For real symmetric matrices X and Y , X > Y (resp., X ≥ Y ) means that the matrix X −Y is positive (resp., nonnegative) definite. X ⊥ denotes a basis for the null-space of X. In , 0n and 0m·n denote n × n identity matrix, n × n and m × n zero matrices, respectively. E{·} stands for the mathematical expectation operator. · refers to the Euclidean vector norm or the induced matrix norm. diag{· · ·} denotes the block diagonal matrix. For any vectors xi ∈ Rm (i = 1, 2, . . . , n), col{x1 , x2 , . . . , xn } ∈ Rmn means the column vector; i.e., [xT1 xT2 . . . xTn ]T . ⊗ denotes the notation of Kronecker product.

2

Problem Statements

Consider the following discrete-time CDNs with interval time-varying delays in the coupling term yi (k + 1) = f (yi (k), yi (k − h(k))) + c

N 

gij Γyj (k − h(k)) (i = 1, 2, . . . , N ).

(1)

j=1

Here, N is the number of couple nodes, n is the number of state of each node, yi (k) = [yi1 (k) yi2 (k) . . . yin (k)]T ∈ Rn is the state vector of the ith node. f : Rn → Rn is a vectorvalued function describing the dynamics of an individual node. The constant c > 0 means the coupling strength. The delay h(k) is a time-varying function satisfying 0 ≤ hm ≤ h(k) ≤ hM , where hm and hM are known positive integers. Γ = [γij ]n×n is the inner-coupling matrix of nodes, in which γij = 0 means two coupled nodes are linked through their ith and jth state variables, otherwise γij = 0. G = [gij ]N ×N is the outer-coupling matrix of the network, in which gij is defined as follow: if there is a connection between node i and node j (j = i), then gij = gji = 1; otherwise, gij = gji = 0 (j = i), and the diagonal elements of matrix G are defined by gii = −

N 

gij = −

j=1,i=j

N 

gji (i = 1, 2, . . . , N ).

(2)

j=1,i=j

In order to investigate the synchronization of discrete-time CDNs with interval time-varying delays in the coupling term (1), we introduce the following definition and lemmas. Definition 1 (Li and Chen [5] ). The discrete-time delayed dynamical networks (1) is said to achieve asymptotic synchronization if y1 (k) = y2 (k) = · · · = yN (k) = s(k) as t → ∞, 4

(3)

where s(k) ∈ Rn is a solution of an isolated node, satisfying s(k + 1) = f (s(k), s(k − h(k))). Lemma 1 (Yue and Li [12] ). Consider the network (1). Let 0 = λ1 > λ2 ≥ · · · ≥ λN be the eigenvalues of the outer-coupling matrix G. If the following N − 1 linear delayed difference equations are asymptotically stable about their zero solution xl (k + 1) = Jxl (k) + Jd xl (k − h(k)) + cλl Γxl (k − h(k)) (l = 2, 3, . . . , N ),

(4)

where J and Jd are the Jacobian of f (x(k), x(k − h(k))) at s(k) and s(k − h(k)), respectively. Then the synchronized states (3) are asymptotically stable.

Furthermore, in order to design a controller, we take a non-fragile feedback controller of the following form: ui (k) = (K + ΔK(k))(xi (k) − s(k)),

(5)

where K ∈ Rn is the controller gain to be designed and ΔK(k) represents the following additive gain perturbation ΔK(k) = DF (k)E with D ∈ Rn×p and E ∈ Rp×n are known constant matrices, and uncertain matrix F (k) satisfying F T (k)F (k) ≤ Ip . At this time, it is assumed that the controller parameter is changed by the following assumption. Assumption 1. The controller parameter is randomly changed. This means that, ρ(k) is a random process representing the information changing process of the perturbation in controller; that is, the perturbation in controller is described by the following Binomial sequence ρ(k) = m, if the perturbation in controller, and ΔK(k), is changed by mΔK(k), where l is number of changes, m = 0, 1, 2, . . . , l, ρ0 is probability of change in one term and ρ(k) satisfies E{ρ(k)} = lρ0 , E{ρ2 (k)} = (lρ0 )2 + lρ0 (1 − ρ0 ). At this time, the Binomial sequence is generated by the Matlab function binornd.

5

(6)

Remark 1. In Bernoulli process, a random variable belongs to the index set of two elements: {0, 1}. Hence, Bernoulli sequence looks like on-off switch. However, the random variable of Binomial sequences belongs to the index set of multiple elements: {0, 1, 2, . . . , l}, with E{ρ(k)} = lρ0 . So, we have views on this difference between Bernoulli and Binomial sequences. For example, when l = 10 and ρ0 = 0.5, at the discrete-time k = 1, if the Binomial variable, ρ(1), take 3, then m have 3. This means that the perturbation in controller, ΔK(k), is changed by 3ΔK(k). Also, at the next time k = 2, if ρ(2) = 5, then the the perturbation have 5ΔK(k). Moreover, l is used as the maximum value of change for the perturbation. In this rule, at a discrete-time step, the perturbation is changed. Therefore, when the perturbation affect to the controller, the effect degree of perturbation is changed because the effect in controller is not always the same, and the aforementioned concept from understanding the property of Binomial sequence is the answer to the consideration for the changing effect in controller.

With Lemma 1 and Assumption 1, a model of delayed discrete-time CDNs with the non-fragile feedback controller can be represented as xl (k + 1) = Jxl (k) + Jd xl (k − h(k)) + cλl Γxl (k − h(k)) + (K + ρ(k)ΔK)xl (k) (l = 2, 3, . . . , N ).

(7)

Let us define x(k) = col{x2 (k), x3 (k), . . . , xN (k)} ∈ R(N −1)n , Λ = diag{λ2 , . . . , λN } ∈ R(N −1)n×(N −1)n , where N is the number of nodes. Then, by the use of Kronecker product, the system (7) can be rewritten as the matrix form by x(k + 1) = (IN −1 ⊗ (J + K))x(k) + (IN −1 ⊗ Jd + c(Λ ⊗ Γ))x(k − h(k)) + (IN −1 ⊗ D)p(k) p(t) = (IN −1 ⊗ F (k))q(k), q(k) = (IN −1 ⊗ ρ(k)E)x(k).

(8)

Remark 2. Commonly, in existing works, the study on non-fragile controller design for various systems was considered without randomly occurring perturbations. However, in the real network, there are some perturbations due to change of environment. Hence, in (7), the model of 6

discrete-time CDNs with the perturbations in controller is suggested with (5). Also, when existing these perturbations, its subliminal degree in controller is different according to the random change in the real environment. Therefore, to analysis this problem mentioned above, in this paper, the synchronization problem for new model of discrete-time CDNs with the randomly occurring perturbations in controller is dealt by adopting the property of the Binomial sequence, which is a generalization of the Bernoulli sequence. Moreover, by driving of this model, the random change of real environment will become accessible.

Remark 3. When a controller whose gain is obtained by stabilization criterion is implemented, it is desirable to consider controller gain variations because it is impossible to design the given controller exactly. The reason for this is that the controller is often subject to inaccuracies such as resistance error, A/D and D/A conversion, finite word length, and round-off errors in numerical computation. Then, the controller fragility; in other word, non-fragile controller, issue has introduced in [33-35]. Therefore, it is necessary to consider controller gain variations when their synchronization and stabilization problems of complex networks are considered. In addition to this, while the occurrence probabilities of fragility [20], nonlinearity [36], missing measurement [37] and fault [38] were described by Bernoulli process, to represent the changing effect degree (or strength) of perturbation in controller whenever the fragility occurs, the concept of Binomial process was utilized according to the statements explained from Remarks 1 and 2. From a position of various practical systems, because the degree of fragility in controller is not always the same, it is reasonable to assume that the model (7) with Assumption 1 was proposed as a conceptual model.

The aim of this paper is to design the non-fragile control gain to achieve the synchronization of the system (7). In order to do this, we introduce the following definition and lemma.

Definition 2 (Meng et al. [39] ). The origin of the system (1) is said to be asymptotically stable, if, for any  > 0, there exists δ > 0 such that, if φ(k) < δ, k = −hM , −hM + 1, ..., 0, then x(k) < , for every k ≥ 0 and limk→∞ x(k) = 0. Lemma 2 (Finsler’s lemma [40] ). Let ζ ∈ Rn , Φ = ΦT ∈ Rn×n , and Υ ∈ Rm×n such that

7

rank(Υ) < n. The following statements are equivalent: (i) ζ T Φζ < 0, ∀Υζ = 0, ζ = 0, T

(ii) Υ⊥ ΦΥ⊥ < 0, (iii) ∃F ∈ Rn×m : Φ + F Υ + (F Υ)T < 0.

3

Main Results

In this section, new synchronization criteria for the system (7) will be derived by the use of the Lyapunov method and LMI framework. For the sake of simplicity on matrix representation, ϑi ∈ R9κ×κ , where κ = (N − 1)n, are defined as block entry matrices; e.g., ϑT2 = [0κ Iκ 0κ·7κ ]. The notations of several matrices are defined as: Δx(k) = x(k + 1) − x(k), hm + hM + min{(−1)hm +hM , 0} , hc = 2 ζ(k) = col{x(k), x(k − h(k)), x(k − hm ), x(k − hc ), x(k − hM ), Δx(k), Δx(k − hm ), Δx(k − hc ), Δx(k − hM )}, ⎡ ⊗P I ⊗P I ⊗P I ⊗P I ⎢ N −1 11 N −1 12 N −1 13 N −1 14 ⎢ ⎢  IN −1 ⊗P22 IN −1 ⊗P23 IN −1 ⊗P24 P = ⎢ ⎢ ⎢   IN −1 ⊗P33 IN −1 ⊗P34 ⎣    IN −1 ⊗P44 ⎡ ⎤ IN −1 ⊗Qi,11 IN −1 ⊗Qi,12 ⎦ (i = 1, 2, 3), Qi = ⎣  IN −1 ⊗Qi,22 ⎡ ⎤ IN −1 ⊗Si,11 IN −1 ⊗Si,12 ⎦ (i = 1, 2), Si = ⎣  IN −1 ⊗Qi,22

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

Π1,1 = [ϑ1 ϑ3 ϑ4 ϑ5 ], Π1,2 = [ϑ6 ϑ7 ϑ8 ϑ9 ], Π2 = [ϑ1 ϑ6 ], Π3 = [ϑ3 ϑ7 ], Π4 = [ϑ4 ϑ8 ], Π5 = [ϑ5 ϑ9 ], Ξ1 = (Π1,1 + Π1,2 )P(Π1,1 + Π1,2 )T − Π1,1 PΠT1,1 , Ξ2 = Π2 Q1 ΠT2 − Π3 (Q1 − Q2 )ΠT3 − Π4 (Q2 − Q3 )ΠT4 − Π5 Q3 ΠT5 , Ξ3 = ϑ6 h2m (IN −1 ⊗R1 )ϑT6 + ϑ7 (hc − hm )2 (IN −1 ⊗R2 )ϑT7 + ϑ8 (hM − hc )2 (IN −1 ⊗R3 )ϑT8 , Ξ4 = Π3 (hc − hm )2 S1 ΠT3 + Π4 (hM − hc )2 S2 ΠT4 ,

Ξz,1 = (hc − hm ) ϑ3 (IN −1 ⊗Z1 )ϑT3 − ϑ2 (IN −1 ⊗(Z1 − Z2 ))ϑT2 − ϑ4 (IN −1 ⊗Z2 )ϑT4 , 8

Ξ5,1 = −(ϑ1 − ϑ3 )(IN −1 ⊗R1 )(ϑ1 − ϑ3 )T − (ϑ4 − ϑ5 )(IN −1 ⊗R3 )(ϑ4 − ϑ5 )T , ⎤⎡ ⎤T ⎡ ⎤ ⎡ ϑT3 − ϑT2 IN −1 ⊗M1 IN −1 ⊗(R2 + Z1 ) ϑT3 − ϑT2 ⎦⎣ ⎦ ⎣ ⎦, −⎣ ϑT2 − ϑT4 IN −1 ⊗M1T IN −1 ⊗(R2 + Z2 ) ϑT2 − ϑT4

Ξz,2 = (hM − hc ) ϑ4 (IN −1 ⊗Z3 )ϑT4 − ϑ2 (IN −1 ⊗(Z3 − Z4 ))ϑT2 − ϑ5 (IN −1 ⊗Z4 )ϑT5 , Ξ5,2 = −(ϑ1 − ϑ3 )(IN −1 ⊗R1 )(ϑ1 − ϑ3 )T − (ϑ3 − ϑ4 )(IN −1 ⊗R2 )(ϑ3 − ϑ4 )T ⎤⎡ ⎡ ⎤T ⎡ ⎤ ϑT4 − ϑT2 ϑT4 − ϑT2 IN −1 ⊗M2 IN −1 ⊗(R3 + Z3 ) ⎦⎣ ⎦ ⎣ ⎦, −⎣ ϑT2 − ϑT5 IN −1 ⊗M2T IN −1 ⊗(R3 + Z4 ) ϑT2 − ϑT5 ˆ i = Ξ1 + Ξ2 + Ξ3 + Ξ4 + Ξz,i + Ξ5,i (i = 1, 2), Ξ Υ = (IN −1 ⊗(J − In ))ϑT1 + (IN −1 ⊗Jd + c(Λ ⊗ Γ))ϑT2 − (IN −1 ⊗In )ϑT6 .

(9)

Now, the following theorem is the delay-dependent synchronization criterion of the discretetime CDNs (7) when ui (k) = 0n (i = 1, 2, . . . , N ). Theorem 1. For given positive integers hm , hM and a positive scalar c, the system (7) under ui (k) = 0n is asymptotically synchronous for hm ≤ h(k) ≤ hM , if there exist positive definite matrices Pii ∈ Rn×n (i = 1, 2, 3, 4), Qi,jj ∈ Rn×n (i = 1, 2, 3, j = 1, 2), Si,jj ∈ Rn×n (i, j = 1, 2), Ri ∈ Rn×n (i = 1, 2, 3), any symmetric matrices Zi ∈ Rn×n (i = 1, 2, 3, 4) and any matrices Pij ∈ Rn×n (i = 1, 2, 3, j = 2, 3, 4, i < j), Qi,12 ∈ Rn×n (i = 1, 2, 3), Si,12 ∈ Rn×n (i = 1, 2), Mi ∈ Rn×n (i = 1, 2), satisfying the following LMIs: ˆ i Υ⊥ < 0, (Υ⊥ )T Ξ ⎤ 0κ IN −1 ⊗Zj ⎦ > 0, Si + ⎣ IN −1 ⊗ZjT 0κ ⎤ ⎡ IN −1 ⊗Mi IN −1 ⊗(Ri+1 + Z2i−1 ) ⎦ ≥ 0, ⎣ T IN −1 ⊗Mi IN −1 ⊗(Ri+1 + Z2i ) ⎡

(10) (11)

(12)

where j = 1, 2 when i = 1 and j = 3, 4 when i = 2. Proof. Let us consider the following L-K functional candidate as V (k) = V1 + V2 + V3 + V4 ,

(13)

where V1 = ζ T (k)Π1,1 PΠT1,1 ζ(k), V2 =

k−1 

ζ T (s)Π2 Q1 ΠT2 ζ(s) +

s=k−hm

k−h m −1  s=k−hc

9

ζ T (s)Π2 Q2 ΠT2 ζ(s)

+

k−h c −1 

ζ T (s)Π2 Q3 ΠT2 ζ(s),

s=k−hM −1 

V3 = hm

k−1 

ΔxT (u)(IN −1 ⊗R1 )Δx(u)

s=−hm u=k+s

+(hc − hm )

−h k−1 m −1  

ΔxT (u)(IN −1 ⊗R2 )Δx(u)

s=−hc u=k+s

+(hM − hc )

−h c −1 

k−1 

ΔxT (u)(IN −1 ⊗R3 )Δx(u),

s=−hM u=k+s

V4 = (hc − hm )

−h k−1 m −1  

ζ T (u)Π2 S1 ΠT2 ζ(u)

s=−hc u=k+s

+(hM − hc )

−h c −1 

k−1 

ζ T (u)Π2 S2 ΠT2 ζ(u).

s=−hM u=k+s

The forward differences of V1 and V2 can be calculated as ΔV1 = ζ T (k + 1)Π1,1 PΠT1,1 ζ(k + 1) − ζ T (k)Π1,1 PΠT1,1 ζ(k) = ζ T (k)(Π1,1 + Π1,2 )P(Π1,1 + Π1,2 )T ζ(k) − ζ T (k)Π1,1 PΠT1,1 ζ(k) = ζ T (k)Ξ1 ζ(k),

(14)

ΔV2 = ζ T (k)Π2 Q1 ΠT2 ζ(k) − ζ T (k)Π3 Q1 ΠT3 ζ(k) +ζ T (k)Π3 Q2 ΠT3 ζ(k) − ζ T (k)Π4 Q2 ΠT4 ζ(k) +ζ T (k)Π4 Q3 ΠT4 ζ(k) − ζ T (k)Π5 Q3 ΠT5 ζ(k) = ζ T (k)Ξ2 ζ(k).

With hc =

hm +hM +min{(−1)hm +hM ,0} , 2

(15)

the ΔV3 and ΔV4 can be estimated by the following two

cases: • Case I (hm ≤ h(k) ≤ hc ): First, the ΔV3 can be obtained as k−1 

T

ΔV3 = ζ (k)Ξ3 ζ(k) − hm

ΔxT (s)(IN −1 ⊗R1 )Δx(s)

s=k−hm

−(hc − hm )

k−h m −1 

ΔxT (s)(IN −1 ⊗R2 )Δx(s)

s=k−h(k)

10

k−h(k)−1



−(hc − hm )

ΔxT (s)(IN −1 ⊗R2 )Δx(s)

s=k−hc

−(hM − hc )

k−h c −1 

ΔxT (s)(IN −1 ⊗R3 )Δx(s).

(16)

s=k−hM

To improve the feasible region of synchronization criterion, inspired by the work of [41], the following two zero equalities are introduced with any symmetric matrices Z1 and Z2 : 0 = (hc − hm )xT (k − hm )(IN −1 ⊗Z1 )x(k − hm ) −(hc − hm )xT (k − h(k))(IN −1 ⊗Z1 )x(k − h(k)) k−h m −1 

−(hc − hm )

(ΔxT (s)(IN −1 ⊗Z1 )Δx(s) + 2xT (s)(IN −1 ⊗Z1 )Δx(s)),

(17)

s=k−h(k) T

0 = (hc − hm )x (k − h(k))(IN −1 ⊗Z2 )x(k − h(k)) −(hc − hm )xT (k − hc )(IN −1 ⊗Z2 )x(k − hc ) k−h(k)−1



−(hc − hm )

(ΔxT (s)(IN −1 ⊗Z2 )Δx(s) + 2xT (s)(IN −1 ⊗Z2 )Δx(s)).

(18)

s=k−hc

Adding (17) and (18) leads to 0 = ζ T (k)Ξz,1 ζ(k) −(hc − hm )

k−h m −1 

(ΔxT (s)(IN −1 ⊗Z1 )Δx(s) + 2xT (s)(IN −1 ⊗Z1 )Δx(s))

s=k−h(k) k−h(k)−1

−(hc − hm )



(ΔxT (s)(IN −1 ⊗Z2 )Δx(s) + 2xT (s)(IN −1 ⊗Z2 )Δx(s)).

s=k−hc

Then, by adding (19) into the ΔV4 , we have ΔV4 = ζ T (k)(Ξ4 + Ξz1 )ζ(k) −(hc − hm )

k−h m −1 

k−h(k)−1



⎛

k−h m −1 

⎡

ζ T (s)Π2 ⎝S1 + ⎣

s=k−hc

−(hc − hm )

⎡

ζ T (s)Π2 ⎝S1 + ⎣

s=k−h(k)

−(hc − hm )

⎛

0κ

IN −1 ⊗Z1

IN −1 ⊗Z1T

0κ

0κ

IN −1 ⊗Z2

IN −1 ⊗Z2T

0κ

ΔxT (s)(IN −1 ⊗Z1 )Δx(s)

s=k−h(k) k−h(k)−1

−(hc − hm )



ΔxT (s)(IN −1 ⊗Z2 )Δx(s)

s=k−hc

11

⎤⎞ ⎦⎠ ΠT2 ζ(s) ⎤⎞ ⎦⎠ ΠT2 ζ(s)

(19)

−(hM − hc )

k−h c −1 

ζ T (s)Π2 S2 ΠT2 ζ(s).

(20)

s=k−hM

Here, if the inequality (11) with i = 1 and S2 > 0 hold, then, the ΔV4 can be bounded as ΔV4 ≤ ζ T (k)(Ξ4 + Ξz,1 )ζ(k) −(hc − hm )

k−h m −1 

ΔxT (s)(IN −1 ⊗Z1 )Δx(s)

s=k−h(k) k−h(k)−1

−(hc − hm )



ΔxT (s)(IN −1 ⊗Z2 )Δx(s).

(21)

s=k−hc

One more, by adding the ΔV3 into the upper bound of the ΔV4 and using discrete Jensen’s inequality in [42], we get ΔV3 + ΔV4 ≤ ζ T (k)(Ξ3 + Ξ4 + Ξz,1 )ζ(k) ⎞T ⎛ ⎞ ⎛ k−1 k−1   Δx(s)⎠ (IN −1 ⊗R1 ) ⎝ Δx(s)⎠ −⎝ ⎛ −⎝

s=k−hm k−h c −1 

⎞T

⎛

Δx(s)⎠ (IN −1 ⊗R3 ) ⎝

s=k−hM

s=k−hm k−h c −1 

⎞ Δx(s)⎠

s=k−hM

−Θ(k), where α1 (k) =

hc −h(k) hc −hm ,

(22)

and

⎛ ⎞T ⎛ ⎞ k−h k−h m −1 m −1   1 ⎝ Δx(s)⎠ (IN −1 ⊗(R2 + Z1 )) ⎝ Δx(s)⎠ Θ(k) = 1 − α1 (k) s=k−h(k) s=k−h(k) ⎛ ⎞T ⎛ ⎞ k−h(k)−1 k−h(k)−1   1 ⎝ Δx(s)⎠ (IN −1 ⊗(R2 + Z2 )) ⎝ Δx(s)⎠ . + α1 (k) s=k−hc

s=k−hc

By using reciprocally convex approach in [31], when hm < h(k) < hc , since α1 (k) satisfies 0 < α1 (k) < 1, the following inequality holds for any matrix M1 ⎛ ⎞T ⎛ ⎞ k−h k−h m −1 m −1   α1 (k) ⎝ 0 < Δx(s)⎠ (IN −1 ⊗(R2 + Z1 )) ⎝ Δx(s)⎠ 1 − α1 (k) s=k−h(k) s=k−h(k) ⎞T ⎛ ⎞ ⎛ k−h(k)−1 k−h m −1   Δx(s)⎠ (IN −1 ⊗M1 ) ⎝ Δx(s)⎠ −⎝ ⎛ −⎝

s=k−h(k) k−h(k)−1



⎞T

⎛

Δx(s)⎠ (IN −1 ⊗M1T ) ⎝

s=k−hc

s=k−hc k−h m −1 

⎞ Δx(s)⎠

s=k−h(k)

12

⎛

⎞T

k−h(k)−1

+ which means

1 − α1 (k) ⎝ α1 (k) ⎛

Θ(k) > ⎝



Δx(s)⎠ (IN −1 ⊗(R2 + Z2 )) ⎝

+⎝ ⎛ +⎝ ⎛ +⎝

⎛

Δx(s)⎠ (IN −1 ⊗(R2 + Z1 )) ⎝

k−h m −1 

⎞T

⎞T

k−h(k)−1



⎛

Δx(s)⎠ (IN −1 ⊗M1 ) ⎝

s=k−h(k)



Δx(s)⎠ ,

⎞ Δx(s)⎠

k−h(k)−1

⎛



⎞

Δx(s)⎠

s=k−hc k−h m −1 

⎞ Δx(s)⎠

s=k−h(k)

⎞T

k−h(k)−1

k−h m −1 

s=k−h(k)

Δx(s)⎠ (IN −1 ⊗M1T ) ⎝

s=k−hc



s=k−hc

⎞T

s=k−h(k)

⎞

k−h(k)−1

s=k−hc

k−h m −1 

⎛

⎛

⎛

Δx(s)⎠ (IN −1 ⊗(R2 + Z2 )) ⎝

s=k−hc

k−h(k)−1



⎞

Δx(s)⎠

s=k−hc

= ζ T (k)Ξ5,1 ζ(k).  m −1 It should be noted that when h(k) = hm or h(k) = hM , we have k−h s=k−h(k) Δx(s) = 0κ·1 or k−h(k)−1 s=k−hc Δx(s) = 0κ·1 , respectively. Thus, if the inequality (12) with i = 1 holds, the following inequality can be obtained ΔV3 + ΔV4 ≤ ζ T (k)(Ξ3 + Ξ4 + Ξz,1 + Ξ5,1 )ζ(k).

(23)

From (14) - (23) and by the use of S-procedure [32], ΔV (k) has a new upper bound as ˆ 1 ζ(k). ΔV (k) ≤ ζ T (k)Ξ

(24)

Also, the system (7) with the augmented matrix ζ(k) can be rewritten as Υζ(k) = 0(N −1)n·1 .

(25)

Then, a synchronization condition for the system (7) under ui (k) = 0n is ˆ 1 ζ(k) < 0 subject to (25). ζ T (k)Ξ

(26)

Here, from (i) and (iii) of Lemma 2, the inequality (26) is equivalent to ˆ 1 + X Υ + ΥT X T = Ω1 < 0. Ξ

(27)

where X is any free matrix with appropriate dimension. From (24) to (27), if (27) holds, then there exists positive scalar ε1 such that ΔV (k) ≤ ζ T (k)Ω1 ζ(k) < −ε1 x(k)2 . 13

• Case II (hc ≤ h(k) ≤ hM ): By similar process of Case I, the ΔV3 and ΔV4 are represented as k−1 

ΔV3 = ζ T (k)Ξ3 ζ(k) − hm

ΔxT (s)(IN −1 ⊗R1 )Δx(s)

s=k−hm k−h m −1 

ΔxT (s)(IN −1 ⊗R2 )Δx(s)

−(hc − hm )

s=k−hc

−(hM − hc )

k−h c −1 

ΔxT (s)(IN −1 ⊗R3 )Δx(s)

s=k−h(k) k−h(k)−1

−(hM − hc )



ΔxT (s)(IN −1 ⊗R3 )Δx(s),

(28)

s=k−hM

ΔV4 = ζ T (k)(Ξ4 + Ξz,2 )ζ(k) k−h m −1 

ζ T (s)Π2 S1 ΠT2 ζ(s)

−(hc − hm )

s=k−hc

−(hM − hc )

k−h c −1 

⎛

ζ T (s)Π2 ⎝S2 + ⎣

s=k−h(k) k−h(k)−1

−(hM − hc )



⎛

k−h c −1 

⎡

ζ T (s)Π2 ⎝S2 + ⎣

s=k−hM

−(hM − hc )

⎡ 0κ

IN −1 ⊗Z3

IN −1 ⊗Z3T

0κ

0κ

IN −1 ⊗Z4

IN −1 ⊗Z4T

0κ

⎤⎞ ⎦⎠ ΠT2 ζ(s) ⎤⎞ ⎦⎠ ΠT2 ζ(s)

ΔxT (s)(IN −1 ⊗Z3 )Δx(s)

s=k−h(k) k−h(k)−1

−(hM − hc )



ΔxT (s)(IN −1 ⊗Z4 )Δx(s).

(29)

s=k−hM

At this time, with any symmetric matrices Z3 and Z4 , the following zero equality was utilized in obtaining the ΔV4 . 0 = ζ T (k)Ξz,2 ζ(k) −(hM − hc )

k−h c −1 

(ΔxT (s)(IN −1 ⊗Z3 )Δx(s) + 2xT (s)(IN −1 ⊗Z3 )Δx(s))

s=k−h(k) k−h(k)−1

−(hM − hc )



(ΔxT (s)(IN −1 ⊗Z4 )Δx(s) + 2xT (s)(IN −1 ⊗Z4 )Δx(s)).

s=k−hM

Here, if S1 > 0 and the inequality (11) with i = 2 hold, then, the ΔV4 can be bounded as ΔV4 ≤ ζ T (k)(Ξ4 + Ξz,2 )ζ(k) 14

(30)

−(hM − hc )

k−h c −1 

ΔxT (s)(IN −1 ⊗Z3 )Δx(s)

s=k−h(k) k−h(k)−1

−(hM − hc )



ΔxT (s)(IN −1 ⊗Z4 )Δx(s).

(31)

s=k−hM

As a result, by using discrete Jensen’s inequality and reciprocally convex approach, if the inequality (12) with i = 2 hold, then an upper bound of ΔV3 + ΔV4 is obtained by ΔV3 + ΔV4 ≤ ζ T (k)(Ξ3 + Ξ4 + Ξz,2 )ζ(k) ⎞T ⎛ ⎞ ⎛ k−1 k−1   Δx(s)⎠ (IN −1 ⊗R1 ) ⎝ Δx(s)⎠ −⎝ ⎛ −⎝

s=k−hm

⎞T

k−h m −1 

⎛

Δx(s)⎠ (IN −1 ⊗R2 ) ⎝

s=k−hc

⎛

⎞T

s=k−hm k−h m −1 

⎞

Δx(s)⎠

s=k−hc

⎛ ⎞ k−h k−h c −1 c −1   1 ⎝ Δx(s)⎠ (IN −1 ⊗(R3 + Z3 )) ⎝ Δx(s)⎠ − 1 − α2 (k) s=k−h(k) s=k−h(k) ⎛ ⎞T ⎛ ⎞ k−h(k)−1 k−h(k)−1   1 ⎝ Δx(s)⎠ (IN −1 ⊗(R3 + Z4 )) ⎝ Δx(s)⎠ − α2 (k) s=k−hM

s=k−hM

T

≤ ζ (k)(Ξ3 + Ξ4 + Ξz,2 + Ξ5,2 )ζ(k), where α2 (k) =

(32)

hM −h(k) hM −hc .

From (14), (15) and (28) - (32), ΔV (k) has a new upper bound as ˆ 2 ζ(k). ΔV (k) ≤ ζ T (k)Ξ

(33)

Then, a synchronization condition for the system (7) is ˆ 2 ζ(k) < 0 subject to (25). ζ T (k)Ξ

(34)

Here, also, from (i) and (iii) of Lemma 2, the inequality (34) is equivalent to ˆ 2 + X Υ + ΥT X T = Ω2 < 0. Ξ

(35)

where X is any matrix with appropriate dimension. From (33) to (35), if (35) holds, then there exists positive scalar ε2 such that ΔV (k) ≤ ζ T (k)Ω2 ζ(k) < −ε2 x(k)2 . Therefore, from Case I and Case II, it can be seen that for all discrete time k, if (27) and (35) 15

hold, then ΔV (k) < − min{ε1 , ε2 }x(k)2 . From the Lyapunov stability theory, Definition 2, and (ii) and (iii) of Lemma 2, it can be concluded that if (10) hold, then the system (7) with 

ui (k) = 0n is asymptotically synchronous.

Remark 4. In delay-dependent analysis for stability and stabilization of dynamic systems with time-delays, the most important index for checking the conservatism of delay-dependent criteria is to find maximum delay bounds for guaranteeing the asymptotic stability of the system. One of the most utilizing method to enhance the feasible region of the concerned criteria is to divide delay interval into some subintervals. This method utilizes a Lyapunov-Krasovskii functional which employs redundant state of differential equations shifted delay in time by a fraction of the time delay. However, if delay-partitioning number increases, then the computational burden becomes large, and the solving of the concerned LMIs is much time-consuming. While there are many results which utilized delay-partitioning method for a continuous dynamic system [43-45], only a few results in stability and stabilization for a discrete-time systems are reported [12, 46, 47]. In this paper, as a tradeoff between time-consumption and improvement of the feasible region, the delay-partitioning numbers in the interval [hm , hM ] are chosen as two. Furthermore, in subintervals [hm , hc ] and [hc , hM ], different zero equalities are added to improve the feasible region of Theorem 1 as shown in (19) for [hm , hc ] and in (30) for [hc , hM ]. Theorem 1 provided the synchronization criterion for system (7) under ui (k) = 0n in the LMI framework. Based on the results of Theorem 1, we will design a synchronization controller for the discrete-time system (1) when ui (k) = 0n (i = 1, 2, . . . , N ) with ΔK(k) = 0n . Before deriving this, the notations of several matrices are defined for simple representation of Theorem 2: ¯ 1,1 + Π1,2 )T − Π1,1 PΠ ¯ T , ¯ 1 = (Π1,1 + Π1,2 )P(Π Ξ 1,1 ¯1 − Q ¯ 2 )ΠT3 − Π4 (Q ¯2 − Q ¯ 3 )ΠT4 − Π5 Q ¯ 1 ΠT2 − Π3 (Q ¯ 3 ΠT5 , ¯ 2 = Π2 Q Ξ ¯ 1 )ϑT6 + ϑ7 (hc − hm )2 (IN −1 ⊗R ¯ 2 )ϑT7 + ϑ8 (hM − hc )2 (IN −1 ⊗R ¯ 3 )ϑT8 , ¯ 3 = ϑ6 h2m (IN −1 ⊗R Ξ ¯ 4 = Π3 (hc − hm )2 S¯1 ΠT + Π4 (hM − hc )2 S¯2 ΠT , Ξ 3 4

¯ z,1 = (hc − hm ) ϑ3 (IN −1 ⊗Z¯1 )ϑT3 − ϑ2 (IN −1 ⊗(Z¯1 − Z¯2 ))ϑT2 − ϑ4 (IN −1 ⊗Z¯2 )ϑT4 , Ξ ¯ 1 )(ϑ1 − ϑ3 )T − (ϑ4 − ϑ5 )(IN −1 ⊗R ¯ 3 )(ϑ4 − ϑ5 )T , ¯ 5,1 = −(ϑ1 − ϑ3 )(IN −1 ⊗R Ξ

16

⎡ −⎣

ϑT3 − ϑT2 ϑT2

−

ϑT4

⎤T ⎡ ⎦ ⎣



¯ 2 + Z¯1 ) IN −1 ⊗(R

¯1 IN −1 ⊗M

¯T IN −1 ⊗M 1

¯ 2 + Z¯2 ) IN −1 ⊗(R

⎤⎡ ⎦⎣

ϑT3 − ϑT2 ϑT2

−

ϑT4

⎤ ⎦,

¯ z,2 = (hM − hc ) ϑ4 (IN −1 ⊗Z¯3 )ϑT4 − ϑ2 (IN −1 ⊗(Z¯3 − Z¯4 ))ϑT2 − ϑ5 (IN −1 ⊗Z¯4 )ϑT5 , Ξ ¯ 1 )(ϑ1 − ϑ3 )T − (ϑ3 − ϑ4 )(IN −1 ⊗R ¯ 2 )(ϑ3 − ϑ4 )T ¯ 5,2 = −(ϑ1 − ϑ3 )(IN −1 ⊗R Ξ ⎤⎡ ⎡ ⎤T ⎡ ⎤ T T T T ¯ ¯ ¯ ϑ − ϑ2 ϑ4 − ϑ2 ⊗(R3 + Z3 ) IN −1 ⊗M2 I ⎦⎣ 4 ⎦ ⎣ N −1 ⎦, −⎣ T T T T T ¯ ¯ ¯ ϑ2 − ϑ5 IN −1 ⊗M2 IN −1 ⊗(R3 + Z4 ) ϑ2 − ϑ5 ˆ¯ = Ξ ¯1 + Ξ ¯2 + Ξ ¯3 + Ξ ¯4 + Ξ ¯ z,i + Ξ ¯ 5,i (i = 1, 2), Ξ i ¯ = ϑ1 μ1 + ϑ6 μ2 , Π ¯ = (IN −1 ⊗ (J P¯11 + Y − P¯11 ))ϑT + (IN −1 ⊗ Jd P¯11 + c(Λ ⊗ ΓP¯11 ))ϑT Υ 1 2 −(IN −1 ⊗ P¯11 )ϑT6 .

(36)

Now, the following Theorem is given by the second main result. Theorem 2. For given positive integers hm , hM , a positive scalar c, and any scalars μ1 , μ2 , the system (7) under ui (k) = K(xi (k) − s(k)) is asymptotically synchronous for hm ≤ h(k) ≤ hM , ¯ i,jj ∈ Rn×n (i = 1, 2, 3, j = if there exist positive definite matrices P¯ii ∈ Rn×n (i = 1, 2, 3, 4), Q ¯ i ∈ Rn×n (i = 1, 2, 3), any symmetric matrices Z¯i ∈ Rn×n 1, 2), S¯i,jj ∈ Rn×n (i, j = 1, 2), R ¯ i,12 ∈ Rn×n (i = 1, 2, 3, 4) and any matrices P¯ij ∈ Rn×n (i = 1, 2, 3, j = 2, 3, 4, i < j), Q ¯ i ∈ Rn×n (i = 1, 2), Y ∈ Rn×n , satisfying the LMIs (11), (i = 1, 2, 3), S¯i,12 ∈ Rn×n (i = 1, 2), M (12) and ˆ ¯ Υ} ¯ < 0 (i = 1, 2). ¯ i + sym{Π Ξ

(37)

−1 is Then, the system (7) under ui (k) = K(xi (k) − s(k)) with the synchronization gain K = Y P¯11

asymptotically stable, which means the obtained controller gain K guarantees the asymptotically stable results in a synchronization of states. Proof. To design a gain K, with the same Lyapunov-Krasovskii functional candidate in (13), by using the similar method in (14) - (32) and considering the following zero equality with any matrices X1 and X2 0 = 2ζ T (t)(e1 (IN −1 ⊗X1 ) + e6 (IN −1 ⊗X2 ))Υζ(t), a sufficient condition guaranteeing stability for the system (7) can be ˆ i + sym{(e1 (IN −1 ⊗X1 ) + e6 (IN −1 ⊗X2 ))Υ} Ξ 17

(38)

ˆ i + sym{(e1 μ1 + e6 μ2 )(IN −1 ⊗P11 )Υ} = Ξ < 0 (i = 1, 2),

(39)

At this time, the matrices X1 and X2 were defined as μ1 P11 and μ2 P11 , respectively. Also, to obtain the consensus protocol gain, let us define −1 P¯11 = P11 , Y = K P¯11 ,

P¯ = diag{IN −1 ⊗P¯11 , . . . , IN −1 ⊗P¯11 }T P diag{IN −1 ⊗P¯11 , . . . , IN −1 ⊗P¯11 },    4

¯ i = diag{IN −1 ⊗P¯11 , IN −1 ⊗P¯11 }T Qi diag{IN −1 ⊗P¯11 , IN −1 ⊗P¯11 }, Q S¯i = diag{IN −1 ⊗P¯11 , IN −1 ⊗P¯11 }T Si diag{IN −1 ⊗P¯11 , IN −1 ⊗P¯11 }, ¯j R

T T ¯ i = P¯ T Mi P¯11 , = P¯11 Rj P¯11 , Z¯i = P¯11 Zi P¯11 , M 11

(40)

where i = 1, 2, 3, j = 1, . . . , 6. Then, pre- and post- multiplying inequality (39) by matrix diag{(IN −1 ⊗P¯11 ), . . . , (IN −1 ⊗P¯11 )}    N −1

leads to ˆ ¯ Υ} ¯ < 05N n (i = 1, 2). ¯ i + sym{Π Ξ In conclusion, the condition (41) is equivalent to the LMIs (37).

(41) 

Theorem 2 provided the method of control design for synchronization of the system (7) in the LMI framework. Based on the results of Theorem 2, we will design a non-fragile controller for the discrete-time system (7) under ui (k) = (K + ρ(k)ΔK)(xi (k) − s(k)). This result will be introduced as the following Theorem 3. Theorem 3. For given positive integers hm , hM , l, positive scalar c, ρ0 ∈ [0, 1], and any scalars μ1 , μ2 , the system (7) under ui (k) = (K + ρ(k)ΔK)(xi (k) − s(k)) is asymptotically synchronous ¯ i,jj ∈ for hm ≤ h(k) ≤ hM , if there exist positive definite matrices P¯ii ∈ Rn×n (i = 1, 2, 3, 4), Q ¯ i ∈ Rn×n (i = 1, 2, 3), any symmetric Rn×n (i = 1, 2, 3, j = 1, 2), S¯i,jj ∈ Rn×n (i, j = 1, 2), R matrices Z¯i ∈ Rn×n (i = 1, 2, 3, 4) and any matrices P¯ij ∈ Rn×n (i = 1, 2, 3, j = 2, 3, 4, i < j), ˜ i ∈ Rn×n (i = 1, 2), Y ∈ Rn×n , satisfying ¯ i,12 ∈ Rn×n (i = 1, 2, 3), S¯i,12 ∈ Rn×n (i = 1, 2), M Q the LMIs (11), (12) and ˆ ˜ + sym{Π ˜ Υ} ˜ + Ψ[l,ρ ] < 05κ . Ξ 0 18

(42)

where ˜ 1,2 = [ϑ˜6 ϑ˜7 ϑ˜8 ϑ˜9 ], ˜ 1,1 = [ϑ˜1 ϑ˜3 ϑ˜4 ϑ˜5 ], Π Π ˜ 3 = [ϑ˜3 ϑ˜7 ], Π ˜ 4 = [ϑ˜4 ϑ˜8 ], Π ˜ 5 = [ϑ˜5 ϑ˜9 ], ˜ 2 = [ϑ˜1 ϑ˜6 ], Π Π ¯ 1,1 + Π1,2 )T − Π1,1 PΠ ˜ 1 = (Π1,1 + Π1,2 )P(Π ¯ T1,1 , Ξ ¯1 − Q ¯ 2 )ΠT − Π4 (Q ¯2 − Q ¯ 3 )ΠT − Π5 Q ¯ 1 ΠT − Π3 (Q ¯ 3 ΠT , ˜ 2 = Π2 Q Ξ 2 3 4 5 ¯ 1 )ϑ˜T + ϑ˜7 (hc − hm )2 (IN −1 ⊗R ¯ 2 )ϑ˜T + ϑ˜8 (hM − hc )2 (IN −1 ⊗R ¯ 3 )ϑ˜T , ˜ 3 = ϑ˜6 h2 (IN −1 ⊗R Ξ m 6 7 8 ˜ 4 = Π3 (hc − hm )2 S¯1 ΠT3 + Π4 (hM − hc )2 S¯2 ΠT4 , Ξ   ˜ z,1 = (hc − hm ) ϑ˜3 (IN −1 ⊗Z¯1 )ϑ˜T3 − ϑ˜2 (IN −1 ⊗(Z¯1 − Z¯2 ))ϑ˜T2 − ϑ˜4 (IN −1 ⊗Z¯2 )ϑ˜T4 , Ξ ¯ 1 )(ϑ˜1 − ϑ˜3 )T − (ϑ˜4 − ϑ˜5 )(IN −1 ⊗R ¯ 3 )(ϑ˜4 − ϑ˜5 )T , ˜ 5,1 = −(ϑ˜1 − ϑ˜3 )(IN −1 ⊗R Ξ ⎤⎡ ⎤T ⎡ ⎤ ⎡ T T T T ˜ ˜ ˜ ˜ ¯ ¯ ¯ ϑ − ϑ2 ⊗(R2 + Z1 ) IN −1 ⊗M1 I ϑ3 − ϑ2 ⎦⎣ 3 ⎦ ⎣ N −1 ⎦, −⎣ T T T T T ˜ ˜ ˜ ˜ ¯ ¯ ¯ IN −1 ⊗M1 IN −1 ⊗(R2 + Z2 ) ϑ2 − ϑ4 ϑ2 − ϑ4   ˜ z,2 = (hM − hc ) ϑ˜4 (IN −1 ⊗Z¯3 )ϑ˜T − ϑ˜2 (IN −1 ⊗(Z¯3 − Z¯4 ))ϑ˜T − ϑ˜5 (IN −1 ⊗Z¯4 )ϑ˜T , Ξ 4

2

5

¯ 1 )(ϑ˜1 − ϑ˜3 )T − (ϑ˜3 − ϑ˜4 )(IN −1 ⊗R ¯ 2 )(ϑ˜3 − ϑ˜4 )T ˜ 5,2 = −(ϑ˜1 − ϑ˜3 )(IN −1 ⊗R Ξ ⎤⎡ ⎡ ⎤T ⎡ ⎤ ¯ 3 + Z¯3 ) ¯2 ϑ˜T4 − ϑ˜T2 ϑ˜T4 − ϑ˜T2 IN −1 ⊗M IN −1 ⊗(R ⎦⎣ ⎦ ⎣ ⎦, −⎣ ¯T ¯ 3 + Z¯4 ) IN −1 ⊗M IN −1 ⊗(R ϑ˜T − ϑ˜T ϑ˜T − ϑ˜T 2

5

2

2

5

ˆ˜ = Ξ ˜1 + Ξ ˜2 + Ξ ˜3 + Ξ ˜4 + Ξ ˜ z,i + Ξ ˜ 5,i (i = 1, 2), Ξ i ˜ = ϑ˜1 μ1 + ϑ˜6 μ2 , Π ˜ = (IN −1 ⊗(J P¯11 + Y − P¯11 ))ϑ˜T1 + (IN −1 ⊗Jd P¯11 + c(Λ ⊗ ΓP¯11 ))ϑ˜T2 Υ −(IN −1 ⊗P¯11 )ϑ˜T6 + (IN −1 ⊗D)ϑ˜T10 Ψ[l,ρ0] = ((lρ0 )2 + lρ0 (1 − ρ0 ))ϑ˜1 (IN −1 ⊗E T E)ϑ˜T1 − ϑ˜10 ϑ˜T10 , and ϑ˜i ∈ R10κ×κ (i = 1, 2, . . . , 10) are defined as block entry matrices; for example, ϑ˜T2 = [0N n IN n 0N n·8N n ]. Then, the system (7) under ui (k) = (K + ρ(k)ΔK)(xi (k) − s(k)) with the synchronization gain −1 is asymptotically stable. K = Y P˜11

Proof. Based on the same Lyapunov-Krasovskii functional (13), the deriving process of its new upper bound is very similar to the proof of Theorem 2 excluding the process (43), so it is ˜ omitted. At this time, the augmented vector ζ(k) = col{ζ(k), p(k)} is utilized instead of the ζ(k) used in Theorem 2. Since the relational expression between p(k) and q(k), pT (k)p(k) ≤ q T (k)q(k) holds from the 19

second equality of the system (7), there exist a positive scalar  satisfying the following inequality 0 ≤ (q T (k)q(k) − pT (k)p(k)) ˜ = ζ˜T (k)[{ρ2 (k)ϑ˜1 (IN −1 ⊗E T E)ϑ˜T1 − ϑ˜10 ϑ˜T10 }]ζ(k).   

(43)

Ψ[ρ(k)]

Therefore, from (14) - (43) and by use of S-procedure [32, E{ΔV (k)} has a new non-fragile robust synchronization condition as ˆ˜ + sym{Π ˜ ˜ Υ} ˜ + Ψ[ρ(k)] )ζ(k)} E{ΔV (k)} ≤ E{ζ˜T (k)(Ξ ˆ˜ + sym{Π ˜ ˜ Υ} ˜ + Ψ[l,ρ ] )ζ(k) = ζ˜T (k)(Ξ 0 < 0.

(44)

Here, with (6), the matrix Ψ[lρ0 ] will be obtained by the following process ˜ 0 ≤ E{ζ˜T (k)[{ρ2 (k)(ϑ˜1 (IN −1 ⊗E T E)ϑ˜T1 ) − ϑ˜10 ϑ˜T10 }]ζ(k)} ˜ = ζ˜T (k)[{((lρ0 )2 + lρ0 (1 − ρ0 ))ϑ˜1 (IN −1 ⊗E T E)ϑ˜T1 − ϑ˜10 ϑ˜T10 }]ζ(k).   

(45)

Ψ[l,ρ0 ]



Also, the condition (44) is equivalent to the LMIs (42). This completes our proof.

4

Numerical Examples

In this section, we provide two numerical examples to show the effectiveness of the presented stability criteria in this paper. Example 1. Consider the following 2-order system with 50 nodes in [12] and the inner-coupling matrix Γ = 0.01In yi1 (k + 1) = f (yi1 (k)) + Σ1 (k), yi2 (k + 1) = f (yi2 (k)) + Σ2 (k),

(46)

where 3 (k) − 0.1yi1 (k − h(k)), f (yi1 (k)) = βyi1 (k) − yi1 2 (k)yi2 (k) − 0.2yi1 (k − h(k)) − 0.1yi1 (k − h(k)), f (yi2 (k)) = 0.05yi1 (k) + 0.9yi2 (k) − yi1

Σl (k) = c

N 

gij Γyjl (k − h(k)) (l = 1, 2),

j=1

20

8 6 4 2

xl(k)

0 −2 −4 −6 −8 −10 −12

0

100

200 300 Discrete time k

400

500

Figure 1: State responses with case of 5 ≤ h(k) ≤ 54 when c = 0.1 (Example 1). Table 1: Maximum bounds hM with fixed hm = 5 and various c (Example 1). c

0.1

0.3

0.5

0.7

0.9

1

NoVar

Yue et al. [12]

17

13

10

8

7

7

101 × 49

Theorem 1

54

25

17

13

10

9

115

which is asymptotically stable at the equilibrium point s(k) = 0 and s(k − h(k)) = 0. To analyze the synchronization for the system, the N − 1 linear delayed difference equations (4) are x(k + 1) = (IN −1 ⊗J) x(k) + (IN −1 ⊗Jd + c(Λ ⊗ Γ)) x(k − h(k)) with the Jacobian matrices

⎡

J =⎣

β

⎡

⎤ 0

0.05 0.9

⎦ , Jd = ⎣

−0.1

(47)

⎤ 0

−0.2 −0.1

⎦.

When β = 0.8, the results of the maximum bound hM with fixed hm = 5 and various c obtained by Theorem 1 are listed in Table 1. It can be seen that Theorem 1 in this paper provides larger delay bounds than [12] as listed in Table 1. According to the results in Table 1, one can confirm that the coupling strength, c, affects the dynamic behavior of the CDNs since the maximum delay bounds for guaranteeing synchronization are different according to the value of c. Thus, in view of coupling strength, as the coupling strength c decreases, the difference of maximum 21

9

2.5

x 10

2

xl(k)

1.5

1

0.5

0

−0.5

0

5

10

15 Discrete time k

20

25

30

Figure 2: State responses without u(k). bound between the results of this work and the one in [12] increases. Moreover, from Table 1, the number of decision variable (NoVar) in this work is less than the one in [12] because the work in [12] use the mode-dependent variables. In order to confirm the obtained results with the time-varying delay condition as h(k) = round{29.5 + 24.5 sin( π4 k)}, where round is Matlab function and can be used to generate the integer value of time-delay; that is, 5 ≤ h(k) ≤ 54 when c = 0.1, the simulation result for the state responses of the system (47) are shown in Figure 1. This figure shows the system with the synchronized states converge to zero for given initial values of the state randomly.

Remark 5. According to the work in [25], the general aim of the delay-dependent stability analysis is to develop delay-dependent conditions to provide the maximum delay bound as large as possible, or by using decision variables as few as possible while keeping the same maximum delay bound. Here, the maximum delay bound is used as an index to judge the feasibility region, which is in proportional to the maximum delay bound. In view of this, from Table 1, it should be noted that the existing result listed in Table 1 is more conservative than those of Theorem 1. This means that the proposed Theorem 1 effectively reduces the conservatism or increase the feasibility region of stability criterion. Also, because the few number of decision variable was used in comparison with the existing result, it can be expected that the proposed criterion relieves the computation burden.

Example 2. Recall the system in Example 1. Thus, to design a non-fragile synchroniza-

22

4 3 2

xl(k)

1 0 −1 −2 −3 −4

0

5

10

15 Discrete time k

20

25

30

Figure 3: State responses with u(k) = Kx(k). tion controller with randomly occurring perturbation for the system, consider the following difference equations (7)   ˜ ⊗ Γ) x(k − h(k)) + u(k), x(k + 1) = (IN −1 ⊗J) x(k) + IN −1 ⊗Jd + c(Λ

(48)

where the associated parameters are defined in Example 1 and the outer-matrix ⎡ ⎤ −2 1 0 0 1 ⎢ ⎥ ⎢ ⎥ ⎢ 1 −3 1 1 0 ⎥ ⎢ ⎥ ⎥ ˜=⎢ G ⎢ 0 1 −2 1 0 ⎥. ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 1 −3 1 ⎥ ⎣ ⎦ 1 0 0 1 −2 ˜ = diag{−4.6180, −3.6180, −2.3820, −1.3820}. Moreover, from the matrix G, Λ In this time, when β = 2, c = 0.5, μ1 = 1, μ2 = 1, hm = 1 and hM = 4, the uncontrolled equations (48) are unstable shown in Figure 2. Hence, the controller gain K by applying Theorem 2 with the above condition can be obtained ⎡ ⎤ −1.4572 −0.0940 ⎦. K=⎣ (49) −0.1395 −0.1873 Then, Figure 3 shows that the state responses between the synchronized states converge to zero under the time-delay h(k) = round{2.5 + 1.5 sin( π4 k)} and the controller gain in (49) without randomly occurring perturbations; that is, u(k) = Kx(k). Also, to show the affect of the randomly occurring perturbations in the controller, the results 23

4 3 2 1

xl(k)

0 −1 −2 −3 −4 −5 −6

0

5

10

15 Discrete time k

20

25

30

Figure 4: State responses with u(k) = (IN −1 ⊗(K + ρ(k)ΔK(k)))x(k). with the controller gain K in (49) and the perturbations occurred with l = 10 and ρ0 = 0.7; that is, u(k) = (IN −1 ⊗(K + ρ(k)ΔK(k)))x(k), where D = I2 , F (k) = sin(πk/2) and E = 0.05I2 , are drawn in Figure 4. Here, the values of element in matrix E are permitted beforehand by Theorem 3. Comparing with Figure 3, it can be confirmed that it is necessary to consider the probabilistic property; for example, Binomial sequence, to design the controller in the real environment like as the situation explained in Remark 1. ˜ is obtained as Finally, by applying of Theorem 3, the non-fragile controller gain K ⎡ ⎤ −2.7574 −0.1406 ˜ =⎣ ⎦. K −0.1906 −1.4172

(50)

Also, the simulation results for the state responses of system (48) with the non-fragile controller ˜ + ρ(k)ΔK(k)))x(k) and same condition above are shown in Figure 5. From u(k) = (IN −1 ⊗(K Figure 5, one can confirm that the performance of state is improved when comparing the results in Figure 4. In addition, in Figure 6, the sequence of ρ(k) is drawn l = 10 and ρ0 = 0.7.

5

Conclusions

In this paper, new synchronization criteria for the discrete-time CDNs with interval time-varying delays and randomly occurring perturbations in the controller were proposed. The randomly occurring perturbations in the controller are considered with the concept of the Binomial random process. To drive main results, the suitable Lyapunov-Krasovskii functional fractionizing the delay interval into two subsections and reciprocally convex approach were used to obtain the 24

4 3 2

xl(k)

1 0 −1 −2 −3 −4

0

5

10

15 Discrete time k

20

25

30

.

˜ + ρ(k)ΔK(k)))x(k). Figure 5: State responses with u(k) = (IN −1 ⊗(K 10

9

ρ(k)

8

7

6

5

4

0

5

10

15 Discrete time k

20

25

30

Figure 6: The curve of property in controller perturbations. feasible region of synchronization conditions for the synchronization. Two numerical examples have been given to show the effectiveness and usefulness of the presented methods.

Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (20080062611), and by a Grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea(A100054).

25

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