Partial-information-based consensus of network systems with time-varying delay via sampled-data control

Signal Processing 162 (2019) 97–105

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Partial-information-based consensus of network systems with time-varying delay via sampled-data control Shenghuang He, Yongkang Lu, Yuanqing Wu∗, Yanzhou Li School of Automation, Guangdong University of Technology, Guangzhou 510006, Guangdong, China

a r t i c l e

i n f o

Article history: Received 16 August 2018 Revised 21 February 2019 Accepted 7 April 2019 Available online 8 April 2019 Keywords: Network systems Consensus Sampled-data control Time-varying delay

a b s t r a c t In this paper, a novel decoupling method is introduced to investigate the consensus of network systems with partial coupling on the basis of sampling control. The novel method is enabled us to discuss channels of each node for network systems individually. By constructing several Lyapunov functionals and combining with free weighting approach, the stability of network systems with target node is analyzed. On the other hand, network systems with time-varying delay are considered. Based on Fourier theory, an alternative inequality is presented, which is more precise than the Wirtinger inequalities. This inequality encompasses the Jensen one and leads to manageable linear matrix inequalities conditions (LMIs). Therefore, the consensus of network systems with time-varying delay can be guaranteed. Through two illustrated examples, the effectiveness of the proposed method is demonstrated sufficiently. © 2019 Elsevier B.V. All rights reserved.

1. Introduction With the rapid development of intelligent devices and communication technology, consensus analysis of network systems has been received great attention over the past several decades [1–6]. Network systems are extensively applied in different fields, such as smart power grids [7], unmanned aerial vehicle, industrial applications [8], etc. Network systems can be drawn as a directed graph, containing many correlated nodes. These nodes exhibit several dynamics behaviors, which can be categorized as liner model [9,10], single-integrator [11,12] and double-integrator [13–15]. The mentioned behaviors have the important impact on the consensus of network systems. Nevertheless, it is significant to consider the nonlinear dynamic [16–18], which also improve the results. Consensus problem of heterogeneous multi-agent systems with nonlinear dynamics has been considered in [19]. Nonlinear multi-agent state-delay systems on the basis of adaptive consensus approach have been presented in [20]. Looking through a large number of literatures on the sampling pattern of network systems [21–24], we have witnessed fast increasing interest in sampled-data control systems. It is well known that sampling interval of periodic sampled-data systems is usually assumed as a constant. During the past few decades, the aperiodic sampling has stirred much attention in the design of sampleddata control [25,26]. It is beneficial to reduce the energy, commu-

∗

Corresponding author. E-mail address: [email protected] (Y. Wu).

https://doi.org/10.1016/j.sigpro.2019.04.012 0165-1684/© 2019 Elsevier B.V. All rights reserved.

nication costs and computation by adopting the nonuniform sampling pattern in sampled-data control systems. To deal with the uncertainty sampling interval, an input delay method has been introduced in [27]. A discrete-time system has been received transformed from sampled-data control systems by utilizing the exact integration over a sampling interval, and reduced conservative for stability criteria in [28]. A clock dependent Lyapunov function has been proposed to investigate impulsive system in [29], certain and uncertain case of sampled-data control has been considered by using this method. It should be pointed out that network nodes exchange information with their neighbors to achieve consensus. Nevertheless, information transmission is limited due to environmental disturbance or communication constraints, such as noise and time delay. The each network node has multiple dimensions representing information channel. However, in fact, not all channels can transmit information successfully, and the partial-coupling problem will be caused. The coupling phenomenon can be easily observed in network systems. In other words, information transmissions incompletely appear in different fields, like as ad hoc networks [30]. Multiple dimensions of each node are transmitted through multiple channels. The each network node has various levels of information, and multiple communication channels are used to transmit the corresponding level of information. For example, in [31], level of information is transmitted by multiple communication channels. In addition, time delays may appear in complex networks unavoidably, which also are a source of instability. It is noted that time delays have been catched much attention in various fields [32,33].

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S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

Moreover, many novel methods of evaluating the stability for time delay systems have been built. Such as a descriptor model transformation, free weighting matrix approach, integral inequality technology. In [34], a coupling method has been introduced to deal with partial coupling in complex network, and constant transmission time delay is considered in the local interactions of connected nodes. In [35], Wirtinger-based integral inequality has been presented, which used to time delay systems and sampled-data systems to reduce conservatism of resulting. Motivated by the above facts, in this paper, the consensus problem of network systems with sampled-data control is tackled. An interesting decoupling method is proposed to deal with the phenomenon of partial coupling between any two nodes. The stability criteria is derived, which expressed in terms of LMIs. On the other hand, network systems with time delay are considered. Wirtinger-based integral inequality, which is based on Fourier theory, is used to reduce the conservatism. Presented inequality encompasses Jensen inequality, and depends on the integral of sampling interval or the state over the delay. Therefore, we obtain a new stability criterion for network systems with time delay showed in LMIs. The major contributions of the paper are given as: (1) The dynamics of all nodes for network systems are common, which consist of liner-type, integrator-type, but the special case is nonlinear-type. (2) The partial information exchange among nodes is studied, which is more close to the practical applications. By proposing a new decoupling method, the consensus problem of network systems converts into the stabilization issue of an error system. (3) According to Fourier theory, an alternative inequality is given, which is more precise than the Wirtinger inequalities. This inequality encompasses the Jensen one and leads to maturely LMI conditions. Thus, the consensus of network systems subject to time-varying delay is guaranteed. This article is organized as four parts. Section 2 gives the problem formulation. Sections 3 and 4 represent the central contributions of the paper on exponential stability. Section 5 proposes a stability analysis of network systems subject to time delay. Two mathematical examples are presented in Section 6. Notations: From this paper, Rn represents the n-dimensional Euclidean space. Then, a set of i × j real matrices is denoted by Ri× j . The positive and symmetric matrix P is expressed by P > 0 and P ∈ Rn×n . diag{. . .} indicates block diagonal matrix. I stands for an identity matrix with appropriate dimensions. 0 denotes a zero matrix. The space of function ψ : [a, b] → Rn , the function is completely continuous on [a, b), has a finite limθ →b− ψ (θ ), has square integrable first order derivative, and is denoted by U[a, b) with a the norm ψU = maxθ ∈[a,b] |ψ (θ )| + [ b |ψ˙ (s )|2 ds]1/2 . The notation



A ∗



B C



means

A BT



B . C

Consider the dynamics of network systems with partial coupling nodes, which can be described as follows:



yi j Fi j (x j (t ) − xi (t ))

(2)

In this paper, sampling information is obtained at instant tk , which is maintained by zero-order-hold(ZOH) circuits at the sequence of hold time 0 < t1 < . . . < tk < . . . < limt→∞ tk = +∞. It is supposed that

hk = tk+1 − tk ∈ [h1 , h2 ] ≥ 0,

(3)

the sampling intervals are nonuniform, h2 > 0 and h1 > 0 denote the upper and lower bounds of sampling interval, respectively. Definition 1 [36]. For any initial conditions, the considered network systems with partial coupling can achieve synchronization exponentially, there exist positive scalars ι, ς such that

xi (t ) − v(t ) ≤ ιe−ς t , i = 1, 2, . . . , N

(4)

holds for t > T , where T > 0. Assumption 1. gk ( · ) is nonlinear function and satisfies the Lipschitz condition. The following inequality is true

0<

gk ( a ) − gk ( b ) ≤ bk , a−b

(5)

where k = 1, 2, . . . , n, a and b ∈ R Lemma 1 [37]. Given scalar κ , the following inequality is feasible for any real vector β 1 and β 2

β1T β2 + β2T β1 ≤ κβ1T β1 + κ −1 β2T β2 .

(6)

Lemma 2 [35]. For a given positive-definite matrix G > 0 and all continuous differentiable function f in [m, n] → Rn , then the following inequality holds



n

m

f˙ T (s )G f˙ (s )ds ≥

1 ( f (n ) − f (m ))T G( f (n ) − f (m )) n−m 3 + ζ T Gζ , n−m

where ζ = f (n ) + f (m ) −

2 n−m

n

m

(7)

f (s )ds.

1

1

(δ, W ) = ϑ T RT1 W R1 ϑ + ϑ T RT2 W R2 ϑ . δ 1−δ where n, m are positive integers, a scalar δ ∈ (0, 1). Assume that there exists a matrix Y which satisfies

j∈Ni

+ zi (v(tk ) − xi (tk )), i = 1, 2, . . . , N

v(t ) = Av(t ) + BG(v(t )).

Lemma 3 [38]. For a given positive matrix W ∈ Rn×n , matrices R1 and R2 ∈ Rn×m . Define a vector ϑ ∈ Rm , and the following function is given by

2. Problem formulation

x˙ i (t ) = Axi (t ) + BG(xi (t )) +

 one obtains that yii = j∈N yi j if i = j. Fi j = diag{ fi1j , fi2j , . . . , finj } is i a diagonal matrix, which means information flow between node j and node i with multiple channels. fikj = 1, or 0, which denotes kth channel of any two nodes exchange information or disconnected, where k = 1, 2, . . . , n. zi ≥ 0 is a scalar and denotes the connection strength between ith node and objective node. if zi > 0, node i can obtain the information from target node; otherwise, zi = 0. v(t ) = [v1 (t ), v2 (t ), . . . , vn (t )]T means the state of designed target node, whose dynamics are described in the following differential equation:

0, the following inequality is true

(1)

where subscript i means the ith node of network systems. xi (t ) = [xi1 , xi2 , . . . , xin ]T represents the state of ith node. A ∈ Rn×n is diagonal matrix and matrix B ∈ Rn×n . Define G(xi (t )) = [g1 (xi1 (t )), g2 (xi2 (t )), . . . , gn (xin (t ))]T as the nonlinear function satisfying the global Lipschitz condition. Ni indicates the adjacent nodes of node i. yij is connection weigh, if yij > 0, the node j ∈ Ni realizes signal transferring with node i; otherwise, yi j = 0. Then,



R1 ϑ min (δ, W ) ≥ R2 ϑ δ ∈ ( 0,1 )

T 

W YT

Y W



(8)



W YT

Y W

 >



R1 ϑ . R2 ϑ

(9)

Remark 1. In Lemma 2, due to G > 0, the right term of (7) is definite positive. Obviously, Jensen inequality is adopted in inequality (7). On the other hand, an improvement aspect is presented, i.e., n it can be applied not only to signal m f˙ (s )ds, but also to signal nu f˙ (s )ds. m m

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

3. Main results The difference of node and objective node is state consensus error which is defined as θi (t ) = xi (t ) − v(t ), where θi (t ) = [θi1 (t ), θi2 (t ), . . . , θin (t )]T . The dynamic of error θ i (t) is given as follows:

θi (t ) =Aθi (t ) + BG(xi (t ), v(t )) +

N 

where

⎡

(10)

⎤

g1 (xi1 (t )) − g1 (v1 (t )) ⎢g2 (xi2 (t )) − g1 (v2 (t ))⎥ ⎥. G(xi (t ), v(t )) = G(xi (t )) − G(v(t )) = ⎢ ⎣ ⎦ ... gn (xin (t )) − gn (vn (t )) Keeping in mind that there exists difference between two nodes with partial coupling, a decoupling method is presented to cope with this problem. Each channel is considered based on this method independently. Then, we define the following matrix for eth channel: z f11 z f21 f21 ⎢ Fz = ⎢ . ⎣ .. z fN1 fN1

⎤

... ... .. . ...

z f12 f12 z f22 .. . z fN2 fN2

f1N f1zN f2N f2zN ⎥ .. ⎥ ⎦, . z fNN

z . where e = 1, 2, . . . , n, b = 1, 2, . . . , N, f jzj = − a=1,a =b fba fba T Define θ¯z (t ) = [θ1z (t ), θ1z (t ), . . . , θNz (t )] as the consensus error state vector of zth channel for these nodes. If the condition θ¯z (t ) →

0 is satisfied, the state of network systems can achieve consensus. The differential equation of zth channel is expressed as n 

ˆ b + (F z − Z )θ¯z (tk ), bzc G

(11)

c=1

where

⎡

 ηz1 (t ) = θ¯z (t )T θ¯z (tk )T  ηz2 (t ) = θ¯z (t )T θ¯˙ z (t )T ⎡ S +ST −S −ST 1z

2z

1z

2z

2

Y =⎣

t tk

T θ¯z (s )T ds ,

z1 (hˇ ) =

∗ ∗

1

−k1z I 0

3

η12 + λ12 η22 + λ22

⎡ η11 + ϑ11 ∗ ⎢ ¯ z2 = ⎣  ∗

η12 + ϑ12 η22 + ϑ22

∗ ∗

∗

η13 + λ13 η23 + λ23 η33 + λ33

∗ ∗

∗

∗

η13 + ϑ13 η23 η33

η14 + λ14

⎤

−U2Tz ⎥ , η34 + λ34 ⎦ η44 + λ44

η14 + ϑ14

∗

−U2Tz

η34 η44

⎤ ⎥ ⎦,

S1z + S1T z − S2z − S2T z −S1z − S1T z + S2z + S2T z , z22 = , 2 2 √ √ = [ nW1Tz 0 0 0]T , 2 = [0 nW1Tz 0 0]T ,

z11 =

1 3 = −e−2ς h2 Qz , η11 = ς (Pz + PzT ) − z11 + V1z + V1Tz + az (W1z + W1Tz ) n 

(k1z + k2z )bc I,

c=1

η12 η13 η33 η34

η22 = −W2z − W2Tz , = S2z − S2T z − Rz + V3z − V1z + W1Tz (F z − Z ), = − z22 + Rz + RTz − V3z − V3Tz , η14 = −S3z − U1z + V4z , = −S4z − U3Tz − V4Tz , η44 = − z33 − U4z − U4Tz , = Pz + V2z + azW2z − W1Tz ,

ϑ11 = hˇ (U1z + U1Tz ), ϑ12 = hˇ U2z , ϑ13 = hˇ U3z , ϑ14 = hˇ U4z , ϑ22 = hˇ

h2 Qz , 4

for hˇ ∈ {h1 , h2 }, and given the initial conditions, the stability of network systems with partial coupling can be guaranteed by sampled-data control.

T ¯z (s )T ds , θ tk ⎤

t

θ¯z (tk )T −S2z + S2T z

Proof. The time-dependent Lyapunov functional can be chosen as follows:

S3z

⎦. S4z T S5z + S5z

Va (t ) =

Theorem 1. Given positive scalars h1 , h2 , κ 1z and κ 2z , z = 1, 2, . . . , n, there exist positive matrices Pz , Sz , Qz , and Rz , matrices S1z , S2z , S3z , S4z , S5z , Uz = [U1z U2z U3z U4z ], Vz = [V1z V2z V3z V4z ], Wz = [W1z W2z 0 0] satisfying

¯ z1

⎡ η11 + λ11 ⎢ ∗ ¯ z1 = ⎣ ∗

(14)

λ33 = 2ς hˇ z22 − 4ς hˇ Rz , λ34 = ς hˇ S4ze , λ44 = ς hˇ z33 , η23 = −V2Tz + W2Tz (F z − Z ), z33 = S2z + S2T z

∗

Pz + hˇ z11

z (hˇ ) = ⎣ ∗ ∗

⎤

hˇ VzT 0 ⎥ ⎥ 0 ⎥ < 0, ⎦ 0

λ13 = 2ς hˇ (−S2T z + S2T z ) + hˇ S4T z + 2ς hˇ Rz , λ14 = ς hˇ S3z + hˇ z33 ,

2

⎡

0 −k2z I 0 ∗



λ23 = hˇ (−S2z + S2T z ),

−S1z −S1T z +S2z +S2T z

∗ ∗

h2 hˇ UzT 0 0 −2Sz 0

λ11 = 2ς hˇ z11 + hˇ (S3z + S3T z ), λ12 = hˇ z11 ,

⎤

gb (x1b (t )) − gb (vb (t )) ⎢gb (x2b (t )) − gb (vb (t ))⎥ ⎥. ˆb = ⎢ G .. ⎣ ⎦ . gb (xnb (t )) − gb (vb (t )) Denote



2

−k1z I 0 ∗ ∗

∗

+ N

θ¯˙ z (t ) =az θ¯i (t ) +

1

where

ai j Fi j θi (tk ) − zi θi (tk )

i=1

⎡

⎡ ¯ z2  ⎢ ∗ ⎢ z2 (hˇ ) = ⎢ ∗ ⎣ ∗

99

hˇ (−S2z + S2T z ) hˇ z22 ∗

2 0 −k2z I

Vai (t ),

(12)

where

Va1 (t ) = Va2 (t ) =

n 

n  z=1

< 0,

θ¯z (t )T Pz θ¯z (t ),

n   z=1

Va3 (t ) =

 (13)

t ∈ [tk , tk+1 ),

i=1

z=1

⎤

hˇ S3z hˇ S4z ⎦ > 0, hˇ z33

5 

Va4 (t ) =

n  z=1

0 −t+tk



t

e2ς (s−t ) θ¯˙ z (s )T Sz θ¯˙ z (s )dsdσ ,

t+σ

(tk+1 − t )



t

tk

e2ς (s−t ) θ¯˙ z (s )T Qe θ¯˙ z (s )ds,

(tk+1 − t )ηz1 (t )T Y ηz1 (t ),

(15)

100

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

Va5 (t ) =

n 

(tk+1 − t )(θ¯z (t ) − θ¯z (tk ))T 2Rz θ¯z (tk ).

(16)

z=1

Therefore, (18) and (19) can be developed as follows:

V˙ a2 (t ) + 2ς Va2 (t ) ≤

It is clearly obtained that

n   h2 (t − tk )

2

z=1

Va (t ) ≥ Va1 (t ) + Va4 (t )  h − (t − t ) k+1 T ≥ ηz1 (t ) 2

z ( h 1 ) h2 − h1  (t − t ) − h1 + k+1

z (h2 ) ηz1 (t ). h2 − h1

V˙ a3 (t ) + 2ς Va3 (t ) ≤

From inequality (15), it is obtained that z (h1 ) > 0 and

z (h2 ) > 0. Therefore, we obtain Va (t) > 0. It is obvious that (15) is

+ 2ηz2 (t )T VzT θ¯z (t )

(tk+1 − t )θ¯˙ z (t )T Sz θ¯˙ z (s ) + 2(t − tk )ηz2 (t )T UzT θ¯z (t )  t  − 2ηz2 (t )T UzT θ¯z (s )ds , tk

n 

 2θ¯z (t )T Pz θ¯˙ z (t ) + 2ς θ¯z (t )T Pz θ¯z (t ).

V˙ a2 (t ) + 2ς Va2 (t ) =

n  z=1

n 

u1 (t, tk ).

(18)

h2 ¯˙ θz (t )T Qz θ¯˙ z (t ) 4



(23)

Inspired by the descriptor method and given matrix Wz = [W1z W2z 0 0] with proper dimensions, one has



2ηz2 (t )WzT az θi (t ) +

(tk+1 − t )(t − tk )θ¯˙ z (t )T Qz θ¯˙ z (t )

h2 ¯˙ θz (t )T Qz θ¯˙ z (t ) 4

− 2ηz2 (t )T VzT θ¯z (tk ) .

(17)

z=1

n 

V˙ a3 (t ) + 2ς Va3 (t ) =

+ (t − tk )

z=1

(tk+1 −t )θ¯˙ z (t )T Sz θ¯˙ z (s ) +

(tk+1 − t )

+ (t − tk )ηz2 (t )T e−2ς h2 VzT Sz−1Vz ηz2 (t )

n

z=1

n   z=1

effective Lyapunov functional for the system (10). By the time derivative of Va (t) (a = 1, 2, 3, 4, 5 ) along system (10), we get V˙ a1 (t ) + 2ς Va1 (t ) =

ηz2 (t )T e−2ς h2 UzT Sz−1Uz ηz2 (t )

n 



ˆ c + (F z − Z )θ¯z (tk ) − θ¯˙ i (t ) = 0. bzc G

c=1

z=1 n 

+

(24) u2 (t, tk ).

(19)

z=1

V˙ a4 (t ) + 2Va4 (t ) =

n 

ηz1 (t )T Y ηz1 (t )

z=1

c=1

⎤ θ¯˙ z (t ) +2 (tk+1 − t )ηz1 (t )T Y ⎣ 0 ⎦. z=1 θ¯z (t ) V˙ a5 (t ) + 2Va5 (t ) =

−1 ¯˙ 2z z

⎡

n 

n 

+ nκ (20)

(tk+1 − t )θ¯˙ z (t )2Rz θ¯z (tk )

n 

(tk+1 − t )(θ¯z (t ) − θ¯z (tk ))T 2Rθ¯z (tk ).

u1 (t, tk ) = − u2 (t, tk ) = −





0

tk −t



t

e tk

t

t+σ −2h2

(21)

θ ( ) θ (t ).

n 

θ¯zT (t )

t

(26)

−t (z + 2z (hk )) hk

k+1



t − tk + (z + 3z (hk )) θ¯z (t ), hk where

⎡ η¯11 ⎢∗ z = ⎣ ∗

θ¯˙ z (s )T Sz θ¯˙ z (s )dsdσ ,

η12 η¯ 22 ∗ ∗

∗

θ¯˙ z (s )T Qz θ¯˙ z (s )ds.

η¯ 11 =

It is noted that the free-weighting matrix approach is inspired in [36], for arbitrary matrices Uz and Vz , we can obtain that

h2 (t − tk ) u1 (t, tk ) ≤ ηz2 (t )T e−2ς h2 UzT Sz−1Uz ηz2 (t ) 2 + 2(t − tk )ηz2 (t )T UzT θ¯z (t )  t − 2ηz2 (t )T UzT θ¯z (s )ds,

⎡ λ11 ⎢∗ 2z (hk ) = ⎣ ∗

λ12 λ22

⎡ ϑ11 ⎢ ∗ 3z (hk ) = ⎣ ∗

ϑ12 ϑ22

∗ ∗

∗ ∗

∗

+ h2 e

u2 (t, tk ) ≤ 2ηz2 (t )T VzT θ¯z (t ) − 2η2 (t )T VzT θ¯z (tk ) (22)

−2ς h2

Thus, we obtain

η13 η23 η33 ∗

T nk−1 1z W1zW1z ,

∗

tk

+ (t − tk )ηz2 (t )T e−2ς h2 VzT Sz−1Vz ηz2 (t ).

+ (κ1z + κ )

z=1

z=1

where

θ()

V˙ a (t ) + 2ς Va (t ) ≤

z=1

2ς

θ()

Recalling (17), (20), (21), (23), and (25), t ∈ [tk , tk+1 ), it is easy to calculate that

z=1

+

t W2TzW2z ¯˙ z t n  dc b2zc ¯c t T ¯c 2z T

c=1

−(θ¯z (t ) − θ¯z (tk ))T 2Rz θ¯z (tk )

n 

(25)

  n ˆ c ≤ nκ −1 θ¯z (t )T W1TzW1z θ¯z (t ) 2 θ¯z (t )T W1z + θ¯˙ z (t )T W2z bzc G 1z

z=1

−

≤ dc θ¯c (t )T θ¯c (t ),

and by using Lemma 1, the following inequality is true

2ς (tk+1 − t )ηz1 (t )T Y ηz1 (t )

n 

According to Assumption 1, we obtain

ˆTG ˆ G b c

η14

(27)

⎤

⎥ , η34 ⎦ η44

−U2Tz

T η¯ 22 = nk−1 2z W2zW2z , ⎤ λ13 λ14 λ23 0 ⎥ , λ33 λ34 ⎦ ∗ λ44 ⎤ ϑ13 ϑ14 0 0 ⎥ 0 0 ⎦

∗ UzT Sz−1Uz

0 + e−2ς h2 VzT Sz−1Vz .

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

z + 2z (hk ) =

[θi1 (t ), θi2 (t ), . . . , θin (t )]T . The dynamic of θ i (t) is shown as

h2 − hk (z + 2z (h1 )) h2 − h1 h − h1 + k (z + 2z (h2 )). h2 − h1

θ˙ i (t ) = Aθi (t ) + Y θi (t − h(t )) + BG(xi (t ), v(t ))

Due to z1 (hˇ ) < 0, and combined with Schur complement, we get z + 2z (h1 ) < 0 and z + 2z (h2 ) < 0. Therefore, we conclude that z + 2z (hk ) < 0. Similarly, it is obtained that z + 3z (hk ) < 0. Motivated by the above discussions, we yield V˙ a (t ) + 2ς Va (t ) ≤ 0. By integrating that, it can be obtained that Va (t ) ≤ e−2ς t Va (0 ), t ∈ [tk , tk+1 ). Then, from Lemma 1 of [24], one obtains

θ¯z (t )2 ≤ υθ¯z (tk )2 υ ≤ e−2ς tk Va (0 ) λmin (Pz ) υ = e−2ς t e2ς (t−tk )V1 (0 ) λmin (Pz ) υλmin Pz −ς t ¯ ≤ e 2ς h 2 e  θz ( 0 )  , λmin Pz implies that

θ¯z (t ) ≤ e2ς h2



+

N 

(32)

where θ¯z (t ) = [θ1z (t ), θ1z (t ), . . . , θNz (t )]T is the consensus error state vector of zth channel for these nodes. Suppose that the condition θ¯z (t ) → 0 is satisfied, the network systems can achieve synchronization exponentially. Then, it is easy to obtain the differential equation of zth channel n 

ˆ b + (F z − Z )θ¯z (tk ). bzc G

c=1

(33) For convenience, we denote

(28)

(29)

According to Definition 1, exponential stability of network systems with partial coupling is achieved.  Remark 2. In this paper, we study a decoupling method for network systems, which has two main advantages. In the first place, it is helpful to study channels by building the matrix Fz . In the second place, if the network framework is complex, it is not necessary to establish the Kronecker product, which extremely reduces the computational burden. Remark 3. A series of Lyapunov functionals is constructed. It is obviously found that, before and after sampling instant tk , the terms of V2z , V3z and V5z will disappear, which fully capture the feature of the sampled-data systems and perform effectively the sampling pattern. On the other hand, the augment functional V3z (t) plays an important part in reducing the conservatism of our main results. 4. Network systems with time-varying delay In practical applications of engineering systems, time delay is ubiquity, such as chemical control systems, unmanned aerial vehicles (UAVs), lasers models, etc. Time delays have a great impact on the challenges of communication and information technology, i.e. the stability of network control systems will be impacted. Nevertheless, for some cases, the presence of delay plays a positive impact on the stabilizing effect. Therefore, the network systems with time-varying delay should be considered, the corresponding dynamic is described as follows:



T θ¯zT (t ) θ¯zT (t − h(t )) θ¯zT (t − hM ) ,   T  t −h(t ) T 1 ¯ (s )ds , x˜3z (t ) = h(1t ) tt−h(t ) θ¯zT (s )ds h −h θ z t−h ( t ) M M   x˜2z (t ) =



ei = 0n×(i−1)n

Theorem 2. For 0 ≤ h(t) < hM , dm ≤ h˙ (t ) ≤ dM , we assume that Assumption 1 is satisfied. Given positive constants κ 1z , κ 2z and sampling upper bound h2 . If there exist matrices Qz > 0, Rz > 0, Kz > 0, Pz ∈ R3n×3n > 0, X ∈ R2n×2n , U1z , U2z , and the following inequalities hold



ϒz (h, h˙ ) =

 =

Q˜z ∗

The difference of ith node and objective node is state error, which is θi (t ) = xi (t ) − v(t ), where θi (t ) =

ϒ¯ z (h, h˙ )

J1z −κ1z ∗

∗ ∗



J2z 0 −κ2z



< 0,

(34)

X > 0, Q˜z

(35)

then, network systems subject to partial couplings and timevarying delay is stable. where

R1 = [I

−I

0

0

0

0

0],

R2 = [I

−I

0

0

0

0

0],

R3 = [0

I

−I

0

0

0

0],

R4 = [0

I

I

− 2I

0

0



0],

T

= RT1 RT2 RT3 RT4 ,   ϒ¯ z (h, h˙ ) = T1 Pz 2 − 1 − h˙ (t ) eT2 Rz e2 + eT1 Kz e1 Q˜z = diag(Qz , 3Qz ),

− eT3 Kz e3 + hM eT2 Qz e2 −

1 T

 + h22 eT6 Hz e6 hM

π2

(e7 − e1 )T Hz (e7 − e1 ) + eT1 Rz e1 4 + eT1 U1Tz az e1 + eT1 U1z (F z − Z )e7 − eT1 U1z e6 + eT6 U6Tz az e1 −

(30)

(31)

i = 1, 2, . . . , 7

T ξz (t ) = x˜T2z (t ) x˜T3z (t ) θ¯˙ zT (t ) θ¯zT (tk ) .

yi j Fi j (x j (t )−xi (t ))

where h(t) ∈ [0, hM ] denotes time-varying delay, h˙ (t ) ∈ [dm , dM ] and dm ≤ dM ≤ 1. Define v(t ) = [v1 (t ), v2 (t ), . . . , vn (t )]T as the state vector of designed target node, whose dynamics are described in the following equation

0n×(7−i )n ,

In



j∈Ni

+ zi (v(tk ) − xi (tk )), i = 1, 2, . . . , N

v˙ (t ) = Av(t ) + Y v(t − h(t )) + BG(v(t )).

ai j Fi j θi (tk ) − zi θi (tk ),

i=1

θ¯˙ z (t ) =az θ¯i (t ) + yz θ¯i (t − d (t )) +

υλmin Pz −ς t ¯ e  θz ( 0 )  . λmin Pz

x˙ i (t ) = Axi (t )+Y xi (t − d (t ))+BG(xi (t ))+

101

+ eT6 U2z (F z − Z )e7 − eT6 U2z e6 + (k1z + k2z ) J1z =

√ 

J2z = 0

nU1Tz 0

0 0

0 0

0 0

0 0 √ T nU2z

T

n 

dc bzc eT1 e1 ,

c=1

0 ,

T

0 .

Proof. Let us consider the Lyapunov functional for the error system (33) showed as

Vb (t ) = Vb1 (t ) + Vb2 (t ),

(36)

102

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

where

Vb1 (t ) = x˜T1z (t )Pz x˜1z (t ) +  +



t

t−hM

 +

t

t−hM



t

σ



where t

t −h(t )

z

− eT3 Kz e3 + hM eT2 Qz e2 −

θ¯˙ T (s )Qz θ¯˙ z (s )dsdσ z

θ¯zT (s )Kz θ¯z (s )ds,

(37)

π2

t



tk



where x˜T1z (t ) = θ¯zT (t )

t



tk

θ¯ (tk ) − θ¯ (s )

 t −h(t )

¯T t −h (t ) θz (s )d s

t−hM

+ eT1 yzW1Tz e2 + eT6 yzW2z e2 + (κ1z + κ2z )



θ¯zT (s )ds . Differen-

tiating the functional along the trajectories of (37) yield:



V˙ b1 (t ) = ξzT (t ) T1 Pz 2 + eT1 Rz e1 − (1 − h˙ (t ))eT2 Rz e2 + eT1 Kz e1 − eT3 Kz e3 + hM eT2 Qz e2



V˙ b2 (t ) = ξzT (t ) h22 eT6 Hz e6 −



π2 4

ξz (t ) −



t

t−hM



t

t−hM

 1 RT Q˜z R23 h(t ) 23  1

hM − h(t )

RT45 Q˜z R45



t

t−hM

θ¯˙ T (s )Qz θ¯˙ (s )ds ≤ −

2ξ (t ) +

WzT

n 

ξz (t ).

(40)

1 T ξ (t ) T  ξz (t ). hM z

(41)



az θ¯i (t ) + yz θ¯z (t − h(t ))

ˆ c + (F z − Z )θ¯z (tk ) − θ¯˙ i (t ) = 0, bzc G

(42)

c=1

where Wz = [W1z 0 0 0 0 W2z 0] with proper dimensions. According to Assumption 1, we obtain T ¯ ¯ ˆTG ˆ G b c ≤ dc θc (t ) θc (t )

(43)

  n ˆc 2 θ¯z (t )T U1z + θ¯˙ z (t )T U2z bzc G c=1

≤ nκ

θ()

T T ¯˙ ¯˙ θ ( ) + nκ2−1 z θz (t ) U2zU2z θz (t )

t U1TzU1z ¯z t n  dc b2zc ¯c 2z c=1

+ (κ1z + κ )

θ (t )T θ¯c (t ).

(44)

It can be found from (39), (41), (42) and (44) that we obtain the following inequality

V˙ b (t ) ≤

n  z=1

ζzT (t )ϒz (h, h˙ )ξz (t ),

(49)

5. Numerical examples In this section, we present two numerical examples to illustrate effectiveness of the proposed method for network systems subject to partial couplings. Example 1. Consider the consensus of network systems with partial coupling via sampled-data control, the corresponding system parameters are described as follows:



−1 0 0

(45)

0 −1 0



0 0 , B= −1



0.77 −0.72 −0.91

0.8 −1.56 −0.78



0.25 −0.63 . 0.23

It is to be noted that information exchange happens in arbitrary two nodes which is presented by matrix F = [ fi j ] ∈ RN×M shown as ⎡ ⎤

−0.5

and by using Lemma 1, it can be found that the following inequality is true

T

Vb (t ) < e−αt V (0 ), ∀t > 0.

A=



−1 ¯ 1z z

(48)

Thus, the trajectories of between nodes and target node achieve agreement, which implies that the consensus of network systems with time-varying delay is guaranteed according to Definition 1. 

On the other hand, inspired by the descriptor method, one has T z

(47)

Assume that α = mini=1,2,... {αi } is a positive constant, we have

Then, for given matrices X and  > 0, the following inequality is ensured by Lemma 3 of [38]

−

(46)

Based on Schur complement, it is calculated that we have ϒz (h, h˙ ) < 0 from (34), which implies V˙ b (t ) < 0. Given a small sufficiently scalar α i , then according to Pz > 0 and inequality (35), for t ∈ [tk , tk+1 ) we obtain

Vb (t ) < e−αi (t−tk )Vb (tk ), t ∈ [tk , tk+1 ).

θ¯˙ T (s )Qz θ¯˙ (s )ds ≤ −ξzT (t ) +

dc bzc eT1 e1

c=1

By utilizing the comparison principle, it is obtained that

θ¯˙ zT (s )Qz θ¯˙ z (s )ds,

(39)

−

T T −1 T T + nκ1−1 z e1 U1zU1z e1 + nκ2z e6 U2zU2z e6 .

n 

V˙ b (t ) + αVb (t ) < 0.

 (e7 − e1 )T Hz (e7 − e1 ) ξz (t ).

By Lemma 2, we have

π2

+ eT6 U6Tz az e1 + eT6 U2z (F z − Z )e7 − eT6 U2z e6

T (38)

t

1 T

 + h22 eT6 Wz e6 hM

(e7 − e1 )T Wz (e7 − e1 ) + eT1 Rz e1 4 + eT1 U1Tz az e1 + eT1 U1z (F z − Z )e7 − eT1 U1z e6 −

θ¯˙ zT (s )Hz θ¯˙ z (s )ds − 4   ¯ ¯ Hz θ (tk ) − θ (s ) ds,

Vb2 (t ) = h22

  ϒz (h, h˙ ) = T1 Pz 2 − 1 − h˙ (t ) eT2 Rz e2 + eT1 Kz e1

θ¯ T (s )Rz θ¯z (s )ds

⎢ 0.1 F =⎢ 0 ⎣ 0.2 0.1

0.1 −0.4 0.25 0.2 0

0 0.1 −0.45 0.1 0.2

0.2 0.2 0.1 −0.6 0.1

0.2 0 ⎥ 0.1 ⎥. ⎦ 0.1 −0.4

Dimensions of each network node are defined as 3. Node i sends information to node j by three channels, respectively. Fij is diagonal matrix and denotes the information flow of all channels of node j and node i. It can be obtained that

F12 = diag{0, 1, 0},

F13 = diag{0, 0, 0},

F14 = diag{0, 0, 1},

F15 = diag{1, 0, 0},

F21 = diag{1, 1, 1},

F23 = diag{0, 0, 0},

F24 = diag{0, 0, 0},

F25 = diag{0, 0, 0},

F31 = diag{0, 0, 0},

F32 = diag{1, 0, 1},

F34 = diag{0, 1, 0},

F35 = diag{0, 0, 1},

F41 = diag{0, 0, 1},

F42 = diag{0, 1, 0},

F43 = diag{0, 1, 0},

F45 = diag{1, 1, 0},

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

103

Fig. 1. The information channels for network systems and target node when z = 1.

F51 = diag{1, 0, 0},

F52 = diag{0, 0, 0},

F53 = diag{0, 0, 1},

F54 = diag{1, 1, 0}.

The dynamic of target node is presented in (2). According to Fig. 1, first node connects with target node, it can be separated into two steps to understand. Firstly, target node sends information to node 1. Secondly, information flow between node i and node j by three channels. The diagonal matrix is shown as Z = diag{1 0 0 0 0}. It is noted that a communicational graph is designed, i.e., target node is root and each node contains a spanning tree. In this paper, a novel decoupling method is applied to deal with partial couplings for any two nodes. The matrix F z ∈ RN×N is defined as the zth channel of N nodes. Motivated by the above discussions, we have the matrices Fz which are given as (z = 1, 2, 3 )

⎡−0.2

⎢ 0.1 F1 = ⎢ 0 ⎣ 0 0.1

0 −0.1 0.25 0 0

0 0 −0.25 0 0

⎢ 0.1 F2 = ⎢ 0 ⎣ 0 0

0.1 −0.3 0 0.2 0

0 0 −0.1 0.1 0

⎢ 0.1 F3 = ⎢ 0 ⎣ 0.2 0

0 −0.1 0.25 0 0

0 0 −0.35 0 0.2

⎡−0.1

⎡−0.2

0 0 0 −0.1 0.1 0 0.2 0.1 −0.4 0.1 0.2 0 0 −0.2 0

Fig. 2. The state error of the first channel between nodes and target node. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

⎤

0.2 0 ⎥ 0 ⎥ ⎦ 0.1 −0.2

⎤

0 0 ⎥ 0 ⎥ ⎦ 0.1 −0.1

⎤

Fig. 3. The state error of the second channel between nodes and target node. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0 0 ⎥ 0.1 ⎥. ⎦ 0 −0.2

The nonlinear activation function is given as

⎡ |x (t ) + 1| − |x (t ) − 1| ⎤ i1 i1 ⎤ 2 ⎢ ⎥ g1 (xi1 (t )) ⎢ |xi2 (t ) + 1| − |xi2 (t ) − 1| ⎥ ⎥ G(xi (t )) = ⎣g2 (xi2 (t ))⎦ = ⎢ ⎢ ⎥ 2 ⎣ |x (t ) + 1| − g3 (xi3 (t )) |x (t ) − 1| ⎦ ⎡

i3

i3

2 and it satisfies the Lipschitz condition, where bc = 1, c = 1, 2, 3. The initial state of nodes is randomly selected, and the initial state of the target node is chosen as [3 3 1]T . It is noted that the consensus of the network systems cannot be achieved if control action of target node is loss. The sufficient conditions are shown in Theorem 1. We assume that h1 = 0.0 0 05, the value of sampling intervals is obtained, i.e., h2 = 0.05. By the working of consensus theme, the state error value between target node v(t) and node state is presented in Figs. 2–4. In these figures, red dotted line

Fig. 4. The state error of the third channel between nodes and target node. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

104

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

Fig. 6. The error value between nodes and target node for three channels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. The total state error converges to 0.

stands for error value between the first node state and target node state. Similarly, green dotted line, blue dotted line, and so on, corresponding to the error value. Obviously, the error value tends to zero in a short time by utilizing the consensus theme. Therefore, the state trajectories of each node and target node can achieve consensus.  5 The total state error is defined as e(t ) = i=1  xi (t ) − v (t ) , which is shown in Fig. 5. Noted that networks systems achieve consensus with target node in a brief time. The obtained results demonstrate the effectiveness of the proposed method. Example 2. In this example, network systems with time-varying delay are considered, the corresponding system matrix parameters are given as follows:



A=

0 −1 0

0 0 , B= −1

−1 0 0

0 −1 0

0 0 . −1

 Ad =



−1 0 0





0.77 −0.72 −0.91

0.8 −1.56 −0.78



0.25 −0.63 , 0.23

The nonlinear function is defined in Example 1. Initial value of each node and target node is chosen as x1 (0 ) = [−7 8 − 8]T , x2 (0 ) = [3 10 − 6]T , x3 (0 ) = [1 − 2 − 4]T , x4 ( 0 ) = [6 − 4 4]T , x5 (0 ) = [5 1 8]T , V (0 ) = [−3 6 5]T . We select time-varying delay function as h(t ) = 0.375 + 0.375 × cost. It is noted that h˙ (t ) < 1. Based on the proposed control strategy, the consensus of network systems with target node can be guaranteed. Then, the state error value between nodes and target node is demonstrated in Fig. 6. The red dotted line denotes state error between first node and target node. Similarly, error value between other nodes and target node is designed as green dotted line,blue dotted line and so on. The total error is defined 5 as e(t ) = i=1  xi (t ) − v (t ) . Then, the consensus can be achieved in a short time, the total error value tend to zero which is expressed in Fig. 7. The upper bound of sampling interval is obtained, i.e. h2 = 1.28. The obtained results sufficiently illustrate the effectiveness of proposed approach. 6. Conclusion This paper has investigated the consensus problem of network systems subject to time-varying delay with aperiodic sampled-data control. By using decoupling method, the difficulty of information

Fig. 7. The total state error converges to 0.

exchange among connected nodes has been solved. On the basis of Lyapunov functional approach and free-weighting matrix method, the stability criteria for network systems have been established. The design of Lyapunov functionals has captured the characteristic of sampled-data network systems. The obtained conditions have guaranteed the consensus of network systems with target node. Then, time-varying delay for such system has been considered. Wirtinger-based integral inequality has been proposed, and the efficient consensus strategy has been established. Two numerical examples have been presented to express the validity of the proposed approach.

Acknowledgments This work is supported by the National Natural Science Foundation of China (grant no. 51405430, U1509210, 61473258, 61573322 and 61403113), the National High-Tech Research And Development Program (863) of China (grant no. 2012AA041703) and Public Welfare Technology Application Research Plan of Zhejiang (grant no. 2016C33G2010137), the Zhejiang Provincial Natural Science Foundation of China (grant no. LQ14F030010).

S. He, Y. Lu and Y. Wu et al. / Signal Processing 162 (2019) 97–105

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