Consensus of second-order delayed nonlinear multi-agent systems via node-based distributed adaptive completely intermittent protocols

Applied Mathematics and Computation 326 (2018) 1–15

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Consensus of second-order delayed nonlinear multi-agent systems via node-based distributed adaptive completely intermittent protocols Hongjie Li a,∗, Yinglian Zhu a, Liu jing a, Wang ying b a b

College of Mathematics and Information and Engineering, Jiaxing University, Zhejiang 314001, PR China College of Mathematics and Systems Science, Shandong University of Science and Technology Shandong 266590, PR China

a r t i c l e

i n f o

Keywords: Multi-agent systems Second-order consensus Time delay Adaptive intermittent control Distributed adaptive law

a b s t r a c t The paper discusses second-order consensus problem of nonlinear multi-agent systems with time delay and intermittent communications. Basing on local intermittent information among the agents, an effective control protocol is proposed by node-based distributed adaptive intermittent information, which a time-varying coupling weight to each node in the communication, some novel criteria are derived in matrix inequalities form by resorting to the generalized Halanay inequality. It is proved that second-order consensus can be reached if the measure of communication is larger than a threshold value under the strongly connected and balanced topology. Moreover, consensus problem is also considered for second-order non-delayed nonlinear multi-agent systems. Finally, a simulation example is presented to illustrate the theoretical results. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Over the last few years, the coordination problem of multi-agent systems has attracted considerable attentions due to their extensive application in many field, such as sensor networks, spacecraft formation flying, power grid and so on [1,2]. The consensus problem plays an important role in the area of cooperative control and has been investigated from various perspectives [3–7], where the main task is to design an appropriate protocol based on the local relative information to achieve consensus [8–18]. In the aforementioned literature, many results are based on the assumption that each individual agent is governed by first-order dynamics [8–11], however, second-order consensus algorithms will contribute to the study of more complicated dynamics, where all the agents are governed by position and velocity states, thus helping engineering implement the consensus algorithms in many real-world networked multi-agent systems. Therefore, there is a growing interest focusing on the second-order consensus problem. In [12], some sufficient conditions are obtained for achieving second-order consensus under directed communication topology. It is shown that second-order consensus may fail to be reached even if the communication topology has a spanning tree, and some additional conditions should to be satisfied for achieving second-order consensus [18–23], which are somewhat different from those in multi-agent systems with first-order dynamics. Recently, it can be seen that the real and imaginary parts of the eigenvalues of the Laplacian matrix play key roles in reaching con-

∗

Corresponding author. E-mail address: [email protected] (H. Li).

https://doi.org/10.1016/j.amc.2018.01.005 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

Fig. 1. Illustration for aperiodic intermittent communication.

Fig. 2. Node-based adaptive protocol.

sensus [5,22,24–26]. Firstly, there is a common assumption in the existing literature that each agent can receive the state’s information between its neighbors and itself all the time, which is not always appropriate in some real situations due to the unreliability of communication channels, failure of physical devices and recovery of the sensing devices from failures, thus it is reasonable to assume that each agent can sense its neighbors only in intermittently [27–33], where the consensus for the first-order and second-order multi-agent systems is investigated with intermittent information transmissions, which is shown that consensus can be reached if the communication time duration is larger than their corresponding threshold value. Due to the intermittent information transmissions, as a discontinuous control, intermittent control is introduced and activated during certain nonzero time intervals, but is off during other time intervals, which has been widely used in engineering fields for its practical and easy implementation [34–38], such as manufacturing, transportation and communications. Secondly, adaptive strategies to appropriately tune the strengths of the interconnections among network nodes have been proposed [39–41]. The synchronization of complex topologies using coupling of time-varying strength is numerically investigated and made a comparison between fixed and varying coupling strength [40], it can be seen that the fixed coupling strength is larger than those needed in practice. The strength is given in advance to guarantee the performance of a network in the worst cases, but the worst cases are seldom happen in practical systems [42], therefore, it may be conservative, adaptive strategies are proposed and can effectively overcome these shortcomings, which can appropriately tune the

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

Fig. 3. The communication topology of multi-agent systems.

Fig. 4. The position state trajectories of agents xi (t) (i = 1, 2, . . . , 7 ) under the adaptive intermittent control.

Fig. 5. The velocity state trajectories of agents vi (t ) (i = 1, 2, . . . , 7 ) under the adaptive intermittent control.

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H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

strength with the dynamic evolution of the network. Some distributed adaptive laws are designed on the weights of the network and uniformly bound is achieved for second-order multi-agent systems [43–49], the distributed adaptive consensus protocol updated the coupling weight for each node, which is node-based adaptive consensus protocol. In [48,49], the edge-based adaptive consensus protocol is used for the consensus of multi-agent systems with general linear and Lipschitz nonlinear dynamics [48]. In [49], the authors investigate the distributed consensus problem of second-order multi-agent systems with nonlinear dynamics for both the cases without and with a leader, the edge-based adaptive protocol which assigns a time-varying coupling weight to each edge in the communication is proposed. Thirdly, it well known that time delay is ubiquitous in information spreading due to the finite speed of signal transmission, thus time delay should be considered in the consensus problem for multi-agent systems [50–53]. Therefore, it is still a challenging problem to design an effective distributed adaptive intermittent protocol for reaching second-order consensus in delayed nonlinear multi-agent systems with intermittent communications, the purpose of this paper is to close this gap. In summary, the main contributions of this paper can be summarized as follows: (1) A novel distributed adaptive intermittent control protocol is designed based on local node-based completely intermittent information. (2) An effective distributed control gains-design strategy has been rarely investigated for consensus in multi-agent systems with second-order delayed nonlinear dynamics. (3) Second-order consensus can be reached if the measure of communication is larger than a threshold value under the strongly connected and balanced topology. (4) The generalized Halanay inequality and an appropriate Lyapunov–Krasovskii functional will be introduced, which play a crucial role in the derivation of our main results. The rest of this paper is organized as follows. In Section 2, problem formulation and some preliminaries are briefly outlined. In Section 3, the main results are derived for second-order consensus. A numerical example is provided to verify the theoretical results in Section 4. Some conclusions are drawn in Section 5. Notation: The following notations are used throughout the paper. Rn denotes the n-dimensional Euclidean space, Rn×m is a set of real n × m matrices, and N represents a set of natural numbers. The notation diag{} stands for a blockdiagonal matrix. The superscript 1n (0n ) indicate the n-dimensional column vector with each entry being 1(0). 1N × N represents the N × N matrix with all entries being 1. Notations . and  denote the Euclidian norm and the Kronecker product, respectively. Suppose that matrix M ∈ Rn×n is a real symmetric matrix, λi (M) (i = 1, 2, . . . , n) is the ith eigenvalue, and λmax (M)(λmin (M)) is the largest (smallest) eigenvalue, and λ2 (M) is the second smallest eigenvalue. 2. Problem formulation and preliminaries Consider a group of N agents with second-order delayed nonlinear dynamics as follows



x˙ i (t ) = vi (t ) v˙ i (t ) = f (t , xi (t ), vi (t )) + g(t, xi (t − τ ), vi (t − τ )) + ui (t )

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

(1)

where xi (t ) ∈ Rn and vi (t ) ∈ Rn represent the position and velocity state vectors of agent i. f (t, ·, · ) ∈ Rn and g(t, ·, · ) ∈ Rn are two continuous vector-valued functions, τ > 0 is time delay. ui (t ) ∈ Rn is the control input acting on agent i. The topology of the system can be described as a graph, which can be represented by g = (v, ε , G ), where v = (v1 , v2 , . . . , vN ) and ε ⊆ v × v denote the set of nodes and the set of directed edges, respectively. In some real situations, neighboring agents may only communicate to each other over some disconnected time intervals due to unreliability of communication channels, failure of physical devices and limitations of sensing ranges, which is shown in Fig. 1. In addition, in order to overcome the drawback taking too large control gains, the adaptive control approach is used to achieve the consensus. Motivated by the above observation, we introduce a new type of distributed node-based adaptive completely intermittent consensus protocol, which assigns a same time-varying weight for all the ingoing edges of each node, as shown in Fig. 2. The controller ui (t)(1 ≤ i ≤ N) can be designed by

ui (t ) =

⎧ ⎨

−α ci (t )

⎩0

N  j=1

li j x j (t ) − β ci (t )

N  j=1

li j v j (t )

t ∈ [tk , tk + δk ) t ∈ [tk + δk , tk+1 )

(2)

( k = 1, 2 . . . )

where α > 0 and β > 0 are coupling strengths, ci (t) (i = 1, 2, . . . , N ) is the time-varying gain of the ith agent. tk+1 − tk > 0 is the control period satisfying 0 < t ≤ tk+1 − tk ≤ t¯ < +∞. δ k denotes the control width satisfying 0 ≤ δ ≤ δk ≤ δ¯ ≤ tk+1 − tk .  L = [li j ]N×N is the Laplacian matrix, satisfying lij ≤ 0 and Lii = − Nj=1, j=i li j . In this paper, we suppose that the topology g of the network is the strongly connected and balanced. The objective of introducing the intermittent adaptive control strategy and the time-varying control gains here is to find some adaptive distributed laws acting on the control gain such that second-order consensus can be reached. Remark 1. It can be seen from (2) that the agent i can only obtain the state information from agent j on time intervals [tk , tk + δk ), no any information is transmitted on time intervals [tk + δk , tk+1 ). Our goal is to find some small δ k to make the second-order consensus can be reached. Remark 2. Different from the existing research results [57], the time-varying strength ci (t) is associated with i, which will adaptive change with the system dynamical evolution. The control period tk+1 − tk and the control width δ k (k = 0, 1, 2 . . . )

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

5

are associated with k, which don’t be required to be constants, and only have the upper and lower bounds, which can be adjusted with the network’s conditions. Before moving on, the following assumption, definition and lemmas play important roles in deriving the main results. Assumption 1. There are some positive constants Lfi and Lgi (i = 1, 2 ), such that for any x1 , y1 , x2 , y2 ∈ Rn , the following inequalities hold 2 

 f (t, x1 , x2 ) − f (t, y1 , y2 ) ≤

L f i xi − yi 

(3)

Lgi xi − yi 

(4)

i=1

g(t, x1 , x2 ) − g(t, y1 , y2 ) ≤

2  i=1

Definition 1. Second-order consensus in multi-agent systems (1) is said to achieve if for any initial condition

lim xi (t ) − x j (t ) = 0

(5)

t→∞

lim vi (t ) − v j (t ) = 0

t→∞

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

(6)

Definition 2. (Completely aperiodic intermittence) [58] the directed communication topology among all agents is called completely aperiodic intermittence, if all agents aperiodically lose contact with the neighboring agents as the time evolves. Lemma 1. [59] Suppose that a directed graph is strongly connected, then 0 is a simple eigenvalue of its Laplacian matrix L, and all the other eigenvalues of L have positive real parts. Lemma 2. [30] A directed graph is balanced if and only if 1N is the left eigenvector of its Laplacian matrix L associated with zero eigenvalue, that is 1TN L = 0TN . Lemma 3. [54] Suppose that continuous function V(t) is non-negative when t ∈ (a − τ , +∞ ) and satisfies the following inequality

V˙ (t ) ≤ −k1V (t ) + k2V (t − τ )

(t ≥ a )

(7)

where k1 and k2 are two positive constants, and k1 > k2 , then the following inequality holds

V (t ) ≤ max

a−τ ≤s≤a

|V (s )|e−r (t−a) = V (a )τ e−r (t−a)

(8)

where r is the unique positive solution of −r = −k1 + k2 erτ . Lemma 4. [55] Suppose that continuous function V(t) is non-negative when t ∈ (a − τ , +∞ ) and satisfies the following inequality

V˙ (t ) ≤ k1V (t ) + k2V (t − τ )

(t ≥ a )

(9)

where k1 and k2 are two positive constants, then the following inequality holds

V (t ) ≤ max

a−τ ≤s≤a

|V (s )|e(k1 +k2 )(t−a) = V (a )τ e(k1 +k2 )(t−a)

(10)

Lemma 5. For some given matrices A, B, C and D with appropriate dimension, the following equalities hold

(i ) (ii ) (iii ) ( iv )

(rA )  B = A  (rB ) (A  B )(C  D ) = (AC )  (BD ) (A + B )  C = A  C + B  C ( A  B )T = AT  BT

where r is a constant. Lemma 6. [56] The following linear matrix inequality



S S = 11 S21



S12 >0 S22

(11)

T , S T T where S11 = S11 12 = S21 , S22 = S22 is appropriate dimensions, the following matrix inequalities hold −1 (i ) S11 > 0, S22 − S21 S11 S12 > 0

(12)

−1 (ii ) S22 > 0, S11 − S12 S22 S21 > 0

(13)

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H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

3. Main result In this section, node-based distributed adaptive intermittent protocols are designed for second-order consensus of delayed nonlinear multi-agent systems.   Let x¯ (t ) = N1 Nj=1 x j (t ) and v¯ (t ) = N1 Nj=1 v j (t ) be the average position state and the average velocity state of all agents, respectively, and xˆi (t ) = xi (t ) − x¯ (t ) and vˆ i (t ) = vi (t ) − v¯ (t ) denote the position and velocity states relative to the average position and velocity of agents in system (1), respectively. In virtue of (1) and (2), it is easy to get the following error dynamical system

⎧˙ xˆ (t ) = vˆ i (t ) ⎪ ⎪ i ⎧ ⎪ ⎪ N  ⎪ ⎪ ⎪ ⎪ ⎪ f (t, xi (t ), vi (t )) − N1 f (t, xk (t ), vk (t )) + g(t, xi (t − τ ), vi (t − τ )) ⎪ ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ N N N ⎪    ⎪ ⎪ ⎪ ⎪ − N1 g(t, xk (t − τ ), vk (t − τ )) − α ci (t ) li j x j (t ) − β ci (t ) li j v j (t ) ⎪ ⎪ ⎪ ⎨ ⎪ j=1 j=1 k=1 ⎪ ⎨ N N N N     ⎪ +α N1 ck (t ) lk j x j (t ) + β N1 ck (t ) lk j v j (t ) t ∈ [tk , tk + δk ) vˆ˙ i (t ) = ⎪ ⎪ ⎪ j=1 j=1 ⎪ k=1 k=1 ⎪ ⎪ ⎪ ⎪ N ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ f (t, xi (t ), vi (t )) − N1 f (t, xk (t ), vk (t )) + g(t, xi (t − τ ), vi (t − τ )) ⎪ ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ ⎩ − N1 g(t, xk (t − τ ), vk (t − τ )) t ∈ [tk + δk , tk+1 ) ⎩ k=1

(14) For simplify, we lead to the following notations

xˆ(t ) = [xˆT1 (t ), xˆT2 (t ), . . . , xˆTN (t )]T

vˆ (t ) = [vˆ T1 (t ), vˆ T2 (t ), . . . , vˆ TN (t )]T f (t, x(t ), v(t )) = [ f T (t, x1 (t ), v1 (t )), f T (t, x2 (t ), v2 (t )), . . . , f T (t, xN (t ), vN (t ))]T g(t, x(t − τ ), v(t − τ )) = [gT (t, x1 (t − τ ), v1 (t − τ )), . . . , gT (t, xN (t − τ ), vN (t − τ ))]T ξ (t ) = [xˆT (t ), vˆ T (t )]T then the system (14) can be recast into a compact form as follows

 ˆ ξ˙ (t ) = F (t , x(t ), v(t )) + G(t, x(t − τ ), v(t − τ )) + (L  In )ξ (t ) + H (t , x(t ), v(t )) F (t , x(t ), v(t )) + G(t, x(t − τ ), v(t − τ ))

where

 F (t, x(t ), v(t )) =

 G(t, x(t − τ ), v(t − τ )) =

 H (t, x(t ), v(t )) =

 Lˆ =

t ∈ [tk , tk + δk ) t ∈ [tk + δk , tk+1 )

(15)



0Nn {(IN − N1 1N×N )  In } f (t , x(t ), v(t ))



0Nn {(IN − N1 1N×N )  In }g(t, x(t − τ ), v(t − τ ))



0Nn IN  h(t, x(t ), v(t )) 0N −α c (t )L



IN −β c (t )L

c (t ) = diag{c1 (t ), c2 (t ), . . . , cN (t )} h(t, x(t ), v(t )) =

α

N N N N   1 1 ck (t ) lk j x j (t ) + β ck (t ) lk j v j (t ) N N k=1

j=1

k=1

j=1

With the above development, the main results are presented in the following. Theorem 1. Suppose that Assumption 1 holds, let τ ≤ min{min{δk }, tk+1 − tk − max{δk }}(k = 0, 1, 2, . . . ), then second-order consensus is achieved under the following distributed adaptive intermittent control protocol

ui (t ) =

⎧ ⎨

−α ci (t )

⎩0

N  j=1

li j x j (t ) − β ci (t )

and the following distributed adaptive laws

N  j=1

li j v j (t )

t ∈ [tk , tk + δk ) t ∈ [tk + δk , tk+1 )

(16)

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15



1 r1 t ¯ i x(t ) + 2β xT (t ) ¯ i v(t ) + ai e {α xT (t ) 2 0

c˙ i (t ) =

β2 T ¯ v (t ) i v(t )} α

t ∈ [tk , tk + δk ) t ∈ [tk + δk , tk+1 )

7

(17)

where ai > 0 (i = 1, 2, . . . , N ), if there exist some positive constants ri (i = 1, 2, 3, 4 ) and an infinite time sequence of uniformly bounded, non-overlapping time intervals [tk , tk+1 ) with t1 = 0 and t ≤ tk+1 − tk ≤ t¯, such that for each time interval [tk , tk+1 ), the following conditions hold



1 =

1 2



2 =



3 =



4 =

δk >

(1 − c¯2 + 12 r1 μ )IN

α r1 L¯ T

α r1 L¯ <0 (3 − c¯4 )IN + 12 β r1 L¯ 1 2

(18)



(5 − 12 r2 μ )IN

− 12 α r2 L¯

− 12 α r2 L¯ T

6 IN − 12 β r2 L¯

<0

(19)



(1 − 12 r3 μ )IN

− 12 α r3 L¯

− 12 α r3 L¯ T

(3 − 12 αλmax (L¯ + L¯ T ))IN − 12 β r3 L¯

<0

(20)



(5 − 12 r4 μ )IN

− 12 α r4 L¯

− 12 α r4 L¯ T

6 IN − 12 β r4 L¯

<0

(21)

rτ + (r3 + r4 )(tk+1 − tk ) rτ + (r3 + r4 )t > r + r3 + r4 r + r3 + r4

(22)

where

L¯ = L + LT

1 = NL¯ max



α (Lg1 + Lg2 ) 2

+ αL f 1 +

1 2 = α 2 λ2 (L¯ T L + LT L¯ ) 2 β (Lg1 + Lg2 ) ¯ 3 = NLmax + βL f2 + 2

αL f 2 2

βL f1 2

+

+

βL f1

2

αL f 2 2

+

1 αλmax (L¯ + L¯ T ) 2

1 2 β λ2 (L¯ T L + LT L¯ ) 2 1 5 = N (α + β )L¯ max Lg1 2 1 6 = N (α + β )L¯ max Lg2 2 ¯ i = ( i + T )  In i

4 =

i = [L¯ i1 , L¯ i2 , . . . , L¯ iN ]T [Li1 , Li2 , . . . , LiN ] μ = αβ c¯λ2 (L¯ T L + LT L¯ ) and r is the positive solution of −r = −r1 + r2 erτ . Proof. Take the following Lyapunov–Krasovskii function candidate as

V (t ) = V1 (t ) + V2 (t )

(23)

where

V1 (t ) =

1 T ξ (t )(  In )ξ (t ) 2

V2 (t ) =

N  i=1

α 2ai

e−r1 t (ci (t ) − cˆi )2

α L¯ ¯ ), L = L + LT , cˆi (i = 1, 2, . . . , N ) are some undetermined sufficiently large positive constants. It will be β L¯ 2 shown that V(t) is a valid Lyapunov function. By using Lemma 6, > 0 is equivalent to μ > 0, β L¯ − αμ L¯ 2 > 0, which are and = (

μIN α L¯ T

satisfied if we select sufficient large μ, such that μ >

α 2 λ2max (L¯ ) , then we can derive V1 (t) ≥ 0. In addition, it is easy to see βλ2 (L¯ )

that V2 (t) ≥ 0, thus one has V(t) ≥ 0 and V (t ) = 0 if and only if ξ (t ) = 0.

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H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

(1) For t ∈ [tk , tk + δk ) with arbitrarily given k ∈ N, the derivative of V1 (t) along the solution (15) can be calculated as follows

ξ T (t )(  In )[F (t, x(t ), v(t )) + G(t, x(t − τ ), v(t − τ )) + (Lˆ  In )ξ (t ) + H (t , x(t ), v(t ))] = ξ T (t )(  In )F (t, x(t ), v(t )) + ξ T (t )(  In )G(t, x(t − τ ), v(t − τ ))

V˙ 1 (t ) =

+

1 T ξ (t )[( Lˆ + LˆT )  In ]ξ (t ) + ξ T (t )(  In )H (t, x(t ), v(t )) 2

(24)

Since L is balanced, thus we have L¯ 1N = 0N×1 , L¯ 1N×N = 0N×N , one gets

ξ T (t )(  In )F (t, x(t ), v(t ))     1 = (α xˆT (t ) + β vˆ T (t ))(L¯  In ) IN − 1N×N  In f (t, x(t ), v(t )) N

= (α xˆT (t ) + β vˆ T (t ))(L¯  In )[ f (t, x(t ), v(t )) − 1N  f (t, x¯ (t ), v¯ (t ))] +(α xˆT (t ) + β vˆ T (t ))(L¯ 1n  f (t, x¯ (t ), v¯ (t ))) 1 (α xˆT (t ) + β vˆ T (t ))[(L¯ 1N×N  In ) f (t, x(t ), v(t ))] N = (α xˆT (t ) + β vˆ T (t ))(L¯  In )[ f (t, x(t ), v(t )) − 1N  f (t, x¯ (t ), v¯ (t ))] −

(25)

The same as (25), we can obtain

ξ T (t )(  In )G(t, x(t − τ ), v(t − τ )) = (α xˆT (t ) + β vˆ T (t ))(L¯  In )[g(t, x(t − τ ), v(t − τ )) − 1N  g(t, x¯ (t − τ ), v¯ (t − τ ))]

(26)

On the other hand, one has

1 T ξ (t )[( Lˆ + LˆT )  In ]ξ (t ) 2  1 T −α 2 (L¯ c (t )L + LT c (t )L¯ T ) = ξ (t ){ μIN − αβ (L¯ c(t )L + LT c(t )L¯ T ) 2 and

μIN − αβ (L¯ c(t )L + LT c(t )L¯ T )  In }ξ (t ) α (L¯ + L¯ T ) − β 2 (L¯ c(t )L + LT c(t )L¯ T )

 ξ T (t )(  In )H (t, x(t ), v(t )) = ξ T (t ){ =

μIN α L¯ T

α L¯ β L¯



  In }

(27)



0Nn 1N  h(t , x(t ), v(t ))

ξ T (t )[α L¯ 1N  h(t, x(t ), v(t )), β L¯ 1N  h(t, x(t ), v(t ))] = 0

(28)

Substituting (25)– (28) into (24), we have

V˙1 (t ) = (α xˆT (t ) + β vˆ T (t ))(L¯  In )[ f (t, x(t ), v(t )) − 1N  f (t, x¯ (t ), v¯ (t ))] +(α xˆT (t ) + β vˆ T (t ))(L¯  In )[g(t, x(t − τ ), v(t − τ )) − 1N  g(t, x¯ (t − τ ), v¯ (t − τ ))] +

1 T ξ (t ){ 2



−α 2 (L¯ c (t )L + LT c (t )L¯ T ) μIN − αβ (L¯ c(t )L + LT c(t )L¯ T )

μIN − αβ (L¯ c(t )L + LT c(t )L¯ T )  In }ξ (t ) α (L¯ + L¯ T ) − β 2 (L¯ c(t )L + LT c(t )L¯ T )

(29)

According to Assumption 1, one derives

α xˆT (t )(L¯  In )[ f (t, x(t ), v(t )) − 1N  f (t, x¯ (t ), v¯ (t ))] =

α

N  N 

L¯ i j (xi (t ) − x¯ (t ))T ( f (t, x j (t ), v j (t )) − f (t, x¯ (t ), v¯ (t )))

i=1 j=1

≤ Nα L¯ max L f 1

N 

xˆi (t )2 +

i=1

N α L¯ max L f 2 N 

2

(xˆi (t )2 + vˆ i (t )2 )

(30)

i=1

and

β vˆ T (t )(L¯  In )[ f (t, x(t ), v(t )) − 1N  f (t, x¯ (t ), v¯ (t ))] =

β

N  N 

L¯ i j (vi (t ) − v¯ (t ))T ( f (t, x j (t ), v j (t )) − f (t, x¯ (t ), v¯ (t )))

i=1 j=1

≤

β NL¯ max L f 2

N 

vˆ i (t )2 +

i=1

where L¯ max = max |Li j |. 1≤i, j≤N

N β L¯ max L f 1 N 

2

i=1

(xˆi (t )2 + vˆ i (t )2 )

(31)

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

9

Noting that (30) and (31), then we have

(α xˆT (t ) + β vˆ T (t ))(L¯  In )[ f (t, x(t ), v(t )) − 1N  f (t, x¯ (t ), v¯ (t ))]

N

N  αL f 2 β L f 1  βL αL ≤ N L¯ max α L f 1 + + xˆi (t )2 + NL¯ max β L f 2 + f 1 + f 2 vˆ i (t )2 2

2

2

i=1

2

(32)

i=1

Similarly, we can get

(α xˆT (t ) + β vˆ T (t ))(L¯  In )[g(t, x(t − τ ), v(t − τ )) − 1N  g(t, x¯ (t − τ ), v¯ (t − τ ))]   N N   N L¯ max (Lg1 + Lg2 ) 2 2 ≤ α xˆi (t ) + β vˆ i (t ) 2



i=1

N L¯ max (α + β ) + Lg1 2

i=1

N 

N 

xˆi (t − τ )2 + Lg2

i=1



α (Lg1 + Lg2 ) 2

 +N L¯ max

+ αL f 1 +

β (Lg1 + Lg2 ) 2

vˆ i (t − τ )2

αL f 2 2

+ βL f2 +

+

βL f1



N βL f1  2 +

2

xˆi (t )2

i=1

N αL f 2  2

vˆ i (t )2

i=1

N N   N L¯ max (α + β ) + Lg1 xˆi (t − τ )2 + Lg2 vˆ i (t − τ )2 2

1 + ξ T (t ){ 2





i=1





i=1

−α L¯ c (t )L + LT c (t )L¯ T μIN − αβ (L¯ c(t )L + LT c(t )L¯ T ) 2

(33)

i=1

Considering (29), (32) and (33), we can get

V˙1 (t ) ≤ N L¯ max



μIN − αβ (L¯ c(t )L + LT c(t )L¯ T )  In }ξ (t ) α (L¯ + L¯ T ) − β 2 (L¯ c(t )L + LT c(t )L¯ T )

(34)

Taking the time derivative V2 (t) along the trajectories of (15), one obtains

V˙ 2 (t ) =

N  i=1

α ai

e−r1 t (ci (t ) − cˆi )c˙ i (t ) − r1

i=1

i=1

− r1

N  i=1

≤

α 2ai

e−r1 t (ci (t ) − cˆi )2

1  ¯ i x(t ) + 2β xT (t ) ¯ i v(t ) + α (ci (t ) − cˆi ){α xT (t ) 2 N

=

N 

α 2ai

β2 T ¯ v (t ) i v(t )} α

e−r1 t (ci (t ) − cˆi )2

1 2 T 1 α xˆ (t )[(L¯ T c(t )L + LT c(t )L¯ )  In ]xˆ(t ) − α 2 c¯λ2 (L¯ T L + LT L¯ )xˆT (t )xˆ(t ) 2 2 1 1 + β 2 vˆ T (t )[(L¯ T c (t )L + LT c (t )L¯ )  In ]vˆ (t ) − β 2 c¯λ2 (L¯ T L + LT L¯ )vˆ T (t )vˆ (t ) 2 2 +αβ xˆT (t )[(L¯ T c (t )L + LT c (t )L¯ )  In ]vˆ (t ) − αβ c¯λ2 (L¯ T L + LT L¯ )xˆT (t )vˆ (t ) −r1

N  i=1

α 2ai

e−r1 t (ci (t ) − cˆi )2

(35)

where

c˜(t ) = diag{c˜1 (t ), c˜2 (t ), . . . , c˜N (t )} cˆ = diag{cˆ1 , cˆ2 , . . . , cˆN } c˜i (t ) = ci (t ) − cˆi c¯ = min {cˆi } 1≤i≤N

According to (34) and (35), one has

V˙ (t ) ≤ (1 − c¯2 )xˆT (t )xˆ(t ) + (3 − c¯4 )vˆ T (t )vˆ (t ) + 5 xˆT (t − τ )xˆ(t − τ ) +6 vˆ T (t − τ )vˆ (t − τ ) + [μ − αβ c¯λ2 (L¯ T L + LT L¯ )]xˆT (t )vˆ (t ) − r1

N  i=1

α 2ai

e−r1 t (ci (t ) − cˆi )2

(36)

10

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

Let μ = αβ c¯λ2 (L¯ T L + LT L¯ ), we have



V˙ (t ) ≤

ξ (t ) T

1 − c¯2



0  INn 3 − c¯4

0

   T ξ (t ) + ξ (t − τ ){ 5 0

0

6

 INn }ξ (t − τ )

1 1 + r1 ξ T (t )(  In )ξ (t ) − r2 ξ T (t − τ )(  In )ξ (t − τ ) 2 2 1 1 T − r1 ξ (t )(  In )ξ (t ) + r2 ξ T (t − τ )(  In )ξ (t − τ ) 2 2 N N   α −r1 t α −r1 (t−τ ) −r1 e (ci (t ) − cˆi )2 + r2 e (ci (t − τ ) − cˆi )2 2ai 2ai i=1

i=1

ξ T (t )( 1  In )ξ (t ) + ξ T (t − τ )( 2  In )ξ (t − τ ) − r1V (t ) + r2V (t − τ ) < −r1V (t ) + r2V (t − τ ) ≤

(37)

(2) For t ∈ [tk + δk , tk+1 ) with arbitrarily given k ∈ N, taking the time derivative V(t) along the trajectories of (15), along the similar line, one gets



V˙ (t ) ≤ 1 xˆT (t )xˆ(t ) + 3 −



1 αλmax (L¯ + L¯ T ) vˆ T (t )vˆ (t ) 2

+5 xˆT (t − τ )xˆ(t − τ ) + 6 vˆ T (t − τ )vˆ (t − τ ) +

N  i=1

α ai

e−r1 t (ci (t ) − cˆi )c˙ i (t ) − r1

N  i=1



α 2ai

e−r1 t (ci (t ) − cˆi )2



1 ≤ 1 xˆ (t )xˆ(t ) + 3 − αλmax (L¯ + L¯ T ) vˆ T (t )vˆ (t ) 2 T

+5 xˆT (t − τ )xˆ(t − τ ) + 6 vˆ T (t − τ )vˆ (t − τ )

ξ T (t )( 3  In )ξ (t ) + ξ T (t − τ )( 4  In )ξ (t − τ ) + r3V (t ) + r4V (t − τ ) < r3V (t ) + r4V (t − τ ) ≤

(38)

When t ∈ [0, δ 1 ), by using Lemma 3 and (37), we have

V (t ) ≤ max |V (s )|e−rt = V (0 )τ e−rt

(39)

−τ ≤s≤0

where r is the positive solution of −r = −r1 + r2 erτ . When t ∈ [δ 1 , t2 ), according to Lemma 4 and (38), one has

V (t ) ≤

max

δ1 −τ ≤s≤δ1

|V (s )|e(r1 +r2 )(t−δ1 ) ≤ V (0 )τ e−r (δ1 −τ )+(r3 +r4 )(t−δ1 )

(40)

Taking t = t2 , we can obtain

V (t2 ) = lim− V (t ) = V (0 )τ e−r (δ1 −τ )+(r3 +r4 )(t2 −δ1 ) = V (0 )τ e− 2 t →t2

(41)

where 2 = r (δ1 − τ ) − (r3 + r4 )(t2 − δ1 ). Next, we will prove that

V (tm )τ ≤ V (0 )τ e

−

m +1 i=2

i

(42)

holds for any positive integer m. When m = 2, we can obtain

V (t2 )τ ≤ max |V (s )|e−r (δ1 −τ ) e(r3 +r4 )(t2 −δ1 ) = V (0 )τ e− 2 −τ ≤s≤0

(43)

Suppose that the inequality (42) holds when m = k − 1, next we will prove (42) holds for m = k.

V (tk )τ ≤ =

max

tk−1 +δk−1 −τ ≤s≤tk−1 +δk−1

|V (s )|e(r3 +r4 )(tk −tk−1 −δk−1 )



max

tk−1 −τ ≤s≤tk−1

|V (s )|

 max

tk−1 +δk−1 −τ ≤s≤tk−1 +δk−1

= V (tk−1 )τ e− k+1 ≤ V (0 )τ e

−

k +1 i=2

e

−r (t−tk−1 )

e(r3 +r4 )(tk −tk−1 −δk−1 )

i

thus (42) holds for any positive integer m. For any t > 0, there exists a positive integer k0 , such that tk0 ≤ t ≤ tk0 +1 , then we have

(44)

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

11

When t ∈ [tk0 , tk0 + δk0 ), one has

V (t ) ≤

|V (s )|e

max

tk −τ ≤s≤tk 0

−r (t−tk )

≤ max |V (s )|e

0

−τ ≤s≤0

0

−

k +1 i=2

i



≤ V (0 )τ e−k ≤ V (0 )τ e−k t¯ t

(45)

where = min { i } = (r + r3 + r4 )δ − rτ − (r3 + r4 )t¯ > 0. 2≤i≤k+1

When t ∈ [tk0 + δk0 , tk0 +1 ), we have

V (t ) ≤

0

≤

|V (s )|e(r3 +r4 )(t−tk0 −δk0 )

max

tk +δk −τ ≤s≤tk +δk 0

0

|V (s )|e

max

tk −τ ≤s≤tk 0

0

−r (δk −τ )+(r3 +r4 )(t−tk −δk ) 0

0

0

0

= V (0 )τ e

−

k0 +2



i=2

i

≤ max |V (s )|e

−

−τ ≤s≤0

k0 +1



i=2

i

e− k0 +2



≤ V (0 )τ e−(k0 +1) = V (0 )τ e− t¯ t

(46)

then we have

V (t ) < V (0 )τ e− t¯ t

(47)

which means second-order consensus can be achieved. The proof is completed.



Remark 3. In [45], the authors proposed a node-based adaptive consensus protocol, which assign a time-varying coupling weight to each node. However, the topology is undirected graph, and the information can be transmitted continuously among the agents. In this paper, the communication topology is strongly connected and balanced directed graph, moreover each agent is assumed to obtain the information between own and the neighbor’s on some disconnected time intervals due to communication constraints. Remark 4. The edge-based adaptive protocol is proposed in [49], which assigns a time-varying coupling weight to each edge in the communication. We design a new type of distributed node-based adaptive completely intermittent consensus protocol, which assign a time-varying coupling weight to each node. Remark 5. It can be seen from (47) that second-order consensus can be achieved with an exponential convergence rate t¯ , since = (r + r3 + r4 )δ − rτ − (r3 + r4 )t¯, then we can obtain

t¯

= ( r + r3 + r4 )

δ t¯

−r

τ t¯

− r3 − r4

from the above equation, we can know the convergence rate may be increased by enlarging the control width δ or decreasing the delay τ , but larger δ or smaller τ imply the high cost of control. Therefore, a trade-off has to be made between the convergence rate and the control cost. Remark 6. In Theorem 1, by using LMI toolbox in MATLAB, the upper-bound delay can be obtained, then it can be concluded that second-order consensus can be achieved for all τ ∈ [0, τ 0 ], that is, the consensus achievement is robust to τ ∈ [0, τ 0 ]. Remark 7. From (22), we can obtain the lower of the control width δ k , that is, for all k, δ k is not less than second-order consensus can be achieved.

rτ +(r3 +r4 )t r+r3 +r4 ,

By some simple computation, Theorem 1 can be simplified as follows Corollary 1. Suppose that Assumption 1 holds, let τ ≤ min{min{δk }, tk+1 − tk − max{δk }} (k ∈ N ), then second-order consensus is achieved under the following distributed adaptive intermittent control protocol

ui (t ) =

⎧ ⎨

−α ci (t )

⎩0

N  j=1

li j x j (t ) − β ci (t )

N 

li j v j (t )

j=1

t ∈ [tk , tk + δk )

and the following adaptive update laws

c˙ i (t ) =

⎧ ⎨1 2 0

⎩



ai e

r1 t

¯ i x(t ) + 2β xT (t ) ¯ i v(t ) + α x (t ) T

(48)

t ∈ [tk + δk , tk+1 )

β2 T ¯ v (t ) i v(t ) α

 t ∈ [tk , tk + δk )

(49)

t ∈ [tk + δk , tk+1 )

where ai > 0 (i = 1, 2, . . . , N ), if there exist some positive constants ri (i = 1, 2, 3, 4 ) and an infinite time sequence of uniformly bounded, non-overlapping time intervals [tk , tk+1 ) with t1 = 0, such that for each time interval [tk , tk+1 ), k ∈ N, the following conditions hold

max{1 − c¯2 , 3 − c¯4 } +

1 r1 λN ( ) < 0 2

(50)

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H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15





max 1 , 3 −

1 1 αλmax (L¯ + L¯ T ) − r3 λ2 ( ) < 0 2 2

(51)

max{5 , 6 } −

1 min{r2 , r4 }λ2 ( ) < 0 2

(52)

δk >

rτ + (r3 + r4 )(tk+1 − tk ) rτ + (r3 + r4 )t > r + r3 + r4 r + r3 + r4

(53)

where the related parameters can be found in Theorem 1. Remark 8. Compared with Theorem 1, the conditions in Corollary 1 are easily verified. Let g(t, ·, · ) = 0, then delayed nonlinear multi-agent systems (1) will degenerate into nonlinear multi-agent systems in the following.



x˙ i (t ) = vi (t ) v˙ i (t ) = f (t , xi (t ), vi (t )) + ui (t )

(54)

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

along the similar line of the above analysis, the distributed adaptive intermittent control protocol and consensus conditions can be derived as follows. Theorem 2. Suppose that Assumption 1 holds, then second-order consensus is achieved under the following distributed adaptive intermittent control protocol



ui (t ) =

−α ci (t ) 0

N

j=1 li j x j

(t ) − β ci (t )

N

j=1 li j

v j (t )

t ∈ [tk , tk + δk ] t ∈ (tk + δk , tk+1 )

(55)

and the update laws are given as

⎧ ⎨1

c˙ i (t ) =

⎩

  β2 T ¯ T T ¯ ¯ ai e α x (t ) i x(t ) + 2β x (t ) i v(t ) + v (t ) i v(t ) 2 α r1 t

0

t ∈ [tk , tk + δk ]

(56)

t ∈ (tk + δk , tk+1 )

where ai > 0 (i = 1, 2, . . . , N ), if there exist some positive constants ri (i = 1, 2 ), c¯ and an infinite time sequence of uniformly bounded, non-overlapping time intervals [tk , tk+1 ) with t1 = 0, such that for each time interval [tk , tk+1 ), k ∈ N, the following conditions hold





¯ 1 IN ,  ¯ 2 IN + 1 α (L¯ + L¯ )T + 1 r1 − 1 c¯λ2 (L¯ T L + LT L¯ )diag{α 2 IN , β 2 IN } < 0 diag  2 2 2 ¯ 1 IN ,  ¯ 2 IN } − diag{

δk > where

1 r2 < 0 2

(58)

r2 (t − tk ) r1 + r2 k+1

(59)

  ¯ 1 = N L¯ max α L f + 1 α L f + 1 β L f  1 2 1 2

(57)

2

  ¯ 2 = N L¯ max β L f + 1 β L f + 1 α L f  2 1 2 2

2

Remark 9. It is noted that there are some desirable results about consensus of multi-agent systems by the distributed adaptive protocols, but the communication topology is assumed to be undirected, for general directed communication topology, the Laplacian matrices are generally asymmetric, which lead to some trouble for the selection of appropriate Lyapunov function and construction of adaptive consensus protocol. Thus, how to design fully distributed adaptive protocols for multiagent systems with general directed communication topology is still a challenging work. 4. Simulations In this section, a simulation example is presented to demonstrate the effectiveness of the established criterion for the consensus of multi-agent systems (1). Example 1. The interaction diagraph of multi-agent systems (1) is shown in Fig. 3, where the weights are 1 on the edges, it is obvious that the interaction diagraph is strongly connected and balanced. Let the nonlinear functions f (t, ·, · ) = g(t, ·, · ) = 2cos(2t ) + tanh(0.001xi (t )) + tanh(0.001vi (t )) ∈ R(i = 1, 2, . . . , 7 ), and τ = 0.001, by Assumption 1, one obtains L f 1 = Lg1 =

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

13

Fig. 6. The trajectories of adaptive intermittent controller ui (t )(i = 1, 2, . . . , 7 ).

Fig. 7. The trajectory of adaptive intermittent control gain ci (t) (i = 1, 2, . . . , 7 )under the updating law.

0.001, L f 2 = Lg2 = 0.001. Let α = 1, β = 10, r1 = 0.099, r2 = 0.038, r3 = 0.165, r4 = 0.038, tk+1 − tk = 1 (k = 0, 1, 2, . . . ) and t1 = 0, by using Corollary 1, we can obtain the control width δ k > 0.7689. In simulation, taking δk = 0.8, it is assumed that the agents communicate only when t ∈ ∪k∈N [k, k + 0.8 ), then consensus can be achieved. The position and velocity state trajectories of all agents are shown in Figs. 4 and 5 with stochastic initial condition, respectively. The curve of adaptive intermittent control is shown in Fig. 6, it is noted that the control ui (t ) = 0(i = 1, 2, . . . , 7 ) in time interval ∪k∈N [k + 0.8, k + 1 ). Fig. 7 gives the trajectory of adaptive intermittent control gain ci (t) (i = 1, 2, . . . , 7 ) under the updating laws, it is easy to see when t ∈ ∪k∈N [k + 0.8, k + 1 ), the control gain ci (t ) = 0 , the coupling strength ci (t) will quickly converge to the steady value due to it with an exponential rate. The effectiveness of consensus criteria proposed are verified very well.

14

H. Li et al. / Applied Mathematics and Computation 326 (2018) 1–15

5. Conclusions In this paper, an effective distributed control gains-design strategy and a new kind of distributed control protocol are designed for second-order multi-agent systems with delayed nonlinear dynamics. Based on local information between its own and the neighbor’s on some disconnected time intervals, node-based adaptive completely intermittent consensus protocol which assigns a same time-varying weight for all the ingoing edges of each node in the communication is proposed for the case without the leader. It is proved that second-order consensus can be reached if the measure of communication is larger than a threshold value under the strongly connected and balanced topology. It is possible to extend the main results to the more complicated cases such as the high-order multi-agent systems with a switching topology, or with the weaker connectivity that the graph contains a spanning tree, which are the future research topics. Acknowledgments This work was jointly supported by National Natural Science Foundation of China under Grant No. 11301226 and 61572014, Zhejiang Provincial Natural Science Foundation of China under Grant No. LY17F030020, LY16F020028, LY15F030021 and LQ15F010 0 08. Jiaxing science and technology project under Grant No.2016AY13011 and 2016AY13013. Shandong Provincial Natural Science Foundation, China NO. ZR2016FM48 Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.amc.2018.01.005 References [1] R. Olfati-Saber, A. Fax, R.M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE 95 (1) (2007) 215–233. [2] J. Wang, D. Cheng, X. Hu, Consensus of multi-agent linear dynamic systems, Asian J. Control 10 (2) (2008) 144–155. [3] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control 49 (9) (2004) 1520–1533. [4] P. 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