Proceedings of the 36th Chinese Control Conference July 26-28, 2017, Dalian, China
Containment control for multi-agent systems via intermittent sampled data algorithm Hongxiao Zhang,Jinrong Li,Li Ding,Zhiwei Liu Department of Automation, School of Power and Mechanical Engineering, Wuhan University, Wuhan, 430072, P. R. China E-mail:
[email protected] Abstract: In this paper, the containment control problem in second-order multi-agent systems for static and dynamic leaders under directed and undirected topologies are studied. We propose the periodic intermittent containment control algorithm, which both reduces the load of updating rates of controller and cuts down the working time of the controller in each sampling interval. Some necessary and sufficient conditions depending on the eigenvalues of the Laplacian matrix associated with the communication graph, the communication width, sampling period and the gain parameters, are obtained to guarantee the containment control. Finally, some simulations are conducted to verify the effectiveness of the proposed algorithms. Key Words: Containment Control, Multi-agent Systems, Intermittent Sampled Data
1
Introduction
Multi-agent systems, fuelled from their broad applications in physics, aerospace engineering and systems biology, have attracted ever-increasing interest in the past decade. Cooperative control is concerned with engineered systems that can be characterized as a collection of decision-making components with locally sensed information, limited intercomponent communications, and processing capabilities, all seeking to achieve a collective objective. For the past few years, as a result of the rapid developments of computer science and sensing/communication technologies, distributed cooperative control of multi-agent systems has made great progress. Applications of cooperative control in biological, physical and engineering systems include consensus[1–4], rendezvous [5], flocks [6, 7], autonomous vehicle systems [8] and formation of robotic systems [9]. Leader-following consensus, which is also categorized as target tracking or pinning control problem, has been studied in many recent works [10–12]. In conclusion, all the followers will ultimately track the leader on condition that the whole network topology formed by the leader and the followers contains a spanning tree. Generally, the leader-following consensus is that all the followers will not eventually arrive at consensus, but will enter the convex hull formed by all the leader agents in the presence of multiple leaders. In [13], the containment control protocol was designed to drive a group of mobile robots to a convex region spanned by the leader agents when the communication topology is undirected and fixed. Then the result was generalised to the case of directed network topology in [14] and further extended to the case when each agent has a general linear dynamics in [15]. Some discontinuous containment control protocols invoking sign functions without velocity measurements were also proposed in [16] and [17]. It is worth noting that most of the aforementioned works on consensus problems were focused on the continuous time control, which requires the continuous communication among agents and information updated continuously by controllers. These control laws may infeasible or impractical in This work is supported by National Natural Science Foundation (NNSF) of China under Grants 61403284 and 61673303.
many resources. To deal with this issue, intermittent control strategy was proposed. In [18], the authors investigated the first-order consensus of multi-agent systems with nonlinear dynamics and external disturbance via intermittent communication. In [19] and [20], second-order consensus of multiagent systems with and without time delays were studied by using intermittent control. In [21], consensus of multiagent system via adaptive intermittent control was investigated. The advantage of intermittent control is that it can shorten the working time of the controllers, but the deficiency is that the information updating rates of controllers cannot be reduced. For the sake of reducing the load of controllers updates, the authors studied the consensus of second-order multi-agent systems via periodic sampled data control, and some necessary and sufficient conditions for reaching consensus were obtained in [22]. Motivated by the above discussions, we will extend the intermittent sampled data algorithms to achieve containment control in second-order multi-agent systems. The main contributions of this paper can be summarized as follows. (1) Intermittent sampled data control protocol, which both shortens the working time of controllers and reduces the update rates of controllers compared with continuous control, is designed for second-order multi-agent systems. (2) A necessary and sufficient condition is established for achieving containment control with directed and undirected communications. (3) The sampled data control strategy can be regarded as a special case of intermittent sampled control. The outline of this paper is shown as follows: we first establish some of the basic notations. We then, in Section 2, give some preliminaries and the problem formulation. Then the main results are in Section 3. Some simulation results are presented in Section 4 to illustrate the given analyses. Finally, we conclude the whole paper briefly in Section 5. The symbols used in this paper are given: R and C denote the sets of real numbers and complex numbers, respectively. Let In be the n × n identity matrix, 1n = [1, 1, · · ·, 1]T . 0n×m is the n × m matrix with all elements zero (or simply 0 if no confusion arises). For λ ∈ C, Re(λ), Im(λ) and |λ| denote the real part, imaginary part and modulus of the complex number λ. triag{λ1 , λ2 , · · ·, λn } denotes the upper triangular matrix with diagonal elements λ1 , λ2 , · · ·, λn .
8078
For square matrix A ∈ Rn×n , |A| denotes the determinant of square matrix A. For A = [aij ] ∈ Rm×n and B = [bij ] ∈ Rp×q , the Kronecker product of A and B is defined as following ⎛ ⎞ a11 B . . . a1n B ⎜ ⎟ .. .. mp×nq .. A⊗B =⎝ . ⎠∈R . . . am1 B
2
···
Lemma 1 L11 is invertible if and only if the directed graph G has a directed spanning forest. Moreover, if L11 is invertible, we can obtain that all the eigenvalues of L11 have positive real parts and all the elements of L−1 11 are nonnegative. From [23], we can obtain the following lemma. Lemma 2 The complex polynomial R(s) = s2 + as + b , where a ∈ C and b ∈ C, is Hurwitz stable if and only if Re(a) > 0 and Re(a)Im(a)Im(b) + Re2 (a)Re(b) − Im2 (b) > 0.
amn B
Preliminaries and Problem Formulation
In this section, we summarise some definitions and results from matrix theory and graph theory which are needed in our analysis. 2.1
Definition 1 A subset C of RN is said to be convex if (1 − λ)x + λy ∈ C whenever x ∈ C, y ∈ C and 0 < λ < 1.
Preliminaries
The convex hull of a finite set of points x1 , x2 , ..., xq ∈ RN (q is a positive integer) is the minimal convex set containing all points in {x1 , x2 , ..., xq }. We use co{x1 , x2 , ..., xq } to denote it, that is
Let G = {V, E} be a directed graph consisting of the set of nodes V = {1, 2, · · ·, n}, the set of edges E ∈ V × V, and a weighted adjacency matrix A = [aij ], i, j = 1, 2, ..., n. If (j, i) ∈ E, then aij > 0, otherwise, aij = 0 and the diagonal elements of A are zero, that is, aii = 0. Node j is a neighbor of node i if (j, i) ∈ E. A directed path from node i to node j is a sequence of edges (i, s1 ), (s2 , s3 ), · · ·, (sk , j), where (i, s1 ), (s2 , s3 ), · · ·, (sk , j) ∈ E. A directed tree is a directed graph, where there exists a root connecting with any other agents by exactly one directed path. A directed forest is a directed graph consisting of one or more directed trees, but none of the nodes are in common. A directed spanning forest of a directed graph G is a directed forest formed by edges that connect all the nodes in the directed graph G. The Laplacian matrix of graph G donates as: L = [lij ] ∈ Rn×n , lii = n j=1 aij and lij = −aij for i = j.
co{x1 , x2 , ..., xq } =
⎧ q ⎨ ⎩
j=1
aj xj |aj ∈ R, aj ≥ 0,
q j=1
aj =1
⎫ ⎬ ⎭
Definition 2 The containment control is said to be achieved asymptotically in system (1) if for any initial conditions, xi (t) ∈ co{xm+1 , xm+2 , ..., xn }, vi (t) ∈ co{vm+1 , vm+2 , ..., vn } as t → +∞, i ∈ F. Remark 1 The containment control is said to be achieved asymptotically in system (1) means that the position and velocity of each follower will converge into the convex hull formed by the positions and velocities of the leaders as t → +∞, respectively. For example, a group of agents need to move from one place to another place. Some agents (called leaders) equipped with necessary sensors can detect and clear the potential obstacles. Other agents (called followers), which are not equipped the necessary sensors, will stay in the moving safety range formed by the leaders.
2.2 Problem Formulation Consider a second-order multi-agent system with n agents. We assume that there are m(m < n) followers and n − m leaders. An agent is called a leader if the agent does not receive any information from others. Otherwise, it is called a follower. The set of followers is denoted as F = {1, 2, · · ·, m} and the set of leaders is denoted as L = {m + 1, m + 2, · · ·, n}. Let V = F ∪ L = {1, 2, · · ·, n} be the set of all the agents. The communication topology among the n agents is represented by a directed graph G. The dynamics of the multi-agent system are given by x˙ i (t) = vi (t), v˙ i (t) = ui (t), i ∈ F, (1) x˙ i (t) = vi (t), v˙ i (t) = 0, i ∈ L,
3
Main results
An intermittent containment control algorithm is proposed as follows
where xi (t), vi (t) ∈ R are the position and velocity states of the ith agent, respectively. ui (t) ∈ R is its control input. All the results still hold for xi (t), vi (t), ui (t) ∈ R by using the property of Kronecker product. From the definitions of the leader and the follower, the Laplacian matrix of the directed graph G can be partitioned as
L12 L11 , (2) L= 0(n−m)×m 0(n−m)×(n−m) where L11 is a m × m matrix and denotes the communications among the followers. L12 is a m × (n − m) matrix and denotes the communications among the followers and leaders. From the property of the Laplacian matrix, we can obtain the following lemma.
⎧ lij (xj (tk ) − xi (tk )) −α ⎪ ⎪ ⎨ j∈N i −β lij (vj (tk ) − vi (tk )), t ∈ [tk , tk + θ) , ui (t) = ⎪ j∈Ni ⎪ ⎩ 0, t ∈ [tk + θ, tk+1 ) (3) Where t ∈ [tk , tk+1 ), i ∈ F, k = 1, 2, ..., α > 0, β > 0 are gain parameters to be determined. h = tk+1 − tk > 0 is the sampling period, θ is the communication width and 0< 0 1 θ < h. Let εi (t) = (xi (t), vi (t))T , A = ,B = 0 0
0 0 . When i ∈ F, α β
8079
⎧ n ⎨ Aε (t) − l Bε (t ), t ∈ [t , t + θ) i ij j k k k ε˙i (t) = . j=1 ⎩ Aεi (t), t ∈ [tk + θ, tk+1 ) (4)
.
Let εF (t) = (ε1 (t), ε2 (t), ..., εm (t))T and system (4) can be rewritten as the following form ⎧ ⎨ (Im ⊗ A)εF (t) −[L11 L12 ] ⊗ Bε(tk ), t ∈ [tk , tk + θ) . (5) ε˙F (t) = ⎩ (Im ⊗ A)εF (t), t ∈ [tk + θ, tk+1 ) When i ∈ L, ε˙L (t) = (In−m ⊗ A)εL (t), t ∈ [tk , tk+1 ).
(6)
Combine (5) with (6), we can obtain (In ⊗ A)ε(t) − (L ⊗ B)ε(tk ), t ∈ [tk , tk + θ) ε(t) ˙ = . (In ⊗ A)ε(t), t ∈ [tk + θ, tk+1 ) (7) Then ε(tk + θ) = t +θ e(In ⊗A)θ ε(tk ) − tkk e(In ⊗A)(tk +θ−τ ) (L ⊗ B)ε(tk )dτ = (I2n + θ(In ⊗ A) − L ⊗ (θB + 12 θ2 AB))ε(tk ). (8) And ε(tk+1 ) = (I2n + (h − θ)(In ⊗ A))ε(tk + θ).
(9)
Combining (8) and (9), the evolution of the controlled system can be expressed by the following discrete system ε(tk+1 ) = W ε(tk ).
(10)
where W = I2n +h(In ⊗A)−L⊗(θB+(hθ− 12 θ2 )AB), k = 1.2, ...
W12 W11 , W = 0 W22 1 W11 = I2m + hIm ⊗ A − L11 ⊗ (θB + (hθ − θ2 )AB), 2 1 W12 = −L12 ⊗ (θB + (hθ − θ2 )AB), 2 W22= I2(n−m) + hIn−m ⊗ A. From equality (10), we can obtain ε(tk ) = W k ε(t0 ). (11)
k Φk−1 W11 k−1 Where W k = , Φk−1 = W11 W12 + k 0 W22 k−2 k−1 W11 W12 W22 + ... + W12 W22 . According to some calculations, we can obtain −1 k k Φk−1 = W11 ((L−1 11 L12 ) ⊗ I2 ) − ((L11 L12 ) ⊗ I2 )W22 .
Rewrite the above expression (11) in the follower and leader components, we can obtain
k εF (tk ) εF (t0 ) + Φk−1 εL (t0 ) W11 , (12) = k εL (tk ) W22 εL (t0 )
Lemma 3 The multi-agent system using the intermittent control algorithm achieves containment control for dynamic leaders asymptotically if and only if the directed graph G has a spanning forest and ρ(W11 ) < 1. Proof 1 (Proof of Lemma 3) Motivated by the work in [14], we obtain the following proof. From the property of the Laplacian matrix L and Lemma 1, we can obtain L1n = 0n . According to equality (2), we can obtain L11 1m + L12 1n−m = 0m and −(L−1 11 L12 )1n−m = 1m . Also, all the elements of L12 are non-positive and all the elements −(L−1 11 L12 ) are nonnegative. Therefore all the elements −(L−1 11 L12 ) are nonnegative. Let xL (t) = [xm+1 (t), xm+2 (t), ..., xn (t)]T denote the positions of the leaders. Hence, each element of m −(L−1 11 L12 )xL (t) ∈ R is in the convex hull formed by all the elements of xL (t). The case of the velocity counterpart is similar. Suppose that the condition ρ(W11 ) < 1 is satisfied. Besides, we get the equality (13). When tk → +∞, that is, k → +∞, we can obtain k W11 → 0. According to the equality (13), we can obtain εF (tk ) = limtk →∞ (−(L−1 11 L12 ) ⊗ I2 )εL (tk ). It is obvious that the final positions and velocities of all the followers are determined by the convex hull formed by the positions and velocities of all the leaders, respectively. According to Definition 2, we can obtain that the multi-agent system (1) using the impulsive algorithm (3) realizes containment control. Suppose that the containment control is realized but the condition is not satisfied. According to equality (13), when tk → +∞, k → +∞, the final states of the followers are not only affected by the states of the leaders but also affected by the states of the followers. However, from Definition 2, we can obtain that the containment control is realized means the position and velocity of each follower will converge to the convex hull formed by the positions and velocities of the leaders as t → +∞, respectively. That is, the final states of the followers can only be affected by the states of the leaders. There is a contradiction. Theorem 1 The multi-agent system (1) using intermittent containment control algorithm (3) realizes containment control asymptotically if and only if the directed graph G has a directed spanning forest and the following conditions should be satisfied a>0 (14) abd + a2 c − d2 > 0 where
where
εL (tk ) = [εm+1 (tk ), εm+2 (tk ), ..., εn (tk )]T , k = 1, 2, .... Obviously, we can obtain − ((L−1 11 L12 ) ⊗ I2 )εL (tk ).
a = Re( 2β−αθ αh ), b = Im( 2β−αθ αh ), 4−λi αθ(h−θ)−2λi βθ c = Re( ), λi αhθ i βθ ), d = Im( 4−λi αθ(h−θ)−2λ λi αhθ
and λi , i = 1, 2, · · ·, m, denote all the eigenvalues of L11 .
εF (tk ) = [ε1 (tk ), ε2 (tk ), ..., εm (tk )]T ,
k (εF (t0 ) + ((L−1 εF (tk ) =W11 11 L12 ) ⊗ I2 )εL (t0 ))
(15)
(13)
Proof 2 (Proof of Theorem 1) According to Lemma 3, we just need to prove ρ(W11 ) < 1. Let J be the Jordan form associated with L11 . That is L11 = P JP −1 , where P is a m × m non-singular matrix and J = triag {λ1 , λ2 , ..., λm }.Because of W11 = I2m + hIm ⊗ A − L11 ⊗ (θB + (hθ − 12 θ2 )AB), we can obtain Y =
8080
I2m + hIm ⊗ A − J ⊗ (θB + (W11 )AB), where Y = (P −1 ⊗ I2 )W11 (P ⊗ I2 ). So, we can obtain the eigenvalues of W11 by solving the equation |sI2m − Y | = 0. That is, m di (s) = 0, where i=1
s − 1 + λi α(hθ − 1 θ2 ) 2 di (s) = λi αθ
ϳ ϭ
ϱ
ϲ
Ϯ
λi β(hθ − 12 θ2 ) − h . s − 1 + λi βθ
We need to determine Schur stability of the polynomial:
ϭϬ
ϯ
ϰ
ϴ
ϵ
Fig. 1: Undirected interaction graph
1 di (s) = s + (λi α(hθ − θ2 ) + λi βθ − 2)s 2 1 2 +1+ λi αθ − λi βθ, i = 1, 2, ..., m. 2 2
ϳ ϭ
Let s = [(σ + 1)/(σ − 1)] and we can obtain
ϱ
ϲ
Ϯ
σ+1 ri (σ) = (σ − 1)2 di ( ) σ−1 ri (σ) = λi αhθσ 2 + (2λi βθ − λi αhθ2 )σ +4 − λi αθ(h − θ) − 2λi βθ.
ϭϬ
ϯ
ϰ
ϴ
ϵ
Fig. 2: Directed interaction graph
Hence 4 − λi αθ(h − θ) − 2λi βθ 2β − αθ σ+ . αh λi αhθ (16) The polynomial di (s) is Schur stable if and only if Ri (σ) is Hurwitz stable. According to Lemma 2, the complex polynomial Ri (σ) is Hurwitz stable if and only if conditions (14) and (15) are satisfied. Ri (σ) = σ 2 +
1200
time t
800
Remark 2 When k → +∞ and tk → +∞, the final positions and velocities of followers are εF (tk ) → −((L−1 11 L12 ) ⊗ I2 )εL (tk ). Theorem 1 holds for the cases with only one leader in the systems and the containment control achieves means that the positions and velocities of the followers will track the position and velocity of the one leader as tk → +∞. Corollary 1 The multi-agent system (1) using intermittent containment control algorithm (3) achieves containment control asymptotically if and only if the communications among the followers are undirected and for each follower, there exists at least one leader which has a directed path to that follower in the graph G, and the following conditions should be satisfied 2β θ< , (17) α 4 − 2λi βθ h < min ( + θ), (18) 1≤i≤m λi αθ where λi , i = 1, 2, · · ·, m, denote all the eigenvalues of L11 .
4
Simulations
In this section, we give some examples to verify the theoretical analysis. Consider the system (1) and the undirected and directed communication graphs are shown in Figs.1 and 2, respectively. The agents labeled as 7, 8, 9 and 10 are the leaders and the agents labeled as 1, 2, 3, 4, 5 and 6 are the followers. In Figs.1, the arrows indicate the follower agent can receive the information of the corresponding leader agent.
follower1 follower2 follower3 follower4 follower5 follower6 leader7 leader8 leader9 leader10
1000
600 400 200 0 −50 0 50 100 150 yi(t)
100
−100
0
−200
−300
xi(t)
Fig. 3: Positions of all the agents using algorithm (3) with static leaders, where α = 0.5 and β = 0.5.
Example 1 Suppose that the leaders are static. When the communications are undirected and the topology is shown in Fig.1, Using the algorithm (3), we can obtain the simulation results shown in Fig. 3. According to the conditions in Corollary 1, α and β can be arbitrary positive number and we choose α = 0.5 and β = 0.5. From inequality (17), communication width θ < 2 and we can obtain sampling period from inequality (18). So we let θ = 0.1 and h = 0.2 to satisfy the conditions. In Fig. 3, we can obtain that the final positions of the followers converge into the convex hull formed by the positions of the static leaders and Fig. 4 shows that the velocities of all the followers converge to zero in the end. Example 2 Suppose that the leaders are dynamic (vi (0) = (0.5, 0.5), i ∈ L). When the communications are undirected and the topology is shown in Fig. 1, Using the algorithm (3), we can obtain the simulation results shown in Fig. 5 and
8081
we let θ = 0.1 and h = 0.2 to satisfy the conditions. In Fig. 5 and 6, we can obtain that the final positions and velocities of the followers converge into the convex hull formed by the leaders in the end, respevtively.
150 follower1 follower2 follower3 follower4 follower5 follower6 leader7 leader8 leader9 leader10
100
v(t)
50
5
In this paper, containment control of second-order multiagent systems with multiple leaders via intermittent sampled data algorithms has been investigated. A necessary and sufficient condition, which is related to the coupling gains, the sampling period, the communication width and the structure of the networks, is obtained for achieving containment control, which guarantees that the states of all the followers will converge into the convex hull formed by the states of the leaders asymptotically. Besides, when the communications among the followers are undirected, some much simpler conditions are obtained. Then we give some simulations to verify the effectiveness of our theoretical analysis. In the future, we will focus the containment control of swarm problem.
0
−50
−100
0
100
200
300
400
500
time t
Fig. 4: Velocities of all the agents using algorithm (3) with static leaders, where α = 0.5 and β = 0.5
References
1200 follower1 follower2 follower3 follower4 follower5 follower6 leader7 leader8 leader9 leader10
1000
time t
800 600 400 200 0 400 200 0 yi(t)
−200
−200
−150
−100
−50
0
50
150
100
xi(t)
Fig. 5: Positions of all the agents using algorithm (3) with dynamic leaders, where α = 0.15 and β = 0.15.
80
follower1 follower2 follower3 follower4 follower5 follower6 leader7 leader8 leader9 leader10
60
40
v(t)
20
0
−20
−40
−60
Conclusions
0
200
400
600
800
1000
time t
Fig. 6: Velocities of all the agents using algorithm (3) with dynamic leaders, where α = 0.15 and β = 0.15.
6. According to the conditions in Corollary 1, α and β can be arbitrary positive number and we choose α = 0.15 and β = 0.15. From inequality (17), communication width θ < 2 and we can obtain sampling period from inequality (18). So
[1] Li Ding, Pian Yu, Zhiwei Liu, and Zhihong Guan. Consensus and performance optimisation of multi-agent systems with position-only information via impulsive control. IET Control Theory & Applications, 7(1):16–24, 2013. [2] Li Ding, Pian Yu, Zhiwei Liu, Zhihong Guan, and Gang Feng. Consensus of second-order multi-agent systems via impulsive control using sampled hetero-information . Automatica, 49(9):2881–2886, 2013. [3] Zhiwei Liu, Xinghuo Yu, Zhihong Guan, Bin Hu, and Chaojie Li. Pulse-modulated intermittent control in consensus of multiagent systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016. [4] Zhiwei Liu, Zhihong Guan, Xuemin Shen, and Gang Feng. Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements. IEEE Transactions on Automatic Control, 57(10):2639–2643, 2012. [5] Hongwei Zhang, Frank L Lewis, and Zhihua Qu. Lyapunov, adaptive, and optimal design techniques for cooperative systems on directed communication graphs. IEEE Transactions on Industrial Electronics, 59(7):3026–3041, 2012. [6] Pian Yu, Li Ding, Zhiwei Liu, and Zhihong Guan. Leader– follower flocking based on distributed event-triggered hybrid control. International Journal of Robust and Nonlinear Control, 26(1):143–153, 2016. [7] Felipe Cucker and Steve Smale. Emergent behavior in flocks. IEEE Transactions on Automatic Control, 52(5):852–862, 2007. [8] Wei Ren, Randal W Beard, and Ella M Atkins. Information consensus in multivehicle cooperative control. IEEE Control Systems, 27(2):71–82, 2007. [9] Francesco Bullo, Jorge Cortes, and Sonia Martinez. Distributed control of robotic networks: a mathematical approach to motion coordination algorithms. Princeton University Press, 2009. [10] Yuebing Hu, Housheng Su, and James Lam. Adaptive consensus with a virtual leader of multiple agents governed by locally lipschitz nonlinearity. International Journal of Robust and Nonlinear Control, 23(9):978–990, 2013. [11] Ping Wang and Yingmin Jia. Brief paper-distributed containment control of second-order multi-agent systems with inherent non-linear dynamics. IET Control Theory & Applications, 8(4):277–287, 2014.
8082
[12] Wei Zhu and Daizhan Cheng. Leader-following consensus of second-order agents with multiple time-varying delays. Automatica, 46(12):1994–1999, 2010. [13] Meng Ji, Giancarlo Ferrari-Trecate, Magnus Egerstedt, and Annalisa Buffa. Containment control in mobile networks. IEEE Transactions on Automatic Control, 53(8):1972–1975, 2008. [14] Huiyang Liu, Guangming Xie, and Long Wang. Necessary and sufficient conditions for containment control of networked multi-agent systems. Automatica, 48(7):1415–1422, 2012. [15] Huiyang Liu, Guangming Xie, and Long Wang. Containment of linear multi-agent systems under general interaction topologies. Systems & Control Letters, 61(4):528–534, 2012. [16] Yongcan Cao, Wei Ren, and Magnus Egerstedt. Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks. Automatica, 48(8):1586–1597, 2012. [17] Hong Zhou, Zhenhua Wang, Zhiwei Liu, Wenshan Hu, and Guoping Liu. Containment control for multi-agent systems via impulsive algorithms without velocity measurements. IET Control Theory & Applications, 8(17):2033–2044, 2014. [18] Guanghui Wen, Zhisheng Duan, Zhongkui Li, and Guanrong Chen. Consensus and its 2-gain performance of multi-agent systems with intermittent information transmissions. International Journal of Control, 85(4):384–396, 2012. [19] Guanghui Wen, Zhisheng Duan, Wei Ren, and Guanrong Chen. Distributed consensus of multi-agent systems with general linear node dynamics and intermittent communications. International Journal of Robust and Nonlinear Control, 24(16):2438–2457, 2014. [20] Na Huang, Zhisheng Duan, and Yu Zhao. Consensus of multiagent systems via delayed and intermittent communications. IET Control Theory & Applications, 9(1):62–73, 2015. [21] Hongjie Li and Hongye Su. Distributed consensus of multiagent systems with nonlinear dynamics via adaptive intermittent control. Journal of the Franklin Institute, 352(10):4546– 4564, 2015. [22] Wenwu Yu, Lei Zhou, Xinghuo Yu, Jinhu L¨u, and Renquan Lu. Consensus in multi-agent systems with second-order dynamics and sampled data. IEEE Transactions on Industrial Informatics, 9(4):2137–2146, 2013. [23] Wenwu Yu, Wei Xing Zheng, Guanrong Chen, Wei Ren, and Jinde Cao. Second-order consensus in multi-agent dynamical systems with sampled position data. Automatica, 47(7):1496– 1503, 2011.
8083
