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Second-Order Global Consensus in Multiagent Networks With Random Directional Link Failure Huaqing Li, Xiaofeng Liao, Senior Member, IEEE, Tingwen Huang, Wei Zhu, and Yanbing Liu
Abstract— In this paper, we consider the second-order globally nonlinear consensus in a multiagent network with general directed topology and random interconnection failure by characterizing the behavior of stochastic dynamical system with the corresponding time-averaged system. A criterion for the secondorder consensus is derived by constructing a Lyapunov function for the time-averaged network. By associating the solution of random switching nonlinear system with the constructed Lyapunov function, a sufficient condition for second-order globally nonlinear consensus in a multiagent network with random directed interconnections is also established. It is required that the second-order consensus can be achieved in the time-averaged network and the Lyapunov function decreases along the solution of the random switching nonlinear system at an infinite subsequence of the switching moments. A numerical example is presented to justify the correctness of the theoretical results. Index Terms— Global consensus, multiagent network, nonlinear dynamics, random switching, second-order consensus.
I. I NTRODUCTION ISTRIBUTED coordination of multiagent systems has been intensively studied in recent years. The consensus problem as one of the most fundamental research topics in the field of coordination control of multiagent systems has attracted considerable attention over the past few years due to its extensive applications in cooperative control of mobile autonomous robots, design of distributed sensor networks, spacecraft formation control, and other areas [1]–[3]. Consensus means that the states of all agents reach an agreement on a common value of interest using local information of each agent’s neighbors. The consensus problem for agents with first-order dynamics has recently been investigated from various perspectives [4]–[11], [30], [31].
D
Manuscript received September 17, 2013; revised December 24, 2013 and April 7, 2014; accepted April 22, 2014. Date of publication May 16, 2014; date of current version February 16, 2015. This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant XDJK2014C117, in part by the National Natural Science Foundation of China under Grant 60973114, Grant 61170249, and Grant 61003256, in part by the Research Fund of Preferential Development Domain for the Doctoral Program, Ministry of Education of China, under Grant 201101911130005, and in part by the National Priority Research Project under Grant NPRP 4-1162-1-181 through the Qatar National Research Fund, Qatar. H. Li and X. Liao are with the College of Electronics and Information Engineering, Southwest University, Chongqing 400715, China (e-mail:
[email protected];
[email protected]). T. Huang is with Texas A&M University at Qatar, Doha 23874, Qatar (e-mail:
[email protected]). W. Zhu is with the College of Mathematics and Physics, Chongqing 400065, China (e-mail:
[email protected]). Y. Liu is with the College of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2320274
In many practical situations, agents such as unmanned aerial vehicles and mobile robots can be controlled directly by their accelerations rather than by their velocities [12], [13]. Hence, there has been an increasing research interest on the consensus problems of second-order multiagent systems, where the agents are governed by both position and velocity states. Ren and Atkins [14] proposed several control algorithms for second-order consensus under directed graphs. It has been shown that the systems described by second-order integrators might not achieve consensus even if the directed graph has a directed spanning tree. Yu et al. [15], Zhu et al. [16], and Li et al. [17] presented some necessary and sufficient conditions to ensure the second-order consensus under a directed graph. It is now known that in most cases, secondorder consensus can be reached in multiagent systems if the coupling control gains and the spectra of the Laplacian matrix satisfy some additional conditions, which are somewhat different from those in multiagent systems with first-order dynamics [6]–[8]. In addition, in reality, the agents might be governed by some nonlinear terms, and the agents usually have time-varying intrinsic velocities rather than constant ones, even after a velocity consensus has been reached. In this case, the agents are not only affected by the interaction among neighboring agents, but also by their own intrinsic nonlinear dynamics [18]–[21]. Yu et al. [18] studied the leaderless consensus problem for second-order multiagent systems with intrinsic nonlinear dynamics under directed graphs. The work is extended in [19] to the leader-following tracking case, in [20] to a connectivity-preserving control algorithm, and in [21] to the robust leader-following consensus in a finite time. The connectivity of a graph plays a key role in the behavior of multiagent systems. In practice, however, the connectivity of a graph may vary over time, and their interaction topology may also be changing dynamically with time. Therefore, the problem of second-order consensus of multiagent systems with nonlinear dynamics and switching topology deserves more attention. In general, the extension of consensus algorithms for multiagent systems from first-order to second-order is nontrivial except in the case that all topologies in the switching sequence have directed spanning trees. Guo et al. [22] investigated the flocking problem of leader-following multiagent systems in directed graphs with switching topology. From the results obtained, it is found that when the multiagent networks run on topologies with isolated agents, the proposed techniques in [22] need to add several directed edges between the leader agent and the corresponding isolated agents to constantly guarantee the connectivity of the underlying interaction topology. Thus, the results essentially require all
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directed topologies in the switching sequence have directed spanning trees. To this end, Li et al. [23] considered the second-order consensus multiagent systems with nonlinear dynamics and random switching directed topology. Making use of the orthogonal decomposition technique and local linearization method, two criteria for almost surely secondorder nonlinear consensus for the nontime-delay coupling and time-delay coupling are, respectively, derived. Meanwhile, Li et al. [24] further discussed the second-order consensus of multiagent systems with nonlinear dynamics and arbitrarily fast switching directed topology using the generalized matrix measure and tools from contraction and circle analysis. The theoretical results in [23] and [24] require that the inherent nonlinear term of all individual agents are continuously differentiable to meet the need of local linearization, and thus are limited to second-order local rather than global consensus. Inspired by the previous studies, this paper addresses the problem of second-order globally nonlinear consensus in multiagent networks with directed topology and random switching interconnections, in which there may exist some isolated agents in certain topology during the random switching sequence. This topic is not only significant, but also fundamental, which has not been fully addressed so far. This is because in such case, the methods for analyzing the stability of switched systems, e.g., the common Lyapunov function method or the multiple Lyapunov functions method, fail to deal with the stability of the resultant error dynamical system associated with the random switching network. We resort to the Lyapunov function designed for the corresponding timeaveraged network to further capture the convergence characteristics of the multiagent systems associated with the random switching directed network. First, by constructing a suitable Lyapunov function for the time-averaged network, a criterion for the second-order global consensus is derived. Then, by associating the solution of random switching nonlinear system with the constructed Lyapunov function, we analytically investigate the exponential stability of the resultant error system corresponding to the random switching nonlinear system. A sufficient condition for second-order globally nonlinear consensus in a multiagent network with random directed interconnections is established. The results obtained require that the second-order consensus can be achieved in the time-averaged network and the designed Lyapunov function decreases along the solution of the random switching nonlinear system at an infinite subsequence of the switching moments. From the results, we find that even though the network is not always connected instantaneously in time, sufficient information could be propagated through the dynamical network to guarantee the second-order globally nonlinear consensus. Finally, a numerical example is presented to show the effectiveness of the designed distributed interaction protocols and the correctness of the theoretical analysis. This paper is organized as follows. Some concepts and knowledge of graph theory are given in Section II. The problem to be solved is also formulated. Section III systematically investigates the criteria for reaching the secondorder globally nonlinear consensus of multiagent networks with the time-averaged directed topology and the random
switching directed topology, respectively. An illustrative example is presented in Section IV. Section V draws the conclusions of this paper. II. P RELIMINARY AND P ROBLEM F ORMULATION A. Graph Theory Let G = (V, E, We ) be a weighted directed graph of order N, with the set of nodes V = {v1 , v2 , . . . , v N }, the set of edges E ⊆ V × V , and a weighted adjacent matrix We = (Weij ) N×N . A directed edge in G is denoted by ei j = (vi , v j ). If there is a directed edge from node vi to node v j , it is said that node vi can reach node v j and we assign it an edge weight Weji > 0; otherwise Weji = 0. We assume that there are no self-loops or multiple edges in G, i.e., Weii = 0 for i = 1, 2, . . . , N. G has a directed spanning tree if there exists at least one node called the root, which has a directed path to all the other nodes in G. The Laplacian matrix denoted by L = (L i j ) N×N of G is defined as L i j = −Weij for i = j and L ii = Nj=1, j =i Weij , where i, j = 1, 2, . . . , N. In order to represent a random switching directed graph G(t) = (V, E, W (t)), we allocate an edge probability matrix P = ( pi j ) N×N for G. The existence of a directed edge ei j in G(t) between a pair of nodes vi and v j is randomly determined and is independent of other edges with probability pi j satisfying 0 ≤ pi j < 1. An information link is referred to as a potential link when the associated edge probability is positive. We define N(N − 1) independent Bernoulli random variables δi j s, i, j = 1, 2, . . . , N, i = j, as follows: δi j = 1 with probability pi j and δi j = 0 with probability 1 − pi j , in which each random variable δi j is associated with the edge ei j . Therefore, at time t, the weighted adjacency matrix W (t) = (Wi j (t)) N×N of G(t) can be determined as Wi j (t) = Weij δi j for i = j , and Wii (t) = 0, where i, j = 1, 2, . . . , N. Correspondingly, denote L(t) = (L i j (t)) N×N the time-varying Laplacian matrix associated with the random directed graph G(t). The sampling space (all possible topologies) of all such kind of random directed graphs in this framework is represented by the set M. Let tk s, k ∈ N, be the random switching moments appearing in the running process of the dynamical network G(t) during t ∈ [0, +∞). Suppose that the random switching instants satisfy t0 < t1 < · · · < tk−1 < tk < · · · , limk→∞ tk = +∞ and 0 < 1 = inf k {tk − tk−1 } ≤ supk {tk − tk−1 } = 2 < +∞. Define tk = tk − tk−1 , k ∈ N. In this paper, we assume that the sequence {tk }∞ k=1 consists of independent and identically distributed random variables. In addition, the time and topology switchings are independent of each other. Denote G¯ as the time-averaged graph of G(t). The Laplacian matrix of the time-averaged graph, denoted as L¯ = ( L¯ i j ) N×N , can be computed in elements by L¯ i j = − pi j Weij for i = j , and L¯ ii = Nj=1 pi j Weij , i, j = 1, 2, . . . , N. B. Problem Formulation Suppose that the multiagent network under consideration consists of N agents, which dynamically updates their states based on local information exchange. Each isolated agent is governed by second-order nonlinear dynamics. At time t,
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567
all agents are randomly interconnected pairwise according to the edge probability matrix using the weighted, directional position, and velocity information based on the dynamical topology of the directed random switching network G(t). Therefore, a more general version of multiagent networks of linearly coupled second-order dynamical systems can be formulated as follows: ⎧ x˙i (t) = vi (t) ⎪ ⎪ ⎪ N ⎪ ⎪ ⎨v˙i (t) = f (x i (t), vi (t), t) + α Wi j (t)B(x j (t)−x i (t)) (1) j =1 ⎪ ⎪ N ⎪ ⎪ ⎪ +β Wi j (t)B(v j (t) − vi (t)), i = 1, 2, . . . , N ⎩ j =1
where x i (t) = (x i1 (t), . . . , x in (t))T ∈ R n and vi (t) = (vi1 (t), . . . , vin (t))T are the position and velocity variable vectors of agent i , respectively. f (x i (t), vi (t), t) = ( f 1 (x i (t), vi (t), t), . . . , f n (x i (t), vi (t), t))T : R n × R n × R + → R n represents a continuous but not necessarily differentiable vector-valued function, which models the inherent nonlinear dynamics of the uncoupled agent i . α > 0 and β > 0 represent the position and velocity coupling strengths between any two agents in G(t). B ∈ R n×n denotes the inner coupling configuration between the agents. The notations used throughout this paper are quite standard. Let R + be the set of positive real numbers. The symmetric part of a matrix C ∈ R m×m is indicted with sym(C) = 1/2(C + C T ). N refers to the set of all nonnegative integers. Letting x ∈ R + , indicate x the largest nonnegative integer which is smaller than x. Definition 1: The second-order globally nonlinear consensus in the multiagent dynamical network (1) with random switching directed topology is considered to be achieved if, for any initial conditions x i (t0 ), vi (t0 ) ∈ R N , limt →∞ x i (t) − x j (t) = 0 and limt →∞ vi (t)− v j (t) = 0, i = j, ∀i, j = 1, 2, . . . , N. In what follows, for i = 2, 3, . . . , N, let X i1 (t) = x i (t) − x 1 (t) and Vi1 (t) = vi (t) − v1 (t) be the position and velocity differences between agent i and agent 1, respectively, in the multiagent dynamical network G(t). For convenience, T (t), . . . , X T (t))T , V¯ (t) = (V T (t), . . . , ¯ we define X(t) = (X 21 21 N1 T T ¯ VN1 (t)) and F( X (t), V¯ (t), t) = ( f T (x 2 (t), v2 (t), t) − f T (x 1 (t), v1 (t), t), . . . , f T (x N (t), v N (t), t) − f T (x 1 (t), v1 (t), t))T, the following error dynamical system in compact vector form can be derived from (1)
I(N−1)n X˙¯ (t) O(N−1)n X¯ (t) = αSW (t) ⊗ B β SW (t) ⊗ B V¯ (t) V¯˙ (t) ⎡ ⎢ ⎢ ⎢ ⎢ SW (σ (t)) = ⎢ ⎢ ⎢ ⎣
−W12 (t) −
j =2
W2 j (t)
W32 (t) − W12 (t) .. . W N2 (t) − W12 (t)
+
0(N−1)n ¯ F( X(t), V¯ (t), t)
(2)
where (3), as shown at the bottom of the page, holds. Correspondingly, for a multiagent network associated with the time-averaged directed topology corresponding to the ¯ we can obtain the following error Laplacian matrix L, dynamical system:
I(N−1)n O(N−1)n X˙¯ E (t) X¯ E (t) = α S¯ W ⊗ B β S¯ W ⊗ B V¯ E (t) V˙¯ E (t)
0(N−1)n + (4) F( X¯ E (t), V¯ E (t), t) where S¯ W ⎡ ¯ −W12 − W¯ 2 j W¯ 23 − W¯ 13 j =2 ⎢ ⎢ W¯ − W¯ W¯ 3 j −W¯ 13 − 32 12 ⎢ ⎢ j =3 =⎢ .. .. ⎢ ⎢ . . ⎣ ¯ W N2 − W¯ 12 W¯ N3 − W¯ 13
···
W¯ 2N − W¯ 1N
···
W¯ 3N − W¯ 1N
⎤
⎥ ⎥ ⎥ ⎥ ⎥ .. ⎥ .. ⎥ . . ⎦ W¯ N j · · · −W¯ 1N − j = N
(5) with W¯ = (W¯ i j ) N×N , W¯ i j = Weij Pi j , i, j = 1, 2, . . . , N. In addition T X¯ E (t) = X TE21 (t), . . . , X TE N1 (t) and T T V¯ E (t) = VE21 (t), . . . , VETN1 (t) where X Ei1 (t) = x Ei (t) − x E1 (t) and VEi1 (t) = v Ei (t) − v E1 (t) i = 2, 3, . . . , N. T F( X¯ E (t), V¯ E (t), t) = ( f (x E2 (t), v E2 (t), t) − f T (x E1 (t), v E1 (t), t), . . . , f T (x E N (t), v E N (t), t) − f T (x E1 (t), v E1 (t), t))T. In the following, we present some assumptions and lemmas which will be used to derive our main results. Assumption 1 [19]: For the nonlinear function f in (1), there exist two constant matrices M 1 = (wi j )n×n and M 2 =
W23 (t) − W13 (t) ··· −W13 (t) − W3 j (t) · · · j =3
.. . W N3 (t) − W13 (t)
W2N (t) − W1N (t) W3N (t) − W1N (t)
.. . . · · · −W1N (t) − W N j (t) ..
j = N
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(3)
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(m i j )n×n , in which wi j ≥ 0, m i j ≥ 0 such that f (x, v, t)− f (y, z, t) ≤
n
the following conditions hold:
(wi j |x j − y j | + m i j |v j −z j |)
j =1
i = 1, 2, . . . , n ∀x, y, v, z ∈ R
n
∀t ≥ 0.
(6)
Remark 1: Note that Assumption 1 is a Lipschitz-type condition satisfied by many well-known nonlinear dynamical systems. By the classical method of differential equation theory, Assumption 1 can guarantee the existence and uniqueness of the solution for the random switching nonlinear system (2) and the time-averaged system (4). In addition, for all t ≥ t0 , f (0, 0, t) = 0 holds, which makes the second-order consensus possible. Lemma 1 [12]: The Laplacian matrix L has a simple eigenvalue (zero) and all the other eigenvalues have positive real parts if and only if the underlying network G has a directed spanning tree. Lemma 2 [25]: Suppose that the network G¯ associated with time-averaged Laplacian matrix L¯ has a directed spanning tree. Then, all the eigenvalues of matrix − S¯ W are the same as those ¯ of L¯ except for the zero eigenvalue, i.e., λi (− S¯ W ) = λi+1 ( L) for i = 1, 2, . . . , N − 1. Lemma 3 [26]: Let k ∈ R and X, Y, P, and Q be matrices with appropriate dimensions. Then, the following statements hold:
Let C = diag{c1 , c2 , . . . , c N−1 }, D = diag{d1 , d2 , . . . , d N−1 }, where ci and di , i = 1, 2, . . . , N − 1, are positive real numbers. Using the two matrices M 1 and M 2 , which are the same as those defined in Assumption 1, we can define the following parameters: ρ1 = ρ2 = ρ3 =
max
1≤i≤N−1
max
1≤ j ≤n
max
1≤i≤N−1
max
1≤ j ≤n
max
1≤i≤N−1
max
1≤ j ≤n
n 2ε 2(1−ε) w j k + m 2ε di j k + wkj k=1 n 2(1−ε) m kj di
k=1 n 2(1−ε) wkj ci k=1
ρ4 =
max
1≤i≤N−1
max
1≤ j ≤n
ci
n
w2ε jk
+ m 2ε jk
+ m 2(1−ε) kj
We are now in the position to state our main results with regard to the second-order global consensus in multiagent systems with nonlinear dynamics and directed fixed topology. Theorem 1: If there exist two positive definite diagonal matrices C and D, two suitable positive coupling strengths α and β such that the conditions in (H1) hold, then the zero solution of the second-order nonlinear system (4) is globally asymptotically stable. Equivalently, the second-order globally nonlinear consensus in a multiagent network with fixed directed topology corresponding to the time-averaged Laplacian matrix L¯ can be achieved asymptotically. Proof: Construct the following Lyapunov function:
1 X¯ (t) (7) V ( X¯ E (t), V¯ E (t), t) = X¯ TE (t), V¯ ET (t) ¯ E VE (t) 2
A D ⊗ In
D ⊗ In C ⊗ In
with matrices A, C, and D the same as those defined in (H1). First, we show that the Lyapunov function (7) is valid, which implies > 0. By Schur complement theorem [19], the positive definiteness of matrix can be guaranteed by combining conditions (H1) (a) and (b). The time derivative of (7) along the solution of (4) produces the following results: V˙ ( X¯ E (t), V¯ E (t), t) = X¯ TE (t)A V¯ E (t)+ V¯ ET (t) (D ⊗ In ) V¯ E (t) +α X¯ TE (t)[(D ⊗ In ) ( S¯ W ⊗ B)] X¯ E (t) +β X¯ TE (t)[(D ⊗ In ) ( S¯ W ⊗ B)]V¯ E (t) +α V¯ ET (t)[(C ⊗ In ) ( S¯ W ⊗ B)] X¯ E (t) +β V¯ ET (t)[(C ⊗ In ) ( S¯ W ⊗ B)]V¯ E (t) + X¯ TE (t) (D ⊗ In ) F( X¯ E (t), V¯ E (t), t)
and
III. M AIN R ESULTS
=
3) (X ⊗ Y )(P ⊗ Q) = (X P) ⊗ (Y Q); 4) (X ⊗ Y )T = X T ⊗ Y T.
(b) C ⊗ In − (D ⊗ In ) A−1 (D ⊗ In ) > 0; ρ1 + ρ3 I(N−1)n < 0; (c) αsym((D ⊗ In ) ( S¯ W ⊗ B)) + 2 ρ2 +ρ4 I(N−1)n < 0. (d) D ⊗ In +βsym((C ⊗ In ) ( S¯ W ⊗ B))+ 2
where
1) (k X) ⊗ Y = X ⊗ (kY ); 2) (X + Y ) ⊗ P = X ⊗ P + Y ⊗ P;
(a) A = −βsym((D ⊗ In ) ( S¯ W ⊗ B)) − αsym((C ⊗ In ) ( S¯ W ⊗ B)) > 0;
k=1
with ε ∈ [0, 1]. To this end, we further make the following assumption. (H1) Suppose that there exist two positive constants α, β and two positive definite diagonal matrices C and D such that
+V¯ ET (t) (C ⊗ In ) F( X¯ E (t), V¯ E (t), t) = α X¯ TE (t)[(D ⊗ In ) ( S¯ W ⊗ B)] X¯ E (t) +V¯ ET (t) (D ⊗ In ) V¯ E (t) +β V¯ ET (t)[(C ⊗ In ) ( S¯ W ⊗ B)]V¯ E (t)
+ X¯ TE (t) (D ⊗ In ) F( X¯ E (t), V¯ E (t), t) +V¯ ET (t) (C ⊗ In ) F( X¯ E (t), V¯ E (t), t). (8)
By Assumption 1 and the algebraic inequality 2μ|x y| ≤ μ2ε x 2 + μ2(1−ε) y 2 , ∀μ ≥ 0, x, y ∈ R, ε ∈ [0, 1], [19], (9), as
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569
shown at the top of the next page, can be derived
the following results after some manipulation:
X¯ TE (t) (D ⊗ In ) F( X¯ E (t), V¯ E (t), t)
V˙ ( X¯ E (t), V¯ E (t), t) ≤ X¯ TE (t) ρ1 +ρ3 I(N−1)n X¯ E (t)+ V¯ ET (t) × αsym((D⊗ In ) ( S¯ W ⊗ B))+ 2 ρ2 +ρ4 ¯ × D⊗ In+βsym((C ⊗ In ) ( SW ⊗ B))+ I(N−1)n V¯ E (t). 2 (11)
=
N−1
di (x (i+1)e (t) − x 1e (t))T ( f (x (i+1)e (t), v(i+1)e (t), t)
i=1
− f (x 1e (t), v1e (t), t)) =
N−1 n i=1 j =1
j j di x (i+1)e (t)−x 1e (t) ( f j (x (i+1)e (t), v(i+1)e (t), t) − f j (x 1e (t), v1e (t), t))
≤
N−1 n n i=1 j =1 k=1
j j di x (i+1)e (t) − x 1e (t)
k k k k (t) − x 1e (t) + m j k v(i+1)e (t) − v1e (t) × w j k x (i+1)e
≤
N−1 n n 2 1 2ε j j di w j k x (i+1)e (t) − x 1e (t) 2 i=1 j =1 k=1
2(1−ε) k x (i+1)e (t)
+w j k +
2
N−1 n n 2 1 2ε j j di m j k x (i+1)e (t) − x 1e (t) 2 i=1 j =1 k=1
≤
k − x 1e (t)
k 2 k v(i+1)e (t) − v1e +m 2(1−ε) (t) jk
n N−1 n 2 j 1 j di w2ε j k x (i+1)e (t) − x 1e (t) 2 i=1 j =1 k=1
+
N−1 n n j 2 1 j di m 2ε j k x (i+1)e (t) − x 1e (t) 2 i=1 j =1 k=1
+
+
1 2 1 2
n n N−1 i=1 j =1 k=1 n n N−1
1 2
N−1 ⎨ i=1
2(1−ε) j v(i+1)e (t)
di m kj
i=1 j =1 k=1
⎧
≤
2 j 2(1−ε) j x (i+1)e (t) − x 1e (t) di wkj
⎩
di
n n
j
− v1e (t)
2
2(1−ε)
2ε w2ε j k + m j k + wkj
j =1 k=1
j 2 j × x (i+1)e (t) − x 1e (t) ⎧ ⎫ n N−1 n ⎬ 1 ⎨ 2(1−ε) j 2 j m kj + v(i+1)e (t) − v1e (t) di ⎩ ⎭ 2 i=1
j =1 k=1
ρ2 ρ1 (9) ≤ X¯ E (t) 2 + V¯ E (t) 2 . 2 2 After a similar derivation process as that for obtaining inequality (9) and noting the definitions of ρ3 and ρ4 , we can obtain the following results: V¯ ET (t) (C ⊗ In ) F( X¯ E (t), V¯ E (t), t) ρ3 ρ4 ≤ X¯ E (t) 2 + V¯ E (t) 2 . (10) 2 2 Substituting inequalities (9) and (10) into (8) and one obtains
By conditions H1 (c) and (d), we have V˙ ≤ 0 for all X¯ E (t) and V¯ E (t). In addition, V˙ = 0 if and only if X¯ E (t) = 0 and V¯ E (t) = 0. Therefore, the set M = {( X¯ TE (t), V¯ ET (t))T | X¯ E (t) = V¯ E (t) = 0} is the largest invariant set contained in the set D = {( X¯ TE (t), V¯ ET (t))T |V˙ = 0} for the nonlinear dynamical system (4). According to LaSalle’s invariance principle [19], all trajectories of system (4) starting from any initial condition will finally approach the set M as t → ∞, i.e., X¯ E (t) → 0 and V¯ E (t) → 0 as t → ∞. This means that the second-order globally nonlinear consensus in a multiagent network associated with the underlying interaction topology corresponding to the time-averaged Laplacian matrix L¯ can be achieved asymptotically. This completes the proof. Remark 2: According to Lemma 2 and the knowledge from algebraic graph theory [27], L¯ has a zero eigenvalue with algebraic multiplicity m if and only if the fixed topology of graph G¯ associated with the Laplacian matrix L¯ has m connectivity branches. This also means that the algebraic multiplicity of eigenvalue zero of the matrix S¯ W equals to m − 1 (when the algebraic multiplicity of eigenvalue zero of matrix S¯ W equals to 0, we claim that S¯ W does not have zero eigenvalue). In fact, conditions (c) and (d) in (H1) imply that matrix S¯ W does not have zero eigenvalue. Otherwise, the matrices βsym((D ⊗ In )( S¯ W ⊗ B)) and αsym((C ⊗ In )( S¯ W ⊗ B)) have zero eigenvalues, which further indicate that (H1) (c) and (d) do not hold. Furthermore, (H1) implies that there exists a directed spanning tree in the underlying fixed time¯ averaged topology associated with the Laplacian matrix L. Therefore, the connectivity of a graph is a necessary condition for consensus in multiagent systems. In addition, in Theorem 1 as well as the following Theorem 2, we omit the proof of existence and uniqueness of the solution to nonlinear systems (2) and (4). Under the Lipschitz-type condition as described in Assumption 1, this can be proved using the classic differential equation theory. In what follows, we will study the second-order global consensus in multiagent networks with nonlinear dynamics and random switching directed interconnections. For the case that all topologies of random dynamical network G(t) contain directed spanning trees, e.g., the connectivity of network can be maintained constantly, the results can be obtained trivially by mimicking the analysis procedure for fixed topology. However, when there are some isolated agents in certain topologies during the random switching process, these isolated nodes fail to receive neighbors’ state information to further update their current states. Thus, it is possible that these isolated agents may deviate the consensus manifold as time goes by. From the theoretical perspective, this can also be viewed as that the
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coefficient matrix of the resultant error dynamical system (2) is a singular matrix when the associated topology does not contain a directed spanning tree. Thus, the classical methods for dealing with the switched system, i.e., common Lyapunov function method or multiLyapunov functions method, are no longer valid. However, in general, the time-averaged network of the random switching dynamical network G(t) may contain a directed spanning tree. It is natural to think of using the Lyapunov function constructed for the nonlinear system associated with time-averaged network to further characterize the convergence properties of the corresponding random switching nonlinear system. Therefore, by associating it with the solution of the random switching dynamical network G(t), we can make use of the constructed Lyapunov function for the time-averaged network to analyze the second-order global consensus in G(t). For convenience, let y(t) = ( X¯ T (t), V¯ T (t))T and V (y(t), t) = 1/2y T (t) y(t), where is the same as that defined in (7). We further make the following assumption. (H2) Suppose there exists an infinite subsequence composed of some fixed moments in the random switching sequence, ∗ < tk∗ < · · · , satisfying denoted by t0 = t1∗ < t2∗ < · · · < tk−1 ∗ limk→+∞ tk = +∞. In addition, there exists a finite positive ∗ − t ∗ ≤ q and V (y(t ∗ ), t ∗ ) − integer q such that ti+1 2 i i+1 i+1 ∗ ∗ V (y(ti ), ti ) ≤ − p y(ti∗ ) 2 , where i = 1, 2, . . . , and p > 0. We are now in the position to state our main results for achieving the second-order globally nonlinear consensus in the multiagent network (1) with directed topology and random switching interconnections. Theorem 2: Suppose that the conditions in (H1) and (H2) hold. The globally exponential stability of zero solution of random switching nonlinear system (2) can be guaranteed. Equivalently, the second-order globally nonlinear consensus in the multiagent network (1) with directed topology and random switching interconnections can be reached. Proof: It follows from (H1) that Theorem 1 holds and the second-order globally nonlinear consensus in time-averaged network of random switching network G(t) can be achieved. In addition, if is associated with the solution of the random switching dynamical network G(t) with (7), the positive definiteness of Lyapunov function V (y(t), t) can also be ensured. In the following, we will make use of the properties (H2) of V (y(t), t) to further guarantee the exponential stability of zero solution of random switching nonlinear system (2). According to algebraic theory, we can obtain 1/ 2λmin ( ) y(t) 2 ≤ V (y(t), t) ≤ 1/2λmax ( ) y(t) 2 . From (H2), we have that there exists a k0 ∈ N such that tk∗0 − t0 < q2 . For simplicity, let ˆ X(t), ¯ F( V¯ (t), t) =
O(N−1)n I(N−1)n αSW (σ (t)) ⊗ B β SW (σ (t)) ⊗ B ˜ X¯ (t), V¯ (t), t) + F(
where ˜ X¯ (t), V¯ (t), t) = F(
0(N−1)n ¯ F( X(t), V¯ (t), t)
.
y(t) (12)
When t ∈ [t0 , tk∗0 ], one has t ˆ X(s), ¯ F( V¯ (s), s)ds. y(t) = y(t0 ) +
(13)
t0
If s ∈ [t0 , t], by computation, we have ˆ X¯ (s), V¯ (s), s) ≤ K y(s) F( where K = K 1 + K 2 , with K 1 = (max {α, β})2 A1 + 1 and K2 = in which
2 max {B1 , C1 }
(14)
(15)
(16)
T A1 = λmax SW (σ (t))SW (σ (t)) λmax (B T B) ! " B1 = max max w2j k 1≤ j ≤n 1≤k≤n
and C1 = max
! " max m 2j k .
1≤ j ≤n 1≤k≤n
For more details, please refer to the Appendix. Thus, based on the Bellman–Growall formula [28], we can obtain the following results from (13): $ # y(t) ≤ y(t0 ) e K (t −t0 ) , t ∈ t0 , tk∗0 (17) where tk∗0 is a switching instant in the subsequence defined in (H2). Since 1 % %2 V y tk∗0 , tk∗0 ≤ λmax % y tk∗0 % 2 and % %2 V y tk∗0 +1 , tk∗0 +1 − V y tk∗0 , tk∗0 ≤ − p % y tk∗0 % < 0 it can be derived that ∗ ∗ V y tk0 +1 , tk0 +1 ≤ 1 −
2p V y tk∗0 , tk∗0 λmax ( )
(18)
for t > tk∗0 . Due to 0 ≤ V (y(tk∗0 +1 ), tk∗0 +1 ) < V (y(tk∗0 ), tk∗0 ), we have 0 ≤ 1 − 2 p/λmax ( ) < 1. Repeat this computation procedure m ∗ times [as determined in the (20)], we have
m ∗ 2p V y tk∗0 +m ∗ , tk∗0 +m ∗ ≤ 1− V y tk∗0 , tk∗0 . (19) λmax ( ) Let φ = λmax ( )/λmin ( ) and select &
'
( λmin ( ) 2p m ∗ = ln ln 1 − λmax ( ) λmax ( ) we can desire from (19) that
m ∗ % ∗ % %2 %2 2p %y t % ≤ 1 − φ % y tk∗0 % k0 +m ∗ λmax ( ) and from (H2) that 1−
2p λmax ( )
(20)
(21)
m ∗ φ < 1.
(22)
LI et al.: SECOND-ORDER GLOBAL CONSENSUS IN MULTIAGENT NETWORKS
Define λ=
) ln 1 −
2p λmax ( )
*
>0 (23) −q2 by some computations, the following relationship can be easily established, that is:
m ∗ 2p −λ tk∗ +m ∗ −tk∗ −λm ∗ q2 0 0 1− =e ≤e . (24) λmax ( ) Combining (21)–(24), it can be derived that ) * % ∗ % % − λ2 tk∗ +m ∗ −tk∗ % %y t % % y t ∗ %. 0 0 ≤ φe (25) k0 +m ∗ k0 Then, for all n ∈ N, by a similar argument as that leading to (21), we can obtain
nm ∗ % ∗ % %2 %2 2p %y t % ≤ 1 − φ % y tk∗0 % . (26) k0 +nm ∗ λmax ( ) By the definitions of (20) and (23), one further has inequality * )
nm ∗ 2p ∗ −λ tk∗ +nm ∗ −tk∗ 0 0 1− = e−λnm q2 ≤ e (27) λmax ( ) holds. Therefore,∀n ∈ N, the following inequality holds: ) * % ∗ % % − λ2 t ∗ −tk∗ % k0 +nm ∗ %y t % %y t ∗ % . 0 ≤ (28) φe k0 +nm ∗ k0 On the other hand, for all t ≥ tk∗0 , there exists n 0 ∈ N such that for t ∈ [tk∗0 +n0 m ∗ , tk∗0 +(n0 +1)m ∗ ), we have t % % ˆ X¯ (s), V¯ (s), s)ds y(t) ≤ % y tk∗0 +n0 m ∗ % + F( % % ≤ % y tk∗0 +n0 m ∗ % + ≤e
) K t −tk∗
*
0 +n 0 m
) − λ2 tk∗
∗
tk∗
0 +n 0 m
∗
t
K y(s) ds
tk∗
∗ 0 +n 0 m
% ∗ %y t
% %
k0 +n 0 m ∗ ) * ∗ −tk K t −tk∗ +n m∗
*
% ∗ % e φ % y tk0 % ≤e ) * % % ∗ − λ t∗ −t ∗ (29) ≤ e 2 k0 +n0 m ∗ k0 e K m q2 φ % y tk∗0 % 0 +n 0
0
0
0m
∗
since t − tk∗0 +n0 m ∗ ≤ m ∗ q2 . For t ∈ [tk∗0 +n0 m ∗ , tk∗0 +(n0 +1)m ∗ ), we have tk∗0 +n0 m ∗ − tk∗0 ≥ t − m ∗ q2 − tk∗0 . Thus, the following inequality holds: % − λ (t −tk∗ ) λ m ∗ q2 K m ∗ q2 % 0 e2 y(t) ≤ e 2 e φ % y tk∗0 % . (30) For any t > tk∗0 , since tk∗0 − t0 < k0 q2 , we have t − tk∗0 + t0 > t − k0 q2 . Combining this with y(tk∗0 ) ≤ y(t0 ) e K k0 q2 , it follows from (30) that + , λ λ (m ∗ +k0 )+K (m ∗ +k0 ) q2 y(t) ≤ e− 2 (t −t0 ) e 2 φ y(t0 ) , t ≥ tk∗0 . (31) When t ∈ [t0 , tk∗0 ), by (17) and e
− λ2 (t −t0 )
we have y(t) ≤ e
− λ2 (t −t0 )
+
e +
e
λ 2
λ 2
(m ∗ +k0 )+K m ∗
, q2
(m ∗ +1)+K (m ∗ +1)
φ≥1
,
q2
φ y(t0 )
t0 ≤ t < tk∗0 . (32)
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Therefore, under the two assumptions (H1) and (H2), the exponential stability of the zero solution of random switching nonlinear system (2) can be established with a convergence rate λ/2 > 0. This implies that the second-order globally nonlinear consensus in the multiagent dynamical network (1) with directed topology and random switching interconnections can be achieved exponentially. The proof is thus completed.
IV. N UMERICAL E XAMPLE In this section, a simulation example is used to demonstrate the correctness of our theoretical results. It is assumed that there are five agents flying in a 3-D space. All the agents share their position and velocity states over a dynamically random switching directed time-varying network G(t). In addition, the framework of the random switching directed network G(t) can be determined by ⎡
0 ⎢ 0.2800 ⎢ We = ⎢ ⎢ 0.1103 ⎣ 0.0976 0.1557
0.3264 0 0.4127 0.2423 0.4345
0.3044 0.4230 0 0.2180 0.4466
⎤ 0.3194 0.3233 ⎥ ⎥ 0.0151 ⎥ ⎥ (33) 0.2939 ⎦ 0
0.1514 0.1687 0.2496 0 0.0974
and ⎡
0 ⎢ 0.7235 ⎢ P =⎢ ⎢ 0.6992 ⎣ 0.6714 0.7145
0.6080 0 0.6335 0.7108 0.6272
0.7193 0.7399 0 0.6521 0.7392
0.6894 0.6630 0.7042 0 0.7137
⎤ 0.6644 0.7035 ⎥ ⎥ 0.6768 ⎥ ⎥. (34) 0.6794 ⎦ 0
After the linear couplings, the dynamical behavior of agent i can be determined by ⎧ x˙i (t) = vi (t), ⎪ ⎪ ⎪ N ⎪ ⎪ ⎨v˙i (t) = f (vi (t)) + α Weij (m(t))B(x j (t) − x i (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+β
N j =1
j =1
(35)
Weij (m(t))B(v j (t) − vi (t))
where i = 1, 2, . . . , 5, x i (t) = (x i1 (t), x i2 (t), x i3 (t))T represents the i th agent’s position and vi (t) = (vi1 (t), vi2 (t), vi3 (t))T represents its velocity in a 3-D space. Thus, f (vi ) = (a(vi2 − g(vi1 )), vi1 − vi2 + vi3 , −bvi2 )T with g(vi1 ) = cvi1 + 0.5(d − c)(|vi1 + 1| − |vi1 − 1|). In particular, the isolated second-order oscillator (α = β = 0) exhibits a chaotic attractor at the parameters a = 10, b = 18, c = 1/4, and d = −1/3 (see [29]). All the agents’ inner coupling constant matrix B are assumed to be I3 . Let the coupling strengths be α = 25 and β = 50. Suppose that the dwell time of each topology in the random switching sequence is randomly and independently distributed over the interval [0.1, 0.2], i.e., 1 = 0.1 and 2 = 0.2. In addition, we take D = C = I4 and ε = 0.5. By numerical
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Fig. 1. Position and velocity states of five agents in dynamically switching networkG(t). (a) Position states. (b) Velocity states.
computations, one has ⎛
⎞
25/3 10 0 1 1⎠ M2 = ⎝ 1 0 18 0 2 1 55 , 3, 18 = 18.3333 ρ1 = ρ2 = max 3 2 1 83 , 32, 19 = 32, λmin (A) = 51.1097 ρ3 = 0, ρ4 = max 3 λmin (C ⊗ I3 − (D ⊗ I3 ) A−1 (D ⊗ I3 )) = 0.9804
ρ1 +ρ3 ¯ I12 = −7.8699 λmax αsym (D ⊗ I3 ) ( SW ⊗ B) + 2
M 1 = O3 ,
and λmax
ρ2 + ρ4 ¯ I12 D ⊗ I3 + βsym (C ⊗ I3 ) ( SW ⊗ B) + 2 = −7.9065.
This implies that the positive definiteness of matrix as defined in (7) can be guaranteed, and thus the conditions in (H1) are satisfied. In the simulation, we use the Runge–Kutta method to solve the differential equations by taking step size 0.0025. The initial position and velocity states of all agents are
Fig. 2. (a) Time evolution of the Euclidean norm of time-varying Laplacian matrix L(t). (b) Time evolution of Re(λ2 (L(t))).
randomly selected from the interval [0, 8] and [0, 5], respectively. The time evolutions of position and velocity states of all the agents are shown in Figs. 1(a) and (b), respectively. As shown in the figures, the second-order nonlinear consensus in the random switching multiagent directed network G(t) is achieved. Even though the network is not always connected instantaneously, all agents can still fly in a 3-D space with the same position and velocity states after the second-order consensus has been achieved. In order to illustrate the change of network topology structure in the random switching sequence, we depict the time evolution of the Euclidean norm of the time-varying Laplacian matrix L(t) corresponding to G(t), which is shown in Fig. 2(a). In addition, we also provide the time evolution of Re(λ2 (L(t))) [the real part of the second minimum eigenvalue of L(t)], which is shown in Fig. 2(b). From Fig. 2(b), we can observe that in some time intervals, the value Re(λ2 (L(t))) is equal to zero, which indicates that there are no directed spanning trees in topologies running in these time intervals, i.e., there exist some isolated agents in these topologies. Fig. 3 shows that the time evolution of logarithm with regard to V (y(t), t) = 1/2y T (t) y(t), where y(t) = ( X¯ T (t), V¯ T (t))T . It can be observed that the time derivative of V (y(t), t) has positive and negative values.
LI et al.: SECOND-ORDER GLOBAL CONSENSUS IN MULTIAGENT NETWORKS
Fig. 3.
Time evolution of logarithm with regard to V (y(t), t).
This is because in the random switching sequence, the random switching multiagent network (1) undergoes some topologies, which do not contain directed spanning trees. In this case, as described previously, the exponential stability associated with the resultant error dynamical system of (1) can also be established. From the simulation results, we found that even though the time-varying network topology is not always connected instantaneously in time, sufficient information could also be propagated through the dynamical network to guarantee the second-order globally nonlinear consensus. The underlying mechanism that makes consensus still possible even though some links between agents are broken within a period of time, is useful in particular domains, such as swarms, autonomous vehicles formation, or attitude adjustment of man-made satellites. Remark 3: Different from the results in [23] and [24], the inherent nonlinear function of the agent under studied only needs to be continuous but not necessarily differentiable. Thus, the results are in the sense of global consensus. A sufficient condition for reaching the second-order globally nonlinear consensus in a multiagent network both with directed topology and random switching interconnections is obtained and the consensus can also be achieved at an exponential convergence rate estimated analytically. It is required that the designed Lyapunov function decreases along the solution of the nonlinear random switching system at an infinite subsequence of switching moments. To reach the second-order nonlinear consensus in networks of multiple agents with directed topologies
O(N−1)n αSW (σ (t)) ⊗ B
573
and random switching connections, a relatively conservative condition, i.e., real-time measurements of position and velocity states of all agents, is needed. This condition can effectively guide us to regulate the coupling strengths to overcome the irregular switching and finally achieves consensus. Remark 4: For a multiagent network, the dynamical behavior of the coupled agents may be different from that of the isolated ones upon strong linear couplings. For example, the whole coupled network may be divergent even if the states of the isolated agent are bounded. This may be resulted from the sufficiently large coupling strengths or undesirable switching topologies. It is worth mentioning that even if the network dynamics change qualitatively due to these factors, the second-order nonlinear consensus in a multiagent network with directed topologies and random switching connections can also be achieved. Therefore, the results obtained in this paper are more general. Here, we omit the simulation results due to the limited space. V. C ONCLUSION The second-order consensus problem in a multiagent network with inherent nonlinear dynamics and random switching directed interconnections has been investigated. It is found that the second-order global consensus of multiagent systems with random switching topologies can be achieved as long as the consensus can be realized in the corresponding time-averaged network and the designed Lyapunov function decreases along the solution of the nonlinear random switching system at an infinite subsequence of switching moments. Although the result obtained seems to be slightly conservative, it can effectively instruct one to regulate the coupling strengths to overcome the adverse effect caused by the random switching. The results of this paper can also provide some insights on how the consensus is achieved even though the potential network topology is disconnected simultaneously. There are still a number of related interesting problems deserving further investigation. For example, it is desirable to study: 1) consensus of agents with different nonlinear dynamics; 2) consensus of agents with time-varying delay couplings; 3) cluster consensus; and 4) consensus with the communication constraints, such as packet loss, channel noise, limited bandwidth, and so on. Some of them will be investigated in near future. A PPENDIX By some computation step, we have (A1), as shown at the bottom of the page.
T
I(N−1)n I(N−1)n O(N−1)n β SW (σ (t)) ⊗ B αSW (σ (t)) ⊗ B β SW (σ (t)) ⊗ B 2 T
T (σ (t))S (σ (t)) ⊗ B T B αβ SW α SW (σ (t))SW (σ (t)) ⊗ B T B W = T (σ (t))S (σ (t)) ⊗ B T B αβ S T (σ (t))SW (σ (t)) ⊗ B T B I + β 2 SW W 2 TW
T T α SW (σ (t))SW (σ (t)) ⊗ B B αβ SW (σ (t))SW (σ (t)) ⊗ B T B I ≤ + T (σ (t))S (σ (t)) ⊗ B T B T (σ (t))S (σ (t)) ⊗ B T B 0 αβ SW β 2 SW W W
0 I
(A1)
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The maximum eigenvalue of the above matrix is bounded by T K 1 = (max{α, β})2 λmax SW (σ (t))SW (σ (t)) λmax(B T B) +1. (A2) Moreover, the inequality (A3) holds, as shown at the bottom of the page, F( X¯ E (t), V¯ E (t), t) 2 N−1 = f (x (i+1)e (t), v(i+1)e (t), t) − f (x 1e (t), v1e (t), t) 2 = ≤
i=1 n N−1
f j (x (i+1)e (t), v(i+1)e (t), t)− f j (x 1e (t), v1e (t), t)2
i=1 j =1 N−1 n n
i=1 j =1 k=1
=
n n N−1 i=1 j =1 k=1
+
k k w j k x (i+1)e (t) − x 1e (t) 2 k k +m j k v(i+1)e (t) − v1e (t)
2 2 k k w j k x (i+1)e (t) − x 1e (t)
n n N−1 i=1 j =1 k=1
≤
n n N−1 i=1 j =1 k=1
2 k k +m 2j k v(i+1)e (t) − v1e (t)
k (t) + 2w j k m j k x (i+1)e k k k (t)v(i+1)e (t) − v1e (t) −x 1e
2 k k 2w2j k x (i+1)e (t) − x 1e (t)
2 k k +2m 2j k v(i+1)e (t) − v1e (t) ≤ K 22 ( X¯ E (t) 2 + V¯ E (t) 2 ) = K 22 y(t) 2 (A3) where K 2 = 2 max
max
! " max w2j k
1≤ j ≤n 1≤k≤n
max
! " max m 2j k .
1≤ j ≤n 1≤k≤n
ˆ X¯ (s), V¯ (s), s) ≤ K y(s) where K Therefore, F( K1 + K2.
=
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and constructive comments, based on which the presentation of this paper has been greatly improved. R EFERENCES [1] W. Ren, R. Beard, and E. Atkins, “Information consensus in multivehicle cooperative control: Collective group behavior through local interaction,” IEEE Control Syst. Mag., vol. 27, no. 2, pp. 71–82, Apr. 2007. [2] W. Yu, G. Chen, Z. Wang, and W. Yang, “Distributed consensus filtering in sensor networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 6, pp. 1568–1577, Jun. 2009. [3] S. S. Pereira and A. Pages-Zamora, “Consensus in correlated random wireless sensor networks,” IEEE Trans. Signal Process., vol. 59, no. 12, pp. 6279–6284, Dec. 2011. [4] J. Lu, J. Kurths, J. Cao, N. Mahdavi, and C. Huang, “Synchronization control for nonlinear stochastic dynamical networks: Pinning impulsive strategy,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 285–292, Feb. 2012.
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Huaqing Li received the B.S. degree from the College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, China, and the Ph.D. degree from the College of Computer Science and Technology, Chongqing University, Chongqing, China, in 2009 and 2013, respectively. He is currently an Associate Professor with the College of Electronics and Information Engineering, Southwest University, Chongqing. His current research interests include nonlinear dynamical systems, bifurcation and chaos, neural networks, and consensus of multiagent systems.
Xiaofeng Liao (A’10–M’10–SM’12) received the B.S. and M.S. degrees in mathematics from Sichuan University, Chengdu, China, in 1986 and 1992, respectively, and the Ph.D. degree in circuits and systems from the University of Electronic Science and Technology of China, Chengdu, in 1997. He was a Post-Doctoral Researcher with Chongqing University, Chongqing, China, from 1999 to 2001. He was a Research Associate with the Chinese University of Hong Kong, Hong Kong, from 1997 to 1998, where he was a Research Associate from 1999 to 2000, a Senior Research Associate from 2001 to 2002, and a Research Fellow from 2006 to 2007. From 1999 to 2012, he was a Professor with Chongqing University. He is currently a Professor with Southwest University, Chongqing, where he is the Dean of College of Electronics and Information Engineering. His current research interests include neural networks, nonlinear dynamical systems, bifurcation and chaos, and cryptography.
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Tingwen Huang received the B.S. degree from Southwest University, Chongqing, China, in 1990, the M.S. degree from Sichuan University, Chengdu, China, in 1993, and the Ph.D. degree from Texas A&M University, College Station, TX, USA, in 2002. He is a Professor with Texas A&M University at Qatar, Doha, Qatar. He has expertise in chaotic dynamical systems, neural networks, optimization and control, and traveling wave phenomena. He has authored more than 70 peer-reviewed journal papers. He is an Editor of the five volumes of the Proceedings of the 19th International Conference on Neural Information Processing published in Lecture Notes in Computer Science by Springer. He is also an Editor for a book Advances in Intelligent and Soft Computing (Springer). His research on chaotic dynamical systems received Qatar National Priority Research Project supported by the Qatar Research Fund. Dr. Huang currently serves as an Editorial Board Member for four international journals, the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS , Cognitive Computation, Advances in Artificial Neural Systems, and Intelligent Control and Automation.
Wei Zhu received the Ph.D. degree in applied mathematics from Sichuan University, Chengdu, China, in 2007. He is currently a Professor with the College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, China, and a Visiting Scholar with the Polytechnic Institute of New York University, Brooklyn, NY, USA. His current research interests include switched systems, consensus of multiagent systems, and stability theory of functional differential equations.
Yanbing Liu received the M.S. degree in computer application from the Beijing University of Posts and Telecommunications, Beijing, China, and the Ph.D. degree from the University of Electronic Science and Technology of China, Chengdu, China. He is a Professor and Ph.D. Supervisor with Chongqing University of Posts and Telecommunications, Chongqing, China. His current research interests include cloud computing and security of computer network system.
