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...irected topologies are developed. Lemma 1 (Ren and Beard (2005)). The Laplacian matrix L has a simple eigenvalue 0 and all the other eigenvalues have positive real parts if and only if the directed network has a directed spanning tree. For the linear model (5), eigenvalues of the matrix L are very important in convergence analysis. Suppose that λij (i = 1, 2, . . . ,N, j = 1, 2) and µi (i = 1, 2, . . . ,N) are eigenvalues of W. Yu et al. / Automatica 46 (2010) 1089–1095 1091L and the Laplacianmatrix L, respectively. First, some relationships between the eigenvalues of L and L are reviewed (Ren, 2008; Ren & Atkins, 2005). Letλ be an eigenvalue ofmatrix L. Then, one has det(λI2N−L) = 0. Note that det(λI2N − L) = det ( λIN −IN αL λIN + βL ) = det ( λ2IN + (α + βλ)L ) = N∏ i=1 ( λ2 + (α + βλ)µi ) = 0. Hence, λi1 = −βµi + √ β2µ2i − 4αµi 2 , λi2 = −βµi − √ β2µ2i − 4αµi 2 , i = 1, 2, . . . ,N. (6) From (6), it is easy to see that L has a zero eigenvalue of algebraic multiplicity m if and only if L has a zero eigenvalue of algebraic multiplicity 2m. In the sequel, for the sake of simplicity, we simply write algebraic multiplicity as multiplicity. Lemma 2. Second-order consensus in multi-agen...

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...bject Identifier 10.1109/TSMCB.2009.2031624 Recently, much progress has been made in the study of collective behaviors in multiagent dynamical systems, such as consensus [10], [11], [14], [19], [21], =-=[22]-=-, [24]–[26], [40], synchronization [1], [4], [15]–[18], [29]–[38], [41]–[43], and swarming and flocking [20], [27], [28]. The consensus problem usually refers to the problem of how to reach an agreeme...

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...54, NO. 6, JUNE 2009 1331 where and are, respectively, the position and velocity of the th vehicle, and is the control input. A consensus algorithm for (1) is studied in [9], =-=[18]-=- as (2) where is the sth entry of weighted adjacency matrix associated with weighted directed graph , and is a positive damping g...

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...nd switching topologies. A passivity-based design framework is proposed in [13] to achieve group coordination. The consensus problems for networks of double- and high-order integrators are studied in =-=[14]-=-, [15], [16]. Readers are referred to the recent surveys [1], [2] for a relatively complete coverage of the literature on consensus. This paper considers the distributed consensus problems for multi-a...

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... on prescribing the dependence of the agreed-upon value on the initial conditions or agreement-law design, see [4]. Finally, cursory studies of consensus under delay [1–3, 21] and actuator saturation =-=[6]-=- are available. Noting that the ongoing research on consensus is progressing toward models of increasing generality (from first- [1–5] to second-order and integrator-chain internal dynamics [7, 13, 14...

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...t each channel), and thus gave conditions for synchronization through control in this case. More recently, numerous other dynamical network tasks such as formation, agreement, and alignment [6], [12]–=-=[15]-=- have been addressed in essentially similar ways. Development of stabilizing network controllers has also been pursued under the heading of decentralized control. In a seminal work [16], Wang and Davi...

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...rtain global criteria of common interest. Recently, much progress has been achieved in the study of collective behaviors of multiagent dynamical systems, such as consensus [4], [9], [10], [13], [18], =-=[21]-=-–[24], [33], [34], [36], synchronization [1], [5], [6], [14], [16], [17], [20], [27], [29], [31], [32], [35], [37], [38], and swarming and flocking [19], [25], [26]. Many investigations have been devo...