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Digraph Laplacians and Multi-Agent Systems

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Digraph Laplacians and Multi-Agent Systems On graph theoretic results underlying the analysis of consensus in multi-agent systems Pavel Chebotarev1 Institute of Control Sciences of the Russian Academy of Sciences 65 Profsoyuznaya Street, Moscow 117997, Russia Key words: consensus algorithms, cooperative control, flocking, graph Laplacians, networked multi-agent systems The objective of this note is to give several comments regarding the paper [1] published in the Proceedings of the IEEE. As stated in the Introduction of [1], “Graph Laplacians and their spectral properties […] are important graph-related matrices that play a crucial role in convergence analysis of consensus and alignment algorithms.” In particular, the stability properties of the distributed consensus algorithms & xi (t) = åaij (t)(x j (t) - xi (t)), i =1,...,n (1) jÎN i for networked multi-agent systems are completely determined by the location of the Laplacian eigenvalues of the network. The convergence analysis of such systems is based on the following lemma [1, p. 221]: Lemma 2: (spectral localization) Let G be a strongly connected digraph on n nodes. Then rank(L) = n − 1 and all nontrivial eigenvalues of L have positive real parts. Furthermore, suppose G has c ³ 1 strongly connected components, then rank(L) = n − c. Here, L = [lij] is the Laplacian matrix of G, i.e., L = D – A, where A is the adjacency matrix of G, and D is the diagonal matrix of vertex out-degrees. Four comments need to be made concerning this lemma. First, the last statement of the lemma is not correct. Indeed, recall that the strongly connected components (SCC) of a digraph G are its maximal strongly connected subgraphs. For instance, if G is a converging tree, i.e., G is a directed tree with root r such that every vertex of G can be linked to r via a directed path, and n > 1, then G has c = n strongly connected components, but rank(L) = n − 1 > n − c = 0. The statement under consideration becomes valid if one replaces strongly connected components with weakly connected components (WCC) and additionally requires that these WCC’s are strong. A weakly connected component of G is a maximal subgraph of G whose vertices are mutually reachable by violating the edge directions. A more general correct statement results by substituting, in the same place, sink SCC’s, where a sink strongly connected component is an SCC having no edges directed outwards. This result was proved in [2] as well as some other Laplacian related results applicable to the cooperative control. Second, the proof of the rank property (the first statement of Lemma 2) is attributed in [1] to [3]. Let me note that a stronger fact was proved earlier in [2]. More specifically, Proposition 11 of [2] states that rank(L) = n – d, where d is the so-called in-forest dimension of G, i.e., the minimum possible number of converging trees in a spanning converging forest of G. It was also shown (Proposition 6) that the in-forest dimension of G is equal to the number of its sink SCC’s and that the forest dimension of a strongly connected digraph is one (Proposition 7). Consequently, for a strongly connected digraph, rank(L) = n − 1, which coincides with the first statement of Lemma 2. In addition, according to Proposition 8, “the forest dimension of a digraph is no less than its number of weak components2 and does not exceed the number of its strong components and the number of its unilateral components.” 1 E-mail: [email protected]; [email protected] . 2 A weak component = a weakly connected component; a strong component = a strongly connected component. Third, Remark 1 given after the proof of Lemma 2 says3: “Lemma 2 holds under a weaker condition of existence of a directed spanning tree for G. […] This type of condition on existence of directed spanning trees have appeared in [4]–[6].” Here, by Lemma 2 the authors conceivably mean the conclusion that rank(L) = n − 1. Let us observe that the existence of a directed spanning tree for G implies that d = 1, so this statement follows from Proposition 11 of [2]. Fourth, the statement of Lemma 2 that “all nontrivial eigenvalues of L have positive real parts” holds true in the general case, and not only for strongly connected digraphs or digraphs that contain directed spanning trees. This was shown in [7, Proposition 9]. In Section II.C of [1] a discrete-time counterpart of the consensus algorithm (1) is considered: n xi (k +1) = xi (k) + e åaij (x j (k) - xi (k)), i = 1,...,n , (2) j=1 where e > 0 is the step size. In the matrix form, (2) is represented as follows: x(k+1)= Px(k), (3) where P = I -e L is referred to in [1] as the Perron matrix with parameter e of G. The matrices P = I -e L were studied in [2] and [7]; in particular, (i) of Lemma 3 in [1] coincides with Proposition 12 of [2]. The asymptotic behavior of the process (3) is determined by the properties of the sequence P, P2, P3,…. If the stochastic matrix P is primitive, i.e., it has only one eigenvalue with modulus 1, k T then, as stated in Lemma 4 of [1], limk ®¥ P = vw , where v and w are the right and left eigenvectors of P corresponding to the eigenvalue 1, respectively, with a normalization that provides vT w =1. In the case of a general nonnegative Perron matrix P, the sequence P, P2, P3,… need not have a limit, so the long-run m transition matrix P¥ = lim m-1 Pk is considered. The matrix P¥ always exists and, by the m®¥ åk =1 Markov chain tree theorem proved in [8], [9], it coincides with the normalized matrix J of maximal in- forests of G. J is the eigenprojector of L; by Proposition 11 of [2] rank (J ) = d, where d is the in-forest dimension of G. The columns of J are the eigenvectors of L corresponding to the eigenvalue 0; consequently, they determine the consensus trajectories of the process (1) and the flocking trajectories [10]. The elements of J were characterized in Theorems 2' and 3 of [2]. An algebraic method for calculating J was presented in [7]. As has been shown above, [2] and [7] contained a number of results on the Laplacians of directed graphs which were useful for the cooperative control of multi-agent systems. A number of additional results were presented in [11] and [12]. Some of them are surveyed in [13]. In January 2001 Alex Fax, one of the authors of [1], sent me a message, where he asked about the eigenstructure of digraph Laplacians and requested to send copies of related papers. During the subsequent correspondence, later in 2001, I sent him [2] and [7]. Recently, I was pleased to familiarize myself with [1] and to learn that our early results proved to be useful in the analysis of consensus and cooperation algorithms of decentralized control. However, I was surprised that, instead of references to [2] and [7], this article contained references to papers published several years later. 3 The bibliographic references are redirected here to the list of references of this note. REFERENCES [1] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE. vol. 95, pp. 215–233, Jan. 2007. [2] R. P. Agaev and P. Yu. Chebotarev, “The matrix of maximum out forests of a digraph and its applications,” Automation and Remote Control, vol. 61, pp. 1424-1450, Sept. 2000. [3] R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, Sep. 2004. [4] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, pp. 988–1001, June 2003. [5] L. Moreau, “Stability of multiagent systems with time-dependent communication links,” IEEE Trans. Autom. Control, vol. 50, pp. 169–182, 2005. [6] W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. Autom. Control, vol. 50, pp. 655–661, 2005. [7] R. P. Agaev and P. Yu. Chebotarev, “Spanning forests of a digraph and their applications,” Automation and Remote Control, vol. 62, pp. 443–466, March 2001. [8] A. D. Wentzell and M. I. Freidlin, “On small random perturbations of dynamical systems,” Russian Mathematical Surveys, vol. 25, no. 1, pp. 1–55, 1970. [9] T. Leighton and R. L. Rivest, “The Markov chain tree theorem,” Computer Science Technical Report MIT/LCS/TM–249, Laboratory of Computer Science, MIT, Cambridge, Mass., 1983. [10] J. J. P. Veerman, G. Lafferriere, J. S. Caughman, and A. Williams, “Flocks and formations,” J. Statistical Physics, vol. 121, no. 5–6, pp. 901–936, 2005. [11] P. Chebotarev and R. Agaev, “Forest matrices around the Laplacian matrix,” Linear Algebra and Its Applications, vol. 356, pp. 253–274, 2002. [12] R. Agaev and P. Chebotarev, “On the spectra of nonsymmetric Laplacian matrices,” Linear Algebra and Its Applications, vol. 399, pp. 157–168, 2005. [13] P. Yu. Chebotarev and R. P. Agaev, “Coordination in multiagent systems and Laplacian spectra of digraphs,” Automation and Remote Control, vol. 70, no. 3, March 2009. In press. .
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