1. bookVolume 25 (2015): Issue 2 (June 2015)
Journal Details
License
Format
Journal
eISSN
2083-8492
First Published
05 Apr 2007
Publication timeframe
4 times per year
Languages
English
access type Open Access

A generalization of the graph Laplacian with application to a distributed consensus algorithm

Published Online: 25 Jun 2015
Page range: 353 - 360
Received: 06 Jan 2014
Journal Details
License
Format
Journal
eISSN
2083-8492
First Published
05 Apr 2007
Publication timeframe
4 times per year
Languages
English
Abstract

In order to describe the interconnection among agents with multi-dimensional states, we generalize the notion of a graph Laplacian by extending the adjacency weights (or weighted interconnection coefficients) from scalars to matrices. More precisely, we use positive definite matrices to denote full multi-dimensional interconnections, while using nonnegative definite matrices to denote partial multi-dimensional interconnections. We prove that the generalized graph Laplacian inherits the spectral properties of the graph Laplacian. As an application, we use the generalized graph Laplacian to establish a distributed consensus algorithm for agents described by multi-dimensional integrators.

Keywords

Bauer, P.H. (2008). New challenges in dynamical systems: The networked case, International Journal of Applied Mathematics and Computer Science 18(3): 271-277, DOI: 10.2478/v10006-008-0025-8.10.2478/v10006-008-0025-8Search in Google Scholar

Cai, K. and Ishii, H. (2012). Average consensus on general strongly connected digraphs, Automatica 48(11): 2750-2761.10.1016/j.automatica.2012.08.003Search in Google Scholar

Fax, J.A. and Murray, R.M. (2004). Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control 49(9): 1465-1476.10.1109/TAC.2004.834433Search in Google Scholar

Gantmacher, F.R. (1959). The Theory ofMatrices, Chelsea, New York, NY.Search in Google Scholar

Jadbabaie, A., Lin, J. and Morse, A.S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control 48(6): 988-1001.10.1109/TAC.2003.812781Search in Google Scholar

Khalil, H.K. (2002). Nonlinear Systems, Second Edition, Prentice Hall, Englewood Cliffs, NJ.Search in Google Scholar

Mohar, B. (1991). The Laplacian spectrum of graphs, in Y. Alavi, G. Chartrand, O. Ollermann and A. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Wiley, New York, NY.Search in Google Scholar

Moreau, L. (2005). Stability of multi-agent systems with time-dependent communication links, IEEE Transactions on Automatic Control 50(2): 169-182.10.1109/TAC.2004.841888Search in Google Scholar

Olfati-Saber, R., Fax, J.A. and Murray, R.M. (2007). Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE 95(1): 215-233.10.1109/JPROC.2006.887293Search in Google Scholar

Priolo, A., Gasparri, A., Montijano, E. and Sagues, C. (2014). A distributed algorithm for average consensus on strongly connected weighted digraphs, Automatica 50(3): 946-951.10.1016/j.automatica.2013.12.026Search in Google Scholar

Ren, W. and Beard, R.W. (2005). Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control 50(5): 655-661.10.1109/TAC.2005.846556Search in Google Scholar

Shamma, J. (2008). Cooperative Control of Distributed Multi- Agent Systems, Wiley, New York, NY.10.1002/9780470724200Search in Google Scholar

Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I. and Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles, Physical Review Letters 75(6): 1226-1229.10.1103/PhysRevLett.75.1226Search in Google Scholar

Zhai, G., Okuno, S., Imae, J. and Kobayashi, T. (2009). A matrix inequality based design method for consensus problems in multi-agent systems, International Journal of Applied Mathematics and Computer Science 19(4): 639-646, DOI: 10.2478/v10006-009-0051-1.10.2478/v10006-009-0051-1Search in Google Scholar

Zhai, G., Takeda, J., Imae, J. and Kobayashi, T. (2010). Towards consensus in networked nonholonomic systems, IET Control Theory & Applications 4(10): 2212-2218. 10.1049/iet-cta.2009.0658Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo