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Mitigation of complex behavior over networked systems: analysis of spatially invariant structures. (English) Zbl 1360.93059
Summary: In this paper, we consider a simple distributed averaging system, which incorporates various communication constraints including delays, noise, and link failures. It has been shown by the authors [“Distributed averaging under constraints on information exchange: emergence of Lévy flights”, IEEE Trans. Autom. Control 57, No. 10, 2435–2449 (2012; doi:10.1109/TAC.2012.2186093)], that such networked system generates a collective Lévy flight behavior when part of the system loses Mean Square (MS) stability. We focus on spatially invariant architectures to gain more insights into how model parameters affect emergence of this complex scale-invariant behavior, and to seek structures robust to communication constraints. Specifically, we develop a computational expression for checking MS stability, which is scalable with the number of unreliable links. We derive the closed form formulas from this expression in the limiting case of zero and large delays, and in the case of large number of nodes. In the limit of large delays, we derive various results that are independent of the network size and its specific interconnections. We find that small inter-agent coupling improves the robustness of the system. Networks with larger connectivity tend to be more fragile in the presence of fading connections for fixed inter-agent coupling. That gossiping improves the robustness and that the lattice is the most robust among the spatially invariant systems with generalized circulant interconnections.

MSC:
93A14 Decentralized systems
93A15 Large-scale systems
93E15 Stochastic stability in control theory
Software:
SumTools
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[1] Bamieh, B.; Paganini, F.; Dahleh, M., Distributed control of spatially invariant systems, IEEE Transactions on Automatic Control, 47, 1091-1118, (2002) · Zbl 1364.93363
[2] Boyd, S.; Ghosh, A.; Prabhakar, B.; Shah, D., Randomized gossip algorithms, IEEE Transactions on Information Theory, 52, 6, 2508-2530, (2006) · Zbl 1283.94005
[3] Brockmann, D.; Hufnagel, L.; Geisel, T., The scaling laws of human travel, Nature, 439, 462-465, (2006)
[4] Davis, P. J., (Circulant matrices, A Wiley-interscience publication, pure and applied mathematics, (1979), John Wiley & Sons New York, Chichester, Brisbane)
[5] Elia, N., Remote stabilization over fading channels, Systems & Control Letters, 54, 3, 237-249, (2005) · Zbl 1129.93498
[6] Elia, N. (2006). Emergence of power laws in networked control systems. In Proc. of the 45-th IEEE conf. on decision and control, December (pp. 490-495).
[7] Fax, J. A.; Murray, R. M., Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, 49, 9, 1465-1476, (2004) · Zbl 1365.90056
[8] Horn, R. A.; Johnson, C. R., Matrix analysis, (1987), Cambridge Univ. Press Cambridge, UK
[9] Huang, M.; Dey, S.; Nair, G. N.; Manton, J. H., Stochastic consensus over noisy networks with Markovian and arbitrary switches, Automatica, 46, 10, 1571-1583, (2010) · Zbl 1204.93107
[10] Huang, M.; Manton, J. H., Stochastic consensus seeking with noisy and directed inter-agent communication: fixed and randomly varying topologies, IEEE Transactions on Automatic Control, 55, 1, 235-241, (2010) · Zbl 1368.94002
[11] Jadbabaie, A., Motee, N., & Barahona, M. (2005). On the stability of the Kuramoto model of coupled nonlinear oscillators. In Proc. of the American control conference, June (pp. 4296-4301).
[12] Kar, S.; Moura, J., Distributed consensus algorithms in sensor networks with imperfect communication: link failures and channel noise, IEEE Transactions on Signal Processing, 57, 1, 355-369, (2009) · Zbl 1391.94263
[13] Koepf, W., Hypergeometric summation: an algorithmic approach to summation and special function identities, (1998), Vieweg Braunschweig, Germany · Zbl 0909.33001
[14] Li, L., Alderson, D., Doyle, J. C., & Willinger, W. (2005). Towards a theory of scale-free graphs: definition, properties, and implications (extended version). October [Online]. Available: http://arxiv.org/abs/cond-mat/0501169. · Zbl 1103.05082
[15] Mandelbrot, B., The variation of certain speculative prices, Journal of Business, 36, 394-419, (1963)
[16] Mantegna, R.; Stanley, H., Scaling behavior in the dynamics of an economic index, Nature, 376, 46-49, (1995)
[17] Nedic, A.; Ozdaglar, A., Distributed subgradient methods for multi-agent optimization, IEEE Transactions on Automatic Control, 54, (2009) · Zbl 1367.90086
[18] Olfati-Saber, R. (2007). Distributed Kalman filtering for sensor networks. In Proc. of the 46th IEEE conf. on decision and control, December (pp. 5492-5498).
[19] Olfati-Saber, R.; Murray, R. M., Consensus problems in network of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49, 9, 1520-1533, (2004) · Zbl 1365.93301
[20] Patterson, S., & Bamieh, B. (2008). Distributed consensus with link failures as a structured stochastic uncertainty problem. In 46-th Allerton conference on communication, control, and computing (pp. 623-627).
[21] Patterson, S.; Bamieh, B.; El Abbadi, A., Convergence rates of distributed average consensus with stochastic link failures, IEEE Transactions on Automatic Control, 55, 4, 800-892, (2010) · Zbl 1368.94198
[22] Reynolds, A. M.; Smith, A. D.; Menzel, R.; Greggers, U.; Reynolds, D. R.; Riley, J. R., Displaced honey bees perform optimal scale-free search flights, Ecology, 88, 8, 1955-1961, (2007)
[23] Tahbaz Salehi, A.; Jadbabaie, A., Necessary and sufficient conditions for consensus over random networks, IEEE Transactions on Automatic Control, 53, 3, 791-796, (2008) · Zbl 1367.90015
[24] Tsitsiklis, J.N. (1984). Problems in decentralized decision making and computation. Ph.D. Thesis. Massachusetts Institute of Technology.
[25] Wang, J., & Elia, N. (2008). Mean square stability of consensus over fading networks with nonhomogeneous communication delays. In Proc. of the 47th IEEE conf. on decision and control, December (pp. 4614-4619).
[26] Wang, J.; Elia, N., Consensus over networks with dynamic channels, International Journal of Systems, Control and Communications, 2, 1, 275-297, (2010), Special Issue on: Information Processing and Decision Making in Distributed Control Systems
[27] Wang, J., & Elia, N. (2010b). Control approach to distributed optimization. In48-th Allerton conference on communication, control, and computing, September 29-October 1.
[28] Wang, J.; Elia, N., Distributed averaging under constraints on information exchange: emergence of Lévy flights, IEEE Transactions on Automatic Control, 57, 10, 2435-2449, (2012) · Zbl 1369.93704
[29] Xiao, L.; Boyd, S.; Kim, S.-J., Distributed average consensus with least-mean-square deviation, Journal of Parallel and Distributed Computing, 67, 33-46, (2007) · Zbl 1109.68019
[30] Xiao, L., Boyd, S., & Lall, S. (2005). A scheme for robust distributed sensor fusion based on average consensus. In Proc. int. conf. information processing in sensor networks (pp. 63-70). Los Angeles, CA. April.
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