Abstract
In this paper, the problems of stability and decentralized control are studied for a class of linear coupled dynamical networks with Fornasini–Marchesini second local state-space dynamics. Necessary and sufficient stability conditions are obtained for a class of linear network composed by \(N\) identical nodes. Effects of the interconnection on stability of network are presented by eigenvalues of the topological matrix, and the effectiveness of interconnection on network stability is pointed out. Moreover, the decentralized control laws are presented for two types of linear regular networks: star-shaped coupled networks and globally coupled networks in detail. The relationships between the stability of a network and the stability of its corresponding nodes are studied. It is shown that some nodes must be made stable in order to stabilize the whole network in some cases. However, the detailed relationship is needed to be further investigated.
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Bender, J. G. (1991). An overview of systems studies of automated highway systems. IEEE Transactions on Vehicle Technology, 40(1), 82–99.
Curtin, T. B., Bellingham, J. G., Catipovic, J., & Webb, D. (1993). Autonomous oceanographic sampling networks. Oceanography, 6(3), 86–94.
Duan, Z. S., Wang, J. Z., Chen, G. R., & Huang, L. (2007). Special decentralized control problems in discrete-time interconnected systems composed of two subsystems. System and Control Letters, 56(3), 206–214.
Duan, Z. S., Wang, J. Z., Chen, G. R., & Huang, L. (2008). Stability analysis and decentralized control of a class of complex dynamical networks. Automatica, 44(4), 1028–1035.
Ebihara, Y., Ito, Y., & Hagiwara, T. (2006). Exact stability analysis of 2-D systems using LMIs. IEEE Transaction on Automatica Control, 51(9), 1509–1513.
Fax, J. A., & Murray, R. M. (2004). Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control, 49(9), 1465–1476.
Fornasini, E., & Marchesini, G. (1978). Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory, 12(1), 59–72.
Fornasini, E., & Marchesini, G. (1980). Stability analysis of 2-D systems. IEEE Transaction on Circuits and System, 27(12), 1210–1217.
Galkowski, K., Rogers, E., Xu, S. Y., Lam, J., & Owens, D. H. (2002). LMIs-A fundamental tool in analysis and controller design for discrete linear repetitive processes. IEEE Transaction on Circuits System. I: Fundamental Theory and Applications, 49(6), 768–778.
Galkowski, K., Xu, S. Y., Lam, J., & Lin, Z. P. (2003). LMI approach to state-feedback stabilization of multidimensional systems. International Journal of Control, 76(14), 1428–1436.
Hinamoto, T. (1993). 2-D Lyapunov equation and filter design based on the Fornasini–Marchesinni second model. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 40(2), 102–110.
Huang, L., & Duan, Z. S. (2003). Complexity in control science. Acta Automatica Sinica, 29(5), 748–754.
Lin, Z. P., & Bruton, L. T. (1989). BIBO stability of inverse 2-D digital filters in the presence of nonessential singularities of the second kind. IEEE Transactions Circuits and Systems, 36(2), 244–254.
Liu, X. J., & Zou, Y. (2010). A consensus problem for a class of vehicles with 2-D dynamics. Multidimensional Systems and Signal Processing, 21(4), 373–389.
Liu, X. J., & Zou, Y. (2012). Stability analysis for a special class of dynamical networks. International Journal of Systems Science, 43(10), 1950–1957.
Ooba, T. (2000). On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 47(8), 1263–1265.
Ooba, T. (2001). Stabilization of multidimensional systems using local state estimators. Multidimensional Systems and Signal Processing, 12(1), 49–61.
Packard, A., & Doyle, J. C. (1993). The complex structured singular value. Automatica, 29(1), 71–109.
Schaub, H., Vadali, S. R., & Junkins, J. L. (2000). Spacecraft formation flying control using mean orbital elements. Journal of the Astronautical Sciences, 48(1), 69–87.
Sharma, R. K., & Ghose, D. (2009). Collision avoidance 505 between UAV clusters using swarm intelligence techniques. International Journal of Systems Science, 40(5), 521–538.
Steven, H. S. (2001). Exploring complex networks. Nature, 410(6825), 268–276.
Wang, X. F. (2002). Complex networks: Topology, dynamics, and synchronization. International Journal of Bifurcation and Chaos, 12(5), 885–916.
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393(6684), 440–442.
Yamaguchi, H., Arai, T., & Beni, G. (2001). A distributed control scheme for multiple robotic vehicles to make group formations. Robotics and Autonomous Systems, 36(4), 125–147.
Yao, J., Wang, W. Q., & Zou, Y. (2013). The delay-range-dependent robust stability analysis for 2-D state-delayed systems with uncertainty. Multidimensional Systems and Signal Processing, 24(1), 87–103.
Zou, Y. (2010). Investigation on mechanism and extension of feedback: An extended closed-loop coordinate control of interconnection-regulation and feedback. Journal of Nanjing University of Science and Technology, 34(1), 1–7.
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This paper is sponsored by National Natural Science Foundation of P. R. China under Grant 61174038, 11102072, and the Fundamental Research Funds for the Central Universities JUSRP21104.
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Liu, X., Zou, Y. Stability analysis for a class of complex dynamical networks with 2-D dynamics. Multidim Syst Sign Process 25, 531–540 (2014). https://doi.org/10.1007/s11045-012-0216-1
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DOI: https://doi.org/10.1007/s11045-012-0216-1