第3期 戴丽：双向通信无人机集群领航顶点选取方法 ·491· 34个。随着k的增加，k(k≥4)关键集的情况异常 [11]OLESKY DD,TSATSOMEROS M,VAN DEN 复杂。但是，这又是一个十分有意义的问题。比 DRIESSCHE P.Qualitative controllability and uncontrol- 如，从数值实验可以看出，仅依据本文给出的定 lability by a single entry[J].Linear algebra and its applic- 理2~5求出有限特殊的关键集，CSA算法就能在 ations.1993.187:183-194. 大多数情况下以90%以上的概率搜索到领航集。 [12]JI Meng,EGERSTEDT M.A graph-theoretic characteriz- 另外，从图论角度给出领航集的充要条件，最 ation of controllability for multi-agent systems[Cl//Pro- ceedings of 2007 American Control Conference.New 小领航顶点数的上、下界等也都是值得进一步研 York.NY,USA.2007:4588-4593. 究的问题。 [13]RAHMANI A,JI Meng,MESBAHI M,et al.Controllab- 参考文献： ility of multi-agent systems from a graph-theoretic per- spective[J].SIAM journal on control and optimization, [1]DORRI A,KANHERE SS,JURDAK R.Multi-agent sys- 2009,48(1y162-186 tems:a survey[J].IEEE access,2018,6:28573-28593. 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[6] MOUSAVI S S, CHAPMAN A, HAERI M, et al. Null space strong structural controllability via skew zero for￾cing sets[C]//Proceedings of 2018 European Control Con￾ference. Limassol, Cyprus, 2018: 1845−1850. [7] MOUSAVI S S, HAERI M, MESBAHI M. On the struc￾tural and strong structural controllability of undirected net￾works[J]. IEEE transactions on automatic control, 2018, 63(7): 2234–2241. [8] CHAPMAN A, MESBAHI M. On strong structural con￾trollability of networked systems: a constrained matching approach[C]//Proceedings of 2013 American Control Con￾ference. Washington, DC, USA, 2013: 6126−6131. [9] TREFOIS M, DELVENNE J C. Zero forcing number, constrained matchings and strong structural controllabil￾ity[J]. Linear algebra and its applications, 2015, 484: 199–218. [10] OLESKY D D, TSATSOMEROS M, VAN DEN DRIESSCHE P. Qualitative controllability and uncontrol￾lability by a single entry[J]. Linear algebra and its applic￾ations, 1993, 187: 183–194. [11] JI Meng, EGERSTEDT M. A graph-theoretic characteriz￾ation of controllability for multi-agent systems[C]//Pro￾ceedings of 2007 American Control Conference. New York, NY, USA, 2007: 4588−4593. [12] RAHMANI A, JI Meng, MESBAHI M, et al. Controllab￾ility of multi-agent systems from a graph-theoretic per￾spective[J]. SIAM journal on control and optimization, 2009, 48(1): 162–186. [13] MARTINI S, EGERSTEDT M, BICCHI A. Controllabil￾ity analysis of multi-agent systems using relaxed equit￾able partitions[J]. International journal of systems, con￾trol and communications, 2010, 2(1/2/3): 100–121. [14] AGUILAR C O, GHARESIFARD B. Almost equitable partitions and new necessary conditions for network con￾trollability[J]. Automatica, 2017, 80: 25–31. [15] JI Zhijian, WANG Zidong, LIN Hai, et al. Interconnec￾tion topologies for multi-agent coordination under leader￾follower framework[J]. Automatica, 2009, 45(12): 2857–2863. [16] JI Zhijian, YU Haisheng. A new perspective to graphical characterization of multiagent controllability[J]. IEEE transactions on cybernetics, 2017, 47(6): 1471–1483. [17] WANG Long, JIANG Fangcui, XIE Guangming, et al. Controllability of multi-agent systems based on agree￾ment protocols[J]. Science in China series F: information sciences, 2009, 52(11): 2074–2088. [18] JIANG Fangcui, WANG Long, XIE Guangming, et al. On the controllability of multiple dynamic agents with fixed topology[C]//Proceedings of the 2009 Conference on American Control Conference. St. Louis, Missouri, USA, 2009: 5665−5670. [19] HAMDAN A M A, NAYFEH A H. Measures of modal controllability and observability for first-and second-or￾der linear systems[J]. Journal of guidance, control, and dynamics, 1989, 12(3): 421–428. [20] OLSHEVSKY A. Minimal controllability problems[J]. IEEE transactions on control of network systems, 2014, 3(1): 249–258. [21] PASQUALETTI F, ZAMPIERI S, BULLO F. Controllab￾ility metrics, limitations and algorithms for complex net￾works[J]. IEEE transactions on control of network sys￾tems, 2014, 1(1): 40–52. [22] SUMMERS T H, CORTESI F L, LYGEROS J. On sub￾modularity and controllability in complex dynamical net￾works[J]. IEEE transactions on control of network sys- [23] 第 3 期 戴丽：双向通信无人机集群领航顶点选取方法 ·491·