摘要
定位问题是无线传感器网络中最基本的问题之一,本文研究的重点是定位问题中的可定位性问题和定位算法问题。对于任一给定网络,并非其中所有节点的位置都可以通过距离测量计算出相应的位置。可定位性问题就是判断一个网络或者网络中的某个节点,在给定的距离约束下,是否可以唯一确定其位置。从图论角度出发,可以得到图的可定位性条件;从解的存在性出发,可以得到代数的可定位性条件。在识别出可定位节点之后,如何来计算这些节点的坐标,尤其是,如何只利用局部信息来计算坐标,就是定位算法所要关心的问题。
本文首先利用图的刚性理论,研究了二维平面下传感器网络的可定位性条件以及定位算法。我们给出了一组以簇集为单位的可定位性的充要条件。基于这组条件,我们提出了一种新的序贯式的、以簇为单位的定位算法。该算法从一组锚节点集合出发,序贯式的扩展这个集合,直到所有的可定位节点都被识别出来。该算法可以通过分布式的方式来实现,只需要利用局部信息就可以判断以簇为单位的节点的可定位性。当距离测量带噪音时,该定位算法可以通过求解一个最小二乘的优化问题来计算节点坐标。我们给出了针对此优化算法的一种新的初始化方法,来提高定位精度,并且获得了更好的鲁棒性。
接下来,本文还研究了如何利用质心坐标来分布式的计算传感器节点的位置。质心坐标的绝对值可以完全由节点间的距离测量计算得来。使用质心坐标,可以将距离与坐标间的非线性约束转化成线性约束,并且,可以得到一个线性系统来计算节点坐标。但是,引入质心坐标之后,需要面临两个问题:首先,如何只利用距离信息,来判定质心坐标的符号;其次,如何设计一个稳定器,使得计算坐标的线性系统能够保证收敛。在本文中,我们给出了利用距离信息来判定符号的准则,并且,还给出了一个稳定器存在的充分条件,以及如何用集中式或者局部集中式的方法来设计这个稳定器的方法。除此之外,我们还从图的角度和代数角度,分别给出了网络可定位性的充要条件。
最后,我们解决了如何利用相对位置信息来对传感器网络中的节点进行分布式定位的问题。当每一个传感器节点都拥有一个独立的局部坐标系,并且可以测量到一定距离和角度范围内的邻居节点的相对位置时,一个传感器网络可以被建模成有向图。从解的存在性角度,我们给出了这一类传感器网络可定位性的代数条件。同时,从代数图论的角度出发,利用有向图的性质,我们发现这一类传感器网络能够被定位,等价于每个节点相对于锚节点来说都应该是2-可达的。基于此,对于一个满足可定位条件的传感器网络,我们给出了如何利用它的局部信息,来分布式的、迭代的计算节点的全局坐标。这一定位算法的收敛同样依赖于一个对角稳定器的设计。我们证明了,该对角稳定器的存在依赖于有向图的拓扑性质。我们定义了一个一般性的框架,并且证明,在此框架下,只要网络所对应的有向图中每个点都是对于锚节点2-可达的,那么几乎一定存在一个对角稳定器使得算法收敛。
Localization is the fundamental problem in wireless sensor network. This thesis focus on the localizability problem and localization algorithm in wireless sensor network. For a randomly given network, not all of the nodes are localizable by using the obtained distance measurements. The localizability problem is to judge whether the localization of a network or a single node can be uniquely determined under the constraints of distance measurements. From the view of graph theory and the existence of solution, we can obtain the graphical and algebraic localizable conditions, respectively. After that, how to compute the positions of these nodes, especially when using local distance information only, is concerned by the localization algorithm.
This thesis studies the localization problem for sensor nodes in a two dimen-sional sensor network using only distance measurements between sensor nodes. We explore the localizability conditions for up to four nodes at a time and provide a necessary and sufficient condition. Based on our new localizability conditions, a new sequential cluster-based localization algorithm is proposed which starts from a set of anchor nodes and sequentially expands this set until all localizable nodes are covered. An alternative distributed implementation of this algorithm is also given using local computations and communication to reduce computational complexity. For noisy measurements, our localization algorithm is implement-ed as a nonlinear least squares optimization problem and we provide a novel initialization method to offer high localization accuracy and good robustness.
This thesis also studies the problem of determining the sensor locations in a large sensor network using relative distance (range) measurements only. By in-troducing the barycentric coordinate, we can transform the nonlinear constraints between distance measurements and positions into a linear constraints. Moreover, the positions can be computed through a linear dynamic system. But we need to determine the sign patterns of the barycentric coordinate by using distance measurements only. Also, we need to design a stabilizer to guarantee the conver- gence of the linear system. A criterion is developed to address this issue based on the available distance measurements. A new distributed iterative algorithm is proposed using a diagonal stabilizer to guarantee the asymptotic localization for sensor nodes. We give a condition on the existence of such a stabilizer and show how to design it in both centralized and distributed manners. A necessary and sufficient condition for the localizability of the network is also given.
At last, we explore the localization problem of sensor networks using relative position measurements. It is assumed that relative positions are measured in local coordinate frames of individual sensors, for which different sensors may have different orientations of their local frames and the orientation errors with respect to the global coordinate frame are not known. A new necessary and sufficient condition is developed for localizability of such sensor networks that are modeled as directed graphs. That is, every sensor node should be2-reachable from the anchor nodes. Moreover, for a localizable sensor network, a distributed, linear, and iterative scheme based on the graph Laplacian of the sensor network is developed to solve the coordinates of the sensor network in the global coordinate frame. We define a generic framework, under which there almost surely exists a diagonal stabilizer to guarantee the convergence if and only if the network is2-reachable.
引文
[1]孙利民.无线传感器网络.清华大学出版社有限公司,2005.
[2]R.J. Fontana and S.J. Gunderson. Ultra-wideband precision asset location system. In Proceedings of the IEEE Conference on Ultra Wideband Systems and Technologies, pages 147-150. IEEE,2002.
[3]H. Cho and S.W. Kim. Mobile robot localization using biased chirp-spread-spectrum ranging. IEEE Transactions on Industrial Electronics, 57(8):2826-2835,2010.
[4]D. Estrin, R. Govindan, J. Heidemann, and S. Kumar. Next century chal-lenges:Scalable coordination in sensor networks. In Proceedings of the 5th annual ACM/IEEE international conference on Mobile computing and networking (MobiCom), pages 263-270. ACM,1999.
[5]S. Teller, J. Chen, and H. Balakrishnan. Pervasive pose-aware applications and infrastructure. IEEE Computer Graphics and Applications,23(4):14-18,2003.
[6]C.Y. Chong and S.P. Kumar. Sensor networks:evolution, opportunities, and challenges. Proceedings of the IEEE,91 (8):1247-1256,2003.
[7]V. Raghunathan, A. Kansal, J. Hsu, J. Friedman, and M. Srivastava. Design considerations for solar energy harvesting wireless embedded systems. In Proceedings of the 4th international symposium on Information processing in sensor networks (IPSN), page 64. IEEE,2005.
[8]S. Roundy, D. Steingart, L. Frechette, P. Wright, and J. Rabaey. Power sources for wireless sensor networks. In Wireless sensor networks, pages 1-17. Springer,2004.
[9]S. He, J. Chen, F. Jiang, G. Yau, D.K.Y.and Xing, and Y. Sun. Energy provisioning in wireless rechargeable sensor networks. In Proceedings of the 30th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pages 2006-2014. IEEE,2011.
[10]V.C Gungor, B. Lu, and G.P. Hancke. Opportunities and challenges of wireless sensor networks in smart grid. IEEE Transactions on Industrial Electronics,57(10):3557-3564,2010.
[11]S. Sun, L. Xie, W. Xiao, and Y.C. Soh. Optimal linear estimation for systems with multiple packet dropouts. Automatica,44(5):1333-1342,2008.
[12]N. Xiao, L. Xie, and L. Qiu. Feedback stabilization of discrete-time net-worked systems over fading channels. IEEE Transactions on Automatic Control,57(9):2176-2189,2012.
[13]T. Li, M. Fu, L. Xie, and J. Zhang. Distributed consensus with limit-ed communication data rate. IEEE Transactions on Automatic Control, 56(2):279-292,2011.
[14]B. Hull, K. Jamieson, and H. Balakrishnan. Mitigating congestion in wire-less sensor networks. In Proceedings of the 2nd international conference on Embedded networked sensor systems (SenSys), pages 134-147. ACM,2004.
[15]Z. Luo. Universal decentralized estimation in a bandwidth constrained sensor network. IEEE Transactions on Information Theory,51(6):2210-2219,2005.
[10]B. Sinopoli, C. Sharp, L. Schenato, S. Schaffert, and S.S. Sastry. Distribut-ed control applications within sensor networks. Proceedings of the IEEE, 91(8):1235-1246,2003.
[17]R. Olfati-Saber and J.S. Shamma. Consensus filters for sensor networks and distributed sensor fusion. In Proceedings of the 44th IEEE Conference on Decision and Control and 2005 European Control Conference. (CDC-ECC'05),pages 6698-6703. IEEE,2005.
[18]P. Wan and M.D. Lemmon. Event-triggered distributed optimization in sensor networks. In Proceedings of the International Conference on Infor-mation Processing in Sensor Networks (IPSN), pages 49-60. IEEE,2009.
[19]P. Lin, C. Qiao, and X. Wang. Medium access control with a dynamic duty cycle for sensor networks. In IEEE Wireless Communications and Networking Conference (WCNC), volume 3, pages 1534-1539. IEEE,2004.
[20]L. Shi and H. Zhang. Scheduling two gauss-markov systems:An optimal solution for remote state estimation under bandwidth constraint. IEEE Transactions on Signal Processing,60(4):2038-2042,2012.
[21]P. Tabuada. Event-triggered real-time scheduling of stabilizing control tasks. IEEE Transactions on Automatic Control,52(9):1680-1685,2007.
[22]P. Juang, H. Oki, Y. Wang, M. Martonosi, L. Peh, and D. Rubenstein. Energy-efficient computing for wildlife tracking:Design tradeoffs and early experiences with zebranet. In ACM Sigplan Notices, pages 96-107. ACM, 2002.
[23]G.L. Foresti and L. Snidaro. A distributed sensor network for video surveil-lance of outdoor environments. In Proceedings of the International Confer-ence on Image Processing, volume 1, pages 1-525. IEEE,2002.
[24]M. Maroti, B. Kusy, G. Simon, and A. Ledeczi. The flooding time, synchro-nization protocol. In Proceedings of the 2nd international conference on Embedded networked sensor systems, pages 39-49. ACM,2004.
[25]S. Li, Y. Liu, and X. LI. Capacity of large scale wireless networks under gaussian channel model. In Proceedings of the 14th ACM international conference on Mobile computing and networking (MobiCom), pages 140-151. ACM,2008.
[26]F. Ellingor, R. Eickhoff, R. Giorlich, J. Huttner, A. Ziroff, S. Wehrli, T. Ussmuller, J. Carls, V. Subramanian, M. Krcmar, et al. Local positioning for wireless sensor networks. In IEEE Globecom Workshops,2007, pages 1-6. IEEE,2007.
[27]M. Vossiek, L. Wiebking, P. Gulden, J. Wieghardt, C. Hoffmann, and P. Heide. Wireless local positioning. IEEE Microwave Magazine,4(4):77-86, 2003.
[28]L. Mo, Y. He, Y. Liu, J. Zhao, S.-J. Tang, X. LI, and G. Dai. Canopy closure estimates with greenorbs:sustainable sensing in the forest. In Proceedings of the 7th ACM Conference on Embedded Networked Sensor Systems (SenSys), pages 99-112. ACM,2009.
[29]刘云浩.物联网导论.科学出版社,2011.
[30]刘云浩.从普适计算、CPS到物联网:下一代互联网的视界.中国计算机学会通讯,5(12):66-69,2009.
[31]L. Atzori, A. Iera, and G. Morabito. The internet of things:A survey. Computer Networks.54(15):2787-2805,2010.
[32]E.A. Lee. Cyber physical systems:Design challenges. In Proceedings of the 11th IEEE International Symposium on Object Oriented Real-Time Dis-tributed Computing (ISORC), pages 363-369. IEEE,2008.
[33]N. Patwari. Location estimation in sensor networks. PhD thesis, The University of Michigan,2005.
[34]N. Patwari, J.N. Ash, S. Kyperountas, A.O. Hero Ⅲ, R.L. Moses, and N.S. Correal. Locating the nodes:cooperative localization in wireless sensor networks. IEEE Signal Processing Magazine,22(4):54-69,2005.
[35]Y. Diao, M. Fu, and H. Zhang. An overview of range detection techniques for wireless sensor networks. In Proceedings of the 8th World Congress on Intelligent Control and Automation (WCICA), pages 1150-1155. IEEE, 2010.
[36]T.S. Rappaport, J.H. Reed, and B.D. Woerner. Position location using wireless communications on highways of the future. IEEE Communications Magazine,34(10):33-41,1996.
[37]D. Niculescu and B. Nath. Ad hoc positioning system (aps). In Proceedings of the IEEE Global Telecommunications Conference 2001 (GLOBECOM), volume 5, pages 2926-2931. IEEE,2001.
[38]R.L. Moses, D. Krishnamurthy, and R.M. Patterson. A self-localization method for wireless sensor networks. EURASIP Journal on Applied Signal Processing, pages 348-358,2003.
[39]N. Patwari and A.O. Hero Ⅲ. Using proximity and quantized rss for sensor localization in wireless networks. In Proceedings of the 2nd ACM inter-national conference on Wireless sensor networks and applications, pages 20-29. ACM,2003.
[40]A. Catovic and Z. Sahinoglu. The cramer-rao bounds of hybrid toa/rss and tdoa/rss location estimation schemes. IEEE Communications Letters, 8(10):626-628,2004.
[41]D. Niculescu and B. Nath. Error characteristics of ad hoc positioning sys-tems (aps). In Proceedings of the 5th ACM international symposium on Mobile ad hoc networking and computing (MobiCom), pages 20-30. ACM, 2004.
[42]S.Y. Seidel and T.S. Rappaport.914 mhz path loss prediction models for indoor wireless communications in multifloored buildings. IEEE Transac-tions on Antennas and Propagation,40(2):207-217,1992.
[43]N.B. Priyantha, A. Chakraborty, and H. Balakrishnan. The cricket location-support system. In Proceedings of the 6th annual international conference on Mobile computing and networking (MobiCom), pages 32-43. ACM,2000.
[44]C. Peng, G. Shen, Y. Zhang, Y. Li, and K. Tan. Beepbeep:a high accuracy acoustic ranging system using cots mobile devices. In Proceedings of the 5th international conference on Embedded networked sensor systems (SenSys), pages 1-14. ACM,2007.
[45]A. Nasipuri and K. Li. A directionality based location discovery scheme for wireless sensor networks. In Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications, pages 105-111. ACM,2002.
[46]J. Gao, R. Sion, and S. Lederer. Collaborative location certification for sensor networks. ACM Transactions on Sensor Networks (TOSN),6(4):30, 2010.
[47]王福豹,史龙,and任丰原.无线传感器网络中的自身定位系统和算法.软件学报,16(5),2005.
[48]尚志军,曾鹏,and于海斌.无线传感器网络节点定位问题.计算机科学,31(10):35-38,2004.
[49]Y. Liu, Z. Yang, X. Wang, and L. Jian. Location, localization, and lo-calizability. Journal of Computer Science and Technology,25(2):274-297, 2010.
[50]彭宇and王丹.无线传感器网络定位技术综述.电子测量与仪器学报,25(5):389-399,2011.
[51]D.K. Goldenberg, A. Krishnamurthy, W.C. Maness, Y.R. Yang, A. Young, A.S. Morse, and A. Savvides. Network localization in partially localiz-able networks. In Proceedings of the 24th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pages 313-326. IEEE,2005.
[52]J. Aspnes, T. Eren, D.K. Goldenberg, A.S. Morse, W. Whiteley, Y.R. Yang, B.D.O. Anderson, and P.N. Belhumeur. A theory of network localization. IEEE Transactions on Mobile Computing,5(12):1663-1678,2006.
[53]Z. Yang and Y. Liu. Understanding node localizability of wireless ad-hoc networks. In Proceedings of the 29th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pages 1-9. IEEE, 2010.
[54]G. Laman. On graphs and rigidity of plane skeletal structures. Journal of Engineering mathematics,4(4):331-340,1970.
[55]J.B. Saxe. Embeddability of weighted graphs in k-space is strongly np-hard. In Proceedings of the 17th Allerton Conference on Communication, Control and Computing, pages 480-489. SIAM,1979.
[56]Y. Yemini. Some theoretical aspects of position-location problems. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science, pages 1-8. IEEE,1979.
[57]R. Connelly. Generic global rigidity. Discrete (?) Computational Geometry, 33(4):549-563,2005.
[58]D.J. Jacobs and B. Hendrickson. An algorithm for two-dimensional rigid-ity percolation:the pebble game. Journal of Computational Physics, 137(2):346-365,1997.
[59]B. Hendrickson. Conditions for unique graph realizations. SIAM Journal on Computing,21(1):65-84,1992.
[60]B. Jackson and T. Jordan. Commoted rigidity matroids and unique real-izations of graphs. Journal of Combinatorial Theory, Series B,94(1):1-29, 2005.
[61]P. Gupta and P.R. Kumar. Critical power for asymptotic connectivity in wireless networks. In Stochastic analysis, control, optimization and appli-cations, pages 547-566. Springer,1998.
[62]F. Xue and P.R. Kumar. The number of neighbors needed for connectivity of wireless networks. Wireless networks,10(2):169-181,2004.
[63]M.D. Penrose. On k-connectivity for a geometric random graph. Random Structures (?) Algorithms,15(2):145-164,1999.
[64]L. Lovasz and Y. Ycrnini. On generic rigidity in the plane. SIAM Journal on Algebraic Discrete Methods,3(1):91-98,1982.
[65]X. Li, P. Wan, Y. Wang, and C. Yi. Fault tolerant deployment and topology control in wireless ad hoc networks. Wireless Communications and Mobile Computing,4(1):109-125,2004.
[66]B. Jackson and T. Jordan. A sufficient connectivity condition for generic rigidity in the plane. Discrete Applied Mathematics,157(8):1965-1968, 2009.
[67]T. Eren, D.K. Goldenberg, W. Whiteley, Y.R. Yang, A.S. Morse, B.D.O. Anderson, and P.N. Belhumeur. Rigidity, computation, and randomization in network localization. In Proceedings of the 23rd Annual Joint Confer-ence of the IEEE Computer and Communications Societies (INFOCOM). volume 4, pages 2673-2684. IEEE,2004.
[68]Z. Yang, Y. Liu, and X. LI. Beyond trilateration:On the localizability of wireless ad hoc networks. IEEE/ACM Transactions on Networking, 18(6):1806-1814,2010.
[69]S.J. Gortler, A.D. Healy, and D.P. Thurston. Characterizing generic global rigidity. American journal of mathematics,132(4):897-939,2010.
[70]S.J. Gortler and D.P. Thurston. Characterizing the universal rigidity of generic frameworks. arXiv preprint arXiv:1001.0172,2009.
[71]W. Whiteley. Some matroids from discrete applied geometry. Contemporary Mathematics,197:171-312,1996.
[72]T.S. Tay and W. Whiteley. Generating isostatic frameworks. Structural Topology,1985.
[73]R. Connelly. Rigidity and energy. Inventiones Mathematicae,66(1):11-33, 1982.
[74]B.D.O. Anderson, I. Shames, G. Mao, and B. Fidan. Formal theory of noisy sensor network localization. SIAM Journal on Discrete Mathematics, 24(2):684-698,2010.
[75]A. Youssef and M. Youssef. A taxonomy of localization schemes for wireless sensor networks. In Proceedings of the International Conference on Wireless Networks,2007.
[76]R. Nagpal, H. Shrobe, and J. Bachrach. Organizing a global coordinate system from local information on an ad hoc sensor network. In Information Processing in Sensor Networks, pages 333-348. Springer,2003.
[77]K. Whitehouse, C. Karlof, and D. Culler. A practical evaluation of radio signal strength for ranging-based localization. ACM Mobile Computing and Communications Review,11(1):41-52,2007.
[78]D.K. Goldenberg, P. Bihler, M. Cao, J. Fang, B.D.O. Anderson, A.S. Morse, and Y.R. Yang. Localization in sparse networks using sweeps. In Proceed-ings of the 12th annual international conference on Mobile computing and networking (MobiCom), pages 110-121. ACM,2006.
[79]L. Doherty, Pister S. J.K., and L. El Ghaoui. Convex position estimation in wireless sensor networks. In Proceedings of the 20th Annual Joint Confer-ence of the IEEE Computer and Communications Societies (INFOCOM), volume 3, pages 1655-1663. IEEE,2001.
[80]Y. Shang, W. Rumi, Y. Zhang, and M. Fromherz. Localization from connec-tivity in sensor networks. IEEE Transactions on Parallel and Distributed Systems,15(11):961-974,2004.
[81]马震,刘云,and沈波.分布式无线传感器网络定位算法mds-map.通信学报,29(6):57-62,2008.
[82]P.J.F. Groenen. The majorization approach to multidimensional scaling: some problems and extensions. DSWO Press Leiden,1993.
[83]J.O. Ramsay. Some statistical approaches to multidimensional scaling data. Journal of the Royal Statistical Society. Series A (General), pages 285-312, 1982.
[84]R.L. Moses, D. Krishnamurthy, and R. Patterson. An auto-calibration method for unattended ground sensors. In Proceedings of the IEEE Interna-tional Conference on Acoustics, Speech, and Signal Processing (ICASSP), volume 3, pages 2936-2941. IEEE,2002.
[85]J.A Costa, N. Patwari, and A.O. Hero Ⅲ. Distributed weighted-multidimensional scaling for node localization in sensor networks. ACM Transactions on Sensor Networks (TOSN),2(1):39-64,2006.
[86]S. Capkun, M. Hamdi, and J-P. Hubaux. Gps-free positioning in mobile ad-hoc networks. In Proceedings of the 34th Annual Hawaii International Conference on System Sciences, pages 10-16. IEEE,2001.
[87]D. Moore, J. Leonard, D. Rus, and S. Teller. Robust distributed network localization with noisy range measurements. In Proceedings of the 2nd international conference on Embedded networked sensor systems (SenSys), pages 50-61. ACM,2004.
[88]Z. Lin, B. Francis, and M. Maggiore. Distributed control and analysis of coupled cell systems. VDM Publishing,2008.
[89]C.D. Godsil and G. Royle. Algebraic graph theory. Springer New York, 2001.
[90]B.D.O. Anderson, P.N. Belhumeur, T. Eren, D.K. Goldenberg, A.S. Morse, W. Whiteley, and Y.R. Yang. Graphical properties of easily localizable sensor networks. Wireless Networks,15(2):177-191,2009.
[91]J.M. Hendrickx, B.D.O. Anderson, and V.D. Blondel. Rigidity and per-sistence of directed graphs. In Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference 2005(CDC-ECC'05), pages 2176-2181. IEEE,2005.
[92]B.D.O. Anderson, C. Yu, B. Fidan, and J. Hendrickx. Rigid graph control architectures for autonomous formations. IEEE Control Systems Magazine, 28(6):48-63,2008.
[93]Y. Diao, M. Fu, and H. Zhang. Comparison of node localization methods for sensor networks. In Proceedings of the 12th International Conference onControl, Automation, Robotics and Vision (ICARCV), pages 235-239. IEEE,2012.
[94]H. Coxeter. Introduction to geometry. John Wiley\& Sons, Inc.,1969.
[95]M.J. Sippl and H.A. Scheraga. Cayley-menger coordinates. Proceedings of the National Academy of Sciences,83(8):2283-2287,1986.
[96]Y. Diao, Z. Lin, M. Fu, and H. Zhang. A new distributed localization method for sensor networks. arXiv preprint arXiv:1301.2648,2013.
[97]X. Wang, J. Luo, Y. Liu, S. Li, and D. Dong. Component-based localiza-tion in sparse wireless networks. IEEE/ACM Transactions on Networking (TON),19(2):540-548,2011.
[98]Y. Diao, M. Fu, Z. Lin, and H. Zhang. Sequential cluster-based localization for sensor networks. submitted to IEEE/ACM Transaction on Networking.
[99]S. Bandyopadhyay and E.J. Coyle. An energy efficient hierarchical clus-tering algorithm for wireless sensor networks. In Proceedings of the 22nd Annual Joint Conference of the IEEE Computer and Communications (IN-FOCOM), volume 3, pages 1713-1723. IEEE,2003.
[100]O. Younis and S. Fahmy. Distributed clustering in ad-hoc sensor networks: A hybrid, energy-efficient, approach. In Proceedings of the 23rd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), volume 1. IEEE,2004.
[101]M. Youssef, A. Youssef, and M. Younis. Overlapping multihop clustering for wireless sensor networks. IEEE Transactions on Parallel and Distributed Systems,20(12):1844-1856,2009.
[102]沈波,张世永,and钟亦平.无线传感器网络分簇路由协议.软件学报,17(7):1588-1600,2006.
[103]Y. Diao, M. Fu, and H. Zhang. New results on node localizable conditions for sensor networks. In Proceedings of the IEEE 51st Annual Conference on Decision and Control (CDC), pages 2726-2731. IEEE,2012.
[104]C. Yu, B. Fidan, and B.D.O. Anderson. Principles to control autonomous formation merging. In Proceedings of the 2006 American Control Confer-ence (ACC), pages 7-pp. IEEE,2006.
[105]D.P. Bertsekas. A new class of incremental gradient methods for least squares problems. SI AM Journal on Optimization,7(4):913-926,1997.
[106]A. Berg and T. Jordan. A proof of connelly's conjecture on 3-connected generic cycles. Journal of Combinatorial Theory. Series B,2002.
[107]U.A. Khan, S. Kar, and J.M.F. Moura. Distributed sensor localization in random environments using minimal number of anchor nodes. IEEE Transactions on Signal Processing,57(5):2000-2016,2009.
[108]P. Biswas and Y. Ye. Somidefinite programming for ad hoc wireless sensor network localization. In Proceedings of the 3rd international symposium on Information processing in sensor networks (IPSN), pages 46-54. ACM, 2004.
[109]M. Rabbat and R. Nowak. Distributed optimization in sensor networks. In Proceedings of the 3rd international symposium on Information processing in sensor networks (IPSN), pages 20-27. ACM,2004.
[110]K. Dantu and G.S. Sukhatme. Contour detection using actuated sensor networks. In Proceedings of the 1st international conference on Embedded networked sensor systems (SenSys), pages 310-311. ACM,2003.
[111]N.B. Priyantha. The cricket indoor location system. PhD thesis, Mas-sachusetts Institute of Technology,2005.
[112]C.S. Ballantine. Stabilization by a diagonal matrix. Proceedings of Amer-ican Mathematical Soceity,25(4):728-734,1970.
[113]L. Wang, Z. Han, and Z. Lin. Formation control of directed multi-agent networks based on complex laplacian. In Proceedings of the 51st IEEE Conference on Decision and Control (CDC). IEEE,2012.
[114]Z. Lin, W. Ding, G. Yan, C. Yu, and A. Giua. Leader-follower formation via complex laplacian. Automatica,2013.
[115]B.H. Cheng, L. Vandenberghe, and K. Yao. Distributed algorithm for node localization in wireless ad-hoc networks. ACM Transactions on Sensor Networks (TOSN),6(1):8,2009.
[116]吴正平,关治洪,and吴先用.基于一致性理论的多机器人系统队形控制.控制与决策,22(11):12411244,2007.
[117]朱战霞and袁建平.无人机编队飞行问题初探.飞行力学、21(2):5762,2003.
[118]张洪华and林来兴.卫星编队飞行相对轨道的确定.宇航学报,23(6):77-81,2002.
[119]S.L. Collier and D.K. Wilson. Performance bounds for passive sensor arrays operating in a turbulent medium:Plane-wave analysis. The Journal of the Acoustical Society of America,113:2704,2003.
[120]P. Yang, R.A. Freeman, G.J. Gordon, K.M. Lynch, S.S. Srinivasa, and R. Sukthankar. Decentralized estimation and control of graph connectivity for mobile sensor networks. Automatica,46(2):390-396,2010.
[121]H. Wymeersch, J. Lien, and M.Z. Win. Cooperative localization in wireless networks. Proceedings of the IEEE,97(2):427-450,2009.
[122]T. Biyikoglu and J. Leydold. Algebraic connectivity and degree sequences of trees. Linear Algebra and its Applications,430(2):811-817,2009.
[123]L. Lovasz. Singular spaces of matrices and their application in combina-torics. Bulletin of the Brazilian Mathematical Society,20(1):87-99,1989.
[124]W.K. Chen. Applied graph theory:graphs and electrical networks. North-Holland Amsterdam,1976.