基于一致性滤波算法的传感器网络估计问题研究
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摘要
无线传感器网络(Wireless Sensor Network)是当前一个前沿的热点研究领域,有着广泛的应用前景。由于单传感器自身的能量、存储和处理能力的局限性,带来了稳定性差、可靠性较低、检测精度不高的挑战性问题,信息融合技术成为解决上述问题的有效手段,也引起了学者的极大关注。信息融合技术是一个多学科高度集成的热点研究领域,本文针对无线传感器网络自身特点,将无线传感器网络和信息融合算法相结合,对一致性滤波融合算法进行了研究。
论文提出了一种基于一致性的分布式滤波算法,针对实际应用中存在的网络丢包问题,重点研究了有丢包时的分布式滤波算法,通过理论分析给出了估计误差系统收敛的充分条件。通过仿真实验将本文算法与前人滤波算法在理想状况和有丢包状况下进行比较研究,表明本算法具有较优的滤波效果；探讨了丢包率对算法的影响,提出了三种改进方案,仿真表明每种方案都能较好地改善丢包问题；在保持一定精度条件和降低网络能耗的前提下,将该算法引入牵制网络结构中,讨论了牵制度大节点和牵制随机节点两种牵制策略,并通过仿真实验比较相应策略的跟踪精度及网络同步化性能；针对非线性系统,提出了一种基于一致性的UKF算法,仿真结果表明与传统EKF和UKF相比,一致性UKF具有良好的效果。
Wireless sensor networks (WSN) is currently a top research field with a wide range of applications. Since the limitations of single sensor's energy, storage and processing bring challenging issues including low stability, poor reliability and precision, information fusion technology can be an effective approach to address these problems and has caused great concern of scholars. Information fusion technology is a highly integrated multi-disciplinary research field and in this paper, for the characteristics of wireless sensor networks, we study the Kalman filter fusion algorithm based on consensus by combining wireless sensor networks with information fusion algorithm.
In the first place, this paper introduces a consensus-based distributed filtering algorithm. Aiming to the problem of network packet-dropping in the practical application, we focus on the distributed filtering algorithm with packet-dropping. By the theoretical analysis, we give a sufficient condition for the convergence of the estimation error system. We compare our algorithm with a classical filtering one by some simulations in ideal and packet-dropping cases, respectively. The results show that our algorithm does better filtering in the packet-dropping case. In response to this problem, three schemes of improvement are presented and simulation results show the effectiveness. Secondly, in order to reduce the network energy consumption under the circumstances of maintaining certain precision, the proposed algorithm is introduced into contain network structure. Two contain strategies are put forward, containing big degree nodes and random node. Their tracking precision and network synchronization are compared through simulation. Finally, a consensus-based UKF algorithm applying in nonlinear system is introduced. The result of comparing traditional EKF and UKF with our algorithm indicates the latter has good effect.
论文提出了一种基于一致性的分布式滤波算法,针对实际应用中存在的网络丢包问题,重点研究了有丢包时的分布式滤波算法,通过理论分析给出了估计误差系统收敛的充分条件。通过仿真实验将本文算法与前人滤波算法在理想状况和有丢包状况下进行比较研究,表明本算法具有较优的滤波效果；探讨了丢包率对算法的影响,提出了三种改进方案,仿真表明每种方案都能较好地改善丢包问题；在保持一定精度条件和降低网络能耗的前提下,将该算法引入牵制网络结构中,讨论了牵制度大节点和牵制随机节点两种牵制策略,并通过仿真实验比较相应策略的跟踪精度及网络同步化性能；针对非线性系统,提出了一种基于一致性的UKF算法,仿真结果表明与传统EKF和UKF相比,一致性UKF具有良好的效果。
Wireless sensor networks (WSN) is currently a top research field with a wide range of applications. Since the limitations of single sensor's energy, storage and processing bring challenging issues including low stability, poor reliability and precision, information fusion technology can be an effective approach to address these problems and has caused great concern of scholars. Information fusion technology is a highly integrated multi-disciplinary research field and in this paper, for the characteristics of wireless sensor networks, we study the Kalman filter fusion algorithm based on consensus by combining wireless sensor networks with information fusion algorithm.
In the first place, this paper introduces a consensus-based distributed filtering algorithm. Aiming to the problem of network packet-dropping in the practical application, we focus on the distributed filtering algorithm with packet-dropping. By the theoretical analysis, we give a sufficient condition for the convergence of the estimation error system. We compare our algorithm with a classical filtering one by some simulations in ideal and packet-dropping cases, respectively. The results show that our algorithm does better filtering in the packet-dropping case. In response to this problem, three schemes of improvement are presented and simulation results show the effectiveness. Secondly, in order to reduce the network energy consumption under the circumstances of maintaining certain precision, the proposed algorithm is introduced into contain network structure. Two contain strategies are put forward, containing big degree nodes and random node. Their tracking precision and network synchronization are compared through simulation. Finally, a consensus-based UKF algorithm applying in nonlinear system is introduced. The result of comparing traditional EKF and UKF with our algorithm indicates the latter has good effect.
引文
[1]郁文贤,雍少为,郭桂蓉.多传感器信息融合技术述评[J].国防科技大学学报.1994,16(3)：1-8
[2]B.S.Y. Rao, H.F.Durrant-Whyte, J. A.Sheen. A fully decentralized multi-sensor system for tracking and surveillance[J]. International Journal of Robotics Research.1993,12(1): 20～44
[3]C.W.Sanders, E.C.Tacker, T.D.Linton. A new class of decentralized filters for interconnected systems[J]. IEEE Transactions on Automatic Control.1974,19(3): 259～262
[4]A.Iftar. Decentralized estimation and control with overlapping input, state and output decomposition[J]. Automatica.1993,29(2):511～516
[5]孙利民,李建中,朱红松,陈渝.无线传感器网络[M].北京：清华大学出版社,2005
[6]何友,王国宏,陆大鑫等.多传感器信息融合及应用[M].北京：电子工业出版社,2000
[7]J. A. Benediktsson and P. H. Swain. Consensus theoretic classification methods[J]. IEEE Transactions on Systems, Man and Cybernetics.1992,22(4):688～704
[8]S. H. Strogatz. From Kuramoto to Crawford:Exploring the onset of synchronization in populations of coupled oscillators[J]. Physica D.2000,143:1-20
[9]R. Sepulchre, D. Paley and N. Leonard. Collective motion and oscillator Synchronization[C]. Proc. Block Island Workshop Cooperative Control, Block Island, RI.Jun.2003
[10]A. Papachristodoulou and A. Jadbabaie. Synchronization in oscillator networks: Switching topologies and non-homogeneous delays[C].44th IEEE Conference. 2005:5692～5697
[11]H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable flocking of mobile agents, Part I: Fixed topology[J]. Proc. IEEE Conference on Decision and Control, Maui, Hawaii USA. 2003,2:2010～2015
[12]H. Tanner. Flocking with Obstacle Avoidance in Switching Networks of Interconnected Vehicles[C]. IEEE International Conference Robotics and Automation.New Orleans LA,April 26-May 1,2004:3006～3011
[13]V. Gupta, B. Hassibi, and R. Murray. Stability analysis of stochastically varying formations of dynamic agents[C]. Proc. of CDC.2003:504～509
[14]S. Roy, A. Saberi, and K. Herlugson. Formation and alignment of distributed sensing agents with double-integrator dynamics and actuator saturation[C]. Sensor Network Applications, IEEE Press.2004
[15]J. A. Fax and R. M. Murray. Information flow and cooperative control of vehicle formations[J].IEEE Trans. On Automatic Control.2004,49:1465～1476
[16]Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical conditions for formation control of unicycles[J].IEEE Trans. On Automatic Control.2005,50:121～127
[17]V. Gupta, B. Hassibi, and R. Murray. Stability analysis of stochastically varying formations of dynamic agents[J]. Proc. of CDC.2003:504～509
[18]S. Roy, A. Saberi, and K. Herlugson. Formation and alignment of distributed sensing agents with double-integrator dynamics and actuator saturation[C]. Sensor Network Applications, IEEE Press.2004
[19]J. Lin, A. S. Morse, and B. D. O. Anderson. The multi-agent rendezvous problem[C]. Proc. of CDC.2003:1508～1513
[20]J. Cortes, S. Martinez, F. Bullo. Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions[C]. IEEE Trans. on Automatic Control.July 2004
[21]X. Wang and G. Chen. Complex networks:Small-world, scale-free and beyond[J].IEEE Circuits Systems Magazine.2003,3:6-20
[22]Xiao L and Boyd S. Fast linear iterations for distributed averaging[J].Systems and Control Letters.2004,53(1):65～78
[23]D J Watts and S H Strogatz.Collective dynamics of "small-world" networks[J].Nature.1998,393:440～442
[24]R. Olfati-Saber and J S Shamma. Consensus filters for sensor networks and distributed sensor fusion[C]. Proceedings of the IEEE Conference on Decision and Control.2005:6698～6703
[25]R. Olfati-Saber. Distributed Kalman Filter with Embedded Consensus Filters[C].Proc of the joint CDC-ECC Conference.December 2005
[26]Rao B S Y Durrant-Whyte H F and Sheen J A.A fully decentralized multi-sensor system for tracking surveillance [J].International Journal of Robotics Research.1993, 12(1):20～44
[27]R.Olfati.Saber and R M Murray. Consensus problems in networks of agents with switching topology and time-delays[J].IEEE Transactions on Automatic Control.2004,49(9):1520～1533
[28]W. Ren and R. W. Beard. A survey of Consensus Problems in Multi-agent Coordination[C]. Proc of the 2005 American Control Conference.2005:1859～1864
[29]W. Ren and R. W. Beard. Consensus seeking in multi-agent systems under dynamically changing interaction topologies[J]. IEEE Trans. on Automatic Control.2005,50: 655～660
[30]R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays[J]. IEEE Trans. on Automatic Control. 2004,49:1520～1533
[31]R. Olfati-Saber and R. M. Murray. Consensus Protocols for Networks of Dynamic Agents[C]. Proc of the 2003 American Control Conference. June 2003
[32]A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules[J]. IEEE Trans. On Automatic Control.2003,48: 988～1001
[33]J. A. Fax and R. M. Murray. Information flow and cooperative control of vehicle formations[J]. IEEE Trans. On Automatic Control.2004,49:1465～1476
[34]L. Moreau. Stability of multi-agent systems with time-dependent communication links[J]. IEEE Trans.on Automatic Control.2005,50:169～182
[35]W. Ren and R. W. Beard. Coordination variables and consensus building in multiple vehicle systems, Cooperative Control [J]. Springer-Verlag Series:Lecture Notes in Control and Information Sciences.2004,309:171～188
[36]Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical conditions for formation control of unicycles[J]. IEEE Trans. On Automatic Control.2005,50:121～127
[37]W. Ren and R. W. Beard.Consensus seeking in multi-agent systems under dynamically changing interaction topologies[J]. IEEE Trans. on Automatic Control.2005,50: 655～660
[38]R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays[J]. IEEE Trans. on Automatic Control.2004,49: 1520～1533
[39]J. Wolfowitz. Products of indecomposable, aperiodic, stochastic matrices[J]. Proc of the American Mathematical,Society.1963,15:733～736
[40]R. Olfati-Saber. Ultrafast consensus in small-world networks[C]. Proc of the 2005 American Control Conference.2005:2371～2378
[41]Wei Ren, Randal W. Beard, Derek B. Kingston. Multi-agent Kalman Consensus with Relative Uncertainty [C]. Proc of the 2005 American Control Conference. 2005:1865～1870
[42]Derek B. Kingston, Wei Ren, Randal W. Beard. Consensus Algorithms are Input-to-State Stable[J]. Proc of the 2005 American Control Conference.2005:1686～1690
[43]Jacobs, O. L. R. Introduction to Control Theory,2nd Edition[M]. Oxford University Press, 1993
[44]R. Olfati-Saber. Distributed Kalman filtering for sensor networks[C]. In Proceedings of the 46th IEEE Conference on Decision and Control. New Orleans, USA:IEEE. 2007:5492～5498
[45]S S Stankovic, M S Stankovic, D M Stipanovic. Consensus based overlapping decentralized estimation with missing observations and communications faults[J]. Automatica.2009,45:1397～1406
[46]R. Olfati-Saber. Distributed Kalman filtering for sensor networks[C]. In Proceedings of the 46th IEEE Conference on Decision and Control. New Orleans, USA:IEEE.2007: 5492～5498
[47]T J Richardson, R H Kwong. On positive definite solutions to the algebraic Riccati equation[J]. Systems and Control Letters.1986,7:99～104
[48]B Sinopoli, L Schenato, Franceschetti M, Poolla K, Jordan M I, Sastry S S. Kalman filtering with intermittent observations[J]. IEEE Transactions on Automatic Control.2004, 49:1453～1464
[49]L Epstein M, L Shi, A Tiwari, R M Murray. Probabilistic performance of state estimation across a lossy network[J]. Automatica.2008,44:3046-3053
[50]张冬梅,俞立,周明华.具有快变时延和丢包的网络控制系统镇定.控制理论与应用[J].2008,25：480～484
[51]S L Sun, L H Xie, W D Xiao, N Xiao. Optimal filtering for systems with multiple packet dropouts[J]. IEEE Transactions on Circuits and Systems-Ⅱ:Express Briefs.2008,55: 695-699
[52]S L Sun, Xie L H, W D Xiao, Y C Soh. Optimal linear estimation for systems with multiple packet dropouts[J]. Automatic.2008,44:1333～1342
[53]S L Sun, W D Xiao. Optimal full-order and reduced order estimators for discrete-time systems with multiple packet dropouts[J]. IEEE Transactions on Signal Processing.2008, 56:4031-4038
[54]郭戈,王宝凤.多丢包不确定离散系统的鲁棒Kalman滤波[J].自动化学报.2009,36(5)：767～772
[55]R A Horn, C R Johnson. Matrix Analysis[M]. UK:Cambridge University Press,1985
[56]C Peskin.Mathematical Aspects of Heart Physiology [C]. Courant Institute of Mathematics and Science, New York.1977
[57]R O Grigoriev, M C Cross, H G Schuster. Pinning Control of Spatiotemporal Chaos[J]. Physical Review Letters.1997,79:2795～2798
[58]N Parekh, S Parthasarathy, S Sinha. Global and local control of spatiotcmporal chaos in coupled map lattices[J].Physical Review Letters.1997, (81):1401～1404
[59]X Wang, G Chert.Pinning control of scale-free dynamical networks[J]. Physics A.2002, 310:521～531
[60]Veronique Delouille,Ramesh Neelsh Neelamani,Richard G Baraniuk. Robust Distributed Estimation Using the Embedded Subgraphs Algorithm[J].IEEE Transactions on Signal Processing.2006,54:2998～3010
[61]Erik B Sudderth, Michael I Mandel,William T Freeman.Visual Hand Tracking Using Nonparametrie Belief Propagation[R].MIT Laboratory for Information & Decision Systems Technical Report 2603,MAY 2004.Presented at the 2004 IEEE CVPR Workshop On Generative Model Based Visio.2004:1～9
[62]Neeta Trivedi,N Balakrishnan. Graphical Models for Distributed Inference in Wireless Sensor Networks[C].2009 Third International Conference on Sensor Technologies and Applications.2009:596～603
[63]Alexander T Ihler,John W Fisher,Randolph L Moses. Nonparametric Belief Propagation for Sensor Self-Calibration[C].2004:861～864
[2]B.S.Y. Rao, H.F.Durrant-Whyte, J. A.Sheen. A fully decentralized multi-sensor system for tracking and surveillance[J]. International Journal of Robotics Research.1993,12(1): 20～44
[3]C.W.Sanders, E.C.Tacker, T.D.Linton. A new class of decentralized filters for interconnected systems[J]. IEEE Transactions on Automatic Control.1974,19(3): 259～262
[4]A.Iftar. Decentralized estimation and control with overlapping input, state and output decomposition[J]. Automatica.1993,29(2):511～516
[5]孙利民,李建中,朱红松,陈渝.无线传感器网络[M].北京：清华大学出版社,2005
[6]何友,王国宏,陆大鑫等.多传感器信息融合及应用[M].北京：电子工业出版社,2000
[7]J. A. Benediktsson and P. H. Swain. Consensus theoretic classification methods[J]. IEEE Transactions on Systems, Man and Cybernetics.1992,22(4):688～704
[8]S. H. Strogatz. From Kuramoto to Crawford:Exploring the onset of synchronization in populations of coupled oscillators[J]. Physica D.2000,143:1-20
[9]R. Sepulchre, D. Paley and N. Leonard. Collective motion and oscillator Synchronization[C]. Proc. Block Island Workshop Cooperative Control, Block Island, RI.Jun.2003
[10]A. Papachristodoulou and A. Jadbabaie. Synchronization in oscillator networks: Switching topologies and non-homogeneous delays[C].44th IEEE Conference. 2005:5692～5697
[11]H. G. Tanner, A. Jadbabaie, and G. J. Pappas. Stable flocking of mobile agents, Part I: Fixed topology[J]. Proc. IEEE Conference on Decision and Control, Maui, Hawaii USA. 2003,2:2010～2015
[12]H. Tanner. Flocking with Obstacle Avoidance in Switching Networks of Interconnected Vehicles[C]. IEEE International Conference Robotics and Automation.New Orleans LA,April 26-May 1,2004:3006～3011
[13]V. Gupta, B. Hassibi, and R. Murray. Stability analysis of stochastically varying formations of dynamic agents[C]. Proc. of CDC.2003:504～509
[14]S. Roy, A. Saberi, and K. Herlugson. Formation and alignment of distributed sensing agents with double-integrator dynamics and actuator saturation[C]. Sensor Network Applications, IEEE Press.2004
[15]J. A. Fax and R. M. Murray. Information flow and cooperative control of vehicle formations[J].IEEE Trans. On Automatic Control.2004,49:1465～1476
[16]Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical conditions for formation control of unicycles[J].IEEE Trans. On Automatic Control.2005,50:121～127
[17]V. Gupta, B. Hassibi, and R. Murray. Stability analysis of stochastically varying formations of dynamic agents[J]. Proc. of CDC.2003:504～509
[18]S. Roy, A. Saberi, and K. Herlugson. Formation and alignment of distributed sensing agents with double-integrator dynamics and actuator saturation[C]. Sensor Network Applications, IEEE Press.2004
[19]J. Lin, A. S. Morse, and B. D. O. Anderson. The multi-agent rendezvous problem[C]. Proc. of CDC.2003:1508～1513
[20]J. Cortes, S. Martinez, F. Bullo. Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions[C]. IEEE Trans. on Automatic Control.July 2004
[21]X. Wang and G. Chen. Complex networks:Small-world, scale-free and beyond[J].IEEE Circuits Systems Magazine.2003,3:6-20
[22]Xiao L and Boyd S. Fast linear iterations for distributed averaging[J].Systems and Control Letters.2004,53(1):65～78
[23]D J Watts and S H Strogatz.Collective dynamics of "small-world" networks[J].Nature.1998,393:440～442
[24]R. Olfati-Saber and J S Shamma. Consensus filters for sensor networks and distributed sensor fusion[C]. Proceedings of the IEEE Conference on Decision and Control.2005:6698～6703
[25]R. Olfati-Saber. Distributed Kalman Filter with Embedded Consensus Filters[C].Proc of the joint CDC-ECC Conference.December 2005
[26]Rao B S Y Durrant-Whyte H F and Sheen J A.A fully decentralized multi-sensor system for tracking surveillance [J].International Journal of Robotics Research.1993, 12(1):20～44
[27]R.Olfati.Saber and R M Murray. Consensus problems in networks of agents with switching topology and time-delays[J].IEEE Transactions on Automatic Control.2004,49(9):1520～1533
[28]W. Ren and R. W. Beard. A survey of Consensus Problems in Multi-agent Coordination[C]. Proc of the 2005 American Control Conference.2005:1859～1864
[29]W. Ren and R. W. Beard. Consensus seeking in multi-agent systems under dynamically changing interaction topologies[J]. IEEE Trans. on Automatic Control.2005,50: 655～660
[30]R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays[J]. IEEE Trans. on Automatic Control. 2004,49:1520～1533
[31]R. Olfati-Saber and R. M. Murray. Consensus Protocols for Networks of Dynamic Agents[C]. Proc of the 2003 American Control Conference. June 2003
[32]A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules[J]. IEEE Trans. On Automatic Control.2003,48: 988～1001
[33]J. A. Fax and R. M. Murray. Information flow and cooperative control of vehicle formations[J]. IEEE Trans. On Automatic Control.2004,49:1465～1476
[34]L. Moreau. Stability of multi-agent systems with time-dependent communication links[J]. IEEE Trans.on Automatic Control.2005,50:169～182
[35]W. Ren and R. W. Beard. Coordination variables and consensus building in multiple vehicle systems, Cooperative Control [J]. Springer-Verlag Series:Lecture Notes in Control and Information Sciences.2004,309:171～188
[36]Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical conditions for formation control of unicycles[J]. IEEE Trans. On Automatic Control.2005,50:121～127
[37]W. Ren and R. W. Beard.Consensus seeking in multi-agent systems under dynamically changing interaction topologies[J]. IEEE Trans. on Automatic Control.2005,50: 655～660
[38]R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays[J]. IEEE Trans. on Automatic Control.2004,49: 1520～1533
[39]J. Wolfowitz. Products of indecomposable, aperiodic, stochastic matrices[J]. Proc of the American Mathematical,Society.1963,15:733～736
[40]R. Olfati-Saber. Ultrafast consensus in small-world networks[C]. Proc of the 2005 American Control Conference.2005:2371～2378
[41]Wei Ren, Randal W. Beard, Derek B. Kingston. Multi-agent Kalman Consensus with Relative Uncertainty [C]. Proc of the 2005 American Control Conference. 2005:1865～1870
[42]Derek B. Kingston, Wei Ren, Randal W. Beard. Consensus Algorithms are Input-to-State Stable[J]. Proc of the 2005 American Control Conference.2005:1686～1690
[43]Jacobs, O. L. R. Introduction to Control Theory,2nd Edition[M]. Oxford University Press, 1993
[44]R. Olfati-Saber. Distributed Kalman filtering for sensor networks[C]. In Proceedings of the 46th IEEE Conference on Decision and Control. New Orleans, USA:IEEE. 2007:5492～5498
[45]S S Stankovic, M S Stankovic, D M Stipanovic. Consensus based overlapping decentralized estimation with missing observations and communications faults[J]. Automatica.2009,45:1397～1406
[46]R. Olfati-Saber. Distributed Kalman filtering for sensor networks[C]. In Proceedings of the 46th IEEE Conference on Decision and Control. New Orleans, USA:IEEE.2007: 5492～5498
[47]T J Richardson, R H Kwong. On positive definite solutions to the algebraic Riccati equation[J]. Systems and Control Letters.1986,7:99～104
[48]B Sinopoli, L Schenato, Franceschetti M, Poolla K, Jordan M I, Sastry S S. Kalman filtering with intermittent observations[J]. IEEE Transactions on Automatic Control.2004, 49:1453～1464
[49]L Epstein M, L Shi, A Tiwari, R M Murray. Probabilistic performance of state estimation across a lossy network[J]. Automatica.2008,44:3046-3053
[50]张冬梅,俞立,周明华.具有快变时延和丢包的网络控制系统镇定.控制理论与应用[J].2008,25：480～484
[51]S L Sun, L H Xie, W D Xiao, N Xiao. Optimal filtering for systems with multiple packet dropouts[J]. IEEE Transactions on Circuits and Systems-Ⅱ:Express Briefs.2008,55: 695-699
[52]S L Sun, Xie L H, W D Xiao, Y C Soh. Optimal linear estimation for systems with multiple packet dropouts[J]. Automatic.2008,44:1333～1342
[53]S L Sun, W D Xiao. Optimal full-order and reduced order estimators for discrete-time systems with multiple packet dropouts[J]. IEEE Transactions on Signal Processing.2008, 56:4031-4038
[54]郭戈,王宝凤.多丢包不确定离散系统的鲁棒Kalman滤波[J].自动化学报.2009,36(5)：767～772
[55]R A Horn, C R Johnson. Matrix Analysis[M]. UK:Cambridge University Press,1985
[56]C Peskin.Mathematical Aspects of Heart Physiology [C]. Courant Institute of Mathematics and Science, New York.1977
[57]R O Grigoriev, M C Cross, H G Schuster. Pinning Control of Spatiotemporal Chaos[J]. Physical Review Letters.1997,79:2795～2798
[58]N Parekh, S Parthasarathy, S Sinha. Global and local control of spatiotcmporal chaos in coupled map lattices[J].Physical Review Letters.1997, (81):1401～1404
[59]X Wang, G Chert.Pinning control of scale-free dynamical networks[J]. Physics A.2002, 310:521～531
[60]Veronique Delouille,Ramesh Neelsh Neelamani,Richard G Baraniuk. Robust Distributed Estimation Using the Embedded Subgraphs Algorithm[J].IEEE Transactions on Signal Processing.2006,54:2998～3010
[61]Erik B Sudderth, Michael I Mandel,William T Freeman.Visual Hand Tracking Using Nonparametrie Belief Propagation[R].MIT Laboratory for Information & Decision Systems Technical Report 2603,MAY 2004.Presented at the 2004 IEEE CVPR Workshop On Generative Model Based Visio.2004:1～9
[62]Neeta Trivedi,N Balakrishnan. Graphical Models for Distributed Inference in Wireless Sensor Networks[C].2009 Third International Conference on Sensor Technologies and Applications.2009:596～603
[63]Alexander T Ihler,John W Fisher,Randolph L Moses. Nonparametric Belief Propagation for Sensor Self-Calibration[C].2004:861～864