摘要
利用矩阵半张量积,论文将逻辑动态系统表示为代数状态空间动力系统.并在这个框架下,讨论并解决或部分解决了切换布尔网络的能控能观性,混合值逻辑系统的干扰解耦,以及基于混合值逻辑的队列控制问题.论文第一章,作为预备知识介绍了布尔网络,队列控制,半张量积的背景知识和研究现状.第二章介绍了布尔网络的模型,半张量积定义和性质以及将逻辑运算形式的布尔网络动态方程代数化的方法.第三章研究了切换布尔网络的能控性和能观性.利用矩阵的半张量积,切换布尔控制网络的动态方程可转换为代数形式.随后提出了MIS (model-input-state)矩阵并研究了其相关性质.MIS矩阵包含了模型、输入、状态映射的完整信息.然后是切换布尔网络能控性的充要条件.能控条件下,我们给出了点到点之间的控制和切换律设计算法.关于切换系统的能观性,我们先给出了一个充分条件.然后,在能控性假设下,我们也给出了能观性的充要条件.在此章最后一节我们研究了高阶切换布尔网络的能控性.我们给出了它的两种代数形式.基于第二种代数形式得到了能控性的充要条件.第四章,我们研究混合值逻辑网络的干扰解耦问题.利用质因子分解定理,我们得到唯一的混合值逻辑网络的Y_友好子空间.对于干扰解耦问题的可解性我们得到了充要条件,并且找到了一个新的算法,可以得到一个系统所有存在的干扰解耦控制器.这个方法比现存的其它算法有更好的操作性,我们用一个算例验证了这一点.第五章中,我们用混合值逻辑的方法研究了队列控制问题.首先,给出一个改进的寻找混合值逻辑网络轨迹的算法.然后首次为队列控制问题建立基于混合值逻辑的离散时间模型,并给出状态反馈控制器.我们还提出了部分队列控制的概念,并给出数学描述.最后,本文详细讨论了一个十分有趣的例子,来进一步解释部分队列问题的应用.
In Chapter1of this dissertation, Boolean network and semi-tensor product are introduced for preliminary. The controllability and observability are investigated for a class of switched Boolean control networks (SBCNs) in Chapter2. Using semi-tensor product of matrices, the dynamics of an SBC-N can be transformed into an algebraic form. The model-input-state matrix of an SBCN is introduced and studied for the first time. This matrix contain-s complete information of the model-input-state mapping. A necessary and sufficient condition for the controllability of SBCN is obtained. The corre-sponding control and switching law which drive a point to a given reachable point are designed. A sufficient condition for the observability of an SBCN is also given. Moreover, under the assumption of controllability, one neces-sary and sufficient condition is derived for the observability. Controllability of higher order SBCNs is investigated in this chapter. Two algebraic forms of higher order SBCNs are derived. A necessary and sufficient condition of controllability for higher order SBCNs is obtained. Finally, two illustrative examples are given to show the validity of the main results. In Chapter3, we investigate the disturbance decoupling problems (DDPs) of mix-valued logical control networks (MLCNs) via semi-tensor product method. By us-ing prime factor decomposition, a unique Y-friendly subspace of mix-valued logical networks (MLNs) is derived. A necessary and sufficient condition is obtained for the solvability of DDPs. An algorithm is proposed to find all disturbance decoupling controllers. This approach is more operational than the existing results in which the Y-friendly subspace is not unique. Finally an illustrative numerical example is given to show the effectiveness of the proposed method. In Chapter4, the formation control (FC) problem is investigated via mix-valued logic approach. First, a trajectory tracking algo-rithm of MLCN is proposed. A new formulation of FC problems is established and a feedback controller is designed to solve FC problems. The mathemati-cal description of partial-formation control (PFC) problems is then designed as a structure of logical networks. An interesting practical example of PFC is also presented and discussed in detail.
引文
[1]H. Kestler, C.Wawra, B. Kracher, and M. Kuhl, "Network modeling of signal transduction:establishing the global view," Bioessays, vol.30, no.11-12, pp. 1110-1125,2008.
[2]T. Lee, N. Rinaldi, F. Robert, D. Odom, Z. Bar-Joseph, G. Gerber, N. Hannett, C. Harbison, C. Thompson, and I. Simon, "Transcriptional regula-tory networks in Saccharomyces cerevisiae," Science, vol.298, no.5594, pp. 799-804,2002.
[3]S. Cho, S. Park, B. Lee, and D. Park., "Protein-protein interaction network-s:from interactions to networks," Journal of biochemistry and molecular biology, vol.37, no.1, pp.45-52,2004.
[4]N. Duarte, S. Becker, N. Jamshidi, I. Thiele, M. Mo, T. Vo, R. Srivas, and B.O.Palsson, "Global reconstruction of the human metabolic network based on genomic and bibliomic data," Proceedings of the National academy of Sciences of the United States of America, vol.104, no.6, pp.1777-1782, 2007.
[5]L. Sanchez and D. Thieffry, "Logical Analysis of the Drosophila Gap-gene System," Journal of theoretical Biology, vol.211, no.2, pp.115-141,2001.
[6]S. A. Kauffman, The Origins of Order:Self-organization and Selection in Evolution. New York:Oxford University Press,1993.
[7]A. Sackmann, M. Heiner, and I. Koch, "Application of Petri net based anal-ysis techniques to signal transduction pathways," BMC Bioinformatics, vol. 7, no.1, pp.1-17,2006.
[8]L. Glass, "Combinatorial and topological methods in nonlinear chemical kinetics," The Journal of chemical physics, vol.63, no.4, pp.1325-1335, 1975.
[9]M. M. Waldrop, Complexity.北京：三联书店,1997.
[10]S. Kauffman, "Metabolic stability and epigenesis in randomly constructed genetic nets.," Theoretical Biology, vol.22, no.3, pp.437-440,1969.
[11]M. M. Waldrop, At Home in the Universe,长沙：湖南科技出版社,2003.
[12]M. Aldana, "Boolean dynamics of networks with scale-free topology," Phys-ica D:Nonlinear Phenomena, vol.185, no.1, pp.45-66,2003.
[13]J. Heidel, H. Maloney, C. Farrow, and J. A. Rogers, "Finding cycles in synchronous Boolean networks with applications to biochemical systems," International Journal of Bifurcation and Chaos, vol.13, no.3, pp.535-552, 2003.
[14]R. Albert and A. Barabasi, "Dynamics of complex systems:scaling laws or the period of Boolean networks," Physical Review Letters, vol.84, no.24, pp.5660-5663,2000.
[15]T. Akutsu, S. Miyano, and S. Kuhara, "Inferring qualitative relations in genetic networks and metabolic pathways," Bioinformatics, vol.16, no.8, pp.727-734,2000.
[16]S. Huang and D. E. Ingber, "Shape-dependent control of cell growth, dif-ferentiation, and apoptosis:switching between attractors in cell regulatory networks," Experimental Cell Research, vol.261, no.1, pp.91-103,2000.
[17]S. Huang, "Regulation of cellular states in mammalian cells from a genome wide view," Gene Regulation and Metabolism, pp.181-220,2002.
[18]I. Shmulevich, E. Dougherty, S. Kim, and W. Zhang, "Probabilistic Boolean networks:a rule-based uncertainty model for gene regulatory networks," Bioinformatics, vol.18, no.2, pp.261-274,2002.
[19]T. Akutsu, M. Hayashida, W. Ching, and M. Ng, "Control of Boolean net-works:hardness results and algorithms for tree structured networks.," Bioin-formatics, vol.244, no.4, pp.670-679,2007.
[20]杨萌,“含不确定性的布尔网络动力学分析及控制问题研究,”北京大学工学院博士毕业论文,2013.
[21]D. Cheng, H. Qi, and Z. Li, Analysis and Control of Boolean Networks-a Semi-tensor Product Approach. London:Springer,2011.
[22]F. Li, J. Sun, and Q. Wu, "Observability of Boolean control networks with state time delays," IEEE Transactions on Neural Networks, vol.22, no.6, pp.948-954,2011.
[23]F. Li and J. Sun, "Controllability of Boolean control networks with time delays in states," Automatica, vol.47, no.3, pp.603-607,2011.
[24]R. Alonso-Sanz and L. Bull, "Boolean networks with memory," International Journal of Bifurcation and Chaos, vol.18, no.12, pp.3799-3814,2008.
[25]A. Graudenzi, R. Serra, M. Villani, C. Damiani, A. Colacci, and S. Kauff-man, "Dynamical properties of a Boolean model of gene regulatory network with memory," Journal of Computational Biology, vol.18, no.10, pp.1291-1303,2011.
[26]S. Kim, H. Li, E. Dougherty, N. Cao, Y. Chen, M. Bittner, and E. Suh, "Can Markov chain models mimic biological regulation?," Journal of Biological Systems, vol.10, no.4, pp.337-357,2002.
[27]F. Li and J. Sun, "Stability and stabilization of multivalued logical network-s,"Nonlinear Analysis:RealWorld Applications, vol.12, no.6, pp.111-126, 2011.
[28]W. Ching, S. Zhang, M. Ng, and T. Akutsu, "An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks," BMC Bioinformatics, vol.23, no.12, pp.1511-1518,2007.
[29]R. Pal. A. Datta, and E. Dougherty, "Optimal infinite-horizon control for probabilistic Boolean networks," IEEE Transactions on Signal Processing, vol.54, no.6, pp.2375-2387,2006.
[30]R. Layek, A. Datta, R. Pal, and E. Dougherty, "Adaptive intervention in probabilistic Boolean networks," BMC Bioinformatics, vol.25, no.16, pp. 2042-2048,2009.
[31]D. Bryce, M. Verdicchio, and S. Kim, "Planning interventions in biological networks," ACM Transactions on Intelligent Systems and Technology, vol. 1, no.2, p.11,2010.
[32]P. Li, J. Lam. and Z. Shu, "On the transient and steady-state estimates of interval genetic regulatory networks," Systems. Man, and Cybernetics, Part B:Cybernetics, vol.40, no.2, pp.336-349,2010.
[33]D. CHENG, H. QI, and Y. ZHAO, "Analysis and Control of Boolean Net-works:A Semi-tensor Product Approach," ACT A AUTOMATICA SINICA, vol.37, no.5, pp.529-540,2011.
[34]H. Chen and J. Sun, "A new approach for global controllability of higher order Boolean control network," Neural Networks, vol.39, pp.2-17,2013.
[35]F. Li and J. Sun, "Controllability and optimal control of a temporal Boolean network," Neural Networks, vol.34, pp.10-17,2012.
[36]D. Cheng, H. Qi, and Z. Li, "Model construction of Boolean network via observed data," IEEE Transactions on Neural Networks, vol.22, no.4, pp. 525-536,2011.
[37]A. Garg, Ⅰ. Xenarios, L. Mendoza, and G. D. Micheli, "Model construction of Boolean network via observed data," Research in Computational Molecular Biology, pp.62-76,2007.
[38]A. Garg, A. D. Cara, I. Xenarios, L. Mendoza, and G. D. Micheli, "Syn-chronous versus asynchronous modeling of gene regulatory networks," BMC Bioinformatics, vol.24, no.17, pp.1917-1925,2008.
[39]T. Balch and R. C. Arkin, "Behavior-based formation control for multirobot team," IEEE Transactions on Robotics and Automation, vol.14, no.6, pp. 926-939,1998.
[40]J. Fredslund and M. J. Mataric, "A general local algorithm for robot forma-tions," IEEE Transaction on Robotics and Automation, vol.18, no.5, pp. 837-846,2002.
[41]R. W. Beard, J. Lawton, and F. Y. Hadaegh, "A coordination architecture for spacecraft formation control," IEEE Transaction on Control Systems Technology, vol.9, no.6, pp.777-790,2001.
[42]W. Ren and R. W. Beard, "Formation feedback control for multiple space-craft via virtual structures," IEEE Proc Control Theory and Applications, vol.15, no.3, pp.357-368,2004.
[43]F. Bauer, J. Bristow, and D. Folta, "Satellite formation flying using an innovative autonomous control system environment," Proc of the AIAAGNC Conf in New Orleans,1997.
[44]J. P. Desai, J. O. rowski, and V. Kumar, "Controlling formations of multiple mobile robots," IEEE Int Conf on Robotics and Automation, pp.2864-2869, 1998.
[45]H. Yamaguchi and J. W. Burdick, "A symptotic stabilization of multiple nonholonomic mobile robots forming group formations," IEEE Int Conf on Robotics and Automation, pp.3573-3580,1998.
[46]R. W. Hogg, A. L. Rankin, and S. I. Roumeliotis, "Algorithms and sensors for small robot path following," Proc 2002 IEEE Int Conf on Robotics and Autom.at.ion, pp.3850-3857,2002.
[47]G. A. S. Pereira, A. K. Das, and V. Kumar, "Formation control with config-uration space constraints," Proc of the IEEE or RJS Int Conf on Intelligent Robots and Systems, pp.2755-2760,2003.
[48]D. J. Naffin and S. Gaurav, " Negotiated formations," Int Conf on Intelligent Autonomous Systems, pp.181-190,2004.
[49]A. K. Das, R. Fierro, and V. Kumar, " A vision based formation control framework," IEEE Trans on Robotics and Automation, vol.18, no.5, pp. 813-825,2002.
[50]曹志强,“未知环境下多机器人协调与控制的队形问题研究,”北京:中国科学院自动化研究所,2002.
[51]C. W. Reynolds, "Flocks, herds, and schools:A distributed behavioral mod-el," Computer Graphics., vol.21, no.4, pp.25-34,1987.
[52]A. Jadbabaie, J. Lin, and A. S. Morse, "Coordination of groups of mobile autonomous agents using nearest neighbor rules," IEEE Trans on Automatic Control, vol.48, no.6, pp.988-1001,2003.
[53]H. G. Tanner, A. Jadbabaie, and J. George. "Pappas Coordination of mul-tiple autonomous vehicles." 11th Mediterranean Conf on Control and Au-tomation,2003.
[54]T. Balch and M. Hybinette, "Social potentials for scalable multi-robot for-mations," Proc IEEE Int Conf on Robotics and Automation, vol.1, pp. 73-80,2000.
[55]P. Kostelnik, M. Samulka, and M. Janosik, "Scalable multi-robot formations using local sensing and communication," Proc of the 3rd Int Workshop on Robot Motion and Control, pp.319-324,2002.
[56]J. Lawton, B. Young, and R. Beard, "Adecent ralized approach to elemen-tary formation maneuvers," IEEE Int Conf on Robotics and Automation, vol.3, pp.2728-2733,2000.
[57]董胜龙,陈卫东,席裕庚,“多移动机器人编队的分布式控制系统,”机器人,vol.22,no.6,pp.433-438,2000.
[58]M. A. Lewis and K. H. Tan, "High precision formation control of mobile robots using virtual structures autonomous," Autonomuous Robots, no.4, pp.387-403,1997.
[59]B. J. Young, R. W. Beard, and J. M. Kelsey, "A control scheme for improving multi-vehicle formation maneuvers," American Control Conf, pp.704-709, 2001.
[60]J. P. Desai, "A graph theoretic approach for modeling mobile robot team formations," J of Robotic Systems, vol.19, no.11, pp.511-525,2002.
[61]R. Fierro and A. K. Das, "A modular architecture for formation control," Proc of the 3rd Int Workshop on Robot Motion and Control, pp.285-290, 2002.
[62]R. Fierro and A. K. Das, "A modular architecture for formation control," Proc of the 3rd Int Workshop on Robot Motion and Control, pp.285-290, 2002.
[63]R. Olfati-Saber and R. W. Murray, "Distributed cooperative control of mul-tiple vehicle formations using structural potential functions," Proc of the 15th IFAC World Congress,2002.
[64]H. G. Tanner, G. J. Pappas, and V. Kumar, "Leader to formation stability," IEEE Trans on Robotics and Automation, vol.20, no.3, pp.443-455,2004.
[65]A. F. J and R. M. Murray, "Graph laplacians and stabilization of vehicle formations," Proc of 15th IFAC Conf,2002.
[66]J. A. Fax and R. M. Murray, "Information flow and cooperative control of vehicle formations," Proc of 15th IFAC Conf,2002.
[67]J. A. Fax and R. M. Murray, "Information flow and cooperative control of vehicle formations," IEEE Tran on Automatic Control, vol.49, no.9, pp. 1465-1476,2004.
[68]Y. Ketema and G. Balas, "Formation stability with limited information exchange between vehicles," Proc of the American Control Conf, vol.1, pp. 290-295,2003.
[69]Khatib, "Real time obstacle avoidance for maipulators and mobile robots," Int J of R obotics R esearch, vol.5, no.1, pp.90-98,1986.
[70]N. E. Leonard and E. Fiorelli, "Virtual leaders and artificial potentials and coordinated control of groups," Proc of the 40th IEEE Int Conf on Decision and Control, vol.3, pp.2968-2973,2001.
[71]P. Ogren, E. Fiorelli, and N. E. Leonard, "ormations with a mission:Stable coordination of vehicle group maneuvers," Proc of the 15th Int Symposium on Mathematical Theory of Networks and Systems,2002.
[72]J. S. Baras, X. Tan, and P. Hovareshti, "Decentralized control of autonomous vehicles," Proc of the 42nd IEEE Conf on Decision and Control, pp.1532-1537,2003.
[73]W. B. Dunbar, "Model predictive control:Extension to coordinated multi-vehicle formations and real-time implementation," Pasadena:California In-stitute of Technology,2001.
[74]苏治宝,陆际联,“基于行为法队形保持中的队形反馈,”机床与液压：,no.3,pp.167-169,2003.
[75]郭雷,“评“矩阵的半张量积：一个便捷的新工具”,”科学通报,vol.56,no.32,pp.2662-2663,2011.
[76]D. Cheng, An Introduction to Semi-tensor Product of Matrices and Its Ap-plications. Springer,2011.
[77]梅生伟,刘锋,薛安成,电力系统暂态分析中的半张量积方法.北京：清华大学出版社,2010.
[78]W. Hwang and D. Lee, "Cell signaling dynamics analysis in leukemia with switching Boolean networks," The Fourth International Conference on Com-putational Systems Biology, pp.168-175,2010.
[79]Z. Li, Y. Zhao, and D. Cheng, "Structure of higher order Boolean networks,' Joural of Graduate Univesity of Chinese Academy of Sciences, vol.28, no. 4, pp.431-447,2011.
[80]F. Li and J. Sun, "Controllability of higher order Boolean control networks," Applied Mathematics and Computation, vol.219, no.1, pp.158-169,2012.
[81]Y. Zhao and D. Cheng, "Controllability and stabilizability of probabilistic logical control networks," 2012 IEEE 51st Annual Conference on Decision and Control (CDC), pp.6729-6734,2012.
[82]D. Cheng, "Disturbance decoupling of Boolean control networks," IEEE Transactions on Automatic Control, vol.56, no.1, pp.1-10,2011.
[83]M. Yang, R. Li, and T. Chu, "Controller design for disturbance decoupling of Boolean control networks," Automatica, available online 24 November 2012.
[84]D. Cheng and H. Qi, "A linear representation of dynamics of Boolean net-works," IEEE Transactions on Automatic Control, vol.55, no.10, pp.2251-2258,2010.
[85]Z. Liu and Y. Wang, "Disturbance decoupling of mix-valued logical networks via the semi-tensor product method," Automatica. vol.48, no.8, pp.1839-1844,2012.
[86]D. CHENG and Y. GUO, "Advances on switched systeIns," Control Theory and Application, vol.22, no.6, pp.954-960,2005.
[87]T. OOBA and Y. FUNAHASH, "On a common quadratic Lyapunov function for widely distant system," IEEE Transaction on Automatic Control, vol. 42, no.12, pp.1697-1699,2003.
[88]L. VU and D. LIBERZON, "Common Lyapunov functionsfor afamilies of commuting nonlinear systems," Systems and Control Letter, vol.54, no.5, pp.405-416,2005.
[89]L. VU and D. LIBERZON, "Supervisory control of families of linear set-point controllers part 1:exact matchin," IEEE Transaction on Au-tomatic Control, vol.41, no.9, pp.1413-1431,1996.
[90]A. BALLUCHI, M. D. BENEDETFO, and C. PONELLO, "Hybrid control in automative applications:the cut-off control," Automatica, vol.35, no.3, pp.519-535,1999.
[91]Z. SUN, "Stabilization and insensitivity of switched linear system," IEEE Transaction on Automatic Control, vol.49, no.7, pp.1133-1137,2004.
[92]Y. MORI, T. MORI, and Y. KUROEY, "On a class oflinearconstant systems which have a common quadratic Lyapunov function," Proc of the 37th IEEE Conyon Decision and Control, vol.3, pp.2808-2809,1998.
[93]Y. MORI, T. MORI, and Y. KUROEY, "Some new subclasses of system-s having acommon quadratic Lyapunovfunctionand co mparison of known subclasse," Proc of the 40th IEEE Conyon Decision and Control, vol.3, pp. 2179-2180,2001.
[94]K. NARENDA and J. BALAKRISHNAN, "A common Lyapunov function for stable LTI systems with commuting A matrice," IEEE Transaction on Automatic Control, vol.39, no.12, pp.2469-2471,1994.
[95]L. Zhang, D. Cheng, and J. LIU, "StabifiTation oflinear switched systems," Asian J of Control, vol.5, no.4, pp.476-484,2003.
[96]L. Zhang, D. Cheng, and J. LIU, "On a switching control scheme for non-linear systems with ill-defined relativedegre," Systems and Control Letter, vol.47, no.2, pp.149-166,2002.
[97]A. JADBABAIE, J. LIN, and S. MORSE, "Coordination of groups of Ino-bile autonomous agents using nearest neighbor rules," IEEE Transaction on Automatic Control, vol.48, no.6, pp.988-1001,2003.
[98]程代展,陈翰馥,“从群集到社会行为控制,”科技导报,no.8,pp.4-7,2004.
D. LIBERZON, J. HESPANHA:and A. MORSE, "Stability of switched systems:a Lie-algebraic condition," Systems and Control Letter, vol.37, no.3, pp.117-122,1999.
[100]Z. SUN and S. S. GE, "Anlaysis and synhtesis of siwtched linear control systems;" Automatica, vol.41, no.2, pp.181-356,2005.
[101]Y. Zhao, H. Qi, and D. Cheng, "Input-state incidence matrix of Boolean control networks and its applications," Systems and Control Letters, vol.59, no.12, pp.767-774,2010.
[102]D. Cheng and Y. Zhao, "Identification of Boolean control networks," Auto-matica, vol.47, no.4, pp.702-710,2011.
[103]D. Cheng, Y. Zhao, and X. Xu, "Mix-valued logic and its applications," Journal of Shandong University (Natural Science), vol.46, no.10, pp.32-44,2011.
[104]T. M. Apostol, Mathematical Analysis. Second Edition. Addison-Wesley, 1974.
[105]J. Yao and J. Feng, "Comments on disturbance decoupling of Boolean con-trol networks," IEEE Transactions on Automatic Control, vol.56, no.12, pp.3001-3002,2011.
[106]A. K. Das, R. Fierro, V. Kumar, J. P. Ostrowski, and C. J. T. J. Spletzer, "A vision-based formation control framework," IEEE Transactions on Robotics and Automation, vol.18, no.5, pp.813-825,2002.
[107]X. Chen, A. Serrani, and H. Ozbay, "Control of leader-follower formations of terrestrial uavs," Proceedings of the 42nd IEEE Conference on Decision and Control, vol.1, pp.498-503,2003.
[108]D. J. Stilwell and B. E. Bishop, "Platoons of underwater vehicles," IEEE Control Systems Magzine, vol.20, no.6, pp.45-52,2000.
[109]D. P. Scharf, F. Y. Hadaegh, and S. R. Ploen, "A survey of space formation flying guidance and control (part 2)," in Proceedings of the American Control Conference, vol.4, (Boston, Massachusetts), pp.2976-2985,2004.
[110]J. G. Bender, "An overview of systems studies of automated highway system-s," IEEE Transactions on Vehicular Technology, vol.40, no.1, pp.82-99, 1991.
[111]J. A. Fax and R. M. Murray, "Information flow and cooperative control of vehicle formations," IEEE Transactions on Automatic Control, vol.49, no. 9, pp.1465-1476,2004.
[112]J. Rodrigues, D. figueira, C. Neves, and I. Ribeiro, "Leader-following graph-based distributed formation control," Proc. Robotica, pp.71-77,2008.
[113]N. Han, X. Luo, and X. Guan, "Distributed Hx formation control for discrete-time multiagent systems," IEEE Transactions on Automatic Con-trol, pp.4436-4441,2010.
[114]D. Cheng and Y. Zhao, "Normal form of general logic mappings," in Pro-ceedings of the 30th Chinese Control Conference, (Yantai, China), pp.6368-6373,2011.
[115]M. Egerstedt and X. Hu, "Formation constrained multi-agent control," Robotics and Automation, vol.17, no.6, pp.947-951,2011.
[116]Y. Zhao and D. Cheng, "On controllability and stabilizability of probabilistic Boolean control networks," Science China Information Sciences, pp.1-14, 2013.
[117]M. Aldana, "Boolean dynamics of networks with scale-free topology," Phys-ica D:Nonlinear Phenomena, vol.185, no.1, pp.45-66,2003.
[118]B. Drossel, T. Mihaljev, and F. Greil, "Number and length of attractors in a critical kauffman model with connectivity one," Physical Review Letters, vol.94, no.8, pp.88701-88704,2005.
[119]R. Albert and H. Othmer, "The topology and signature of the regulatory interactions predict the expression pattern of the segment polarity genes in drosophila melanogaster," Theoretical Biology, vol.223, no.1, pp.1-18, 2003.
[120]R. Pal, A. Datta, M. L. Bittner, and E. R. Dougherty, "Optimal Infinite-horizon control for probabilistic Boolean networks," IEEE Trans. Signal Process, vol.54, pp.2375-2387,2006.
[121]T. Akutsu, M. Hayashida, and M. N. W. Ching, "Control of Boolean net-works:hardness results and algorithms for tree structured networks." The-oretical Biology, vol.244, no.4, pp.670-679,2007.
[122]F. Li and J. Sun, "Observability analysis of Boolean control networks with impulsive effects," IET Control Theory Application, vol.5, no.14, pp.1609-1616,2011.
[123]D. Cheng, "Semi-tensor product of matrices and its applications-a survey, in:Proc.4th international congress of chinese mathematicians," (Hangzhou), pp.641-668,4th International Congress of Chinese Mathe-maticians, Higher Edu. Press,2007.
[124]K. D. Do and J. Pan. "Nonlinear formation control of unicycle-type mobile robots," Robotics and Autonomous Systems, vol.55, no.3, pp.191-204, 2007.
[125]H. D. Jong, "Modeling and simulation of genetic regulatory systems:a literature review," Journal of computational biology, vol.9, no.1, pp.67-103,2010.
[126]P. K. C. Wang, "Navigation strategies for multiple autonomous mobile robots moving in formation," J of Robotic Systems,, vol.8, no.2, pp.177-195,1991.
[127]P. Pirjanian, "Behavior coordination mechanisms state of the art," LA:U-niversily of Southern California,1999.
[128]韩学东,洪炳熔,孟伟,“多机器人任意队形分布式控制研究,”机器人,vol.25,no.1,pp.66-72,2003.