摘要
集中供热系统是城市重要的基础设施,因其具有节能、环保和供热质量好等优点,在“节能减排”的大环境下将得到更广泛的应用。供热管网作为集中供热系统中的重要部分,具有规模大、结构复杂、投资巨大以及社会影响重大等特点,因此倍受关注。目前,供热管网水力工况计算、运行调度以及可靠性分析等几个方面的研究较多,而供热管网阻力特性辨识方面的研究较少。如果无法准确获得供热管网实际运行条件下的阻力系数,供热管网水力工况计算以及运行调度等研究大都只能停留在理论计算模拟与定性分析上。计算与分析的结果往往与实际运行情况不符,从而大大降低研究成果的实用价值。目前在为数不多的供热管网阻力系数辨识研究中,对供热管网辨识模型的研究还比较少,并且辨识方法单一,辨识准确程度不高、辨识计算耗时较大。因此,对供热管网阻力系数辨识问题开展深入研究有着重要的理论意义和实用价值。
本文针对供热管网阻力系数辨识问题进行了较为深入的研究,提出了一些行之有效的辨识研究方法。首先利用矩阵方程形式表达供热管网阻力系数辨识问题,并结合系统能观测性理论,指出供热管网节点压力均可观测是满足供热管网阻力系数辨识能观测性的必要条件。在此基础上提出多水力工况条件下满足供热管网阻力系数辨识能观测性的充分条件以及该条件下的辨识求解方法,为供热管网阻力系数辨识的理论研究及实验测试提供了一种新的研究途径。
考虑现有供热管网的实际观测条件一般不能满足辨识能观测性的要求,本文结合线性方程组的解分析理论和广义逆矩阵理论,提出基于广义逆矩阵理论辨识供热管网阻力系数的方法。该方法可以应用于一般观测条件下的供热管网阻力系数辨识问题的研究,其中利用逐步线性化的迭代方法求解辨识方程组的广义逆解,并以方程组解中的阻力系数值作为辨识结果,逼近供热管网阻力系数的实际值。而后,将基于广义逆矩阵理论辨识供热管网阻力系数的方法应用于辨识算例。
本文提出在可能获取更多的观测数据的条件下,可以利用部分管段(一般为一组链支管段)阻力系数初始值作为辨识的已知条件,辨识得到供热管网阻力系数。考虑到利用空间供热管网模型可以更直接地反映供热管网实际情况,利用分块矩阵表示空间供热管网阻力系数辨识问题,在求解辨识问题的同时结合空间供热管网的特点提出适用于空间供热管网阻力系数辨识的链支管段选择方案。
结合聚类分析理论提出了基于反映供热管网阻力系数变化的管段流量和节点压力观测点优化布置方法。基于矩阵论中的矩阵对矩阵微分运算法则以及供热管网阻力系数变化对管段流量和节点压力影响的线性分析结果,在上述研究基础上,提出利用管段流量和节点压力观测点相对灵敏度研究观测点优化布置问题,并且根据供热管网设计值计算相对灵敏度。
最后,利用供热系统多功能试验台及某集中供热管网观测数据进行供热管网阻力系数模拟辨识研究,说明不同供热管网观测条件下的阻力系数辨识方法的实现过程,表明本文所提出的辨识方法具有较高的辨识计算效率和较好的实际工程应用前景。
本文基于系统能观测性理论指出多水力工况条件下供热管网阻力系数辨识满足能观测性条件的可能性,在满足辨识能观测性的条件下可以直接求解阻力系数辨识问题。考虑到实际供热管网中压力和流量观测点的布置情况,提出基于广以逆矩阵理论辨识供热管网阻力系数的方法,以辨识方程组的广义逆解作为供热管网阻力系数辨识结果,该结果与实际值一般比较接近。在具备更加完善观测条件的情况下,可以利用基于一组链支管段阻力系数值的辨识方法得到更加接近实际值的供热管网阻力系数辨识结果。在上述研究基础上,可以依据观测点相对灵敏度选择合适的观测数据进行供热管网阻力系数辨识。本文的研究成果为进一步研究供热管网阻力系数辨识问题及实际工程应用奠定了理论基础,具有一定前导性。
Centralized heating systems have become important modern municipal infrastructures and will be applied widely when we focus on energy saving and emission reduction for those energy saving, environmental protection and good heating quality. Heat-supply network is an important part of centralized heating system, whose further study has drawn great attention because of its large scale, complicated structure, enormous investment and significant social impact. Nowadays the study of heat-supply networks focuses on hydraulic loading condition computation, reliability analysis, operation and regulation, but there is less study on pipe friction parameters identification of heat-supply networks. However, if actual pipe friction parameters can not be identified, the study mentioned above would just be theoretic simulation or qualitative analysis. In real-world engineering, theoretic simulation results can not always accord with practical operation conditions, so the application value could be reduced seriously. In the study of pipe friction parameters identification, there are a few identification models and identification methods of heat-supply networks pipe friction parameters. Identification accuracy is not high, and consuming time of identification computation is great. These deep reserarches of pipe friction parameters identification have theoretical significance and practical value.
This thesis focuses on pipe friction parameters identification of heat-supply networks, and proposes some effective identification methods. Pipe friction parameters identification is expressed as matrix equations. Combined with system observability theory, it is proposed that a necessary condition for observability of pipe friction parameters which is all the nodes pressure heads values are available. Then, a sufficient condition for observability of pipe friction parameters is proposed under multiple hydraulic loading conditions, which provides a new approach to theoretic analysis and field measurement of pipe friction parameters.
Considering the observability conditions of pipe friction parameters identification can not be satisfied under common measured conditions, based on solution analysis and pseudo-inverse matrix theories of linear equations, an identification method based on pseudo-inverse matrix theory is proposed. This identification method can be applied under the common measured conditions. This thesis presents a progressive linearization iteration methodology for solving the pseudo-inverse solution of the identification equations, and considers the pipe friction parameters values in the pseudo-inverse solution as the identification results which are approximate to real pipe friction parameters values. Then, this identification method is applied in a case study.
Furthermore, if more known operational parameters of heat-supply networks can be obtained, a method by using the initial values of some pipe friction parameters (a group of link pipe friction parameters generally) can be used in pipe friction parameters identification. Considering spacial heat-supply network models can represent heat-supply networks more accurately, we presents pipe friction parameters identification problem of spacial heat-supply networks by the form of partitioned matrixes. In the identification process, combined with characteristics of spacial heat-supply networks, an option method of link pipes which is suitable for spacial heat-supply networks is proposed.
Combined with clustering theory, it is proposed that an optimal arrangement method of pipe flow measured points and node pressure head measured points based on reflecting the change of pipe friction parameters. By using matrix differential calculation method, the linear results of pipe flows and node pressure heads reflected by the change of pipe friction parameters can be studied. Then, measured points optimal arrangement is studied by pipe flows and node pressure heads relative sensitivity which can be expressed by design values of heat-supply networks.
Finally, the pipe friction parameters identification methods are applied on a multi-function experiment platform and a centralized heating system. By using field measured values, the identification processes are introduced under different field measured conditions, which can show the identification method has higher efficiency and better prospect of real-world engineering.
It is proposed that pipe friction parameters identification of heat-supply networks may satisfy observability conditions under multiple hydraulic loading conditions based on system observability theory. When pipe friction parameters observability conditions can be satisfied, a new approach for solving pipe friction parameters identification directly is proposed. Considering the real-world nodes pressure heads and pipes flows measured conditions, an identification method based on pseudo-inverse matrix theory is proposed, which can be used in real-world pipe friction parameters identification in heat-supply networks. We consider the pipe friction parameters values in the pseudo-inverse solution as the identification results which are approximate to real pipe friction parameters values. When more known operational parameters of heat-supply networks can be obtained, we could calculate pipe friction parameters which are more closed to their real values by using a group of link pipe friction parameters initial values. On this basis, some measured values with high relative sensitivity can be used. The research results of the thesis which can be used in future study are theoretical foundation of heat-supply network pipe friction parameters identification and real-world application.
引文
[1] Olsson L C. Local District Heating Systems [J]. Doktorsavhandlingar vid Chalmers Tekniska Hogskola, 2001, 35(1727):173-177.
[2]曾享麟,蔡启林,解鲁生,等.欧洲集中供热的发展[J].区域供热, 2002 (1):1-8.
[3]邹平华.借鉴俄罗斯经验积极发展我国集中供热事业[J].暖通空调, 2000, 30(4):33-37.
[4]李娥飞.俄罗斯暖通情况简介[J].暖通空调, 2005, 35:297-299.
[5]辛坦.欧洲发达国家供热改革经验[J].建设科技, 2008, 23:40-41.
[6] Gabrielaitiene I, Bohm B, Sunden B. Modelling Temperature Dynamics of a District Heating System in Naestved, Denmark—A Case Study [J]. Energy Conversion and Management, 2007, (48):78-86.
[7] Karin S. District Heat Supply in Germany in the Year 2001 [J]. Euroheat and Power, 2003, 32(2):28-35.
[8]王士军,李德英,阎全英.法国城市集中供热简介[J].区域供热, 2002, (3):29-31.
[9]冯慧明.热力站供热系统自控策略的探究[J].区域供热, 2010, (2):18-20.
[10]王建军,于黎明,刘甲锟,等.多热源热网自动化控制系统研究[J].区域供热, 2010, (1):15-18.
[11]供热术语标准[S]. CJJ55-93.北京:中国计划出版社, 1994:15-20.
[12] Fuchs H, Frommhold W, Poggemann R. Acoustic Leak Detection on District Heating Pipelines [J]. Technisches Messen, 1991, 58(2):47-60.
[13] Li X M, Visier J C, Vuezr N H. A Neural Network Prototype for Fault Detection and Diagnosis of Heating Systems [J]. ASHPRAE Trans., 1997, 103(1):634-644.
[14] Li X M, Vuezr N H, Visier J C. Development of a Fault Diagnosis Method for Heating Systems Using Neural Networks [J]. ASHRAE Trans., 1996, 102(1):607-614.
[15]周志刚.供热管网阻力特性的辨识研究[D].哈尔滨:哈尔滨工业大学博士学位论文, 2006:31-39.
[16]韩伟国,江亿,郭非.多种供热供暖方式的能耗分析[J].暖通空调, 2005, 35(11):106-110.
[17]夏天昌.系统辨识—最小二乘法[M].北京:国防工业出版社,1984:21-54.
[18] Astrom K, Eykhoff P. System Identification-A survey [J]. Automatica, 1971, 7:123-167.
[19] Goodwin G C, Payne R L. Dynamic System Identification: Experiments Design and Data Analysis [M]. Amsterdam:Academic Press, 1977:2-5.
[20] Ljung L. System Identificatin-Theory for the User [M]. New Jersey:Prentice Hall, Inc., 1987:1-11.
[21]方崇德,萧德云.过程辨识[M].北京:清华大学出版社, 1988:133-156.
[22]丁峰,肖德云.多变量系统状态空间模型的递阶辨识[J].控制与决策, 2005, 20(8):848-853.
[23]袁平.多变量系统辨识方法比较研究[D].无锡:江南大学硕士学位论文, 2008:9-12.
[24]程兆林,黄民懿,马树萍.线性系统的状态最小二乘估计[C]//山东济南:中国控制与决策学术年会论文集, 1996:289-293.
[25] Holland J H. Concerning Efficient Adaptive Systems [J]. Self-Organizing Systems, 1962:215-230.
[26] Varol Y, Avci E, Koca A, et al. Prediction of Flow Fields and Temperature Distributions due to Natural Convection in a Triangular Enclosure using Adaptive-Network-Based Fuzzy Inference System (ANFIS) and Artificial Neural Network (ANN) [J]. International Communications in Heat and Mass Transfer, 2007, 34(7):887-896.
[27] Bremermann H J. Optimization through Evolution and Recombination in Self-Organizing Systems [M]. New York:Spartan Books, 1962:93-106.
[28] Fogel L J, Owens A J, Walsh M J. Artificial Intelligence through Simulated Evolution [M]. New York:John Wiley, 1966:42-45.
[29] Fraser A S. Simulation of Genetic System by Automatic Digital Computers. Introduction [J]. Australian Journal of Biology, 1957, 10:484-491.
[30] Barricelli N A. Symbiogenetic Evolution Processes Realized by Artificial Methods [J]. Methodos, 1957, 9:143-182.
[31]王小平,曹立明.遗传算法—理论、应用与软件实现[M].西安:西安交通大学出版社, 2002:1-16.
[32] Bagley J D. The Behavior of Adaptive System which Employ Genetic and Coorelation Algorithm [D]. Michigan : Dissertation for the Doctoral Degree in the University of Michigan, 1967:68-76.
[33] Hollstien R B. Artifical Genetic Adaption in Computer Control Systems [D]. Michigan: Dissertation for the Doctoral Degree in the University of Michigan, 1971:71.
[34]陈国良,王熙法,庄镇泉.遗传算法及其应用[M].北京:人民邮电出版社, 1996:1-3.
[35]维纳N.控制论(或关于在动物和机器中控制和通信的科学)[M].郝季仁,译.北京:北京大学出版社, 2007:9-19.
[36]钱学森,宋健.工程控制论[M].修订版.北京:科学出版社, 1983:125-130.
[37]曹慧哲.管网慢变流水力计算及水击波动过程优化控制研究[D].哈尔滨工业大学博士学位论文, 2008:5-23, 24-32.
[38] Simpson A, Elhay S. Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model [J]. Journal of Hydraulic Engineering, 2011, 137(6):696-700.
[39] Pudar R S, Liggett J A.Leaks in Pipe Networks [J]. Journal of Hydraulic Engineering, 1992, 118(7):1031-1046.
[40] Liggett J A, Chen L C. Inverse Transient Analysis in Pipe Network [J]. Journal of Hydraulic Engineering, 1994, 120(8):934-955.
[41] Vitkovsky J P, Simpson A R, Lambert M F. Leak Detection and Calibration Using Transients and Genetic Algorithms [J]. Journal of Water Ressources Planning and Management, 2000, 126(4):262-265.
[42] Axworthy D H, Karney B W. Valve Closure in Graph-Theoretical Models for Slow Transient Network Analysis [J]. Journal of Hydraulic Engineering, 2000, 126(4):304-309.
[43] Nash G A, Karney B W. Efficient Inverse Transient Analysis in Series Pipe Systems [J]. Journal of Hydraulic Engineering, 1999, 125(7):761-764.
[44] Walski T M. Case Study: Pipe Network Model Calibration Issues [J]. Journal of Water Resources Planning and Management, 1986, 112(2):238-249.
[45] Walski T M. Technique for Calibration Network Models [J]. Journal of Water Resources Planning and Management, 1983, 109(4):360-372.
[46] Bhave P R. Calibration Water Distribution Network Models [J]. Journal of Environmental Engineering, 1988, 114(1):120-136.
[47] Ormsbee L E, Wood D J. Explicit Pipe Network Calibration [J]. Journal of Water Resources Planning and Management, 1986, 112(2):166-182.
[48] Boulos P F, Wood D J. Explicit Calibration of Pipe-Network Parameters [J]. Journal of Hydraulic Engineering, 1990, 116(11):1329-1344.
[49] Ormsbee L E. Implicit Network Calibration [J]. Journal of Water Resources Planning and Management, 1989, 115(2):243-257.
[50] Rahal C M, Sterling M H, Coulbeck B. Parameter Tuning for Simulation Models of Water Distribution Networks [J]. Institution of Civil Engineers, 1980, 69(2):751-762.
[51] Savic D A, Walters G A. Genetic Algorithm Techniques for Calibrating Network Models [M]. Centre for Systems and Control Engineering. Devon, UK:University of Exeter, 1995:203-212.
[52] Lansey K, Basnet C. Parameter Estimation for Water Distribution Networks [J]. Journal of Water Ressources Planning and Management, 1991, 117(1):126-144.
[53] Lansey K, Shorbagy W E, Ahmed I, et al. Calibration Assessment and Data Collection for Water Distribution Networks [J]. Journal of Hydraulic Engineering, 2001, 127(4):270-279.
[54] Boulos P F, Ormsbee L E. Explicit Network Calculation for Multiple Loading Conditions [J]. Journal of Civil Engineering Systems, 1991, 8(3):153-160.
[55] Ferreri G B, Napoli E, Tumbiolo A. Calibration of Roughness in Water Distribution Networks [C]//Proceeding of 2nd International Conference on Water Pipeline Systems. Edinburgh, UK:BHRA Group, 1994:379-396.
[56] Datta R S, Sridharan K. Parameter Estimation in Water-Distribution Systems by Least Squares [J]. Journal of Water Ressources Planning and Management, 1994, 120(4):405-422.
[57] Reddy P V, Sridharan K, Rao P V. WLS Method for Parameter Estimation in Water Distribution Networks [J]. Journal of Water Ressources Planningand Management, 1996, 122(3):157-164.
[58] Bush C, Uber J. Sampling Design Methods for Water Distribution Model Calculation [J]. Journal of Water Ressources Planning and Management, 1998, 124(6):334-344.
[59] Schaetzen W B, Walters G A, Savic D A. Optimal Sampling Design for Model Calibration Using Shortest Path Genetic and Entropy Algorithms [J]. Urban Water, 2000, 2(2):141-152.
[60] Kapelan Z S, Savic D A, Walters G A. Multiobjective Sampling Design for Water Distribution Model Calibration [J]. Journal of Water Ressources Planning and Management, 2003, 129(6):466-479.
[61] Kapelan Z S, Walters G A, Savic D A. A Hybrid Inverse Transient Model for Leakage Detection and Roughness Calibration in Pipe Networks [J]. Journal of Hydraulic Engineering, 2003, 41(5):481-492.
[62] Vitkovsky J P, Simpson A R, Lambert M F. Leak Detection and Calibration of Water Distribution Systems Using Transients and Genetic Algorithms [C]// Proceedings of 29th Annual Water Resources Planning and Management Conference. Arizona, USA:ASCE, 1999:1-9.
[63] Ainola L, Koppe T, Tiiter K. Water Network Model Calibration Based on Grouping Pipes with Similar Leakage and Roughness Estimates [C]// Proceedings of Joint Conference on Water Resources Engineering and Water Resources Planning & Management. Minneapolis:ASCE, 2000:50-56.
[64] Lingireddy S, Lindell E. Optimal Network Calibration Model Based on Genetic Algorithms [C]//Proceedings of 29th Annual Water Resources Planning and Management Conference. USA:ASCE, 1999:1-8.
[65] Kapelan Z S, Savic D A, Walters G A. Calibration of Water Distribution Hydraulic Models Using a Bayesian-Type Procedure [J]. Journal of Hydraulic Engineering, 2007, 133(8):927-936.
[66] Xu C C, Tickle K S. Parameter Dimension Estimation for Water Distribution Networks [C]//The 27th Congress of the International Association for Hydraulic Research. San Francisco, California:ASCE, 1997:435-440.
[67]信昆仑,程声通,刘遂庆.实数型编码遗传算法校核管道摩阻系数[J].中国给水排水, 2004, 20(9):68-70.
[68]段焕丰,俞国平.改进遗传算法校核管道摩阻系数[J].管道技术与设备, 2005, 3:14-16.
[69] Ormsbee S. Lingireddy S. Calibrating Hydraulic Network Models [J]. Journal of America Water Works Association, 1997, 89(2):42-50.
[70] Ormsbee L E. Implicit Network Calibration [J]. Journal of Water Ressources Planning and Management, 1989, 115(2):243-257.
[71]路琦.给水管网模式复核[J].中国给水排水, 1989, 1:21-23.
[72]毕美华,肖立川,茅建波.网络法用于系统仿真及其阻力参数的确定[J].石油化工高等学校学报, 2000, 13(1):68-72.
[73]袁一星,张志军.供水管网校核模型参数估计与求解方法的研究[J].给水排水, 2005, 31(9):105-111.
[74]于景洋,沈致和.基于偏最小二乘回归法的管网测压点压力宏观模型[J].安徽建筑工业学院学报, 2006, 14(4):31-34.
[75]陈宇辉.给水管网动态模型维护与校验方法研究[D].上海:同济大学博士学位论文. 2006:154-155.
[76]张洪国,袁一星,赵洪宾.给水管网动态模型中管道阻力系数的组合灰色推定方法[J].哈尔滨建筑大学学报, 1998, 31(5):59-63, 58.
[77]梁永图.采用辨识方法确定管道沿程摩阻系数[J].油气储运, 2006, 25(9):28-63.
[78]才建,李晓萍,宫敬,等.含蜡原油管道摩阻系数的辨识计算[J].石油化工高等学校学报, 2007, 20(2):50-53.
[79]吕谋,张士乔,李卫红.配水管网测压点的动态组合预测方法[J].系统工程理论与实践, 2003, 3(4):139-144.
[80]陈森发,徐南荣,仲伟俊.确定城市给水管网测点的决策方法[J].系统工程理论与实践, 1988, 18(9):1.
[81]王训俭,王增义.论给水管网压力测点的选择[J].中国给水排水, 1989,5(3):9-12.
[82]黄廷林,丛海兵.给水管网测压点优化布置的模糊聚类法[J].中国给水排水, 2001,17:50-52.
[83]丛海兵,黄廷林.测压点优化布置及状态估计在西安市给水管网中的应用[J].西安建筑科技大学学报, 2003, 35(1):40-43.
[84]张宏伟,张丽,梁建文.给水管网压力测点的布置方法[J].中国给水排水, 2003, 9(3):52-55.
[85]郭思元,刘遂庆,陈嵘.给水管网压力测点的优化布置[J].中国给水排水, 2004, 12:82-84.
[86]谢春利.给水系统的测点配置与状态估计[D].成都:四川大学硕士学位论文, 2001:7-10.
[87]周敏.西宁市给水管网测压点优化布置及状态估计[D].西安:西安建筑科技大学硕士学位论文, 2004:3.
[88] Pasha M F. Uncertainty Analysis and Calibration of Water Distribution Quality Models [D]. Arizona:Dissertation for the Doctoral Degree in the University of Arizona, 2006:86-91.
[89] Bush C A, Uber J G. Sampling Design Methods for Water Distribution Model Calibration [J]. Journal of Water Resource Planning and Management, 1998, 124(6):334-344.
[90] Vitkovsky J P, Liggett J A, Simpson A R, et al. Optimal Measurement Site Locations for Inverse Transient Analysis in Pipe Networks [J]. Journal of Water Resource Planning and Management, 2003, 126(6):480-492.
[91] Jiang Y, Qin X Z. Hydraulic Process Fault Diagnosis and Parameter Identification in District Heating Networks [J]. ASHRAE Transactions, 2000, 106(2):284-291.
[92]秦绪忠,江亿.区域热网管网阻力系数的在线辨识与故障诊断[J].清华大学学报, 2000, 40(2):81-85.
[93]刘笑驰,蔡瑞忠,吕崇德.大型蒸汽管网的在线仿真研究[J].系统仿真学报, 2002, 14(3):397-402.
[94]周志刚,邹平华,谈和平.基于模糊聚类分析的热网流量测点优化布置[J].深圳大学学报(理工版), 2008, 25(4):392-396.
[95] Vitkovsky J P, Simpson A R. Calibration and Leak Detection in Pipe Networks Using Inverse Transient Analysis and Genetic Algorithms [R]. Adelaide:Research Report of the University of Adelaide, 1997:155-157.
[96]石兆玉.流体网络的分析与综合[R].北京:清华大学热能工程系, 1993:16-22.
[97] Butman S, Sivan R. On Cancellations, Controllability and Observability [J]. IEEE Transactions Automatic Control, 1964:317-328.
[98] Ogata K. Modern Control Engineering [M]. New Jersey : PearsonEducation, Inc., 2002:1-6.
[99]陈哲.现代控制理论基础[M].北京:冶金工业出版社, 1987:3-11.
[100]王晓霞.多热源环状热水管网故障工况及可靠性研究[D].哈尔滨:哈尔滨工业大学博士学位论文, 2004:21-46.
[101] Wang X X, Liu M J, Zou P H. Effect of Failure Scheduling Scheme to Reliability of Complicated Heat-supply Network [C]//The 2nd International Conference on Built Environment and Public Health. Shantou, China:China Environmental Science Press, 2004:455-461.
[102] Wang X X, Liu M J, Zou P H. Reliability Assessment Method of Complicated Heat-supply Network Based on Failure Effect Analysis [C]//The 5th International Decentralized Energy and Combined Heat & Power Conference. Shantou, China:China Environmental Science Press, 2004:436-443.
[103]郑宝东.线性代数与空间解析几何[M].北京:高等教育出版社, 2003:65-81.
[104]李剑,李细林,苏泳涛,等.矩阵分析与应用习题解答[M].北京:清华大学出版社, 2007:1-23.
[105]程云鹏,张凯院,徐仲.矩阵论[M].第三版.西安:西北工业大学出版社, 2010:308-345.
[106]董增福.矩阵分析[M].第二版.哈尔滨:哈尔滨工业大学出版社, 2005:201-226.
[107] Sankaran S G, and Beex A A. Convergence Behavior of Affine Projection Algorithms [J]. IEEE Trans Signal Processing, 2000, 48(4):1086-1096.
[108] Mbekhta M, and Suciu L. Generalized Inverses and Similarity to Patial Isometries [J]. Journal of Mathematical Analysis and Applications, 2011, Inpress, Accepted Manuscript.
[109] Falco D D, Pennestri E, Vita L. Investigation of the Influence of Pseudoinverse Matrix Calculations on Multibody Dynamics Simulations by Means of the Udwadia-Kalaba Formulation [J]. Journal of Aerospace Engineering, 2009, 22(4):365-372.
[110] Hunter J J. Generalized Inverses and their Application to Applied Probability Problems [J]. Linear Algebra and its Applications, 1982, 45(6):157-198.
[111]贺平,孙刚.供热工程[M].第三版.北京:中国建筑工业出版社, 1993:156-175.
[112]陈华伟钟化兰,靳蕃.基于双向权值调整算法的神经网络非线性系统辨识[J].铁道学报, 2007, 29(5):48-53.
[113]徐宁寿.系统参数估计的递推最小范数解及其用于递推最小二乘算法的启动[J].北京工业大学学报, 1991, 17(3):15-23.
[114] Zhang X D. Matrix Analysis and Applications [M]. Beijing:Tsinghua University Press, 2004:85-95.
[115]吴华安.矩阵变量的矩阵值函数的导数[J].武汉理工大学学报, 2004, 28(3):410-412.
[116]江亿.管网可调性和稳定性的定量分析[J].暖通空调, 1997, 27(3):1-7.
[117]刘丁酉.矩阵分析[M].武汉:武汉大学出版社, 2003:194-200.
[118]刘孟军.热网变量的随机分析及其随机优化设计研究[D].哈尔滨:哈尔滨工业大学博士学位论文, 2006:34-63.
[119]何清.模糊聚类分析理论与应用研究进展[J].模糊系统与数学, 1998, 12(2):89-94.
[120]杨小兵.聚类分析中若干关键技术的研究[D].杭州:浙江大学博士学位论文, 2005:51.
[121]李红涛.基于GIS的城市公交网络服务水平聚类分析[D].西安:西安建筑科技大学硕士学位论文, 2005:50-51.
[122]刘洪.面向供电质量提高的城市电网专项规划研究[D].天津:天津大学博士学位论文, 2009:19-21.
[123] Qian W N, Zhou A Y. Analyzing Popular Clustering Algorithms from Different Viewpoints [J]. Journal of Software, 2002, 8:1382-1394.
[124] Zhang R. Research on the Data Clustering Techniques [J]. Computer Engineering and Applications, 2002, 16:145-170.
[125]黄永锋,刘同明.聚集式聚类分析方法及其应用[J].华东船舶工业学院学报(自然科学版), 2002, 4:33-37.
[126]高新波.模糊聚类分析及其应用[M].西安:西安电子科技大学出版社, 2004, 12(2):37-42.
[127]董吉文,周劲,杨秀丽.兼容值贴近和形贴近的新统计量的研究[J].微电子学与计算机, 2005, 22(5):192-194.
[128] Spiegel M R, Stephens L J. Theory and Problems of Statistics [M]. TheThird Edition. Boston:McGraw-Hill Companies, Inc., 1999:45-49.
[129]盛骤,谢式千,潘承毅.概率论与数理统计[M].第四版.北京:高等教育出版社, 2009:88-113.