基于椭球不确定性的平差模型与算法
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摘要
测量平差模型中的参数通常存在一些不确定的附加信息或先验信息,充分利用它们可以对部分参数进行约束,从而保证参数解的唯一性和稳定性。本文利用椭球集合描述不确定性,建立了一个新的带有椭球不确定性的平差模型。以两个椭球交集的外接椭球的特征矩阵的迹最小平差准则,分析了不确定度的传播规律,给出了带有椭球不确定性的平差方法。最后,通过算例验证了算法的有效性,说明了平差解与带权混合估计的关系。
In surveying adjustment models, there usually is some uncertain additional information or prior information on parameters, which can constraint on the parameters, and guarantee uniqueness and stability of parameters solution.In this paper, ellipsoidal sets are used to describe uncertainty, so an adjustment model with ellipsoidal uncertainty is established. The minimization in matrix trace of circumscribed ellipsoid with two ellipsoid intersections is regarded as a proposed adjustment criterion, the propagation law of uncertainty is analyzed, and the adjustment method with ellipsoid uncertainty is given. Finally, a numerical example is given to test and verify the effectiveness of the proposed algorithm, and the relation between the adjustment result and the weighted mixed estimation is illustrated.
In surveying adjustment models, there usually is some uncertain additional information or prior information on parameters, which can constraint on the parameters, and guarantee uniqueness and stability of parameters solution.In this paper, ellipsoidal sets are used to describe uncertainty, so an adjustment model with ellipsoidal uncertainty is established. The minimization in matrix trace of circumscribed ellipsoid with two ellipsoid intersections is regarded as a proposed adjustment criterion, the propagation law of uncertainty is analyzed, and the adjustment method with ellipsoid uncertainty is given. Finally, a numerical example is given to test and verify the effectiveness of the proposed algorithm, and the relation between the adjustment result and the weighted mixed estimation is illustrated.
引文
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[24]孙先仿,范跃祖,宁文如.有界误差模型的一种结构辨识方法[J].自动化学报,1999,25(2):242-246.SUN Xianfang,FAN Yuezu,NING Wenru.A structure identification method for bounded-error models[J].ActaAutomatica Sinica,1999,25(2):242-246.
[25]杨婷,杨虎.椭球约束与广义岭型估计[J].应用概率统计,2003,19(3):232-236.YANG Ting,YANG Hu.Ellipsoidal restriction and generalized ridge estimation[J].Chinese Journal of Applied Probability and Statistics,2003,19(3):232-236.
[26]DURIEU C,WALTER,POLYAK B.Multi-input multioutput ellipsoidal state bounding[J].Journal of Optimization Theory and Applications,2001,111(2):273-303.
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[2]宋迎春,谢雪梅,陈晓林.不确定性平差模型的平差准则与解算方法[J].测绘学报,2015,44(2):135-141.DOI:10.11947/j.AGCS.2015.20130213.SONG Yingchun,XIE Xuemei,CHEN Xiaolin.Adjustment criterion and algorithm in adjustment model with uncertain[J].Acta Geodaetica et Cartographica Sinica,2015,44(2):135-141.DOI:10.11947/j.AGCS.2015.20130213.
[3]王志忠,陈丹华,宋迎春.具有不确定性平差算法[J].测绘学报,2017,46(7):334-340.DOI:10.11947/j.AGCS.2017.20160522.WANG Zhizhong,CHEN Danhua,SONG Yingchun.An algorithm in adjustment model with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2017,46(7):334-340.DOI:10.11947/j.AGCS.2017.20160522.
[4]ELDAR Y C,BECK A,TEBOULLE M.A minimax Chebyshev estimator for bounded error estimation[J].IEEE Transactions on Signal Processing,2008,56(4):1388-1397.
[5]GARULLI A,VICINO A,ZAPPA G.Optimal induced-norm and set membership state smoothing and filtering for linear systems with bounded disturbances[J].Automatica,1999,35(5):767-776.
[6]GARULLI A,VICINO A,ZAPPA G.Conditional central algorithms for worst case set-membership identification and filtering[J].IEEE Transactions on Automatic Control,2000,45(1):14-23.
[7]BAI Erwei,FU Minyue,TEMPO R,et al.Convergence results of the analytic center estimator[J].IEEE Transactions on Automatic Control,2000,45(3):569-572.
[8]梁礼明,钟敏.集员辨识理论发展及算法综述[J].自动化技术与应用,2007,26(11):7-9,18.LIANG Liming,ZHONG Min.A survey of the set membership identification[J].Techniques of Automation and Applications,2007,26(11):7-9,18.
[9]ALAMO T,BRAVO J M,REDONDO M J,et al.A setmembership state estimation algorithm based on DCprogramming[J].Automatica,2008,44(1):216-224.
[10]BRAVO J M,ALAMO T,REDONDO M J,et al.An algorithm for bounded-error identification of nonlinear systems based on DC functions[J].Automatica,2008,44(2):437-444.
[11]WALTER,KIEFFER M.Guaranteed nonlinear parameter estimation in knowledge-based models[J].Journal of Computational and Applied Mathematics,2007,199(2):277-285.
[12]OTANEZ P G,CAMPBELL M E.Bounded switched linear estimator for smooth nonlinear systems[J].IEEE Transactions on Control Systems Technology,2007,15(2):358-368.
[13]JOACHIM D,DELLER J R.Multiweight optimization in optimal bounding ellipsoid algorithms[J].IEEE Transactions on Signal Processing,2006,54(2):679-690.
[14]CHERNOUSKO F,POLYAK B.Special issue on the set membership modelling of uncertainties in dynamical systems[J].Mathematical and Computer Modelling of Dynamical Systems,2005,11(2):123-124.
[15]SCHWEPPE F C.Recursive state estimation:unknown but bounded errors and system inputs[J].IEEE Transactions on Automatic Control,1968,13(1):22-28.
[16]FOGEL E.System identification via membership set constraints with energy constrained noise[J].IEEE Transactions on Automatic Control,1979,24(5):752-758.
[17]FOGEL E,HUANG Y F.On the value of information in system identification-bounded noise case[J].Automatica,1982,18(2):229-238.
[18]梁礼明,吴莉,李钟侠.一种改进型椭球外定界集员辨识算法[J].自动化技术与应用,2009,28(11):11-13,50.LIANG Liming,WU Li,LI Zhongxia.An improved setmembership identification algorithm based on ellipsoid outside[J].Techniques of Automation and Applications,2009,28(11):11-13,50.
[19]黄一,陈宗基,魏晨.最小迹扩展集员估计[J].中国科学:信息科学,2010,40(4):526-538.HUANG Yi,CHEN Zongji,WEI Chen.Least trace extended set-membership filter[J].Scientia Sinica Informationis Sciences,2010,40(4):526-538.
[20]周波,戴先中.自适应噪声定界的改进集员辨识算法[J].控制理论与应用,2012,29(2):167-171.ZHOU Bo,DAI Xianzhong.Improved set-membership identification algorithm with adaptive noise bounding[J].Control Theory&Applications,2012,29(2):167-171.
[21]刘大杰,华慧.GIS位置不确定性模型的进一步探讨[J].测绘学报,1998,27(1):45-49.DOI:10.3321/j.issn:1001-1595.1998.01.007.LIU Dajie,HUA Hui.The more discussion to the modeling uncertainty of line primitives in GIS[J].Acta Geodaetica et Cartographica Sinica,1998,27(1):45-49.DOI:10.3321/j.issn:1001-1595.1998.01.007.
[22]范爱民,郭达志.误差熵不确定带模型[J].测绘学报,2001,30(1):48-53.DOI:10.3321/j.issn:1001-1595.2001.01.010.FAN Aimin,GUO Dazhi.The uncertainty band model of error entropy[J].Acta Geodaetica et Cartographica Sinica,2001,30(1):48-53.DOI:10.3321/j.issn:1001-1595.2001.01.010.
[23]蓝悦明,陶本藻.以点位误差描述线元位置不确定性的误差带方法[J].测绘学报,2004,33(4):289-292.DOI:10.3321/j.issn:1001-1595.2004.04.002.LAN Yueming,TAO Benzao.End points accuracy based error band method for determination of a line segment position uncertainty[J].Acta Geodaetica et Cartographica Sinica,2004,33(4):289-292.DOI:10.3321/j.issn:1001-1595.2004.04.002.
[24]孙先仿,范跃祖,宁文如.有界误差模型的一种结构辨识方法[J].自动化学报,1999,25(2):242-246.SUN Xianfang,FAN Yuezu,NING Wenru.A structure identification method for bounded-error models[J].ActaAutomatica Sinica,1999,25(2):242-246.
[25]杨婷,杨虎.椭球约束与广义岭型估计[J].应用概率统计,2003,19(3):232-236.YANG Ting,YANG Hu.Ellipsoidal restriction and generalized ridge estimation[J].Chinese Journal of Applied Probability and Statistics,2003,19(3):232-236.
[26]DURIEU C,WALTER,POLYAK B.Multi-input multioutput ellipsoidal state bounding[J].Journal of Optimization Theory and Applications,2001,111(2):273-303.
[27]SCHAFFRIN B,TOUTENBURG H.Weighted mixed regression[J].Zeitschrift für Angewandte Mathematik und Mechanik,1990,70(6):T735-T738.