﻿ 基于椭球不确定性的平差模型与算法
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1. 有色金属成矿预测与地质环境监测教育部重点实验室(中南大学), 湖南 长沙 410083;
2. 中南大学地球科学与信息物理学院, 湖南 长沙 410083;
3. 中南林业科技大学土木工程学院, 湖南 长沙 410004

Adjustment model and algorithm based on ellipsoid uncertainty
SONG Yingchun1,2, XIA Yuguo1,2, XIE Xuemei1,2,3
1. Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring(Central South University), Ministry of Education, Changsha 410083, China;
2. School of Geosciences and Info-Physics, Central South University, Changsha 410083, China;
3. School of Civil Engineering, Central South University of Forestry and Technology, Changsha 410004, China
Abstract: 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.
Key words: uncertainty    ellipsoid constraint    adjustment model    ill-posed problem    set membership estimation

1 有界椭球不确定性平差模型

(1)

(2)

L=AX是相容方程组，取X0使得L=AX0。当L=AX不相容时，取X0=XLS=(ATA)-1ATL，这时，LAX0，利用式(1)有

e的有界不确定性也可以近似地表示为

(3)

X带有椭球约束先验信息，X的可行空间可以用下面的椭球集合来表示

(4)

(5)

(6)

E=E(e)∩E(c, Q)是参数向量的可行解集。

2 带有椭球不确定性约束的集员估计

(7)

(8)
(9)
(10)
(11)
(12)
 图 1 两个椭球交的最小外包椭球 Fig. 1 The minimum circumscribed ellipsoid with two ellipsoid intersections

(13)
(14)

3 ρa的计算方法

(15)

(16)
(17)

(18)

(19)

(20)

(21)

a在满足式(20)和式(21)的条件下，直接利用搜索算法(a从0开始到1止，通过增量Δa，逐步搜索得到使tr(PU)达到最小的a)求出a的值，从而求出参数估计值PU

4 算例分析

(22)

(23)
(24)

 图 2 算例1中的误差椭圆，X的约束椭圆及解的不确定性椭圆 Fig. 2 Error ellipse, constrained ellipse of X and the uncertainty ellipse of solution in example 1

 图 3 算例2中的误差椭圆，X的约束椭圆及解的不确定性椭圆 Fig. 3 Error ellipse, constrained ellipse of X and the uncertainty ellipse of solution in example 2

 点名 真实坐标 近似坐标 P3 P4 P5 P6 P3 P4 P5 P6 x/m 53 743.151 48 681.398 43 767.234 40 843.239 53 743.674 48 680.496 43 768.794 40 840.905 y/m 61 003.810 55 018.270 57 968.590 64 867.876 61 006.568 55 018.806 57 966.087 64 870.541

 边号 观测边长/m 1 45.075 2 5 222.056 3 5 187.391 4 7 838.867 5 5 483.162 6 5 731.756 7 5 438.383 8 7 493.316 9 8 884.603 10 8 839.687

(25)

(26)
(27)

(1) 可看作是带有椭球约束不确定信息平差模型式(25)的解，此解包含的不确定度可以用椭球

(2) 从加权混合估计的角度来看，因为计算得到a=0.052 8，说明参数约束先验信息式(27)在参数估计中的作用更大。这也正好说明当模型出现病态时，利用参数先验信息可以改善其病态性。

(3) 令

 真值 最小二乘方法 截断奇异值法 岭估计法(L曲线法) 本文算法 -0.523 0 -1.347 3 -0.535 0 -0.552 7 -0.510 7 -2.758 0 6.162 5 -2.372 9 -2.263 9 -2.677 4 0.902 0 10.443 9 1.358 5 1.382 6 1.034 0 -0.536 0 -0.364 5 -0.530 7 -0.497 5 -0.540 6 -1.560 0 2.869 2 -1.331 3 -1.328 4 -1.480 0 2.503 0 -5.908 4 2.067 1 2.080 8 2.368 2 2.334 0 -5.331 9 1.907 1 1.935 5 2.184 4 -2.665 0 -16.214 1 -3.387 3 -3.328 9 -2.907 2 0.004 3 1.686 1×10-5 0.001 2 0.004 4 0.306 8 m 0 504.044 1 1.303 2 1.308 9 0.129 7

(4) 本文算法中不确定度的最小化是通过求椭球最小特征矩阵的迹来实现的，也可以通过最小化的椭球体积(对应的是特征矩阵的行列式最小)，相关的算法可参看文献[8]。

(5) 算法中，a=0.052 8是一个近拟值。a从0开始，通过增量Δa=0.000 1，逐步搜索得到使tr(PU)达到最小的a

(6) 对于病态模型的其他算法，如表 3中的截断奇异值算法和岭估计算法，它们是利用数学原理来处理病态系数矩阵，不能有效地利用先验信息，计算的结果不如本文的算法。更重要的是，本文算法不仅能给出参数估计的值，而且还能对参数估计的不确定度进行估计。

5 结束语

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 [1] 葛旭明, 伍吉仓. 误差限的病态总体最小二乘解算[J]. 测绘学报, 2013, 42(2): 196–202. GE Xuming, WU Jicang. A regularization method to ill-posed total least squares with error limits[J]. Acta Geodaetica et Cartographica Sinica, 2013, 42(2): 196–202. [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. 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. 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. DOI:10.1109/TSP.2007.908945 [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. DOI:10.1016/S0005-1098(98)00212-X [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. DOI:10.1109/9.827352 [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. DOI:10.1109/9.847746 [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. DOI:10.3969/j.issn.1003-7241.2007.11.003 [9] ALAMO T, BRAVO J M, REDONDO M J, et al. A set-membership state estimation algorithm based on DC programming[J]. Automatica, 2008, 44(1): 216–224. DOI:10.1016/j.automatica.2007.05.008 [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. DOI:10.1016/j.automatica.2007.05.026 [11] WALTER É, KIEFFER M. Guaranteed nonlinear parameter estimation in knowledge-based models[J]. Journal of Computational and Applied Mathematics, 2007, 199(2): 277–285. DOI:10.1016/j.cam.2005.07.039 [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. DOI:10.1109/TCST.2006.886436 [13] JOACHIM D, DELLER J R. Multiweight optimization in optimal bounding ellipsoid algorithms[J]. IEEE Transactions on Signal Processing, 2006, 54(2): 679–690. DOI:10.1109/TSP.2005.861893 [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. DOI:10.1080/13873950500067296 [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. DOI:10.1109/TAC.1968.1098790 [16] FOGEL E. System identification via membership set constraints with energy constrained noise[J]. IEEE Transactions on Automatic Control, 1979, 24(5): 752–758. DOI:10.1109/TAC.1979.1102164 [17] FOGEL E, HUANG Y F. On the value of information in system identification-bounded noise case[J]. Automatica, 1982, 18(2): 229–238. DOI:10.1016/0005-1098(82)90110-8 [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 set-membership identification algorithm based on ellipsoid outside[J]. Techniques of Automation and Applications, 2009, 28(11): 11–13, 50. DOI:10.3969/j.issn.1003-7241.2009.11.003 [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. 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. 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. 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. DOI:10.3969/j.issn.1001-4268.2003.03.002 [26] DURIEU C, WALTER É, POLYAK B. Multi-input multi-output ellipsoidal state bounding[J]. Journal of Optimization Theory and Applications, 2001, 111(2): 273–303. DOI:10.1023/A:1011978200643 [27] SCHAFFRIN B, TOUTENBURG H. Weighted mixed regression[J]. Zeitschrift für Angewandte Mathematik und Mechanik, 1990, 70(6): T735–T738.
http://dx.doi.org/10.11947/j.AGCS.2019.20170611

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#### 文章信息

SONG Yingchun, XIA Yuguo, XIE Xuemei

Adjustment model and algorithm based on ellipsoid uncertainty

Acta Geodaetica et Cartographica Sinica, 2019, 48(5): 555-562
http://dx.doi.org/10.11947/j.AGCS.2019.20170611