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Distance coefficient-Fisher information criterion for optimal sensor placement
DONG Xiaoyuan, PENG Zhenrui, YIN Hong, DONG Haitang
School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
Abstract: For damage parameters identification, when the traditional Fisher information criterion is used for optimal sensor placement, the measuring points are susceptible to gathering in a local sensitivity area, which results in information redundancy and this is not conducive to damage location. To avoid aggregation of the measuring points and improve the ability of the damage location, the distance coefficient, which reflects the degree of information independence, was used first to correct the Fisher information matrix, and then the measuring points were obtained by maximizing the determinant of the modified information matrix using a sequential algorithm. The method was employed to design the optimal sensor configuration for a simple 16-DOF chain mass-spring model. The results show the method can effectively avoid the aggregation of measuring points and solve the problem of information redundancy.
Key words: optimal sensor placement     damage detection     Fisher information matrix     information redundancy     Euclidean distance     sensitivity analysis     modal analysis

1 传感器优化布置问题的数学模型

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

 (9)
2 Fisher信息准则及Fisher信息矩阵的计算 2.1 Fisher信息准则

 (10)

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2.2 Fisher信息矩阵的计算

 (14)

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3 距离系数-Fisher信息准则

 (18)

ai(t) 为雅可比矩阵∇θX(t) 的第i行，由于Σ(t) 为对角阵，可将Fisher信息矩阵表达为各个自由度贡献之和的形式，即

 (19)

 (20)

 (21)

 (22)

 (23)

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4 距离系数-Fisher信息准则下的传感器优化布置算法

1) 分别计算每个候选自由度对应的Fisher信息矩阵的行列式，记录行列式值最大的候选自由度，作为第一个测点位置。

2) 假设已经确定了m个测点位置，且m < N0，此时还剩余Nd-m个候选自由度。

①根据式 (21) 分别计算第m个测点与剩余Nd-m个候选自由度的信息阵之间的欧氏距离dkn，下标k表示第k个候选自由度，记录Nd-m个距离中的最大值dmax，并由式 (22) 得到标准化的欧氏距离Dkm

②当m=1时，只有一个已选测点，所以Rk=Dk1；当m>1时，为保证式 (23) 成立，比较RkDkm，若Dkm < Rk，则更新Rk=Dkm，否则Rk不变。

③将第k个候选自由度增加到已选测点集合中，计算该m+1个测点所对应的有效信息阵的行列式，计算公式为

 (25)

3) 重复2)，确定剩余传感器布置位置，直至确定全部N0个测点位置，得到基于距离系数-Fisher信息准则的传感器优化布置方案。

5 数值算例

 图 1 16自由度剪切型弹簧—质量模型 Fig. 1 16 DOFs shear type spring-mass model
5.1 模型简介

 图 2 模型前四阶振型 Fig. 2 First 4 mode shapes of the spring-mass model

 图 3 δK1的Fisher信息 Fig. 3 Fisher information of δK1
5.2 灵敏度分析

 图 4 结构位移响应对损伤参数的灵敏度 Fig. 4 Sensitivity of structural displacements with respect to damage parameters
5.3 传感器优化布置方案

 图 5 基于传统Fisher信息准则的传感器优化布置方案 Fig. 5 Scheme of optimal sensor placement based on traditional Fisher information criterion

Fisher信息矩阵包含结构响应对损伤参数的灵敏度信息，最大化Fisher信息阵的行列式得到的传感器测点满足对损伤参数的敏感性要求。在图 5中，测点分布于结构响应对损伤参数的敏感区，从而能够最大程度上发现损伤的发生，但测点在悬臂顶端聚集，并不满足损伤参数可观性的要求，即不利于损伤定位。根据工程经验，在实际布设传感器时，力求将传感器分散布置，避免局部过分集中现象[17]

 图 6 基于距离系数-Fisher信息准则的传感器优化布置方案 Fig. 6 Scheme of optimal sensor placement based on distance coefficient-Fisher information criterion

6 结论

1) Fisher信息准则通过使Fisher信息矩阵的范数最大化，使得估计误差的协方差矩阵最小，即最小化结构损伤参数的估计误差。

2) 基于传统Fisher信息准则的传感器布置方案中，测点聚集在结构响应对损伤参数的敏感区，产生信息冗余。

3) 基于距离系数-Fisher信息准则得出的传感器布置方案空间分布更加合理，测点大部分位于结构响应对损伤参数的敏感区，同时相互之间保持一定的距离，避免了信息冗余问题，在能够判定损伤发生的同时，有利于损伤定位。

4) 需要进一步深入研究，比较本文方法与其他传感器优化布置方法对实际结构的损伤识别效果。

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DOI: . 10.11992/tis.201604026

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#### 文章信息

DONG Xiaoyuan, PENG Zhenrui, YIN Hong, DONG Haitang

Distance coefficient-Fisher information criterion for optimal sensor placement

CAAI Transactions on Intelligent Systems, 2017, 12(1): 32-37
. http://dx.doi.org/10.11992/tis.201604026