以Laplace 分佈之最大似然估算於結構模態識別

The Structural Modal Identification via Maximum Likelihood Estimation with Laplace Distribution

蔡文魁(Wen-Kuei Tsai)

指導教授 ： 洪士林 ； 林昌佑

國立交通大學/工學院/土木工程系所/碩士(2018年)

摘要

在土木機械領域，常以連續型的MCK運動方程式表示動力反應，而在真實量測中大多只能測量到有限自由度的反應，有鑑於此則改以ARX線性模型去近似結構的動力反應。真實情形與近似模型通常不會完全符合，之間的差值稱作殘差，倘若ARX模型對於真實的運動行為近似度高，則殘差的產生會來自於資料的噪音。而在ARX的模型參數估計假設殘差為噪音分布為高斯分佈時，能以最小二乘法求得其參數估計值，倘若殘差為其他分佈時，最小二乘法的參數估計會受到影響。本篇將以Laplace分布來近似殘差的分布，又由於以數值方法無有效推導出Laplace分布的算法，本篇以最佳化演算法─粒子群演算法來搜尋近似解，並探討最小二乘法與粒子群演算法兩者間的差異，殘差與噪音之間的關聯性，與假設不同殘差分佈下對模態識別的影響。本篇的方法可以用於模態識別，但由於PSO在高維度的問題搜尋力不章而使得結果無規律性，偶有比最小二乘法更加的識別模態產生。再根據數值模擬所得的假說，則能說明影響動力反應的噪音為粉紅噪音與布朗噪音之間。

關鍵字

最大似然法則 ； Laplace分布 ； 粒子群演算法 ； ARX模型 ； 殘差分析 ； 結構物模態識別

並列摘要

In civil and mechanical engineering domains, the structural dynamic responds are usually expressed as continuous dynamic equation with parameters M, K, and C. However, only a few degree of freedoms of dynamic response can be measured in real cases. Hence, a linear ARX model can be utilized to approximate of real structural dynamic responses and the difference between real and approximate responses is called residual. If the approximates close to the real reactions, the residual would be caused by the noise of data mostly. The parameters of the ARX model are commonly obtained via least square method and the residual of model is the formula of Gaussian white noise. If the residual model is other noise forms, the parameters of ARX model cannot be estimated by least square appropriately. The aim of this work is a Laplace distribution model is employed as residual model and the parameters of ARX are estimated via particle swarm optimization (PSO) approach. Numerical simulation results and experiment measured responses are employed to verify the proposed approach. The differences of approximates obtained via PSO and least square methods are compared based on Laplace distribution and Gaussian white noise residual models. The results reveal that the proposed approach of the thesis is useful for mode identification. In addition, according to the hypothesis of the numerical simulation study, structural dynamic respond is mostly affected by the pink noise or Brownian noise.

並列關鍵字

Maximum Likelihood ； Laplace Distribution ； Particle Swarm Optimization ； Autoregressive eXogenous model ； Residual Analysis ； Structural Modal Identification

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