數種濾波技術於線性系統識別之可行性探討

Abstract

本研究旨在探討數種常用濾波技術對於非時變系統與時變系統之系統識別模型參數的影響。本研究選用三種濾波技術來進行比較，分別是卡爾曼濾波演算法(Kalman filtering algorithm)、離散小波濾波法(discrete wavelet de-noising)，以及中位值濾波器(median filter)。研究中將數值模擬地震反應分為非時變系統與時變系統兩大部分：非時變系統中，將數值模擬七個自由度的系統之地震反應，利用連續小波轉換建構ARX (Autoregressive with exogenous input) 模式，進一步估算其系統模態參數。為探討不同噪訊含量的訊號中，經由上述各種濾波法處理後的差異性，我們將數值模擬地震反應加入不同比例的白色噪訊，使其噪訊比分別為10%、20%、30%、40%，再利用濾波法對含噪訊之地震反應進行濾波處理，而濾波方法對於提升系統識別之效率的可行性，乃取決於識別出其準確之模態參數時所需的最小階數。時變系統部分，本研究採用數值模擬五個自由度的系統受地震反應時的訊號，不同於非時變系統，此系統的勁度與模態的阻尼比與時間相關，利用移動最小平方差法(Moving Least Square method，MLS)架構TVARX模型( Time Varying Autoregressive with exogenous input)，識別其系統之瞬時模態特性。對於時變系統，其研究方法與探討非時變系統雷同，將上述的濾波技術應用於處理時變系統中其噪訊比分別為1%、3%、5%的地震反應，用來評估其濾波技術對於改善系統識別之效能的可行性。

The main purpose of this study is to investigate the effects that several popular de-noising techniques on identifying the modal parameters of time invariant and time varying linear systems. There are three de-noising techniques under consideration, and they are Kalman filtering algorithm, discrete wavelet de-noising, median filter. When studying invariant time systems, we numerically simulate acceleration responses of a seven degrees of freedom (DOF) system under earthquake. A continuous wavelet transform combined with ARX (Auto-regressive with exogenous input) model is applied to process the acceleration responses and estimate the modal parameters of the system. White noise with different noise levels (noise-to-signal ratio (NSR) =10%, 20%,30%, 40%) is added to the acceleration responses. The de-noising techniques are employed to the noisy responses. The feasibility of these de-noising techniques on handling the noisy data is determined by the least order of ARX yielding accurate modal parameter identification. When investigating time varying systems, we numerically simulate acceleration responses of a five DOF system under earthquake. Their stiffness and modal damping ratios are time dependent. A moving least square method is adopted to construct TVARX model (time varying auto-regressive with exogenous input) and to identify the instantaneous modal parameters of the five DOF system. Similar to the study for a time invariant system those de-noising techniques are utilized to process the noisy acceleration responses with NSR equal to 1%, 3%, and 5%, and their effectiveness on improving the modal identification is evaluated.