80 R. Yao and S.N. Pakzad adverse effect of measurement noise on the accuracy of estimated modal parameters [16]. In this chapter, the sensitivity of two damage features based on AR modeling due to measurement noise is studied and an analysis methodology is proposed. The two methods are the Mahalanobis distance [17] of AR coefficients and the Cosh distance [18] of AR model spectra between the baseline state and the current state. The validity of this methodology is supported by simulation results from a 10 DOF bridge model. The paper is organized as follows: Sect. 8.2 contains stepwise derivations regarding the analysis for the sensitivity with respect to measurement noise for both features. In Sect. 8.3, sensitivity analysis is applied to a 10-DOF simulated model and the results are compared with those from direct simulation and theoretical calculation. Conclusions are then made on the efficiency of the algorithms and the effectiveness of the features. 8.2 Measurement Noise Sensitivity for the AR Damage Features Distance measures between characteristics of undamaged and damaged structure state are often adopted as damage features. Damage features examined in this chapter are the Mahalanobis distance of AR coefficients and the Cosh distance of AR model spectra extracted from structural acceleration measurements [14]. The definition of a univariate AR model of order p is [19]: x.t/ D p X jD1 'jx.t j/ C x.t/: (8.1) In this equation, x(t) is the time series to be analyzed, ®j terms are the AR model coefficients, and x(t) is the model residual. Mahalanobis distance, which in this case operates on AR coefficient vectors, is a metric to evaluate the deviation within vectorial Gaussian sample groups [17]. Its definition is stated as below: D2 .®u; ®b/ D ®T u ®T b † 1 b .®u ®b/ : (8.2) where ®u is the feature vector from the unknown structural state and ®b=†b is the mean/covariance of feature vectors from baseline state. When the unknown vector ®u is not generated from the baseline distribution, it is expected that the distance value will increase significantly. From each vector of AR coefficients, corresponding AR spectrum plot can be constructed: S .p/ AR .!/ D 2 e j®.ej!/j 2 D 2 e ˇ ˇ ˇ Xp kD0 'ke j!kˇ ˇ ˇ 2 ; (8.3) where ®0 D1. For feature extraction purposes model residual variance 2 e is not calculated and set to unity, since its value is determined by excitation level. Cosh spectral distance based on AR spectrum estimates can be used as a frequency domain alternative to Mahalanobis distance of AR coefficients: C S; Sb D 1 2N N X jD1 " S !j Sb !j C Sb !j S !j 2 # : (8.4) whereSb is the baseline spectrum, Sis the spectrum from the unknown state, andNis the length of each spectrum vector. An illustration of the procedures through which the features are generated is also given in Fig. 8.1. To get the noise sensitivity of both indices, the analysis is divided into four steps according to the feature extraction procedures. 8.2.1 Theoretical Expression of Structural Response Autocovariance Function (ACF) An AR model is an all-pole system. In [15] it is stated that the ithmode of N degrees-of-freedom system corresponds to a conjugate pair of discrete system poles: zi ; z i De i !ni Ts˙jp1 2 i !ni Ts : (8.5)
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