186 M. Chang and S.N. Pakzad many engineering problems [5, 6]. The canonically arranged input/output responses directly derive transition state matrix. Depending on the type of input, ARMA is further extended to exogenous input/output systems (ARX) [7] and output-only systems (AR) [8]. Stochastic Subspace Identification (SSI) uses the projections of the future responses onto the previous input/output or output responses and shows reliable identification results [9, 10]. Each SID method requires a specific type of data such as impulse response, input/output, or output data. Most methods utilize over-parameterization technique to compensate the rank deficiency and to estimate accurate modal parameters. For these reasons, a comparison study of the performance of these methods is necessary. There is existing literature on the accuracy comparison of SID methods [11, 12]. The computational time and number of operations are estimated as measures of efficiency and conclude that changes in model order, specific modes, and SID method affect the identification results [3, 13]. The presence of uncertainty in vibration response is another issue for modal identification. The measurement noise is considered as an epistemic uncertainty for SID and is possible to increase the identification of computational modes and the distortion of modes [14]. Many studies for sensing devices show that the measurement noise does not have a significant effect on the amplitude of the actual response and can be considered random for modal identification. The effect of the measurement noise depending on the choice of the SID method has been less studied, even though it is critical for accurate estimation of the modal parameters of structural systems. This paper aims to compare the performance of several SID methods based on ERA for output-only systems (ERAOKID, ERA-NExT, and ERA-NExT-AVG). In order to investigate the effect of uncertainty in vibration response, Physical Contribution Ratio (PCR) [1] is adopted as a measure of signal power at sensing nodes. Two examples are used to investigate the effect of uncertainty for modal identification: (1) numerically simulated 4-DOF beam model subject to white noise excitation with several levels of measurement noise and (2) ambient vibration, measured from wireless sensors on Golden Gate Bridge. 18.2 Modal Identification Using ERA-Based Methods Eigensystem Realization Algorithm (ERA) is one of most widely used time-domain system identification methods [2]. ERA uses impulse responses to derive the modal parameters. Consider the state space description of dynamic system in discretized formas: xiC1 DAxi CBui yi DCxi CDui (18.1) In Eq. 18.1, xi 2<2N is the state vector, N is the number of DOF, ui is the input vector, and yi 2< m is the output vector at m locations on the structure (nodes). The coefficients [A, B, C, D] are called the discretized state, input, output, and feed-through matrices, respectively. The at-rest initial state and unit impulse excitation derive the output response in terms of coefficients matrices as yi DCAi 1B which is also known as Markov parameters. The consecutive Markov parameters are used to construct Hankel matrix which uses over-parameterizing technique to compensate the rank deficiency for modal parameter estimation as: Hi 1 D 2 6 6 6 4 yi yiC1 yiCq 1 yiC1 yiC2 yiCq : : : : : : : : : : : : yiCp 1 yiCp yiCpCq 2 3 7 7 7 5 D 2 6 6 6 4 CAi 1B CAi B CAiCq 2B CAi B CAiC1B CAiCq 1B : : : : : : : : : : : : CAiCp 2B CAiCp 1B CAiCpCq 3B 3 7 7 7 5 DPpAi 1Qq .i 1/ (18.2) Singular Value Decomposition (SVD) is applied toH0 to derive transition state matrixA, which is separated into the block matrices Pp andQq as:
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