27 Subspace Algorithms in Modal Parameter Estimation for Operational Modal Analysis: Perspectives and Practices 299 Since the state transition matrixAis constant (the system is time invariant), a number of equations similar to Eq. (27.10) can be formed as shown below. 2 666 4 0 I 0 0 0 0 : : : : : : : : : : : : ’0 ’1 ’m 1 3 777 5 2 666 4 yiC0 yiC1 yiCm 1 yiC1 yiC2 yiCm : : : : : : : : : : : : yiCm 1 yiCm yiC2.m 1/ 3 777 5 D 2 666 4 yiC1 yiC2 yiCm yiC2 yiC3 yiCmC1 : : : : : : : : : : : : yiCm yiCmC1 yiC2m 1 3 777 5 (27.11) For the sake of simplicity, Eq. (27.11) can be written in a compact form as APDF (27.12) which can be solved for Ain a least squares manner as, ADFPT PPT 1 (27.13) The two products in the above equation, FPT, PPT are two Toeplitz matrices comprising covariance functions PPT D 2 666 4 ƒ0 ƒ1 ƒm 1 ƒ 1 ƒ0 ƒm 2 : : : : : : : : : : : : ƒ .m 1/ ƒ .m 2/ ƒ0 3 777 5 ; FPT D 2 666 4 ƒ1 ƒ2 ƒm ƒ0 ƒ1 ƒm 1 : : : : : : : : : : : : ƒ m ƒ .m 1/ ƒ1 3 777 5 It is recognizable that the two matrices FPT, PPT have similar structure as ! H; Hm 1, described in Eq. (27.7). This explains the connection between the traditional approach (based onHDYf YT p) and the approach suggested in this paper. Based on above observations, it can be argued that there is no fundamental difference between various formulations of SSI-Cov. The biggest difference is perhaps how the covariance matrices are formed and stacked, and as shown in this paper, this can be done in several ways without having any impact on the final outcome in terms of estimation of modal parameters. 27.2.2 Data Driven Stochastic Subspace Identification Algorithm Formulation of SSI-Cov algorithm assumes that covariance functions are available and raw output data does not play any role irrespective of whether it is available or not. Various formulations shown in previous section assume that raw data is available, but it is easily noticeable that this is not a requirement and, starting with Eq. (27.3) (or Eq. 27.13) the formulations can be arrived at without much difficulty by simply using the covariance functions in case they are available a priori. SSI-Data, on the contrary, makes it mandatory to have the raw data available. This is the oft-quoted difference between the two popular variants of SSI. However, it can be argued that this requirement is driven more by the need to estimate state vectors xwithin controls engineering domain than by requirements associated with modal parameter estimation. Estimation of state vectors is typically done by means of Kalman filter [5–7], which provides optimal prediction of state vector. It is this requirement that necessitates the availability of raw output time data. SSI-Data is based on the concept of projection [5, 10], where future outputs are projected on the past outputs. This projection is defined as Pfp D Yf YT p YpYT p CYp (27.14)
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