300 S. Chauhan The Kalman filter state estimates are given as bXk DC YpYT p CYp (27.15) Using Eqs. (27.3) and (27.4), it is easy to follow that projection defined in Eq. (27.14) can be written as Pfp DOC YpYT p CYp DObXi (27.16) This paves the way for decomposing projection Pfp such that extended observability matrix Oand Kalman filter states bXi can be estimated. This is done by means of singular value decomposition of Pfp. It is noticeable that estimation of extended observability matrix remains same. ODUS 1 2 bXi DS 1 2 VT (27.17) The state transition matrix can now be easily obtained using the expression provided in Eq. (27.6). However, it is common to provide an alternate expression based on the estimated Kalman filter state and its shifted versionbXkC1. Using Eq. (27.1) it is easy to obtain A, in terms of estimated state vectors, as bXkC1 ADbXk (27.18) It is clear from above discussion that the reason for using SSI-Data is estimation of state vectors. However, as has been mentioned, this is not a requirement from modal parameter estimation perspective. More importantly, even after estimation of Kalman filter states, one can still utilize the extended observability matrix for estimating state transition matrix [7]. Thus, if estimating the state vectors is not the goal, it is easy to follow that SSI-Data can be formulated simply on the basis of extended observability matrix. In that case, there is no difference between this approach and the one expressed in previous section that described SSI-Cov. Note that basic formulation of SSI-Cov is based on covariance functions and one cannot estimate state vectors only on the basis of covariance functions. The major distinction between the two variants of SSI is in terms of implementation; unlike SSI-Cov, SSI-Data can be implemented such that calculation of covariance matrices is not required [6]. Otherwise, the only distinction between the two approaches is that in case of SSI-Cov, covariance functions are supposed to be pre-calculated, which is not the case with SSI-Data. SSI-Cov starts directly from the formation of Hankel matrix H(Eq. 27.3) where as in SSI-Data the raw output data is arranged in terms of past and future responses (Eq. 27.2) so that the covariance functions are calculated as a part of the algorithm. 27.3 Conclusions This paper reviews the Stochastic Subspace Identification (SSI) algorithm and its two variants (SSI-Cov and SSI-Data) within the framework of modal parameter estimation. One underlining aspect of SSI pointed out in this paper is the fact that the goals of modal parameter estimation stage of operational modal analysis, i.e. estimation of natural frequency, damping and unscaled mode shape, can be achieved through several formulations of SSI. This fact is highlighted by means of an alternate formulation of SSI-Cov presented in this paper. This is done by exploring the relationship between state-space model and high order polynomial model representation of a dynamic system. The paper further emphasizes that, when viewed within the framework of modal parameter estimation, there is not much difference between SSI-Cov and SSI-Data. This is due to the fact that the aim of these variants differ from when they are applied in Controls Engineering domain (where these algorithms were originally developed) to their application in modal analysis domain. It is understandable from the formulation of SSI-Data that one of its primary goal is state estimation, a goal that is not shared by modal parameter estimation. That state transition matrix can be obtained using the estimated states is
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