296 S. Chauhan H Hankel matrix of covariance matrices ƒ Covariance matrix O Extended observability matrix C Extended controllability matrix † State covariance matrix ’ Matrix polynomial coefficients Pfp Projection of future output responses on past output responses bXi Kalman filter state estimate 27.1 Introduction There are several applications, including modal parameter estimation where parametric models are sought. These include signal processing, defect detection, design of control systems and many others. Subspace algorithms belong to the class of system identification algorithms that utilize state-space models of time series for estimating parametric models. The use of these algorithms for parameter estimation in modal analysis is not new. Several popular algorithms, such as Ibrahim Time Domain(ITD) [1, 2] andEigensystem Realization Algorithm(ERA) [3, 4], use state space formulations for estimating modal parameters. The aim of this paper is to understand state space modelling in context of modal parameter estimation, with a focus on one of the most commonly used Operational Modal Analysis (OMA) algorithm, Stochastic Subspace Identification (SSI) [5–7], along with its two variants: Covariance-driven SSI (SSI-Cov) and Data-driven SSI (SSI-Data). In order to understand state-space models in the context of modal analysis, it is vital to state the objective of modal analysis, which is to estimate modal parameters, i.e. natural frequency, damping and mode shape, of a structure. Thus, unlike system identification problem in control system design or forecasting problem in econometrics, the goal of a parameter estimation algorithm is not estimation of matrices A, C(defined in Sect. 27.2) or associated quantities such as state vectors, observability or controllability matrix, etc. but modal parameters. The paper starts with a general discussion on SSI algorithm and describes sequentially its two variants, Covariance-driven SSI (SSI-Cov) and Data-driven SSI (SSI-Data). In Sect. 27.2.1, two separate approaches are provided for the development of SSI-Cov. First, traditional formulation of SSI-Cov is provided along with a modified formulation. Then an alternate formulation is provided that takes inspiration from, and explores, the relationship between the polynomial model and statespace model representation of a dynamic system. Additionally, this section explains how estimation of extended observability matrix in the conventional formulation is not a requirement from modal analysis point of view and desired results can be achieved in comparatively simple steps. The section concludes by noting how various formulations of SSI-Cov simply differ in how the covariance matrices are stacked. Same approach is taken while discussing SSI-Data (Sect. 27.2.2). Given the framework of modal parameter estimation, the aim is to connect the two variants of SSI. In this context, it is explained how the need to estimate state vectors necessitates the availability of raw output data for SSI-Data algorithm and it is emphasized that these requirements do not form a part of modal parameter estimation. The formulation of SSI-Data is then explained in light of this knowledge and it is shown how SSI-Data is not much different from SSI-Cov in the absence of the need to estimate state vectors. 27.2 Stochastic Subspace Identification Algorithm Stochastic Subspace Identification (SSI) is a well-known operational modal analysis (OMA) algorithm [5–7]. It is based on state-space representation of a discrete linear time invariant (LTI) system described as xkC1 DAxk Cwk yk DCxk Cvk (27.1) where yis the vector of measured responses, xis the vector of state variables matrix, Ais the state transition matrix, Cis the output matrix andwandvare process and measurement noise vectors. As is clear from Eq. (27.1), SSI algorithm operates on measured output response data only (input excitation is not measured in OMA). From the point of view of modal parameter estimation, it is the estimation of state transition matrix Athat is most important as eigenvalue decomposition of this matrix reveals modal parameters.
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