5. Dual Estimation and Reduced Order Modeling of Damaging Structures

In this chapter, the dual estimation and reduced order modeling of a damaging structure is studied. In this regard, proper orthogonal decomposition is considered for reduced order modeling in order to find a subspace which optimally captures the dynamics of the system. Through a Galerkin projection, the equations governing the dynamics of the system are projected onto the subspace provided by the proper orthogonal decomposition technique. It is proven that the subspace established by application of the proper orthogonal decomposition is sensitive to changes of the parameters; therefore, it can be profited in the algorithms for estimation of the damage incidence. As for the dual estimation goal, the extended Kalman filter and extended Kalman particle filter are adopted; both filters, in their so-called update stage, make a comparison between the latest observation and the prediction of the state of the system to quantify the required adjustment in the estimation of the state and parameters. In the case of the reduced order modeling, for realization of such a comparison, reconstruction of full state of the system is required, which is obviously possible only if the subspace is known. In this chapter, an adjustment of the dual estimation concept has led to an online estimation of the proper orthogonal modes, components of the reduced stiffness matrix and the states of the structure. This novelty can intuitively help to detect the damage in the structure, locate it and potentially identify its intensity.