Indirect Reconstruction of Structural Responses Based on Transmissibility Concept and Matrix Regularization

A novel method is proposed based on the transmissibility concept and matrix regularization for indirectly measuring the structural responses. The inputs are some measured responses that are obtained via physical sensors. The outputs are the structural responses corresponding to some critical locations where no physical sensors are installed. Firstly, the transmissibility concept is introduced for expressing the relationship between the measured responses and the indirectly measured ones. Herein, a transmissibility matrix is formulated according to the theory of force identification under unknown initial conditions. Then, in order to reduce the size of the transmissibility matrix, structural responses are reshaped in a form of a matrix by using the concept of moving time windows. According to the matrix form of input-output relationship, indirect reconstruction of responses is boiled down to an optimization equation. Since inverse problem may be ill-conditioned, matrix regularization such as F-norm regularization is then recommended for improving the optimization problem. Herein, the penalty function is defined by using a weighted sum of two F-norm values, which correspond to the estimated responses of physical sensors and the ones of the concerned critical locations, respectively. Numerical simulations and experimental studies are finally carried out for verifying the effectiveness and feasibility of the proposed method. Some results show that the proposed method can be applied for indirectly measuring the responses with good robustness.