A linear recursive smoothing method for input and state estimation of vibrating structures

Recursive Bayesian filters have been widely deployed in structural system identification where output-only filters are of higher practicality. Unfortunately, the estimation obtained by instantaneous system inversion via filters can be compromised by an ill-conditionedness of the system, which is a consequence of the architecture of the sensor network. To significantly reduce the ill-conditioning and increase the robustness to available networks, a new recursive smoothing algorithm is proposed for simultaneous input and state estimation of linear systems. Unlike the existing minimum-variance unbiased (MVU) smoothing methods that are restricted to either systems with no direct feedthrough or systems with a full-rank feedforward matrix, the proposed smoothing algorithm is universally applicable to linear systems with and without direct feedthrough as well as those with a rank-deficient feedforward matrix. In addition, no prior knowledge is assumed concerning the statistical characteristics or evolutionary model pertaining to the input. A different indexing of the discrete-time input leads to a distinct linear algebra from the existing MVU smoothing methods. Numerical simulations of an eight-storey shear frame and the Taipei 101 tower in Taiwan are used as case studies, and a thorough comparison including computational cost is established with the Augmented Kalman Filter, Dual Kalman Filter, MVU filters and MVU smoothing methods. It is shown that the incorporation of singular value truncation for system inversion can result in a noticeable improvement in the estimation. A strategy for a more effective online monitoring is also proposed. Across various sensor networks and in the presence of a rank-deficient feedforward matrix, the proposed method could achieve at least 67% noise reduction compared to other filters and at least 30% improvement compared to other smoothing methods.