  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. The proposed smoothing method does not assume any prior knowledge of
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. An eight-storey shear frame
and the Taipei 101 tower in Taiwan are used as case studies, and a thorough
comparison is established with the Augmented 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. Moreover, 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.
