A linear Bayesian filter for input and state estimation of structural systems

This paper proposes a linear recursive Bayesian filter for minimum variance unbiased joint input and state estimation of structural systems. Unlike the augmented Kalman filter (AKF), the proposed filter falls within the category of Bayesian filters in which unknown inputs are estimated without attributing any fictitious input model or statistics. Also, in contrast with the existing algorithms in the latter category, such as the Gillijns and De Moor Filters (GDFs), the developed filter applies to systems with and without direct feedthrough, in particular, systems with a rank‐deficient feedforward matrix. Because of the latter features, the filter is referred to as universal filter (UF) for convenience. The numerical examples show that the UF performs better than the AKF. Due to its structure, the UF does not require the tuning of the hyperparameters for inputs, and therefore the problematic instability of the AKF is not encountered in the case of a large modeling error variance of the input. For systems with direct feedthrough, the error and covariance propagation terms differ due to the distinct state space. Consequently, the UF can enhance estimations due to the well‐conditionedness of the relevant inversion problem. Moreover, the UF can deal with systems with rank‐deficient feedforward matrix where these systems are not covered by GDFs.