An output-only unbiased minimum variance state estimator for linear systems

This study proposes a novel filtering method for unbiased minimum variance state estimation. A key challenge in the development of digital twinning and system identification of operational civil infrastructure is the difficulties associated with measurement of the often nonstationary, and stochastic unknown input to the system, e.g. traffic, earthquake, and wind loads. Herein a novel algorithm is presented for an unbiased output-only state estimation for linear state space systems with direct feedthrough. In this paper, the problem of state estimation for discrete time systems with unknown input, where no previous assumption is made regarding the probability distribution of the input, is addressed. In this regard, a two-stage output-only filter is proposed. The unbiasedness of the estimation is demonstrated, and the error covariance propagation over time is derived. It is shown that the unknown input of the system can be subsequently reconstructed via the system matrices, the measured responses, and the estimated state vector. A closed-form solution is provided for the parameters associated with pseudo-inverse in gain calculations to ensure the numerical stability of the filter. Three case studies including simulated and laboratory experiments for verification, validation, and cross comparison of the proposed method are presented. The first case study corresponds to a simulated shear building subjected to two different load time histories, a parametric sinusoidal load, and an earthquake ground acceleration. The second case study shows the results for a simulated experiment on a steel railway bridge subjected to a concentrated load. The third case study corresponds to a laboratory shear building subjected to ground motion and observed through accelerometers. The results of the experiments show a good agreement between the "true response" and the filter state estimation. Moreover, the proposed method is compared against a state-of-the-art algorithm.