Minimum detectable changes based on linear Bayesian filters

This article proposes a method to assess the minimum detectable changes in system parameters based on Bayesian inference. The approach considers the monitored parameters as random variables, and uses linear Bayesian filters to update them based on system observations. To determine the detectable changes in them, a special quality of Kalman filter is exploited, that is, the fact that the posterior variance is independent of the observations of the changed state, and that it can be calculated based on previous knowledge. To enable an automated change diagnosis, a decision rule is introduced, based on the posterior's probability of exceeding a user-defined threshold. This way, the user can specify an allowable probability of exceedance, and subsequently, this value can be converted into an equivalent detectable change. To cope with large systems and multiple system observations, a functional Kalman Filter is applied, which enables an analytical solution to the Bayesian inverse problem. For proof of concept, three numerical case studies related to structural health monitoring are presented, including a 6-DOF mass-and-spring system, an offshore tower subject to marine growth, and a reinforced concrete bridge affected by seismic damage. The case studies highlight that no data from the changed state is required to accurately evaluate the detectable change. Secondly, the detectability can be evaluated for any measurement quantity (vibrations, inclinations) or extracted damage-sensitive feature (natural frequencies, mode shapes), provided it is sensitive to changes in the system parameters. Furthermore, the detectability can be assessed for various damage scenarios and a wide range of monitored systems, provided that numerical models are available, e.g. physics-based models or surrogate models.