Novel Probabilistic and Bayesian Approaches to Uncertainty Quantification for Operational Modal Analysis

The analysis of dynamic systems is of fundamental importance to countless engineering applications. The default way to analyse linear structural dynamic systems is through a methodology known as modal analysis. Since its introduction over 50 years ago, experimental modal analysis (EMA) has presented engineers with a principled way of recovering the modal properties of dynamic systems from observations of the excitation and structural response. However, the task of EMA becomes difficult when access to, or generation of, a controlled input force is limited. This is often the case for large engineering structures or structures in hard-to-replicate environments. To solve this problem, engineers have developed a new suite of methods known collectively as operational modal analysis (OMA), that rely on the natural excitation of a structure to obtain the modal properties. Over the last 20 years, a broad range of OMA algorithms have become available to the dynamicist, with Stochastic Subspace Identification (SSI) emerging as one of the leading methods across industry and academia owing to its unrivalled performance in many scenarios.

Despite their success, several research challenges remain in the field of OMA. Of these challenges, the handling of uncertainty is of key interest. It is well understood that measured data contain uncertainties that obscure the underlying signal one intends to measure. The effect of these uncertainties on the identification process in OMA algorithms are seldom considered, nay quantified, but are known to result in variations in the modal properties. Variations in recovered modal information that arise from uncertainty can pose a significant risk; modal information is frequently used to inform safety-critical decision-making in areas such as Structural Health Monitoring (SHM).

One way to address this challenge is to develop OMA methodologies capable of quantifying the uncertainty over the modal properties. Access to the uncertainty provides level of insight into the variability of the parameters given the available data and chosen methodology. In general Bayesian methods are a popular form of uncertainty quantification that combine a description of ones prior belief, with the likelihood of observed data, to obtain an estimate for the uncertainty as a posterior distribution. Despite some existing approaches to uncertainty quantification for OMA, no Bayesian formulation of the SSI algorithm currently exists in the literature, where posterior distributions over the modal properties are obtainable. In light of this fact, the current thesis aims to address this shortcoming.

At the centre of this work lies a novel probabilistic interpretation of the SSI algorithm. This form is achieved through the direct replacement of the identification mechanism at the core of SSI — canonical correlation analysis (CCA) — with its probabilistic equivalent. This now probabilistic algorithm unlocks the ability to apply Bayesian hierarchical modelling techniques that can better represent noise in the identification process and incorporate prior information. In this work three unique algorithms are presented. The first is a statistically robust SSI algorithm capable of robust identification of the modal parameters when faced with atypical observations in the measured time series. The second is a Bayesian approach to the SSI algorithm. By incorporating weakly informative proper prior information over the identification mechanism (forming Bayesian CCA), posterior distributions over the modal properties can be recovered. The last algorithm is an efficient form of the Bayesian SSI algorithm, achieved using stochastic approximation techniques, known to speed up the computation by reducing the number of operations over the entire dataset.

The algorithms and results contained within this thesis take positive steps towards viewing OMA as a problem in probabilistic and Bayesian inference; one where uncertainty can be principally considered and quantified, to provide a more informed description of a dynamic system’s behaviour from observed data. Such descriptions provide additional information about a system and its measurement that can ultimately facilitate more-informed and confident decision-making where modal information is to be considered.