Integrating Bayesian Filtering and Gaussian Process Modeling for Identification of Structural Systems under Misspecified Dynamics

Bayesian filtering methods are an interesting avenue for structural health monitoring because they enable joint estimation of dynamic states and model parameters from noisy dynamics data, yielding a physics-based model of the structural system in its current condition necessary for prognosis. However, filtering methods such as the unscented Kalman filter (UKF) require knowledge of the functional form of the dynamics of the system, including potential nonlinearities, and model discrepancies can yield divergence of the filter and parameter estimates. This work proposes a framework that integrates a Gaussian process within the UKF to represent the unmodeled or missing dynamics, mitigating the impact of inaccurate modeling on the estimation of the physical parameters that characterize the known portion of the dynamics. First, we show that under some conditions on the measurements, integrating a probabilistic model to account for the uncertain portion of the dynamics yield heteroskedasticity and correlation between the process and measurement noise terms in the system’s equations, requiring appropriate modifications of the UKF formulation. We then present a method to learn the Gaussian process (GP)-based model error in a online fashion, which requires (1) careful consideration of the heteroskedastic uncertainty in both the input and output training data, and (2) a downsampling technique to avoid increase in computational time over time. We illustrate the behavior of our proposed algorithm on numerical examples, considering imperfectly known damping restoring forces. Using these examples we illustrate potential nonidentifiability issues that arise when performing joint state–parameter–model error estimation, and discuss the importance of appropriate GP prior design in this setting.