[Paper Review] Input-State-Parameter-Noise Identification and Virtual Sensing in Dynamical Systems: A Bayesian Expectation-Maximization (BEM) Perspective

Daniz Teymouri, Omid Sedehi+2 more|arXiv (Cornell University)|Jul 5, 2022

Structural Health Monitoring Techniques53 references22 citations

TL;DR

This paper proposes a Bayesian Expectation-Maximization (BEM) framework for joint input-state-parameter-noise identification in dynamical systems, using an augmented state-space model with Kalman filtering and smoothing. It enables uncertainty quantification, stabilizes low-frequency drifts via dummy input observations, and automatically calibrates noise covariance matrices through iterative EM updates, achieving accurate and robust estimation of states, inputs, parameters, and noise with posterior uncertainty characterization in numerical and experimental tests.

ABSTRACT

Structural identification and damage detection can be generalized as the simultaneous estimation of input forces, physical parameters, and dynamical states. Although Kalman-type filters are efficient tools to address this problem, the calibration of noise covariance matrices is cumbersome. For instance, calibration of input noise covariance matrix in augmented or dual Kalman filters is a critical task since a slight variation in its value can adversely affect estimations. The present study develops a Bayesian Expectation-Maximization (BEM) methodology for the uncertainty quantification and propagation in coupled input-state-parameter-noise identification problems. It also proposes the incorporation of input dummy observations for stabilizing low-frequency components of the latent states and mitigating potential drifts. In this respect, the covariance matrix of the dummy observations is also calibrated based on the measured data. Additionally, an explicit formulation is provided to study the theoretical observability of the Bayesian estimators, which helps characterize the minimum sensor requirements. Ultimately, the BEM is tested and verified through numerical and experimental examples, wherein sensor configurations, multiple input forces, and abrupt stiffness changes are investigated. It is confirmed that the BEM provides accurate estimations of states, input, and parameters while characterizing the degree of belief in these estimations based on the posterior uncertainties driven by applying a Bayesian perspective.

Motivation & Objective

• To address the challenge of simultaneous estimation of inputs, states, physical parameters, and noise characteristics in dynamical systems.
• To overcome the sensitivity of Kalman filters to noise covariance matrix calibration, which critically affects estimation accuracy.
• To stabilize low-frequency drifts in state estimates when only acceleration measurements are available, using input dummy observations.
• To provide a systematic, data-driven method for noise covariance matrix calibration through Bayesian inference.
• To establish theoretical observability conditions for partially known systems with unknown inputs using Lie derivatives.

Proposed method

• Proposes a Bayesian Expectation-Maximization (BEM) framework for joint estimation of inputs, states, parameters, and noise covariance matrices.
• Uses an augmented state-space model combining states, inputs, and parameters into a latent vector governed by a joint dynamics model.
• Incorporates input dummy observations with calibrated covariance to stabilize low-frequency components and mitigate drifts in state estimates.
• Employs an extended Kalman filter (EKF) for real-time filtering and a fixed-point smoother for improved state estimation accuracy.
• Applies the EM algorithm to iteratively update noise covariance matrices using explicit formulations derived from posterior expectations.
• Derives a closed-form observability condition using Lie derivatives to determine minimum sensor requirements for system identification.

Experimental results

Research questions

• RQ1 How can noise covariance matrices be automatically calibrated in coupled input-state-parameter estimation problems without relying on heuristic tuning?
• RQ2 What is the impact of low-frequency drifts in state estimates when only acceleration measurements are available, and how can they be mitigated?
• RQ3 How can the observability of a partially known dynamical system with unknown inputs be theoretically assessed?
• RQ4 What is the role of input dummy observations in stabilizing state estimation and improving convergence in the BEM framework?
• RQ5 How does the proposed BEM method compare to conventional Kalman filtering approaches in terms of accuracy and robustness under uncertain noise conditions?

Key findings

• The BEM framework successfully estimates states, inputs, parameters, and noise covariances with high accuracy in both numerical and experimental examples.
• The inclusion of input dummy observations with calibrated covariance significantly reduces low-frequency drifts in state estimates, especially when only acceleration data is available.
• The steady-state initialization algorithm for noise covariance matrices improves convergence speed and stability of the main BEM algorithm.
• Theoretical observability analysis via Lie derivatives provides a clear criterion for determining the minimum number of sensors required for system identification.
• Posterior uncertainty quantification is effectively captured, enabling confidence assessment of all estimated quantities.
• The method achieves robust performance across multiple input forces and abrupt stiffness changes, demonstrating its applicability to real-world structural health monitoring.