Bayesian system identification and dynamic virtualization using incomplete noisy measurements

This study presents the application of Bayesian Expectation-Maximization (BEM) methodology to coupled state-input-parameter estimation in both linear and nonlinear structures. The BEM is built upon a Bayesian foundation, which utilizes the EM algorithm to deliver accurate estimates for latent states, model parameters, and input forces while updating noise characteristics effectively. This feature allows for quantifying associated uncertainties using response-only measurements. The proposed methodology is equipped with a recursive backward-forward Bayesian estimator that provides smoothed estimates of the state, input, and parameters during the Expectation step. Next, these estimates help calculate the most probable values of the noise parameters based on the observed data. This adaptive approach to the coupled estimation problem allows for real-time quantification of estimation uncertainties, whereby displacement, velocity, acceleration, strain, and stress states can be reconstructed for all degrees-of-freedom through virtual sensing. Through numerical examples, it is demonstrated that the BEM accurately estimates the unknown quantities based on the measured quantities, not only when a fusion of displacement and acceleration measurements is available but also in the presence of acceleration-only response measurements

Asymptotic expansions for reliabilities and moments of uncertain dynamic systems

An asymptotic approximation is developed for evaluating the probability integrals which arise in the determination of the reliability and response moments of uncertain dynamic systems subject to stochastic excitation. The method is applicable when the probabilities of failure or response moments conditional on the system parameters are available, and the effect of the uncertainty in the system parameters is to be investigated. In particular, a simple analytical formula for the probability of failure of the system is derived and compared to some existing approximations, including an asymptotic approximation based on SORM methods. Simple analytical formulas are also derived for the sensitivity of the failure probability and response moments to variations in parameters of interest. Conditions for which the proposed asymptotic expansion is expected to be accurate are presented. Since numerical integration is only computationally feasible for investigating the accuracy of the proposed method for a small number of uncertain system parameters, simulation techniques are also used. A simple importance sampling method is shown to converge much more rapidly than straightforward Monte-Carlo simluation. Simple structures subjected to white noise stochastic excitation axe used to illustrate the accuracy of the proposed analytical approximation. Results from the computationally efficient perturbation method are also included for comparison. The results show that the asymptotic method gives acceptable approximations, even for systems with relatively large uncertainty, and in most cases, it outperforms the perturbation method

Bayesian Optimal Sensor Placement for Virtual Sensing and Strain Reconstruction

A Bayesian optimal sensor placement (OSP) framework is presented for virtual sensing in structures using output-only vibration measurements. Particularly, this probabilistic OSP scheme aims to enhance the reconstruction of dynamical responses (e.g., accelerations, displacements, strain, stresses) for updating structural reliability and safety, as well as fatigue lifetime prognosis. The OSP framework is formulated using information theory. The information gained from a sensor configuration is defined as the Kullback-Liebler divergence (KL-div) between the prior and posterior distributions of the response quantities of interest (QoI). The Gaussian nature of the response estimate for linear models of structures is employed, and the information gain is characterized in terms of the reconstruction error covariance matrix. A Kalman-based input-state estimation technique is integrated within an existing OSP strategy, aiming to obtain estimates of response QoI and their uncertainties. The design variables include the location, type and number of sensors. Heuristic algorithms are used to solve optimization problem and provide computationally efficient solutions. The effectiveness of the method is demonstrated using an example from structural dynamics

A New Online Bayesian Approach for the Joint Estimation of State and Input Forces using Response-only Measurements

In this paper, a recursive Bayesian-filtering technique is presented for the joint estimation of the state and input forces. By introducing new prior distributions for the input forces, the direct transmission of the input into the state is eliminated, which allows removing low-frequency error components from the predictions and estimations. Eliminating such errors is of practical significance to the emerging fatigue monitoring methodologies. Furthermore, this new technique does not require a priori knowledge of the input covariance matrix and provides a powerful method to update the noise covariance matrices in a real-time manner. The performance of this algorithm is demonstrated using one numerical example and compared it with the state-of-the-art algorithms. Contrary to the present methods which often produce unreliable and inaccurate estimations, the proposed method provides remarkably accurate estimations for both the state and input.Financial support from the Hong Kong research grants councils under grant numbers 16234816 and 16212918 is gratefully appreciated. The last author gratefully acknowledges the European Commission for its support of the Marie Sklodowska Curie program through the ETN DyVirt project (GA 764547). This paper is completed as a part of the second authors PhD dissertation conducted jointly at Sharif University of Technology and the Hong Kong University of Science and Technology. The second author would like to gratefully appreciate kind support and supervision of Professor Fayaz R. Rofooei at Sharif University of Technology

Quantification of Aleatory Uncertainty in Modal Updating Problems using a New Hierarchical Bayesian Framework

Identification of structural damage requires reliable assessments of damage-sensitive quantities, including natural frequencies, mode shapes, and damping ratios. Lack of knowledge about the correct value of these parameters introduces a particular sort of uncertainty often referred to as epistemic uncertainty. This class of uncertainty is reducible in a sense that it can be decreased by enhancing the modeling accuracy and collecting new information. On the contrary, such damage-sensitive parameters might also have intrinsic randomness arising from unknown phenomena and effects, which gives rise to an irreducible category of uncertainty often referred to as aleatory uncertainty. The present Bayesian modal updating methodologies can produce reasonable quantification of the epistemic uncertainties, while they often fail to account for the aleatory uncertainties. In this paper, a new multilevel (hierarchical) probabilistic modeling framework is proposed to bridge this significant gap in uncertainty quantification and propagation of structural dynamics inverse problems. Since multilevel model calibration schemes establish a complicated model structure associated with additional parameters and variables, their computational costs are often considerable, if not prohibitive. To reduce the computational costs, the modal updating procedure is simplified using a second-order Taylor expansion approximation. This approximation is combined with a Markov chain Monte-Carlo (MCMC) sampling method to compute marginal posterior distributions of quantities of interest. The proposed framework is illustrated using one simple experimental example. As a result, it is demonstrated that the proposed framework surpasses the present Bayesian modal updating methods as it accounts for both the aleatory and epistemic uncertainties.Financial support from the Hong Kong research grants councils under grant numbers 16234816 and 16212918 is gratefully appreciated. The last author gratefully acknowledges the European Commission for its support of the Marie Sklodowska Curie program through the ETN DyVirt project (GA 764547). This paper is completed as a part of the second authors PhD dissertation conducted jointly at Sharif University of Technology and the Hong Kong University of Science and Technology. The second author would like to gratefully appreciate kind support and supervision of Professor Fayaz R. Rofooei at Sharif University of Technology. We would also like to express our sincere appreciation to Professor Chih-chen Chang for generously sharing sensors, prototypes, and laboratory facilities

Comparative Value of Simple Inflammatory Markers in the Prediction of Left Ventricular Systolic Dysfunction in Postacute Coronary Syndrome Patients

Objectives. We sought to assess the comparative value of inflammatory markers on the occurrence of left ventricular systolic dysfunction (LVSD) after an acute coronary syndrome (ACS). Methods. During 2006–2008, 760 patients with an ACS were enrolled. C-reactive protein (CRP) and white blood cell (WBC) count were measured during the first 12 hours of hospital admission. Results. CRP levels and WBC count were significantly higher in those who developed LVSD compared to those who did not. The analysis revealed that a 10 mg/dL increase of CRP levels and a 1000/μL increase in WBC are associated with a 6% and a 7% increase in the likelihood of developing LVSD, respectively. Furthermore, WBC count at entry and CRP have almost the same predictive value for development of LVSD after an ACS (R2 = 0.109 versus R2 = 0.093). Conclusions. Serum CRP levels and WBC count at entry are almost equally powerful independent predictors of LVSD, after an ACS

An analytical perspective on Bayesian uncertainty quantification and propagation in mode shape assembly

Assembling local mode shapes identified from multiple setups to form global mode shapes is of practical importance when the degrees of freedom (dofs) of interest are measured separately in individual setups or when one expects to exploit the computational autonomous capabilities of different setups in full-scale operational modal test. The Bayesian mode assembly methodology was able to obtain the optimal global mode shape as well as the associated uncertainties by taking the inverse of the analytically derived Hessian matrix of the negative log-likelihood function (NLLF) (Yan and Katafygiotis, 2015) [1]. In this study, we investigate how the posterior uncertainties existing in the local mode shapes obtained from different setups propagate into the global mode shapes in an explicit manner by borrowing a novel approximate analysis strategy. The explicit closed-form approximation expressions are derived to investigate the effects of various data parameters on the posterior covariance matrix of the global mode shapes. Such quantitative relationships, connecting the posterior uncertainties with global mode shapes and the data information, offer a better understanding of uncertainty propagation over the process of mode shape assembly. The posterior uncertainty of the global mode shapes is inversely proportional to ‘normalized data length’ and the ‘frequency bandwidth factor’, and propositional to ‘noise-to-environment’ ratio and damping ratio. Validation studies using field test data measured from the Metsovo bridge located in Greece provide a practical verification of the rationality of the theoretical findings of uncertainty quantification and propagation analysis in Bayesian mode shape assembly