A Bayesian smoothing for input‐state estimation of structural systems

Abstract

Instantaneous output‐only inversion of a system with delayed appearance of input influences on the measured outputs via filtering methods suffer from intensive amplification of the observation noise in the estimated quantities due to the ill‐conditionedness. To remedy this issue, in this paper, a new unbiased recursive Bayesian smoothing method is developed for input‐state estimation of linear systems without direct feedthrough to reduce estimation uncertainty through an extended observation equation. By minimizing input and state estimation error variance, the optimal smoothing input and state gain matrices are derived. Moreover, a new efficient method is proposed for the recursive calculation of correlation of state estimation error with modeling and observation noise vectors.

A Smoothing EKF-UI-WDF Method for Simultaneous Identification of Structural Systems and Unknown Seismic Inputs without Direct Feedthrough

“…From equation (35), it is found that any value of K k+1 is acceptable for the unbiased condition once 􏽢 Z k|k+N−1 is unbiased, i.e., (E[􏽢 e Z k ] � 0).…”

“…Tis addition improves the quality of the estimation at the current step since the addition is from the later steps. Actually, such a technique is the so-called smoothing that has been adopted by researchers [28,[31][32][33][34][35]. Recently, Ebrahimzadeh et al [35] proposed a method for joint identifcation of structural state and unknown inputs by combining smoothing technique, but structural parameters need to be known prior.…”

“…Actually, such a technique is the so-called smoothing that has been adopted by researchers [28,[31][32][33][34][35]. Recently, Ebrahimzadeh et al [35] proposed a method for joint identifcation of structural state and unknown inputs by combining smoothing technique, but structural parameters need to be known prior. Maes et al [31] proposed a method of joint identifcation of input, state and parameter by combining smoothing and a two-step fltering for linear structural systems.…”

A modal‐based Kalman filtering framework for mode extraction and decomposition of damped structures

“…To overcome these limitations, a modal identification framework based on a novel mode decomposition technique that identifies modal response and mode shape by the Kalman filter defined in modal space is proposed in this study. In general, Kalman filtering has been mainly applied to estimate the state variable using the measured response and further extended to reconstruct the external load using the inverse problem (e.g., Gillijns & De Moor, 2007;Gordon et al, 1993;Hassanabadi et al, 2022). Peng et al (2019), Hwang et al (2009), Kalman (1960), Kang et al (2012), Lei et al (2019), andNiu et al (2015) proposed a modal-based Kalman filter in conjunction with the optimum sensor placement method for the response reconstruction and excitation estimation of structures by using noisy acceleration and strain measurements.…”

“…To overcome these limitations, a modal identification framework based on a novel mode decomposition technique that identifies modal response and mode shape by the Kalman filter defined in modal space is proposed in this study. In general, Kalman filtering has been mainly applied to estimate the state variable using the measured response and further extended to reconstruct the external load using the inverse problem (e.g., Gillijns & De Moor, 2007; Gordon et al., 1993; Hassanabadi et al., 2022). Peng et al.…”