Recursive principal component analysis for model order reduction with application in nonlinear Bayesian filtering

A linear Bayesian filter for input and state estimation of structural systems

“…The zero‐mean white noise vector $${{\bf{w}}_k} \in {{\mathbb{R}}^n}$$ represents modeling error. It should be emphasized that unlike Equation () that is expressed in terms of $${{\bf{d}}_k}$$, in the derivation of GDFs (Gillijns & De Moor, 2007a, 2007b), the process equation was formulated by choosing a different approximation of input with a ZOH assumption, that is, $${{\bf{d}}_{k - 1}}$$, leading to $${{\bf{x}}_k} = {{\bf{A}}_{k - 1}}{{\bf{x}}_{k - 1}} + {{\bf{G}}_{k - 1}}{{\bf{d}}_{k - 1}} + {{\bf{w}}_{k - 1}}$$, wherein the input in the process equation lags one step behind the input in the observation equation.…”

“…In these circumstances, an estimation technique that can treat systems without direct feedthrough will be needed. To cover systems with direct feedthrough (WDF), another algorithm was proposed by Gillijns and De Moor (2007b) for joint input and state estimation; this filter requires a full‐rank feedforward matrix, that is, the minimum number of acceleration measurements must be equal or greater than the number of unknown inputs. For convenience in referencing, a Gillijns–De Moor filter is abbreviated as GDF and these abbreviations are employed hereafter: GDF‐WNDF for the filter in Gillijns and De Moor (2007a) and GDF‐WDF for the filter in Gillijns and De Moor (2007b).…”

A Bayesian smoothing for input‐state estimation of structural systems

“…The major drawback of the latter methods is that the true characteristics of the loading statistics and its evolving model are oftentimes unknown. To cope with this issue, an output‐only minimum variance unbiased (MVU) Bayesian filter was developed by Gillijns and De Moor (2007b), which is applicable to systems modeled by state‐space equations with direct feedthrough. The aforementioned MVU filter was recently employed and extended to different structural system identification applications (Eftekhar Azam et al., 2017; Lourens, et al., 2012; Maes et al., 2016; Maes, Iliopoulos, et al, 2016; Wan et al, 2018).…”

“…Gillijns and De Moore (2007a) developed another MVU filter for systems without direct feedthrough. While the statistical properties of the presumed observation and modeling noise vectors are similar to the filter for systems with direct feedthrough, its structure and analytical derivations are substantially different (Gillijns & Moor, 2007b). In the systems with direct feedthrough, by receiving the observations at step $$k$$, the input at step $$k$$ can be estimated, as the observations contain information on the input at the current step $$k$$.…”

“…Concerning structural systems with direct feedthrough, Maes et al. (2018) showed that the instantaneous system inversion by MVU filtering technique proposed by Gillijns and De Moore (2007b) may lead to an ill‐conditioned problem for structural system identification. This ill‐conditionedness significantly affects the input reconstruction quality and notably amplifies the measurement (observation) noise.…”