A technique to Improve the Empirical Mode Decomposition in the Hilbert-Huang Transform

The Hilbert-based time-frequency analysis has promising capacity to reveal the time-variant behaviors of a system. To admit well-behaved Hilbert transforms, component decomposition of signals must be performed beforehand. This was first systematically implemented by the empirical mode decomposition (EMD) in the Hilbert-Huang transform, which can provide a time-frequency representation of the signals. The EMD, however, has limitations in distinguishing different components in narrowband signals commonly found in free-decay vibration signals. In this study, a technique for decomposing components in narrowband signals based on waves’ beating phenomena is proposed to improve the EMD, in which the time scale structure of the signal is unveiled by the Hilbert transform as a result of wave beating, the order of component extraction is reversed from that in the EMD and the end effect is confined. The proposed technique is verified by performing the component decomposition of a simulated signal and a free decay signal actually measured in an instrumented bridge structure. In addition, the adaptability of the technique to time-variant dynamic systems is demonstrated with a simulated time-variant MDOF system

Structural Health Monitoring by Recursive Bayesian Filtering

A new vision of structural health monitoring (SHM) is presented, in which the ultimate goal of SHM is not limited to damage identification, but to describe the structure by a probabilistic model, whose parameters and uncertainty are periodically updated using measured data in a recursive Bayesian filtering (RBF) approach. Such a model of a structure is essential in evaluating its current condition and predicting its future performance in a probabilistic context. RBF is conventionally implemented by the extended Kalman filter, which suffers from its intrinsic drawbacks. Recent progress on high-fidelity propagation of a probability distribution through nonlinear functions has revived RBF as a promising tool for SHM. The central difference filter, as an example of the new versions of RBF, is implemented in this study, with the adaptation of a convergence and consistency improvement technique. Two numerical examples are presented to demonstrate the superior capacity of RBF for a SHM purpose. The proposed method is also validated by large-scale shake table tests on a reinforced concrete two-span three-bent bridge specimen

Large-Scale Shake Table Test Verification of Bridge Condition Assessment Methods

Methods that identify structural component stiffness degradation by pre- and postevent low amplitude vibration measurements, based on a linear time-invariant (LTI) system model, are conceptually justified by examining the hysteresis loops the structural components experience in such vibrations. Two large-scale shake table experiments, one on a two-column reinforced concrete (RC) bridge bent specimen, and the other on a two-span three-bent RC bridge specimen were performed, in which specimens were subjected to earthquake ground motions with increasing amplitude and progressively damaged. In each of the damaged stages between two strong motions, low amplitude vibrations of the specimens were aroused, and the postevent component stiffness coefficients were identified by optimizing the parameters in a LTI model. The stiffness degradation identified is consistent with the experimental hysteresis, and could be quantitatively related to the capacity residual of the components

Baseline Models for Bridge Performance Monitoring

A baseline model is essential for long-term structural performance monitoring and evaluation. This study represents the first effort in applying a neural network-based system identification technique to establish and update a baseline finite element model of an instrumented highway bridge based on the measurement of its traffic-induced vibrations. The neural network approach is particularly effective in dealing with measurement of a large-scale structure by a limited number of sensors. In this study, sensor systems were installed on two highway bridges and extensive vibration data were collected, based on which modal parameters including natural frequencies and mode shapes of the bridges were extracted using the frequency domain decomposition method as well as the conventional peak picking method. Then an innovative neural network is designed with the input being the modal parameters and the output being the structural parameters of a three-dimensional finite element model of the bridge such as the mass and stiffness elements. After extensively training and testing through finite element analysis, the neural network became capable to identify, with a high level of accuracy, the structural parameter values based on the measured modal parameters, and thus the finite element model of the bridge was successfully updated to a baseline. The neural network developed in this study can be used for future baseline updates as the bridge being monitored periodically over its lifetime

Low Rank Directed Acyclic Graphs and Causal Structure Learning

Despite several important advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse. In particular, the recent formulation of structure learning as a continuous optimization problem proved to have considerable advantages over the traditional combinatorial formulation, but the performance of the resulting algorithms is still wanting when the target graph is relatively large and dense. In this paper we propose a novel approach to mitigate this problem, by exploiting a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model. We establish several useful results relating interpretable graphical conditions to the low rank assumption, and show how to adapt existing methods for causal structure learning to take advantage of this assumption. We also provide empirical evidence for the utility of our low rank algorithms, especially on graphs that are not sparse. Not only do they outperform state-of-the-art algorithms when the low rank condition is satisfied, the performance on randomly generated scale-free graphs is also very competitive even though the true ranks may not be as low as is assumed

And\^o dilations for a pair of commuting contractions: two explicit constructions and functional models

One of the most important results in operator theory is And\^o's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an And\^o dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of And\^o's dilation theorem by giving both Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction $T$ into the product of two commuting contractions. Unlike the one-variable case, two minimal And\^o dilations need not be unitarily equivalent. However, we show that the compressions of the two And\^o dilations constructed in this paper to the minimal dilation spaces of the contraction $T_1T_2$, are unitarily equivalent. In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}\to 0$ strongly, an And\^o dilation was constructed recently in \cite{D-S-S}, which, as this paper will show, is a corollary to the Douglas-type construction. We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a pure contraction.Comment: 24 page