Fault Diagnosis for a PDE Suspended-Bridge Model With Kalman Filter and Statistical Decision Making

Abstract

This article proposes a fault diagnosis method for the partial differential equation (PDE) model that describes the interaction between a suspended-bridge and a moving vehicle. Using semidiscretization and the finite differences approach, and considering a grid of N equidistant points on the bridge's longitudinal axis, the PDE that describes the spatiotemporal dynamics of the bridge is decomposed into a set of N ordinary differential equations (ODEs). Thus, a state-space model of the bridge and vehicle interaction is obtained comprising N+2 ordinary differential equations. Equivalently, one can obtain a matrix description of this state-space model of dimension 2N+4. Moreover, by denoting transformed control inputs, this state-space model is finally written in the canonical Brunovsky form which is known to be observable. For the latter state-space representation of the bridge-vehicle system in the canonical form, state estimation is performed with the use of Kalman filtering. Next, by comparing the outputs of the Kalman Filter with the real-time outputs of the bridge-vehicle system, the residuals sequence is generated. It is demonstrated that the sum of the squares of the residuals vectors weighted by the inverse of their covariance matrix stands for a stochastic variable, which follows the χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> distribution. Thus, by exploiting the 96% or the 98% confidence intervals of the χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> distribution one can define fault thresholds which show in an almost infallible manner, whether the bridge retains its structural integrity or whether it has been subjected to a fault.