Eigenvector Assignement for Damage Localization With Invariant Eigenvalues

Abstract

A recently proposed scheme infers the spatial location of structural damage by interrogating the damage-induced shifts in assigned closed-loop system eigenvalues. The concept is to render the assigned eigenvalues invariant to model-based damage pattern postulates using eigenstructure assignment with left eigenvectors. In an idealized setting, damage is then confined to the pattern for which the damage-induced shifts in the assigned eigenvalues are zero. In practice, invariance is unattainable due to disturbances and the inevitable discrepancy between the physical system and its model representation, thus damage is located when the induced eigenvalue shifts are minimized. This paper explores how the assignment of the associated left eigenvectors affects the methodological robustness and proposes three different eigenvector assignment procedures for promoting it. The first procedure, which constitutes an optimization problem, explicitly maximizes the sensitivity to damage outside the postulated patterns, while the second procedure seeks to maximize the noted sensitivity in a direct setting based on singular value decomposition (SVD). The third procedure, which is also cast as an optimization problem, seeks to promote robustness by minimizing the uncertainties on the assigned eigenvalues. The proposed procedures are tested in a numerical example, and their individual viability, in terms of damage resolution promotion and computational efficiency, is discussed.