Eigenperturbation-based Detection of Features of Interest and the Importance of Stochastic Differential Equations in Understanding the Behaviour and Control of Dynamical Systems

The domain of SHM has known many developments in recent years. Among those, the improvement of damage detection algorithms was significant. That was especially the case for eigen-perturbation algorithms, like first order eigen-perturbation (FOP) which uses recursive rules to update an eigendecomposition. From such a result, the eigendecomposition of a multidimensional structural system's accelerations can be updated with reduced cost. Their impact in detecting damage from acceleration signals was well established through simulations and experiments. It was nonetheless observed that FOP had limitations in terms of accuracy over time and limited performance with mild to moderate damping ratios. This motivated the research presented here where it was proposed to increase the accuracy of such an algorithm through the development of higher order eigen-perturbation algorithms. These algorithms are subsequently tested with various structural and dynamical systems where their error performance and computational efficiency were quantified to show positive and significant gains. Alternatively, numerical simulations are a core pillar of the tools used to parameterize, analyse and make decisions regarding the design and control of structural systems. For this purpose, stochastic differential equations (SDEs) have been used to simulate structural systems in an alternative to avoid the use of an inconsistent call to ordinary calculus. The theory of stochastic calculus has been leveraged in many a case of random excitation. This is not yet the case of Pendulum Tuned Mass Dampers (PTMDs). It was therefore proposed to explore the realm of capabilities provided by stochastic calculus to simulate a structure under random excitation equipped with a PTMD. This was performed in many forms, with a three dimensional PTMD, a two dimensional PTMD equipped with an Shape Memory Alloy (SMA) wire and a 2-degree of freedom (DOF) building system. It further allowed tuning and optimal control of these systems. Both of these works, damage detection and stochastic simulation were lastly linked through the implementation of a damage detection and retuning process for a 2-DOF and PTMD system experiencing instantaneous damage