Data-driven full-field vibration response estimation from limited measurements in real-time using dictionary learning and compressive sensing

Full-field online sensing provides dense spatial information of vibrating structures in real-time. Such compact sensing is essential to pinpoint the location of possible damage and for real-time active/semi-active control of structures where the vibration responses are necessary for controller feedback. To accurately harness these dense sensor time history, contact-based sensors need to be installed in a dense way that is cost-inefficient and infeasible for a real-life scenario. Computer vision-based technologies like Digital Image Correlation (DIC), and edge tracking methods are capable of capturing dense responses. However, when the structures are in operating condition, long-term implementation of such techniques could be expensive and, at times, impractical. In this paper, to address these problems, we propose a framework to estimate full-field vibration responses from the time histories of a few sensors. In this framework, we use the compressive sensing technique in the spatial domain, where the full-field spatial signal is recovered from a handful of sensors for a definite time instant and repeating this procedure for all the time instants will lead to online full-field response estimation. This framework does not require any prior information on the structural model, making it entirely data-driven. This framework learns the spatial basis functions needed for compressive sensing operations from the training data using the Dictionary learning technique. We also address the optimal/minimum number of sensors required to accurately estimate the dense responses, which is based on the Singular Value Decomposition (SVD) of the basis matrix. This technique primarily applies to Linear Time-Invariant (LTI) systems, and implementation to Linear Time-Varying (LTV) systems could be explored in future work. We validate the proposed method numerically on a simply supported beam (1D system) and a simply supported plate (2D system) and experimentally on a cantilever beam. The reconstruction accuracy in the proposed full-field sensing shows excellent potential in health monitoring and control of aerospace, mechanical, and civil systems.