Atomic Norm Minimization for Modal Analysis from Compressive Measurements
One analytical technique for assessing the health of a structure such as a building or bridge is to estimate its mode shapes and frequencies via vibrational data collected from the structure. A change in a structure's modal parameters could be indicative of damage. Due to the considerable time and expense required to perform manual inspections of physical structures, and the difficulty of repeating these inspections frequently, there is a growing interest in developing automated techniques for structural health monitoring (SHM) based on data collected in a wireless sensor network (WSN) [1] [2] [3] In order to save energy and extend battery life, it is desirable to reduce the dimension of data that must be collected and transmitted in the WSN. In recent work [4] Park et al. provided a rigorous analysis of a singular value decomposition (SVD) based technique for estimating the structure's mode shapes in free vibration without damping. As a means of compression, this work considered both random sampling in time and multiplication by random matrices. While promising, the SVD based algorithm requires orthogonality of the mode shapes and offers only approximate, not exact recovery. Recently, atomic norm minimization (ANM) based approaches for line spectrum estimation have been shown to be efficient and powerful for exactly recovering unobserved samples and identifying off grid frequencies in both single measurement vector (SMV) [5] [6] and multiple measurement vector (MMV) [7] [8] scenarios. With a sufficient number of random samples and sufficiently well separated frequencies, exact frequency localization can be guaranteed. In particular, theoretical guarantees have been established for random sampling in time when the sampling times are synchronous [7] and asynchronous [8] across the sensors. However, these guarantees assume randomness of the mode shapes, which is not physically plausible. Moreover, these guarantees suggest that sample complexity per sensor will increase as the the number of sensors increases, which is both undesirable and contrary to intuition. In this work, we consider the modal analysis problem when data is compressed at each sensor via multiplication by a random matrix. We show that ANM can perfectly recover modal parameters even when the mode shapes are not orthogonal. We provide new theoretical analysis on the sample complexity of this scheme. In particular, our theory does not require randomness of the mode shapes, and it shows that the sample complexity per sensor will decrease as the the number of sensors increases. Our theory can be interpreted as an extension of the SMV treatment in [6] to the MMV scenario.