Generalized sensor-to-sensor transmissibility operators: Theory, identification, and applications in soft sensing and modal estimation

Sensor-to-sensor transmissibility operators are mathematical objects that relate two subsets of system outputs. A transmissibility operator can be used along with one subset of outputs to predict the other subset of outputs of the underlying system without knowledge of a model of the underlying system or the excitation signal acting on it. Transmissibility operators have been used in applications including fault detection, virtual sensing, state estimation, and system identification. Standard transmissibility formulations assume that the number of transmissibility inputs equals the dimension of the excitation signal acting on the underlying system. However, since transmissibilities operate in environments with unknown inputs, estimating the dimension of the excitation signal can be challenging. Moreover, numerical evidence from previous research shows that the predicted outputs obtained using transmissibility operators become more accurate as the number of transmissibility inputs increases. In this paper, we introduce a more general mathematical representation of transmissibility operators that allows the number of transmissibility inputs to exceed the excitation dimension, hence the term generalized transmissibility operators. We further show that the determinant of the difference between two generalized transmissibility operators constructed between the same outputs but under different input locations can be used to determine the poles of the underlying system, outperforming existing time- and frequency-domain transmissibility-based modal estimation techniques. The framework is validated through numerical pole estimation of a mechanical structure and experimental soft sensing of an acoustic system, demonstrating improved accuracy and robustness over existing approaches.