On recursive proper orthogonal decomposition via perturbation theory with applications to distributed sensing in cyber-physical systems

Distributed sensing of cyber-physical systems has become feasible with recent developments in sensor technology, wireless communication and distributed computing. Distributed sensing generates huge amounts of data from the events occurring in the physical side, which should be promptly reflected in the cyber side so that actions can be made timely by the computing systems. Due to the dense temporal-spatial distribution of the measured data, great challenges have been posed in terms of data storage, information processing and communications. The proper orthogonal decomposition (POD) method is a powerful tool to extract dominant information from distributed observational data, which has been widely used in signal processing and pattern analysis of fluid turbulence. The classical POD method implements dominant information extraction when the entire data set is known. However, in real-time measurements, new data is collected and incorporated into the historic data set at each sampling time. We propose a recursive proper orthogonal decomposition (rPOD) method based on the operator perturbation theory, where the accumulative truncation error can be controlled by a gradient search algorithm. This method is illustrated with two state-of-the-art problems governed by the heat conduction equation (1D) and the Navier-Stokes equations (2D) respectively.