Compressive sensing-moving horizon estimator for distributed loads

The compressive sensing-moving horizon estimator (CS-MHE) was recently proposed for joint state/input estimation. It exploits the moving horizon estimator potential to correlate a model with measurements within a finite length time window sliding over time, while considering the involved stochastic phenomena. Furthermore, compressive sensing principles allow the observation of a large number of input locations with minimal instrumentation. This paper summarizes the CS-MHE approach and its implementation, and then extends it with a complex formulation which enables the use of complex representations such as Fourier shape functions in order to model an input signal. Therefore, we address the main challenges arising when handling complex components within an optimization framework. To assess the validity of the proposed approach, we consider a numerical test case that involves a cantilever beam subjected to a periodic load distributed in time, modeled by complex Fourier components and applied at a known position on the structure. The resulting problem is recast as a second order cone program (SOCP) and it is solved by commercial software. The proposed handling of complex representations such as Fourier components can be instrumental to estimate forces and torques in rotating machinery, whose nature is quasi-periodic.