Joint state/input estimation with a Fourier dictionary for the input representation: effect of spectral leakage

Abstract

The compressive sensing-moving horizon estimator (CS-MHE) is an approach for joint state/input estimation. It integrates compressive sensing principles into a moving horizon estimator, enabling to exploit shape functions to model an input, resulting in better observability and wider input bandwidth in comparison to other input models. With the final aim of using the CS-MHE for the estimation of forces and torques in rotating machinery which exhibit some form of periodicity, the authors have recently investigated Fourier shape functions. A first experimental validation showed very accurate estimation under the hypothesis of no spectral leakage, i.e., the MHE window, the sampling rate and the Fourier dictionary match a known input periodicity, with the input consisting of few sinusoidal components. This paper discusses the problem of spectral leakage that can result if the MHE window does not match the signal periodicity. In particular, we show how to remove the link between the MHE window and a Fourier dictionary, we discuss how the autocorrelation can be employed to detect the periodicity and we list further possible alternatives to enhance a sparse solution. The discussion is supported by numerical and experimental investigations.