Compressive sensing-moving horizon estimator for combined state/input estimation: an observability study

The compressive sensing-moving horizon estimator (CS-MHE) is a recently proposed approach for combined state/input estimation. It exploits the intrinsic capability of a moving horizon estimator of minimizing the noise while correlating a model with measurements, together with compressive sensing principles that allow the observation of a large number of input locations for a small set of measurements. The CS-MHE has been shown able to estimate the states and an unknown force impulse on a linear time-invariant mechanical system, in terms of input magnitude, time and position, provided that the input is sparse. This paper summarizes the CS-MHE approach and discusses observability, focusing on the differences between the CS-MHE, a MHE with no assumptions on the input and a MHE in which inputs are described by a random walk model. This comparison illustrates the benefit of exploiting known information about the input behavior and allows to define an observability threshold on input sparsity for the CS-MHE.