Estimation of unknown inputs, with application to fault detection, via partial hierarchical observation

In this manuscript we show that neither left invertibility is a sufficient condition nor strong detectability is a necessary condition for estimating the unknown inputs. Here we show that the necessary and sufficient condition for estimating the unknown inputs is that the invariant zeros of the system not belonging to the set of unobservable eigenvalues have negative real part. For estimating the unknown inputs, we propose a method of design based on the partial state observation, which is made via a hierarchical scheme where a second order (super-twisting) sliding mode is used. This is also applied for the case of fault detection and isolation. We give a numerical example to illustrate the method of design proposed here.