Exact state estimation for linear systems with unknown inputs based on hierarchical super-twisting algorithm

A robust hierarchical observer is designed for linear time invariant systems with unknown bounded inputs under conditions of strong observability, providing exact state estimation. The main condition for designing the state estimator is the, so-called, strong observability condition. The supertwisting (second-order sliding mode) algorithm is used in each step of the hierarchy; the continuity of the supertwisting output injection allows to reconstruct a vector formed by some full column rank matrix premultiplied by the state vector, and that vector is obtained in of finite time and without any sort of filtration. For the case when the unknown inputs are considered as constant uncertain parameters, the continuous version of the least-square method is developed. Two numerical examples illustrate the efficiency of the suggested technique.