Utility of high-order sliding mode differentiators for dynamical left inversion problems

This study highlights some crucial issues related to dynamical left inversion problem for non-linear systems, that is the recovery of the applied input function from the knowledge of the output and its derivatives. In particular, it is shown that the presence of non-smooth unknown inputs and non-involutivity of the forced fields´ distribution constitutes an obstruction to conducting a left inversion. As a solution to this problem, the authors propose a novel left-invertible form that explicitly considers smooth as well as non-smooth inputs and for which they design a high-order sliding mode observer to reconstruct the states and the control inputs. Simulations results are provided to validate the performance of the proposed approach.