Semi-global Fixed-time State Estimation and unknown Input Reconstruction via First-order Sliding Mode observers with Delay

This paper addresses the state estimation and unknown input reconstruction in strongly observable linear systems under the so-called matching conditions. Our proposal extends previous approaches to predefined-time state estimation, which involved the use of a pair of time-delay observers, adapting them to the framework of sliding mode observation in the presence of unknown inputs. As a result, a novel sliding mode-based observation algorithm is developed, which exhibits desirable properties such as global finite-time convergence or semi-global fixed-time convergence. Notably, the method is also capable of reconstructing the unknown inputs without requiring any low-pass filtering or continuous approximations of the discontinuous observer signals. The convergence properties are proven through a formal stability analysis and validated through numerical simulations.