A Generalization of Ackermann’s Formula for the Design of Continuous and Discontinuous Observers

This paper proposes a novel design algorithm for nonlinear state observers for linear time-invariant systems. The approach is based on a well-known family of homogeneous differentiators and can be regarded as a generalization of Ackermann’s formula. The method includes the classical Luenberger observer as well as continuous or discontinuous nonlinear observers, which enable finite time convergence. For strongly observable systems with bounded unknown perturbation at the input the approach also involves the design of a robust higher order sliding mode observer. An inequality condition for robustness in terms of the observer gains is presented. The properties of the proposed observer are also utilized in the reconstruction of the unknown perturbation and robust state-feedback control.