An SOS method for the design of continuous and discontinuous differentiators

Given a (differentiable) signal, it is an important task for many applications to estimate on line its derivatives. Some well-known algorithms to solve this problem include the (continuous) high-gain observers and (discontinuous) Levant's exact differentiators. With exception of the linear high-gain observers, a systematic design of the gains of nonlinear differentiators is, in general, a difficult task and an open research field. In this work, we propose a novel method for the gain-tuning of a family of homogeneous differentiators which estimate the time-derivatives of a signal in finite-time. We show that the stability analysis and the gain-tuning of such differentiators can be done under a unified Lyapunov function framework, and it is converted to a sum of squares problem, that can be solved using LMIs, much in the same spirit of the linear systems.