Output feedback stabilization for SISO nonlinear systems with an observer in the original coordinates

We address the problem of designing a stabilizing output feedback, via the separation principle. Our aim is to propose a more usable technique. The system can be written in any coordinates and is supposed to be locally uniformly observable. Starting form a known state feedback we do one step of backtepping to have access to the input derivative. This is sufficient to design a high gain observer in the original coordinates that we modify to prevent peaking and constrain the estimated state to remain in the observability domain.