Practical observers for uncertain affine output injection systems

For nonlinear systems in output injection feedback form with a scalar uncertainty on the nonlinear term, we give necessary and sufficient conditions for the observability with unknown input. The known approach of building an observer decoupled from the disturbances is not possible. Under the additional assumption that the uncertainty is bounded, we build practical observers whose error converges exponentially towards an arbitrary small neighborhood of the origin, and interval observers giving guaranteed time-varying bounds on the state variables. We illustrate these results on a biological model of a structured population.