Parallelotopic and practical observers for non-linear uncertain systems

For a class of dynamical systems, with uncertain non-linear terms considered as unknown inputs, we give suffcient conditions for observability. We show also that there does not exist any exact observer independent of the unknown inputs. Under the additional assumption that the uncertainty is bounded, we build practical observers whose error converges exponentially towards an arbitrarily small neighbourhood of the origin. Under the hypothesis that bounds are available for the uncertain terms, we build parallelotopic observers providing time-varying bounds for the state variables, even when the system is not observable for unknown inputs. These results are illustrated with a biological model of a structured population.