State observers for nonlinear systems with smooth/bounded input

It is known that for affine nonlinear systems the drift-observability prope rty (i.e. observability for aero input) is not sufficient to guarantee the existence of an asymptotic observer for any input. Many authors studied str uctural conditions that ensure uniform observability of nonlinear systems ( i.e. observability for any input). Conditions are available that define cla sses of systems that are uniformly observable. This work considers the problem of state observation with exponential error rate for smooth nonlinear systems that meet or not conditions of uniform o bservability. In previous works the authors showed that drift-observability together with a smallness condition on the input is sufficient to ensure e xistence of an exponential observer. Here it is shown that drift-observabil ity implies a kind of local uniform observability, that is observability fo r sufficiently small and smooth input. For locally uniformly observable sys tems two observers are presented: an exponential observer that uses derivat ives of the input functions; an observer that does not use input derivative s and ensures exponential decay of the observation error below a prescribed level (high-gain observer). The construction of both observers is straight forward. Moreover the state observation is provided in the original coordin ate system. Simulation results close the paper.