State observers for nonlinear systems with smooth/bounded input

It is known that for affine nonlinear systems the drift-observability property (i. e. observability for zero input) is not sufficient to guarantee the existence of an asymptotic observer for any input. Many authors studied structural conditions that ensure uniform observability of nonlinear systems (i. e. observability for any input). Conditions are available that define classes 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 observability. In previous works the authors showed that drift-observability together with a smallness condition on the input is sufficient to ensure existence of an exponential observer. Here it is shown that drift-observability implies a kind of local uniform observability, that is observability for sufficiently small and smooth input. For locally uniformly observable systems two observers are presented: an exponential observer that uses derivatives of the input functions; an observer that does not use input derivatives and ensures exponential decay of the observation error below a prescribed level (high-gain observer). The construction of both observers is straightforward. Moreover the state observation is provided in the original coordinate system. Simulation results close the paper.