Extension of the observability rank condition to nonlinear systems driven by unknown inputs

This paper investigates the unknown input observability problem in the nonlinear case under the assumption that the unknown inputs are differentiable functions of time (up to a given order). The goal is not to design new observers but to provide simple analytic conditions in order to check the weak local observability of the state. The analysis starts by providing a new definition of indistinguishable states in the case of unknown inputs. Then, in order to separate the effect of the known inputs from the effect of the unknown inputs on the system outputs, the state is augmented. This allows us to obtain the extension of basic properties, which hold in the case of known inputs. Starting from these properties, the paper provides a sufficient analytic condition for the state observability, which is called the extended observability rank condition. The proposed approach is used to derive the observability properties of two systems. The former is very simple while the latter is very complex and describes the fusion of visual and inertial measurements.