On the Semi-Global Stability of an EK-Like Filter

This letter proposes to apply the Kalman-like observer paradigm to general nonlinear systems by linearization along the estimated trajectory, similarly to an Extended Kalman Filter. The main difference is that the quadratic Riccati equation is replaced by a linear Lyapunov equation which can be solved and explicitly related to a determinability Gramian. This allows to show by Lyapunov analysis and without any ad-hoc assumption on the Riccati solution, that the resulting observer, called Extended Kalman-like Filter, can be made semi-globally convergent if the input is actively used to a) stabilize the (unknown) true trajectory, b) sufficiently excite the determinability of the linearized systems along the (known) estimated trajectory. A class of systems where this compromise can be reached is provided.