On nonlinear observers with approximately linear error dynamics

We discuss the design of observers for autonomous nonlinear systems with one output. New necessary and sufficient conditions for existence of a solution to the observer error linearization problem are given. In the case when an exact linearization of the observer error dynamics is impossible, a simple least-squares scheme is proposed that yields an observer whose error dynamics is approximately linear. An algorithm for observer construction is provided. An example of observer design for a mass-spring system with nonlinear spring and dynamic friction forces is given.