Existence and design of observers for nonlinear systems: Relation to distance to unobservability

This paper presents a systematic design methodology and some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The existence of an asymptotically stable observer is shown to be guaranteed when the distance to unobservability is larger than the Lipschitz constant of the nonlinear system. An analytical solution for the observer is provided in this case. A methodology for the use of a coordinate transformation is then developed so as to reduce the Lipschitz constant and increase the distance to unobservability in the new coordinates. The methodology is directly applicable to the important class of feedback linearizable systems. The developed theory is used successfully in the design of an observer for a flexible joint robotic system.