Nonlinear observers for autonomous Lipschitz continuous systems

This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct observers of a remarkably simple structure: A linear state variable filter followed by a nonlinearity. The observer is established by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist, and by giving an explicit expression for its nonlinearity. It is demonstrated that the existence conditions are satisfied, and hence our results include a new observer which is not high-gain, for the wide class of smooth systems. It is shown that the observer can as well be designed to realize an arbitrary, finite accuracy rather than ultimate exactness. On a compact region of the state space, this requires only observability of the system. A corresponding numerical design procedure is described, which is easy to implement and computationally feasible for low order systems.