Observer Design for Non-Globally Lipschitz Nonlinear Systems Using Hilbert Projection Theorem

This letter deals with observer design for a class of Lipschitz nonlinear systems. Specifically, we propose a mathematically rigorous technique to handle systems having non-globally Lipschitz properties on the whole set R-n. The unique assumption made on the nonlinearity is for it to be Lipschitz on a compact convex set Omega subset of R-n, in which lives the system state. The idea consists in extending the nonlinear function to become globally Lipschitz on the whole space R-n. Such an extension is performed by using the famous Hilbert projection theorem, which generalizes some existing results in the literature. The projection is then involved in the observer structure to overcome the non-satisfaction of the global property by the original nonlinear function. More importantly, to overcome the conservatism related to the boundedness of the system states, an extension to systems having only some bounded states is proposed under different but less conservative assumptions. It is shown that all the previous observer design methods in the literature that rely on a global Lipschitz property can be applied straightforwardly without changing their synthesis conditions.