On the design of approximate state estimators for nonlinear systems

State estimation for continuous-time, nonlinear dynamic systems with Lipschitz nonlinearities is considered. A class of estimators composed of a prediction term and an innovation term is defined, where the innovation function belongs to a suitable smoothness class and has to be determined in such a way as to minimize an estimation cost, represented by the L-infinity norm of the estimation error. Since the admissible innovation functions belong to an infinite-dimensional functional space, the minimization of such a cost represents a functional optimization problem, difficult to solve in a general setting. Approximating the innovation function by a family of parametrized nonlinear approximators allows one to reduce the original functional optimization problem to a sequence of nonlinear programming problems.