An approximate solution to optimal Lp state estimation problems

We consider optimal estimation problems characterized by a state vector with i) dynamics described via a differential equation with Lipschitz nonlinearities, ii) partial information provided via a Lipschitz nonlinear mapping, and iii) an L_p norm measure of the estimation error to be minimized. An approximate solution of such optimal estimation problem is searched for by restricting the optimization to parameterized nonlinear approximators such as feedforward neural networks. The parameters of a feedforward neural network are the neural weights. This approach entails a constrained nonlinear programming problem, whose constraints are given by the dynamic and measurement equations, and the conditions guaranteeing the stability of the estimation error. To optimize the parameters values of neural networks, an algorithm is developed that is based on appropriate sampling of the state and error spaces. Choices of the sample points are devised based on the notion of dispersion, which allow one to obtain an approximate solution of the optimal estimation problem by a small sample complexity.