Nonlinear Estimation for a Class of Systems

This paper considers nonlinear estimation problems for classes of models, and employs relative entropy to describe the uncertainty classes. Two optimization problems are formulated on general Banach spaces, and their solutions are sought: 1) when the transition probability between the signal to be estimated X and the measurement Y or stochastic kernel is unknown, and 2) when the joint probability induced by the random variables (RVs) X, Y is unknown. For both problems, the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution. The results include existence of the optimal measures using weak convergence techniques, and properties associated with the estimate of the true distribution. Classical examples are chosen to illustrate the applicability of the results.