Estimation theory for nonlinear models and set membership uncertainty

This chapter studies the problem of estimating a given function of a vector of unknowns, called the problem element, by using measurements depending non-linearly on the problem element and affected by unknown but bounded noise. Assuming that both the solution sought and the measurements depend polynomially on the unknown problem element, a method is given to compute the axis-aligned box of minimal volume containing the feasible solution set, i.e., the set of all unknowns consistent with the actual measurements and the given bound on the noise. The center of this box is a point estimate of the solution, which enjoys useful optimality properties. The sides of the box represent the intervals of possible variation of the estimates. Important problems, like parameter estimation of exponential models, time series prediction with ARMA models and parameter estimates of discrete time state space models, can be formalized and solved by using the developed theory.