Estimation of variables and parameters in discrete time non-linear systems

A new method of estimating variables and parameters in nonlinear stochastic systems described by difference equations is given. The estimates are determined using the minimum mean square error criterion in the class of functions of observations satisfying certain difference equations. An exact solution of this problem is given based on using the difference equation for the characteristic function of the corresponding random variables similar to the author's equation for the one-dimensional characteristic function of a random process determined by a stochastic differential equation. The main feature of the new method is the computational simplicity of determining estimates which is reduced to applying a recursive formula. All the complicated calculations involved are based only on prior data and can be done while designing estimating algorithms or filters. As a special case the theory developed yields known results of Kalman on linear estimation of variables in linear systems.