Stochastic Estimates of Nonlinear Dynamic Systems

Abstract

Stochastic estimates of nonlinear dynamic systems including hysteretic structural systems are formulated in the form of the Ito stochastic differential equations. Based on the theory of continuous Markov vector process, differential forms of conditional probability density functions given observations during a finite time interval are presented by making use of the Kolmogorov differential operators and innovations process. Differential forms of conditional expectations of an arbitrary differentiable function of state variables are also presented for filtering, smoothing and prediction problems arising in a general class of nonlinear dynamic systems. From these Ito-Dynkin formulas, truncated conditional moment equations are derived by introducing an approximate conditional probability density functions expressed in the form of a finite mixed type series expansion in terms of different sets of orthogonal polynomials. Special cases of the general results obtained in the present study are found to coincide with the known results for nonlinear filtering as well as linear filtering and smoothing. The solution techniques to obtain conditional probability density functions as well as optimal stochastic estimators associated with filtering and smoothing are described for a general class of nonlinear dynamic systems.