Third-order continuous-discrete filtering for a Stochastic Duffing System

The intent of this paper is to construct higherorder ‘continuous-discrete’ filtering equations for a Stochastic Duffing System, noisy non-linear dynamical system. The complexity associated with higher-order filtering for the ‘continuous state-discrete measurement system’ is chiefly attributed to two factors. First, the continuous-discrete filtering algorithm, a two-stage estimation procedure, encompasses the prediction algorithm un-accounting observation terms. The filtering algorithm at the discrete-time instant accounts observation correction terms. As a result of these, the continuous-discrete filtering involves two different sets of conditional mean and conditional variance evolution equations. On the other hand, the higher-order filtering for the ‘continuous state-continuous measurement system’ involves one set of conditional mean and conditional variance evolution equations. Secondly, deriving the higher-order filtering equations, e.g. the third-order at the observation under ‘nearly’ Gaussian assumptions, require replacing the sixth-order the conditional moment with conditional variance terms. The relationship involves the concept of the conditional characteristic function as well as greater permutation terms resulting lengthier expressions. On the other hand, the third-order ‘continuous’ filtering equations require replacing the fourth-order conditional moment with conditional variance terms. Subsequently, the efficacy of the filtering equations of this paper is examined on the basis of its comparison with extended Kalman filtering and true state trajectories.