Nonlinear Estimation in Polynomial Chaos Framework

In this paper we present two nonlinear estimation algorithms that combine generalized polynomial chaos theory with higher moment updates and Bayesian framework. Polynomial chaos theory is used to predict the evolution of uncertainty of the nonlinear random process. In the first estimation algorithm, higher order moment updates are used to estimate the posterior non Gaussian probability density function of the random process. The moments are updated using a linear gain. In the second method, Bayesian update rule is used to determine the posterior probability density function. Since polynomial chaos is a method of moments, it does not directly yield the needed probability density functions. The density function is determined from the moments using Gaussian mixture models and maximum entropy optimization. The nonlinear estimation algorithms are applied to the duffing oscillator system with initial condition uncertainty and its performance is compared with an estimator based on extended Kalman filtering (EKF) framework. We observe that the proposed estimators outperform the EKF based estimator when measurements are not available very frequently, thus highlighting the need for nonlinear estimator in such scenarios.