Schmidt-Kalman Filter with Polynomial Chaos Expansion for State Estimation

Errors due to uncertain parameters of dynamical systems can result in deterioration of state estimation performance or even filter divergence sometimes using a conventional Kalman filter algorithm. Even worse, these parameters cannot be measured accurately or are unobservable for many applications. Hence, estimating parameters along with state variables would not achieve satisfactory performance. To handle this problem, the Schmidt-Kalman filter (SKF) was introduced to compensate for these errors by considering parameters' covariance, with an assumption of only Gaussian distributions. This paper introduces a new SKF algorithm with polynomial chaos expansion (PCE-SKF). Within the framework of PCE, the dynamical system is predicted forward with an ability to quantify non-Gaussian parametric uncertainties as well. More specifically, the a priori covariance of both the state and parameters can be propagated using PCE, followed by the update step of SKF formulation. Two examples are given to validate the efficacy of the PCE-SKF. The state estimation performance by PCE-SKF is compared with the extended Kalman filter, SKF, unscented Kalman filter and unscented Schmidt-Kalman filter. It is implied that the covariance propagation using PCE leads to more accurate state estimation solutions in comparison with those based on linear propagation or unscented transformation.