NIRK-based Cholesky-factorized square-root accurate continuous-discrete unscented Kalman filters for state estimation in nonlinear continuous-time stochastic models with discrete measurements

Abstract

This paper further advances the idea of accurate Gaussian filtering towards efficient unscented-type Kalman methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the differential equations evolving sigma points utilized in computations of the predicted mean and covariance in time-propagations of the Gaussian distribution are solved accurately, i.e. with negligible error. The latter allows the total error of the unscented Kalman filtering technique to be reduced significantly and gives rise to the novel accurate continuous-discrete unscented Kalman filtering algorithm. At the same time, this algorithm is rather vulnerable to round-off and numerical integration errors committed in each state estimation run because of the need for the Cholesky decomposition of covariance matrices involved. Such a factorization will always fail when the covariance's positivity is lost. This positivity lost issue is commonly resolved with square-root filtering implementations, which propagate not the full covariance matrix but its square root (Cholesky factor) instead. Unfortunately, negative weights encountered in applications of the accurate continuous-discrete unscented Kalman filter to high-dimensional stochastic systems preclude from designing conventional square-root methods. In this paper, we address this problem with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined numerically in an air traffic control scenario.