Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems

Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.