Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems

Abstract

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

Multiple-Object Space Surveillance Tracking Using Finite-Set Statistics

“…is handled by the use of the AEGIS algorithm [12] n times (one for each conventional PDF). When a sensor is providing data, either zero or m > 0 measurements are received.…”

“…The particular GM technique used for the approximation is the adaptive entropy-based Gaussian-mixture information synthesis (AEGIS) method [11,12], which is an estimation approach for nonlinear continuous dynamical systems. AEGIS implements a GM model representation of the probability density function that is adapted online via splitting of the GM components whenever an entropy-based detection of nonlinearity-induced distortions of the Gaussian components is triggered during the forward propagation of the PDF.…”

Rapid uncertainty propagation and chance‐constrained path planning for small unmanned aerial vehicles

“…17,18 Other nonlinear UQ methods include unscented transformation, 19,20 polynomial chaos expansions, 21 and Gaussian mixture models. [22][23][24] A thorough survey of many other UQ methods was provided by Luo nd Yang, 25 though none of these were deemed appropriate for our use due to our models' large number of states, the presence of stochastic inputs, and the need for rapid computations. For that reason, the linear covariance (LC) propagation method is an attractive choice for our problem.…”

Parallel Computation of Trajectories Using Graphics Processing Units and Interpolated Gravity Models

“…The evolution of the difference in ARORA, VITTALDEV, AND RUSSELL the entropy from Eq. (6) of the two uncertainty propagation methods shows the nonlinearity, and therefore presence of any non-Gaussianity [66]. The entropy difference between the linear EKF method and the nonlinear second-order divided difference (DD2) method [67] is…”