Optimal Rank-1 Directional State Transition Tensors

TL;DR

This paper introduces an optimal rank-1 approximation method for state transition tensors that enhances accuracy and efficiency in nonlinear uncertainty quantification, especially in aerospace applications.

Contribution

It develops a novel optimal directional state transition tensor construction by solving a tensor eigenpair problem, improving over previous methods.

Findings

Increased approximation accuracy of state transition tensors.

Enhanced Gaussian moment propagation in nonlinear flight scenarios.

Outperforms previous directional tensor methods in efficiency.

Abstract

An optimal rank-1 approximation of state transition tensors was developed as an efficient alternative to state transition tensors for nonlinear uncertainty quantification. While previous directional state transition tensors used the dominant right singular subspace of the state transition matrix to construct a reduced-dimension representation of the state transition tensors, optimal directional state transition tensors are constructed to maximize the information retained in a rank-1 approximation of the state transition tensors in the Frobenius-norm sense. The optimal rank-1 directional state transition tensor is found by solving a tensor z-eigenpair problem of the "square" of the state transition tensor. This construct leads to increased approximation accuracy of the state transition tensors and improved Gaussian moment propagation for nonlinear flight scenarios like aerocapture.