Optimal Rank-1 Directional State Transition Tensors

TL;DR

This work introduces optimal rank-1 directional state transition tensors (R1-ODSTTs) by maximizing information retention in the Frobenius-norm sense through a tensor z-eigenpair problem on the square of the STT. Compared to the original R1-DSTTs, ODSTTs select order-specific input directions, improving approximation accuracy for nonlinear dynamics and Gaussian moment propagation, especially in aerocapture and CR3BP scenarios. The authors derive the optimal rank-1 partially symmetric approximation, compare against R1-DSTTs, and demonstrate improved fidelity and a rigorous error bound, while acknowledging a higher computational cost due to tensor eigenproblem solutions. The results suggest ODSTTs are a strong first choice for rank-1 DSTTs in nonlinear applications, with future work extending to rank-k approximations.

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.

Figures (3)

• Figure 1: Normalized Frobenius Norm Error.
• Figure 2: Normalized Frobenius Norm Error of Nondimensional R1-DSTT Propagated Covariance.
• Figure 3: Norm of perturbation propagation error using R1-ODSTTs.