Applications of Induced Tensor Norms to Guidance Navigation and Control

TL;DR

This paper introduces tensor operator norms to bound errors in linear control and estimation methods, enabling efficient analysis of their performance and applicability in guidance, filtering, and nonlinear systems.

Contribution

It presents a novel approach using tensor norms based on eigenvalues to analyze and bound errors in linear and higher-order approximations, improving computational efficiency and understanding.

Findings

Tensor norms provide accurate error bounds faster than sampling methods.

Eigenvalue-based tensor norms help determine the applicability of linear approximations.

The approach applies to guidance, filtering, and nonlinear dynamical systems.

Abstract

Linear methods are ubiquitous for control and estimation problems. In this work, we present a number of tensor operator norms as a means to approximately bound the error associated with linear methods and determine the situations in which that maximum error is encountered. An emphasis is placed on induced norms that can be computed in terms of matrix or tensor eigenvalues associated with coefficient tensors from higher-order Taylor series. These operator norms can be used to understand the performance and range of applicability of an algorithm exploiting linear approximations in different sets of coordinates. We examine uses of tensor operator norms in the context of linear and higher-order rendezvous guidance, coordinate selection for a filtering measurement model, and to present a unified treatment of nonlinearity indices for dynamical systems. Tensor norm computations can offer insights into these problems in one to two orders of magnitude less time than similarly accurate sampling methods while providing more general understanding of the error performance of linear or higher-order approximations.