Assessment of validity of linear covariance analysis with unscented methods

Heflin, Lucas Benjamin

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https://hdl.handle.net/2142/115800 Copy

Description

Title

Assessment of validity of linear covariance analysis with unscented methods

Author(s)

Heflin, Lucas Benjamin

Issue Date

2022-04-29

Director of Research (if dissertation) or Advisor (if thesis)

Putnam, Zachary R

Department of Study

Aerospace Engineering

Discipline

Aerospace Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

• Aerospace Engineering
• Linear Covariance Analysis
• Monte Carlo
• Unscented Methods
• Linearization

Abstract

Linear covariance analysis is a computationally efficient method of uncertainty analysis that relies on linearization of system dispersions about a nominal nonlinear trajectory. However, this linearization makes linear covariance analysis only valid for a local neighborhood about the nominal trajectory in which the dispersion dynamics are sufficiently linear. The boundary of this neighborhood and what qualifies as “sufficiently linear” for engineering applications is poorly quantified. In practice, to validate linear covariance analysis, a more computationally intensive Monte Carlo simulation analysis is conducted. This thesis assesses the effectiveness of leveraging unscented covariance propagation and a novel statistical sample heuristic to estimate the validity of a linear covariance analysis applied to a nonlinear system. The mean and nominal states and the respective covariances of the unscented covariance propagation and the linear covariance analysis are compared using univariate hypothesis testing of means and variances, a univariate goodness-of-fit test with the Kolmogorov-Smirnoff statistic, and multivariate Hotelling’s Z2 statistic. These statistics are locally optimized to produce the sample heuristic, a metric that will be shown to be correlated with linearization validity for a given confidence level. The sample heuristic is evaluated by comparison of results to an equivalent Monte Carlo analysis for three nonlinear scenarios: the Cartesian-to-polar transformation, a single-order integrator, and a simplified hypersonic entry system. It is shown that the sample heuristics with the unscented methods are a good predictor of linear covariance analysis validity with the univariate goodness-of-fit tests and multivariate Z2 test being the strongest predictors. These tests are robust to singularities that cause false-negatives in the univariate heuristics due to a singular covariance matrix. The thesis is then concluded with a discussion on applications and future work. Applications include quantifying a mesh of initial conditions that, when propagated with linear covariance analysis, can be aggregated to produce posterior statistics much more efficiently and accurately than a Monte Carlo analysis. Similarly, the sample heuristic could be used to trigger a “branching” of the linear covariance analysis to account for bifurcations in the system dynamics. Future work includes assessing the impact of different sigma-point generation schemes in the unscented propagation on the efficacy of the sample heuristic, applying more advanced statistics, such as multivariate goodness-of-fit tests, to the sample heuristic, analysis of stochastic systems with integrated guidance, navigation, and control, and computational studies of algorithm efficiency. Developing analytical higher-order methods to validate linear covariance analysis as an alternative to the unscented methods presented is also discussed.

Graduation Semester

2022-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Lucas B. Heflin

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