Doctoral Dissertations

Keywords and Phrases

Angular uncertainty propagation; Polynomial chaos; Rogers-Szego-chaos


“The state of a dynamical system will rarely be known perfectly, requiring the variable elements in the state to become random variables. More accurate estimation of the uncertainty in the random variable results in a better understanding of how the random variable will behave at future points in time. Many methods exist for representing a random variable within a system including a polynomial chaos expansion (PCE), which expresses a random variable as a linear combination of basis polynomials.

Polynomial chaos expansions have been studied at length for the joint estimation of states that are purely translational (i.e. described in Cartesian space); however, many dynamical systems also include non-translational states, such as angles. Many methods of quantifying the uncertainty in a random variable are not capable of representing angular random variables on the unit circle and instead rely on projections onto a tangent line. Any element of any space V can be quantified with a PCE if V is spanned by the expansion’s basis polynomials. This implies that, as long as basis polynomials span the unit circle, an angular random variable (either real or complex) can be quantified using a PCE.

A generalization of the PCE is developed allowing for the representation of complex valued random variables, which includes complex representations of angles. Additionally, it is proposed that real valued polynomials that are orthogonal with respect to measures on the real valued unit circle can be used as basis polynomials in a chaos expansion, which reduces the additional numerical burden imposed by complex valued polynomials. Both complex and real unit circle PCEs are shown to accurately estimate angular random variables in independent and correlated multivariate dynamical systems”--Abstract, page iii.


DeMars, Kyle J.

Committee Member(s)

Darling, Jacob E.
Hosder, Serhat
Paige, Robert L.
Pernicka, Henry J.


Mechanical and Aerospace Engineering

Degree Name

Ph. D. in Aerospace Engineering


Missouri University of Science and Technology

Publication Date

Summer 2020


x, 110 pages

Note about bibliography

Includes bibliographic references (pages 103-110).


© 2020 Christine Louise Schmid, All rights reserved.

Document Type

Dissertation - Open Access

File Type




Thesis Number

T 11756

Electronic OCLC #