  As uncertainty considerations become increasingly important aspects of
concurrent plant and control optimization, it is imperative to identify and
compare the impact of uncertain control co-design (UCCD) formulations on their
associated solutions. While previous work developed the theory for various UCCD
formulations, their implementation, along with an in-depth discussion of the
structure of UCCD problems, implicit assumptions, method-dependent
considerations, and practical insights, is currently missing from the
literature. Therefore, in this study, we address some of these limitations by
focusing on UCCD formulations, with an emphasis on optimal control structures,
and uncertainty propagation techniques. Specifically, we propose three optimal
control structures for UCCD problems: (i) open-loop multiple-control (OLMC),
(ii) multi-stage control (MSC), and (iii) open-loop single-control (OLSC).
Stochastic in expectation UCCD (SE-UCCD) and worst-case robust UCCD (WCR-UCCD)
formulations, which are motivated by probabilistic and crisp representations of
uncertainties, respectively, are implemented for a simplified strain-actuated
solar array case study. Solutions to the OLMC SE-UCCD problem are obtained
using two uncertainty propagation techniques: generalized Polynomial Chaos
expansion (gPC) and Monte Carlo simulation (MCS). The OLMC and MSC WCR-UCCD
problems are solved by leveraging the structure of the linear program, leading
to polytopic uncertainties. To highlight the importance of uncertainty in
early-stage design, the closed-loop reference-tracking response of the systems
is also investigated. Insights from such studies underscore the role of the
control structure in managing the trade-offs between risk and performance, as
well as meeting problem requirements. The results also emphasize the benefits
of efficient uncertainty propagation techniques for dynamic optimization
problems.
