A case study comparing both stochastic and worst-case robust control co-design under different control structures

TL;DR

The paper investigates uncertainty in uncertain control co-design by comparing stochastic in expectation ($SE$-$UCCD$) and worst-case robust ($WCR$-$UCCD$) formulations under three control structures: open-loop multiple-control ($ ext{OLMC}$), multi-stage control ($ ext{MSC}$), and open-loop single-control ($ ext{OLSC}$). It demonstrates SASA as a simple benchmark, applying Monte Carlo Simulation ($ ext{MCS}$) and generalized Polynomial Chaos ($ ext{gPC}$) for forward uncertainty propagation, and uses polytopic uncertainty for $WCR$ with a nested coordination strategy to manage computational complexity. Key findings show that SE-UCCD with OLMC can closely match MCS results using $ ext{gPC}$ with substantial time savings, while MSC and WCR-UCCD provide notable improvements in performance and robustness at the cost of higher computation; worst-case designs yield quick, consistent closed-loop behavior but can be overly conservative. The work provides practical guidance for selecting UCCD formulations in early-stage design and highlights avenues for extending uncertainty handling to probabilistic constraints, path-dependent disturbances, and scalable implementations. Overall, the study underscores the critical role of control structure and uncertainty propagation in balancing risk and performance in dynamic co-design.

Abstract

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.

Figures (11)

• Figure 1: Comparison of OLSC, MSC, and OLMC structures with $\mathcal{U}_{a}$ describing the admissible control set, and $\Delta t_{r}$ being the robust horizon.
• Figure 2: Illustration of OLMC, and OLSC structures for an arbitrary problem.
• Figure 3: Illustration of steps involved in uaing gPC for uncertainty propagation in a two-dimensional uncertain problem: Step 1: Create the $n_x$-variate gPC basis functions $\Phi_{m}(\cdot)$ using the tensor product of univariate basis functions selected from Table \ref{['Tab:Polyrep']}, Step 2: Find quadrature nodes $\bm{x}_{j}$ and weights $\alpha_{w_{j}}$ by selecting a certain number of collocation nodes in each dimension, Step 3: solve the deterministic problem at all of the quadrature nodes to find $y(t,\bm{x}_{j})$ where ${y}$ is any desired uncertain output such as states or controls, and Step 4: estimate statistics of the output using Eq. (\ref{['Eq:gPC9']}).
• Figure 4: Illustration of the original and simplified strain-actuated solar array systems.
• Figure 5: Open-loop multiple-control (OLMC), stochastic in expectation UCCD (SE-UCCD) solution using MCS and gPC in comparison to the deterministic CCD solution.