This article explores various uncertain control co-design (UCCD) problem formulations. While previous work offers formulations that are method-dependent and limited to only a handful of uncertainties (often from one discipline), effective application of UCCD to real-world dynamic systems requires a thorough understanding of uncertainties and how their impact can be captured. Since the first step is defining the UCCD problem of interest, this article aims at addressing some of the limitations of the current literature by identifying possible sources of uncertainties in a general UCCD context and then formalizing ways in which their impact is captured through problem formulation alone (without having to immediately resort to specific solution strategies). We first develop and then discuss a generalized UCCD formulation that can capture uncertainty representations presented in this article. Issues such as the treatment of the objective function, the challenge of the analysis-type equality constraints, and various formulations for inequality constraints are discussed. Then, more specialized problem formulations such as stochastic in expectation, stochastic chance-constrained, probabilistic robust, worst-case robust, fuzzy expected value, and possibilistic chance-constrained UCCD formulations are presented. Key concepts from these formulations, along with insights from closely-related fields, such as robust and stochastic control theory, are discussed, and future research directions are identified.

This article explores various uncertain control co-design (UCCD) problem formulations. While previous work offers formulations that are method-dependent and limited to only a handful of uncertainties (often from one discipline), effective application of UCCD to real-world dynamic systems requires a thorough understanding of uncertainties and how their impact can be captured. Since the first step is defining the UCCD problem of interest, this article aims at addressing some of the limitations of the literature by identifying possible sources of uncertainties in a general UCCD context and then formalizing ways in which their impact is captured through problem formulation alone (without having to immediately resort to solution strategies). We first develop a universal UCCD formulation and discuss its fundamental elements. Issues such as the treatment of the objective function, the challenge of the analysis-type equality constraints, and various formulations for inequality constraints are discussed. Then, more specialized problem formulations such as the risk-neutral and risk-averse stochastic, worst-case robust, probabilistic robust, fuzzy expected value, and possibilistic chance-constrained UCCD formulations are presented. Key concepts from these formulations, along with insights from closely-related fields, such as robust and stochastic control theory, are discussed, and future research directions are identified.

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 has 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 proposing two optimal control structures for UCCD problems that we refer to as the open-loop single-control (OLSC) and open-loop multiple-control (OLMC). Next, we implement the stochastic in expectation UCCD (SE-UCCD) and worst-case robust UCCD (WCR-UCCD) for a simplified strain-actuated solar array (SASA) case study. For the implementation of SE-UCCD, we use generalized Polynomial Chaos expansion and benchmark the results against Monte Carlo Simulation. Next, we solve a simple SASA WCR-UCCD through OLSC and OLMC structures. Insights from such implementations indicate that constructing, implementing, and solving a UCCD problem requires an in-depth understanding of the problem at hand, formulations, and solution strategies to best address the underlying co-design under uncertainty questions.