Obscured by terminology: Hidden parallels in direct methods for open-loop optimal control

Active research on optimal control methods comprises the developments of research groups from various fields, including control, mathematics, and process systems engineering. Although there is a consensus on the classification of the main solution methods, different terms are often used for the same method. For example, solving optimal control problems with control discretization and embedded state integration may be called sequential method or direct single shooting. Equally severely, the same term may be used ambiguously: Is control vector parameterization a synonym for control discretization or for direct single shooting? Both misleading distinctions and ambiguity complicate the scientific discourse. Thus, we delineate standard terms from open-loop optimal control in this tutorial. More precisely, we formulate and challenge hypotheses on the terminology of direct methods, i.e., solution methods using control discretization combined with state integration and/or state discretization. In particular, we point out the parallel of the embedded state integration with a numerical integration scheme and the reduced-space formulation of approaches using state discretization. Taking a step further towards integrated scheduling and control problems, we additionally investigate the similarities and differences between the discrete-time solution of optimal control problems and optimal quasi-steady operation. In this context, we also hint on the discrete-time representation in scheduling which refers to the handling of controls rather than the handling of process dynamics. Rather than concluding with the “correct” term to use, this tutorial concludes with recommendations on how to avoid misunderstandings in the versatile research community.