Unification of closed-loop scheduling and control: State-space formulations, terminal constraints, and nominal theoretical properties

Due to increases in both connectivity of process systems and availability of computing power, there is growing interest in the integration of online scheduling and control. While significant progress has been made on formulations and optimization strategies for this integrated problem, obtaining a single solution is not sufficient for a practical implementation: because of disturbances and the inherent finite-horizon scope, it eventually becomes necessary to reoptimize considering the new state of the plant. When this loop is closed, there is an unfortunate gap in knowledge with respect to theoretical properties possessed by the resulting evolution of the system. Thus, to facilitate an analysis of the online behavior, we present a unified state-space formulation for integrated scheduling and control. We then use recent theory from economic model predictive control to derive terminal constraints that can be added to the problem to provide nominal recursive feasibility and a bound on closed-loop performance. Via examples, we demonstrate the improved performance of these techniques relative to naive closed-loop implementation strategies. These developments put closed-loop scheduling, economic-optimizing control, and the integrated problem within a common framework, and they can hopefully serve to guide the development of alternative decomposition strategies that still enjoy the desired nominal theoretical properties while avoiding unexpected pathological behavior.