ARRTOC: Adversarially Robust Real-Time Optimization and Control

TL;DR

ARRTOC tackles the problem of Real-Time Optimization set-points that remain operable under control-layer disturbances by embedding Adversarially Robust Optimization at the RTO layer. It formalizes a constrained robust framework with an uncertainty set $\mathcal{U}$, including ellipsoidal generalizations $\mathcal{U} = \left\{\boldsymbol{\Delta}\boldsymbol{x} \middle| \sum_i \frac{(\Delta x_i)^2}{\Gamma_i^2} \le 1\right\}$, and solves $\min_{\boldsymbol{x}} \max_{\boldsymbol{\Delta}\boldsymbol{x}\in\mathcal{U}} f(\boldsymbol{x}+\boldsymbol{\Delta}\boldsymbol{x})$ subject to $\max_{\boldsymbol{\Delta}\boldsymbol{x}\in\mathcal{U}} h_j(\boldsymbol{x}+\boldsymbol{\Delta}\boldsymbol{x}) \le 0$, using a tutorial-style ARRTOC procedure with neighbourhood cost/constraint exploration and SOCP-based robust local moves. The authors demonstrate ARRTOC on an illustrative 2D problem and on two industrially relevant processes—a continuous bioreactor and a multi-loop evaporator—showing improved operability and, in some cases, greater profitability (e.g., up to 50% improvement in RTO objectives) when considering control-layer robustness. Key contributions include the practical integration of constrained ARO into the RTO layer, a flexible ellipsoidal uncertainty framework with per-state bounds $\Gamma_i$, and a robust local-move algorithm that simultaneously respects safety constraints and improves worst-case performance. The work highlights ARRTOC’s potential to offload robustness requirements from resource-constrained controllers onto the RTO layer, enabling better overall process performance and reliability.

Abstract

Real-Time Optimization (RTO) plays a crucial role in the process operation hierarchy by determining optimal set-points for the lower-level controllers. However, at the control layer, these set-points may be difficult to track due to challenges in implementation as a result of disturbances, measurement noise, and actuator performance limitations. To address this, in this paper, we present the Adversarially Robust Real-Time Optimization and Control (ARRTOC) algorithm. ARRTOC addresses this issue by finding set-points which are both optimal and inherently robust to implementation errors at the control layers. ARRTOC draws inspiration from adversarial machine learning, offering a novel constrained Adversarially Robust Optimization (ARO) solution applied to the RTO layer. By integrating controller design with RTO, ARRTOC enhances overall system performance and robustness by ensuring the chosen set-points are tailored to the underlying controller designs. To validate our claims, we present three case studies: an illustrative example, a bioreactor case study, and a multi-loop evaporator process. The proposed approach is found to improve RTO objectives, such as profit, by as much as $50\%$ in some case studies compared to RTO formulations which ignore the performance of the control layers.

Figures (16)

• Figure 1: A simple one dimensional example depicting the nominal objective function as a black curve and the worst-case objective function ($\Gamma = 0.5$) as a red curve. The nominal optimum is depicted as a black cross and the adversarially robust optimum is depicted as a red cross.
• Figure 2: (a): A depiction of the neighbourhood of a point $\mathbf{\hat{x}}$. (b): Neighbourhood (green hue) of a point, depicted as a black cross, for the simple one dimensional example from section \ref{['constrained:prob_def']}.
• Figure 3: Both figures depict infeasible regions in blue and red. (a): There are no constraint violations in the neighbourhood of the point i.e. it is feasible under perturbations. The algorithm's goal is to reduce the worst-case cost. (b): The neighbourhood of point $\mathbf{\hat{x}}$ intersects with an infeasible region, rendering the point infeasible under perturbations. The algorithm's goal is to find a descent direction to guide the iterate back into the feasible region.
• Figure 4: A successful application of a neighbourhood exploration. Global maximizers, from Eq. (\ref{['eq:unconstrained_worst_possible_neighbours']}), are indicated by light grey dashed arrows. Descent directions based on these maximizers is shown as a light grey cone with dashed lines. No knowledge of these is assumed. Instead, the inner-maximization problem is solved to give local solutions, represented as black arrows. Crucially, the set of descent directions derived from the local maximizers (dark grey cone with solid lines) is a subset of the desired descent directions based on the global maximizers.
• Figure 5: A contour plot of the nominal optimization problem from Eq. (\ref{['eq:illustrative_nominal']}). The global optimum is depicted as a blue cross while the adversarially robust optimum, given $\Gamma = 0.3$, is depicted as a red cross.