Explainable LP-MPC: Shadow Price Contributions Reveal MV-CV Pairings

Abstract

Large industrial LP-MPC (Linear Program-Model Predictive Control) systems contain tens to hundreds of manipulated variables (MVs) and controlled variables (CVs) for honoring constraints while operating at plant-wide economic optima. The LP is a white-box model, yet for a number of reasons, abnormal behaviors in industrial controllers are often difficult to rationalize. We introduce a novel, post-hoc LP explainability method by recasting the role of shadow prices in the LP solution as an attribution mechanism for MV-CV relationships. The core idea is that CV shadow prices are not just intrinsic properties of the optimal solution, but an aggregate of the cost sensitivities contributed by MVs in enforcing CV constraints, which is then resolved into one-to-one MV-CV pairings using a linear sum assignment solution. The proposed MV-CV pairing framework serves as a practical explainability tool for online LP-MPC systems, enabling practitioners to diagnose suboptimal constraints and verify alignment of the controller's behavior with its original design intent and historical constraints.