Optimal control for integrated emission management in diesel engines

Integrated Emission Management (IEM) is a supervisory control strategy that minimises operational costs (consisting of fuel and AdBlue) for diesel engines with an aftertreatment system, while satisfying emission constraints imposed by legislation. In most work on IEM, a suboptimal heuristic real-time implementable solution is used, which is based on Pontryagin's Minimum Principle (PMP). In this paper, we compute the optimal solution using both PMP and Dynamic Programming (DP). As the emission legislation imposes a terminal state constraint, standard DP algorithms are sensitive to numerical errors that appear close to the boundary of the feasible sets. Therefore, we propose two extensions to existing DP methods, which use an approximation of the forward reachable sets to reduce the grid size over time and an approximation of the backward reachable sets to avoid the aforementioned numerical errors. Using a simulation study of a cold-start World Harmonised Transient Cycle for a Euro-VI engine, we show that the novel extension to the DP algorithm yields the best approximation of the optimal cost, when compared to existing DP methods. Furthermore, we show that PMP yields almost the same results as DP, and that the real-time implementable solution only deviates approximately 0.08-0.16% from the optimal solution.