An Equivalent Emission Minimization Strategy for Causal Optimal Control of Diesel Engines

One of the main challenges during the development of operating strategies for modern diesel engines is the reduction of the CO₂ emissions, while complying with ever more stringent limits for the pollutant emissions. The inherent trade-off between the emissions of CO₂ and pollutants renders a simultaneous reduction difficult. Therefore, an optimal operating strategy is sought that yields minimal CO₂ emissions, while holding the cumulative pollutant emissions at the allowed level. Such an operating strategy can be obtained offline by solving a constrained optimal control problem. However, the final-value constraint on the cumulated pollutant emissions prevents this approach from being adopted for causal control. This paper proposes a framework for causal optimal control of diesel engines. The optimization problem can be solved online when the constrained minimization of the CO2 emissions is reformulated as an unconstrained minimization of the CO₂ emissions and the weighted pollutant emissions (i.e., equivalent emissions). However, the weighting factors are not known a priori. A method for the online calculation of these weighting factors is proposed. It is based on the Hamilton–Jacobi–Bellman (HJB) equation and a physically motivated approximation of the optimal cost-to-go. A case study shows that the causal control strategy defined by the online calculation of the equivalence factor and the minimization of the equivalent emissions is only slightly inferior to the non-causal offline optimization, while being applicable to online control.