Optimal Diesel engine calibration using convex modelling of Pareto frontiers

The optimal control of Diesel engines remains a challenging task. On the one hand, the number of control inputs is high, resulting in a large optimisation problem. On the other hand, low fuel consumption and low nitrogen oxides (NOx) emissions are conflicting objectives. This means there is no single best solution, but rather a set of Pareto optimal solutions. In this paper, we tackle the steady-state engine calibration problem by directly modelling the Pareto frontiers. This way, the degrees of freedom are reduced, resulting in a much simpler problem. Moreover, because the Pareto frontiers are (close to) convex, we are able to describe them by a convex function. We use lossless constraint relaxations to reformulate the problem as a convex optimisation problem. Solving this problem requires very little computation time and yields the globally optimal solution. The optimal control inputs can be retrieved from the optimal solution in a straightforward manner. We present experimental results to demonstrate the practical feasibility and effectiveness of the proposed approach. Furthermore, we show how the methodology can be readily extended to calculate application-specific calibrations that are tailored to typical in-use operation. Steady-state as well as transient measurements from the engine test-bench prove that significant fuel savings are achievable, while keeping the NOx emissions below the same limit.