Dynamic programming method for electrified vehicles

Dynamic programming (DP) is a numerical technique that enables solving all types of optimal control problems. In this article, two main problems will be addressed while using the DP technique. The first one is related to a fuel consumption minimization problem, where only the power distribution between the primary (engine or a fuel cell system) and secondary power source (electric machine with battery pack) is optimized without loss of genericity. The second problem deals with a similar problem, however, here the power distribution is extended with the optimization of the engine on/off state and the gear position. In the last part of the article explains by introducing two examples on how to implement optimal controls solutions in an online manner. Thereto, predictive frameworks will be discussed based on the model-predictive control (MPC) method as first example and are extended using vehicle velocity optimization as second example. The main contribution in this article for the reader is on the understanding of how to implement the DP algorithm for: (i) a (relatively simple) continuous optimal control problem; and (ii) for a hybrid optimal control problem with discrete and continuous dynamics with mixed constraints on states and inputs. Finding a solution to the latter problem is far from trivial due to its non-convex character and here we show how to design a solution that is close to global optimal using a combination of DP and the Pontryagin's minimum principle (PMP) Finally, the last contribution is on understanding (iii) how to implement combinations of the DP-PMP technique for online control.