Convex relaxations in the optimal control of electrified vehicles

When controlling the energy flow among multiple power sources in electrified powertrains, the typically non-convex set of the original formulation is relaxed to a convex super-set, and the problem is then approached by the means of convex optimization. In this paper we show that when using the backward simulation approach, where vehicle velocity is equal to the reference velocity, the global optimum of the original non-convex problem can be obtained by solving a relaxed convex problem. When vehicle velocity is kept as a state in the problem, in the so called forward simulation approach, we provide a condition for which, when satisfied, the solution of the relaxed problem will provide the solution of the non-relaxed problem.