Hybrid Vehicle Energy Management: Singular Optimal Control

Hybrid vehicle energy management is often studied in simulation as an optimal control problem. Under strict convexity assumptions, a solution can be developed using Pontryagin's minimum principle. In practice, however, many engineers do not formally check these assumptions resulting in the possible occurrence of so-called unexplained “numerical issues.” This paper intends to explain and solve these issues. Due to the binary controlled-state variable considered (e.g., switching on/off an internal combustion engine) and the use of a lookup table with linear interpolation (e.g., engine fuel consumption map), the corresponding Hamiltonian function can have multiple minima. Optimal control is not unique. Moreover, it is defined as being singular. Consequently, an infinite number of optimal state trajectories can be obtained. In this paper, a control law is proposed to easily construct a few of them.