Control Optimization of Discontinuous Hybrid Systems Using Embedding

In this paper, a framework for the approximation of optimal controls for a class of hybrid systems that exhibit discontinuities in the states on a change of the active subsystem is proposed. Via a discretization, the hybrid optimal control problem is first transcribed to a nonlinear program with integer constraints. The integer constraints are then relaxed and the height of the jump is approximated by a function that is continuous but nonsmooth with respect to the discrete optimization variables. The resulting nonsmooth nonlinear program can in many cases be solved using standard variable metric methods. A blackbox method is used to provide gradients, where the model is nonsmooth. The proposed method is demonstrated on the problem of an optimal energy management of a hybrid vehicle with different drive modes.