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Efficient Optimal Control of Hybrid Systems with Conditioned Transitions

This paper considers an optimal control problem for discrete time hybrid systems with affine continuous dynamics and polyhedral guard sets. The control goal is to steer the system optimally from an initial state into a target region over a finite number of time steps. In contrast to optimal control of piecewise affine systems, the setting in this paper conditions the transitions to guard sets and external triggers. The paper proposes an efficient encoding of these logic conditions, and it shows that the original problem can be exactly cast into a Mixed Integer Programming problem without approximation. In addition, the proposed encoding method leads to a relatively low number of value combinations of binary variables to be explored in the search procedure, and thus reduces the practical complexity of the optimization with respect to the transition dynamics.