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# Learning solutions to hybrid control problems using Benders cuts

*Proceedings of the 2nd Conference on Learning for Dynamics and Control*, PMLR 120:118-126, 2020.

#### Abstract

Hybrid control problems are complicated by the need to make a suitable sequence of discrete decisions related to future modes of operation of the system. Model predictive control (MPC) encodes a finite-horizon truncation of such problems as a mixed-integer program, and then imposes a cost and/or constraints on the terminal state intended to reflect all post-horizon behaviour. However, these are often ad hoc choices tuned by hand after empirically observing performance. We present a learning method that sidesteps this problem, in which the so-called N-step Q-function of the problem is approximated from below, using Bendersâ€™ decomposition. The function takes a state and a sequence of N control decisions as arguments, and therefore extends the traditional notion of a Q-function from reinforcement learning. After learning it from a training process exploring the state-input space, we use it in place of the usual MPC objective. We take an example hybrid control task and show that it can be completed successfully with a shorter planning horizon than conventional hybrid MPC thanks to our proposed method. Furthermore, we report that Q-functions trained with long horizons can be truncated to a shorter horizon for online use, yielding simpler control laws with apparently little loss of performance.