  Generalized Disjunctive Programming (GDP) provides an alternative framework
to model optimization problems with both discrete and continuous variables. The
key idea behind GDP involves the use of logical disjunctions to represent
discrete decisions in the continuous space, and logical propositions to denote
algebraic constraints in the discrete space. Compared to traditional
mixed-integer programming (MIP), the inherent logic structure in GDP yields
tighter relaxations that are exploited by global branch and bound algorithms to
improve solution quality. In this paper, we present a general GDP model for
optimal control of hybrid systems that exhibit both discrete and continuous
dynamics. Specifically, we use GDP to formulate a model predictive control
(MPC) model for piecewise-affine systems with implicit switching logic. As an
example, the GDP-based MPC approach is used as a supervisory control to improve
energy efficiency in residential buildings with binary on/off, relay-based
thermostats. A simulation study is used to demonstrate the validity of the
proposed approach, and the improved solution quality compared to existing
MIP-based control approaches.
