  We analyze a time-coarsening strategy for model predictive control (MPC) that
we call diffusing-horizon MPC. This strategy seeks to overcome the
computational challenges associated with optimal control problems that span
multiple timescales. The coarsening approach uses a time discretization grid
that becomes exponentially more sparse as one moves forward in time. This
design is motivated by a recently established property of optimal control
problems that is known as exponential decay of sensitivity. This property
states that the impact of a parametric perturbation at a future time decays
exponentially as one moves backward in time. We establish conditions under
which this property holds for a constrained MPC formulation with linear
dynamics and costs. Moreover, we show that the proposed coarsening scheme can
be cast as a parametric perturbation of the MPC problem and thus the
exponential decay condition holds. We use a heating, ventilation, and air
conditioning plant case study with real data to demonstrate the proposed
approach. Specifically, we show that computational times can be reduced by two
orders of magnitude while increasing the closed-loop cost by only 3%.
