Real-Time Minimization of Average Error in the Presence of Uncertainty and Convexification of Feasible Sets

We consider a two-level discrete-time control framework with real-time constraints where a central controller issues setpoints to be implemented by local controllers. The local controllers implement the setpoints with some approximation and advertize a prediction of their constraints to the central controller. The local controllers might not be able to implement the setpoint exactly, due to prediction errors or because the central controller convexifies the problem for tractability. In this paper, we propose to compensate for these mismatches at the level of the local controller by using a variant of the error diffusion algorithm. We give conditions under which the minimal (convex) invariant set for the accumulated-error dynamics is bounded, and give a computational method to construct this set. This can be used to compute a bound on the accumulated error and hence establish convergence of the average error to zero. We illustrate the approach in the context of real-time control of electrical grids.