Kullback-Leibler-Quadratic Optimal Control

This paper presents approaches to mean-field control, motivated by distributed control of multi-agent systems. Control solutions are based on a convex optimization problem, whose domain is a convex set of probability mass functions (pmfs). The main contributions follow: 1. Kullback-Leibler-Quadratic (KLQ) optimal control is a special case, in which the objective function is composed of a control cost in the form of Kullback-Leibler divergence between a candidate pmf and the nominal, plus a quadratic cost on the sequence of marginals. Theory in this paper extends prior work on deterministic control systems, establishing that the optimal solution is an exponential tilting of the nominal pmf. Transform techniques are introduced to reduce complexity of the KLQ solution, motivated by the need to consider time horizons that are much longer than the inter-sampling times required for reliable control. 2. Infinite-horizon KLQ leads to a state feedback control solution with attractive properties. It can be expressed as either state feedback, in which the state is the sequence of marginal pmfs, or an open loop solution is obtained that is more easily computed. 3. Numerical experiments are surveyed in an application of distributed control of residential loads to provide grid services, similar to utility-scale battery storage. The results show that KLQ optimal control enables the aggregate power consumption of a collection of flexible loads to track a time-varying reference signal, while simultaneously ensuring each individual load satisfies its own quality of service constraints. Keywords: Mean field games, distributed control, Markov decision processes, Demand Dispatch.