Network Entropy and Data Rates Required for Networked Control

Abstract

We consider the problem of making a set of states invariant for a network of controlled systems. We assume that the subsystems, initially uncoupled, must be interconnected through controllers to be designed with a constraint on the data rate obtained by every subsystem from all other subsystems. As a measure for the smallest data rate arriving at a fixed subsystem, above which the overall system is able to achieve the control goal, we introduce the notion of subsystem invariance entropy. Moreover, we associate with a network of $n$ subsystems, a closed convex subset of ${\mathbb{R}}^{n}$ encompassing all possible combinations of data rates within the network that guarantee the existence of corresponding feedback strategies for making a given set invariant. The extremal points of this convex set can be regarded as Pareto-optimal data rates for the control problem, expressing a tradeoff between the data rates required by different systems. For linear systems and for synchronization of chaos, these quantities are characterized.