On the Nearest Quadratically Invariant Information Constraint

TL;DR

This paper investigates methods to find the nearest quadratically invariant constraints for decentralized control problems, enabling convex reformulations when the original constraints are not invariant, with specific algorithms for delay and sparsity constraints.

Contribution

It introduces convex optimization approaches to find closest quadratically invariant constraints, including algorithms for delay and sparsity cases where invariance does not hold.

Findings

Convex formulation for delay-based constraints

Algorithm for closest superset under sparsity constraints

Discussion on methods to find close subsets

Abstract

Quadratic invariance is a condition which has been shown to allow for optimal decentralized control problems to be cast as convex optimization problems. The condition relates the constraints that the decentralization imposes on the controller to the structure of the plant. In this paper, we consider the problem of finding the closest subset and superset of the decentralization constraint which are quadratically invariant when the original problem is not. We show that this can itself be cast as a convex problem for the case where the controller is subject to delay constraints between subsystems, but that this fails when we only consider sparsity constraints on the controller. For that case, we develop an algorithm that finds the closest superset in a fixed number of steps, and discuss methods of finding a close subset.