M.C. Rotkowitz and N.C. Martins
On the Nearest Quadratically Invariant Constraint
IEEE Transactions on Automatic Control, vol. 57, no. 5, pp. 1314-1319, May 2012.

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.

Notes

The numerical examples, as well as some of the discussion on finding the closest set or subset for sparsity constraints, had to be cut to meet the length requirements. Above is a link to the paper as it appeared in TAC, as well as to a full version as posted on arXiv. The full version is recommended.