Convexity of Decentralized Controller Synthesis

In decentralized control problems, a standard approach is to specify the set of allowable decentralized controllers as a closed subspace of linear operators. This then induces a corresponding set of Youla-Kucera parameters. Previous work has shown that quadratic invariance of the controller set implies that the set of Youla-Kucera parameters is convex. In this technical note, we prove the converse. We thereby show that the only decentralized control problems for which the set of Youla-Kucera parameters is convex are those which are quadratically invariant. We further show that under additional assumptions, quadratic invariance is necessary and sufficient for the set of achievable closed-loop maps to be convex. We give two versions of our results. The first applies to bounded linear operators on a Banach space and the second applies to (possibly unstable) causal LTI systems in discrete or continuous time.