Decentralized Control via Gröbner Bases and Variable Elimination

We consider the problem of optimal decentralized controller synthesis. There are several classes of such problems for which effective solution methods are known, including the quadratically invariant one. In this technical note, we use Gröbner bases and elimination theory to characterize all closed-loop maps achievable by forming a feedback loop with decentralized controllers. We show that this approach allows solution of a wide class of optimal decentralized control problems; it includes not only quadratically invariant problems under a technical condition but also some other problems which are not quadratically invariant.