A seminal result in decentralized control is the development of fixed modes by Wang and Davison in 1973 — that plant modes which cannot be moved with a static decentralized controller cannot be moved by a dynamic one either, and that the other modes which can be moved can be shifted to any chosen location with arbitrary precision. These results were developed for perfectly decentralized, or block diagonal, information structure, where each control input may only depend on a single corresponding measurement. Furthermore, the results were claimed after a preliminary step was demonstrated, omitting a rigorous induction for each of these results, and the remaining task is nontrivial.

In this paper, we consider fixed modes for arbitrary information structures, where certain control inputs may depend on some measurements but not others. We provide a comprehensive proof that the modes which can be altered by a static controller with the given structure can be moved by a dynamic one to any chosen location with arbitrary precision, thus generalizing and solidifying the second part of Wang and Davison's result. A previous paper discussed the first part.

This shows that a system can be stabilized by a linear time-invariant controller with the given information structure as long as all of the modes which are fixed with respect to that structure are in the left half-plane; an algorithm for synthesizing such a stabilizing decentralized controller is then distilled from the proof.

We would recommend instead reading the full paper "Constructive Stabilization and Pole Placement by Arbitrary Decentralized Architectures", which addresses both parts of the fixed mode results, as well as the algorithm, refining and generalizing the results and proofs from this paper and the prior paper.