  Optimal decentralized controller design is notoriously difficult, but recent
research has identified large subclasses of such problems that may be
convexified and thus are amenable to solution via efficient numerical methods.
One recently discovered sufficient condition for convexity is quadratic
invariance (QI). Despite the simple algebraic characterization of QI, which
relates the plant and controller maps, proving convexity of the set of
achievable closed-loop maps requires tools from functional analysis. In this
work, we present a new formulation of quadratic invariance that is purely
algebraic. While our results are similar in flavor to those from traditional QI
theory, they do not follow from that body of work. Furthermore, they are
applicable to new types of systems that are difficult to treat using functional
analysis. Examples discussed include rational transfer matrices, systems with
delays, and multidimensional systems.
