An Algebraic Approach to the Control of Decentralized Systems

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 paper, we present a new formulation of QI 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.