An algebraic framework for quadratic invariance

In this paper, we present a general algebraic framework for analysing decentralized control systems. We consider systems defined by linear fractional functions over a com-mutative ring. This provides a general algebraic formu-lation and proof of the main results of quadratic invari-ance, as well as naturally covering rational multivariable systems, systems with delays, and multidimensional sys-tems. The approach extends to the extended class of internally quadratically invariant systems.