Connection Between Unimodular Interaction Constraint for Decentralized Controllers and Small Gain Bounds

Decentralized controllers for a stable plant can be designed as if the plant were decoupled provided that the individual controller's Youla parameter satisfies a norm bound imposed by the off diagonal plant operators. This bound is derived from the unimodular interaction constraint associated with the set of stabilizing decentralized controllers. One result from such a bound is the quantification of the notion of weak coupling, used as a condition for the nonsingular perturbation design approach to decentralized control. Using a robust stability framework, the above bound can be related to a small gain bound. Specifically, in the two channel case, any Youla parameters which satisfy the small gain bound will satisfy the above bound. The converse is not true, making this bound less conservative than small gain bound. However, in a fundamental sense either framework (robust stability or unimodular interaction constraint) produce the same stability constraint for the set of stable plants. Therefore, either point of view can be used to derive these bounds. These results are extended to the multi-channel case.