A Convex Characterization of Multidimensional Linear Systems Subject to SQI Constraints

This technical note proposes a convex characterization of the set of all stable closed-loop linear systems that are obtained from a given plant, which may be multidimensional, by interconnecting it in feedback with controllers that satisfy a certain pre-selected constraint. We take an approach that is particularly useful when a doubly-coprime factorization of the plant is difficult to obtain, and a stable stabilizing controller may not exist (the plant is not strongly-stabilizable) or when one may be difficult to find; most related work requires one of these. We adopt the so-called coordinate-free approach, which, unlike Youla's parametrization, does not rely on a doubly-coprime factorization of the plant. We show that if constraints which satisfy a condition called strong quadratic invariance (SQI) are imposed on the controllers then the set of all stable closed-loop multidimensional linear systems has a convex representation, and norm-optimal control problems can be cast in convex form. Although the SQI condition is in general slightly stronger than quadratic invariance (QI), which was developed in related work, they are equivalent for common classes of problems arising in decentralized control.