Noncommutative convexity vs LMI'S

Most linear control problems convert directly to matrix inequalities, MIs. Many of these are badly behaved but a classical core of problems convert to linear matrix inequalities (LMIs). in many engineering systems problems convexity has all of the advantages of a LMI. Since LMIs have a structure which is seemingly much more ridged than convexity, there is the hope that a convexity based theory will be less restrictive than LMIs. A dimensionless MI is a MI where the unknowns are matrices and appear in the formula in a manner which respects matrix multiplication. This holds for most of the classic MIs of control theory. The results presented here suggest the surprising conclusion that for dimensionless MIs convexity offers no greater generality than LMIs. in fact, we prove, for a class of model situations, that a convex dimensionless MI is equivalent to an LMI.