Convex matrix inequalities versus linear matrix inequalities

Most linear control problems lead directly to matrix inequalities (MIs). Many of these are badly behaved but a classical core of problems are expressible as 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 rigid than convex MIs, there is the hope that a convexity based theory will be less restrictive than LMIs. How much more restrictive are LMIs than convex MIs?. There are two fundamentally different classes of linear systems problems: dimension free and dimension dependent. A dimension free MI is a MI where the unknowns are -tuples of matrices and appear in the formulas in a manner which respects matrix multiplication. Most of the classic MIs of control theory are dimension free. Dimension dependent MIs have unknowns which are tuples of numbers. The two classes behave very differently and this survey describes what is known in each case about the relation between convex MIs and LMIs. The proof techniques involve and necessitate new developments in the field of semialgebraic geometry.