  The set of controllers stabilizing a linear system is generally non-convex in
the parameter space. In the case of two-parameter controller design (e.g. PI
control or static output feedback with one input and two outputs), we observe
however that quite often for benchmark problem instances, the set of
stabilizing controllers seems to be convex. In this note we use elementary
techniques from real algebraic geometry (resultants and Bezoutian matrices) to
explain this phenomenon. As a byproduct, we derive a convex linear matrix
inequality (LMI) formulation of two-parameter fixed-order controller design
problem, when possible.
