Structural instability of a class of decoupling solutions

This short paper is concerned with a class of decoupling solutions for linear dynamic systems of the type
\dot{x}=Ax+Bu, z_{i}=D_{i}x, i = 1,2,...,K
, with controller given by
u= Fx + g_{1}\upsilon{1} + ... + g_{K}\upsilon{K}
. The structural stability of solutions in this class is studied from a geometric viewpoint. The solution set defined here consists of points representing the controller gains
F
and
\{g_{1},...,g_{K}\}
. It is found that most decoupling problems that are practically solvable-via known computational procedures-yield solution sets which are proper hypersurfaces in the control parameter spaces; hence, they form structurally unstable solutions in the sense that a solution will generally be forced out of this class when the system undergoes even quite small parameter variations.