Employing Kronecker Canonical Form for LMI-Based $H_{\infty}$ Synthesis Problems

The Kronecker canonical form (KCF) can be employed when solving
$H_{\infty}$
synthesis problem. The KCF structure reveals variables that can be eliminated in the semidefinite program that defines the controller. The structure can also be used to remove states in the controller without sacrificing performance. In order to find the KCF structure, we can transform the relevant matrices to a generalized upper triangular (Guptri) form using orthogonal transformations. Thus, we can avoid finding the KCF structure explicitly, which is a badly conditioned problem.