The Modified Optimal $\mathcal{H}_\infty$ Control Problem for Descriptor Systems

The $\mathcal{H}_\infty$ control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions are derived for systems of arbitrary index. These conditions are formulated in terms of deflating subspaces of even matrix pencils containing only the parameters of the original system. It is shown that this approach leads to a more numerically robust and efficient method in computing the optimal value $\gamma$ in contrast to other methods such as the widely used Riccati- and linear matrix inequality (LMI)-based approaches. The results are illustrated by a numerical example.