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Robust numerical methods for robust control

Benner, Peter; Byers, Ralph; Mehrmann, Volker; Xu, Hongguo

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

We present numerical methods for the solution of the optimal $H_{\infty}$ control problem. In particular, we investigate the iterative part often called the $\gamma$-iteration. We derive a method with better robustness in the presence of rounding errors than other existing methods. It remains robust in the presence of rounding errors even as $\gamma$ approaches its optimal value. For the computation of a suboptimal controller, we avoid solving algebraic Riccati equations with their problematic matrix inverses and matrix products by adapting recently suggested methods for the computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils. These methods are applicable even if the pencil has eigenvalues on the imaginary axis. We compare the new method with older methods and present several examples.