Optimal $R{H}_2$-- and $R{H}_{\mathrm{\infty }}$--Approximation of Unstable Descriptor Systems

Stability perserving is an important topic in approximation of systems, e.g.
model reduction. If the original system is stable, we often want the approximation to be stable. But even if an algorithm preserves stability the resulting system could be unstable in practice because of round-off errors. Our approach is approximating this unstable reduced system by a stable system. More precisely, we consider the following problem. Given an unstable linear time-invariant continuous-time descriptor system with transfer function $G$, find a stable one whose transfer function is the best approximation of $G$ in the spaces $R{H}_2$ and $R{H}_{\mathrm{\infty }}$, respectively. Explicit optimal solutions are presented under consideration of numerical issues.