Optimal $RH_2$-- and $RH_\infty$--Approximation of Unstable Descriptor Systems

TL;DR

This paper develops methods for optimally approximating unstable descriptor systems with stable ones in $RH_2$ and $RH_ fty$ norms, ensuring stability preservation despite numerical errors, with explicit solutions provided.

Contribution

It introduces explicit optimal approximation techniques for unstable descriptor systems in $RH_2$ and $RH_ fty$ norms, addressing numerical stability issues.

Findings

Explicit optimal solutions for stable approximation are derived.

The methods effectively handle numerical round-off errors.

The approach ensures stability preservation in practical applications.

Abstract

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 $RH_2$ and $RH_\infty$, respectively. Explicit optimal solutions are presented under consideration of numerical issues.