Subspace Method for the Estimation of Large-Scale Structured Real Stability Radius

We consider the autonomous dynamical system $x' = Ax$, with $A \in \mathbb{R}^{n\times n}$. This linear dynamical system is said to be asymptotically stable if all of the eigenvalues of A lie in the open left-half of the complex plane. In this case, the matrix A is said to be Hurwitz stable or shortly a stable matrix. In practice, stability of a system can be violated because of arbitrarily small perturbations such as modeling errors. In such cases, one deals with the robust stability of the system rather than its stability. The system above is said to be robustly stable if the system, as well as all of its arbitrarily small perturbations, are stable. To measure the robustness of the system subject to perturbations, a quantity of interest is the stability radius or in other words distance to instability. In this paper we focus on the estimation of the structured real stability radius for large-scale systems. We propose a subspace framework to estimate the structured real stability radius and prove that our new method converges at a quadratic rate in theory. Our method benefits from a one-sided interpolatory model order reduction technique, in a sense that the left and the right subspaces are the same. The quadratic convergence of the method is due to the certain Hermite interpolation properties between the full and reduced problems. The proposed framework estimate the structured real stability radius for large-scale systems efficiently. The efficiency of the method is demonstrated on several numerical experiments. Key words. real stability radius, structured, large-scale, projection, singular values, Hermite interpolation, model order reduction, greedy search.