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# Fast Approximation of
the Stability Radius and the H_{∞} Norm for Large-Scale
Linear Dynamical Systems

Abstract:

The stability radius and the H_{∞} norm are
well-known quantities in the robust analysis of linear dynamical
systems with output feedback. These two quantities, which are
reciprocals of each other in the simplest interesting case,
respectively measure how much system uncertainty can be tolerated
without losing stability, and how much an input disturbance may be
magnified in the output. The standard method for computing them, the
Boyd-Balakrishnan-Bruinsma-Steinbuch algorithm from 1990, is globally
and quadratically convergent, but its cubic cost per iteration makes
it inapplicable to large-scale dynamical systems. We present a new
class of efficient methods for approximating the stability radius and
the H_{∞} norm, based on iterative methods to find
rightmost points of spectral value sets, which are generalizations of
pseudospectra for modeling the linear fractional matrix
transformations that arise naturally in analyzing output feedback. We
also discuss a method for approximating the real structured stability
radius, which offers additional challenges. Finally, we describe our
new public-domain MATLAB toolbox for low-order controller synthesis,
HIFOOS (H-infinity fixed-order optimization --- sparse). This offers a
possible alternative to popular model order reduction techniques by
applying fixed-order controller design directly to large-scale
dynamical systems.