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Real and complex stability radii: A survey. (English) Zbl 0729.93056

Control of uncertain systems, Proc. Int. Workshop, Bremen/Ger. 1989, Prog. Syst. Control Theory 6, 119-162 (1990).
Summary: [For the entire collection see Zbl 0723.00049.]
We introduced [Syst. Control Lett. 7, 1-10 (1986; Zbl 0631.93064), ibid. 8, 105-113 (1986; Zbl 0626.93054)] complex and real stability radii as robustness measures for stable matrices A under complex and real perturbations of the form \(A\to A+D\Delta E\), where D, E are given and \(\Delta\) is unknown. In this paper we give a survey of the results obtained so far. Although our main reference is to continuous-time systems we also present some results for the discrete-time case.
We first review the basic properties and characterizations of the stability radii emphasizing the remarkable differences between the complex and the real radius. In particular we discuss the role they play when the class of disturbances is enlarged to include time-varying, nonlinear and dynamic parameter perturbations. In a further generalization we consider arbitrary affine perturbations of the matrix A. Finally we study the problem of maximizing the complex stability radius by linear state feedback.

MSC:

93D09 Robust stability