Stability radii of positive linear difference equations under affine parameter perturbations (Q1855956)

The authors study the robustness of positive linear difference delay equations under arbitrary affine perturbations of the form \[ x(k+1)=(A_{\nu}+D_{\nu}\Delta_{\nu}E_{\nu})x(k)+\ldots +(A_0+D_0\Delta_0E_0)x(k-\nu),\quad k=\nu, \nu+1,\ldots \] where \(\nu\in\mathbb{N}\), \(A_i\in\mathbb{R}^{n\times n}_+\) are nonnegative square coefficient matrices of the nonperturbed equation, \(D_i\), \(E_i\) are given matrices specifying the structure of the perturbations and \(\Delta_i\) are unknown disturbance matrices whose sizes are measured by their operator norms, \(i=0,\ldots,\nu\). The aim of the paper is a generalization of the previous results of \textit{D. Hinrichsen} and \textit{N. K. Son} [Int. J. Robust Nonlinear Control 8, 1169-1188 (1998; Zbl 0918.93036)] established for equations without delay. A formula for the complex stability radius is obtained. It is proved that complex, real and nonnegative stability radii are the same for block-diagonal perturbations. The new results are illustrated by simple second order examples.