On structured singular values and robust stability of positive systems under affine perturbations (Q1423560)

Robust stability results are presented without proof for nonnegative systems under structured perturbations. Let \(\mathbb{F}^{l\times q}\) be the space of \(l\times q\) matrices over \(\mathbb{F}=\mathbb{R}\), \(\mathbb{F}=\mathbb{C}\) or \(\mathbb{F}=\mathbb{R}_+= [0,\infty)\), and let \({\mathcal D}\subset \mathbb{C}^{l\times q}\) be a given set. Then two typical results are as follows: 1) Let \(M\in \mathbb{R}^{q\times l}_+\) and \({\mathcal D}\) be a cone. Then the \(\mu\)-structured singular values of \(M\) relative to \({\mathcal D}\), \({\mathcal D}_{\mathbb{R}}={\mathcal D}\cap \mathbb{R}^{l\times q}\) and \({\mathcal D}_+= {\mathcal D}\cap \mathbb{R}^{l\times q}_+\), are equal; 2) Let the Schur stable matrix \(A\in \mathbb{R}^{n\times n}_+\) be subject to perturbations \(A\mapsto A+ D\Delta E\), where \(D\in \mathbb{R}^{n\times l}_+\), \(E\in \mathbb{R}^{l\times n}_+\) and \(\Delta\in{\mathcal D}\). Then the stability radii of \(A\) relative to \({\mathcal D}\), \({\mathcal D}_{\mathbb{R}}\) and \({\mathcal D}_+\), coincide.