Robust \(D\)-stability (Q1861729)

Thirteen conditions by Johnson for \(D\)-stability of matrices are extended to robust (or strong) \(D\)-stability, i.e. \(D\)-stability in the presence of a tiny perturbation. By \(D\)-stability of a given matrix \(A\), it is meant stability (all the eigenvalues in the left half of the complex plane) of the product \(DA\) for any positive diagonal scaling matrix \(D\). Note that in the robust control literature, the terminology \(D\)-stability is sometimes used to refer to the location of the roots of the matrix \(A\) in some given region \(D\) of the complex plane (e.g. a sector or a disk, to ensure closed-loop performance), which is a totally different problem. Robust \(D\)-stability is studied when the nominal matrix \(A\) is affected by additive norm-bounded perturbation \(A+G\), with norm \((G) < \alpha\) and \(\alpha\) is a tiny positive scalar. A disappointing feature of the obtained conditions is that none of them explicitly involve the measure \(\alpha\) of the considered perturbation.