Structured singular values and their application in computing eigenvalue backward errors of the Rosenbrock system matrix

The structured singular values (aka the -values) are essential in analyzing the stability of control systems and in the structured eigenvalue perturbation theory of matrices and matrix polynomials. In this paper, we study the -value of a matrix under block-diagonal structured perturbations (full blocks but possibly rectangular). We provide an explicit expression for the -value and also obtain a computable upper bound in terms of minimizing the largest singular value of a parameter-dependent matrix. This upper bound equals the -value when the perturbation matrix has no more than three blocks on the diagonal. We then apply the -value results in computing eigenvalue backward errors of a Rosenbrock system matrix corresponding to a rational matrix function when some or all blocks of the Rosenbrock system matrix are subject to perturbation. The results are illustrated through numerical experiments.