First and second order sensitivity of the singular value decomposition

We develop algebraic expressions for the first- and second-order sensitivities of the singular value decomposition of a general complex matrix. These algebraic results have been verified by numerical methods. Owing to the increasing analytical and computational applications of singular value analysis of dynamical systems, these results have many potential applications. To illustrate a typical family of applications in design of robust controllers, we consider a stabilizing class of output feedback controllers, and set up an approach to minimize the condition number of the matrix of closed-loop eigenvectors. As specific numerical illustrations, we use the formulation as an integral part of a control design algorithm to design controls for a low dimensioned example (6th order, 4 outputs, 2 inputs) and a moderately high dimensioned example (40th order, 6 outputs, 3 inputs).

First and second order sensitivity of the singular value decomposition Academic Article