Complex-analytic theory of the \(\mu\)-function
The determinant of the multivariable return difference Nyquist map, essential in defining the complex structured singular value, is considered as a holomorphic function defined on a polydisk of uncertainty. The key property of holomorphic functions of several complex variables that is crucial in the authors' argument is that it is an open mapping. From this single result only, it is shown that, in the diagonal perturbation case, all preimage points of the boundary of the Horowitz template are included in the distinguished boundary of the polydisk. In the block-diagonal perturbation case, where each block is norm-bounded by one, a preimage of the boundary is shown to be a unitary matrix in each block. Finally, some algebraic geometry, together with the Weierstrass preparation theorem, allow to show that the deformation of the crossover under (holomorphic) variations of ``certain'' parameters is continuous. In this way, the authors show that, by relying solely on the theory of holomorphic functions of several complex variables, one can rederive in a self-contained manner the key features of the complex \(\mu\)-function analysis, derive strong results, and remove the diagonal, multilinear, even lumped parameter assumptions, avoiding any kind of programming argument. It is noted in the conclusion that, in the real-smooth case, the singularity set is generically a network of curves forming a smooth manifold, instead of a set of isolated points. Furthermore, the image of a singular curve crossing \(0+ j0\) could create lack of continuity, while in the complex case it does not. To analyze the discontinuity of the real \(\mu\) associated with a bifurcation of the crossover in a neighborhood of a singular point, the authors propose to use the Malgrange preparation theorem, instead of the Weierstrass preparation theorem.
multivariable return difference Nyquist map
holomorphic functions of several complex variables
scientific article; zbMATH DE number 1348145
scientific article