Robust control, multidimensional systems and multivariable Nevanlinna-Pick interpolation

TL;DR

This paper reviews the evolution of robust control and Nevanlinna-Pick interpolation, highlighting their interconnected developments in multidimensional and multivariable systems over recent decades.

Contribution

It clarifies the preserved connections between $H^$ control theory and Nevanlinna-Pick interpolation in advanced multidimensional and multivariable contexts.

Findings

Established links between $H^$ control and Nevanlinna-Pick in multidimensional systems.

Reviewed advances in robust control for structured uncertainties.

Connected multivariable Nevanlinna-Pick interpolation to control theory.

Abstract

The connection between the standard $H^\infty$-problem in control theory and Nevanlinna-Pick interpolation in operator theory was established in the 1980s, and has led to a fruitful cross-pollination between the two fields since. In the meantime, research in $H^\infty$-control theory has moved on to the study of robust control for systems with structured uncertainties and to various types of multidimensional systems, while Nevanlinna-Pick interpolation theory has moved on independently to a variety of multivariable settings. Here we review these developments and indicate the precise connections which survive in the more general multidimensional/multivariable incarnations of the two theories.