Robust Stability of Plants With Mixed Uncertainties and Quantitative Feedback Theory

Abstract

Quantitative Feedback Theory (QFT) has often been criticized for lack of a rigorous mathematical theory to support its claims. Yet it is known to be a very effective design methodology. In this paper, we re-examine QFT and state several results that confirm the validity of this highly effective framework proposed by Horowitz. Also provided are some additional insights into the QFT methodology that may not be immediately apparent. We consider three important fundamental questions: (i) whether or not a QFT design is robustly stable, (ii) does a robust stabilizer exist, and (iii) does a controller assuring robust QFT performance exist. The first two are obvious precursors for synthesizing controllers for performance robustness. We give necessary and sufficient conditions that unambiguously resolve the question of robust stability under mixed uncertainty, thereby, confirming that a properly executed QFT design is automatically robustly stable. Also given is a sufficiency condition for a robust stabilizer to exist which is derived from the well known Nevanlinna-Pick theory in classical analysis. Finally, we give a sufficiency theorem for the existence of a QFT controller and deduce that when the uncertain plant set is minimum phase with no unstructured uncertainty there always exists a controller satisfying robust performance specifications in the sense of QFT.