A Simplified Robust Circle Criterion Using the Sensitivity-Based Quantitative Feedback Theory Formulation

Abstract

The circle criterion provides a sufficient condition for global asymptotic stability for a specific class of nonlinear systems, those consisting of the feedback interconnection of a single-input, single-output linear dynamic system and a static, sector-hounded nonlinearity. Previous authors (Wang et al, 1990) have noted the similarity between the graphical circle criterion and design bounds in the complex plane stemming from the Quantitative Feedback Theory (QFT) design methodology. The QFT formulation has specific advantages from the standpoint of controller synthesis. However, the aforementioned approach requires that plant uncertainty sets (i.e., “templates”) be manipulated in the complex plane. Recently, a modified formulation for the QFT linear robust performance and robust stability problem has been put forward in terms of sensitivity function bounds. This formulation admits a parametric inequality which is quadratic in the open loop transfer function magnitude, resulting in a computational simplification over the template-based approach. In addition, the methodology admits mixed parametric and nonparametric plant models. The disk inequality which results represents a much closer analog of the circle criterion, requiring only scaling and a real axis shift. This observation is developed in this paper, and the methodology is demonstrated in this paper via feedback design and parametric analysis of a quarter-car active suspension model with a sector nonlinearity.