The Gap Metric: Robustness of Stabilization of Feedback Systems

In this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open-loop systems. We elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply. By admissible uncertainties we mean those which preserve closed-loop stability and a specified small tolerance on the I/O behavior of a feedback system. We show that admissible uncertainties are precisely those which are constrained in the gap. Finally, we conclude that any metric which preserves a continuous relationship between open-loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric.