Necessary and sufficient conditions for convergence in the gap metric

In this paper we use the gap metric to study the convergence properties of systems which are not necessarily stable. This convergence will be called the "generalized" convergence (compared to the "normal" convergence in the norm metric for stable systems). We develop necessary and sufficient conditions for generalized convergence, and provide geometrical interpretations of these conditions in terms of Nyquist diagram. Several examples for demonstrating the results of this paper are provided.