Sufficiency of vertex matrix check for robust stability of interval matrices via the concept of Qualitative Robustness

Abstract

This paper revisits the issue of robust stability analysis of linear interval parameter matrices, which used to be a highly active research topic in the eighties and nineties. The reason for this revived interest in this topic is that the recent research by the authors on Qualitative Stability, a topic of interest in the field of population/community dynamics in ecology is shown to shed considerable insight with possible new results in the robust stability of matrix families. Thus in this paper, we expand on the two notions of robustness introduced recently by the authors, namely `Qualitative Robustness' and `Quantitative Robustness' and investigate their interdependence. Specifically, it is shown that for a class of matrix families with specified `Qualitative Robustness' indices, it is sufficient to check the stability of only `vertex' matrices (i.e. an extreme point solution) to guarantee the robust stability of the entire interval matrix family. This is indeed deemed important and significant because with this result, we can easily identify for which `interval matrix families' we need to resort to more sophisticated stability check algorithms, and for which families we can get away with a `vertex matrix check' (i.e. an `extreme point solution'). It turns out that this class of `qualitative stable' matrices that admit `vertex solution' for its `quantitative robustness' is quite large. Thus the results of this paper offer new insight into the nature of interactions and interconnections in a matrix family on its robust stability. Encouraged by the results of this paper, continued research is underway in using this interdependence of `qualitative robustness' and `quantitative robustness' in the design of robust controllers for engineering systems.