Pointwise gap metrics on transfer matrices

A family of metrics, pointwise gap metrics, in the space of real rational matrices of fixed size is developed and used to study open-loop and closed-loop stability robustness of linear time-invariant finite-dimensional continuous-time systems. It is shown that pointwise gap metrics have the desired qualitative properties for the study of stability robustness. Necessary and sufficient conditions on the robustness are obtained in terms of the radii of the pointwise gap metric balls centered at the nominal plant and/or the nominal controller. Comparison with the gap metric and the graph metric is made. All of these metrics induce the same topology. Many of the quantitative properties of pointwise gap metrics are the same as those of the gap metric, although they differ in value. Pointwise gap metrics, in the scalar case, have a very simple expression which is potentially useful for accessing the relationship between the uncertainty of physical parameters and uncertainty measured by pointwise gap metrics