Pointwise gap metrics on transfer matrices

A new family of metrics, called pointwise gap metrics, in the space of real rational matrices of fixed size is developed. These metrics are then used to study open-loop and closed-loop stability robustness of lumped 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 open- and closed-loop stability robustness are obtained in terms of the radii of the pointwise gap metric balls centered at the nominal plant and/or the nominal controller. The new metric is compared with the available metrics, e.g. the gap metric and the graph metric. All these metrics induce the same topology. It is shown that many of the quantitative properties of pointwise gap metrics are the same as those of the gap metric, although they differ in value. In the scalar case they have a very simple expression which is potentially useful for accessing the relationship between the uncertainty of physical parameters and the uncertainty measured by pointwise gap metrics