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 in this paper. These metrics are then 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 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. Comparison of the new metric with the available metrics, e.g., the gap metric and the graph metric, is made. All of these metrics induce the same topology. Surprisingly, 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. AA notable distinct property of pointwise gap metrics is that in the scalar case, they have aa very simple expression which is potentially useful to access the relationship between the uncertainty of physical parameters and uncertainty measured by pointwise gap metrics.