The Symmetric Rouche's Form for Robustness

The objective of this work is to bring to control literature the strengthened and symmetric version of the classical Rouche's form. This symmetric form has been discovered by Glicksberg [1], [2] and was first employed in [3] to determine a robust dead time for closed-loop stability. In this note we motivate the need for Rouche's forms as a tool for the analysis of linear time-invariant control systems. A robustness criterion is given and its corresponding geometrical interpretation is obtained. A simplified implementation of the symmetric form is obtained and its relation to passivity is highlighted. Finally, an application to time delay systems is given.