Counterexamples to extremality conjectures on step response analysis of interval plants

The main focus is on step responses of interval plants. Interval plants are useful models of uncertain systems because many worst-case analyses of these models are simple to carry out. For example, robust stability of an interval plant can be determined by only the four Kharitonov vertices of the denominator polynomial; the maximum peak of the Bode magnitude plot can be found using just 16 special plant vertices. These 16 vertices are connected by 32 special line segments. Most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out by using just the segments. From these results, it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses. Examples are presented showing that these conjectures are not true