math.OC — Optimization and Control
A Sixth-Order Hurwitz Interval Polynomial Family Without Degree-Five Weak Strictly Positive Real Numerator
We construct an explicit sixth-order Hurwitz-stable interval polynomial family with independent closed coefficient intervals that admits a common strictly positive real (SPR) numerator of degree six, yet admits no common weak strictly positive real (WSPR) numerator of degree five. More strongly, no nonzero real polynomial of degree at most five makes the real part of the associated transfer functions nonnegative across the family at all real frequencies. This counterexample shows that, for sixth-order interval polynomial families, robust Hurwitz stability does not guarantee the existence of a strictly proper common SPR or WSPR numerator of degree five, and equal numerator degree six is necessary in general.