ABSTRACT
In the following it is shown that the argument of a Hurwitz polynomial f(s)= h(s2 ) + sg(s 2) in the frequency domain f(jω) = h(−ω 2 ) + jωg(ω 2) as well as the argument of the modified functions f 1 * ( ω ) = h ( − ω 2 ) + j g ( ω 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067702/c6b3ff05-4348-45f5-97a1-acfe807326ca/content/eq2761.tif"/> and f 2 * ( ω ) = h ( − ω 2 ) + j ω 2 g ( ω 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067702/c6b3ff05-4348-45f5-97a1-acfe807326ca/content/eq2762.tif"/> are monotonically increasing functions of ω whereby ω varies between 0 and ∞.
Also it is shown that given a Schur polynomial f(z) = h(z) + g(z), where h(z) and g(z) are the symmetric and the antisymmetric parts of f(z) respectively, the same monotony property can be obtained for an auxiliary function defined as f ( e j θ ) = 2 e j n θ / 2 f * ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067702/c6b3ff05-4348-45f5-97a1-acfe807326ca/content/eq2763.tif"/> . The argument of f * ( θ ) = h * ( θ ) + j g * ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067702/c6b3ff05-4348-45f5-97a1-acfe807326ca/content/eq2764.tif"/> as well as the argument of the modified functions f 3 * ( θ ) − h * ( θ ) cos θ 2 + j g * ( θ ) sin θ 2 and f 4 ∗ ( θ ) = h ∗ ( θ ) sin θ 2 + j g ∗ ( θ ) cos θ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067702/c6b3ff05-4348-45f5-97a1-acfe807326ca/content/eq2765.tif"/>
are monotonically increasing functions of θ whereby θ varies between 0 and π. These results can be directly applied to the robust stability problem.
