Fall 2001 16.31196 Now assume that H has no eigenvalues on the jw-axis has no poles there I-g g has no zeros there So 1-G g has no zeros on the jw-axis, and we also know that I-G*G→I>0asu→( since d=0 Together, these imply that I-G(juG(jw)=I-GG>0 V For a SISo system, this condition(I-G*G>0 is equivalent to G o<1 which is true iff Gmax= max G(j)<1 Can use state-space tools to test if a generic system has a gain less that 1, and can easily re-do this analysis to include the boundFall 2001 16.31 19–6 • Now assume that H has no eigenvalues on the jω-axis, ⇒ S = [I − G˜G] −1 has no poles there ⇒ I − G˜G has no zeros there • So I − G˜G has no zeros on the jω-axis, and we also know that I − GG → I > 0 as ω → ∞ (since D = 0). – Together, these imply that I − GT (−jω)G(jω) = I − G G > 0 ∀ ω • For a SISO system, this condition (I − GG > 0) is equivalent to |G(jω)| < 1 ∀ ω which is true iff Gmax = max ω |G(jω)| < 1 • Can use state-space tools to test if a generic system has a gain less that 1, and can easily re-do this analysis to include the bound γ