10.3.Structured Robust Stability and Performance 2 Proof.(x)Suppose sups c 1a(G(s)03.Then det(I1G(s)△(s)2≤or all s2C+)f+g whenever k△k <4B,i.e.,the system is robustly stable.Now it is sufficient to show that sup 1△(G(s)×sup1a(G(jw). 0R. It is clear that suPμ△(G(s)×su2μ△(G(s)6sP4△(G(jw) s C Now suppose sups CA(G(s))>,then by the definition ofu,there is an so 2 C+)f+g and a complex structured△such that(△)<W3 and det(I1G(so)△)×≤ This implies that there is a≤0g0+and≤<a0 such that det(I1G(jg)a:△)× ≤This in turn implies that u△(G(jg)>3 since(a△)<W3.In other words, sups CA(G(s))0 supo(G(jw)).The proofis complete. (5 Suppose supo R 1A(G(jw))>B.Then there is a <<wo <such that 1 ((jwo))>B.By Remark t there is a complex Ac 2 A that each full block has rank u and (Ac)<such that I1 G(jwo)Ac is sin}ular.Next,usin}the same construction used in the prooo the small ain theorem (Theorem eu),one can find a rational△(s)such that k△(s)k+×△c)<W3,△(jwo)×△c,and△(s)desta bilizes the sy stem. 口 Hence,the peak value on the u plot ofthe frequency response determines the size o perturbations that the loop is robustly stable afainst. Remar 10.4 The internal stability with closed ball of uncertainties is more compli- cated.The ollowin}example is shown in Tits and Fan Consider <1 G(s)×s+μ and△×s)I2.Then 8盟1a(G(u》×8盟w+4 ×1△(G(js)×h On the other hand,1(G(s))<ufor all s 2 ss2 C+,and the only matrices in the orm ofr x I2 with 0 ufor which det(I1G(ST)x≤ are the compler matrices +j2.Thus,clearly,(I1 G(s)(s))1 2 RH+for all real rational A(s)x (s)I2 with kok 0 u since A(s)must be feal.This shows that Structured Robust Stability and Performance ￾ Proof  Suppose sup s￾C ￾￾Gs   Then detI ￾ Gss   for all s C   fg whenever kk  ￾  ie the system is robustly stable Now it is sucient to show that sup s￾C ￾￾Gs sup ￾R ￾￾Gj It is clear that sup s￾C ￾Gs sup s￾C ￾Gs sup  ￾Gj Now suppose sups￾C ￾Gs   then by the denition of ￾ there is an so C   fg and a complex structured  such that   ￾  and detI ￾ Gso  This implies that there is a      and    ￾ such that detI ￾ Gj   This in turn implies that ￾Gj   since    ￾  In other words sups￾C ￾Gs  sup ￾Gj The proof is complete  Suppose sup￾R ￾￾Gj   Then there is a   o   such that ￾￾Gjo   By Remark ￾￾ there is a complex c that each full block has rank ￾ and c  ￾  such that I ￾ Gjoc is singular Next using the same construction used in the proof of the small gain theorem Theorem ￾ one can nd a rational s such that ksk c  ￾  joc and s destabilizes the system ￾ Hence the peak value on the ￾ plot of the frequency response determines the size of perturbations that the loop is robustly stable against Remark  The internal stability with closed ball of uncertainties is more compli cated The following example is shown in Tits and Fan ￾ Consider Gs ￾ s  ￾ ￾  ￾￾ ￾  and  sI  Then sup ￾R ￾￾Gj sup ￾R ￾ jj  ￾j ￾￾Gj ￾ On the other hand ￾￾Gs  ￾ for all s   s C  and the only matrices in the form of  I with j j  ￾ for which detI ￾ G  are the complex matrices jI Thus clearly I ￾ Gss￾ RH for all real rational s sI with kk  ￾ since  must be real This shows that