6.8.Problems 97 1.Letw(s)heascalar weightirg fircticn asumedinRH.Defire e=kw(I+PK)4 8=kK(I+PK)4e. so e neses say,cstubarce attenaticnards neeres say,crtd effcrt Derive the folloirg ireuality,that shois that e ards carrct bthe snall si- mltarecusly ingereral:Fareery reso5 0 lw(so)I<e+w(so)lomin[P(so)]8. 2.Ifwearty odcsturlarceateraticnat aparticlar frererc,yo nigt quessthat wereedhigh crticller gainat that frepuercy.Fir-withj-rct apdle cf P(s),ardsuppcse e:=omazl(I+PK)4(j-)]<1. Derivealcuer brdfrminK(j-).This ler burdshculdlc up ase0.Problems   Let w s be a scalar weighting function assumed in RH￾ Dene  kw I  P Kk￾   kK I  P Kk￾ So measures say disturbance attenuation and  measures say control e ort Derive the fol lowing inequality that shows that and  cannot both be smal l si multaneously in general For every Re s￾   jw s￾j   jw s￾jmin P s￾  If we want very good disturbance attenuation at a particular frequency you might guess that we need high control ler gain at that frequency Fix with j not a pole of P s and suppose  max I  P K j  Derive a lower bound for min K j This lower bound should blow up as