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156 CHAPTER 10.DESIGN FOR PERFORMANCE Step 3 Choose r so that the oo-norm of 100(s-2)2(s+7) (s+1)A (Ts+1)2 is 1.The norm is computed for decreasing values of r: oo-Norm 10- 199.0 10-2 19.97 1.997 10 0.1997 So take T 10-4. Step 4 Q(s)= (s+1)(s+7) (10-4s+1)2 Step 5 C(s)=104 (s+1)3 s(s+7)(10-4s+2) This section concludes with a result stated but not proved in Section 6.2.It concerns the performance problem where the weight Wi sat isfies w1≤w≤w2 else Thus the criterion WiSo<1 is equivalent to the conditions |S(jw)‖<e,w1≤w≤w2 S(jw)<6,else. (10.1) Lemma 2 If P-is stable,then for every e>0 and 6>1,there erists a proper C such that the feedback system is internally stable and (10.1)holds.I Proof The idea is to approximately invert P over the frequency range [0,w2]while rolling off fast enough at higher frequencies.From Theorem 5.2 again,the formula for all internally stabilizing proper controllers is C X+MQ Q∈S. Y-NQ' For such C S=M(Y-NQ). (10.2) Now fix e>0 and 6>1.We may as well suppose that e<1.Choose c>0 so small that clMY‖o<e, (10.3) The assumption on P in Lemma 2 is slightly stronger than necessary;see the statement in Section 6.2. CHAPTER DESIGN FOR PERFORMANCE Step Choose so that the norm of s s s s is The norm is computed for decreasing values of Norm So take Step Q s s s s Step C s s s s s This section concludes with a result stated but not proved in Section It concerns the performance problem where the weight W satises jW j j else Thus the criterion kWSk is equivalent to the conditions jS j j jS j j else Lemma If P is stable then for every and there exists a proper C such that the feedback system is internal ly stable and holds Proof The idea is to approximately invert P over the frequency range while rolling o fast enough at higher frequencies From Theorem again the formula for all internally stabilizing proper controllers is C X MQ Y NQ Q S For such C S M Y NQ Now x and We may as well suppose that Choose c so small that ckM Y k The assumption on P in Lemma is slightly stronger than necessary see the statement in Section