154 CHAPTER 10.DESIGN FOR PERFORMANCE The first of these is bounded above by eGloo,and the second by l1-J‖xmax|G(jw)小. idil Since 1-J川∞≤1l+JI川o=2, we have IG1-∞≤mx{Gle,2nx1G(joj川. This holds for T sufficiently small.But the right-hand side can be made arbitrarily small by suitable choice of e and w because 1 im max|Gw)川=lGGo)川=0. 1→00Ww1 We conclude that for every 6>0,if r is small enough,then G(1-J)川≤d. This is the desired conclusion. We'll develop the design procedure first with the additional assumption that P is stable.By Theorem 5.1 the set of all internally stabilizing Cs is parametrized by the formula Q C=1-PQ° Q∈S. Then WiS is given in terms of Q by W1S=W1(1-PQ). To make WiSlloo<1 we are prompted to set Q=P-1.This is indeed stable,by assumption,but not proper,hence not in S.So let's try Q=P-1J with J as above and the integer k just large enough to make p-J proper (ie.,k equals the relative degree of P).Then W1S=W(1-J), whose oo-norm is 1 for sufficiently small T,by Lemma 1. In summary,the design procedure is as follows. Procedure:P and P-!Stable Input:P,Wi Step 1 Set k =the relative degree of P. Step 2 Choose r so small that ‖W(1-J)川o<1, where 1 J(s)=8+1 CHAPTER  DESIGN FOR PERFORMANCE The rst of these is bounded above by kGk￾ and the second by k J k￾ max ￾ jGjj Since k J k￾  kk￾  kJ k￾   we have kG J k￾  max ￾ kGk￾  max ￾ jGjj  This holds for suciently small But the right hand side can be made arbitrarily small by suitable choice of  and ￾ because lim ￾￾ max ￾ jGjj  jGjj   We conclude that for every   if is small enough then kG J k￾   This is the desired conclusion ￾ Well develop the design procedure rst with the additional assumption that P is stable By Theorem  the set of all internally stabilizing Cs is parametrized by the formula C  Q P Q Q  S Then W￾S is given in terms of Q by W￾S  W￾ P Q To make kW￾Sk￾ we are prompted to set Q  P ￾  This is indeed stable by assumption but not proper hence not in S So lets try Q  P ￾J with J as above and the integer k just large enough to make P ￾J proper ie k equals the relative degree of P  Then W￾S  W￾ J  whose  norm is for suciently small  by Lemma  In summary the design procedure is as follows Procedure P and P ￾ Stable Input P  W￾ Step Set k  the relative degree of P  Step  Choose so small that kW￾ J k￾ where J s   s  k