48 CHAPTER.UNCERTAINTY AND ROBUSTNESS Fix A4In what follows,functions are evaluated at an arbitrary point w,but this is suppressed to simplify notation4 We have 1=I1+△W2T1△W2T|≤|1+△W2T|+IW2T and t herefore 11IW2T\≤|1+△W2T|≤ This implies that This and (444)yield 1+△W2T <1≤ (→)Assume that W2Tlloo 1 and WiS 1+△W2T <1H△≤ (45) Pick a frequency w where WiS 11 W2T is maximum4 Now pick A so that 11IW2T=|1+△W2T≤ The idea here is that A(w)should rotate W2(w)T(w)so that A(w)W2(w)T(w)is negative real4 The details of how to construct such an allowable A are omitted4 Now we have W S 11W2T WiS 1+△W2T WiS 1+△W2T So from this and (445)there follows (44)4 Test (443)also has a nice graphical interpretation4 For each frequency w,construct two closed disks:one with center 11,radius Wi(w);the other with center L(w),radius W2(w)L(w)4 Then (443)holds i.for each w these two disks are disjoint (Figure 446)4 The robust performance condition says that the robust performance level 1 is achieved4 More generally,let's say that robust performance level a is achieved if WiS W2Tlloo 1 and 1+△W2T <aH△≤ Noting that at every frequency WiS WiS max 1+△W2T 11W2T CHAPTER UNCERTAINTY AND ROBUSTNESS Fix  In what follows functions are evaluated at an arbitrary point j but this is suppressed to simplify notation We have   jW￾T W￾T jjW￾T j  jW￾T j and therefore  jW￾T jjW￾T j This implies that WS  jW￾T j ￾  WS W￾T ￾ This and  yield WS W￾T ￾     Assume that kW￾T k￾   and WS W￾T ￾     Pick a frequency  where jWSj  jW￾T j is maximum Now pick  so that  jW￾T j  jW￾T j The idea here is that j should rotate W￾jT j so that jW￾jT j is negative real The details of how to construct such an allowable  are omitted Now we have WS  jW￾T j ￾  jWSj  jW￾T j  jWSj jW￾T j WS W￾T ￾ So from this and  there follows  ￾ Test  also has a nice graphical interpretation For each frequency  construct two closed disks one with center  radius jWjj the other with center Lj radius jW￾jLjj Then  holds i for each  these two disks are disjoint Figure  The robust performance condition says that the robust performance level  is achieved More generally lets say that robust performance level  is achieved if kW￾T k￾   and WS W￾T ￾    Noting that at every frequency max jj WS W￾T  jWSj  jW￾T j