4.3.ROBUST PERFORMANCE 49 W W2L Figure 4.6:Robust performance graphically. we get that the minimum a equals WiS 1-1w2T (4.6) Alternatively,we may wish to know how large the uncertainty can be while the robust perfor- mance condition holds.To do this,we scale the uncertainty level,that is,we allow A to satisfy Alloo<B.Application of Theorem 1 shows that internal stability is robust iff BW2Tloo<1. Let's say that the uncertainty level B is permissible if BW2Tlloo 1 and WiS 1+△W2Tl∞ <1,△ Again,noting that WIS WS max △<1 1+B△W2T 1-3W2T we get that the maximum B equals W2T 1-wS/. Now we turn briefly to some related problems. Robust Stability for Multiple Perturbations Suppose that a nominal plant P is perturbed to p=P1+4W2 1+△W with Wi,W2 both stable and A1,A2 both admissible.The robust stability condition is WiS+W2T<1, which is just the robust performance condition in Theorem 2.A sketch of the proof goes like this: From the fourth entry in Table 4.1,for fixed A2 the robust stability condition for varying A is 1 ROBUST PERFORMANCE   r   r ￾  jW￾Lj jWj L  Figure  Robust performance graphically we get that the minimum  equals WS  jW￾T j ￾  Alternatively we may wish to know how large the uncertainty can be while the robust perfor mance condition holds To do this we scale the uncertainty level that is we allow  to satisfy kk￾  Application of Theorem  shows that internal stability is robust i k W￾T k￾   Lets say that the uncertainty level is permissible if k W￾T k￾   and WS W￾T ￾    Again noting that max jj WS   W￾T  jWSj  jW￾T j  we get that the maximum equals W￾T  jWSj ￾ Now we turn briey to some related problems Robust Stability for Multiple Perturbations Suppose that a nominal plant P is perturbed to P  P ￾W￾ W with W W￾ both stable and  ￾ both admissible The robust stability condition is kjWSj  jW￾T jk￾   which is just the robust performance condition in Theorem A sketch of the proof goes like this From the fourth entry in Table  for xed ￾ the robust stability condition for varying  is W      ￾W￾L ￾