4.3.ROBUST PERFORMANCE 47 Summary of Robust Stability Tests Table 4.1 summarizes the robust stability tests for the cther uncertainty models. Pert urbation Condition (1+△W2)P W2T<1 P+△W2 W2CS]loo 1 P(1+△W2P) W2PSIloo <1 P1+△W2) W2Slloo 1 Table 4.1 Note that we get the same four transfer functions-T,CS,PS,S-as we did in Section 3.4.This should not be too surprising since (up to sign)these are the only closed-locp transfer functions for a unity feedback SISO sy stem. .3 Robust Performance Now we look into performance of the perturbed plant.Suppose that the plant transfer function belongs to a set P.The general notion of robust performance is that internal stability and per formance,of a specified type,should hold for all plants in P.Again we focus on multiplicative pert urbat ions. Recall that when the nominal feedback system is internally stable,the nominal performance condition is WiSlloo 1 and the robust stability condition is W2Tlloo<1.If P is pert urbed to (1+AW2)P,S is perturbed to 1 1+(1+△W2)Z=1+△wW27T≤ Clearly,the robust performance condition should therefore be WiS W2Tlloo 1 and 1+△W2T <1,△≤ Here A must be allowable.The next theorem gives a test for robust performance in terms of the funct ion sHW1(s)S(s)j+jW2(s)T(s) which is denoted W Sj+2Tj Theorem 2 A necessary and sufficient condition for robust performance is W1Sj+JW2T<1≤ (4.3) Proof ()Assume (4.3),or equivalently, W2Tllo and <1≤ (4.4) ROBUST PERFORMANCE  Summary of Robust Stability Tests Table  summarizes the robust stability tests for the other uncertainty models Perturbation Condition   W￾P kW￾T k￾   P  W￾ kW￾CSk￾   P   W￾P  kW￾P Sk￾   P   W￾ kW￾Sk￾   Table  Note that we get the same four transfer functionsT  CS P S Sas we did in Section  This should not be too surprising since up to sign these are the only closedloop transfer functions for a unity feedback SISO system ￾ Robust Performance Now we look into performance of the perturbed plant Suppose that the plant transfer function belongs to a set P The general notion of robust performance is that internal stability and per formance of a specied type should hold for all plants in P Again we focus on multiplicative perturbations Recall that when the nominal feedback system is internally stable the nominal performance condition is kWSk￾   and the robust stability condition is kW￾T k￾   If P is perturbed to   W￾P  S is perturbed to      W￾L  S W￾T Clearly the robust performance condition should therefore be kW￾T k￾   and WS W￾T ￾    Here  must be allowable The next theorem gives a test for robust performance in terms of the function s  jWsSsj  jW￾sT sj which is denoted jWSj  jW￾T j Theorem A necessary and sucient condition for robust performance is kjWSj  jW￾T jk￾    Proof  Assume  or equivalently kW￾T k￾ and WS  jW￾T j ￾