Fal!2001 16.3120-8 Robust stability Tests From the nyquist plot, we developed a measure of the "closeness of the loop transfer function(LTF)to the critical point dlj) 1+ LNga) N Magnitude of nominal sensitivity transfer function S(s) Based on this result, the test for robust stability is whether 1TN)}=-(u 1+LNu) EoGjw ) Magnitude bound on the nominal complementary sensitiv ity transfer function T(s) Recall that S(s)+T(s,1 Proof: With d ju)=1+ LN(ja), criterion of interest for robust stability is whether the possible changes to the ltF Lpla)-) exceed the distance from the ltf to the critical point dlj)= 1+ Lnga) Because if it does, then it is possible that the modeling error could change the number of encirclements Actual system could be unstableFall 2001 16.31 20—8 Robust Stability Tests • From the Nyquist Plot, we developed a measure of the “closeness” of the loop transfer function (LTF) to the critical point: 1 1 = |d(jω)| |1 + LN(jω)| , |SN(jω)| — Magnitude of nominal sensitivity transfer function S(s). • Based on this result, the test for robust stability is whether: ¯ ¯ ¯ LN(jω) ¯ 1 ¯ ¯ |TN(jω)| = ¯ ¯ < ∀ω 1 + LN(jω) |E0(jω)| — Magnitude bound on the nominal complementary sensitiv￾ity transfer function T(s). — Recall that S(s) + T(s) , 1 • Proof: With d(jω)=1 + LN(jω), criterion of interest for robust stability is whether the possible changes to the LTF |Lp(jω) − LN(jω)| exceed the distance from the LTF to the critical point |d(jω)| = |1 + LN(jω)| — Because if it does, then it is possible that the modeling error could change the number of encirclements ⇒ Actual system could be unstable