Page - 110 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

E. Simplex Method by Nelder-Mead wP bP ibP 2nP 1nP 4nP 3nP P¯ 1P 2P Figure E.1.: Nelder-Mead method for an optimization problem with nP = 2, based on [Obe12] In general, the reflection parameter is set toα= 1. If f (Pb)<f (Pn1)<f (Pib), then Pw is replaced by Pn1. Furthermore, if f (Pn1)<f (Pb), then the expansion point Pn2 = P¯+β ( P¯−Pw ) (E.4) is calculated, where the expansion parameter has to fulfil the condition β > α. If f (Pn2)<f (Pn1), then Pw is replaced by Pn2; otherwise, Pn1 is used. If f (Pn1)>f (Pib), then the contraction points Pn3 or Pn4 are calculated. The condi- tion f (Pn1)>f (Pw) leads to the contraction point Pn3 = P¯+γ ( P¯−Pw ) (E.5) and condition f (Pn1)<f (Pw) leads to Pn4 = P¯−γ ( P¯−Pw ) , (E.6) where inbothcasesγ<α. Iff (Pn3)<f (Pw)orf (Pn3)<f (Pw), thenPw is replaced by Pn3 or Pn4; otherwise, the simplex is shrunk around Pb reading Pi= Pi+Pb 2 . (E.7) In general, the parametersα= 1,β= 2 and γ= 0.5 are used. These steps are repeated until oneof theexit conditions is fulfilled. Thefirstoneconsiders thechangeof thenorm of two consecutive parameter vectors reading∥∥is−1P− isP∥∥≤εP, (E.8) 110