Page - 80 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

6. Upper Level Controller Parameter Identification Theinput fortheACCcontroller iscalculatedusingeqs. (2.22)to(2.24). Theparameters of eq. (2.22)are found inchapters6.2.1and6.2.2. The inter-vehicledistanceusedatzero speed is set to s0 = 1.978m for all simulated manoeuvres. The parameter τset has to be determined for each manoeuvre, using the method described in chapter 6.2.2. This is necessary due to the significant variation of the following behaviour of different drivers. The output of the simulations is the state vector at simulation time tk reading x˜k=   vs˜x,kvv˜x,k va˜x,k   , (6.18) wherethelongitudinalpositionduringthesimulationiscalculatedusingvs˜x,k= ∫ tk 0 vv˜xdt. At time tk, the error between measurement and simulation is defined by e˜k=   vsx,kvvx,k vax,k   − x˜k. (6.19) The goal during the identification process is to minimize the cost function reading JDF = ∑ k e˜Tk e˜k, (6.20) which is the sum of the squared errors over all time steps tk between measurements and simulation. Toperformtheoptimization, theNelder-Mead-Method isused,whichhasthe advantagethatgradientsof thecost functionwithrespect to thesearchedparametersare notneeded, [Obe12]. Adetaileddescriptionof thealgorithmused is given inappendixE. The output of this method is sensitive to the initial parametersP1,0 toP4,0. Ingeneral, theNelder-Mead-AlgorithmperformsanoptimizationwiththeoutputPi∈<. For the special case thatPi has either an upper or a lower bound or both, a parameter transformationhas tobeperformedto includethis limitation intheoptimizationprocess. The transformation rules read Pi=Pi,min+ Pi,max−Pi,min 1+e−P′i for Pi,min<Pi<Pi,max, (6.21) Pi=Pi,max−e−P′i for Pi<Pi,max and (6.22) Pi=Pi,min+e P′i for Pi,min<Pi. (6.23) Figure 6.9 illustrates the functions eqs. (6.21) to (6.23), where eq. (6.21) is a sigmoid function. The minimization can be solved by plugging eqs. (6.21) to (6.23) in the op- timization problem and varying P ′i instead of Pi. At the end of the process, the back transformation has to be done with the help of eqs. (6.21) to (6.23). One important advantage of this transformation is that a certain step width atP ′i leads to small steps near the limits ofPi,max orPi,min. 80