Page - 30 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

2. Adaptive Cruise Control y u k +1k1−k cN+k pN+k· ·· · ·· k,desy Figure 2.18.: MPC optimization at time step k, adapted from [Ali10] Here, the symbol ek+j|k is the predicted state vector at time stepk+j, generated at the present time step k. The same is done for the output equations yk+1|k= C Ae ek+C b ∆uk yk+2|k= C Ae ek+1 +C b ∆uk+1 = = C A2e ek+C Aeb ∆uk+C b ∆uk+1 yk+3|k= C A3e ek+C A 2 eb ∆uk+C Aeb ∆uk+1 +C b ∆uk+2 ... yk+Np|k= C A Np e ek+C A Np−1 e b ∆uk+C A Np−2 e b ∆uk+1 + · ·· +C A Np−Nc e b ∆uk+Nc−1. (2.33) Figure 2.18 shows a graphical illustration of the process described above. The controller seeks to minimize the error ydes−y. Therefore, the output is calculated forNp time steps, beginning at the present step k. The control variableuk is predicted forNc steps (note thatNc≤Np). In eqs. (2.32)and (2.33), the new control variable ∆uj=uj−uj−1 is introduced. Only the first control variable ∆uk is applied to the system. In the next step, the optimization starts from the beginning. The column vectors Yk= [ yTk+1|k y T k+2|k · ·· yTk+Np|k ]T (2.34) and ∆Uk= [ ∆uk ∆uk+1 ∆uk+2 · ·· ∆uk+Nc−1 ]T (2.35) are introduced to have a more compact notation of eqs. (2.32) and (2.33), which reads Yk=F ek+Φ ∆Uk. (2.36) 30