Page - 29 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

2.4. ACC Controller The time derivative from eq. (2.23) reads e˙r= sr˙︸︷︷︸ er˙ −vax τset. (2.29) Assuming that the vehicle follows the desired acceleration without any delay, τlong = 0 in eq. (2.25), then vax = ades. In this case, the vax of eq. (2.29) can be inserted in eq. (2.28), which leads to e˙r=−ker. This differential equation shows that the distance error er will drive asymptotically to zero if k> 0, assuming that the vehicle can follow the desired acceleration without any delay. This boundary condition should be used to find the parameter k. 2.4.3.2. Model Predictive Control Unless otherwise indicated, this chapter is based on [Ali10] and [CA07]. The Model Predictive Control (MPC) method uses the linear state space model, which describes the vehicle with eq. (2.27), to minimize a cost function to find the control variable u. The main advantage of MPC is that constraints for the control variable can be included in the optimization process. The continuous vehicle model of eq. (2.27) can be written as a discrete state-space model ek+1 =   1 T 00 1 −T 0 0 1− Tτlong   ︸ ︷︷ ︸ Ae ek+   00 T τlong   ︸ ︷︷ ︸ b uk, (2.30) where the parameter T is the sampling time of the controller. The output equation of the system reads yk= [ 1 0 0 0 1 0 ] ︸ ︷︷ ︸ C ek, (2.31) whereC is the output matrix. At time stepk, the controller calculates the predictedNp state vectors, which read ek+1|k= Ae ek+b ∆uk ek+2|k= Ae ek+1 +b ∆uk+1 = = A2e ek+Aeb ∆uk+b ∆uk+1 ek+3|k= A3e ek+A 2 eb ∆uk+Aeb ∆uk+1 +b ∆uk+2 ... ek+Np|k= A Np e ek+A Np−1 e b ∆uk+A Np−2 e b ∆uk+1 + · ··+ +A Np−Nc e b ∆uk+Nc−1. (2.32) 29