Page - 34 - 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 0 −40 40 [m]re[m/s]r˙e 20 0 −20 0 100 200 =0t chatter −50 Figure 2.20.: Trajectory and sliding surface from the simulation in chapter 2.4.4 tomeetthis requirement, [Ada09b,SEFL14]. Theparametersqandkhavetobepositive. The time derivative of the surface reads s˙ = ∂s∂xSMC x˙SMC. Using this formulation, eqs. (2.44) and (2.46), the control law ends up as ades=−r T ASMC xSMC+qsign(r T xSMC)+k r T xSMC rT b . (2.47) The vector r and the scalars q and k are used to tune the controller. Note that in the case of eq. (2.25), only the last component of b is not equal to zero. Therefore, the corresponding component of r also cannot be zero, or else a division by zero will occur. The term with the signum functionmainly causes the chatter aroundthedefined surface. If eq. (2.44) is changed to s˙=−ks, meaning q= 0, the chatter can be minimised, as Drew did in [Dre02]. It will not work in general because the error will not drive to zero in finite time. Nevertheless, he achieved satisfying results. Instead of setting q= 0, the signum function may be replaced by a sigmoid function. This minimises chatter, but there is the same problem of asymptotic convergence towards zero. Therefore a real sliding on the defined surface is not possible. However, its performance is close to the system with the input of eq. (2.47), and therefore it is called Quasi Sliding Mode. Figure 2.20 shows the trajectory and the sliding surface for the simulation done in chapter 2.4.4. When the trajectory is near the surface and runs through it, it changes the side of the plane very often, which leads to the aforementioned chatter. 34