Page - 71 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis as presented earlier in this chapter. Instead ofµmax, the investigated parameter is then the initial longitudinal speed vx,0. The initial longitudinal speed vx,0 can be written as z1(0), as it is the initial state of the first state variable z1. Then, it does not explicitly appear in the right-hand side fl of the differential equation system z˙l, [DG76], and fc in Equation 4.3 is a zero vector. Thus, p˙ is then given by p˙=J ·p. (4.12) According to Section 4.4 (cf. Equation 4.7), the initial values of p with regard to the initial longitudinal velocity read p1(0) = 1, (4.13) pl 6=1(0) = 0. (4.14) Whencalculatingthesensitivitieswithrespecttovx,0,nequations forphavetobesolved for all simulations. Since the sensitivities with respect to µmax were already available, anotherapproachwaschosen. Like the investigationof the influenceof time-dependency in Section 4.3.2, the influence of different initial longitudinal velocities on the outcome of the sensitivities with respect toµmax was investigated with simulations that start at different vx. Figure 4.4 shows that the selected sensitivities depend on the initial longitudinal ve- locity vx,0. With increasing initial speed, the sensitivity ∂vx/∂µ also increases for both pure longitudinal accelerations of 0.45 and 2.45 m/s2. With a higher dynamic excitation (e.g. longitudinal acceleration), the sensitivities also increase. The absolute value of the sensitivity ∂ωfl/∂µ shows similar behaviour. It has to be mentioned that the vehicle model does not consider maximum longitudinal velocity of the vehicle or limitations of engine torque. It is obvious that there is a dependency on the initial longitudinal speedvx,0. Althoughtheabsolutevalueof the sensitivities changeswith the longitudinal speed, it can be assumed that the influence of the velocity affects all state variables to a comparable extent. It seems that the influence could be compensated with a quadratic dependence of the longitudinal speed, for example. In any event, for the observer pro- posed in Section 5, it is not necessary to know the exact values of the sensitivities. It is more important to know their influence in comparison to the other state variable’s sensitivities in a certain region in the parameter space. The results of the sensitivity analysis are only used to select the state variables to be observed in the estimation al- 71