Page - 65 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis 4.1.2. Initial values of sensitivities According to Dickinson and Gelinas, the value for the initial sensitivity pl(0) is defined by its limit pl(0) = lim ∆cm→0 zl(cm+∆cm,0)−zl(cm,0) ∆cm , (4.4) [DG76]. Depending on whether a parameter is an explicit initial condition of any of the state variables zl or not, two cases can be distinguished. If parameter cm is not an initial condition of any of the state variables zl, the term zl(cm+∆cm,0)−zl(cm,0) in Equation 4.4 is zero, and all l initial sensitivities are therefore given by pl(0) =0. (4.5) For the secondcase, it is assumedthatparameter cm is anexplicit initial conditionof the n-th state variable zn. Thus, the term zn(cm+∆cm,0)−zn(cm,0) = ∆cm in Equation 4.4 results in pl(0) =1. Thus, the initial sensitivities read pl=n(0) = 1, (4.6) pl 6=n(0) = 0. (4.7) 4.1.3. Sensitivities of functions related to state variables In addition to the sensitivity of the state variables, the sensitivity of other variables that depend on these state variables can be of interest. As mentioned in Section 2.2.2, there are existing methods that focus on the vehicle’s side slip angle β. In addition, the self-aligning torques at the tires have proven to be a good measure for µmax in previous investigations, e.g. Hsu et al., [HLGG06]. As with the propagation of error, thesensitivitypeofe=f [u(cm),w(cm)]withrespect totheparametercm canbewritten by its first order Taylor series pe= ∂e ∂cm = ∂e ∂u · ∂u ∂cm + ∂e ∂w · ∂w ∂cm . (4.8) WithinEquation4.8, theterms∂e/∂uand∂e/∂wcanbeinterpretedasweightingfactors. So, by knowing the sensitivity of state variables, the sensitivity of other variables that are related to state variables can be expressed. Applied to the side-slip angle β= arctan vy vx ≈ vy vx forβ<<1 rad, (4.9) 65