Page - 66 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis its sensitivity based on the relation in Equation 4.8 is given by pβ= ∂β ∂µ = ∂β ∂vy · ∂vy ∂µ + ∂β ∂vx · ∂vx ∂µ . (4.10) With ∂β/∂vy= 1/vx and ∂β/∂vx=vy/v 2 x, the sensitivity ofβ toµ max reads pβ= 1 vx ·pvy+ vy v2x ·pvx, (4.11) including ∂vy/∂µ=pvy and ∂vx/∂µ=pvx. 4.2. Numerical implementation Dickinson and Gelinas propose an approach in which the model and the sensitivity equations are solved simultanoeusly, [DG76]. This procedure is only possible when the Jacobian J is calculated as the partial derivative of an analytical function. Although possible, the many dependencies within the model equations (as shown in Section 4.3) complicate the calculation of an analytical derivative. An alternative method for cal- culating both the Jacobian J and fc is given by automatic differentiation (AD). AD is neither symbolic nor numerical approximation. It enables the calculation of derivatives accurate to working precision at arbitrary points. Theoretically, it can be applied to ev- ery function described in a computable program that can execute elementary arithmetic operations (e.g. additions) and elementary functions (e.g. sine functions). An auto- mated procedure based on the chain rule for derivatives is then applied to this function, and it calculates the desired derivatives, [BH00]. In this work, the automatic differentia- tion toolbox Adimat developed for the computer language Matlab was used, [BBL+02]. The calculation of the Jacobian J and fc using AD requires the nmodel equations to be at least one time differentiable. This does not account for the calculation of the longitudinal slip sx, see Equation 2.3, where both absolute value and a distinction of cases (e.g. finding the maximum out of two values) have to be considered. It is also necessary to calculate an absolute value for the calculation of combined tire forces, see Section 3.3.1. Thus, these mathematical functions are numerically approximated by the functions shown in Appendix C.1. Finally, the steps to calculate the sensitivities p are summarized. In a first step, the differential equation z˙ of the non-linear vehicle model is solved for z. In a second step, the JacobianJand the derivative fc are calculated using automatic differentiation. 66