Page - 64 - in Maximum Tire-Road Friction Coefficient Estimation

4 Sensitivity Analysis for each input factor have to be known. For the proposed problem, local methods are suitable, as they are typically modelled as initial-value ordinary differential equations (ODE). Usually, these methods require the calculation of partial derivatives, see Section 4.1.1. Local methods have shortcomings when dealing with the influence of the uncer- tainty of model parameters, which does not limit the proposed application. The goal is to calculate the (linear) first-order local sensitivity coefficients of the non-linear model derived in Section 3 with respect to the parameterµmax. This work focuses on a quasi- direct approach, which means that a direct method based on Equation 4.2 is used, but with the Jacobian J to be calculated with automatic differentiation (AD) rather than using thederivativeofanalytical functions, [CSST00], seealsoSection4.1.1. Although it wouldbepossible to calculatehigherorder sensitivitieswith this approach, it is assumed that this would not increase the accuracy considerably, [DG76]. 4.1.1. Basic theory of direct sensitivity analysis It is assumed that the investigated model is given by a non-linear, time-dependent ODE in the form of z˙ = f(z1, ...,zn,t,c) with l= 1, ...,n elements in state variable vector z andm= 1, ...,o elements in parameter vector c. The linear sensitivities of the given model with respect to a parameter cm then readp= ∂z ∂cm . For each of the oparameters, nODE have to be solved for z. The sensitivity pl for the l-th state variable zl can be found as the time integral of p˙l= ∂ ∂t(pl). Applying both the chain rule of differentiation and the rule for interchanging the order of differentiation for mixed partials, p˙l= ∂ ∂t ( ∂zl ∂cm ) = ∂ ∂cm ( ∂zl ∂t ) , (4.1) and the sensitivity system is thus given by p˙l= ∂fl ∂cm + n∑ d=1 ∂fl ∂zd · ∂zd ∂cm (4.2) with fl being the right-hand side of z˙l, [DG76]. Using the Jacobian J, whose (l,d)-th element is ∂fl∂zd, and fcbeing the sensitivityof right-handside f with respect toparameter cm, Equation 4.2 reads p˙= fc+J ·p. (4.3) 64