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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 zË™ = 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
Maximum Tire-Road Friction Coefficient Estimation
- Title
- Maximum Tire-Road Friction Coefficient Estimation
- Author
- Cornelia Lex
- Publisher
- Verlag der Technischen Universität Graz
- Location
- Graz
- Date
- 2015
- Language
- English
- License
- CC BY-NC-ND 3.0
- ISBN
- 978-3-85125-423-5
- Size
- 21.0 x 29.7 cm
- Pages
- 189
- Category
- Technik