Page - 92 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator a deterministic function, the model therefore has to be treated as a stochastic process characterised by its probability distributions. Both w(k) and v(k) are assumed to be independent white noises with a priori known probability density functions (PDF), e.g. from past measurements, [Sim06]. PDF describe the probabilistic characteristics of a variableuby giving its probability within a rangea≤u<b. PDF are standardised and can only comprise values between 0≤ p(u)≤ 1, [Wat06, p.35]. The recursive Bayesian state estimator combines both the state estimation of a model parameter given in the form of Equation 5.1 and the prior knowledge of the state variables’ probabilistic char- acteristics using a Bayesian framework. The basis is Bayes’ theorem, which enables the calculation of the conditional probability p of a state, written as p(u|w). It describes the probability that an event uwill occur under the condition that an eventw arose prior. Bayes’ theorem requires a priori knowledge of the probabilities p(u) and p(w), which describe the probability of the occurrence of the individual events u andw. In addition, the conditional probability p(w|u) has to be known, which, unlike the desired conditional PDF, describes the conditional probability of an eventw given the eventu, [Bau07, p.21]. Thus, p(u|w) is finally given by p(u|w) = p(w|u) ·p(u) p(w) . (5.3) 5.1.1. The recursive Bayesian state estimator The recursive Bayesian state estimator applies Bayes’ theorem, see Equation 5.3, to the state model given in Equations 5.1 and 5.2. According to Simon, [Sim06], the two main steps to be solved for each time step k are: 1. Prediction step: The a priori PDF of the current state x(k) is calculated using the Chapman- Kolmogorov equation given by p(x(k)|z(k−1)) = ∫ p(x(k)|x(k−1)) ·p(x(k−1)|z(k−1)) dx(k−1). (5.4) The term p(x(k)|x(k−1)) is known when both the state model equation for f(k) and the PDF of the process noise w(k) are known. From time step k≥ 2, the second term p(x(k−1)|z(k−1)) is known from the update step of the last time step. For k= 1, an initial value p(x(0)|z(0)) = p(x(0)) has to be assumed based on the PDF of its initial state p(x(0)). 92