Page - 93 - in Maximum Tire-Road Friction Coefficient Estimation

5 Tire/road friction estimator 2. Update step: The a posteriori PDF is given by p(x(k)|z(k)) = p(z(k)|x(k)) ·p(x(k)|z(k−1))∫ p(z(k)|x(k)) ·p(x(k)|z(k−1)) dx(k). (5.5) The termp(z(k)|x(k)) in Equation 5.5, which is called the measurement likelihood func- tion, [Wat06, p.70], can be calculated when both the measurement equation h(k) and the PDF of the measurement noise v(k) are known. Assuming that the measurement noise is Gaussian, the likelihood p(z(k)|x(k)) is proportional to a multivariate Gaussian distribution given by p(z(k)|x(k))∝ 1 (2pi) n 2 · |S|(12) e−(z(k)−h(x(k))) T·S−1·(z(k)−h(x(k))), (5.6) forann-dimensionalmeasurementequationz(k)withmeasurementnoisev(k)vN(0,S), wherein S denotes the covariance matrix of the measurement noise, [Sim06]. Only for some cases is there an analytical solution for Equations 5.4 and 5.5. In the special case of linear functions of f(k) and h(k), and when v(k) and w(k) are additive, independent and Gaussian, the solution of the recursive Bayesian state estimator is the Kalman fil- ter, [Sim06], which has also been proposed for friction potential estimation in several publications, see Section 2.2.2. A further development of the recursive Bayesian state estimator is the particle filter that is presented in Section 5.2. An adapted form of a particle filter was applied by Ray, [Ray97], see Section 2.2.2 for the observed variables and Section 5.2 for the particle filter adaptation used. 5.2. Particle filtering According to Simon, the particle filter originated from a numerical implementation of the recursive Bayesian state estimator, [Sim06]. It is a non-linear state estimator that, unlike an extended Kalman Filter, for example, does not need to linearize the non-linear state equations at the working point for each time step before it can be solved, [Sim06]. The recursive Bayesian state estimator considers only one initial state vector x(k) that is to be estimated and that is given with its initial PDF p(x(0)), see Section 5.1.1. In contrast, the particle filter considers h= 1, ...,N particles for each of the l initial state vectors x+h(0) that are generated based on the initial PDF p(x(0)) of each state vector x. These particles are then re-sampled based on the relative likelihood of each of theN different particles in order to obtain the most likely states. One of the shortcomings of 93