where \(z_{k}\in\) \(R^{n_{z}}\) is an observation vector with\(n_{z}\) dimension; \(h\) is a measurement function representing the
relationship between the states and observations; \(w_{k}\) is the
measurement noise sequence, which is generally considered as an
independent and random vector.
With the state-transition equation (5) and the measurement equation (6),
PF recursively estimates system states at each time when an observation
becomes available. For the initial time, the PF estimate process can be
divided into four steps. First, PF uses the given initial distribution
to create N equally-weighted samples, called particles. These
particles are generally represented by\(\left\{x_{0}^{i}\right\}_{i=1}^{N}\). Second, the weights of all
particles are updated by comparing the simulated measurement\(z_{0}^{i}=h(x_{0}^{i},n_{k})\) to the observed \(z_{0}\). The
particle whose agreement between simulation and observation is higher
will be given a larger weight. Third, the estimated state\(\hat{x_{0}}\) will be calculated. The collection of weighted particles\(\left\{\left(x_{0}^{i},\omega_{0}^{i}\right)\right\}_{i=1}^{N}\)can approximate the posterior distribution of the state variable under
the condition of the given observation by normalizing the updated
weights. The estimated state \(\hat{x_{0}}\) is calculated as
(Moradkhani et al., 2005):