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):