## 2 Preliminaries of State Estimation Methods Using Kalman Filter
### Particle Filter (PF)  
When processes and systems are non-Gaussian, the particle filter proves to be a very efficient predictive tool to represent and manage uncertainty and take account of the stochasticity of the process. The PF allows information to be processed from multiple measurement sources that are fused logically. However, there are a few drawbacks; e.g., the prediction results in PF greatly depend upon the number of particles or samples. Oneis more likely to obtain a close prediction of the estimated value to the actual value if there is a greater number of particles.  
Let the discrete state space model design be  
\[P_{t+1} = g(x_{t},u_{t},w_{t}),\] \[y_{t} = f(x_{t})+e_{t}. \tag{32}\]  
The state vector is given by \(x_{t}\in\mathbb{R}^{n}\), and \(u_{t}\) and \(\mathbb{Y}=\{y_{i}\}_{i=1}^{t}\) are the input vector and measurement vector, respectively. With probability density functions (PDFs) for process \(p_{w}(w)\) and measurement noises \(e_{t}\), and for the Bayesian inference [48], the filtering density given by \(p(x_{t}|\mathbb{Y}_{t})\), as well as the non-linear posterior prediction density represented as \(p(x_{t+1}|\mathbb{Y}_{t})\), are given as  
\[x_{t+1}|\mathbb{Y}_{t}=\int_{\mathbb{R}^{n}}p(x_{t+1}|x_{t})p(x_{t}|\mathbb{Y} _{t})dx_{t}, \tag{33}\]  
\[p(x_{t}|\mathbb{Y}_{t})=\frac{p(y_{t}|x_{t})p(x_{t}|\mathbb{Y}_{t-1})}{p(y_{t }|\mathbb{Y}_{t-1})}. \tag{34}\]  
An approximated solution for Equations (33) and (34) (the discrete-time Bayesian estimation problem) has been provided by [PERSON] et al. [5] and [PERSON] et al. [49] through a constant update in the posterior filtering density. The weight and location assigned to each particle represents the density in the state-space region because, unlike UKF, the particle filter estimates the filtering density \(p(x_{t}|\mathbb{Y}_{t})\) with a large set of weighted (\(\psi_{t}^{(i)}\)) samples (particles) \(\{x_{i}^{(i)}\}_{i=1}^{N}\) such that the sum of all weights is unity. The locations and corresponding weights of each particle are updated with each new measurement, and the corresponding weights and particles then can be used to solve Bayesian equations approximately; the re-sampling of samples is performed to avoid divergence [49]. The steps involved in the calculation of PF are (i) the generation of N samples \(\{x_{i}^{(i)}\}_{i=1}^{N}\) from \(p(x_{0})\); (ii) computation of weights \(\psi(i)_{t}=\psi(i)_{t-1}\frac{p(y_{t}|x_{t}^{(i)})p(x_{t}^{(i)}|x_{t-1}^{(i) })}{p^{prop}(x_{t}^{(i)}|x_{t-1}^{(i)},y_{t})}\), and normalisation of weights \(\bar{\psi}_{t}^{(i)}=\psi_{t}^{(i)}/\sum_{i=1}^{N}\psi_{t}^{(i)},i=1,\ldots,N\); (iii) generation of predictions from he proposed distribution \(X_{t+1}^{(i)}\sim p^{prop}(x_{t+1}^{(i)}|x_{t}^{(i*)},y_{t+1}),i=1,\ldots,N\); and (iv) increment of \(t\) and repeated performance of step (ii).  
The minimum mean square estimate (MMSE) (35) is often chosen to estimate each time \(t\); that is,  
\[\hat{x}_{t|t}=\mathbb{E}(x_{t})=\int_{\mathbb{R}^{n}}x_{t}p(x_{t}|\mathbb{Y} _{t})dx_{t}\approx\sum_{i=1}^{N}\psi_{t}^{(i)}X_{t}^{(i)}, \tag{35}\]  
but there are also other available choices of estimation, such as the ML-estimate. From [5], we conclude that \(N\rightarrow\infty\), meaning that the estimated PDF converges into the true value.