*3.4. PFE SOC Brief Presentation*

There is a substantial similarity between the non-linear estimator PFE SOC [10,11] and the first two SOC estimators presented in the previous subsections, i.e., AEKF SOC and AUKF SOC, due to the same "prediction-corrector" structure identified in all three. Therefore, it is easy to anticipate that the PFE SOC estimator updates in a "recursively way" the state estimate and then finds the innovations driving a stochastic process based on a sequence of observations (measurement output dataset), as is shown in detail in the original work [11]. In [11] it is stated that the PFE SOC estimator accomplishes this objective by "a sequential Monte Carlo method (bootstrap filtering), a technique

for implementing a recursive Bayesian filter by Monte Carlo simulations", which is also mentioned in [4]. After the initialization stage of the algorithm, in the second stage (i.e., "the prediction phase"), the state estimates of the process are used to predict and to "smooth" the stochastic process. As a result of the prediction, innovations are useful for estimating the parameters of the linear or nonlinear dynamic model [4,11]. The basic idea disclosed in [4] is that any probability distribution function (pdf) of a random state variable *x* can be approximated by a set of samples (particles), similar to what sigma points do in the AUKF SOC estimator developed in Section 3.2. Each particle has one set of values for each process state variable *x*. The novelty of the PFE SOC estimator is its ability to represent any arbitrary pdf, even if for non-Gaussian or multi-modal pdfs [4,11]. In conclusion, the nonlinear design of the SOC PFE estimator has a similar approach to that of the AUKF SOC design, as long as a local linearization technique is not required, as in the case of AEKF SOC, or "any raw functional approximation" [4,11]. Furthermore, the PFE SOC "can adjust the number of particles to match available computational resources, so a trade-o ff between accuracy of estimate and required computation" [11]. Moreover, it is "computationally compliant even with complex, non-linear, non-Gaussian models, as a trade-o ff between approximate solutions to complex nonlinear dynamic model versus exact solution to approximate dynamic model" [11]. To ge<sup>t</sup> a better insight into this estimation technique, the original paper [11] can be accessed. Since, the current research work follows the same PFE design procedure steps as in [11], our focus is directed only at the implementation aspects.
