6Particle Filter
6.1 Introduction
The optimal nonlinear filtering problem relies on recursively calculating the posterior distribution. However, except for special cases, it is impossible to find closed‐form solutions. To elaborate, when unlike Gaussian distributions, the corresponding probability distribution functions (PDFs) cannot be described by well‐known mathematical functions, the whole PDF must be considered in the calculations. Regarding the fact that dealing with the whole PDF for a single variable would be similar to working with an infinite‐dimensional vector, it is computationally intractable to compute the distributions of interest for general non‐Gaussian cases. Therefore, function approximation techniques must be used to reasonably approximate the corresponding distributions in order to implement a suboptimal Bayesian filter, which is computationally tractable, for the general case of nonlinear non‐Gaussian dynamic systems. We can look at the problem from another perspective as well. For systems with severe nonlinearity, function approximation based on local linearization such as Taylor series expansion used in the extended Kalman filter (EKF) is not adequate. An idea for handling such systems is that approximating a probability distribution may be easier than approximating a nonlinear function. Following this line of thinking, numerical methods are deployed to approximate the corresponding distributions. A popular numerical approximation scheme for this purpose ...