Particle Gaussian Mixture Filters for Nonlinear Non-Gaussian Bayesian Estimation

Nonlinear filtering is the problem of estimating the state of a stochastic nonlinear dynamical system using noisy observations. It is well known that the posterior state estimates in nonlinear problems may assume non-Gaussian multimodal probability densities. We present an unscented Kalman-particle hybrid filtering framework for tracking the three dimensional motion of a space object. The hybrid filtering scheme is designed to provide accurate and consistent estimates when measurements are sparse without incurring a large computational cost. It employs an unscented Kalman filter (UKF) for estimation when measurements are available. When the target is outside the field of view (FOV) of the sensor, it updates the state probability density function (PDF) via a sequential Monte Carlo method. The hybrid filter addresses the problem of particle depletion through a suitably designed filter transition scheme. The performance of the hybrid filtering approach is assessed by simulating two test cases of space objects that are assumed to undergo full three dimensional orbital motion. Having established its performance in the space object tracking problem, we extend the hybrid approach to the general multimodal estimation problem. We propose a particle Gaussian mixture-I (PGM-I) filter for nonlinear estimation that is free of the particle depletion problem inherent to most particle filters. The PGM-I filter employs an ensemble of randomly sampled states for the propagation of state probability density. A Gaussian mixture model (GMM) of the propagated PDF is then recovered by clustering the ensemble. The posterior density is obtained subsequently through a Kalman measurement update of the mixture modes. We prove the convergence in probability of the resultant density to the true filter density assuming exponential forgetting of initial conditions by the true filter. The PGM-I filter is capable of handling the non-Gaussianity of the state PDF arising from dynamics, initial conditions or process noise. A more general estimation scheme titled PGM-II filter that can also handle non-Gaussianity related to measurement update is considered next. The PGM-II filter employs a parallel Markov chain Monte Carlo (MCMC) method to sample from the posterior PDF. The PGM-II filter update is asymptotically exact and does not enforce any assumptions on the number of Gaussian modes. We test the performance of the PGM filters on a number of benchmark filtering problems chosen from recent literature. The PGM filtering performance is compared with that of other general purpose nonlinear filters such as the feedback particle filter and the log homotopy based particle flow filters. The results also indicate that the PGM filters can perform at par with or better than other general purpose nonlinear filters such as the feedback particle filter (FPF) and the log homotopy based particle flow filters. Based on the results, we derive important guidelines on the choice between the PGM-I and PGM-II filters. Furthermore, we conceive an extension of the PGM-I filter, namely the augmented PGM-I filter, for handling the nonlinear/non- Gaussian measurement update without incurring a large computational penalty. A preliminary design for a decentralized PGM-I filter for the distributed estimation problem is also obtained. Finally we conduct a more detailed study on the performance of the parallel MCMC algorithm. It is found that running several parallel Markov chains can lead to significant computational savings in sampling problems that involve multi modal target densities. We also show that the parallel MCMC method can be used to solve global optimization problems

A Bayesian Filtering Algorithm for Gaussian Mixture Models

A Bayesian filtering algorithm is developed for a class of state-space systems that can be modelled via Gaussian mixtures. In general, the exact solution to this filtering problem involves an exponential growth in the number of mixture terms and this is handled here by utilising a Gaussian mixture reduction step after both the time and measurement updates. In addition, a square-root implementation of the unified algorithm is presented and this algorithm is profiled on several simulated systems. This includes the state estimation for two non-linear systems that are strictly outside the class considered in this paper

Improved Gaussian mixture probability hypothesis density for tracking closely spaced targets

Probability hypothesis density (PHD) filter is a suboptimal Bayesian multi-target filter based on random finite set. The Gaussian mixture PHD filter is an analytic solution to the PHD filter for linear Gaussian multi-target models. However, when targets move near each other, the GM-PHD filter cannot correctly estimate the number of targets and their states. To solve the problem, a novel reweighting scheme for closely spaced targets is proposed under the framework of the GM-PHD filter, which can be able to correctly redistribute the weights of closely spaced targets, and effectively improve the multiple target state estimation precision. Simulation results demonstrate that the proposed algorithm can accurately estimate the number of targets and their states, and effectively improve the performance of multi-target tracking algorithm

Multiple Space Object Tracking Using A Randomized Hypothesis Generation Technique

In order to protect assets and operations in space, it is critical to collect and maintain accurate information regarding Resident Space Objects (RSOs). This collection of information is typically known as Space Situational Awareness (SSA). Ground-based and space-based sensors provide information regarding the RSOs in the form of observations or measurement returns. However, the distance between RSO and sensor can, at times, be tens of thousands of kilometers. This and other factors lead to noisy measurements that, in turn, cause one to be uncertain about which RSO a measurement belongs to. These ambiguities are known as data association ambiguities. Coupled with uncertainty in RSO state and the vast number of objects in space, data association ambiguities can cause the multiple space object-tracking problem to become computationally intractable. Tracking the RSO can be framed as a recursive Bayesian multiple object tracking problem with state space containing both continuous and discrete random variables. Using a Finite Set Statistics (FISST) approach one can derive the Random Finite Set (RFS) based Bayesian multiple object tracking recursions. These equations, known as the FISST multiple object tracking equations, are computationally intractable when solved in full. This computational intractability provokes the idea of the newly developed alternative hypothesis dependent derivation of the FISST equations. This alternative derivation allows for a Markov Chain Monte Carlo (MCMC) based randomized sampling technique, termed Randomized FISST (R-FISST). R-FISST is found to provide an accurate approximation of the full FISST recursions while keeping the problem tractable. There are many other benefits to this new derivation. For example, it can be used to connect and compare the classical tracking methods to the modern FISST based approaches. This connection clearly defines the relationships between different approaches and shows that they result in the same formulation for scenarios with a fixed number of objects and are very similar in cases with a varying number of objects. Findings also show that the R-FISST technique is compatible with many powerful optimization tools and can be scaled to solve problems such as collisional cascading

Towards joint communication and sensing (Chapter 4)

Localization of user equipment (UE) in mobile communication networks has been supported from the early stages of 3rd generation partnership project (3GPP). With 5th Generation (5G) and its target use cases, localization is increasingly gaining importance. Integrated sensing and localization in 6th Generation (6G) networks promise the introduction of more efficient networks and compelling applications to be developed