From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.

ABSTRACT Multiagent systems aim to accomplish highly complex learni ng tasks through decentralised consensus seeking dynamics and thei r use has garnered a great deal of attention in the signal processing a nd computational intelligence societies. This article examines the behaviour of multiagent networked systems with nonlinear filtering/l earning dynamics. To this end, a general formulation for the actions of an agent in multiagent networked systems is presented and cond itions for achieving a cohesive learning behaviour is given. Impor tantly, application of the so derived framework in distributed and f ederated learning scenarios are presented. Index T erms -- Multiagent systems, nonlinear dynamics, distributed learning, federated learning, 1. INTRODUCTION Traditionally, signal processing and learning techniques have been concerned with single agent operations [1,2].

The introduction of automated flight control and management systems have made possible aircraft designs that sacrifice arodynamic stability in order to incorporate stealth technology intro their shape, operate more efficiently, and are highly maneuverable. Therefore, modern flight management systems are reliant on multiple redundant sensors to monitor and control the rotations of the aircraft. To this end, a novel distributed quaternion Kalman filtering algorithm is developed for tracking the rotation and orientation of an aircraft in the three-dimensional space. The algorithm is developed to distribute computation among the sensors in a manner that forces them to consent to a unique solution while being robust to sensor and link failure, a desirable characteristic for flight management systems. In addition, the underlying quaternion-valued state space model allows to avoid problems associated with gimbal lock. The performance of the developed algorithm is verified through simulations.

Motivated by the need for accurate frequency information, a novel algorithm for estimating the fundamental frequency and its rate of change in three-phase power systems is developed. This is achieved through two stages of Kalman filtering. In the first stage a quaternion extended Kalman filter, which provides a unified framework for joint modeling of voltage measurements from all the phases, is used to estimate the instantaneous phase increment of the three-phase voltages. The phase increment estimates are then used as observations of the extended Kalman filter in the second stage that accounts for the dynamic behavior of the system frequency and simultaneously estimates the fundamental frequency and its rate of change. The framework is then extended to account for the presence of harmonics. Finally, the concept is validated through simulation on both synthetic and real-world data.