Event-Triggered Algorithms for Leader-Follower Consensus of Networked Euler-Lagrange Agents

This paper proposes three different distributed event-triggered control algorithms to achieve leader-follower consensus for a network of Euler-Lagrange agents. We firstly propose two model-independent algorithms for a subclass of Euler-Lagrange agents without the vector of gravitational potential forces. By model-independent, we mean that each agent can execute its algorithm with no knowledge of the agent self-dynamics. A variable-gain algorithm is employed when the sensing graph is undirected; algorithm parameters are selected in a fully distributed manner with much greater flexibility compared to all previous work concerning event-triggered consensus problems. When the sensing graph is directed, a constant-gain algorithm is employed. The control gains must be centrally designed to exceed several lower bounding inequalities which require limited knowledge of bounds on the matrices describing the agent dynamics, bounds on network topology information and bounds on the initial conditions. When the Euler-Lagrange agents have dynamics which include the vector of gravitational potential forces, an adaptive algorithm is proposed which requires more information about the agent dynamics but can estimate uncertain agent parameters. For each algorithm, a trigger function is proposed to govern the event update times. At each event, the controller is updated, which ensures that the control input is piecewise constant and saves energy resources. We analyse each controllers and trigger function and exclude Zeno behaviour. Extensive simulations show 1) the advantages of our proposed trigger function as compared to those in existing literature, and 2) the effectiveness of our proposed controllers.Comment: Extended manuscript of journal submission, containing omitted proofs and simulation

Event-Triggered Consensus and Formation Control in Multi-Agent Coordination

The focus of this thesis is to study distributed event-triggered control for multi-agent systems (MASs) facing constraints in practical applications. We consider several problems in the field, ranging from event-triggered consensus with information quantization, event-triggered edge agreement under synchronized/unsynchronized clocks, event-triggered leader-follower consensus with Euler-Lagrange agent dynamics and cooperative event-triggered rigid formation control. The first topic is named as event-triggered consensus with quantized relative state measurements. In this topic, we develop two event-triggered controllers with quantized relative state measurements to achieve consensus for an undirected network where each agent is modelled by single integrator dynamics. Both uniform and logarithmic quantizers are considered, which, together with two different controllers, yield four cases of study in this topic. The quantized information is used to update the control input as well as to determine the next trigger event. We show that approximate consensus can be achieved by the proposed algorithms and Zeno behaviour can be completely excluded if constant offsets with some computable lower bounds are added to the trigger conditions. The second topic considers event-triggered edge agreement problems. Two cases, namely the synchronized clock case and the unsynchronized clock case, are studied. In the synchronized clock case, all agents are activated simultaneously to measure the relative state information over edge links under a global clock. Edge events are defined and their occurrences trigger the update of control inputs for the two agents sharing the link. We show that average consensus can be achieved with our proposed algorithm. In the unsynchronized clock case, each agent executes control algorithms under its own clock which is not synchronized with other agents' clocks. An edge event only triggers control input update for an individual agent. It is shown that all agents will reach consensus in a totally asynchronous manner. In the third topic, we propose three different distributed event-triggered control algorithms to achieve leader-follower consensus for a network of Euler-Lagrange agents. We firstly propose two model-independent algorithms for a subclass of Euler-Lagrange agents without the vector of gravitational potential forces. A variable-gain algorithm is employed when the sensing graph is undirected; algorithm parameters are selected in a fully distributed manner with much greater flexibility compared to all previous work concerning event-triggered consensus problems. When the sensing graph is directed, a constant-gain algorithm is employed. The control gains must be centrally designed to exceed several lower bounding inequalities which require limited knowledge of bounds on the matrices describing the agent dynamics, bounds on network topology information and bounds on the initial conditions. When the Euler-Lagrange agents have dynamics which include the vector of gravitational potential forces, an adaptive algorithm is proposed. This requires more information about the agent dynamics but allows for the estimation of uncertain agent parameters. The last topic discusses cooperative stabilization control of rigid formations via an event-triggered approach. We first design a centralized event-triggered formation control system, in which a central event controller determines the next triggering time and broadcasts the event signal to all the agents for control input update. We then build on this approach to propose a distributed event control strategy, in which each agent can use its local event trigger and local information to update the control input at its own event time. For both cases, the trigger condition, event function and trigger behaviour are discussed in detail, and the exponential convergence of the formation system is guaranteed

Opinion Dynamics and the Evolution of Social Power in Social Networks

A fundamental aspect of society is the exchange and discussion of opinions between individuals, occurring in mediums and situations as varied as company boardrooms, elementary school classrooms and online social media. This thesis studies several mathematical models of how an individual’s opinion(s) evolves via interaction with others in a social network, developed to reflect and capture different socio-psychological processes that occur during the interactions. In the first part, and inspired by Solomon E. Asch’s seminal experiments on conformity, a novel discrete-time model of opinion dynamics is proposed, with each individual having both an expressed and a private opinion on the same topic. Crucially, an individual’s expressed opinion is altered from the individual’s private opinion due to pressures to conform to the majority opinion of the social network. Exponential convergence of the opinion dynamical system to a unique configuration is established for general networks. Several conclusions are established, including how differences between an individual’s expressed and private opinions arise, and how to estimate disagreement among the private opinions at equilibrium. Asch’s experiments are revisited and re-examined, and then it is shown that a few extremists can create “pluralistic ignorance”, where people believe there is majority support for a position but in fact the position is privately rejected by the majority of individuals! The second part builds on the recently proposed discrete-time DeGroot–Friedkin model, which describes the evolution of an individual’s self-confidence (termed social power) in his/her opinion over the discussion of a sequence of issues. Using nonlinear contraction analysis, exponential convergence to a unique equilibrium is established for networks with constant topology. Networks with issue-varying topology (which remain constant for any given issue) are then studied; exponential convergence to a unique limiting trajectory is established. In a social context, this means that each individual forgets his/her initial social power exponentially fast; in the limit, his/her social power for a given issue depends only on the previously occurring sequence of dynamic topology. Two further related works are considered; a network modification problem, and a different convergence proof based on Lefschetz Fixed Point Theory. In the final part, a continuous-time model is proposed to capture simultaneous discussion of logically interdependent topics; the interdependence is captured by a “logic matrix”. When no individual remains attached to his/her initial opinion, a necessary and sufficient condition for the network to reach a consensus of opinions is provided. This condition depends on the interplay between the network interactions and the logic matrix; if the network interactions are too strong when compared to the logical couplings, instability can result. Last, when some individuals remain attached to their initial opinions, sufficient conditions are given for opinions to converge to a state of persistent disagreement