Stability and Controllability of Double Integrator Consensus Systems in Heterogeneous Networks

Abstract

A consensus system is a set of communicating agents that agree on a variable of interest using only local communication. Some possible uses for consensus systems include smart grids, wireless sensor networks, social networks and various applications of unmanned aerial vehicles such as surveillance, mapping and environmental monitoring. If the agents have double integrator dynamics, then the consensus system is called a double integrator consensus system. In this thesis a double integrator consensus system in continuous time is studied. The agents are assumed to be communicating their position and velocity information along some possibly different, weighted directed communication networks that are modeled by weighted directed graphs. The mathematical model of the system has the form ẍ(t) = −Lxx(t) −βLẋẋ(t), (S1) where x : R+ → R and ẋ : R+ → R are the collected positions and velocities of the agents, Lx and Lẋ are the Laplacians of the communication graphs and β ∈ R+ {0} is a gain. A stability concept for such systems is derived in the thesis. The system (S1) is said to be consensus stable if all of its velocity differences approach zero in time, and asymptotically consensus stable, if all of its position differences tend to zero. The standard assumption made in the literature, that the communication networks are the same, i.e. homogeneous, for both information, is too strong in many cases. Thus, the algorithm is investigated under the assumption that the communication graphs may be different and possibly even disconnected. Necessary and sufficient conditions for consensus stability and asymptotic consensus stability are presented in the special case that the communication graphs are weighted and undirected. These conditions are subsequently relaxed to some cases of weighted directed graphs. Moreover, we consider the convergence rate of the system and the final convergence value. Following the stability analysis, it is shown how the autonomous consensus system (S1) can be transformed into a control system by introducing an external control input to a subset of the agents. The resulting system is given by ẍ(t) = −Axx(t) −A f ẋẋ(t) +Bu(t), (S2) where x : R+ → Rf and ẋ : R+ → Rf are the states of the controlled, “follower” agents, u : R+ → Rl is the control input