Stability Analysis Of Swarms

The swarm research is one of the most challenging directions in the systems and control field. The swarm is a complex system with the characteristics of emergence, decentralized control and local interactions. The research of swarm is inspired by the collective phenomena of natural social swarms with the goals of understanding the natural laws of the swarm's behaviors and the applications of such behaviors to networked artificial agents such as multi-robots. The research of swarm has great significance in both control theory and engineering. New results and theories continually emerge with many important topics that need to be further explored.This thesis provides an in-depth understanding on the stability and modeling of swarms, which are fundamental yet challenging issues in this research field. The main contributions in this thesis are in three aspects:1) For the first-order continuous swarm in the Lagrangian approach, the topology of the swarm is formed by the attractive/repulsive interactions between agents. We investigate the dynamics and stability of the swarm with a general directed and weighted coupling topology and give the stablility boundary that is characterized by the eigen-structure of the coupling matrix. We prove that in n -th Euclidean space, if the topology of the underlying swarm is strongly connected, then the swarm is stable in the sense that all agents will globally and exponentially converge to a hyper-ellipsoid in a finite time, both in open space and in profiles, whether the center of the hyper-ellipsoid is moving or not. The swarm boundary and convergence rate reveal the quantitative relationship between the swarm's behaviors and characteristics of the coupling topology.2) For the second-order continuous swarm that is composed of multiple inertial agents in the Lagrangian approach, the swarm has both velocity and position couplings. Its stability and dynamics have great relevance to both the coupling topologies and the coupling gains, which is different from the first-order case. We investigate the stability of the system in the case that the velocity coupling may not be equal to the position coupling and the two couplings may be non-balanced. We give the sufficient stability conditions of the system with proofs when one of the velocity and position couplings is strongly connected. Our theorems may serve as the methods of designing the gains of the swarm. Furthermore, we show that the previous stability result is a special case of our results under the assumptions of the coupling matching and balanced constraint.3) For the research of the swarm in the discrete approach, our main contribution is the development of the so-called adaptive velocity model. Though collective behaviors of biological swarms have attracted significant interest in recent years, much attention and efforts in the discrete approach have only been focused on constant speed models in which all agents are assumed to move at the same constant speed. In our adaptive velocity model, each agent adjusts not only its moving direction but also its speed according to the degree of direction consensus among its local neighbors at each time step. One important feature of the adaptive velocity model is that the speeds of all agents are adaptively tuned to the same maximum constant speed after a short transient process. We investigate the emergence of the adaptive velocity swarm with the comparisons of the classic constant model. The adaptive velocity strategy can greatly enhance the emergence of the swarm, and thus provides a powerful mechanism for coordinated motion in biological and technological multi-agent systems.