Stability and controllability of multi-agent systems overstochastically-evolving topology

Abstract

[ES] El comportamiento colectivo de los sistemas multi-agente está determinado por el comportamiento dinámico de cada individuo sistema, así como del modelo de interacción entre los agentes que a menudo se modela mediante un gráfico. En muchas aplicaciones de sistemas multi-agente, el patrón de interacción no es fijo y evoluciona con el tiempo pudiendo seguir modelos no deterministas. Por lo tanto, los conceptos clásicos de estabilidad y rendimiento deben ampliarse para investigar las propiedades de esta clase especial de sistemas multi-agente. La principal contribución de este proyecto es considerar una topología de acoplamiento (semi) aleatorio para los sistemas individuales, tal que las interacciones entre cada dos agentes ocurrirán de forma estocástica, es decir, cada interacción existe de acuerdo con algunas distribuciones de probabilidad. Sobre esta topología variable en el tiempo, el trabajo analizará y aplicará los análisis de estabilidad y controlabilidad necesarios para dichos sistemas. El mencionado modelo con topología aleatoria es de notable importancia y de aplicación en el dominio de la potencia sistemas, red de osciladores, y también recientemente en neurología.

[EN] Stability of stochastic multi-agent systems is an increasingly frequent topic for investigations in recent years due to its broad range of applications in the areas of distribution of electricity, neurology, oscillator networks and others. It is in the wide range of applications where the importance of the topic is. Under various randomness models, stability analysis of stochasticallyevolving multi-agent systems requires further research to study stronger and less conservative stability guarantees for such systems. In this Master Thesis, the major goal is to study the stability of multi-agent systems under a random interaction topology. The interconnections of the dynamical systems, that are modelled by random topologies change randomly according to either a Bernoulli distribution or a Markov process. Prior to the study of the stability, some preliminaries have been provided. First, a literature review has been carried out, where the reader may find what is the current state of the arts. Then, brief introductions to graph theory and stochastic stability have been provided to present some of the basic concepts used in the stability analysis. Finally, the closed-loop model of the multi-agent system under random interactions (topology) is derived to describe the dynamics of the randomly-evolving multi-agent system. Starting from the Drift Criteria, a known criterion for the stability of stochastic systems, a stability theorem is proposed for stochastic multi-agent systems where the randomness of the edge connections is governed by a Bernoulli distribution. This proposed theorem guarantees the stochastic stability of the multi-agent system if some conditions dependent on the system dynamics, the network connections and the randomness over the connections hold. In addition, the theorem provides an upper and a lower boundary for the convergence of the trajectory of the states. Afterwards, a connection between the eigenvalues of the corresponding Laplacian matrix and the stability of the system is derived, but only for some special cases. The results obtained for the random Bernoulli distribution over the network are then extended to networks where the randomness of the connections is governed by a Markov process. Finally, three simulation setups have been developed to demonstrate the validity of the theoretical results. In the first simulation setup, the systems in a multi-agent system are modelled by the nodes of a graph and their physical interactions are represented by the edges between the corresponding nodes. Then random interactions are modelled by giving probabilities to the edges to either exist or not. The lower-bound on the probability (Bernoulli probability) for which the system holds the stability conditions were computed. The system is simulated considering different probability values both inside and outside the range for which the stability is guaranteed. In the second simulation, the control gain is taken as the variable of the system while the system dynamics and the probabilities over the connections are given. The range of control gains that make the system stable are computed. The system is then simulated with different gain values both inside and outside the stability range. The results of the simulations 1 and 2 have reinforced the statements made in the theoretical part. In the third simulation, an LQR controller has been computed for the system by using a novel method for stochastic systems.