Consensus in multi-agent systems with time-delays

Different consensus problems in multi-agent systems have been addressed in this thesis. They represent improvements with respect to the state of the art. In the first part of the thesis in luding Chapters 2, 3, and 4, the state of the art of the representation and stability analysis of consensus problems, time-delay systems, and sampled-data systems have been presented. Novel contributions have been illustrated in Chapters 5-8. Particularly, in Chapter 5 we reported the results of Zareh et al. (2013b), where we investigated the consensus problem for networks of agents with double integrator dynamics affected by time-delay in their coupling. We provided a stability result based on the Lyapunov-Krasovskii functional method and a numerical proc edure based on an LMI condition which depends only on the algebraic connectivity of the considered network topologies, thus reducing greatly the computational complexity of the procedure. Obviously, this result implies the existence of a minimum dwell time such that the proposed consensus protocol is stable for slow swit things between network topologies with suffient algebraic connectivity. Future work will involve actually computing such a dwell time by adopting a multiple Lyapunov function method and evaluating the worst case sider only delayed relative measurements instead of delayed absolute values of the neighbors' state variables. The results of Zareh et al. (2013a) were addressed in Chapter 6, in which a on- tinuous time version of a consensus on the average protocol for arbitrary strongly connected directed graphs is proposed and its convergence properties with respect to time delays in the local state update are characterized. The convergenc e properties of this algorithm depend upon a tuning parameter that an be made arbitrary small to prove stability of the networked system. Simulations have been presented to corroborate the theoretical results and show that the existenc e of a small time delay an a tually improve the algorithm performance. Future work will include an extension of the mathematical characterization of the proposed algorithm to consider possibly heterogeneous or time-varying delays. In Chapter 7 we proposed a PD-like consensus algorithm for a second-order multi- agent system where, at non-periodic sampling times, agents transmit to their neighbors information about their position and veloc ity, while each agent has a perfect knowledge of its own state at any time instant. Conditions have been given to prove onsensus to a ommon xed point, based on LMIs verification. Moreover, we also show how it is possible to evaluate an upper bound on the de ay rate of exponential convergence of stable modes. In Chapter 8, mainly based on our paper Zareh et al. (2014b), we considered the same problem as in Chapter 7. The main contribution consists in proving consensus to a common fixed point, based on LMIs verification, under the assumption that the network topology is not known and the only information is an upper bound on the connectivity. Two are the main directions of our future research in this framework. First, we want to compute analytically an upper bound on the value of the second largest eigenvalue of the weighted adjacency matrix that guarantees consensus, as a function of the other design parameters. Second, we plan to study the case where agents do not have a perfect knowledge of their own state.