Robust consensus for multi-agent systems over unbalanced directed networks

In this paper, a distributed consensus protocol is proposed for discrete-time single-integer multi-agent systems with measurement noises under general fixed directed topologies. The time-varying control gains satisfying the stochastic approximation conditions are introduced to attenuate noises, thus the closed-loop multi-agent system is intrinsically a linear time-varying stochastic difference system. Then the mean square consensus convergence analysis is developed based on the Lyapunov technique, and the construction of the Lyapunov function especially does not require the typical balanced network topology condition assumed for the existence of quadratic Lyapunov function. Thus, the proposed consensus protocol can be applicable to more general networked multi-agent systems, particularly when the bidirectional and/or balanced information exchanges between agents are not required. Under the proposed protocol, it is proved that the state of each agent converges in mean square to a common random variable whose mathematical expectation is the weighted average of agents’ initial state values; meanwhile, the random variable’s variance is bounded.