Self-tuning consensus on directed graph in the case of time-varying nonhomogeneous input gains

In this paper, the problem of self-tuning of coupling parameters in multi-agent systems is considered. Agent dynamics are described by a discrete-time double integrator with time-varying nonhomogeneous input gain. The coupling parameters defining the strength of agents interactions are locally self-tuning by each node based on the velocities of its neighbors. The cost function is equal to the square of the local error between the agent velocity and the weighted average of the velocities of interacting neighbors. So, the proposed algorithm is the normalized gradient algorithm which is minimized the square of the local error between the agent velocity and the one step delayed average of the velocities of its neighbors. Provided that the underlying graph is strongly connected, it is shown that the sequence of the inter-agent coupling parameters generated by the proposed algorithm is convergent. Also, assuming the suitable initial condition on coupling parameters, it is proved that the network achieves average consensus. In other words, the agent velocities converge toward the average of the initial velocities values. Furthermore, the distance among agents converges to a finite limit. Simulation results illustrate effectiveness of the proposed method.