On dual-rate consensus under transmission delays

In this paper, we investigate the problem of dual-rate consensus under transmission delays, where the control updates happen at a faster rate than the measurements being received. We assume that the measurements are delayed by a fixed delay and show that for all delays and rates, the system reaches a consensus if and only if the communication graph of the agents is connected and the control gain is chosen in a specific interval. Based on these results we dive deeper into the convergence properties and investigate how the convergence changes when we change the rate for sending measurements. We observe that in certain cases there exists a sweet spot for choosing the sampling rate of the measurements, which can improve the convergence to the consensus point. We then formulate an optimization problem to find a sampling rate to improve the convergence speed and provide a necessary and sufficient condition for the existence of a finite optimizer of this problem. Our results are verified with numerical simulations.