Accelerating Consensus for Systems on Star and Bipartite Networks Using Multi-Tap Memory

Consensus, as one of the most fundamental tasks of multi-agent systems, can be accelerated by introducing the memory term into the control protocol. It has been demonstrated that on any given network, the one-tap node memory can accelerate the convergence rate, while the two-tap node memory cannot. Then, a problem arises: can the rate of consensus be further improved by adding more taps of memory? By using a novel method based on the Routh stability criterion, this brief shows that more taps of memory can further accelerate the convergence rate on special networks with star or bipartite structure. Specially, explicit formulas for the convergence rate and control parameters are derived to prove that three-tap and five-tap node memory can accelerate the convergence rate, but four-tap node memory cannot. In addition, it is found by extensive simulations that six-tap memory cannot accelerate the rate, but seven-tap memory can. Finally, a conjecture is proposed that the optimal convergence rate can be further improved when the memory taps progress from $2k$ to $2k+1$ , where $k\in \mathbb {N}$ .