  Previous researches have shown that adding local memory can accelerate the
consensus. It is natural to ask questions like what is the fastest rate
achievable by the $M$-tap memory acceleration, and what are the corresponding
control parameters. This paper introduces a set of effective and previously
unused techniques to analyze the convergence rate of accelerated consensus with
$M$-tap memory of local nodes and to design the control protocols. These
effective techniques, including the Kharitonov stability theorem, the Routh
stability criterion and the robust stability margin, have led to the following
new results: 1) the direct link between the convergence rate and the control
parameters; 2) explicit formulas of the optimal convergence rate and the
corresponding optimal control parameters for $M \leq 2$ on a given graph; 3)
the optimal worst-case convergence rate and the corresponding optimal control
parameters for the memory $M \geq 1$ on a set of uncertain graphs. We show that
the acceleration with the memory $M = 1$ provides the optimal convergence rate
in the sense of worst-case performance. Several numerical examples are given to
demonstrate the validity and performance of the theoretical results.
