Decentralised final value theorem for discrete-time LTI systems with application to minimal-time distributed consensus

Abstract

Abstract-In this study, we consider an unknown discretetime, linear time-invariant, autonomous system and characterise, the minimal number of discrete-time steps necessary to compute the asymptotic final value of a state. The results presented in this paper have a direct link with the celebrated final value theorem. We apply these results to the design of an algorithm for minimal-time distributed consensus and illustrate the results on an example.

Decentralised minimum-time average consensus in digraphs

“…Considering the iteration in (2) with weight matrix P , it is easy to show (e.g., using the techniques in [21]) that…”

“…Throughout the work we assume that nodes have enough storage ability to store at least as many values as necessary and the resulting defective matrix. Nevertheless, it has been shown in [21] that the number of steps required for predicting y and x are less than 2n where n is the number of nodes in the network.…”

“…Also, [21], [22] proposed a distributed algorithm with which any arbitrarily chosen agent can compute its asymptotic final consensus value in minimaltime. However, none of the aforementioned works is able to distributively compute -using only local information at each iteration -the average consensus value in a finite number of steps in a digraph.…”

“…Using the weights in this matrix, average consensus is reached via ratio consensus, i.e., two iterations (with appropriately chosen initial conditions) that run simultaneously so that the average can be obtained at each node by taking the ratio of the two values (maintained at each node for each of the two iterations). (b) A decentralized minimum-time consensus (not average consensus) algorithm suggested in [21], [22] that allows any node to compute the consensus value of the whole network in finite time using minimum number of successive values of its own state history. More specifically, an algorithm is developed in which the minimum number of steps is obtained by checking a rank condition on a Hankel matrix of local observations.…”

Distributed minimum-time weight balancing over digraphs

“…Finite-time consensus was proposed in [3]- [5], while [6] proposed a method to compute the asymptotic consensus value in finite-time. Also, [7] proposed a distributed algorithm with which any arbitrarily chosen agent can compute its asymptotic consensus value in minimal time.…”