Finite-time consensus in directed switching network topologies and time-delayed communications

Abstract

There are many practical situations where it is desirable or even required to achieve stable convergence in the finite-time domain. In this paper, a simple distributed continuous-time protocol is introduced that guarantees finite-time consensus in networks of autonomous agents. Protocol convergence in weighted directed/undirected and fixed/switching networks is explored based on a Lyapunov analysis. The stability of the system and the solvability of the consensus algorithm are proved for network topologies that contain a spanning tree frequently enough over contiguous time intervals. The decision value for different topologies and for multi-rate integrator agents is investigated, and a novel approach is proposed to determine the leader subgroup of agents. Communication time-delay and chattering phenomenon in the system are assessed, and additionally some protocols with Lipschitz righthand sides are introduced. Herein, all proposed consensus strategies use a limited-gain control input to account for the physical limitation of control actuation devices, which, in general, are subject to amplitude saturation.

On predefined-time consensus protocols for dynamic networks

“…Finally, since we have shown in the proof of Theorem 4 that, if the topology is static and connected, (12) goes to zero in a fixed-time bounded by the constant 1 κλ 2 (Q)β(n) 2−d is the active topology, then, under this scenario, (16) also goes to zero in a fixed-time bounded by the same constant, because if x is such that (12) is zero then also (16) is zero. Since, in the switching case, (16) is still decreasing and continuous, then it follows that (16) goes to zero in a fixed-time lower or equal than the lowest time such that there exists a connected topology X l , such that the sum of time intervals in which X l has been active is greater than 1 κλ 2 (Q)β(n) 2−d . Since this upper bound is independent of the initial state of the agents, then fixed-time convergence is obtained under switching topologies.…”

“…Using the stability results of the switching systems [10] it can be shown that such protocols reach a consensus even on dynamic networks by arbitrarily switching between highly connected graphs [1,11]. Consensus protocols with enhanced convergence properties have been suggested based on finite-time [12,13], and fixed-time [14,15] stability theory.In [16][17][18][19][20][21] continuous and discontinuous protocols with finite-time convergence were proposed. However, the convergence-time is an unbounded function of the initial conditions.…”

Single Bit and Reduced Dimension Diffusion Strategies Over Distributed Networks

“…In the first case, we have , e.g., in the figure. In this case, no update is needed, , since satisfies both conditions (5), (6) and . In the second case, we have , e.g., in the figure.…”

“…For this case, i.e., , we only need to project to the half hyper sphere (shown as a half circle in two dimensions in Fig. 1), which corresponds to the constraints (5) and (6). This projection can be readily accomplished by first projecting to the vector space perpendicular to and then scaling the projected vector to have unit norm.…”

“…In [4], a quantized parameter estimate is exchanged among nodes to avoid infinite precision in the transmission. In [5], the sign of the innovation sequence in the context of decentralized estimation and in [6], the relative difference between the states of the nodes in a consensus network are exchanged using a single bit of information. Here, we substantially compress the exchanged information, even to a single bit, and perform local adaptive operations at each node to recover the full information vector.…”