The distributed biased min-consensus protocol revisited: pre-specified finite time control strategies and small-gain based analysis

TL;DR

This work addresses the need for convergence of the distributed biased min-consensus protocol (DBMC) within user-defined time horizons. It introduces two control schemes—practical PPT and prescribed-time PT—to achieve finite-time convergence toward Bellman-based stationary values, and establishes global exponential input-to-state stability (expISS) of nominal DBMC under bounded perturbations via small-gain theory. The PPT method guarantees convergence to a controllable neighborhood in time tied to the graph’s effective diameter, while the PT method achieves exact convergence at a prescribed time and ensures trajectory continuity. Simulations on large, varied graphs confirm rapid, robust convergence and demonstrate resilience to edge-weight perturbations and source changes, underscoring practical viability for shortest-path and hierarchical planning applications.

Abstract

Unlike the classical distributed consensus protocols enabling the group of agents as a whole to reach an agreement regarding a certain quantity of interest in a distributed fashion, the distributed biased min-consensus protocol (DBMC) has been proven to generate advanced complexity pertaining to solving the shortest path problem. As such a protocol is commonly incorporated as the first step of a hierarchical architecture in real applications, e.g., robots path planning, management of dispersed computing services, an impedance limiting the application potential of DBMC lies in, the lack of results regarding to its convergence within a user-assigned time. In this paper, we first propose two control strategies ensuring the state error of DBMC decrease exactly to zero or a desired level manipulated by the user, respectively. To compensate the high feedback gains incurred by these two control strategies, this paper further investigates the nominal DBMC itself. By leveraging small gain based stability tools, this paper also proves the global exponential input-to-state stability of DBMC, outperforming its current stability results. Simulations have been provided to validate the efficacy of our theoretical result.

Figures (7)

• Figure 1: Illustration of the necessity of the non-increasing property of $z_i(\cdot)$. In this graph consisting of two nodes, if $z_i(\cdot)$ is not non-increasing, node 1 and node 2 will continuously exchange their initial value $z_1(t_0) = 5$ and $z_1(t_0) = 10$ and min-consensus will never be achieved.
• Figure 2: An undirected graph consisting of 7 non-source nodes and 2 source nodes. In this scenario, node 4 has 2 true parent nodes: node 2 and node 3. The effective diameter of the graph is 3. The longest shortest path of node 6 is $6 \rightarrow 8 \rightarrow 9$, and $6 \in {\cal F}_2$ while $6 \notin {\cal F}_1$.
• Figure 3: Results of applying DBMC using the practical pre-specified finite time control strategy to randomized graphs and line graphs. In all scenarios, state errors drop below the bound given in Theorem \ref{['the:ppt']}. It takes much longer time for line graphs than randomized graphs for the state errors to be bounded, while the time needed for all the cases are less than the theoretical bound provided in Theorem \ref{['the:ppt']}.
• Figure 4: Results of applying DBMC using the pre-specified finite time control strategy to randomized graphs. In all scenarios the state errors converge to 0 within the prescribed time. The protocol achieves a better convergence effect with a larger $h$ or $\gamma$.
• Figure 5: Results of applying nominal DBMC to randomized graphs with varying $\eta$. With $\eta = 1 + 10^{-8}$, the nominal DBMC achieves exponential stability without perturbations, and expISS with the edge weight $w_{ij}(t)$ ranging from $0.8w_{ij}$ to $1.2w_{ij}$. The nominal DBMC can still achieve expISS with a larger $\eta$, manifesting the conservatism of Lyapunov-based analysis.

Theorems & Definitions (36)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Remark 2
• Remark 3
• Theorem 1
• Lemma 1
• Lemma 2
• proof