Consensus over Clustered Networks Using Intermittent and Asynchronous Output Feedback

TL;DR

The paper tackles consensus in clustered networks of identical LTI agents using intermittent and asynchronous output feedback. It develops a hybrid-system model with timer-driven flows and jumps, introduces estimators and a distributed controller, and proves global exponential stability of the consensus set via a Lyapunov function, with ISS robustness to perturbations. Key contributions include a fully distributed intermittent-feedback strategy, nominal well-posedness and a compact GES attractor, and robustness guarantees under non-vanishing measurement noise, validated through a rendezvous simulation. The results enable scalable, resource-conscious coordination in large-scale cyber-physical networks, with potential extensions to nonlinear, heterogeneous, and directed-settings future work.

Abstract

Distributed consensus protocols provide a mechanism for spreading information within clustered networks, allowing agents and clusters to make decisions without requiring direct access to the state of the ensemble. In this work, we propose a strategy for achieving system-wide consensus in the states of identical linear time-invariant systems coupled by an undirected graph whose directed sub-graphs are available only at sporadic times. Within this work, the agents of the network are organized into pairwise disjoint clusters, which induce sub-graphs of the undirected parent graph. Some cluster sub-graph pairs are linked by an inter-cluster sub-graph, where the union of all cluster and inter-cluster sub-graphs yields the undirected parent graph. Each agent utilizes a distributed consensus protocol with components that are updated intermittently and asynchronously with respect to other agents and inter-clusters. The closed-loop ensemble dynamics is modeled as a hybrid system, and a Lyapunov-based stability analysis yields sufficient conditions for rendering the agreement subspace (consensus set) globally exponentially stable. Furthermore, an input-to-state stability argument demonstrates the consensus set is robust to a large class of perturbations. A numerical simulation considering both nominal and perturbed scenarios is provided for validation purposes.

Figures (9)

• Figure 1: Within the undirected graph $\mathcal{G}$, as denoted by the disks and double-headed arrows, there are four sub-graphs induced by clusters and five sub-graphs induced by inter-cluster. Note that solid arrows represent edges within clusters, while dashed arrows represent edges between clusters, i.e., within inter-clusters. The cluster sub-graphs of $\mathcal{G}$ are $\mathcal{G}[\mathcal{V}_1]$, $\mathcal{G}[\mathcal{V}_2]$, $\mathcal{G}[\mathcal{V}_3]$, and $\mathcal{G}[\mathcal{V}_4]$. The inter-cluster sub-graphs of $\mathcal{G}$ are $\mathcal{G}[\mathcal{V}_{12}]$, $\mathcal{G}[\mathcal{V}_{13}]$, $\mathcal{G}[\mathcal{V}_{14}]$, $\mathcal{G}[\mathcal{V}_{24}]$, and $\mathcal{G}[\mathcal{V}_{34}]$ By construction, the union of all sub-graphs returns the original graph $\mathcal{G}$.
• Figure 2: A common inertial reference frame is depicted on the bottom left with origin $0_2$ and right-handed coordinate system defined by the orthogonal unit vectors $\hat{a}$, $\hat{b}$, and $\hat{c}$. The positions of the CM and bow of agent $p$ are given by $c_p$ and $b_p$, respectively. The distance between the CM and bow is denoted by $\ell$. The speed along the unit vector $(b_p - c_p)/\Vert b_p - c_p \Vert$ and heading relative of $\hat{a}$ of agent $p$ are denoted by $v_p$ and $\theta_p$, respectively.
• Figure 3: Illustration of the clustered network used in the nominal and perturbed simulation cases. The nodes and edges of the communication graph $\mathcal{G}$ are represented by the disks and both solid and dashed lines, respectively. Different colored disks connected by solid lines of the same color represent a cluster sub-graph. Different colored disks connected by dashed lines of the same color represent an inter-cluster sub-graph. The planar position of each node is coincident with the initial bow position of the corresponding agent.
• Figure 4: Illustration of the reference bow position trajectories and true bow position trajectories for each agent $p\in\mathcal{V}$. Solid lines represent the trajectories of $\{x_{1,p}\}_{p\in\mathcal{V}}$ (reference bow positions), while dotted lines represent the trajectories of $\{b_p\}_{p\in\mathcal{V}}$ (true bow positions). The $\times$'s and $\bullet$'s depict the initial and final conditions of all trajectories, respectively. Observe rendezvous is achieved in both the reference bow positions $\{x_{1,p}\}_{p\in\mathcal{V}}$ and true bow positions $\{b_p\}_{p\in\mathcal{V}}$.
• Figure 5: Depiction of $\vert\xi\vert_{\mathcal{A}}$ as a function of continuous time, where $\xi=\phi(t,j)$---the particular solution to the hybrid system $\mathcal{H}$ generated in the nominal simulation. The vertical axis uses a logarithmic scale; the horizontal axis uses a linear scale. Based on the plot, the distance of $\xi$ to $\mathcal{A}$ is exponentially converging to $0$.

Theorems & Definitions (15)

• Lemma 1
• proof : Proof of Lemma \ref{['lemma: L and S Identities']}
• Definition 1
• Definition 2
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof