Super-twisting over networks: A Lyapunov approach for distributed differentiation

We study distributed differentiation, where agents in a networked system estimate the average of local time-varying signals and their derivatives under mild assumptions on the agents’ signals and their first and second derivatives. Existing sliding-mode methods provide only local stability guarantees and lack systematic gain selection. By isolating the structural features shared with the super-twisting algorithm and encoding them into an abstract model, we construct a Lyapunov function enabling systematic gain design and proving global finite-time convergence to consensus for the distributed differentiator. Building on this framework, we develop an event-triggered hybrid system implementation using time-varying and state dependent threshold rules and derive minimum inter-event time guarantees and accuracy bounds that quantify the trade-off between estimation accuracy and communication effort.

💡 Research Summary

The paper addresses the problem of distributed differentiation in a network of agents that each measure a local time‑varying signal s_i(t). The goal is for every agent to reconstruct the global average (\bar{s}(t)=\frac{1}{N}\sum_{i=1}^{N}s_i(t)) and its time derivative (\dot{\bar{s}}(t)) using only one‑hop communication. Existing dynamic average consensus (DAC) schemes fail to track the derivative accurately when the signals keep varying, offering only bounded‑error stability. High‑order sliding‑mode (HOSM) approaches such as EDCHO and REDCHO provide exact consensus but suffer from a lack of systematic gain selection and, for REDCHO, only local stability guarantees.

The authors’ key insight is to isolate the structural elements that make the classical super‑twisting algorithm (STA) work: a 3/2‑homogeneous potential function whose gradient yields a 1/2‑power feedback, and a set‑valued sign‑like term that supplies robustness. They embed these features into an abstract super‑twisting model that can accommodate the graph‑dependent coupling of a multi‑agent system. The abstract model is defined by a differential inclusion (6) with a convex, strictly positive‑definite potential U, a homogeneous set‑valued map S, and a bounded disturbance set 𝔇. By choosing ∇U(e₀)= (D Dᵀ e₀)^{1/2} and S(e₀)= (D Dᵀ e₀)^{0}, where D is the incidence matrix of the communication graph, the error dynamics of the distributed differentiator (5) exactly fit the abstract model.

A Lyapunov function of the form V(e₀,e₁)=α‖e₀‖³+β⟨e₁,e₀⟩+γ‖e₁‖² is constructed. Using Clarke’s generalized derivative, the authors show that V̇ ≤ –c₁‖e₀‖^{3/2} –c₂‖e₁‖ + L·sup_{d∈𝔇}‖d‖. Because 𝔇 is a unit‑norm set, the disturbance term is bounded by L. Theorem 7 provides explicit gain conditions (8): k₀ must dominate a supremum involving the graph‑dependent functions Γ and Π, and k₁ must exceed 1/c_S. Under these conditions the origin of the abstract system is finite‑time stable, which directly yields global finite‑time convergence of the distributed differentiator (Theorem 3). Thus the paper upgrades REDCHO’s local stability to a global result and supplies a systematic design recipe for the gains k₀, k₁, γ.

Building on the Lyapunov framework, the authors propose an event‑triggered hybrid implementation. Each agent monitors its local consensus error e_i(t) and a time‑varying, state‑dependent threshold σ_i(t)‖e_i(t)‖_{max}. When the error exceeds the threshold, the agent broadcasts its current estimate (\hat{s}_i^0(t)) to its neighbors; otherwise communication is suppressed. By relating the trigger condition to the Lyapunov decrease, a positive lower bound on inter‑event times (MIET) is derived, guaranteeing avoidance of Zeno behavior and robustness to network delays. The disturbance introduced by the event‑triggering is captured by the set 𝔇, and the analysis shows that the steady‑state error scales as O(σ_max·L). Consequently, smaller thresholds improve accuracy at the cost of higher communication rates, providing a clear quantitative trade‑off.

The paper’s contributions are threefold: (1) a novel abstraction that lifts the super‑twisting structure to graph‑coupled multi‑agent systems, (2) a Lyapunov‑based global finite‑time stability proof with explicit gain selection rules, and (3) an event‑triggered hybrid scheme that reduces communication while preserving the finite‑time convergence and offering explicit performance bounds. These results are directly applicable to sensor networks, power‑grid monitoring, cooperative robotics, and any distributed platform where real‑time derivative information of a global signal is required.