[논문 리뷰] Super-twisting over networks: A Lyapunov approach for distributed differentiation

Rodrigo Aldana-López, Irene Pérez Salesa외 3명|arXiv (Cornell University)|2026. 02. 02.

Distributed Control Multi-Agent Systems인용 수 0

한 줄 요약

이 논문은 이벤트 트리거 하이브리드 구현과 체계적인 이득 설계를 갖춘 네트워크에서의 분산 미분의 전역 유한시간 수렴을 위한 Lyapunov 기반 추상 슈토이팅 프레임워크를 개발한다. 경미한 신호 변화 가정하에 1차 분산 미분기에서의 전역 수렴을 증명하고 이득 및 정확도-통신 트레이드오프에 대한 가이드를 제공한다.

ABSTRACT

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.

연구 동기 및 목표

• Motivate and formalize distributed differentiation as an extension of Dynamic Average Consensus (DAC) where agents estimate the average and its derivative.
• Introduce an abstract super-twisting framework that captures the shared structural features with the classic super-twisting algorithm.
• Derive systematic gain design conditions that guarantee global finite-time convergence for the distributed differentiator.
• Propose an event-triggered hybrid implementation and establish minimum inter-event time and accuracy-communication trade-offs.

제안 방법

• Formulate the distributed differentiation problem on an undirected graph with local signals and their (second) derivatives.
• Proposal a REDCHO-like protocol with correction terms based on differences between local estimates to enforce consensus using only relative information.
• Rewrite the error dynamics in compact form and define consensus errors to study stability.
• Introduce an abstract super-twisting system that encompasses the distributed differentiator error dynamics as a differential inclusion.
• Construct a Lyapunov function combining a convex potential and its convex conjugate to prove finite-time stability under explicit gain conditions.
• Derive explicit gain bounds: k0 and k1 with gamma, ensuring global finite-time convergence; show invariance and Lyapunov decay properties.

실험 결과

연구 질문

• RQ1 Can a Lyapunov-based abstract super-twisting framework yield global finite-time convergence guarantees for distributed differentiation over networks?
• RQ2 What are explicit, constructive gain design conditions ensuring convergence and robustness to bounded disturbances in the distributed differentiator?
• RQ3 How can the distributed differentiator be implemented with event-triggered communication while preserving convergence properties?
• RQ4 How does the network topology (through algebraic connectivity) influence the required gains and convergence rates?

주요 결과

• Global finite-time convergence to consensus for the distributed differentiator is guaranteed with appropriate gains k0 and k1, and gamma can be used to accelerate decay of initialization mismatches.
• A Lyapunov function based on a convex potential and its conjugate provides a tractable route to energy decay and explicit settling-time bounds.
• The abstract super-twisting model captures the essential structure of the distributed differentiator error dynamics, enabling systematic gain computation.
• Convergence of consensus errors implies asymptotic recovery of the average signal and its derivative across agents.
• An invariance argument shows the error dynamics remain within a defined state space, supporting the Lyapunov analysis.
• The framework supports an event-triggered implementation with quantified accuracy vs. communication trade-offs (details in the later sections).