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

TL;DR

By isolating the structural features shared with the super-twisting algorithm and encoding them into an abstract model, this work constructs a Lyapunov function enabling systematic gain design and proving global finite-time convergence to consensus for the distributed differentiator.

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.

Figures (2)

• Figure 1: Simulation of the event-triggered proposed protocol. Top: estimated derivatives $\hat{s}_{i,1}(t)$ (solid) and true average derivative $\dot{\bar{s}}(t)$ (black dashed) for an event-trigger with threshold $\delta=0.02$. Middle: absolute errors $|\hat{s}_{i,1}(t)-\dot{\bar{s}}(t)|$. The theoretical steady state error bound $c_1\sqrt{\delta}$ with $c_1=7.9, \delta=0.02$ is shown (black dashed). Bottom left: steady-state error versus triggering threshold $\delta$ (red dots) with theoretical $c_1\sqrt{\delta}$ bound (black dashed). Here, $\delta=0$ corresponds to the performance of the standard REDCHO redcho. Bottom right: fraction of the total number of events relative to the case of full transmission.
• Figure 2: Comparison of three event thresholds $\delta_{ij}(t)$ in the event-triggered distributed differentiator. Left column: constant threshold $\delta_{ij}(t)=\delta$, yielding practical convergence to a steady state error floor while preserving strictly positive inter-event times. Middle column: vanishing threshold $\delta_{ij}(t)=\delta\exp(-t/2)$, yielding asymptotically exact tracking at the cost of progressively smaller inter-event times. Right column: state dependent threshold $\delta_{ij}(t)=\delta+\sigma|\hat{s}_{i,0}(\tau_{k}^{ij})-\hat{s}_{j,0}(\tau_{k}^{ij})|, \sigma=0.15$, yielding a favorable compromise in which inter-event times remain bounded away from zero along the run, while the steady state error remains small and tunable through $\delta$. Top row: representative maximum tracking error over all agents for each time $t$. Bottom row: inter-event times along the same experiment, with the envelope between maximum and minimum in gray.