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Discontinuities and hysteresis in quantized average consensus
Francesca Ceragioli, Claudio De Persis, Paolo Frasca
TL;DR
The paper studies how uniform quantization affects continuous-time average consensus, including the resulting discontinuities and chattering. It analyzes Krasowskii solutions and introduces a hysteretic quantizer through a hybrid-system framework. The resulting dynamics converge to practical consensus, while the hysteretic design is shown to be well-posed and to avoid chattering, with topology-dependent convergence limits.
Problem
Continuous-time average consensus with quantized communication can produce discontinuous dynamics lacking classical solutions, while static quantization prevents exact consensus.
Method
The paper studies uniform static quantization using Krasowskii solutions and analyzes a hysteretic quantizer as a hybrid system.
Results
The quantized dynamics converge to practical consensus, and the hysteretic system has existence, forward uniqueness, locally finite switching times, convergence, and data-rate estimates.
Takeaways & Limitations
Hysteretic quantization provides a continuous-time consensus framework that addresses chattering while retaining convergence analysis under quantized communication.
Takeaways & Limitations
For systems with more than two agents, hysteretic dynamics can exhibit limit cycles, and relating their size to graph topology remains open.
Abstract
from arXiv · showhide
We consider continuous-time average consensus dynamics in which the agents' states are communicated through uniform quantizers. Solutions to the resulting system are defined in the Krasowskii sense and are proven to converge to conditions of "practical consensus". To cope with undesired chattering phenomena we introduce a hysteretic quantizer, and we study the convergence properties of the resulting dynamics by a hybrid system approach.
1 Introduction
The paper addresses continuous-time average consensus under quantized communication, where discontinuities may prevent classical solutions. It develops Krasowskii and hysteretic approaches to establish convergence, practical consensus, and reduced chattering under data-rate constraints.
- Motivation: Quantization restricts communication to discrete symbols, motivating continuous-time analysis because many applications, including robotic networks, are naturally modeled continuously.
- Problem: Uniform quantization creates discontinuous dynamics for which classical or Carathéodory solutions may not exist, requiring generalized solutions.
- Static quantization: The paper analyzes Krasowskii solutions for weakly connected, weight-balanced graphs and proves existence, boundedness, average preservation, and convergence toward equilibrium-containing sets.
- Hysteretic quantization: For hysteretic quantizers, the paper proves well-posedness through existence, forward uniqueness, and locally finite switching times, then establishes convergence and estimates the required data rate.
- Hysteretic quantization: Static uniform quantization prevents exact consensus, so the paper studies approximations called practical consensus and also develops hysteretic quantization to address chattering.
2 Preliminaries
The preliminaries formulate average consensus as feedback stabilization over weight-balanced graphs and introduce quantized communication as a practically relevant extension. The paper focuses on uniform static and hysteretic quantizers, with the latter designed to prevent chattering.
- 2.1 Graph theory: A weighted graph is represented by agents, directed links, an adjacency matrix, and a Laplacian L = D − A; weight balance means equal in-degree and out-degree at every node.
- 2.2 Feedback consensus dynamics: The agents’ average is x_ave(t) = N^-1 1* x(t), and the consensus objective is formulated as convergence of the states to their average.
- 2.2 Feedback consensus dynamics: Under weak connectivity and weight balance, the non-quantized feedback law u = −Lx provides the standard setting for convergence to average consensus.
- 2.3 Quantized consensus dynamics: A quantizer maps real values to a discrete subset, and quantization may enter communication, sensing, computation, or actuation.
- 2.3 Quantized consensus dynamics: The paper studies uniform static quantization followed by a hysteretic quantizer designed to prevent chattering because continuous-time quantized consensus has received less attention than its discrete-time counterpart.
3 Krasowskii quantized dynamics
The quantized continuous-time consensus dynamics is analyzed using Krasowskii solutions because Carathéodory solutions may fail to exist or be complete. The analysis establishes preservation, convergence, and practical-consensus properties, including network-independent convergence to equilibria on symmetric graphs.
- Solution concept: Carathéodory solutions can fail to exist and may be noncomplete for weakly connected, weight-balanced graphs.This motivates using the more general Krasowskii solution concept.
- Solution concept: Every initial condition admits a complete Krasowskii solution for the quantized dynamics.The existence result is supported by local existence, boundedness, and continuation arguments.
- Fundamental properties: Krasowskii solutions preserve the average of the agents’ states for all time.Average preservation is a key invariant used in subsequent convergence results.
- Graph-dependent convergence results: Quantization error can be made arbitrarily small by decreasing quantization error, but the initial bound scales with ||L|| and the vector dimension.This graph-dependent bound motivates seeking practical-consensus guarantees depending only on quantizer precision.
- Equilibria: On connected symmetric graphs, every Krasowskii solution approaches the equilibrium set, and under an average-value condition it reaches that set in finite time.The equilibrium set depends only on quantizer precision, while finite-time convergence requires xave(0) ≠ (k + 1/2)∆ for every k ∈ Z.
4 Chattering-free quantized dynamics
The hysteretic quantizer replaces discontinuous quantized consensus dynamics with a hybrid model that prevents chattering. Its solutions are complete, preserve the average, and converge toward practical-consensus conditions, although limit cycles can occur.
- 4.1 Hysteretic quantizer: hybrid model: Hysteresis is introduced to overcome sliding modes in the discontinuous dynamics, which are unsuitable for practical implementation.
- 4.1 Hysteretic quantizer: hybrid model: The hysteretic quantizer is modeled as a hybrid system with continuous state evolution in C and discrete quantization updates in D.The state is z=(x,q), with dynamics defined by ż=f(z) in C and z+=g(z) in D.
- 4.2 Hybrid model analysis: Solutions exist, are forward unique and complete, and have continuous-time intervals with nonempty interiors.
- 4.2 Hybrid model analysis: Switching times are locally finite, so hysteresis eliminates chattering; each agent transmits information at most once every T=∆/2.
- 4.2 Hybrid model analysis: The dynamics preserve the average and have equilibria characterized by the quantized states lying in the consensus subspace.The equilibrium set is E=E1∪E2, while average preservation holds for every hybrid state in the solution domain.
- 4.2 Hybrid model analysis: Hysteretic trajectories may converge in finite time to periodic limit cycles rather than the equilibrium set, with directed rings producing quantized-state cycles of amplitude 2∆.The size of such cycles can increase on larger rings, and their relation to graph topology remains open.
- 4.2 Hybrid model analysis: Away from a strip containing the equilibria, the convergence rate is proportional to λ2(Sym(L)), as in non-quantized linear consensus dynamics.
5 Conclusion
The paper studies continuous-time consensus with static and hysteretic quantized communication, using discontinuous-systems and hybrid-systems analyses. It establishes convergence properties and uses hysteresis to prevent chattering, while simulations illustrate the resulting behavior.
- The paper analyzes stabilization of continuous-time consensus dynamics with discontinuous feedback under uniform static and hysteretic quantization.
- The hysteretic quantizer is designed to prevent chattering, and both quantized systems’ convergence properties are studied analytically and through simulations.