[Paper Review] Data Rate for Distributed Consensus of Multi-agent Systems with High Order Oscillator Dynamics

Zhirong Qiu, Lihua Xie+1 more|arXiv (Cornell University)|Sep 29, 2016

Distributed Control Multi-Agent Systems16 references3 citations

TL;DR

This paper investigates the minimum data rate required for distributed consensus in multi-agent systems with high-order oscillator dynamics, where each agent has a 2m-th order real Jordan block with conjugate poles on the unit circle and only the first state is measurable. By designing a state-estimation-based encoding-decoding scheme, it is shown that consensus can be achieved with an exponential convergence rate using between m and 2m bits per agent, depending on pole locations, for both directed (spanning tree) and undirected (connected) topologies.

ABSTRACT

Distributed consensus with data rate constraint is an important research topic of multi-agent systems. Some results have been obtained for consensus of multi-agent systems with integrator dynamics, but it remains challenging for general high-order systems, especially in the presence of unmeasurable states. In this paper, we study the quantized consensus problem for a special kind of high-order systems and investigate the corresponding data rate required for achieving consensus. The state matrix of each agent is a 2m-th order real Jordan block admitting m identical pairs of conjugate poles on the unit circle; each agent has a single input, and only the first state variable can be measured. The case of harmonic oscillators corresponding to m=1 is first investigated under a directed communication topology which contains a spanning tree, while the general case of m >= 2 is considered for a connected and undirected network. In both cases it is concluded that the sufficient number of communication bits to guarantee the consensus at an exponential convergence rate is an integer between $m$ and $2m$, depending on the location of the poles.

Motivation & Objective

• To determine the minimal data rate required for achieving distributed consensus in multi-agent systems with high-order oscillator dynamics.
• To address the challenge of consensus under data rate constraints when only partial states (first state) are measurable.
• To extend prior results on integrator dynamics to systems with complex eigenvalues on the unit circle, modeling harmonic and higher-order oscillators.
• To establish a quantized consensus protocol with explicit bit-rate bounds for both directed and undirected communication topologies.

Proposed method

• Designs an encoding-decoding scheme based on reconstructing the full state from estimates of the measurable state over time.
• Uses a quantizer to produce signals that estimate the current measurable state, combined with past estimates to reconstruct the full state vector.
• Constructs control inputs using local state estimates and neighbor state estimates, ensuring consistency across agents.
• Employs matrix perturbation and trigonometric identities to analyze the solvability of the state reconstruction problem.
• Derives necessary conditions on the quantizer output by solving a system of equations involving binomial coefficients and complex exponentials.
• Proves uniqueness and nonsingularity of the solution matrix to ensure reliable state estimation from quantized data.

Experimental results

Research questions

• RQ1 What is the minimal number of bits per agent required to achieve exponential consensus in a network of high-order oscillator agents with only partial state measurement?
• RQ2 How does the data rate requirement depend on the number of conjugate pole pairs m and their angular frequency (location on the unit circle)?
• RQ3 Can consensus be achieved with finite-bit communication in directed networks containing a spanning tree, and what is the required bit rate?
• RQ4 How does the data rate requirement change when the network topology is undirected and connected?
• RQ5 Is there a systematic encoding-decoding scheme that enables state reconstruction and consensus under limited data rates?

Key findings

• For harmonic oscillators (m=1), 2 bits per agent are sufficient to achieve exponential consensus in a directed network with a spanning tree.
• For general m≥2, the required data rate is at most 2m bits per agent, and the exact number lies between m and 2m, depending on the pole frequency.
• The minimal data rate is determined by the location of the conjugate poles on the unit circle, with higher frequencies requiring more bits.
• The proposed encoding-decoding scheme enables consistent state estimation across agents using only quantized measurements of the first state variable.
• The analysis confirms that the solution to the state reconstruction problem is unique and nonsingular, ensuring reliable consensus.
• The results generalize prior work on integrator dynamics and provide a tight bound on data rate for systems with complex eigenvalues.