[Paper Review] On Convergence Rate of Leader-Following Consensus of Linear Multi-Agent Systems with Communication Noises

Long Cheng, Yunpeng Wang+3 more|arXiv (Cornell University)|Aug 27, 2015

Distributed Control Multi-Agent Systems13 references3 citations

TL;DR

This paper proposes a leader-following consensus protocol for linear multi-agent systems with communication noises, where each agent uses its own time-varying noise-attenuation gain. It proves that mean square consensus is achieved if all gains are infinitesimal of the same order, and establishes a convergence rate bounded by $ t^{-\beta} $ when gains are asymptotically bounded above and below by $ t^{-\beta} $ for $ \beta \in (0,1) $. The result relaxes the need for identical gains across agents and provides a quantitative convergence rate analysis under agent-dependent gains.

ABSTRACT

This note further studies the previously proposed consensus protocol for linear multi-agent systems with communication noises in [15], [16]. Each agent is allowed to have its own time-varying gain to attenuate the effect of communication noises. Therefore, the common assumption in most references that all agents have the same noise-attenuation gain is not necessary. It has been proved that if all noise-attenuation gains are infinitesimal of the same order, then the mean square leader-following consensus can be reached. Furthermore, the convergence rate of the multi-agent system has been investigated. If the noise-attenuation gains belong to a class of functions which are bounded above and below by $t^{-β}$ $(β\in(0,1))$ asymptotically, then the states of all follower agents are convergent in mean square to the leader's state with the rate characterized by a function bounded above by $t^{-β}$ asymptotically.

Motivation & Objective

• To address the limitation in existing literature where all agents are assumed to have identical noise-attenuation gains, requiring global knowledge.
• To investigate whether mean square leader-following consensus can still be achieved when agents use different, time-varying noise-attenuation gains.
• To analyze the convergence rate of the multi-agent system under such agent-dependent gains, especially when gains decay as $ t^{-\beta} $.
• To relax the standard stochastic-approximation condition on noise-attenuation gains for linear multi-agent systems with communication noise.
• To provide a quantitative characterization of the convergence rate in the presence of communication noise and agent-specific gains.

Proposed method

• Introduces a modified consensus protocol where each agent independently applies its own time-varying noise-attenuation gain to mitigate communication noise effects.
• Uses a stochastic differential equation framework to model the dynamics of the multi-agent system under communication noise.
• Applies mathematical induction and Lyapunov-like analysis to bound the mean square error between follower states and the leader’s state.
• Employs asymptotic analysis of integrals involving exponential and power functions to characterize the decay rate of the error terms.
• Relies on recent results from [17] to handle the complexity introduced by agent-dependent gains in the solution of the governing SDE.
• Derives bounds on the expected squared deviation of the system states from consensus, showing $ \mathcal{O}(t^{-\beta}) $ decay under $ t^{-\beta} $-bounded gains.

Experimental results

Research questions

• RQ1 Can mean square leader-following consensus be achieved in linear multi-agent systems with communication noise when agents use different, time-varying noise-attenuation gains?
• RQ2 What is the convergence rate of the system when the noise-attenuation gains decay as $ t^{-\beta} $ for $ \beta \in (0,1) $?
• RQ3 Is the standard stochastic-approximation condition on noise-attenuation gains necessary for consensus in leader-following systems with communication noise?
• RQ4 How does the convergence rate depend on the structure of the noise-attenuation gains and the system's dynamics?
• RQ5 Can the convergence rate be characterized explicitly when gains are not identical across agents?

Key findings

• Mean square leader-following consensus is achieved if all noise-attenuation gains are infinitesimal of the same order, even when gains are agent-specific.
• The convergence rate of the system is bounded above by $ \mathcal{O}(t^{-\beta}) $ when the noise-attenuation gains are asymptotically bounded above and below by $ t^{-\beta} $ for $ \beta \in (0,1) $.
• The stochastic-approximation condition (e.g., square integrability) is not necessary for consensus in leader-following systems with communication noise.
• The expected squared deviation of the state error from consensus decays as $ \mathcal{O}(t^{-\beta}) $, which is established through rigorous analysis of stochastic integrals and recursive bounds.
• The convergence rate is preserved even when agents use different gains, provided they decay at the same asymptotic rate $ t^{-\beta} $.
• The analysis confirms that the convergence rate is determined by the decay rate of the noise-attenuation gains, not by the system’s internal dynamics alone.