Novel average-valued condition for multi-agent systems via a quantized adaptive control protocol

Yufang Wu; Zhenkun Huang; Yufang Wu; Zhenkun Huang

doi:10.3934/nhm.2026071

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1902-1923. doi: 10.3934/nhm.2026071

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Research article

Novel average-valued condition for multi-agent systems via a quantized adaptive control protocol

Yufang Wu ,
Zhenkun Huang ^,

School of Science, Jimei University, Xiamen 361021, China

Received: 15 June 2026 Revised: 05 August 2026 Accepted: 07 September 2026 Published: 14 September 2026

A novel average-valued condition for consensus of multi-agent systems (MASs) is developed via a quantized adaptive control protocol, a framework that simultaneously addresses multiple self-delays, time-varying coupling, and logarithmic quantizers. A adaptive controller is designed, where each follower applies a quantized error signal and a shared adaptive gain that is updated by the quantized sum of all followers' squared errors with a fixed adaptation rate. By constructing a Lyapunov functional and applying a generalized Halanay inequality that exploits the time-average property of the coupling strength, we prove that the adaptive gain is nondecreasing, remains bounded, and converges to a finite limit. A gain threshold, depending explicitly on the average coupling strength and the delay-related Lipschitz constants, is established: If the gain exceeds this threshold, exponential consensus is achieved; otherwise, asymptotic consensus still holds. Numerical simulations on a leader-follower network with Lorenz-type node dynamics and two self-delays validate the theoretical results, including the effect of different adaptation rates, quantization densities, and delay configurations.
- multi-agent system,
- consensus,
- time-varying coupling,
- logarithmic quantizer,
- adaptive control,
- time-delay
Citation: Yufang Wu, Zhenkun Huang. Novel average-valued condition for multi-agent systems via a quantized adaptive control protocol[J]. Networks and Heterogeneous Media, 2026, 21(5): 1902-1923. doi: 10.3934/nhm.2026071

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Abstract

A novel average-valued condition for consensus of multi-agent systems (MASs) is developed via a quantized adaptive control protocol, a framework that simultaneously addresses multiple self-delays, time-varying coupling, and logarithmic quantizers. A adaptive controller is designed, where each follower applies a quantized error signal and a shared adaptive gain that is updated by the quantized sum of all followers' squared errors with a fixed adaptation rate. By constructing a Lyapunov functional and applying a generalized Halanay inequality that exploits the time-average property of the coupling strength, we prove that the adaptive gain is nondecreasing, remains bounded, and converges to a finite limit. A gain threshold, depending explicitly on the average coupling strength and the delay-related Lipschitz constants, is established: If the gain exceeds this threshold, exponential consensus is achieved; otherwise, asymptotic consensus still holds. Numerical simulations on a leader-follower network with Lorenz-type node dynamics and two self-delays validate the theoretical results, including the effect of different adaptation rates, quantization densities, and delay configurations.

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