Practical Consensus of the Multi-Agent Systems on the Two-Layer Networks with Time-Varying Coupling Weights

In practical multi‑agent systems, achieving strict consensus is often challenging due to complex communication environments and time‑varying coupling relationships. Unlike conventional consensus, which requires the system state errors to converge exactly to the origin, practical consensus only requires the errors to converge to a bounded neighborhood of the origin, thereby better satisfying practical engineering requirements. Motivated by the above analysis, this paper investigates the practical consensus problem for two-layer leader-following multi-agent systems, where uncertain communication environments induce bounded time-varying deviations in coupling weights. Taking such bounded deviations of actual coupling weights into consideration, an error dynamics model is established for the time-varying two-layer network. By designing a distributed pinning control protocol and combining tools from graph theory, matrix theory, and strong practical stability theory, a sufficient condition is derived to guarantee practical consensus of the considered system. Numerical simulations are finally carried out to validate the correctness and effectiveness of the theoretical conclusions.