Event-triggered bipartite consensus for multiagent systems with general linear dynamics: An integral-type event-triggered control

We fill a gap in the proofs in the previous works (Wu X, Mu, X. Int J. Robust Nonlin Control. 2020;30:3753U3772; Zhang Z, Lunze J, Wang L. Int J Control. 2020;93:1005-1014; Zhang Z, Wang L. J Robust Nonlin Control. 2018;28:4175U4187; Dai, M-Z, Zhang C, Leung H, Dong P, Li B. IEEE Trans Syst, Man, Cybern: Syst. doi:10.1109/TSMC.2021.3119670) for the consensus using the integral-based event-triggered controls. More precisely, it was inferred for a Lyapunov function V:[0,infinity)-> Double-struck capital R+$$ V:\left[0,\infty \right)\to {\mathbb{R}}_{+} $$ that V(t)$$ \dot{V}(t) $$ is uniformly bounded by showing that V(t)$$ V(t) $$ is uniformly bounded for t >= 0$$ t\ge 0 $$. However, this argument may fail without further information while the boundedness of V(t)$$ \dot{V}(t) $$ is crucially used for applying Barbalat's lemma. The consequence of Barbalat's lemma is that limt ->infinity V(t)$$ {\lim}_{t\to \infty }V(t) $$ which corresponds to the desired consensus result. To overcome this gap, Ma and Zhao (Inform Sci. 2018;457-458:208-221) put an extra condition about the boundedness of measurement error functions inside the proposed integral-based event-triggering protocol. In this article, we propose a new integral-based event-triggering protocol for bipartite consensus problems of the multi-agent systems whose dynamics are described by general linear systems without adding the uniform boundedness of measurement error functions as (Ma Y, Zhao J. Inform Sci. 2018;457-458:208-221) did. Via our new integral-based integral control strategy, we prove that the system achieves the bipartite consensus in asymptotic regime, and provide a complete solution of the freeness of both chattering and genuinely Zeno behaviors. Numerical results are provided supporting the effectiveness of the proposed controller.