This paper presents a novel perspective on fully distributed adaptive protocol design for bipartite consensus under directed communication graphs. It is revealed that the fully distributed protocol is indeed a PI-gain design approach.Based on such PI-gain method, fully distributed observer-based protocols are proposed to solve the consensus problem of nonlinear multi-agent systems over directed cooperative-antagonistic networks, where the neural network approximation is introduced to tackle the unknown nonlinearity. Fully distributed continuous protocols are further presented by modifying the sign function with a decaying variable, which takes the advantage of avoiding chattering problem.Theoretical analysis and numerical simulations are given to support our main results.

This paper studies the bipartite consensus problem with input saturation for general linear represents multi‐agent systems (MASs) with signed digraphs. Based on relative state information among neighbour agents, distributed adaptive protocols and a compensation observer are proposed, wherein both leaderless bipartite consensus and tracking bipartite consensus problems are addressed. For the case only relative output information is available, observer‐based distributed adaptive protocols are also designed, where for each agent, a local observer is given to estimate signed consensus error and a distributed observer is presented to achieve bipartite consensus and generate control input and a compensation observer is designed to handle input saturation. Simulation illustrations are given to explain the feasibility. The control protocols presented in this article depend on only local relative state or output information, without any eigenvalue information of Laplacian matrices associated with signed digraphs, which can be practically used for each agent in a fully distributed way.

In this paper, the bipartite consensus tracking problem of linear multiagent systems on leader‐follower signed directed graph is studied, where the leader's unknown input is taken into consideration. Based on only local relative information of neighboring agents, discontinuous distributed adaptive protocols are proposed, where nonlinear function is introduced for each agent to tackle leader's unknown input. To avoid tremors phenomenon, continuous distributed protocols are further presented to achieve practical bipartite consensus. Both the state feedback and output feedback cases are investigated. Different from existing literature showing bipartite consensus achievement with gauge transformation, this paper reveals a novel property of the signed Laplacian matrix, which plays a key role in simplifying the bipartite consensus demonstration. Two practical examples on synchronization of Chua's circuits are presented to illustrate the effectiveness of the designed protocols.

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