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In this paper, we study the leaderless consensus problem for multipleLagrangian systems in the presence of parametric uncertainties and externaldisturbances under directed graphs. For achieving asymptotic behavior, a robustcontinuous term with adaptive varying gains is added to alleviate the effectsof the external disturbances with unknown bounds. In the case of a fixeddirected graph, by introducing an integrate term in the auxiliary variabledesign, the final consensus equilibrium can be explicitly derived. We show thatthe agents achieve weighted average consensus, where the final equilibrium isdependent on three factors, namely, the interactive topology, the initialpositions of the agents, and the control gains of the proposed controlalgorithm. In the case of switching directed graphs, a model reference adaptiveconsensus based algorithm is proposed such that the agents achieve leaderlessconsensus if the infinite sequence of switching graphs is uniformly jointlyconnected. Motivated by the fact that the relative velocity information isdifficult to obtain accurately, we further propose a leaderless consensusalgorithm with gain adaptation for multiple Lagrangian systems without usingneighbors' velocity information. We also propose a model reference adaptiveconsensus based algorithm without using neighbors' velocity information forswitching directed graphs. The proposed algorithms are distributed in the senseof using local information from its neighbors and using no comment controlgains. Numerical simulations are performed to show the effectiveness of theproposed algorithms.

Distributed Consensus for Multiple Lagrangian Systems with Parametric Uncertainties and External Disturbances Under Directed Graphs