This paper is concerned with the problem of formation-containment on networked systems, with interconnected systems modeled by the Euler-Lagrange equation with bounded inputs and time-varying delays on the communication channels. The main results are the design of control algorithms and sufficient conditions to ensure the convergence of the network. The control algorithms are designed as distributed dynamic controllers, in such a way that the number of neighbors of each agent is decoupled from the bound of the control inputs. That is, in the proposed approach the amplitude of the input signal does not directly increase with the number of neighbors of each agent. The proposed sufficient conditions for the asymptotic convergence follow from the Lyapunov-Krasovskii theory and are formulated in the linear matrix inequalities framework. The conditions rely only on the upper bound of delays and on a subset of the controller parameters, but they do not depend on the model of each agent, which makes it suitable for networks with agents governed by distinct dynamics. In order to illustrate the effectiveness of the proposed method we present numerical examples and compare with similar approaches existing in the literature. K E Y W O R D S Communication delays, Euler-Lagrange systems, formation-containment, input saturation 1 Int J Robust Nonlinear Control. 2020;30:2999-3022. wileyonlinelibrary.com/journal/rnc• the formation-containment is approached taking into account communication delays among agents and input saturation simultaneously, which to the best of authors' knowledge it is a scenario that has not yet been addressed on the literature. Additionally, the proposed distributed control strategy does not depend on the knowledge of the agents inertia matrix nor the Coriolis forces, it also does not rely on the neighbors velocity information. This reduces the communication burden on the network and provides a robust control strategy.Secondly, considering the consensus, a particular case of the main problem investigated in this manuscript, and comparing with those from the literature that primarily study consensus under similar conditions, the following contributions can also be highlighted:• input saturation was previously studied in the context of consensus problem in References 19 and 45, but with some important distinctions from our approach, namely: (a) we now propose a strategy to deal with the input saturation and communication delays simultaneously, (b) different from Reference 45 our dynamic controller does not rely on the knowledge of inertia nor Coriolis matrices and, (c) in contrast with both approaches, as mentioned above, the control strategy proposed here does not depend on relative velocity measurements;• regarding the communication delays among agents, it is noteworthy that the results of Nuño and collaborators 20,21,37,46 seem to contribute with the most notorious conditions on communication delays on the consensus of networked Euler-Lagrange systems. For example, the results in References 33 and 34 hand...
