Lagrangian system control algorithm combined with distributed sliding-mode estimators. A necessary and sufficient condition on the directed graph is presented such that all followers converge to the dynamic convex hull spanned by the dynamic leaders asymptotically. As a byproduct, we show a necessary and sufficient condition on leaderless consensus for networked Lagrangian systems under a directed graph. Numerical simulation results are given to show the effectiveness of the proposed control algorithms.

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This paper considers the containment control problem for uncertain nonlinear multiagent systems under directed graphs. The followers are governed by nonlinear systems with unknown dynamics while the multiple leaders are neighbors of a subset of the followers. Fuzzy logic systems (FLSs) are used to identify the unknown dynamics and a distributed state feedback containment control protocol is proposed. This result is extended to the output feedback case, where observers are designed to estimate the unmeasurable states. Then, an output feedback containment control scheme is presented. The developed state feedback and output feedback containment controllers guarantee that the states of all followers converge to the convex hull spanned by the dynamic leaders. Based on Lyapunov stability theory, it is proved that the containment control errors are uniformly ultimately bounded (UUB). An example is provided to show the effectiveness of the proposed control method.

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This paper studies the containment problem of multi-agent systems with multiple dynamic leaders in both the discrete-time domain and the continuous-time domain. The leader’s motion is described by the n-order polynomial trajectory. This setting makes the practical sense because given some critical points, the leader’s trajectory is usually planned by the polynomial interpolation. In order to drive all followers into the convex hull spanned by leaders, a PIn-type (P and I are short for Proportion and Integration, respectively; In stands for the n-order integral term) containment protocol is proposed. It is theoretically proved that the PIn-type containment protocol is able to solve the containment problem of multi-agent systems where followers are described by not only the single-integrator dynamics but also any-order difference/differential equation. Compared with the previous results on the multi-agent systems with dynamic leaders, the distinguished features of this paper are that: (1) the containment problem is studied not only in the continuous-time domain but also in the discrete-time domain while most existing results only work in the continuous-time domain; (2) to deal with the leader’s n-order polynomial trajectory, existing results require the follower’s dynamics to be n-order integrator while the followers considered this paper can be described by any-order integrator; and (3) the “sign ” function is not employed in

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