Containment Control of Multiagent Systems With Dynamic Leaders Based on a $PI^{n}$ -Type Approach

This paper studies the containment control of multiagent systems (MASs) with multiple dynamic leaders in both continuous-time domain and discrete-time domain. The leaders' motions are described by the n th-order polynomial trajectories. This setting makes practical sense because given some critical points, the leaders' trajectories are usually planned by the polynomial interpolations. In order to drive all followers into the convex hull spanned by the leaders, a PI ⁿ -type containment algorithm is proposed ( P and I are short for proportional and integral, respectively; I ⁿ implies that the algorithm includes up to the n th-order integral terms). It is theoretically proved that the PI ⁿ -type containment algorithm is able to solve the containment problem of MASs where the followers are described by any order integral dynamics. Compared to the previous results on the MASs 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 leaders with the n th-order polynomial trajectories, existing results require the follower's dynamics to be the ( n+ 1)th-order integral while the followers considered in this paper can be described by any-order integral dynamics; 3) the "sign" function is not employed in the proposed algorithm, which avoids the chattering phenomenon; and 4) both disturbance and measurement noise are taken into account. Finally, some simulation examples are given to demonstrate the effectiveness of the proposed algorithm.