SCIENCE CHINA Information Sciences, Volume 60, Issue 11: 110204(2017) https://doi.org/10.1007/s11432-017-9146-1

## Multi-leader multi-follower coordination with cohesion, dispersion, and containment control via proximity graphs

• AcceptedJun 23, 2017
• PublishedOct 10, 2017
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### Abstract

This paper studies the problem of multi-leader multi-follower coordination with proximity-based network topologies. The particular interest is to drive all the followers towards the convex hull formed by the moving leaders while producing cohesion behavior and keeping group dispersion. First, in the case of stationary leaders, we design a gradient-based continuous control algorithm. We show that with this continuous algorithm the control objective can be achieved, and the tracking error bound can be controlled by tuning some control parameters. We apply the continuous control algorithm to the moving leaders case and show that the tracking error bound is related to the velocities of the leaders. However, in this case, the algorithm has one restriction that the velocities of the leaders should depend on neighboring followers' velocities, which might not be desirable in some scenarios. Therefore, we propose a nonsmooth algorithm for moving leaders which works under the mild assumption of boundedness of leaders' velocities. Finally, we present numerical examples to showthe validity of the proposed algorithms.

### Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61473240, 61528301), National Natural Science Foundation of Fujian Province (Grant No. 2017J01119), 111 Project (Grant No. B17048), and State Key Laboratory of Intelligent Control and Decision of Complex Systems.

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• Figure 1

An illustration of the function $r_{ij}$ with the parameters: $r_{\rm~F}=1$, $d_2=0.4$, $a_1=0.84$, $a_2=-1$.

• Figure 2

Snapshots for the stationary leaders case. The leaders are denoted by “$\diamond$”, while the followers are denoted by “*”. The parameters are specified as follows: $r_{\rm~L}=5$, $r_{\rm~F}=1$, $d_1=0.8$, $d_2=0.4$, $a_2=-1$, $a_1=a_2(d_2^2-r_{\rm~F}^2)$, $b_{ik}=1$ for all $i~\in~\mathcal{V}_{\rm~F}$ and $k~\in~\left(\mathcal{N}_i~\cap~\mathcal{V}_{\rm~L}\right)$. (a) $t=0~{\rm~s}$; (b) $t=0.25~{\rm~s}$; (c) $t=0.75~{\rm~s}$; (d) $t=1~{\rm~s}$.

• Figure 3

(Color online) Stationary leaders: the trajectory of $\|x_i-x_j\|$ for $(i,j)~\in~\mathcal{E}(0)$, $i~\in~\mathcal{V}_{\rm~F}$ and $j~\in~\mathcal{V}_{\rm~F}$ with the dashed line $\|x_i-x_j\|=d_1$.

• Figure 4

(Color online) Stationary leaders: the trajectory of $\|x_i-x_j\|$ for $(i,j)~\in~\mathcal{E}(0)$, $i~\in~\mathcal{V}_{\rm~L}$, and $j~\in~\mathcal{V}_{\rm~F}$ with the dashed line $\|x_i-x_j\|=r_{\rm~L}$.

• Figure 5

Snapshots for the moving leaders case. (a) $t=0~{\rm~s}$; (b) $t=0.25~{\rm~s}$; (c) $t=0.75~{\rm~s}$; (d) $t=1~{\rm~s}$.

• Figure 7

(Color online) Moving leaders: the trajectory of $\|x_i-x_j\|$ for $(i,j)~\in~\mathcal{E}(0)$, $i~\in~\mathcal{V}_{\rm~L}$, and $j~\in~\mathcal{V}_{\rm~F}$ with the dashed line $\|x_i-x_j\|=r_{\rm~L}$.

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