SCIENCE CHINA Information Sciences, Volume 61, Issue 5: 052202(2018) https://doi.org/10.1007/s11432-017-9129-y

## Non-fragility of multi-agent controllability

• AcceptedMay 16, 2017
• PublishedDec 7, 2017
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### Abstract

Controllability of multi-agent systems is determined by the interconnection topologies. In practice, losing agents can change the topologies of multi-agent systems, which may affect the controllability. In order to preserve controllability, this paper first introduces the concept of non-fragility of controllability. In virtue of the notion of cutsets, necessary and sufficient conditions are established from a graphic perspective, for almost surely strongly/weakly preserving controllability, respectively. Then, the problem of leader selection to preserve controllability is proposed. The tight bounds of the fewest leaders to achieve strongly preserving controllability are estimated in terms of the diameter of the interconnection topology, and the cardinality of the node set. Correspondingly, the tight bounds of the fewest leaders to achieve weakly preserving controllability are estimated in terms of the cutsets of the interconnection topology. Furthermore, two algorithms are established for selecting the fewest leaders to strongly/weakly preserve the controllability. In addition, the algorithm for leaders' locations to maximize non-fragility is also designed. Simulation examples are provided to illuminate the theoretical results and exhibits how the algorithms proceed.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61375120, 61603288).

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• Figure 3

A weighted graph with 5 nodes.

• Figure 4

A weighted graph with 4 nodes.

• Figure 7

A path with 5 nodes.

• Figure 8

A graph with 8 nodes.

• Figure 9

A graph with 9 nodes.

• Figure 12

A graph consists of 7 nodes.

• Figure 13

A graph consists of 8 nodes.

•

Algorithm 1 Searching for the fewest leaders making the topology ASSPC

Get the diameter of ${G}$, and denote it as $l$;

for $i=\lfloor\frac{l-1}{3}\rfloor+1:\lfloor\frac{n}{2}\rfloor$

Get all combinations of $i$ nodes from ${V}$, and denote them as ${V}_1,\ldots,{V}_{C_n^i}$;

for $j=1:C_n^i$

if $\langle{V}_j,{V}/{V}_j\rangle$ constitutes a bipartite graph then

Output “The topology is ASSPC with the minimal leader set ${V}_j$” and exit.

end if

end for

end for

•

Algorithm 2 Searching for the fewest leaders making the topology at least AS$k$WPC

Let $k&apos;=k$;

if no cutset of ${G}$ contains $k$ or less nodes then

Output “The topology is at least AS$k&apos;$WPC for any selection of leaders” and exit;

end if

while $k\neq1$ do

if there exists a $k$-node cutset of ${G}$ then

Find all the $k$-node cutsets, and denote them as $\mathcal{C}_1,\mathcal{C}_2,\ldots,\mathcal{C}_{s_k}$;

Break;

else

$k=k-1$;

end if

end while

for $l_m=2:n-1$

Denote the selections of $l_m$ nodes from ${G}$ as $V_1,\ldots,V_{C_n^{l_m}}$;

for $j=1:C_n^{l_m}$

if no leader-follower cutset of ${G}$ contains $k$ or less nodes then

Output “The topology is AS$k&apos;$WPC with the leader set ${V}_j$” and exit.

end if

end for

end for

•

Algorithm 3 Leaders' distribution to maximize non-fragility

Get the maximum number of leaders $M$, find all the $M$-node subsets of ${V}$, and denote them as ${V}_1,\ldots,{V}_{C_n^M}$, $\tilde{{V}}=\emptyset$, $\tilde{k}=0$;

for $i=1:C_n^M$

Select ${V}_i$ as the leader set;

if the topology is AS$k$WPC and $k>\tilde{k}$ then

$\tilde{{V}}={V}_i,~\tilde{k}=k$;

end if

end for

if $k==n-M$ then

Run algorithm 1 to search for the fewest leaders making the topology ASSPC, denote the number of the fewest leaders as $m$, $k=n-m$, and update the leader set $\tilde{{V}}$;

end if

Output “The topology is at most AS$k$WPC with no more than $M$ leaders, and achieves AS$k$WPC with the fewest leaders $\tilde{{V}}$”.

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