SCIENCE CHINA Information Sciences, Volume 61 , Issue 5 : 052201(2018) https://doi.org/10.1007/s11432-017-9166-0

## On the structural controllability of distributed systems with local structure changes

• AcceptedJun 21, 2017
• PublishedOct 23, 2017
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### Acknowledgment

This work was supported by National Nature Science Foundation of China (Grant Nos. 61233004, 61590924, 61473184).

• Figure 1

A distributed system.

• Figure 2

Examples for the minimum input theorem. (a) A chain; (b) a tree; (c) a perfect match; (d) a bidirected edge.

• Figure 3

A bidirected edge to two unidirected edges.

• Figure 4

Virtual control inputs. (a) Definition of a virtual control input; (b) a simplest case of perfect match.

• Figure 7

Meaning of the expansion set.

•

Algorithm 1 Distributed algorithm of structural controllability analysis

Given a structured distributed system $S$ as in (3) composed of subsystems $S_i,~i~=~1,\ldots,r$ with graphical form denoted as $G_i(V_i,E_i,E^\ast_i)$;

for $i$ from $1$ to $r$

To subsystem $S_i$, analyze its local structural controllability using the minimum input theorem;

Obtain its maximum matching set $E_i~\ast$, the set of matched vertices $V_i~\ast$, the set of unmatched vertices $\bar{V}_i~\ast$, and the corresponding set of control inputs $U_i$;

end for

for $i$ from $1$ to $r$

Determine the expansion set $D_i$ of $S_i$ according to $E^\ast_i$ containing the interconnection information with its neighbor subsystems;

for each $x_k~x_l~\in~D_i$

If $x_l$ is a vertex with a control input in $S_j$, the control input on $x_l$ is redundant to be removed as a virtual control input;

end for

end for

•

Algorithm 2 Iterative algorithm

Given a structured distributed system $S$ as in (3) composed of subsystems $S_i~(i~=~1,\ldots,r)$ with graphical form denoted as $G_i(V_i,E_i,E^\ast_i)$;

To subsystem $S_1$, analyze its local structural controllability using the minimum input theorem;

for $i$ from $2$ to $r$

To subsystem $S_i$, analyze its local structural controllability using the minimum input theorem;

Obtain its maximum matching set $E_i~\ast$, the set of matched vertices $V_i~\ast$, the set of unmatched vertices $\bar{V}_i~\ast$ and the corresponding set of control inputs $U_i$;

Determine its expansion set $D_i$ based on its neighbor subsystems in subsystems $S_j,~j~=~1,\ldots,i-1$, then find out and remove virtual control inputs in its neighbor subsystems;

for each $S_j(j~\in~\{1,\ldots,i-1\}$ neighbor to $S_i$

Update the expansion sets of $S_j$, then find out and remove virtual control inputs in $S_i$;

end for

end for

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