This paper analyzes the structural controllability of distributed systems, which are composed of many subsystems and have complicated interconnections. Different from traditional methods in centralized systems where global information is required, the method proposed in this paper is based on local structural properties and simplified interconnections, by which the computational burden is highly decreased and the implementation is tractable. Moreover, a necessary condition for global structural controllability is obtained by combining local information. When the structure in any subsystems is changed, only corresponding local information needs to be reevaluated instead of whole distributed systems, which makes the analysis easier. Finally, examples are given to illustrate the effectiveness of our proposed method.
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 5
Condition (1) in Definition 10. (a) $x_1$ is a matched vertex of $S_i$; (b) $x_1$ is an unmatched vertex of $S_i$.
Figure 6
Condition (2) in Definition 10. (a) A common starting vertex; (b) a common ending vertex.
Figure 7
Meaning of the expansion set.
Figure 8
A distributed system composed of three subsystems.
Figure 9
Local structural controllability.
Figure 10
Global structural controllability.
Figure 11
Remove a subsystem.
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 $S_i$, 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 the expansion set $D_i$ of $S_i$ according to $E^\ast_i$ containing the interconnection information with its neighbor subsystems; 

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; 

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; 
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 $S_j,~j~=~1,\ldots,i1$, then find out and remove virtual control inputs in its neighbor subsystems; 

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

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