Key Laboratory of System Control and Information Processing, Ministry of Education of China, Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
Algorithm 1Distributed 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 2Iterative 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 $S_i$;