Figure 1. Input Cacti and Proposed Two-Step Approach.

In (A) we depict a system digraph, and in (B) and (C) two possible disjoint spanning input cacti. Notice that in B one input cactus has nine state vertices and the other six, whereas in (C) one input cactus contains eight state vertices and the other seven. In fact, these are the only two spanning input cacti, so the structural controllability index is equal to eight. Our two-step approach is depicted in (D–F). First, given the state digraph in (D) we consider a partition with at most eight state vertices, leading to two partitions denoted by P₁ and P₂. Secondly, in (E) we find the minimum number of driven nodes that correspond to the roots of a disjoint union of state cacti containing all the vertices in each partition. Finally, we just need to assign inputs to the driven nodes associated with nodes 1 and 11 respectively, as depicted in (F).