  In this paper, we study the network pinning control problem in the presence
of two different types of coupling: (i) node-to-node coupling among the network
nodes and (ii) input-to-node coupling from the source node to the `pinned
nodes'. Previous work has mainly focused on the case that (i) and (ii) are of
the same type. We decouple the stability analysis of the target synchronous
solution into subproblems of the lowest dimension by using the techniques of
simultaneous block diagonalization (SBD) of matrices. Interestingly, we obtain
two different types of blocks, driven and undriven. The overall dimension of
the driven blocks is equal to the dimension of an appropriately defined
controllable subspace, while all the remaining undriven blocks are scalar. Our
main result is a decomposition of the stability problem into four independent
sets of equations, which we call quotient controllable, quotient
uncontrollable, redundant controllable, and redundant uncontrollable. Our
analysis shows that the number and location of the pinned nodes affect the
number and the dimension of each set of equations. We also observe that in a
large variety of complex networks, stability of the target synchronous solution
is de facto only determined by a single quotient controllable block.
