  In this paper, we provide optimal solutions to two different (but related)
input/output design problems involving large-scale linear dynamical systems,
where the cost associated to each directly actuated/measured state variable can
take different values, but is independent of the labeled input/output variable.
Under these conditions, we first aim to determine and characterize the
input/output placement that incurs in the minimum cost while ensuring that the
resulting placement achieves structural controllability/observability. Further,
we address a constrained variant of the above problem, in which we seek to
determine the minimum cost placement configuration, among all possible
input/output placement configurations that ensures structural
controllability/observability, with the lowest number of directly
actuated/measured state variables. We show that both problems can be solved
efficiently, i.e., using algorithms with polynomial time complexity in the
number of the state variables. Finally, we illustrate the obtained results with
an example.
