  Optimal sensing nodes selection in dynamic systems is a combinatorial
optimization problem that has been thoroughly studied in the recent literature.
This problem can be formulated within the context of set optimization. For
high-dimensional nonlinear systems, the problem is extremely difficult to
solve. It scales poorly too. Current literature poses combinatorial submodular
set optimization problems via maximizing observability performance metrics
subject to matroid constraints. Such an approach is typically solved using
greedy algorithms that require lower computational effort yet often yield
sub-optimal solutions. In this letter, we address the sensing node selection
problem for nonlinear dynamical networks using a variational form of the system
dynamics, that basically perturb the system physics. As a result, we show that
the observability performance metrics under such system representation are
indeed submodular. The optimal problem is then solved using the multilinear
continuous extension. This extension offers a computationally scalable and
approximate continuous relaxation with a performance guarantee. The
effectiveness of the extended submodular program is studied and compared to
greedy algorithms. We demonstrate the proposed set optimization formulation for
sensing node selection on nonlinear natural gas combustion networks.
