  A dynamical system is observable if there is a one-to-one mapping from the
system's measured outputs and inputs to all of the system's states. Analytical
and empirical tools exist for quantifying the (full state) observability of
linear and nonlinear systems; however, empirical tools for evaluating the
observability of individual state variables are lacking. Here, a new empirical
approach termed Empirical Individual State Observability (E-ISO) is developed
to quantify the level of observability of individual state variables. E-ISO
first builds an empirical observability matrix via simulation, then applies
convex optimization to efficiently determine the subset of its rows required to
estimate each state variable individually. Finally, (un)observability measures
for these subsets are calculated to provide independent estimates of the
observability of each state variable. Multiple example applications of E-ISO on
linear and nonlinear systems are shown to be consistent with analytical
results. Broadly, E-ISO will be an invaluable tool both for designing active
sensing control laws or optimizing sensor placement to increase the
observability of individual state variables for engineered systems, and
analyzing the trajectory decisions made by organisms.
