Empirical Individual State Observability

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.