Observability for Nonlinear Systems: Connecting Variational Dynamics, Lyapunov Exponents, and Empirical Gramians

Observability quantification is a key problem in dynamic network sciences. While it has been thoroughly studied for linear systems, observability quantification for nonlinear networks is less intuitive and more cumbersome. One common approach to quantify observability for nonlinear systems is via the Empirical Gramian (Empr-Gram) -- a generalized form of the Gramian of linear systems. In this technical note, we produce three new results. First, we establish that a variational form of nonlinear systems (computed via perturbing initial conditions) yields a so-called Variational Gramian (Var-Gram) that is equivalent to the classic Empr-Gram; the former being easier to compute than the latter. Via Lyapunov exponents derived from Lyapunov's direct method, the technical note's second result derives connections between existing observability measures and Var-Gram. The third result demonstrates the applicability of these new notions for sensor selection/placement in nonlinear systems. Numerical case studies demonstrate these three developments and their merits.