Noise-Enabled Observability of Nonlinear Dynamic Systems Using the Empirical Observability Gramian
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While control actuation is well understood to influence the observability of nonlinear dynamical systems, actuation of nonlinear stochastic systems by process noise has received comparatively little attention in terms of the effects on observability. As noise is present in essentially all physically instantiated systems, complete analysis of observability must account for process noise. We approach the problem of process-noise-induced observability through the use of a tool called the empirical observability Gramian. We demonstrate that the empirical observability Gramian can provide a unified approach to observability analysis, by providing sufficient conditions for weak observability of continuous-time nonlinear systems, local weak observability of discrete-time nonlinear systems, and stochastic observability of continuous-time stochastic linear systems with multiplicative noise. The empirical observability Gramian can be used to extend notions of stochastic observability that depend explicitly on linear systems structure to nonlinear stochastic systems. We use Monte Carlo methods to analyze the observability of nonlinear stochastic systems with noise and control actuation.