Nonlinear observability via Koopman analysis: characterizing the role of symmetry (Q2663863)

This manuscript considers the observability of nonlinear systems from Koopman analysis and, in particular, the effect of symmetry on observability. After introducing the mathematical tools on the observability problem and the Koopman operator, the authors examine the representation of the nonlinear system as an infinite-dimensional linear system using independent Koopman eigenfunctions. Although this representation is infinite-dimensional, there exist necessary and sufficient conditions for analyze its observability due to its linearity. These conditions can be checked through the rank of the so-called observability matrix. Next, the authors formulate the nonlinear observability problem in terms of the observability of a transformed infinite-dimensional linear system. Analyzing the observability problem from the Koopman operator provides an analytic relation between discrete symmetries and observability of nonlinear systems. In Section 4, the authors examine structural properties of Koopman eigenvalues, eigenfunctions, and modes of a ``symmetric'' nonlinear system. It is shown that symmetry in the nonlinear dynamics is reflected in the symmetry of the corresponding Koopman eigenfunctions, as well as presence of repeated Koopman eigenvalues. Finally, the authors consider three examples that confirm the application of the proposed results. One of these examples pertain to the observability analysis of coupled nanoelectromechanical systems on a ring topology.