Data-Driven Observer Synthesis for Autonomous Limit Cycle Systems through Estimation of Koopman Eigenfunctions

The signal of system states needed for feedback controllers is estimated by state observers. One state observer design is the Kazantzis-Kravaris/Luenberger (KKL) observer, a generalization of the Luenberger observer for linear systems. The main challenge in applying the KKL design is constructing an injective mapping of the states, which requires solving PDEs based on a first-principles model. This paper proposes a data-driven, Koopman operator-based method for the construction of KKL observers for planar limit cycle systems. Specifically, for such systems, the KKL injective mapping is guaranteed to be a linear combination of Koopman eigenfunctions. Hence, the determination of such an injection is reduced to a least-squares regression problem, and the inverse of the injective mapping is then approximated using kernel ridge regression. The entire synthesis procedure uses solely convex optimization. We apply the proposed approach to the Brusselator system, demonstrating accurate estimations of the system states.