Designing real-time control for nonlinear systems to track periodic orbits can be challenging, especially for complex and coupled dynamical systems (e.g., tethered kites), where the true model parameters are hard to obtain. To address this, we propose a physics-informed Koopman learning approach that incorporates periodic-orbit stability directly into the learned linear predictor. The approach combines the underlying periodic structure of the dynamics along with periodic state-input trajectory data to learn physically consistent linear predictors (in lifted/observable space). We leverage Floquet theory, which studies the stability of periodic orbits in nonlinear systems, to obtain the periodic stability information of a given orbit via a monodromy matrix (i.e., the state transition matrix over an entire period). Subsequently, we impose open-loop monodromy constraints to learn Floquet-parameterized linear predictors using a least-squares Extended Dynamic Mode Decomposition (EDMD) formulation, thereby constraining the search space and aiding in Koopman learning. We then utilize these linear predictors to design a model predictive control (MPC) law for tracking periodic references. Finally, we evaluate the designed framework on a cartpole system under periodic forcing and compare it against a purely data-driven baseline.

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