Constrained Koopman Control Parametrization: Convex Controller Synthesis for Tracking under Input Constraints

Abstract

Koopman operator theory represents nonlinear systems with linear and bilinear terms in the state and input observables, greatly simplifying control design. However, incorporating input constraints into Koopman-based controller synthesis remains challenging, limiting their applicability to constrained control tasks such as tracking. This paper develops a constrained Koopman Control Parametrization (cKoCoP) framework for synthesizing state-feedback controllers that satisfy prescribed input constraints with Lyapunov-certified stability of the identified lifted closed loop. The cKoCoP offers nonlinear state-feedback via linear-in-observables feedback, enabling linear controller design. The proposed controller is obtained through convex optimization: we incorporate ellipsoidal or polytopic input constraints with the S-procedure or Farkas-lemma-based conditions. We also introduce a Control Lyapunov Function (CLF) for trajectory tracking which incorporates input constraints. The approach is validated on two canonical nonlinear systems, a double pendulum and a unicycle, demonstrating data-driven tracking while respecting input constraints. Experiments validate the method's robustness to measurement noise, and runtime improvements over Koopman Model Predictive Control (KMPC) baselines. The results demonstrate that cKoCoP achieves competitive tracking performance with substantially reduced online computational cost, supporting its potential for real-time constrained tracking applications.