A Simulation Preorder for Koopman-like Lifted Control Systems

This paper introduces a simulation preorder among lifted systems, a generalization of finite-dimensional Koopman approximations (also known as approximate immersions) to systems with inputs. It is proved that this simulation relation implies the containment of both the open- and closed-loop behaviors. Optimization-based sufficient conditions are derived to verify the simulation relation in two special cases: i) a nonlinear (unlifted) system and an affine lifted system and, ii) two affine lifted systems. Numerical examples demonstrate the approach.