A Simulation Preorder for Koopman-like Lifted Control Systems
Antoine Aspeel, Necmiye Ozay
TL;DR
The paper addresses the challenge of obtaining finite-dimensional, Koopman-like liftings for nonlinear control by introducing a simulation preorder between lifted systems and proving that $\LS_Y \preceq \LS_Z$ implies $\B_\pi[\LS_Y]\subseteq\B_\pi[\LS_Z]$ and $\B[\LS_Y]\subseteq\B[\LS_Z]$. It then derives optimization-based sufficient conditions to verify the relation in two cases: (i) simulating a nonlinear unlifted system by an affine lifted system via SOS relaxations, and (ii) simulating one affine lifted system by another using an affine refinement map; these results enable comparing lifting choices and control-design utilities. Numerical experiments on a Duffing-type system illustrate the framework and highlight practical considerations like lifting choice and conservatism. Overall, the work unifies Koopman over-approximations, approximate immersions, and hybridizations, and points to future work on less conservative conditions and extensions to continuous-time, hybrid, and Lipschitz nonpolynomial dynamics.
Abstract
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.