Model-Based Koopman-Galerkin Lifting for Guaranteed H-Infinity Output Feedback Control of Nonlinear Systems

Abstract

In this paper, we present a model-based Koopman-Galerkin framework for output-feedback control of nonlinear systems with known dynamics. With a suitable actuator augmentation, the finite-dimensional realization preserves the actuator dynamics and the constant input matrix. The approach is based on an exact decomposition of the lifted dynamics into an LTI system and a finite-order closure residual. It also applies to non-control-affine systems and allows dictionaries that do not span an invariant subspace. Regional bounds are derived for the closure residual, actuator-realization defect, and output-reconstruction error. We use these bounds to formulate a standard ${\mathcal{H}}_{\infty }$ output-feedback problem. The controller is linear and dynamic and is implemented using only the measured tracking error. Neither full-state measurements nor a separate state observer are required. Lifting is performed offline. With the state coordinates retained in the dictionary, we obtain explicit transient bounds on the original state and tracking error. A sufficient condition for regional containment is also given. Under this condition, the nonlinear closed loop is forward complete, and the state and tracking error are uniformly ultimately bounded. The cart-pendulum example illustrates the nonlinear closed-loop guarantees. The proposed output feedback gives lower RMSE and peak tracking error on average than Jacobian-based ${\mathcal{H}}_{\infty }$ feedback, full-state backstepping, and Koopman-based hybrid MPC.

arXiv comment16 pages, 4 figures. Code: https://github.com/kurkcub/RCCG