Model-Based Galerkin Lifting with Exact LTI Decomposition for Guaranteed $\mathcal{H}_\infty$ Output Feedback Control of Nonlinear Systems
In this paper, we present a model-based 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. We retain finite-order nonclosure as an explicit additive residual. The LTI part and this residual describe the lifted nonlinear dynamics exactly. The decomposition does not require an invariant subspace of observables and also applies to non-control-affine systems. We use regional bounds on the closure residual, actuator-realization defect, and output-reconstruction error in the standard $\mathcal{H}_\infty$ design. The proposed controller is a linear dynamic system driven only by the measured tracking error and requires no online lifting. Retaining the state coordinates gives direct bounds on the original state. We use these bounds to derive an a priori containment condition. Under this condition, the nonlinear closed loop is forward complete, and the state and tracking error satisfy explicit transient bounds and are uniformly ultimately bounded. The cart-pendulum example illustrates the nonlinear closed-loop guarantees and compares the controller with full-state backstepping. Although the output-feedback controller uses only the measured tracking error, it achieves almost the same nominal tracking RMSE as backstepping, with a lower peak tracking error.