# Koopman Form of Nonlinear Systems with Inputs

arXiv: https://arxiv.org/abs/2207.12132 · HTML: https://mlsanity.com/paper/2207.12132/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2022.

## Built on by (most cited first, 11 of 11)

- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guarantees](https://mlsanity.com/paper/2405.01292/) (2024)
- [From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma](https://mlsanity.com/paper/2508.07494/) (2025)
- [Invertible Koopman neural operator for data-driven modeling of partial differential equations](https://mlsanity.com/paper/2503.19717/) (2025)
- [MAKO: Meta-Adaptive Koopman Operators for Learning-based Model Predictive Control of Parametrically Uncertain Nonlinear Systems](https://mlsanity.com/paper/2510.09042/) (2025)
- [Data-Driven Predictive Control of Nonholonomic Robots Based on a Bilinear Koopman Realization: Data Does Not Replace Geometry](https://mlsanity.com/paper/2411.07192/) (2024)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Control](https://mlsanity.com/paper/2511.22648/) (2025)
- [An Output Feedback Q-learning Algorithm for Optimal Control of Nonlinear Systems with Koopman Linear Embedding](https://mlsanity.com/paper/2603.29858/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

The Koopman framework proposes a linear representation of finite-dimensional nonlinear systems through a generally infinite-dimensional globally linear embedding. Originally, the Koopman formalism has been derived for autonomous systems. In applications for systems with inputs, generally a linear time invariant (LTI) form of the Koopman model is assumed, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, it can be easily shown that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. Proper theoretical extension for actuated continuous-time systems with a linear or a control-affine input has been worked out only recently, however extensions to discrete-time systems and general continuous-time systems have not been developed yet. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a rather wide class of nonlinear systems in both continuous and discrete time. We prove that the resulting lifted representations give Koopman models where the state transition is linear, but the input matrix becomes state-dependent (state and input-dependent in the discrete-time case), giving rise to a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the dependency of the input matrix contributes to the resulting representation and how well the system behaviour can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach.