Control based on the Koopman operator: A comprehensive review

The Koopman operator provides a powerful linear framework for analyzing nonlinear dynamical systems by lifting their behavior into a higher-dimensional space. This article presents a comprehensive review of the main methodologies for estimating the Koopman operator, with particular attention to forced systems–those influenced by external inputs. The approaches are organized into three primary categories: variants of Dynamic Mode Decomposition (DMD), sparse regression techniques such as Sparse Identification of Nonlinear Dynamics (SINDy), and data-driven methods based on deep neural networks. Building on this foundation, we propose a unified strategy for Koopman operator estimation and its integration into a model predictive control (MPC) framework. Using a multi-tank system as a case study, we show that the Koopman-based MPC yields a response that is twice as fast and significantly more stable under external disturbances–including fault conditions such as leaks–compared to a conventional linearized MPC. These results underscore the Koopman operator’s potential to enhance the modeling, control, and fault diagnosis of complex systems, offering a promising foundation for the development of robust, disturbance-tolerant predictive control architectures.