An attention-based approach for Koopman modeling and predictive control of nonlinear systems

We develop an innovative attention-based deep learning method for constructing Koopman eigenfunctions, addressing the challenge of accurately modeling nonlinear systems for predictive control. This method is motivated by the need to identify a diffeomorphism that precisely delineates the topological conjugacy between nonlinear dynamics and their linearized counterparts. Leveraging the capabilities and suitability of invertible neural networks, we deploy an architecture equipped with conditional affine coupling layers to approximate this diffeomorphism. A pivotal feature is the integration of an attention mechanism within these layers, enhancing the capture of complex interactions across various layers. Additionally, we propose an alternative strategy, which employs an invertible decoding neural network in conjunction with a frozen decoder, to refine the diffeomorphism approximation further. Utilizing Koopman eigenfunctions and eigenvalues obtained from the learned diffeomorphism, we construct a lifted linear system with an optimized input matrix via multi-step error minimization, which facilitates the predictive control of nonlinear systems through an extended robust linear model predictive control framework. Numerical examples and a physical experiment validate the efficacy of our framework, demonstrating superior prediction performance compared to existing dynamic mode decomposition-based methods.