A General Analytical Framework for Fast Solving Nonlinear MPC Problems in the Linear Koopman Space

The Koopman operator stands as a powerful framework to transform nonlinear dynamical systems into equivalent linear ones within a lifted state space. Its application can be extended to nonlinear optimal control problems, enabling their efficient solution in the linear Koopman space. However, a systematic methodology to analytically derive a suitable basis of Koopman observables and handle the operator infinite-dimensionality is still lacking. In this paper, we propose a comprehensive analytical framework to efficiently solve Nonlinear Model Predictive Control (NMPC) problems in the linear Koopman space. We present a general procedure to derive a basis of observables that lifts both the nonlinear prediction model and nonlinear state constraints of NMPC, obtaining a quadratic program in the Koopman lifted space (denoted as Koopman NMPC, in short K-NMPC) that closely approximates the original NMPC solution. Additionally, we propose a general method to arbitrarily reduce the dimensionality of the Koopman lifted space, lowering the K-NMPC complexity and handling the infinite-dimensional case. We validate our K-NMPC approach in simulation, showcasing its solid performance and execution times, which are over ten times lower than classic NMPC.