Solving Nonlinear MPC Problems in the Koopman Lifted Space: The Case Study of Mobile Robot Navigation in Cluttered Environments

The Koopman operator framework allows to transform nonlinear dynamical systems into equivalent linear ones within a higher-dimensional state space. Its application can be extended to nonlinear optimal control problems, enabling their efficient solution in the Koopman lifted space. Here, we present a comprehensive analytical framework to lift general Nonlinear Model Predictive Control (NMPC) problems in the Koopman space, converting them into equivalent quadratic programs (QPs) - referred to as Koopman NMPC (K-NMPC) - that can be solved with superior computational performance. Moreover, we advance analytical Koopman operator methods by proposing an algorithmic procedure to generate an invariant basis of Koopman observables to lift both the nonlinear prediction model and the nonlinear state constraints of NMPC; additionally, we present a general method to arbitrarily reduce the dimensionality of the Koopman lifted space, lowering the K-NMPC complexity and handling the infinite-dimensional case. Our K-NMPC approach is validated through hardware-in-the-loop experiments on the case study of mobile robot navigation in cluttered environments, showcasing its solid performance and a ten-fold reduction in computation times.