Linear model predictive control with lifted bilinear models by Koopman-based approach

Abstract

This study proposes a linear Model Predictive Control (MPC) method that combines high prediction accuracy with low computational cost, using a lifted bilinear model based on Koopman theory. In recent years, there has been a growing interest in approaches to learning prediction models that determine the performance of MPC through lifting linearization and lifting bilinearization based on Koopman theory. In these methods, a linear or bilinear model reflecting nonlinear characteristics of the target system can be obtained by lifting the states to a higher dimensional space with observable functions (observables). In particular, lifting bilinearization provides more accurate models for the nonlinear input affine system, but when combined with MPC, it requires solving a nonlinear optimization problem, which is computationally expensive. Therefore, in this study, we formulate a linear MPC using a lifted bilinear model that can make accurate predictions for the input affine system, thereby realizing an MPC algorithm with high accuracy and low computational cost. In the proposed method, we first formulate a prediction error correction method for the lifted bilinear model by introducing constraints based on the observables and corrective inputs. Furthermore, we propose a linear MPC that can make accurate predictions using the lifted bilinear model by utilizing prior-predicted states based on the optimal solution at the previous time. We evaluate the effectiveness of the proposed method through numerical simulations using a planar drone.