Enhanced Koopman Operator Approximation for Nonlinear Systems Using Broading Learning System

Abstract: Traditional control methods often show limitations in dealing with complex nonlinear systems, especially when it is difficult to accurately obtain the exact system model, and the control accuracy and stability are difficult to guarantee. To solve this problem, the Koopman operator theory provides an effective method to linearise nonlinear systems, which simplifies the analysis and control of the system by mapping the nonlinear dynamics into a high-dimensional space. However, the existing extended dynamical mode decomposition (EDMD) methods suffer from randomness in the selection of basis functions, which leads to bias in the finite-dimensional approximation to the Koopman operator, thus affecting the accuracy of model prediction. To solve this problem, this paper proposes a BLS-EDMD method based on the Broad learning system (BLS) network. The method achieves a high-precision approximation to the Koopman operator by learning more accurate basis functions, which significantly improves the prediction ability of the model. Building on this, we further develop a model predictive controller (MPC) called BE-MPC. This controller directly utilises the high-dimensional and high-precision predictors generated by BLS-EDMD to predict the system state more accurately, thus achieving precise control of the underwater unmanned vehicle (UUV), and its effectiveness is verified by simulation.