Model predictive control of nonlinear dynamical systems based on long sequence stable Koopman network

In recent years, the Koopman method has found numerous applications in the field of nonlinear control due to its ability to map nonlinear states into high-dimensional spaces, thereby transforming nonlinear control problems into linear or bilinear problems. However, Koopman methods based on deep learning suffer from slow convergence, and the Koopman coefficients obtained through iterative processes cannot guarantee long-term prediction stability in the high-dimensional mapped space. To address these issues, we propose a Stable Deep Koopman Network with Model Predictive Control (SDKN-MPC) method for nonlinear control. The SDKN-MPC method utilizes the Stable Koopman Solver Algorithm to solve for a stable Koopman operator. It incorporates neural network training for embedding functions, with both training processes interleaved until convergence is achieved towards a unified stable solution. Subsequently, Model Predictive Control (MPC) is employed to control the high-dimensional linear system mapped through the Koopman operator, yielding high-dimensional desired inputs. These inputs undergo further processing through an auxiliary network to obtain the actual predictive control inputs. The proposed method is subjected to long-term predictive performance testing across multiple typical nonlinear control tasks and is compared with existing deep learning-based approaches. The results demonstrate that our method can extract more effective nonlinear features, converges rapidly, and exhibits superior predictive performance compared to existing methods.