An attention-based approach for Koopman modeling and predictive control of nonlinear systems

Area of Science:

• Control Theory
• Machine Learning
• Dynamical Systems

Background:

• Accurate modeling of nonlinear systems is crucial for effective predictive control.
• Traditional methods struggle with the complexity of nonlinear dynamics.
• Identifying topological conjugacy between nonlinear and linearized systems is a key challenge.

Purpose of the Study:

• To develop an innovative attention-based deep learning method for constructing Koopman eigenfunctions.
• To accurately model nonlinear systems for enhanced predictive control.
• To identify a diffeomorphism for topological conjugacy between nonlinear dynamics and their linearized counterparts.

Main Methods:

• An attention-based deep learning method utilizing invertible neural networks with conditional affine coupling layers.
• Approximation of the diffeomorphism using an attention mechanism to capture complex layer interactions.
• An alternative strategy employing an invertible decoding neural network with a frozen decoder for refined diffeomorphism approximation.

Main Results:

• Construction of Koopman eigenfunctions and eigenvalues from the learned diffeomorphism.
• Development of a lifted linear system with an optimized input matrix via multi-step error minimization.
• Superior prediction performance demonstrated in numerical examples and a physical experiment compared to dynamic mode decomposition-based methods.

Conclusions:

• The proposed attention-based deep learning framework effectively models nonlinear systems for predictive control.
• The method achieves superior prediction accuracy by leveraging Koopman eigenfunctions and an extended robust linear model predictive control framework.
• This approach offers a significant advancement over existing dynamic mode decomposition techniques for nonlinear system analysis and control.