Deep learning for Koopman Operator Optimal Control

Area of Science:

• Complex Systems
• Robotics
• Computational Neuroscience

Background:

• Nonlinear dynamics are prevalent in complex systems.
• The Koopman framework linearizes nonlinear dynamics by lifting states to a higher dimension.
• Identifying coordinate transformation functions is a key challenge.

Purpose of the Study:

• Introduce deep learning for model-free learning of nonlinear dynamics within the Koopman framework.
• Develop novel architectures for improved performance and control.
• Validate the effectiveness of these deep learning approaches for nonlinear system control.

Main Methods:

• Utilized deep learning for model-free learning of nonlinear dynamics.
• Implemented an optimized Linearly Recurrent Encoder Network (LREN).
• Proposed and evaluated a novel Deep Encoder with Initial State Parameterization (DENIS) architecture.
• Developed and described a Double Encoder for Input Nonaffine systems (DEINA) architecture.
• Applied Koopman Model Predictive Control (KMPC) for validation.

Main Results:

• Achieved faster LREN implementation compared to existing methods.
• DENIS demonstrated superior control performance over LREN and matched/exceeded iterative Linear Quadratic Regulator (iLQR).
• DEINA showed potential to outperform existing Koopman frameworks for nonaffine input systems.
• Validated the successful control of nonlinear dynamics using deep learning-based Koopman methods.

Conclusions:

• Deep learning-based Koopman framework offers a promising approach for optimal control of nonlinear dynamics.
• Novel architectures like DENIS and DEINA significantly advance the capabilities of Koopman methods.
• These methods provide effective, data-driven solutions for complex systems in robotics and neuroscience.